Geneviève Baril

Gonzalo Galicia

Claude Deslouis

Nadine Pébère
email: nadine.pebere@ensiacet.fr

Bernard Tribollet

Vincent Vivier

An Impedance Investigation of the Mechanism of Pure Magnesium Corrosion in Sodium Sulfate Solutions

The corrosion behavior of pure magnesium in sodium sulfate solutions was investigated using voltammetry and electrochemical impedance spectroscopy with a rotating disk electrode. The analysis of impedance data obtained at the corrosion potential was consistent with the hypothesis that Mg corrosion is controlled by the presence of a very thin oxide film, probably MgO, and that the dissolution occurs at film-free spots only. This hypothesis was substantiated both by the superposition of the EIS diagrams, obtained for different immersion times and for two Na 2 SO 4 concentrations once normalized, and by use of scanning electrochemical microscopy in the ac mode to sense the local conductivity of the material. On the basis of the electrochemical results, a model was proposed to describe magnesium corrosion at the open-circuit potential. Simulation of the impedance diagrams was in good agreement with the experimental results.

Magnesium is an attractive metal because of its low density and its relative abundance in seawater. Magnesium alloys offer a high potential for use as a lightweight structural material in such transport applications as automobiles and aircraft. 1 However, due to the high reactivity of magnesium, its corrosion resistance is low, which limits applications. In a recent study, the corrosion behavior of as-cast magnesium alloys ͑AM50 and AZ91͒ was investigated in a sodium sulfate solution by electrochemical impedance spectroscopy ͑EIS͒. 2 At the corrosion potential and for short immersion times ͑Ͻ6 h͒, the impedance diagrams presented the same phenomenology as that obtained for pure magnesium ͑three time constants with the same characteristic frequencies͒. 3,4 Thus, for short exposures to the aggressive solution, in spite of the presence of coarse intermetallic particles ͑Mg-Al and Mg-Al-Mn͒ in the matrix of the alloys and of eutectic areas in grain boundaries of AZ91 alloy, the corrosion behavior of the alloys was close to that of pure magnesium. This was explained by the fact that at the beginning of immersion, the coarse intermetallic particles and the eutectic areas remained unaffected due to galvanic coupling between these zones and the grain body. Thus, for short immersion times, the impedance diagrams represent the corrosion process of the grain body because both the coarse particles and the eutectic areas are cathodic and therefore protected against corrosion. Song et al. 5 have already pointed out that the ␣ matrix phase in AZ magnesium alloys follows the same corrosion mechanisms as pure magnesium. Thus, a better understanding of the corrosion mechanisms of pure magnesium is a prerequisite for a comprehensive approach of the mechanisms involved in magnesium alloy corrosion. In fact, although numerous studies have been devoted to pure magnesium, the corrosion mechanism remains unclear.

The anodic dissolution of magnesium is accompanied by an anomalous phenomenon called the negative difference effect ͑NDE͒. It is characterized experimentally by an unexpected increase of the cathodic hydrogen evolution reaction when the anodic overvoltage is increased. To explain the magnesium corrosion and more particularly this NDE phenomenon, several mechanisms have been proposed including the formation of magnesium hydrides, 6 metastable monovalent ions, [7][8][9][10][11][12] and magnesium hydroxides and oxides. [13][14][15][16][17][18][19][20][21][22][23] Song et al. 11,12 recently proposed a new mechanism for the NDE. They explained that the formation of a partially protective layer of Mg͑OH͒ 2 plays an important role in the corrosion behavior of magnesium. They considered the following reactions in their discussion of the NDE

2H + + 2e -→ H 2 ͑cathodic partial reaction͒ ͓ 1͔ 2Mg → 2Mg + + 2e -͑anodic partial reaction͒ ͓ 2͔ 2Mg + + 2H 2 O → 2Mg 2+ + 2 OH -+ H 2 ͑chemical reaction NDE͒ ͓3͔ 2Mg + 2H + + 2H 2 O → 2Mg 2+ + 2 OH -+ 2H 2 ͑overall reaction͒ ͓4͔ Mg 2+ + 2 OH -→ Mg͑OH͒ 2 ͑corrosion product formation͒ ͓5͔
The dissolution mechanism is based on an increase in the film-free areas with increasing anodic potential on which the anodic and cathodic partial reactions ͑Eq. 1 and 2͒ can occur more easily than on the surface covered by a film. In the film-free areas, these authors assumed that magnesium corrosion occurs with the production of univalent ions. For high anodic overpotentials, the rate of the chemical reaction is increased due to the increased concentration of Mg + produced by Reaction 2 and concomitantly, there is an increase in the amount of hydrogen evolved. Song et al. 11,12 mentioned that the protective film is potential dependent in that there is complete film coverage over the whole surface and a low rate of corrosion for potentials below the pitting potential. According to these authors, the pitting potential is slightly more negative than the corrosion potential. In the proposed mechanism, the electrochemical production of divalent magnesium ions was never considered. In addition, the mechanisms of formation and dissolution of the protective film are not taken into account in the corrosion process. The aim of the present work is to report experimental data obtained in sodium sulfate solutions in view of modeling magnesium corrosion. DC and transient electrochemical measurements were complemented by local electrochemical measurements ͓scanning electrochemical microscopy ͑SECM͔͒ to obtain a more complete understanding of the behavior of magnesium in neutral media. On the basis of the experimental results and data taken from the literature, a new model for describing Mg corrosion at the open-circuit potential is presented.

Experimental

Material.-The corrosive medium was a sodium sulfate proanalysis grade solution ͑pH 5.6͒. Two Na 2 SO 4 concentrations were tested, 0.01 and 0.1 M. In a previous paper, 4 it was shown that the presence of O 2 does not influence the corrosion of magnesium. In fact, magnesium corrosion is relatively insensitive to the oxygen concentration because water reduction is the main cathodic reaction involved in the corrosion process. The authors concluded, however, that the protective properties of the film formed on the magnesium surface were dependent on the aeration of the electrolyte due to the presence of CO 2 . They postulated that HCO 3 -increased the rate of dissolution of the corrosion product layer by formation of soluble salts. In the present work, the volume of solution used was 200 mL and the electrolyte was in contact with air. The temperature was maintained at 25°C. The HCO 3 -concentration determined previously under these conditions was about 40 mg L -1 . 4 The working electrode was a rotating disk of 1 cm 2 surface area consisting of the cross section of a cylindrical magnesium rod, provided by Alfa Company. The purity was 99.95%. The body of the rod was covered with a heat-shrinkable sheath, leaving only the cross section of the cylinder in contact with the solution. Prior to any experiment, the electrode was polished with SiC paper down to grade 4000, cleaned in water in an ultrasonic bath, and then dried in warm air. The counter electrode was a platinum grid. A saturated K 2 SO 4 /Hg 2 SO 4 electrode ͑SSE͒ was used as a reference.

Electrochemical measurements.-The polarization curves were plotted under potentiostatic regulation using a Solartron 1286 Electrochemical Interface. They were plotted ͑point by point͒ by steps of 20 mV every 2 min to ensure quasi-steady-state conditions. The anodic and cathodic parts were obtained independently starting from the corrosion potential. The ohmic drop in the electrolyte was dependent on the sodium sulfate concentration. As a consequence, the polarization curves were corrected from the ohmic drop experimentally determined from electrochemical impedance measurements.

The correction on the imposed potential was 80% of the ohmic drop measured at the corrosion potential to avoid instability. Electrochemical impedance measurements were carried out using a Solartron 1250 frequency response analyzer in a frequency range of 65 kHz to a few mHz with eight points per decade. The linearity of the system was checked by varying the amplitude of the ac signal applied to the sample.

SECM.-The SECM apparatus used in this study was a homemade device already described elsewhere. 24,25 It consisted of a 3-axis positioning system ͑VP-25XA, Newport͒ driven by a motion encoder ͑ESP300, Newport͒ allowing a spatial resolution of 100 nm in the three directions. The electrochemical measurements were carried out with a homemade bipotentiostat coupled to a low-noise current-to-voltage converter ͑Femto DLPCA200, FBI Optilas͒ with an adjustable gain ͑10 3 -10 11 V/A͒ and a large bandwidth ͑up to 500 kHz͒. In order to evaluate the electrolyte resistance ͑R e ͒, the high-frequency ͑HF͒ impedance measurement was coupled to the SECM, and the impedance modulus of the electrode tip was measured with an homemade analog device. 24 A sinusoidal voltage ͑frequency f HF of 75-150 kHz and peak-to-peak amplitude V p-p of 10-50 mV͒ delivered by a function generator ͑TG550, TTi͒ was superimposed on the dc voltage of the potentiostat. The output analog signal was sent to the R e -measurement channel that delivered a voltage signal V Re ͑t͒, which was proportional to the amplitude of the ac current and, therefore, was a linear function of the reciprocal of the impedance modulus Mod͑Z HF ͒ at the frequency f HF , according to the relationship

V Re = b 1 Mod͑Z HF ͒ + b 2
This technique, based on impedance-modulus measurements, has been shown to give exact values of R e inasmuch as calibration curves obtained with pure resistors were used to evaluate the constants b 1 and b 2 . The R e variations were then interpreted as local variations of the surface reactivity with a spatial resolution depending on both the electrode dimension and the tip-to-substrate distance. In our case, the spatial resolution was in the micrometer range.

Moreover, it was demonstrated that the electrolyte resistance was dependent on the tip-to-interface distance and also on the nature of the interface; 24,26 R e increased when the ultramicroelectrode ͑UME͒ approached an insulator, whereas it decreased for a conductor. Therefore, it is possible to detect local changes in reactivity or conductivity at the interface by measuring the electrolyte resistance. 26,27 SECM tips consisted of homemade platinum microelectrodes. The 5 m radius platinum wire was sealed into soft glass. The normalized radius of the UME RG = r g /a ͑where a and r g are the radii of the platinum wire and the tip-insulating material, respectively͒ was set between 10 and 20 and was determined by both scanning electron microscope observation and by recording an approach curve at an insulating substrate. 28

Results and Discussion

DC measurements.-Figure 1 illustrates the variation of the free corrosion potential, E corr , of pure magnesium dipped into 0.1 M Na 2 SO 4 solution. During the first 20 h of immersion, E corr increases with time and then it stabilizes at approximately 32 h. This change in corrosion potential was accompanied by an increase of the pH of the medium to a value higher than 10 at the end of the experiment. This pH variation is in agreement with the global reaction of the corrosion process ͑Reaction 4͒. The shift of the corrosion potential in the anodic direction could be linked to an improvement of the protective properties of the film formed on the magnesium surface. In spite of the strong reactivity of magnesium, the time required to reach the stationary state is relatively long.

