Diffusion Impedance in a Thin-Layer Cell - American Chemical Society

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J. Phys. Chem. C 2008, 112, 4626-4634

Diffusion Impedance in a Thin-Layer Cell: Experimental and Theoretical Study on a Large-Disk Electrode E. Remita,†,‡ D. Boughrara,† B. Tribollet,† V. Vivier,*,† E. Sutter,† F. Ropital,‡ and J. Kittel‡ CNRS - UPR 15, Laboratoire Interfaces et Syste` mes Electrochimiques, UniVersite´ Pierre et Marie Curie, Case Courrier 133, 4 Place Jussieu, Paris, F-75005 France, and IFP, BP n°3, 69390 Vernaison, France ReceiVed: October 29, 2007; In Final Form: January 22, 2008

Using an accurate thin-layer cell, the electrochemical impedance corresponding to the diffusion of ferrocyanide ions was measured on a large-disk electrode covered by thin films of electrolyte. Experimental impedance diagrams obtained in a large frequency range (100 kHz-5 mHz) were in good agreement with those expected in the case of a spatially restricted linear diffusion impedance. For electrolyte film thicknesses lower than 100 µm, an increase of the low-frequency limit of the real part of the impedance and a decrease of the high-frequency phase angle were however observed experimentally as the electrolyte film thickness decreases. This specific behavior, in disagreement with the classical linear diffusion theory, was explained by the potential distribution existing in the thin-layer cell and was modeled using an original transmission line. In a very low-frequency range (lower than 1 mHz), an additional time constant was also evidenced. In consistency with previous work of Gabrielli et al.1 on microelectrodes, this feature was ascribed to the radial contribution of diffusion processes existing within a thin-layer cell.

Introduction cells2-5

are experimental devices of interest to Thin-layer perform microanalysis,4 in situ photospectroscopy measurements during electrochemical reactions,6 or to simulate some specific corrosion situations.7-9 They are usually achieved by confining a thin electrolyte layer (the thickness of which is lower than a few hundreds of micrometers) between the surface of a working electrode and a parallel waterproof wall which is mechanically set to reach the desired thickness.3 The most classical thin-layer cells 2-5,7-10 have a cylindrical geometry and involve large disk electrodes (with a radii of several millimeters). The geometrical accuracy is generally limited by the difficulty to achieve a strict parallelism between the electrode surface and the wall ensuring the electrolyte confinement.10,11 Using a positioning control procedure based on the measurement of the electrolyte resistance, parallelism errors could however be accurately quantified and then minimized in these cells.10 Another possibility consists of using an ultramicroelecrode12 (UME) as a working electrode to reduce the dimension of the thin-layer cell. Such a thin-layercell configuration was investigated recently by Gabrielli et al.1 with a scanning electrochemical microscope (SECM) used in feedback mode.13 In a thin-layer cell, the residual convection (“natural convection”) that always exists in a large volume of solution14 becomes negligible.10,15,16 When the migration processes of the electroactive species can also be neglected (for instance in the presence of a supporting electrolyte), the mass transport is purely achieved by diffusion and is governed by Fick’s laws

J ) - D∇c * Corresponding author. E-mail: [email protected]. † CNRS. ‡ IFP.

(1)

∂c ) - D∇2c ∂t

(2)

where J is the flux of the diffusing species and D and c are the diffusion coefficient and the concentration of the electroactive species, respectively. For one-dimensional diffusion processes, Fick’s equations usually admit simple analytical solutions. Such solutions have been proposed already for various cell configurations.11,15-17 Conversely, solving Fick’s equations in cylindrical coordinates leads generally to very tedious numerical calculations,18 and thus, cylindrical thin-layer cells are commonly assimilated to single dimension systems in order to simplify models.11,15,16 Electrochemical impedance spectroscopy19,20 (EIS) is a powerful electrochemical technique allowing the mass transport of electroactive species in the vicinity of electrodes to be investigated.21 In thin-layer-cell configuration, the mass transport impedance can be calculated from Fick’s equations (eqs 1 and 2). Assuming that diffusion processes occur only along the normal direction of the electrode surface8,22 (1D diffusion), analytical expression of the diffusion impedance can be obtained. However, for cylindrical geometry, this diffusion impedance cannot be expressed rigorously with simple analytical expression and should be calculated numerically.1 In any case, the diffusion impedance depends obviously on the boundary conditions considered in the calculations22-29 (see the next section). Moreover, boundary conditions can also account for a time constant distribution within the layer of ionic conductor covering the electrode27,28 or for the kinetics of homogeneous chemical reactions coupled to the diffusion processes.23-25 From a more general point of view, the work of Keddam et al.7,8 also demonstrated that the impedance measured in a thinlayer-cell configuration is significantly modified by the existence of a radial potential distribution within the liquid film confined over the electrode surface. Taking the case of the charge-transfer process occurring on a corroding Fe-Ni alloy disk-electrode

