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High-frequency impedance spectroscopy of fast electrode reactions
J. Electroanal. Chem., 229 (1987) 407-421 Elsevier Sequoia S.A., Lausamte - Printed
HIGH-FREQUENCY FtEACl-IONS l
G. SCH6NE
407 in The Netherlands
IMPEDANCE
SPECTROSCOPY
OF FAST ELECTRODE
and W. WIESBECK
Institute of Microwave
Techniques, University of Karlsruhe, Karlsruhe (F.RG.)
W.J. LORENZ Institute of Physical Chemistry and Electrochemistry, (Received
23rd January
Unrversity of Karlsruhe, Karlsruhe (F R. G.)
1987)
ABSTRACT A new technique of high-frequency impedance spectroscopy working in the frequency range between 1 kHz and 5 MHz has been developed. The method is characterized by a special electrochemical cell with two working electrodes, a newly designed dc control being completely ac-decoupled, a highly precise impedance measurement technique (ICEP = Impedance Conversion and Error Processing) and quantitative calculation of the electrodynamic field distribution in the cell. The accuracy of the measurements is better than 0.05%. The method was tested on the fast Ag/Ag+ system. The results obtained are in quantitative agreement with a theoretical transfer function based on a 2-D inhomogeneous surface model.
(I) INTRODUCTION
The charge-transfer kinetics of fast electrode reactions are characterized by a high exchange current density, i,, corresponding to a low charge-transfer resistance, R,.Until now, fast electrode reactions have been investigated mainly in the time domain using large-signal pulse perturbation [l-4]. However, this transient technique is relatively inaccurate because of low reproducibility and restricted resolution of the system response at high frequencies. Moreover, electrochemical systems have a non-linear signal transfer and, therefore, large-signal perturbation gives a system response representing the dynamic system behaviour only qualitatively. On the other hand, conventional electrochemical impedance spectroscopy (EIS) cannot be applied to analyze the kinetics of fast systems in the high-frequency domain due to the following reasons:
In honour Institute.
l
of Professor
0022-0728/87/$03.50
H. Gerischer
on the occasion
0 1987 Elsevier Sequoia
S.A.
of his retirement
as Director
of the Fritz-Haber
Fig. 1. Block diagram of Conventional Electrochemical Impedance Spectroscopy (EIS). (G) Generator, (PC) personal computer, (WE) working electrode, (CE) counter-electrode, (RE) reference electrode, (S) high-frequency probe, (FRA) frequency response analyzer, Cp) potentiostat.
(i) Impedance measurements including a usual three-electrode cell coupled with a commercial potentiostat and a frequency response analyzer (FRA) as shown in Fig. 1 are restricted to a maximum frequency of f = 50 kHz. At higher frequencies, the potentiostat causes phase shifts and amplitude deformations in order to establish stability. Moreover, the accuracy of the FRA is too low to separate exactly R, from the electrolyte resistance R,, when R, e R,. (ii) The resonance frequency 2nfr = w, = (R,C,,)-’ of fast charge transfer reactions with an exchange current density of i, 2 1 A cme2 is in the range f, a 250 kHz, assuming a double-layer capacitance of Co, = 25 X lop6 F cmp2. (iii) The electrolyte resistance, R,, is considered to be real and constant even at high frequencies. In this paper, the development of a high-frequency ‘impedance spectroscopy HF-EIS) is described which allows the exact determination of the kinetics of fast electrochemical reactions. Here, the following problems had to be solved [5-71: (i) Design of a new electrochemical cell and a suitable potentiostat for dc control. (ii) Extension of measurements to higher frequencies covering the range lo3 S -I
s-‘.