The polarization curves obtained after two preliminary hold times at the corrosion potential, E corr , are reported in Fig. 2. Independent of the immersion time, the anodic part of the curves can be split into three domains: ͑i͒ The first domain starting from E corr ͑low anodic overpotential͒ shows linear increase of the current density with the potential in Tafel coordinates. The calculated slope was lower for 32 h of immersion ͑Ϸ85 mV dec -1 ͒ than for 3 h 30 min ͑Ϸ106 mV dec -1 ͒. The current density decreased as the hold time at E corr increased. ͑ii͒ The second domain is marked by an abrupt increase in the current density around -1.85 V/SSE. This potential value was independent of the immersion time and can be attributed to the change of the dominant dissolution reaction from Reaction 2 to one that involves direct formation of Mg 2+ . 29 ͑iii͒ For larger potential values ͑Ͼ-1.85 V/SSE͒, a current plateau, whose height was independent of the immersion time, characterizes the third domain. The current reached 10 mA cm -2 . Observations after anodic polarization showed that the electrode surface was covered by a thick film of white crystals, which were identified as Mg͑OH͒ 2 by X-ray diffraction. [START_REF] Baril | Proceedings of the International Symposium on Environmental Degradation of Materials and Corrosion Control in Metals[END_REF] In the cathodic range, the current densities were slightly lower after 32 h of immersion at the corrosion potential than after 3 h 30 min. It can be considered that the cathodic branch displays a linear behavior with the semilog coordinates. The slope is particularly high around 300 mV dec -1 . The origin of the steepness of the slope can be attributed to hydrogen release which modifies the electrode surface. As a matter of fact, numerous bubbles were formed on the magnesium surface during cathodic sweeps. Thus, the influence of the immersion time on the polarization curves results in both a decrease of the current density in a large domain around the corrosion potential and a shift of the corrosion potential towards higher values.

When the electrode rotation rate was varied, the cathodic curves were superimposable. Thus, the cathodic process is not controlled by diffusion in the liquid phase. For the anodic part, in the second domain and on the plateau, the curves are independent of the electrode rotation rate. The anodic curves were only poorly dependent on the electrode rotation rate in the first domain starting from E corr ͑about 100-150 mV near E corr ͒. Thus, the mechanism of pure magnesium corrosion in sulfate solutions is not controlled by the diffusion in solution. The effect of the electrode rotation rate was in good agreement with previous results. 4 Electrochemical impedance and SECM measurements.-Figure 3a shows the impedance diagrams obtained for different hold times at E corr in 0.1 M Na 2 SO 4 . All these diagrams are characterized by two well-defined capacitive loops, at high and medium frequencies, followed by an inductive loop in the lower frequency domain. The increase in immersion time at E corr led to a size increase in both high-and medium-frequency capacitive loops and also in the inductive loop. After 32 h of immersion, a stationary state was reached, and no significant variation of the EIS response was observed. In Fig. 3b, the EIS spectra plotted in Fig. 3a are normalized by dividing the real and the imaginary parts of the diagram by the maximum value of the impedance for each diagram. It can be seen that all the diagrams can be merged into a single diagram.

Figure 4a presents the impedance diagrams obtained for two Na 2 SO 4 concentrations. The shape of the diagrams was independent of the Na 2 SO 4 concentration. The increase of the Na 2 SO 4 concentration led to a decrease of the size of the three loops observed on the impedance diagrams. In this case again, when the impedance is normalized, the diagrams are merged into a single spectrum. Moreover, similar behavior has already been observed for the AZ91 magnesium alloy in a Na 2 SO 4 solution for different values of the pH; [START_REF] Baril | Proceedings of the EUROCORR Symposium[END_REF] the diagrams presented the same shape and were reduced to a single diagram following the same procedure.

The merging of the diagrams ͑Fig. 3b and 4b͒ indicates that the corrosion mechanism is independent of the immersion time and the concentration of the medium. Thus, the merging of the diagrams can be explained by a change of the bare parts of the surface. In agreement with the mechanism proposed by Song et al., 11,12 Mg corrosion is proposed to occur only in the film-free areas, and MgO is proposed to act as a protective film.

To verify this hypothesis, SECM experiments were performed using electrolyte resistance as an electrochemical sensor to visualize cracks and holes above the Mg surface after 4 and 24 h of immersion in 0.5 M Na 2 SO 4 solution ͑Fig. 5͒. As the oxide layer thickness is in the nanometer range, the use of a 5 m radius microelectrode tip to scan the surface implies immunity to the surface topography, and the electrolyte resistance variations should be interpreted as variations of the local electronic conductivity of the surface only. Moreover, the ac impedance images were plotted using the dimensionless R Norm quantity, that is, the ratio R e /R ϱ , where R ϱ is the electrolyte resistance far from the substrate. Figure 5a reveals that after 4 h of immersion, the Mg surface is already inhomogeneous, and thus, the oxide film has already begun to be formed, partially covering the Mg surface. Then, progressively, the insulating ͑MgO͒ layer covers larger areas of the metal surface. Figure 5b corresponds to the SECM measurement performed after 24 h of immersion over the same surface area. The influence of this MgO layer is clearly seen, and the small spots correspond to domains on which the electrolyte resistance is lower than R ϱ . These domains thus correspond to the active sites for Mg dissolution. These two images are in full agreement with the hypothesis of surface variations of the active domains with time to describe Mg dissolution. However, from these data, it is difficult to extract the exact values of surface variations because the area scanned ͑300 ϫ 300 m͒ only accounts for a small portion of the immersed sample.

The MgO layer is formed progressively. The formation of MgO corresponds to the following equilibrium

MgO + H 2 O Mg͑OH͒ 2 ͓6͔
The thick and porous layer of Mg͑OH͒ 2 is formed following Eq. 5. This layer is homogeneous and covers the entire Mg surface. 4 Thus, it can be assumed that active zones are moving on the magnesium surface. In other words, active zones become passive, and new active zones are created near the passive ones. Even if the presence of active zones corresponds to localized corrosion, it is not highly localized corrosion, such as pitting corrosion for aluminum and aluminum alloys, where the pit initiation is followed by a propagation step. Here, the corrosion must be interpreted in terms of a more uniform type of corrosion as evidenced by the presence of the thick porous corrosion products layer. This layer impedes the visualization of both the presence of the very thin layer of MgO and the active areas.

Schematic representation of the interface and equivalent circuit.-The electrochemical results obtained for pure magnesium in Na 2 SO 4 solutions revealed behaviors which seem contradictory with its high reactivity: ͑i͒ the stationary state is only obtained after several hours of immersion and ͑ii͒ the impedance measurements could be made to very low frequencies ͑few mHz͒ without any dispersion. These impedance diagrams have been shown to be consistent with the Kramers-Kronig relations. [START_REF] Orazem | [END_REF] In addition, when the immersion time increased, E corr was shifted in the anodic direction, the current density around E corr decreased, and the impedance values increased. These results can only be explained by the presence of a protective film on the Mg surface which controls the corrosion process and whose protective properties increase with time.

On the basis of the experimental results and literature data, a schematic representation of the interface is presented in Fig. 6. Two different layers are present on the Mg surface, a thin barrier film of MgO in contact with the metal and a relatively thick porous film of Mg͑OH͒ 2 on top of the barrier film. The Mg͑OH͒ 2 layer coats the MgO film and the film-free areas. The Mg dissolution occurs in the film-free areas and the Mg 2+ ions produced at the interface diffuse through the porous Mg͑OH͒ 2 layer. The corresponding elements of the equivalent electrical circuit are the faradaic impedance ͑Z f ͒ in parallel with the double-layer capacitance ͑C d ͒. The MgO layer is characterized by a film capacitance ͑C f ͒ in parallel with a film resistance ͑R f ͒. The MgO oxide film is assumed to be very protective, and thus R f is very high with respect to Z f . Therefore, R f has no influence on the impedance measurements and is represented as a dashed line in Fig. 6. The film-free areas are small in comparison with the surface covered by MgO ͑Fig. 5͒ and then C d is negligible with respect to C f . This C f value can be extracted from the highfrequency loop, which is analyzed by considering a constant phase element ͑CPE͒ in parallel with a resistance. The origin of this CPE was recently explained using local electrochemical impedance spectroscopy and attributed to the distribution of the resistance associated with the geometry of the disk electrode ͑2D distribution͒. 33 The corresponding capacitance was deduced from Brug's formula. 34 In- dependent of both the exposure time and of the sodium sulfate concentration, the capacitance has the same value of around 10 F/cm 2 . From the capacitance value, the thickness of the protective layer ͑d͒ can be evaluated from the relationship d = 0 S/C, where S is the surface area ͑S = 1 cm 2 ͒, 0 the vacuum permittivity ͑ 0 = 8.85 10 -14 F/cm͒, and the relative dielectric constant of magnesium oxide ͑ = 10͒. 35 The calculated value is very low, about 1 nm. This low thickness and the presence of Mg͑OH͒ 2 may explain why the MgO oxide layer was not often observed by surface analyses and was not taken into account to describe the Mg corrosion processes. 11,12 The change of the impedance with immersion time or Na 2 SO 4 concentration is due to a variation of free surface area which can explain both the observation that all impedance diagrams are identical after normalization ͑Fig. 3b and 4b͒ and also that the polarization curves are not significantly modified by the hold time at E corr ͑Fig. 2͒.

Kinetics model.-The appearance of the medium-frequency capacitive loop suggests diffusion impedance. The impedance diagrams were identical when the electrode rotation rate was changed ͑not reported͒, and thus this loop can be attributed to diffusion through the porous layer of Mg͑OH͒ 2 . The presence of an inductive loop was independent of immersion time and of Na 2 SO 4 concentration. The impedance spectra of magnesium always presented an inductive loop at low frequencies. 12,36,37 The inductive loop is present at the beginning of immersion when the thick film is not present on the Mg surface. Inductive loops are generally ascribed to the existence of relaxation processes of adsorbed species. 38,39 A model is proposed which takes into account the different processes discussed above. In the frequency range corresponding to the impedance measurements, the film-free areas are considered to be constant for each measurement. The presence of the inductive loop in the low-frequency range involves an adsorbed species, and, as Mg 2+ ion production occurs in two steps, Mg + is assumed to be the absorbed intermediate. Mg + is the simpler species which can be involved in the Mg corrosion as an intermediate. The mass-transport loop in the medium-frequency range corresponds to the diffusion of Mg 2+ through the porous Mg͑OH͒ 2 layer; therefore, Mg 2+ reacts at the interface.

On the film-free areas, the two reactions considered were Mg --→ To take into account the NDE, a chemical reaction similar to reaction 3 was also introduced

Mg ads + + H 2 O --→ k 3 Mg 2+ + OH -+ 1 2 H 2 ͓9͔
The high-frequency capacitive loop corresponds to the chargetransfer resistance of Reactions 7 and 8, in parallel to the MgO film capacitance. The charge-transfer resistance involves cathodic and anodic reactions. This kinetic model corresponds to the free areas and, in agreement with experimental observations ͑Fig. 4͒, does not involve the nature of the electrolyte.