10.1021/jp710407a CCC: $40.75 © 2008 American Chemical Society Published on Web 03/05/2008

Diffusion Impedance in a Thin-Layer Cell (10 mm in diameter) as an example, these authors provided experimental evidence that such potential distribution effects exist in cylindrical thin-layer cells involving large disk electrodes. This effect was ascribed to the ohmic drop that takes place within the liquid film covering the electrode surface. To our best knowledge, such radial potential distribution effect was however never taken into account until now by models dealing with the diffusion impedance in thin-layer-cell configurations. At a different scale, Gabrielli et al.1 investigated recently the particular case of a cylindrical thin-layer cell involving platinum microdisk electrodes of 10 µm in diameter (the radius of the confined zone being equal to 50 µm). By coupling EIS measurements and 2D finite element method (FEM) modeling, they demonstrated that the bidimensional character of mass transport cannot be neglected in a cylindrical thin-layer cell. Indeed, in accordance with the predictions of their model, they showed experimentally that, in addition to the spherical diffusion impedance associated with the microelectrode geometry, the radial diffusion existing within the cell also appeared in the mass transport impedance as an additional low-frequency contribution. This work evidenced the limits of use of the classical 1D diffusion models,7,8,22 which neglect by nature radial mass transport and consequently lead to an inaccurate calculation of the impedance response in cylindrical thin-layer cells. Surprisingly, such EIS investigations of mass transport have never been performed in thin-layer cells involving large disk electrodes and, more generally, very few papers7-9 reported impedance measurements in such electrochemical cells. The problem is however of interest because even by standing at the micrometric scale Gabrielli et al.1 showed that the size effect associated with the characteristic dimensions of the thin-layer cells (thickness, electrode diameter, and diameter of the insulating part surrounding the electrode) could affect drastically the diffusion impedance. Moreover, the nature of the mass transport is different in the vicinity of a large disk electrode (planar diffusion) and in the vicinity of a microelectrode (spherical diffusion). Thus, in the case of a thin-layer cell involving a large disk electrode, a diffusion impedance response different from the one measured on a microelectrode could be expected. At last, even if it was never investigated rigorously in the case of the diffusion impedance, the potential distribution effect evidenced by Keddam et al.7,8 in cylindrical thin-layer cells involving large disk electrodes may also affect the diffusion impedances measured in these kinds of cells. In this context, the aim of this work is to investigate mass transport in cylindrical thin-layer cells involving large disk electrodes using EIS. In addition to this experimental approach, a transmission line model was devised to take into account the radial potential drop existing within the cell in the theoretical diffusion impedance calculation. Conversely to the analytical model proposed previously by Keddam et al.,8 the cylindrical geometry of the electrochemical cell was rigorously taken into account in this new mathematical development. In this way, modified 1D spatially restricted diffusion impedances were calculated in a first step by assuming that mass transport proceeds only normally to the electrode surface. In a second step, both normal and radial mass transport contributions were considered. In the last section of this paper, the agreement between experiments and theory is discussed. Experimental Section The cylindrical thin-layer cell used in this study is sketched in Figure 1. In order to guarantee a good mechanical stability during measurements, we supported this setup with an antivi-

J. Phys. Chem. C, Vol. 112, No. 12, 2008 4627

Figure 1. Scheme of the cylindrical thin-layer cell used for the impedance measurements.