(iv) Calculation of the electrodynamic field distribution in the electrochemical cell to separate the frequency-dependent electrolyte impedance go(w) from the total measured impedance z,(w). This non-conventional impedance measurement technique (called ICEP, Impedance Conversion and Error Processing) was checked on the fast system Ag/Ag+, which has an extremely high exchange current density depending on the surface structure of the substrate and the silver ion concentration in the electrolyte [8-151. Recently, conventional impedance measurements showed that this system is purely transport-controlled up to frequencies of about 5 X lo4 [5-7,16-201. S -’ taking into account 2-D surface inhomogeneity (II) CELL DESIGN
AND
POTENTIOSTATIC
DC CONTROL
In contrast to conventional impedance measurement techniques, two identical working electrodes, WE1 and WE2, with equivalent surface preparation and thus
I
1
/
Cell
Fig. 2. Concept of High-Frequency working electrodes. Fig. 3. Coaxial
ring-disc
configuration
Impedance
Spectroscopy
(HF-EIS)
with
a configuration
of two
of both electrodes.
the same interphase impedances, are used (Fig. 2). The frequency response analyser is coupled directly to the working electrodes, measuring the sum of the interphase impedances at WEl, WE2, and the impedance of the bulk solution in between. A slow ac-decoupled potential-control system regulates the identical dc potential of both WEs vs. the reference electrode, RE. This concept eliminates the above-mentioned potentiostat errors in the frequency range above 1 kHz. The inner and outer conductor of a coaxial line are used as ring and disc working electrodes WE1 and WE2 (Fig. 3). A great advantage of this configuration is the well-defined contact between both electrodes and the FRA, allowing an exact external error correction (cf. Section 111.2). Moreover, the surface preparation of the WEs can be carried out reproducibly and nearly identically. The new dc potential-control system is based on two current sources decoupled up to 10 MHz (Fig. 4). These sources have a high output impedance of more than 100 k!G!in the whole frequency range from 1 kHz to 10 MHz. The current of source 2 is controlled by a slow comparator levelling the dc voltage between WE1 and WE2 to zero. The current of source 1 is controlled by a slow PI regulator (Proportional Integral regulator) to adjust a defined potential of the WEs vs. RE.
Fig. 4. New concept of dc potential control of the working electrodes using voltage-controlled sources. (LP) Low-pass filter, (S,, S,) current sources, (D) differential amplifier.
current
410 (III) IMPEDANCE
MEASUREMENT
The measurement of the unknown impedance _Z, between the working electrode WE and the bulk solution, usually called the electrode impedance, faces three major difficulties: (i) _Z, cannot be measured separately as the bulk resistance & is always in series. (ii) _Z, is small compared to _Z,. (iii) Systematic and statistical errors are nearly of the same order of magnitude as _Z In the following, ways are shown to overcome these problems. The prime ideas are (1) to compensate the bulk impedance &, resulting in an impedance conversion and (2) to eliminate errors by calibration and integration. The procedure developed according to these ideas is called Impedance Conversion and Error Processing (ICEP). (III. 1) Impedance conversion The idea behind impedance conversion is to increase the accuracy of a given commercial Network Analyzer (NWA) by a factor of 5.. .25 inserting an interface between LZ, and the NWA. According to Fig. 5, this interface, the so-called “Impedance Converter”, subtracts a precisely known, real impedance _Z, from the cell impedance z,, compensating nearly the entire real part of &. The resulting residual impedance _Z, can be determined by the NWA with the analyzer’s standard accuracy. Adding the values of the small, not very precisely measured impedance _Z, to the large, precisely known compensation impedance _Zk, leads to a very accurate result for the interesting cell impedance z,. Figure 6 shows the block diagram of the impedance converter. It is based on a bridge, but has an additional circuit to subtract that part of the voltage at z, which is in phase with the current. This circuit consists of a power divider, deriving an in-phase internal reference signal from the generator’s source voltage, a variable attenuator reducing the reference signal by one of 128 available steps, and a differential amplifier which subtracts the residual output voltage of the attenuator from the voltage at z,.
Fig. 5. Real-part correction applied to &. residual measuring value.
(gx)
uncompensated, (Zk) compensated part of &,
(z,)
411
Ref.
Int
‘.‘. \
.
r
Variable
A(tenuation
k I
K
I.
-I-
Fig. 6. Impedance
Conversion
and Error Processing (ICEP) method.