On the basis of this reaction model, ac-impedance diagrams were calculated and simulated by assuming that the adsorbate Mg ads + obeys Langmuir's isotherm and that the rate constant of electrochemical reactions are exponentially potential dependent ͑Tafel's law͒. Each reaction ͑index i͒ has a normalized rate constant K i corresponding to its rate constant k i by

K i = k i exp͓b i ͑E -E 0 ͔͒ ͓10͔
where b i is the activation coefficient with the potential, and E 0 is an origin potential ͑arbitrarially chosen͒. Moreover, the chemical step 9 is supposed to be potential independent. Assuming that the maximum number of sites per surface unit which can be occupied by the adsorbate Mg ads + is ␤, the mass and charge balances are expressed as functions of the fraction of the surface coverage, , by the adsorbed species by

␤ d dt = K 1 ͑1 -͒ -K 2 ␤ + K 22 C Mg 2+ ͑0͒ -k 3 ␤ ͓11͔ DC Mg 2+ ͑0͒ ␦ = K 2 ␤ -K 22 C Mg 2+ ͑0͒ + k 3 ␤ ͓12͔
and

I F = FA͓K 1 ͑1 -͒ + K 2 ␤ -K 22 C Mg 2+ ͑0͒ ͔ ͓ 13͔
where F is Faraday's constant, A is the free electrode surface area, ␦ is the thickness of the Mg͑OH͒ 2 layer, and

I F is the faradic current. C Mg 2+

͑0͒

is the concentration of the Mg 2+ ion at the electrode interface and D is the diffusion coefficient of Mg 2+ ͑in the film͒. Mg 2+ diffusion is supposed to occur only in the Mg͑OH͒ 2 layer, which is in agreement with the fact that experimental data are independent of the electrode rotation rate. At the steady state, and C Mg 2+ ͑0͒ are thus given by =

K 1 ͩ D ␦ + K 22 ͪ K 1 ͩ D ␦ + K 22 ͪ + ͑K 2 + k 3 ͒ D ␦ ␤ ͓14͔ and
C Mg 2+ ͑0͒ = ␤͑K 2 + k 3 ͒ D ␦ + K 22 ͓15͔
respectively.

The faradic impedance Z F is thus calculated by linearizing the mathematical expressions 11-13 for small sine wave perturbations, i.e.

͓␤j + K 1 + ␤͑K 2 + k 3 ͔͒⌬ = ͓͑1 -͒K 1 b 1 -␤K 2 b 2 -C Mg 2+ ͑0͒ K 22 b 22 ͔⌬V + K 22 ⌬C Mg 2+ ͑0͒ ͓16͔ and
⌬I F AF = ͑K 2 ␤ -K 1 ͒⌬ + ͓͑1 -͒K 1 b 1 + ␤K 2 b 2 + C Mg 2+ ͑0͒ K 22 b 22 ͔⌬V -K 22 ⌬C Mg 2+ ͑0͒ ͓17͔
Because the specie Mg 2+ diffuses towards the electrode surface, the resulting concentration perturbation ⌬C Mg 2+

͑0͒

is obtained from the usual Warburg impedance, N͑͒, with a diffusion layer of finite thickness. The resulting impedance can be expressed as

Z F = ⌬V ⌬I F = ͩ1 + K 22 N͑͒ ͭ 1 -͑K 2 ␤ -K 1 ͒ ͓␤j + K 1 + ␤͑K 2 + k 3 ͔͒ ͮͪ AF ͭ ͑r 1 -r 2 ͒͑K 2 ␤ -K 1 ͒ ͓␤j + K 1 + ␤͑K 2 + k 3 ͔͒ + ͑r 1 + r 2 ͒ ͮ ͓18͔ with r 1 = ͑1 -͒K 1 b 1 and r 2 = K 2 b 2 ␤ + K 22 b 22 C Mg 2+

͑0͒

Finally, the overall impedance Z involves the contribution of Z F and of the capacitance C d and C f ͑Fig. 6͒ reduced to C f ͑see before͒

Z = R e + Z F 1 + jZ F C f ͓19͔
where R e is the electrolyte resistance. An example of a simulated diagram is given in Fig. 7. It is characterized by the three loops previously described. The timeconstant characteristic of each semicircle and the magnitude of the overall diagram are fully consistent with experimental data. However, these three loops are defined by six variables and the kinetics model involved at least eleven parameters, thus only six relations between the kinetics parameters can be obtained by a fitting procedure. For this reason, this study was only limited to an impedance simulation. Nevertheless, Eq. 18 can be represented as

Z F = R tͫ 1 + K 22 N͑͒ - 2Sr 1 K 22 N͑͒ + ͑r 1 -r 2 ͒S ͑r 1 -r 2 ͒S + ͑r 1 + r 2 ͒ ͬ ͓20͔ with S = ͑K 2 ␤ -K 1 ͒ ␤j + K 1 + ␤͑K 2 + k 3 ͒ and R t = 1 r 1 + r 2
This impedance can thus be expressed as a sum of three terms

Z F = R t + R t K 22 N͑͒ - R t ͓2r 1 K 22 N͑͒ + ͑r 1 -r 2 ͔͒͑K 2 ␤ -K 1 ͒ ͑r 1 + r 2 ͒␤j + K 1 + ␤͑K 2 + k 3 ͒ + ͑r 1 -r 2 ͒͑K 2 ␤ -K 1 ͒

͓21͔

The latter expression clearly demonstrates the contribution of the charge-transfer resistance, the diffusion loop, and the inductive loop.

It should also be mentioned that the value taken by the surface coverage, , remains low at the corrosion potential and was estimated to have a value as low as 0.5% by the previous numerical calculations.

Conclusion

The corrosion of pure magnesium in sodium sulfate solutions was studied by voltammetry, EIS, and SECM techniques. On the basis of our experimental results and taking into account literature data, the following points have been established.

1. Magnesium corrosion is controlled by the presence of a very thin MgO oxide film, and dissolution occurs only at the bare parts of this film. Both the MgO and the film-free areas are covered by a thick porous layer of Mg͑OH͒ 2 .

2. The invariance of the EIS diagrams, once normalized by the maximum value of the real part, leads to the conclusion that the corrosion mechanism is independent of the electrolyte concentration and of the immersion time. It can be described as the exchange of two electrons in two successive electrochemical steps where the adsorption of the intermediate ͑Mg + ͒ ads is followed by two parallel paths; one is chemical and gives rise to the NDE and the other is electrochemical leading to Mg 2+ .

3. The calculation of the impedance diagram from the proposed corrosion mechanism was shown to be in good agreement with experimental data.

Centre National de la Recherche Scientifique assisted in meeting the publication costs of this article.

Figure 1 .

 1 Figure 1. Corrosion potential of pure magnesium as a function of time in a 0.1 M Na 2 SO 4 solution; electrode rotation rate: 240 rpm.

Figure 2 .

 2 Figure 2. Current-voltage curves plotted after two preliminary hold times at the corrosion potential in a 0.1 M Na 2 SO 4 solution: ͑᭺͒ 3 h 30 min and ͑b͒ 32 h; electrode rotation rate: 1000 rpm.

Figure 3 .

 3 Figure 3. ͑a͒ Electrochemical impedance diagrams plotted after different hold times at E corr in a 0.1 M Na 2 SO 4 solution and with an electrode rotation rate of 240 rpm: ͑᭺͒ 1 h 30 min, ͑b͒ 21 h, and ͑ᮀ͒ 32 h. ͑b͒ Normalized impedance diagrams.

Figure 4 .

 4 Figure 4. ͑a͒ Electrochemical impedance diagrams plotted for two Na 2 SO 4 concentrations and an electrode rotation rate of 240 rpm: ͑᭺͒ 0.1 M and ͑b͒ 0.01 M. ͑b͒ Normalized impedance diagrams.

Figure 5 .

 5 Figure 5. ͑Color online͒ Electrolyte resistance images performed using a 10 m diameter Pt tip over Mg surface after ͑a͒ 4 and ͑b͒ 24 h of immersion in a 0.5 Na 2 SO 4 solution, at 120 kHz and with 30 mV peak-to-peak amplitude. The tip velocity was 10 m/s.

Figure 6 .

 6 Figure 6. Schematic representation of the interface and of the equivalent circuit.

Figure 7 .

 7 Figure 7. Simulated impedance diagram for the following parameters: b1 = 20 V -1 , b 2 = 10 V -1 , b 22 = 10 V -1 , k 1 = 10 -9 cm -2 s -1 , k 2 = 1.5 ϫ 10 -2 cm -2 s -1 ,k 22 = 1.5 ϫ 10 -3 cm -2 s -1 , k 3 = 5 ϫ 10 -4 cm -2 s -1 , ␦ = 20 m, D = 3 ϫ 10 -6 cm -2 s -1 , C dl = 20 F cm -2 , and ␤ = 1.5 ϫ 10 -3 mol cm -2 .


Vicky Mei

Wen Huang

Vincent Vivier

Mark E Orazem

Nadine Pébère

Bernard Tribollet

The Apparent Constant-Phase-Element Behavior of a Disk Electrode with Faradaic Reactions A Global and Local Impedance Analysis

Geometry-induced current and potential distributions modify the global impedance response of a disk electrode subject to faradaic reactions. The problem was treated for both linear and Tafel kinetic regimes. The apparent capacity of a disk electrode embedded in an insulating plane was shown to vary considerably with frequency. At frequencies above the characteristic frequency for the faradaic reaction, the global impedance response has a quasi-constant-phase element ͑CPE͒ character, but with a CPE coefficient ␣ that is a function of both dimensionless frequency K and dimensionless current density J. For small values of J, ␣ approached unity, whereas, for larger values of J, ␣ reached values near 0.78. The calculated values of ␣ are typical of those obtained in impedance measurements on disk electrodes. For determining the interfacial capacitance, the influence of current and potential distributions on the impedance response cannot be neglected, even if the apparent CPE exponent ␣ has values close to unity. Several methods taken from the literature were tested to determine their suitability for extracting interfacial capacitance values from impedance data on disk electrodes. The best results were obtained using a formula which accounted for both ohmic and charge-transfer resistances.

The geometry of an electrode frequently constrains the distribution of current density and potential in the electrolyte adjacent to the electrode in such a way that both cannot simultaneously be uniform. The primary and secondary current and potential distributions associated with a disk embedded in an insulating plane have been developed by Newman. 1,2 Newman showed that the potential distribution on the disk electrode is not uniform under conditions where the current density is uniform and, conversely, the current distribution is nonuniform under the primary condition where the solution potential is uniform. 2,3 The nonuniform current and potential distribution associated with the disk geometry influences the transient and the impedance response. Nisancioglu and Newman 4,5 modeled the transient response of a disk electrode to step changes in current. The solution to Laplace's equation was performed using a transformation to rotational elliptic coordinates and a series expansion in terms of Legendre polynomials. Antohi and Scherson have recently expanded the solution to the transient problem by expanding the number of terms used in the series expansion. 6 Newman demonstrated that geometry-induced current and potential distributions cause a frequency dispersion that distorts the impedance response of a disk electrode. 7 Nisancioglu showed the extent to which this frequency dispersion causes an error in the values for charge-transfer resistance and interfacial capacitance obtained from impedance data. 8,[START_REF] Nisancioglu | The Measurement and Correction of Electrolyte Resistance in Electrochemical Tests[END_REF] The discussion by Nisancioglu 8,[START_REF] Nisancioglu | The Measurement and Correction of Electrolyte Resistance in Electrochemical Tests[END_REF] and by Newman 7 did not address the common practice of describing nonideal impedance response in terms of constant-phase elements.