brating table. The working electrode was the cross section of a platinum cylinder of 5 mm in diameter, which was sealed into an epoxy resin to form a cylindrical insulating holder (30 mm in diameter). Prior to use, the electrode was successively polished with a 4000 grit silicon carbide paper, cleaned with ethanol, rinsed with deionized water, and dried with a N2 flux. The electrode holder was fixed at the bottom of a glass cell. The glass cell/electrode holder assembly was attached to a micropositioning system that consisted of a bidirectional x-y translation stage mounted on a three-axis rotation platform. Thanks to this cell arrangement, both the position and the spatial orientation of the electrode surface could be reached with an accuracy of 2 × 10-7 degrees on the angular settings and a precision of 1 µm on the linear positioning. Facing the platinum working electrode, a mobile waterproof PTFE cylinder (30 mm in diameter) insured the confinement of the electrolyte. This cylinder was attached to a z-translation stage allowing vertical displacements of the rod with an accuracy as small as 1 µm. The geometry of the confined zone facing the working electrode surface was set according to the positioning procedure proposed in a previous work.10 Thanks to this method, the average electrolyte film thickness was set with a precision estimated at (5 µm. The impedance measurements were carried out using a classical electrochemical setup that consisted of a frequency response analyzer (Solartron FRA-1250) and an electrochemical interface (Solartron 1286) monitored by a personal computer. A standard sulfate electrode (SSE) and a large platinum grid were used as reference and counter electrodes, respectively. These two electrodes were located outside the confined zone facing the working electrode surface. This configuration prevented notably the counter electrode reactions from affecting the local chemistry in the vicinity of the working electrode. All experiments were performed in a 0.01 M Fe(CN)63- + 0.01 M Fe(CN)64- + 0.5 M KCl solution. Solutions were freshly prepared in deionized and twice-distilled water from analyticalgrade chemicals (Sigma). All impedance spectra were recorded at the equilibrium potential of the ferri-ferrocyanide couple.

4628 J. Phys. Chem. C, Vol. 112, No. 12, 2008

Remita et al.

Figure 2. 1D spatially restricted diffusion impedance (calculated according to eq 3 with D ) 6.3 × 10-6 cm2 s-1 and Rd ) 5 × 10-2 Ω cm3 s-1) with the diffusion layer thickness e as a parameter.

Theory As discussed above, if we assume that diffusion processes occur only in the direction normal to the electrode surface8,22 then the theoretical diffusion impedance associated with the transport of electroactive species can be derived analytically in a thin-layer-cell configuration. For a thin-layer cell of thickness e, and for a boundary impermeable to the diffusing species (i.e., (∂c/∂x)x)e ) 0), the 1D spatially restricted diffusion impedance Z1D is given by22

Z1D(ω) ) Rd

e D

cotanh

(x ) e2 jω D

x

(3)

e2 jω D

where ω is the pulsation of the sine wave perturbation, Rd is a scaling factor depending on both the kinetics of the interfacial reactions and the bulk concentration of the electroactive species, and e is the electrolyte film thickness. The Nyquist plots of this spatially restricted diffusion impedance, calculated for Rd ) 0.05 Ω cm3 s-1 and D ) 6.3 × 10-6 cm2 s-1, are presented in Figure 2 with the electrolyte film thickness, e, as a parameter. These diagrams exhibit a usual Warburg behavior for high frequencies (i.e., a straight line with a 45° slope) followed by a capacitive behavior for lower frequencies. From an electrical point of view, the low-frequency behavior is equivalent to a resistor and a capacitor in series. The value of this low-frequency resistor decreases as the electrolyte film thickness decreases. An increase of the characteristic frequencies, which correspond to the transition between the Warburg and the capacitive behavior, is also observed as the electrolyte film thickness decreases. As discussed in the previous section, the existence of a radial potential distribution should be taken into account for modeling suitably the impedance response on a large disk electrode located in a cylindrical thin-layer cell. Thus, a transmission line model

Figure 3. Scheme of the transmission line model (TL model) proposed to take into account the potential drop within the thin-layer cell (lateral view).

(TM model) adapted to the geometry of the cylindrical thinlayer cells was derived. Let us assume that the electrolyte resistivity is homogeneous in the whole liquid film confined in the vicinity of the working electrode surface and that the interfacial impedance of the electrode Zint is homogeneous over the electrode surface (i.e., no reactivity or capacitance distributions exist on the electrode surface). This model sketched in Figure 3, is very similar to that of De Levie,30 which was also used by Keddam et al.8 for describing impedance response of electrodes located in a cylindrical thin-layer cell. However, conversely to the previous work of Keddam et al.,8 our approach takes into account the cylindrical geometry of the thin-layer cell. This was achieved by introducing the dependency on the radial coordinate of the ring shape electrode surface elements dSelec(r) (eq 4) and on the lateral surface of the cylindrical electrolyte film confined within the cell dSlat (eq 5)

dSelec(r) ) 2π rdr

(4)

dSlat(r) ) 2πre

(5)

with r being the radial coordinate and dr being a length element following the radius of the electrode. Under these assumptions, the elementary resistances and admittances (dR and dY) involved in the model (see Figure 3) can be expressed as