In series to the impedance converter a commercial network analyser (NWA) is doing the measurement. A personal computer, HP 9845, controls the generator amplitude and frequency, the degree of attenuation and accepts the displayed voltages at the NWA. By variation of the compensation characterized by the step number k at the beginning of an impedance measurement, the computer system minimizes the voltage _U, at the input of the NWA. Knowing the step number k and the result &, = _V,/U_ of the network analyzer’s measurement, the personal computer is able to calculate the compensation rate & defined as the ratio ( 1Zk 1- Re@,))/Re(&), which is an indicator for the increase of accuracy. Applying an easy matrix algorithm to 46 and & leads to the wanted electrode impedance z,. The accuracy of the impedance measurement of & is limited directly by (i) the inaccuracy of stepwise attenuation, the power divider, the bridge, and the differential amplifier,
Relative A Error (*/d
Fig. 7. Relative measurement
error as a function of the real-part compensation
factor &.
412
(ii) the possible compensation rate fik, (iii) the non-linearity and noise of the differential amplifier, and (iv) the residual errors of the network analyser NWA. The influences of (i) and (ii} can be di~nish~ by a ~mputer-~ded error correction described below and by an optimized compensation state characterized by k, respectively. The influence of (iii) limits the voltage level U_ at z, to the range 0.1 mV G IQ ( G 5 mV. The influence of (iv) on the measuring results is low, as demonstrated in Fig. 7. All experiments were carried out under the condition 0 d I& 1 G 28, giving a small relative error. The absolute error of the ICEP measurement technique without error processing to reduce the influence of (i) amounts to 0.5%. (III. 2) Error processing The computer-aided error processing involves the preceding calibration of systematical errors, i.e. converter and external errors, as well as the elimination of random errors during the measurement by a tracking mechanism, the so-called windowing technique. All correction parameters are stored in the computer system and applied automatically to correct each measured value. To calculate and eliminate internal errors of the impedance converter, the error model shown in Fig. 8 was supplied. Each of the five blocks in Fig. 8 represents a possible error source. The transfer properties of these blocks were assumed to be linear but frequency-dependent due to the applied small signal perturbation. The error equation (1) can be derived from the error model in Fig. 8 by combination of all linear-dependent block parameters
z=__L3Kn+l _L?_L+L,
_x
where &, denote linear independent correction parameters and &
Fig. 8. Error model of the impedance converter. sources. (4) Voltages, (Ampl.) amplitude.
The five blocks indicated
the meas~ement
may be influenced
by error
413
Impedance
Fig. 9. Error model of the external circuit.
value. Both &, and L;, depend on the frequency and the degree of the real-part compensation k. The correction parameters L,, can be calculated according to the matrix algorithm
(2) by measuring three frequency-independent calibration standards _ZX1,L_zX,,, &. One obtains more than 4 X lo4 sets of _Li using 340 discrete frequencies at 128 different k values. Potentiostat, cables and electrode configuration cause external errors which can be represented by an error-f-port model shown in Fig. 9 with a linear transfer behaviour. The corresponding error equation
%=S,t+
S:*rL 1 -
S22IL
contains the reflection factors rL and rE and the scattering parameters giJ of the external error sources. The s,, values can be dete~ed according to eqns. (4) and 15): S;; = \iSzzK
(4)
I
by measuring the three calibration standards mentioned above. During calibration, _r,, are measured at different frequencies, whereas the _T~,~ values are the well-known reflection factors corresponding to the calibration standards z,,. In eqns. (4) and (5) &. and 4” are matrix-normalization factors. Using corrections to both converter and external errors, the overall systematic system error could be reduced to less than 0.05% in the whole frequency range lo3 s-i
414
Region:
m1
0 c] 0
11 III
0
Fig. 10. Windowing technique. (P,) Measured value, (/,) radius of an error region.