The impedance response typically reflects a distribution of reactivity that is commonly represented in equivalent electrical circuits as a constant-phase element ͑CPE͒. [START_REF] Brug | [END_REF][START_REF] Macdonald | Impedance Spectroscopy: Emphasizing Solid Materials and Systems[END_REF][START_REF] Lasia | Modern Aspects of Electrochemistry[END_REF] The impedance associated with a simple faradaic reaction without diffusion can be expressed in terms of a CPE as

Z͑͒ = R e + R t 1 + ͑j͒ ␣ QR t ͓1͔
where the parameters ␣ and Q are independent of frequency. When ␣ = 1, Q has units of a capacitance, i.e., F/cm 2 , and represents the capacity of the interface. When ␣ 1, the system shows behavior that has been attributed to surface heterogeneity [START_REF] Lukacs | [END_REF]14 or to continuously distributed time constants for charge-transfer reactions. [15][16][17][18][19] Independent of the cause of CPE behavior, the phase angle associated with a CPE is independent of frequency. Huang et al. 20 have shown that current and potential distributions induce a high-frequency pseudo-CPE behavior in the global impedance response of an ideally polarized blocking electrode with a local ideally capacitive behavior. In a related work, Huang et al. 21 explored the role of current and potential distribution on the global and local impedance response of a blocking electrode exhibiting a local CPE behavior. They were able to relate the global impedance response to local impedance, and distinctive features of the calculated global and local impedance response were verified experimentally.

Using both global and local impedance measurements on a magnesium alloy, Jorcin et al. 22 have shown that the geometry of a disk in an insulating plane can induce CPE behavior and that this CPE behavior can be associated with a radial distribution of local resistance. The authors proposed that these results could be explained in terms of the numerical and analytic treatment for the impedance response of a disk electrode presented by Newman 7 and by Nisancioglu. 8,[START_REF] Nisancioglu | The Measurement and Correction of Electrolyte Resistance in Electrochemical Tests[END_REF] The objective of the present work was to explore the role of current and potential distribution on the global and local impedance response of an electrode exhibiting a faradaic behavior. A second objective was to describe the role of the resulting geometry-induced frequency dispersion in terms of CPE behavior.

Mathematical Development

The mathematical development followed that presented by Newman. 7 The development in terms of rotational elliptic coordinates, i.e.

r = r 0 ͱ ͑1 + 2 ͒͑1 -2 ͒ ͓ 3͔
was summarized by Huang et al. for blocking electrodes. 20,21 The key difference between the present work and that described by Huang et al. 20,21 was the boundary condition applied at the electrode surface.

The problem was solved for two kinetic regimes. Under linear kinetics, following Newman 7 and Nisancioglu, 8,[START_REF] Nisancioglu | The Measurement and Correction of Electrolyte Resistance in Electrochemical Tests[END_REF] the current density at the electrode surface could be expressed as

i = C 0 ץ ͑V -⌽ 0 ͒ ץ t + ͑␣ a + ␣ c ͒i 0 F RT ͑V -⌽ 0 ͒ = - ץ ⌽ ץ y ͯ y=0 = - r 0 ץ ⌽ ץ ͯ =0 ͓4͔
The assumption of linear kinetics applies for i ¯ i 0 . Under the assumption of Tafel kinetics, the current density at the electrode surface could be expressed as

i = C 0 ץ ͑V -⌽ 0 ͒ ץ t -i 0 expͩ- ␣ c F RT ͑V -⌽ 0 ͒ͪ = - ץ ⌽ ץ y ͯ y=0 = - r 0 ץ ⌽ ץ ͯ =0 ͓5͔
where the current in the Tafel regime was assumed to be cathodic. A similar expression can be developed under assumption of anodic currents. The results presented here are general because the impedance results do not depend on whether the current is anodic or cathodic.

The flux boundary conditions 4 or 5 apply at the electrode surface ͑ = 0͒. The boundary conditions 4 or 5 were written in frequency-domain as K⌽ ˜0,j + J͑V ˜r -⌽ ˜0,r ͒ = -1

ͯ ץ ⌽ ˜r ץ ͯ =0 ͓6͔
and

K͑V ˜r -⌽ ˜0,r ͒ + J⌽ ˜0,j = - 1 ͯ ץ ⌽ ˜j ץ ͯ =0 ͓7͔
for linear and Tafel kinetics, respectively. Here, V ˜r represents the imposed perturbation in the electrode potential referenced to an electrode at infinity, and K is the dimensionless frequency

K = C 0 r 0 ͓8͔
Under the assumption of linear kinetics, valid for i ¯ i 0 , the parameter J was defined to be

J = ͑␣ a + ␣ c ͒Fi 0 r 0 RT ͓9͔
For Tafel kinetics, valid for i ¯ i 0 , the parameter J was defined to be a function of radial position on the electrode surface as

J͑͒ = ␣ c F͉i ¯͉͑͒r 0 RT ͓10͔
where i ¯͑͒ was obtained from the steady-state solution as

i ¯͑͒ = -i 0 exp ͭ -␣ c F RT ͓V ¯⌽ ¯0͔͑͒ ͮ ͓11͔
The local charge-transfer resistance for linear kinetics can be expressed in terms of parameters used in Eq. 9 as

R t = RT i 0 F͑␣ a + ␣ c ͒ ͓12͔
The local charge-transfer resistance for Tafel kinetics can be expressed in terms of parameters used in Eq. 11 as

R t = RT i ¯͑͒␣ c F ͓13͔
For linear kinetics, R t was independent of radial position, but under Tafel kinetics, as shown in Eq. 13, R t depended on radial position. From a mathematical perspective, the principal difference between the linear and Tafel cases was that J and R t were held constant for the linear polarization, whereas for the Tafel kinetics, J and R t were functions of radial position determined by solution of the nonlinear steady-state problem.

The relationship between the parameter J and the charge-transfer and ohmic resistances can be established using the high-frequency limit for the ohmic resistance to a disk electrode obtained by Newman, 1 i.e.

R e = r 0 4 ͓14͔
where R e has units of ⍀ cm 2 . The parameter J can therefore be expressed in terms of the ohmic resistance R e and charge-transfer resistance R t as

J = 4 R e R t ͓15͔
Large values of J are seen when the ohmic resistance is much larger than the charge-transfer resistance, and small values of J are seen when the charge-transfer resistance dominates.

The resulting set of equations were solved under the assumption of a uniform capacitance C 0 using the collocation package PDE2D developed by Sewell. [START_REF] Sewell | The Numerical Solution of Ordinary and Partial Differential Equations[END_REF] Calculations were performed for differing domain sizes, and the results reported here were obtained by extrapolation to an infinite domain size.

Results and Discussion

In a previous article in this series, Huang et al. 20 presented a notation that addressed the concepts of a global impedance, which involved quantities averaged over the electrode surface; a local interfacial impedance, which involved both a local current density and the local potential drop V ˜-⌽ ˜0͑r͒ across the diffuse double layer; a local impedance, which involved a local current density and the potential of the electrode V ˜referenced to a distant electrode, and a local ohmic impedance, which involved a local current density and potential drop ⌽ ˜0͑r͒ from the outer region of the diffuse double layer to the distant electrode. The corresponding list of symbols is provided in Table I of Huang et al. 20 The local impedance z can be represented by the sum of local interfacial impedance z 0 and local ohmic impedance z e as

z = z 0 + z e ͓16͔
Huang et al. 20,21 demonstrated for blocking disk electrodes that, while the local interfacial impedance represents the behavior of the system unaffected by the current and potential distributions along the surface of the electrode, the local impedance shows significant frequency dispersion. The local and global ohmic impedances were shown to contain the influence of the current and potential distributions.

While the calculations presented here were performed in terms of solutions of Laplace's equation for a disk geometry, the nature of the electrode-electrolyte interface can be understood in the context of the schematic representation given in Fig. 1. Under linear kinetics, both C 0 and R t were independent of radial position, whereas for Tafel kinetics, 1/R t varied with radial position in accordance with the current distribution presented in Fig. 2.

The calculated results for global impedance, local impedance, local interfacial impedance, and both local and global ohmic impedances are presented in this section.

Global impedance.-The calculated global impedance response is presented in Fig. 3a for J = 0.1 and in Fig. 3b for J = 1.0 with dimensionless frequency K as a parameter. The real and imaginary components are presented in dimensionless form. The impedance results for linear kinetics at J = 0.1 match closely the impedance response

Z r 0 = 1 4 + 1 1/J 1 + jK/J ͓17͔
calculated in terms of the dimensionless groups used in the present work in the absence of frequency dispersion associated with the geometry-induced current and potential distributions. The impedance response for Tafel kinetics differs because the charge-transfer resistance is a function of radial position. The comparison between the impedance for linear kinetics and Eq. 17 for J = 1 shows the distortion of the high-frequency impedance response associated with the influence of current and potential distributions.

The calculated results for linear kinetics in Fig. 3 show good agreement to the corresponding numerical values obtained by Newman. 7 The comparison with Newman's calculations is seen more clearly in the representation of the real and imaginary parts of the impedance response shown in Fig. 4a and4b, respectively. At low frequencies, values for the real part of the impedance differ for impedance calculated under the assumptions of linear and Tafel kinetics, whereas the values of the imaginary impedance calculated under the assumptions of linear and Tafel kinetics are superposed for all frequencies. The slope of the lines presented in Fig. 4b are equal to +1 at low frequencies but differ from -1 at high frequencies. The slope of these lines in the high-frequency range can be related to the exponent ␣ used in models for CPE behavior. [START_REF] Orazem | [END_REF] The calculated derivative of log͑Z j /r 0 ͒ with respect to log͑K͒ ͑taken from Fig. 4b͒ is presented in Fig. 5a as a function of K with J as a parameter. At large frequencies, the quantity d log͑Z j /r 0 ͒/d log͑K͒ can be considered to be equal to -␣ where ␣ is the exponent used for models of CPE behavior. The characteristic frequency where the value of slope deviates from unity increases with the dimensionless parameter J. The transition frequencies correspond to the inverse of the R t C 0 time constant and, as seen in Fig. 5b, overlap when given as a function of

K J = C 0 RT i ¯␣c F = R t C 0 ͓18͔
The characteristic frequency for the change in slope is at K/J = 1, i.e., at the frequency max = 1/R t C 0 . The functional dependence of d log͑Z j /r 0 ͒/d log͑K͒ was independent of assumption of either linear or Tafel kinetics. Two characteristic frequencies are, therefore, evident in Fig. 5. The characteristic frequency K/J = 1 is associated with the R t C 0 time constant for the faradaic reaction, and the characteristic frequency for the influence of current and potential distributions K = 1 is associated with the capacitance C 0 and the ohmic resistance given as Eq. 14.