Diffusion Impedance in a Thin-Layer Cell

dR ) Re dr dY )

dr Z0


(6)

From eqs 16 and 18, the local potential u can then be expressed as

(7)

u ) 2A cosh(R r)

where dR is the resistance of a length element of the liquid layer, Re is the electrolyte resistance, dY is the admittance of a ring shape electrode surface element dSelec(r), and Z0 is the impedance of the interface for a radial unit length of electrode. Equations 6 and 7 can be rewritten as

F dr ) dR ) dr 2π re dSlat(r)

(8)

dSelec(r) 2π rdr ) dY ) Zint Zint

(9)

F

du dR

di ) udY

(10)

(11) (12)

where u and du are the local potential within the thin-layer cell and the local ohmic drop in an length element dr of the liquid layer, respectively. Using eqs 8 and 9, eqs 11 and 12 can be rewritten as

F du ) i dr 2π re

(13)

di 2π r u ) dr Zint

(14)

Using eqs 10 and 14, the derivation of eq 13 leads to

d2u 1 du F + u)0 dr2 r dr eZint

(15)

From numerical simulations, we have shown that the term (1/r du/dr) can be neglected in the thin-layer-cell configuration. From a practical point of view, it allows us to obtain an analytical solution to eq 15, which is given by

u ) A exp(R r) + B exp(-R r)

(16)

where A and B are two constants to be determined with boundary conditions and R is the coefficient defined by xF/eZint. The cylindrical symmetry of the cell imposes the boundary condition

[dudr]

(r)0)

)0

(17)

Combination of eqs 16 and 17 yields

A)B

Z)

I)

Moreover, these currents can also be expressed using Ohm’s law according to

i)

By defining U as the potential at the edge of the electrode (U ) u(R0)) and I as the global current flowing through the interface (I ) i(R0)), the impedance of the electrode Z corresponding to the global impedance of the transmission line shown in Figure 3 is given by

U π R02 I

(20)

Using eq 14, the global ac current flowing through the interface is written as

where F is the electrolyte resistivity. The local currents i and di flowing in the radial and normal directions of the electrode surface through the electrolyte layer are linked by the following charge conservation law

di(r) ) i(r + dr) - i(r)

(19)

(18)

∫0R

0

di )

∫0R

0

2π r udr Zint

(21)

The combination of eqs 19 and 21 allows the global current to be expressed as

I)

[

]

1 4π A R0 sinh(R R0) + 2 (1 - cosh(R R0)) Zint R R

(22)

Finally, the impedance of the electrode Z is given by

Z)

[

R02Zint

(

R0 1 1 tanh(RR0) + 2 2 -1 R R cosh(RR0)

)]

(23)

When R f 0 (i.e., when e f +∞, or F f 0) bulk conditions are achieved and it can be noticed that Z f Zint in eq 23. When R f +∞ (i.e., when e f 0 or F f +∞), the cell can actually be reduced to a one-dimensional system and Z is proportional to xZint, which is similar to the De Levie’s impedance of a one-dimensional porous electrode.30 The knowledge of the interfacial impedance Zint is required for the calculation of the global impedance of the electrode given by eq 23. In the case of a blocking electrode (i.e., a thin-layer cell in presence of supporting electrolyte only), a pure capacitance accounts for Zint, and the shape of the impedance diagrams is very similar to that obtained in Figure 2. In the present work, this interfacial impedance corresponds to the diffusion impedance and was calculated using two different assumptions. On one hand, Zint was calculated using the classical analytical expression for a 1D spatially restricted diffusion impedance Z1D (eq 3). On the other hand, the contribution of radial masstransport was also taken into account in the impedance modeling. Hence, the diffusion impedance was calculated numerically at the electrode using an adaptation of the model proposed previously by Gabrielli et al.1 for the case of microelectrodes located in cylindrical thin-layer cells and then introduced in eq 23. Results and Discussion The experimental impedance diagrams measured in the thinlayer cell for relatively thin electrolyte films (thicknesses higher than 115 µm) and for very thin electrolyte films (thicknesses lower than 115 µm) are presented in Nyquist coordinates in Figures 4 and Figure 5, respectively. In bulk conditions (electrolyte film thicknesses larger than several centimeters), the experimental diagrams consist of a straight lines exhibiting a phase angle very closed to 45° (Figure 4), which is consistent


Remita et al.