peaks or electromagnetic field influences, two different correction procedures were applied. Experimental values with high standard deviations are cancelled by a statistical averaging method [21] using at least ten measurements at each frequency. A so-called windowing technique was used additionally in the case of calibration measurements. It is based on the principle of maximum likelihood illustrated in Fig. 10. The reliability of a measurement point can be estimated on the basis of three previous points in a sequence of impedance measurements made at various consecutive frequencies. Windowing builds up three regions of probability around an estimated point in the impedance domain. Depending on the position of the next experimental value, the measurement is either continued or repeated up to 5 times. Thus, the most reasonable result will be taken as the real one. Windowing detects and eliminates more than 99% of the random errors due to the repetition of measurements caused by a disagreement between estimation and measured result. (Iv) ELECTRODYNAMIC FIELD DISTRIBUTION To calculate the true impedance of the electrode interphase Z?(u), it is necessary to determine exactly the electrolyte impedance &n(w), which must be subtracted from the measured and error-corrected cell impedance z,(w). However, z*(o) can be calculated only if the electrodynamic current distribution in the electrochemical cell is known. Previous calculations of the field distribution in electrochemical cells dealt with the assumption of static or inductive fields [22]. In this paper, a real electrodynamic field distribution is presumed, i.e. the field strength depends on time and place. The following assumptions are made to limit the mathematical formalism and the computation time in the calculation of the electrodynamic field distribution: (i) All metal surfaces are considered to be electric walls. (ii) All bulk-isolator surfaces are assumed to be magnetic walls. (iii) Small-signal perturbation is applied. (iv) The bulk electrolyte is considered as homogeneous, lossy and isotropic.
415
Fig. 11. Mathematical
model of the dual-working-electrode
concept in Fig. 2.
(v) The thickness of the interphase is assumed to be very small compared to the bulk dimensions. (vi) The coaxial cable (electrode configuration plus transmission line) and the cell have infinite length. (vii) Only the TEM mode is able to propagate within the coaxial cable. A hypothetic third medium III, the length of which tends to zero, is inserted between the electrode surface and the bulk in order to solve the boundary-value problems (Fig. 11). So the boundaries I/III and III/II consist of electric and magnetic walls only, respectively. As usual, the mathematical treatment starts with the electrodynamic potentials _T defined in eqn. (6): rot -fH = -E’
rotfE=ti -
(6)
-
where @ and @ denote the electric and magnetic fields, respectively. These trodynamic potentials are adjusted by the Lorentz convention expressed as ArE.H + b2_TE.H = 0
elec-
(7)
with the wave number /c, = o u/g,,, where We and c, are the eigenfrequency and the phase velocity of mode m. Due to the coaxial-cable symmetry (cf. Figs. 3, 11) the potential functions can be described more practically in cylinder coordinates p, rp and z as follows: 2
--
a
P ag [ p
aTE.H
1 a2TE*'
L
ap
The solution
T E,H= B”,“(p)
+_----=---+
1
P*
ad
of this differential
a2TE.H L+k2TE.n=0
az2
equation
. gEJy cp) . _z”.“( z)
--
(8)
is given by eqn. (9):
(9)
where 4, 2 and _Z are functions dependent only in one coordinate, as indicated. The solution of the boundary-value problem in medium I (Fig. 11) results in a universal description of the electrodynamic field distribution in a coaxial cable. The same treatment applied to media II and III (Fig. 11) gives the electrodynamic field distribution in a lossy dielectric cylindric waveguide and in a cylindric waveguide with conducting walls, respectively. In order to couple the different fields I/III and III/II, their development in orthogonal functions is appropriate:
I?r is an independent field vector parallel to the boundary I/III or III/II. 1, denotes a set of orthogonal functions and g, represents the corresponding weighting factor, which can be derived from an integration over the entire boundary area “A ” between two of the media (e.g. media I and II as shown in eqn. 11)