Local interfacial impedance.-The calculated local interfacial impedance for Tafel kinetics with J = 1 is presented in Fig. 6 as a function of frequency with normalized radial position on the disk r/r 0 as a parameter. At low frequencies, the local interfacial impedance, for both real and imaginary, is smallest at the periphery and largest at the center of the disk. The variation at low frequencies is less distinguishable for smaller values of J. All the curves in Fig. 6a and b are superposed at frequencies K Ͼ 1.

For the linear kinetics calculation, where J is independent of radial position, the scaled real part of the local interfacial impedance follows

z 0,r r 0 = J ͑J 2 + K 2 ͒ ͓19͔
and the imaginary part of the local interfacial impedance follows

z 0,j r 0 = -K ͑J 2 + K 2 ͒ ͓20͔
Plots similar to Fig. 7 were obtained for the local interfacial impedance calculated under the assumption of linear kinetics, but for linear kinetics the local interfacial impedance was independent of radial position.

The local interfacial impedance for Tafel kinetics with J = 1 is presented in Fig. 7 as a function of normalized radial position with frequency as a parameter. The lines for K = 0.01 and K = 100 are superposed in both real and imaginary parts of the local interfacial impedance ͑Fig. 7a and b, respectively͒. The real part of the impedance, presented in Fig. 7a, shows a distribution at low frequencies due to the variation of the steady-state current density. Under the Tafel kinetics assumption that J is a function of radial position, as shown in Fig. 7, the real and imaginary parts of the local interfacial impedance change around the values given in Eq. 19 and 20 and have minimum values at the periphery of the disk.

Local impedance.-The calculated local impedance for Tafel and linear kinetics with J = 1 is presented in Fig. 8 in Nyquist format with radial position as a parameter. In both cases, the impedance is largest at the center of the disk and smallest at the periphery, reflecting the greater accessibility of the periphery of the disk electrode. Similar results were also obtained for J = 0.1, but the differences between radial positions were much less significant. Inductive loops are observed at high frequencies and these are seen in both Tafel and linear calculations for J = 0.1 and J = 1.0.

The real and imaginary parts of the local impedance for Tafel kinetics with J = 1.0 are presented in Fig. 9a andb, respectively. The real part of the local impedance presented in Fig. 9a reaches asymptotic values at K → 0 and K → 100. The absolute value of the imaginary part presented in Fig. 9b shows the change of sign associated with the inductive features seen in Fig. 8a. The changes in sign occur at frequencies below K = 100, indicating that the inductive loop cannot be attributed to calculation artifacts.

Local ohmic impedance.-The local ohmic impedance z e accounts for the difference between the local interfacial and the local impedances. The calculated local ohmic impedance for Tafel kinetics with J = 1.0 are presented in Fig. 10 in Nyquist format with normalized radial position as a parameter. The results obtained here for the local ohmic impedance are very similar to those reported for the ideally polarized electrode and for the blocking electrode with local CPE behavior. 20,21 At the periphery of the electrode, two time constants ͑inductive and capacitive loops͒ are seen, whereas at the electrode center only an inductive loop is evident. These loops are distributed around the asymptotic real value of 1/4.

The calculated values for real and imaginary parts of the local ohmic impedance are presented in Fig. 11a andb, respectively, as a function of frequency with radial position as a parameter. The local ohmic impedance has only real values at K → 0 and K → ϱ, but in the frequency range 10 -2 Ͻ K Ͻ 100, z e has both real and imaginary components. This range of dimensionless frequency was not dependent on the value of J. The local ohmic impedance obtained for linear kinetics and for different J were similar to the results reported here for J = 1.

The representation of an ohmic impedance as a complex number represents a departure from standard practice, and the related insights constitute a major contribution of the present work. As shown in previous sections, the local impedance has inductive features that are not seen in the local interfacial impedance. These inductive features are implicit in the local ohmic impedance. As similar results were obtained for ideally polarized 20 and blocking electrodes with local CPE behavior, 21 the result cannot be attributed to faradaic reactions and can be attributed only to the ohmic contribution of the electrolyte.

Global interfacial and global ohmic impedance.-The global interfacial impedance for linear kinetics is independent of radial position and is given by

Z 0 = R t 1 + jC 0 R t ͓21͔
The global ohmic impedance Z e is obtained from the global impedance Z by the expression

Z e = Z -Z 0 ͓22͔
The real part of Z e , obtained for linear kinetics, is given in Fig. 12a, and the imaginary part of Z e is given in Fig. 12b as functions of dimensionless frequency K with J as a parameter. In the low frequency range Z e /r 0 is a pure resistance with a numerical value that depends weakly on J. All curves converge in the high frequency range such that Z e /r 0 tends towards 1/4. The imaginary part of the global ohmic impedance shows a nonzero value in the frequency range that is influenced by the current and potential distributions. At high and low frequency limits, the global ohmic impedance defined in the present work is consistent with the accepted understanding of the ohmic resistance to current flow to a disk electrode. The global ohmic impedance approaches, at high frequencies, the primary resistance for a disk electrode ͑Eq. 14͒ described by Newman. 1 This result was obtained as well for ideally polarized 20 and blocking electrodes with local CPE behavior. 21 The global ohmic impedance approaches, at low frequencies, the value for the ohmic resistance calculated by Newman 7 for a disk electrode. Again, this result was seen as well for blocking electrodes. 20,21 The complex nature of both the global and local ohmic impedances is seen at intermediate frequencies. This complex value is the origin of the inductive features calculated for the local impedance and the quasi-CPE behavior found at high frequency for the global impedance.

Interpretation of Impedance Results

In 1987, Nisancioglu estimated the error caused by frequency dispersion in evaluating physical properties such as charge-transfer resistance and capacitance. 8 A parallel analysis is presented here in terms of the commonly used CPE models.

Determination of charge-transfer resistance.-The impedance response of a disk electrode in the absence of frequency dispersion associated with the geometry-induced current and potential distributions can be expressed by Eq. 17. The corresponding charge-transfer resistance evident at low frequencies is given by

R t r 0 = 1 J ͓23͔
The effective global charge-transfer resistance can be estimated from the calculated impedance data according to

R eff r 0 = ͯ Z r r 0 ͯ K=0 - 1 4 ͓24͔
The value of R eff /R t is presented in Fig. 13 as a function of J under the assumption of linear kinetics. The results are in full agreement with those presented in a different format by Nisancioglu. 8 The influence of the frequency dispersion is greatest when J is large, i.e., when the ohmic resistance dominates over the charge-transfer resistance. At J = 100, an error of 75% is seen in the estimation of the charge-transfer resistance.

Determination of capacitance.-The evaluation of interfacial capacitance is perhaps better done in terms of the CPE. The values of ␣ and 1 -␣ obtained from Fig. 5a are presented in Fig. 14 as functions of J. The value of ␣ ranges from 0.98 for J = 0.01 to 0.87 for J = 10, which demonstrates that nonuniform current and potential distributions on a disk electrode can yield high-frequency CPElike behavior. As J becomes small, i.e., as the charge-transfer resistance dominates over the ohmic resistance, ␣ tends toward unity. It is significant that the calculated value of ␣ shown in Fig. 14 corresponds to a range of ␣ that is frequently observed in experiments.

The effective CPE coefficient Q eff for electrochemical systems can be obtained from the imaginary part of the impedance as

Q eff = sinͩ ␣ 2 ͪ -1 Z j ␣ ͓25͔
The value of effective CPE coefficient, scaled by the interfacial capacitance, is presented in Fig. 15 as a function of J. The frequencies reported in Fig. 15 are limited to those that are one decade larger than the characteristic frequency ͑ max or K/J = 1͒ because, in this frequency range, the value of ␣ is well-defined. Figure 15 was developed taking into account the observation, seen in Fig. 5a, that the value of ␣ is dependent on the frequency at which the slope is evaluated. Thus, the value of Q eff reported is that corresponding to the value of ␣ at a given frequency K.

Figure 15 is presented here because, while the dimensions are not exactly that of a capacitance, the CPE coefficient is often assumed to have approximately the same numerical value as the interfacial capacitance. The value of Q eff is a function of frequency. At high frequencies, where frequency dispersion plays a significant role, the effective CPE coefficient Q eff provides an inaccurate estimate for the interfacial capacitance, even for small values of J where ␣ is close to unity. The errors in estimating the interfacial capacitance are on the order of 500% at K = 100.

A number of researchers have explored the relationship between CPE parameters and the interfacial capacitance. Hsu and Mansfeld 25 proposed

C eff = Q͑ max ͒ ␣-1 = Qͩ K max C 0 r 0 ͪ ␣-1

͓26͔

where max ͑or K max ͒ is the characteristic frequency at which the imaginary part of the impedance reaches its maximum magnitude and C eff is the estimated interfacial capacitance. Equation 26 is tested against the input value of interfacial capacitance in Fig. 16 where C 0 is the known interfacial capacitance which was independent of radial position. As described above, Fig. 16 was developed using local frequency-dependent values of ␣ and Q eff . The frequencies reported in Fig. 16 are limited to those that are one decade larger than the characteristic frequency max . While Eq. 26 represents an improvement as compared to direct use of the CPE coefficient Q eff , the errors in estimating the interfacial capacitance depend on both J and K and range between -70 and +100%. Brug et al. [START_REF] Brug | [END_REF] developed a relationship for a blocking electrode between the interfacial capacitance and the CPE coefficient Q as A similar relationship between the interfacial capacitance and the CPE coefficient Q was developed for a faradaic system as

C eff = ͫ Qͩ 1 R e + 1 R t ͪ ͑␣-1͒ ͬ 1/␣ = ͭ Q ͫ 1 R e ͩ1 + J 4 ͪͬ ͑␣-1͒ ͮ 1/␣

͓28͔

Equations 27 and 28 are compared to the expected value of interfacial capacitance in Fig. 17a and b, respectively. Figure 17a andb was developed using local frequency-dependent values of ␣ and Q eff over the same frequency range as is reported in Fig. 15 and 16. The frequencies reported in Fig. 17 are limited to those that are one decade larger than the characteristic frequency max . The error in Eq. 27 is a function of both frequency K and J. While Eq. 27 applies strictly for a blocking electrode, it gives the correct answer for faradaic systems if one chooses to calculate ␣ at frequencies K Ͻ 5 but fails for K Ͼ 5. The dependence on J is reduced significantly when both the ohmic resistance R e and charge-transfer resistance R t are taken into account, and the errors in estimating interfacial capacitance are less than 20%. The correction for R t in Eq. 28 is important for frequencies K Ͼ 5. Of the relationships tested, Eq. 28 provides the best means for estimating interfacial capacitance when frequency dispersion is significant.