Figure 5. (a) Experimental impedance diagrams obtained at the equilibrium potential in a 0.01 M Fe(CN)63- + 0.01 M Fe(CN)64- + 0.5 M KCl solution as a function of the electrolyte film thickness for e e 115 µm; (b) Magnification of the high-frequency range of the diagrams.

Figure 4. (a) Experimental impedance diagrams obtained at the equilibrium potential in a 0.01 M Fe(CN)63- + 0.01 M Fe(CN)64- + 0.5 M KCl solution as a function of the electrolyte film thickness for e g 115 µm; (b) Magnification of the high-frequency range of the diagram.

with the assumption of a linear mass transport normal to the electrode surface. Indeed, such behavior is well predicted and described by the classical 1D diffusion model in the case of a semi-infinite diffusion process (Warburg impedance corresponding to e infinite in eq 3). In the whole range of thin electrolyte film thicknesses investigated (i.e., thicknesses equal or lower than 300 µm), the general shape of the experimental impedance diagrams was changed. In a first approximation, they can be described by a combination of a straight line in the high frequencies followed by a capacitive arc in the low-frequency domain, showing a behavior quite similar to that predicted by 1D spatially restricted diffusion models. However, several significant differences exist between the experimental (Figures 4 and 5) and the theoretical impedance diagrams (Figure 2) calculated from eq 3. First, whatever the range of electrolyte film thicknesses, the 1D spatially restricted diffusion model predicts a decrease of the low-frequency limit of the real part of the impedance as the electrolyte film thickness decreases (see Figure 2). As shown in Figure 4, this prediction was achieved experimentally only for relatively thick electrolyte film thicknesses (thicknesses

Figure 6. Bode representation of the theoretical 1D spatially restricted diffusion impedance (calculated according to eq 3 with Rd ) 0.05 Ω cm2 and D ) 6.3 × 10-6 cm2 s-1) (line) and of the experimental impedance diagrams as a function of the electrolyte film thickness (line + symbols). The electrolyte resistance was subtracted from the impedance.

ranging from 95 to 300 µm), but for smaller film thickness, the low-frequency limit of the real part of the impedance increases as the electrolyte film thickness decreases (Figure 6). In Figure 6, both the experimental phases and the theoretical results were plotted as a function of a dimensionless frequency22



fadim (defined by eq 24) by considering ferro-cyanide ions as the diffusion species (D was taken equal to 6.3 × 10-6 cm2 s-1).

fadim )

fe2 D

(24)

where f is the frequency. Using this dimensionless frequency, all of the Bode diagrams calculated from eq 3 for different electrolyte film thicknesses result in a single curve (solid line in Figure 6). In other words, for a given dimensionless frequency fadim, the phase angle remains independent of the electrolyte thickness. In particular, in the high-frequency range exceeding the critical frequency (fadim > 1), a constant phase angle with a value of 45° is predicted by the 1D model, which was not observed on the experimental results presented in Figure 6. Indeed, as clearly illustrated by the high frequencies magnification of the experimental Nyquist plots in Figures 4b and 5b, and by the Bode plots (Figure 6), the phase angles decrease monotonously as the electrolyte film thickness decreases. Bode plots also indicate that the experimental phase angles are frequency dependent in the high-frequency range. Quantitatively, in the thin electrolyte film conditions investigated in this study (e e 300 µm), the experimental values of the phase angle are systematically lower than 45° in the high frequencies. For instance, for a dimensionless frequency fadim ) 100, the phase angles vary from 37° for a 300 µm electrolyte layer thickness to 24° for a 20 µm electrolyte layer thickness (Figure 6). In order to account for differences between experimental results and the 1D diffusion model, we investigated the influence of the radial potential distribution existing in the liquid layer confined at the electrode surface theoretically using the transmission line model (TL model) developed in the theory section. In a first step, the radial mass transport was neglected and new theoretical impedance diagrams were calculated according to the expression of the 1D modified spatially restricted diffusion impedance (eq 23 with Zint calculated from eq 3). These calculations (Figures 7-9) were performed by taking the values of the diffusion coefficient of Fe(CN)64- and the solution resistivity F at 6.3 10-6 cm2 s-1 and 18 Ω cm, respectively (the latter was measured experimentally), whereas the scaling factor Rd used in the model was taken as 0.05 Ω cm3 s-1. As shown in Figure 7a, the shape of the calculated diagrams is quite similar to the one predicted when the ohmic drop is neglected (Figure 2) for relatively large electrolyte film thicknesses (e g 95 µm). As the thickness decreases in this range, the TL model predicts the experimentally observed decrease of the low-frequency limit of the real part of the impedance, which is in good agreement with the 1D diffusion model. Below a thickness of 95 µm (Figures 8a and b), the trend is however inversed and an increase of the low-frequency limit of the real part of the impedance is predicted by the TL model as the electrolyte film thickness decreases (Figure 8a), which is in very good agreement with experimental results presented in Figure 5. Moreover, it should also be pointed out that the value of the experimental thickness (e ) 95 µm) for which the variation trend of the low-frequency limit of the real part of the impedance is reversed is also predicted very well by the TL model (Figure 8b). In the whole range of thicknesses investigated (20 µm e e e 300 µm), the theoretical phase angles calculated from eq 23 decrease in the high-frequency range (fadim > 1) as the electrolyte film thickness decreases (Figures 7b, 8c, and 9). This prediction from the TL model is also consistent with the experimental measurements. Moreover, the Bode representation of the theoretical results (Figure 9) further evidenced that the experi-