Conclusions

Geometry-induced current and potential distributions modify the global impedance response of a disk electrode subject to faradaic reactions. The calculated local and ohmic impedances are shown to provide insight into the frequency dispersion associated with the geometry of disk electrodes. While the calculated global impedance is purely capacitive, the local impedance has high-frequency inductive loops. The local impedance is influenced by the local ohmic impedance, which has complex behavior near dimensionless frequency K = 1. The imaginary part of both the local and global ohmic impedances is equal to zero at both high and low frequencies where the ohmic impedance has purely resistive character.

The results of the present work support the conclusion offered by Newman 7 that the apparent capacity of a disk electrode embedded in an insulating plane varies considerably with frequency. At frequencies above the characteristic frequency for the faradaic reaction, the global impedance response has a quasi-CPE character but with a CPE coefficient ␣ that is a function of both frequency K and dimensionless exchange current density J. For small values of J, ␣ approached unity, whereas for larger values of J, ␣ reached values near 0.78. The calculated values of ␣ are typical of those obtained in impedance measurements on disk electrodes.

Several methods were tested to determine their suitability for extracting interfacial capacitance values from impedance data on disk electrodes. The best results were obtained using the formula ͑Eq. 28͒ offered by Brug et al. [START_REF] Brug | [END_REF] which accounted for both ohmic and charge-transfer resistances. This work showed that, for determining the interfacial capacitance, the influence of current and potential distributions on the impedance response cannot be neglected, even if the apparent CPE exponent ␣ has values close to unity.

Figure 1 .

 1 Figure 1. Schematic representation of an impedance distribution for a disk electrode where z e represents the local ohmic impedance, C 0 represents the interfacial capacitance, and R t represents the charge-transfer resistance.

Figure 2 .

 2 Figure 2. Secondary current distribution for Tafel polarization at a disk electrode with J as a parameter.

Figure 3 .

 3 Figure 3. Calculated Nyquist representation of the impedance response for a disk electrode under assumptions of ͑solid lines͒ Tafel and ͑dashed lines͒ linear kinetics. Open circles represent the result calculated by Newman. 7 ͑a͒ J = 0.1 and ͑b͒ J = 1.0.

Figure 4 .

 4 Figure 4. Calculated representation of the impedance response for a disk electrode under assumptions of Tafel and linear kinetics and with J as a parameter. Open symbols represent the result calculated by Newman: 7 ͑a͒ real part and ͑b͒ imaginary part.

Figure 5 .

 5 Figure 5. The calculated derivative of log͑Z j /r 0 ͒ with respect to log͑K͒ ͑taken from Fig. 4b͒ with J as a parameter: ͑a͒ as a function of K and ͑b͒ as a function of K/J ͑see Eq. 18͒.

Figure 6 .

 6 Figure 6. Calculated representation of the local interfacial impedance sponse for a disk electrode as a function of dimensionless frequency K under assumptions of Tafel kinetics with J = 1.0: ͑a͒ real part and ͑b͒ imaginary part.Figure 7. Calculated representation of the local interfacial impedance response for a disk electrode as a function of radial position under assumptions of Tafel kinetics with J = 1.0: ͑a͒ real part and ͑b͒ imaginary part.

Figure 7 .

 7 Figure 6. Calculated representation of the local interfacial impedance sponse for a disk electrode as a function of dimensionless frequency K under assumptions of Tafel kinetics with J = 1.0: ͑a͒ real part and ͑b͒ imaginary part.Figure 7. Calculated representation of the local interfacial impedance response for a disk electrode as a function of radial position under assumptions of Tafel kinetics with J = 1.0: ͑a͒ real part and ͑b͒ imaginary part.

Figure 8 .

 8 Figure 8. Calculated representation of the local impedance response for a disk electrode as a function of dimensionless frequency K under assumptions of Tafel kinetics with J = 1.0: ͑a͒ Tafel kinetics and ͑b͒ linear kinetics.

Figure 9 .

 9 Figure 9. Calculated representation of the local impedance response for a disk electrode as a function of dimensionless frequency K with J = 1.0: ͑a͒ real part and ͑b͒ absolute value of the imaginary part.

Figure 10 .

 10 Figure 10. Calculated representation of the local ohmic impedance response for a disk electrode as a function of dimensionless frequency K under assumptions of Tafel kinetics with J = 1.0

Figure 11 .

 11 Figure 11. Calculated representation of the local ohmic impedance response for a disk electrode as a function of dimensionless frequency K under assumptions of Tafel kinetics with J = 1.0: ͑a͒ real part and ͑b͒ imaginary part.

Figure 12 .

 12 Figure 12. Calculated global ohmic impedance response for a disk electrode as a function of dimensionless frequency for linear kinetics for J = 0.1 and J = 1.0: ͑a͒ real part and ͑b͒ imaginary part.

Figure 13 .

 13 Figure 13. The apparent value of R eff /R t obtained from the calculated impedance response at low frequencies as a function of J.

Figure 17 .

 17 Figure 17. Normalized effective capacitance calculated from relationships presented by Brug et al. 10 for a disk electrode as a function of dimensionless frequency K with J as a parameter: ͑a͒ with correction for ohmic resistance R e ͑Eq. 27͒ and ͑b͒ with correction for both ohmic resistance R e and chargetransfer resistance R t ͑Eq. 28͒.

Figure 14 .

 14 Figure 14. The apparent value of 1 -␣ obtained from the calculated impedance response at high frequencies as a function of J.

Figure 15 .

 15 Figure 15. Effective CPE coefficient scaled by the interfacial capacitance as a function of J.

Figure 16 .

 16 Figure 16. Effective capacitance calculated from Eq. 26 and normalized by the input interfacial capacitance for a disk electrode as a function of dimensionless frequency K with J as a parameter ͑see Hsu and Mansfeld. 25 ͒.

Acknowledgment

The authors gratefully acknowledge the financial support of their home institutions, which facilitated this collaboration.

Centre National de la Recherche Scientifique assisted in meeting the publication costs of this article.


Vicky Mei

Wen Huang

Vincent Vivier

Mark E Orazem

Nadine Pébère

Bernard Tribollet

The Apparent Constant-Phase-Element Behavior of an Ideally Polarized Blocking Electrode A Global and Local Impedance Analysis

Two numerical methods were used to calculate the influence of geometry-induced current and potential distributions on the impedance response of an ideally polarized disk electrode. A coherent notation is proposed for local and global impedance which accounts for global, local, local interfacial, and both global and local ohmic impedances. The local and ohmic impedances are shown to provide insight into the frequency dispersion associated with the geometry of disk electrodes. The high-frequency global impedance response has the appearance of a constant-phase element ͑CPE͒ but can be considered to be only an apparent CPE because the CPE exponent ␣ is a function of frequency.

Both primary and secondary current and potential distributions associated with the disk geometry have been developed by Newman. 1,2 The primary distribution applies when the current and potential are governed by the ohmic resistance to current flow in the electrolyte. The secondary distribution applies when both ohmic and kinetic resistances are controlling. In the absence of faradaic reactions and for very short time scales, the primary current distribution associated with the charging of an electrode surface can be expected to follow

i ͗i͘ = 1 2ͱ1 -ͩ r r 0 ͪ 2

͓1͔

where r 0 is the radius of the disk and ͗i͘ is the average current density on the electrode. In the absence of mass-transfer limitations, the transient response of a disk electrode requires the solution of Laplace's equation with flux conditions at the electrode surface. Nisancioglu and Newman 3,4 have developed a solution for the transient response of a faradaic reaction on a nonpolarizable disk electrode to step changes in current. The solution to Laplace's equation was performed using a transformation to rotational elliptic coordinates and a series expansion in terms of Legendre polynomials. Antohi and Scherson have recently expanded the solution to the transient problem by expanding the number of terms used in the series expansion. 5 The solutions described above account for the relaxation of the potential at the electrode surface. [3][4][5] Conversely, the solution presented by Oldham 6 is incorrect because it assumes that the fixedpotential condition at the electrode surface applies for all time scales and that the local ohmic impedance is therefore a real number obtained from the primary current distribution ͑Eq. 1͒.

The impedance response of electrodes rarely show the ideal response expected for single electrochemical reactions. The impedance response typically reflects a distribution of reactivity that is commonly represented in equivalent electrical circuits as a constant phase element ͑CPE͒. [7][START_REF] Macdonald | Impedance Spectroscopy: Emphasizing Solid Materials and Systems[END_REF][START_REF] Lasia | [END_REF] For a blocking electrode, the CPE can be represented as

Z͑f͒ = R e + 1 ͑j2f͒ ␣ Q ͓2͔
where the parameters ␣ and Q are constants. When ␣ = 1, Q has units of a capacitance, i.e., F/cm 2 , and represents the capacity of the interface. When ␣ 1, the system shows behavior that has been attributed to surface heterogeneity 10,11 or to continuously distributed time constants for charge-transfer reactions. [12][13][14][15][16] The phase angle associated with a CPE is independent of frequency.

Using both global and local impedance measurements on a magnesium alloy, Jorcin et al. 17 have shown that the geometry of a disk in an insulating plane can induce CPE behavior and that this CPE behavior can be associated with a radial distribution of local resistance. The authors suggested that these results could be explained in terms of the numerical and analytic treatment for the impedance response of a disk electrode presented in 1970 by Newman. 18 The objective of this work was to explore the role of current and potential distributions on the global impedance response of an ideally polarized electrode and to relate this response to the local impedance. Subsequent papers will address the influence of current and potential distributions on the impedance response of systems exhibiting local CPE behavior 19 ͑i.e., coupled 2D and 3D distributions͒ and faradaic reactions. 20

2D and 3D Distributions

Frequency dispersion leading to CPE behavior can be attributed to distributions of time constants along either the area of the electrode ͑involving only a two-dimensional surface͒ or along the axis normal to the electrode surface ͑involving a three-dimensional aspect of the electrode͒. A 2D distribution could arise from surface heterogeneities such as grain boundaries, crystal faces on a polycrystalline electrode, or other variations in surface properties. The frequency dispersion associated with geometry-induced nonuniform current and potential distributions results from a 2D distribution.

CPE behavior may also arise from a variation of properties in the direction that is normal to the electrode surface. Such variability may be attributed, for example, to changes in the conductivity of oxide layers [21][START_REF] Young | Anodic Oxide Films[END_REF][START_REF] Schiller | [END_REF] or from porosity or surface roughness. 24,25 This CPE behavior is said to arise from a 3D distribution, with the third direction being the direction normal to the plane of the electrode. 17 A schematic representation of a 2D distribution for an ideally polarized disk electrode is presented in Fig. 1a. For a 2D distribution, the circuit parameters, e.g., capacitance and ohmic resistance, could be a function of radial position along the electrode. Integration of the admittance associated with these circuit elements would yield a global impedance with a CPE behavior. The local impedance, in the case of a 2D distribution, would, however, show ideal behavior. A 3D distribution of blocking components in terms of resistors and constant-phase elements is presented in Fig. 1b. Such a system yields a local impedance with a CPE behavior, even in the absence of a 2D distribution of surface properties. If the 3D system shown schematically in Fig. 1b is influenced by a 2D distribution, the local impedance should reveal a variation of CPE coefficients along the surface of the electrode. Thus, local impedance measurements can be used to distinguish whether the CPE behavior arises from a 2D distribution, from a 3D distribution, or from a combined 2D and 3D distribution.

Using both global and local impedance measurements on a disk made of AZ91 magnesium alloy, Jorcin et al. 17 found CPE behavior that was attributed to a 2D distribution, which yielded locally a pure capacitive response coupled with a radial distribution of local resistance. Jorcin et al. 17 have also found CPE behavior on a pure aluminum disk in which the local impedance response showed a CPE which was modified only slightly by an apparent 2D distribution.

Mathematical Development

The steady-state solution for the current distribution at a blocking electrode is that the current is equal to zero. The primary current distribution given as Eq. 1 therefore applies, not at the steady state, but at infinite frequency. This situation differs from the special case of a faradaic system with an ohmic resistance that is much larger than the kinetic resistance and for which Eq. 1 provides the steadystate current distribution.

The object of this work was to calculate, from first principles, the influence of geometry-induced current and potential distributions on the impedance response of a disk electrode. The mathematical development follows that presented by Newman. 18 Laplace's equation in cylindrical coordinates was expressed in rotational elliptic coordinates, i.e. y = r 0 ͓3͔

and

r = r 0 ͱ ͑1 + 2 ͒͑1 -2 ͒ ͓ 4͔
where 0 Յ Յ ϱ and 0 Յ Յ 1. Within the revised coordinate system, the electrode surface at y = 0 and r Յ r 0 can be found at = 0 and 0 Յ Յ 1. The reference electrode and counter electrode located at r → ϱ can be found at → ϱ. The insulating surface of the disk at y = 0 and r Ͼ r 0 is located at = 1 and 0 Ͻ Յ ϱ, and the center line at y Ͼ 0 and r = 0 is located at = 0 and 0 Ͻ Յ ϱ.

Laplace's equation can be expressed in rotational elliptic coordinates as

ץ ץ ͫ ͑1 + 2 ͒ ץ ⌽ ץ ͬ + ץ ץ ͫ ͑1 -2 ͒ ץ ⌽ ץ ͬ = 0 ͓5͔
The potential was separated into steady and time-varying parts as

⌽ = ⌽ ¯+ Re͓⌽ ˜exp͑jt͔͒ ͓6͔
where ⌽ ¯is the steady-state solution for potential and ⌽ ˜is the complex oscillating potential. Thus, Eq. 5 could be written as

2 ץ ⌽ ˜r ץ + ͑1 + 2 ͒ ץ 2 ⌽ ˜r ץ 2 -2 ץ ⌽ ˜r ץ + ͑1 -2 ͒ ץ 2 ⌽ ˜r ץ 2 = 0 ͓7͔ and 2 ץ ⌽ ˜j ץ + ͑1 + 2 ͒ ץ 2 ⌽ ˜j ץ 2 -2 ץ ⌽ ˜j ץ + ͑1 -2 ͒ ץ 2 ⌽ ˜j ץ 2 = 0 ͓8͔
where ⌽ ˜r and ⌽ ˜j refer to the real and imaginary parts of the complex oscillating potential, respectively. The flux boundary condition at the electrode surface ͑ = 0 and 0 Յ Յ 1͒ is

i = C 0 ץ ͑V -⌽ 0 ͒ ץ t = -ͯ ץ ⌽ ץ y ͯ y=0 = - r 0 ͯ ץ ⌽ ץ ͯ =0 ͓9͔
where C 0 is the interfacial capacitance and is the electrolyte conductivity. Equation 9was written in frequency domain as

K⌽ ˜j = - 1 ͯ ץ ⌽ ˜r ץ ͯ =0 ͓10͔ and KV ˜r -K⌽ ˜r = - 1 ͯ ץ ⌽ ˜j ץ ͯ =0 ͓11͔
where V ˜r represents the imposed perturbation in the electrode potential referenced to an electrode at infinity and K is the dimensionless frequency ⌽ ˜j = 0

K = C 0 r 0 ͓12͔ At = 0

͓16͔

The equations were solved under the assumption of a uniform capacitance C 0 using the collocation package PDE2D developed by Sewell. [START_REF] Sewell | The Numerical Solution of Ordinary and Partial Differential Equations[END_REF] Calculations were performed for differing domain sizes, and the results reported here were obtained by extrapolation to an infinite domain size. The equations were also solved in the cylindrical coordinates using a finite-elements package FEMLAB. The results obtained by the two packages were in excellent agreement for dimensionless frequencies K Ͻ 100.

Definition of Terms

The global impedance results presented in the following section can be understood through examination of the local impedance distribution. As there are several types of local impedance at play, discussion of local impedance requires clear notation and definitions.

A schematic representation of the electrode-electrolyte interface is given as Fig. 2, where the block used to represent the local ohmic impedance reflects the complex character of the ohmic contribution to the local impedance response. The impedance definitions presented in Table I differ in the potential and current used to calculate the impedance. To avoid confusion with local impedance values, the symbol y is used to designate the axial position in cylindrical coordinates.

Global impedance.-The global impedance is defined to be

Z = V Ĩ ˜͓17͔
where the complex current contribution is given by

I ˜= ͵ 0 r 0 i ˜͑r͒2rdr ͓18͔
The use of an upper-case letter signifies that Z is a global value. The global impedance may have real and imaginary values designated as Z r and Z j , respectively. The total current could also be represented by I ˜= r 0 2 ͗i ˜͑r͒͘, where the brackets signify the area-average of the current density.

Local impedance.-The term local impedance traditionally involves the potential of the electrode measured relative to a reference electrode far from the electrode surface. [START_REF] Lillard | [END_REF]28 Thus, the local impedance is given by

z = V ĩ ˜͑r͒ ͓19͔
The use of a lower-case letter signifies that z is a local value. The local impedance may have real and imaginary values designated as z r and z j , respectively. The global impedance can be expressed in terms of the local impedance as

Z = ͳ 1 z ʹ -1

͓20͔

Equation 20 is consistent with the treatment of Brug et al. 7 in which the admittance of the disk electrode was obtained by integration of a local admittance over the area of the disk.

Local interfacial impedance.-The local interfacial impedance involves the potential of the electrode measured relative to a reference electrode ⌽ 0 ͑r͒ located at the outer limit of the diffuse double layer. Thus, the local interfacial impedance is given by

z 0 = V ˜-⌽ ˜0͑r͒ i ˜͑r͒ ͓21͔
The use of a lower-case letter again signifies that z 0 is a local value, and the subscript 0 signifies that z 0 represents a value associated only with the surface. The local interfacial impedance may have real and imaginary values designated as z 0,r and z 0,j , respectively.

Local ohmic impedance.-The local ohmic impedance involves the potential of a reference electrode ⌽ ˜0͑r͒ located at the outer limit of the diffuse double layer and the potential of a reference electrode located far from the electrode ⌽ ˜͑ϱ͒ = 0 ͑see Fig. 2͒. Thus, the local ohmic impedance is given by

z e = ⌽ ˜0͑r͒ i ˜͑r͒ ͓22͔
The use of a lower-case letter again signifies that z e is a local value, and the subscript e signifies that z e represents a value associated only with the ohmic character of the electrolyte. The local ohmic The representation of an ohmic impedance as a complex number represents a departure from standard practice, and the related insights constitute a major contribution of the present work. As is shown in subsequent sections, the local impedance has inductive features that are not seen in the local interfacial impedance. As the calculations assumed an ideally polarized blocking electrode, the result is not influenced by faradaic reactions and can be attributed only to the ohmic contribution of the electrolyte.

Global interfacial impedance.-The global interfacial impedance is defined to be

Z 0 = 2 ͩ͵ 0 r 0 1 z 0 ͑r͒ rdr ͪ -1 ͓24͔ or Z 0 = ͳ 1 z 0 ͑r͒ ʹ -1

͓25͔

The use of an upper-case letter signifies that Z 0 is a global value. The global interfacial impedance may have real and imaginary values designated as Z 0,r and Z 0,j , respectively.

Global ohmic impedance.-The global ohmic impedance is defined to be

Z e = Z -Z 0 ͓26͔
The use of an upper-case letter signifies that Z is a value. As is shown in subsequent sections, the global ohmic impedance has a complex behavior in a mid-frequency range ͑near K = 1͒. The global ohmic impedance may have real and imaginary values designated as Z e,r and Z e,j , respectively.

Results and Discussion

The calculated results for global, local, local interfacial, and both local and global ohmic impedances are presented in this section. The results of both the collocation and finite element method ͑FEM͒ methods were in perfect agreement for frequencies K Ͻ 100.

Global impedance.-The global impedance response presented in Fig. 3a shows the influence of frequency dispersion at frequencies K Ͼ 1. This dispersion is seen as a deviation from the vertical line obtained for low frequencies ͑K Ͻ 1͒. The expanded logarithmic representation presented in Fig. 3b shows the agreement with the numerical solutions presented by Newman. 18 The impedance is made dimensionless according to Z/r 0 , in which the units of impedance Z are scaled by unit area, for example, ⍀ cm 2 .

The comparison with Newman's calculations is seen more clearly in the representation of the real and imaginary parts of the impedance response shown in Fig. 4a andb, respectively. As discussed by Orazem et al., 29 the change in the slope of the lines presented in Fig. 4b shows that the impedance response transitions from ideal RC behavior at low frequencies to a CPE-like behavior at frequencies K Ͼ 1. The slope is equal to -␣, the exponent of the CPE presented in Eq. 2. A deviation from Newman's results is seen for both the collocation and FEM calculations for frequencies K Ͼ 100. This error is attributed to a singular perturbation problem that arises at the periphery of the electrode at high frequencies. 18 The value of -␣, which is equal to the slope of the log͑Z j /r 0 ͒ with respect to log͑K͒, is presented in Fig. 5 as a function of dimensionless frequency K. The numerical results obtained by collocation and FEM methods is compared to the asymptotic limit developed by Newman using a singular perturbation approach. The system behaves as an ideal capacitor at low frequencies K Ͻ 1 with ␣ = 1. At frequencies K Ͼ 1, the value of alpha changes to roughly ␣ = 0.85 before beginning a gradual return towards unity. As the slope is not independent of frequency, the frequency dispersion seen at K Ͼ 1 does not represent true CPE behavior.