Figure 7. (a) Theoretical impedance diagrams calculated with the transmission line model by neglecting radial mass transport (eq 23 with Rd ) 0.05 Ω cm2, D ) 6.3 × 10-6 cm2 s-1, F ) 18 Ω cm and Zint calculated according to eq 3) as a function of the electrolyte film thickness for e g 115 µm; (b) Magnification of the high-frequency range of the diagrams.

mentally observed frequency dependence of the phase angle in the high-frequency range (Figure 6) is consistent with the TL model. These results (Figures 7-9) show the significant improvement provided by the TL model for the description of the diffusion impedance in cylindrical thin-layer cells involving large disk electrode and demonstrate how this potential distribution could have a noteworthy impact on the impedance response in a thin-layer cell involving a large disk electrode. Conversely, previous results from Gabrielli et al.1 demonstrated that this potential distribution effect becomes negligible in the case of electrode of micrometric radii located in a thin-layer cell. The fit between experimental measurements and the 1D spatially restricted modified diffusion impedance calculated from the TL model remains, however, imperfect. Thus, the experimental diagrams (Figures 4-6) did not superimpose on the theoretical curves (Figures 7-9) obtained from eq 23 (it was not, however, the purpose of this work and no fitting procedure such as the simplex method was used). A possible explanation for these quantitative differences between the theoretical diagrams calculated from eq 23 and the measurements may be suggested by the conclusions of the previous work from Gabrielli et al.1 Indeed, these authors showed that neglecting radial mass transport leads to inaccurate impedance calculations in the case of cylindrical thin-layer cells involving microelec-


Remita et al.

Figure 8. (a) Theoretical impedance diagrams calculated with the transmission line model by neglecting radial mass transport (eq 23 with Rd ) 0.05 Ω cm2, D ) 6.3 × 10-6 cm2 s-1, F ) 18 Ω cm and Zint calculated according to eq 3) as a function of the electrolyte film thickness for e e 115 µm. (b) Magnification of the low-frequency limit of the real part of the diagrams for thicknesses close to 95 µm. (c) Magnification of the high-frequency range of the diagrams.

Figure 9. Bode representation of the theoretical impedance diagrams calculated with the transmission line model by neglecting radial mass transport (eq 23 with Zint calculated according to eq 3), comparison with the theoretical 1D spatially restricted diffusion impedance (eq 3) Rd ) 0.05 Ω cm2, D ) 6.3 × 10-6 cm2 s-1, F ) 18 Ω cm. The electrolyte resistance was subtracted from the impedance.

trodes. The discrepancy between the measurements (Figures 4-6) and the predictions from the TL model proposed in this paper (Figures 7-9) may be consequently explained by the use

Figure 10. Bode representation of the theoretical impedance diagrams calculated with the transmission line model by taking into account radial mass transport (eq 23 with Zint calculated numerically using an adaptation of the model from Gabrielli et al.,1 D ) 6.3 × 10-6 cm2 s-1 and F ) 18 Ω cm), comparison with experimental measurements obtained at the equilibrium potential in a 0.01 M Fe(CN)63- + 0.01 M Fe(CN)64- + 0.5 M KCl solution. The electrolyte resistance was subtracted from the impedance.

of a too strong approximation consisting in neglecting the radial diffusion processes in eq 23. Experimentally, the occurrence



TABLE 1: List of Symbols symbol

meaning

c D e f fadim I J Rd r dSelec(r)

concentration of the diffusing species diffusion coefficient of the diffusing specie electrolyte film thickness frequency dimensionless frequency (fadim ) fe2/D) global current (I ) i(R0)) flowing through the interface flux of the diffusing specie a scaling factor depending on the kinetic of the interfacial reactions radial coordinate ring shape electrode surface element located at a distance r from the electrode center lateral surface of a cylindrical electrolyte film element of radius r confined within the thin-layer cell length element following the radius of the electrode resistance of a length element of the liquid layer electrolyte resistance through the confined zone disk-electrode radius potential at the edge of the electrode (U ) u(R0)) local potential within the thin-layer cell local ohmic drop in an length element dr of the liquid layer impedance of the electrode interfacial impedance at the electrode surface admittance of a ring shape electrode surface element dSelec(r) impedance of the interface for a radial unit length of electrode xF/eZint electrolyte resistivity pulsation of the sine wave perturbation

dSlat dr dR Re R0 U u du Z Zint dY Z0 R F ω

of such radial diffusion is, however, not obvious in our case. Indeed, Gabrielli et al.1 demonstrate that radial diffusion appears in the impedance measured at a microelectrode as an additional low-frequency contribution. In the frequency range investigated in this work (100 kHz-5 mHz), such a contribution was not clearly evidenced for large disk electrodes. A careful examination of the results reveals, however, that the purely capacitive behavior predicted theoretically at low frequency by both 1D models discussed previously (eqs 3 and 23) was not achieved in this range of frequency. In fact, the experimental Nyquist diagrams (Figures 4 and 5) do not reach clearly a perfect vertical asymptotic behavior predicted by these models at low frequency (Figures 2, 7-8). Similarly, the phase angle of 90° predicted by both models was not observed in the frequency range investigated (Figures 6 and 9). In order to check the possible occurrence of a radial contribution in the impedance measured on large disk electrodes, we performed some measurements with frequency limit as low as 100 µHz (Figure 10). Moreover, radial mass transport was also taken into account in the TL model and theoretical impedance diagrams (Z2D; also plotted in Figure 10) were calculated numerically. Figure 10 shows an additional time constant on the experimental diagrams in the very low-frequency range, which is evidenced by the existence of a maximum of the experimental phase angle in Bode representation. The value of this maximum increases as the electrolyte film thickness decreases, whereas the dimensionless frequency corresponding to this maximum increases as the electrolyte film thickness decreases. This low-frequency behavior remains totally unexplained if considering a 1D diffusion path only (see Figure 9). Diagrams calculated numerically from the 2D model (Figure 10) are also in very good accordance with the experimental results. In the low-frequency range, the additional time constant is totally predicted by the model, and for high frequencies, the TL model proposed in this paper allows us to explain the small (lower than 45°) and frequently distributed values of the phase angle. Thus, these results are consistent with previous work from Gabrielli et al.,1 but the use of the electrode in the millimeter

units mol cm-3 cm2 s-1 cm Hz A mol s-1 cm2 Ω cm3 s-1 cm cm2 cm2 cm Ω Ω cm-1 cm V V V Ω cm2 Ω cm2 Ω-1 Ω.cm cm-1 Ω cm

or centimeter range requires very low frequency (less than 100 µHz in certain cases) to allow the radial mass transport effect to be observed. Conclusions The use of an accurate experimental device allowed the measurement of the electrochemical impedance corresponding to the diffusion of ferro-cyanide ions on a large disk electrode in a cylindrical thin-layer-cell configuration. Even if the diagrams obtained experimentally exhibit some strong similarities to the theoretical diagrams predicted by the classical spatially restricted linear diffusion impedance models, some significant differences between theory and experiments were evidenced. These differences consisted notably in a nonmonotonous evolution of the low-frequency limit of the real part of the experimental diagrams as well as small (lower than 45°) values of the phase angle in the high frequencies. This behavior was ascribed to the potential distribution in the electrolyte film confined at the electrode surface due to ohmic drop. This effect was taken into account by developing a new transmission line model that accounts for the cylindrical geometry of the electrochemical cell. Moreover, an analytical derivation of the impedance was possible by neglecting the radial mass transport within the cell. Under this linear mass transport assumption, the model allowed us to explain qualitatively the experimental results in a large frequency range. However, in order to fit quantitatively the diagrams and to further explain the additional time constant evidenced experimentally in a very low-frequency range (frequencies lower than 1 mHz), the bidimensional character of the mass transport should be taken into account. References and Notes (1) Gabrielli, C.; Keddam, M.; Portail, N.; Rousseau, P.; Takenouti, H.; Vivier, V. J. Phys. Chem B 2006, 110, 20478-20485. (2) Christensen, C. R.; Anson, F. C. Anal. Chem. 1963, 35, 205-209. (3) Hubbard, A. T; Hanson, F. C. In Electroanalytical Chemistry; Bard, A. J., Ed.; Marcel Dekker: New York, 1970. (4) Hubbard, A. T.; Hanson, F. C. Anal. Chem. 1964, 36, 723-726.