The frequency K = 1 at which the current and potential distributions begin to influence the impedance response can be expressed as

f = 2C 0 r 0 ͓27͔
or, in terms of electrolyte resistance, as

f = 1 8C 0 R e ͓28͔
As shown in Fig. 6, this characteristic frequency can be well within the range of experimental measurements. The value /C 0 = 10 4 cm/s can be obtained for a capacitance C 0 = 1 F/cm 2 ͑corresponding to an oxide layer͒ and conductivity = 0.01 S/cm ͑corresponding to a 0.1 M NaCl solution͒. The value /C = 10 3 cm/s can be obtained for a capacitance C 0 = 10 F/cm 2 ͑corresponding to the double layer on an inert metal electrode͒ and conductivity = 0.01 S/cm ͑corresponding to a 0.1 M NaCl solution͒. Figure 6 can be used to show that, by using an electrode that is sufficiently small, the experimentalist may be able to avoid the frequency range that is influenced by current and potential distributions. Local interfacial impedance.-The calculated local interfacial impedance is presented in Fig. 7a as a function of frequency with position as a parameter and in Fig. 7b as a function of position with frequency as a parameter. All four curves indicated in Fig. 7a andb, respectively, are superposed. The results presented in Fig. 7 show that the local interfacial impedance is that associated with a pure capacitive behavior. At all frequencies, z 0,j K/r 0 = 1/ as is expected for an ideal capacitance. The real part of the local interfacial impedance, not shown here, was equal to zero within computational accuracy. Local impedance.-The calculated local impedance response is presented in Fig. 8 in Nyquist format with radial position as a parameter. The dimensionless impedance is scaled to the disk area r 0 2 in order to show the comparison with the asymptotic value of 0.25 for the real part of the dimensionless impedance. The impedance is largest at the center of the disk and smallest at the periphery, reflecting the greater accessibility of the periphery of the disk electrode. Inductive loops are seen at high frequencies, and these were obtained by both methods of calculation.

The real and imaginary parts of the local impedance are presented in Fig. 9a andb, respectively, with radial position as a parameter. The real part of the local impedance presented in Fig. 9a reaches asymptotic values at K → 0 and K → ϱ. The imaginary part presented in Fig. 9b shows the change of sign associated with the inductive features seen in Fig. 8. The changes in sign occur at frequencies well below K = 100, showing that the inductive loop cannot be ascribed to a calculation artifact. The deviation from ideal capacitive behavior for frequencies K Ͼ 1 is similar to that seen in Fig. 4b for the imaginary part of the global impedance.

The radial distribution of the real and imaginary impedance is presented in Fig. 10a andb, respectively, with dimensionless frequency K as a parameter. At high frequencies, e.g., K = 100, the calculated radial distribution of the real part of the local impedance follows the expression z r r 0 ͩ r r 0 ͪ = 0.5ͱ1 -ͩ r r 0 ͪ 2 ͓29͔ derived from Eq. 1 using the expression for the primary resistance 1 in the form

R e = 1 4r 0 r 0 2 ͓30͔
The radial distribution for the imaginary part of the impedance deviates from ideal capacitive behavior for frequencies K Ͼ 1.

Local ohmic impedance.-Following Eq. 23, the local ohmic impedance z e accounts for the difference between the local interfacial and local impedances. As the local interfacial impedance corre-sponds to a pure capacitor, the term z e cannot be a pure electrolyte resistance R e . The calculated local ohmic impedance is presented in Fig. 11 in Nyquist format with radial position as a parameter. The shape of the diagrams are strongly dependent on the position on the electrode. At the periphery of the electrode, two time constants ͑inductive and capacitive loops͒ are seen, whereas at the electrode center only an inductive loop is evident. These loops are distributed around the asymptotic value of 1/4. The calculated values for real and imaginary parts of the local ohmic impedance are presented in Fig. 12a andb, respectively, as a function of frequency, with radial position as a parameter. The local ohmic impedance has only real values at K → 0 and K → ϱ, but in the frequency range 10 -2 Ͻ K Ͻ 100, z e has both real and imaginary components. Figure 12a clearly shows the asymptotic behavior in the low frequency range with values distributed around 1/4.

Global interfacial and global ohmic impedance.-The local interfacial impedance is associated with a pure capacitance that is independent of radial position. Thus, the global interfacial impedance arises from a pure capacitance C 0 in units of F/cm 2 . The global ohmic impedance Z e is obtained from the global impedance Z by the expression

Z e = Z - 1 jC 0 ͓31͔
or, in the dimensionless terms used in the present work The real part of Z e is equal to the real part of Z as given in Fig. 4a. The imaginary part of Z e is given in Fig. 13 as a function of dimensionless frequency K. In the low frequency range Z e is a pure resistance equal to 1.08R e , and in the high frequency range Z e tends towards R e . The imaginary part of the global ohmic impedance shows a nonzero value in the frequency range that is influenced by the current and potential distributions. Figures 4a and13 show that all the effects of the current and potential distribution appears in the global ohmic impedance.

At high and low frequency limits, the global ohmic impedance defined in the present work is consistent with the accepted understanding of the ohmic resistance to current flow to a disk electrode. The global ohmic impedance approaches, at high frequencies, the primary resistance for a disk electrode ͑Eq. 30͒ described by Newman. 1 The global ohmic impedance approaches, at low frequencies, the value for the ohmic resistance calculated by Newman 18 for a disk electrode.

The complex nature of both the global and local ohmic impedances is seen at intermediate frequencies. This complex value is the origin of the inductive features calculated for the local impedance and the quasi-CPE behavior found at high frequency for the global impedance.

Conclusions

The global impedance for an ideally polarized disk electrode is influenced by current and potential distributions at high frequencies. While the local interfacial impedance shows the expected ideally polarized behavior, the local impedance shows inductive behavior at high frequency and ideal behavior at low frequency. The local impedance is influenced by the local ohmic impedance, which has complex behavior near dimensionless frequency K = 1. The imaginary part of both the local and global ohmic impedance is equal to zero at both high and low frequencies where the ohmic impedance has purely resistive character.

The explanation of the frequency dispersion requires a coherent notation for local and global impedance which accounts for global, local, local interfacial, and both the local and global ohmic impedances. A notation is proposed in Table I that extends terms found in the local impedance literature to account for both local ohmic and local interfacial impedances.

The local and global ohmic impedance is shown to provide insight into the frequency dispersion associated with the geometry of disk electrodes. The high-frequency impedance response has the appearance of a CPE but can be considered to be only an apparent CPE because the CPE exponent ␣ is not independent of frequency.

Figure 1 .

 1 Figure 1. Schematic representation of an impedance distribution for a blocking disk electrode where R e represents the ohmic resistance, C 0 represents the interfacial capacitance, and z 0 represents an interfacial impedance corresponding to Eq. 2: ͑a͒ 2D distribution of blocking components in terms of resistors and capacitors and ͑b͒ 3D distribution of blocking components in terms of resistors and CPEs.

Figure 2 .

 2 Figure 2. The location of current potential terms that make up definitions of global and local impedance.

Figure 3 .

 3 Figure 3. Calculated Nyquist representation of the impedance response for an ideally polarized disk electrode: ͑a͒ linear plot showing effect of dispersion at frequencies K Ͼ 1 as a deviation from a vertical line and ͑b͒ logarithmic scale showing agreement with the calculations of Newman.

Figure 4 .

 4 Figure 4. Calculated representation of the impedance response for an ideally polarized disk electrode: ͑a͒ real part and ͑b͒ imaginary part showing agreement with the calculations and asymptotic formula of Newman.

Figure 5 .

 5 Figure5. The slope of log͑Z j /r 0 ͒ with respect to log͑K͒ ͑Fig. 4b͒ as a function of log͑K͒. The results were calculated by the collocation method, by FEM methods, and by using the asymptotic formula of Newman. The value of this slope is equal to -␣.

Figure 6 .

 6 Figure 6. The frequency K =1 at which the current distribution influences the impedance response with /C 0 as a parameter.

Figure 7 .

 7 Figure 7. Calculated imaginary part of the local interfacial impedance: ͑a͒ as a function of frequency with position as a parameter and ͑b͒ as a function of position with frequency as a parameter. All four curves indicated in Fig. 7a and b, respectively, are superposed.

Figure 8 .

 8 Figure 8. The local impedance in Nyquist format with radial position as a parameter.

Figure 9 .

 9 Figure 9. Calculated local impedance with radial position as a parameter: ͑a͒ real part and ͑b͒ imaginary part.

Figure 10 .

 10 Figure 10. Calculated local impedance as a function of radial position: ͑a͒ real part and ͑b͒ imaginary part multiplied by dimensionless frequency K. The solid line is given in Fig. 10b to indicate the axis at z j = 0.

Figure 11 .

 11 Figure 11. The local ohmic impedance in Nyquist format with radial position as a parameter.

Figure 12 .

 12 Figure 12. Calculated values for local ohmic impedance as a function of frequency with radial position as a parameter: ͑a͒ real part and ͑b͒ imaginary part. The solid line is given in Fig. 12b to indicate the axis at z e,j = 0.

Figure 13 .

 13 Figure 13. The imaginary part of the global ohmic impedance, calculated from Eq. 32, as a function of dimensionless frequency.

Table I . Notation proposed for local impedance variables.

 I

	Symbol	Meaning	Units
	Z	global impedance ͑Eq. 17͒	⍀ or ⍀ cm 2
	Z r	real part of global impedance	⍀ or ⍀ cm 2
	Z j	imaginary part of global impedance	⍀ or ⍀ cm 2
	Z 0	global interfacial impedance ͑Eq. 24͒	⍀ or ⍀ cm 2
	Z 0,r	real part of global interfacial impedance	⍀ or ⍀ cm 2
	Z 0,j	imaginary part of global interfacial impedance	⍀ or ⍀ cm 2
	Z e	global ohmic impedance ͑Eq. 26͒	⍀ or ⍀ cm 2
	Z e,r	real part of global ohmic impedance	⍀ or ⍀ cm 2
	Z e,j	imaginary part of global ohmic impedance	⍀ or ⍀ cm 2
	z	local impedance ͑Eq. 19͒	⍀ cm 2
	z r	real part of local impedance	⍀ cm 2
	z j	imaginary part of local impedance	⍀ cm 2
	z 0	local interfacial impedance ͑Eq. 21͒	⍀ cm 2
	z 0,r	real part of local interfacial impedance	⍀ cm 2
	z 0,j	imaginary part of local interfacial impedance	⍀ cm 2
	z e	local ohmic impedance ͑Eq. 22͒	⍀ cm 2
	z e,r	real part of local ohmic impedance	⍀ cm 2
	z e,j	imaginary part of local ohmic impedance	⍀ cm 2
	͗⌽͘	spatial average of potential	V
	⌽ ¯time average or steady-state value of potential	V
	͗i͘	spatial average of current density	A/cm 2
	i ¯time average or steady-state value of current density	A/cm 2
	y	axial position variable	cm

Acknowledgment

The authors gratefully acknowledge the financial support of their home institutions, which facilitated this collaboration.

Centre National de la Recherche Scientifique assisted in meeting the publication costs of this article.