4634 J. Phys. Chem. C, Vol. 112, No. 12, 2008 (5) Hubbard, A. T.; Hanson, F. C. Anal. Chem. 1966, 38, 58-61. (6) Yu, J.-S.; Yang, C.; Fang, H.-Q. Anal. Chim. Acta 2000, 420, 4555. (7) Fiaud, C.; Chahrouri, R.; Keddam, M.; Maurin, G.; Takenouti, H. 8th International Congress on Metallic Corrosion; Mayence, Germany, 1981; Vol. I, pp 18-23. (8) Fiaud, C.; Keddam, M.; Kadri, A.; Takenouti, H. Electrochim. Acta 1987, 32, 445-448. (9) Keddam, M.; Hugot-Le-Goff, A.; Takenouti, H.; Thierry, D.; Arevalo, M. C. Corros. Sci. 1992, 33, 1243-1252. (10) Remita, E.; Sutter, E.; Tribollet, B.; Ropital, F.; Longaygue, X.; Taravel-Condat, C.; Desamais, N. Electrochim. Acta 2007, 52, 7715-7723. (11) Micka, K.; Kratochvilova, K.; Klima, J. Electrochim. Acta 1997, 42, 1005-1010. (12) Fleischmann, M.; Pons, S.; Rolinson, D. R.; Schmidt, P. P. Ultramicroelectrodes; Datatech Systems, Inc.: Morganton, NC, 1987. (13) Bard, A. J.; Mirkin, M. V. Scanning Electrochemical Microscopy; Marcel Dekker: New York, 2001. (14) Amatore, C.; Szunerits, S.; Thoin, L.; Warkocz, J.-S. J. Electroanal. Chem. 2001, 500, 62-70. (15) Anastasijevic, N. A.; Rousar, I. Electrochim. Acta 1988, 33, 11571160. (16) Rousar, I.; Anastasijevic, N. A. Electrochim. Acta 1989, 34, 887888.

Remita et al. (17) Leddy, J.; Zoski, C. G. J. Electroanal. Chem. 2003, 543, 13-22. (18) Galceran, J.; Salvador, J.; Puy, J.; Cecilia, J.; Gavaghan, D. J. Electroanal. Chem. 1997, 440, 1-25. (19) Gabrielli, C. In Physical Electrochemistry. Principles, Methods, and Applications; Rubinstein, I., Ed.; Marcel Dekker: New York, 1995. (20) Macdonald, D. D. Electrochim. Acta 2006, 51, 1376-1388. (21) Gabrielli, C.; Haas, O.; Takenouti, H. J. Appl. Electrochem. 1987, 17, 82-90. (22) Jacobsen, T.; West, K. Electrochim. Acta 1995, 40, 255-262. (23) Contamin, O.; Levart, E.; Schumann, D. J. Electroanal. Chem. 1977, 84, 271-285. (24) Schumann, D.; Contamin, O.; Levart, E. J. Electroanal. Chem. 1977, 84, 287-293. (25) Contamin, O.; Levart, E.; Schumann, D. J. Electroanal. Chem. 1978, 88, 49-56. (26) Buck, R. P. J. Electroanal. Chem. 1986, 210, 1-19. (27) Bisquert, J.; Garcia-Belmonte, G.; Bueno, P.; Longo, E.; Bulhoes, L. O. S. J. Electroanal. Chem. 1998, 452, 229-234. (28) Bisquert, J.; Compte, A. J. Electroanal. Chem. 2001, 499, 112120. (29) Bisquert, J.; Garcia-Belmonte, G.; Bueno, P.; Fabregat-Santiago, F.; Roberto Bueno, P. J. Electroanal. Chem. 1999, 475, 152-163. (30) De Levie, R. AdVances in Electrochemistry and Electrochemical Engineering; Delahay, R., Ed.; Interscience: New York, 1967; Vol. VI.

Diffusion Impedance in a Thin-Layer Cell - American Chemical Society

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