01) The resulting sets of coupled fields I/III and III/II can be combined under elimination of the fields in medium III by a well-known matrix algorithm [23]. The resulting matrix describes the coupling between all modes in media I and II, Using the knowledge about the physics of the problem, one can reduce the described equation (not provided in this text) drastically. First of all, it is taken into account, that only a TEM-mode causes the electrodynamic field distribution in medium I, and no other mode from outside incides into medium II. In a second step, it is assumed that all modes, except the TEM, are highly attenuated in the coaxial cable. So, the residual equation is very small according to
(12) where c(: * describes the amp litude of the incident ( + ) and emitted ( - ) TEM-mode into and from the coaxial cable, and $$, the ~~espon~ng coupling coefficient. Equation (12) describes the reflection of a TEM-mode being incident from the coaxial cable to the boundary I/III. The value s& is the reflection coefficient from which the impedance follows as: (13) &, denotes the well-known characteristic impedance of the coaxial cable. &(w) represents the electrolyte impedance which has to be calculated for each frequency. Now, all the tools necessary to correct the impedance measurements of fast electrode systems have been given. (V) EXPERIMENTAL
RESULTS
AND
DISCUSSION
The newly developed high-frequency impedance spectroscopy described above was tested on the fast electrode system Ag/Ag+ *. Recently, conventional impedance measurements were carried out on poly- and monocrystalline silver electrodes in the frequency range 10e3 s-’
* The experimental
details of the present
measurement
are described
m the Appendix
417
Re(Z-&$/fkm*~lO~*
Re(Z-Z~)/RxmZ~lO-*
Fig. 12. Conventional (0, 0) and high-frequency (A, A) impedance measurements. System: Ag/O.l AgNO, + 1 M HCIO,, qss = - 6.5 mV, R, = 0.17 D cm*. Frequency marks in kHz.
M
Fig. 13. High-frequency impedance measurement. System: Ag/0.05 M AgNO, + 1 M HCIO,, vss = - 10.4 mV, R, = 0.16 Q cm*. Frequency marks in kHz. (- 0 -) Optimum fit.
plained quantitatively on the basis of a 2-D inhomogeneous surface model. In the frequency range studied, the kinetics are determined by non-linear mass transport, reversible metal ion adsorption and surface relaxation of the 2-D domain structure. As expected, the fast charge-transfer step of Ag+ ions could not be detected. The aim of the present measurements is to determine quantitatively the chargetransfer resistance, R,, of the polycrystalline Ag/Ag+ electrode depending on the silver ion concentration in the electrolyte and the electrode potential. As shown recently [16-191, the influence of the hydrodynamic conditions on the impedance spectra vanishes at higher frequencies of about f z lo* s- ‘. Therefore, conventional impedance spectroscopy on RDE and high-frequency impedance spectroscopy on fixed working electrodes are comparable in a small frequency window, lo3 s- ’ Gf < 5 x lo4 s-l. Figure 12 shows the good agreement of both techniques within this range and the indication of a capacitive semicircle in the Nyquist representation at higher frequencies, representing the charge-transfer step. Examples of high-frequency impedance spectra on positively and negatively polarized electrodes are shown in Figs. 13 and 14. The theoretical transfer function of the fast Ag/Ag+ system taking into consideration 2-D surface inhomogeneity, charge transfer, non-linear diffusion, and reversible adsorption of Ag+ ions, is described by [S-7,16-20]: z-‘(s)=
[(l+u)R,+&+c~_Z~]-~+.sC~~+
z sC,[(l+~fx,+r~~~]
-1
-1
1
+ SC,,
(14)
where C,, is the double-layer capacity, C,, the adsorption capacity of ad-atoms, & the Nernst diffusion impedance, _Z, the lateral transport contribution, R, the charge-transfer resistance, (I is an index of 2-D surface inhomogeneity (a = B/(1 e)), and s is jo. This transfer function can be represented by the equivalent circuit
‘.
,l.(lK$
u
2
I
6
L
8
t-----i@
Re(Z-Zn)/hm2~10-3 _-
Fig. 14. High-frequency impedance measurement. System: Ag/0.05 M AgNO, + 1 M HCIO,, mV, R, = 0.16 s2 cm’. Frequency marks in kHz. (- 0 -) Optimum fit.
g,, = 6.5
Fig. 15. Equivalent circuit of the theoretical transfer function.
in Fig. 15. Experimentally, z(w) can be determined from the measured impedance z,(w) by subtracting z,(w). At high frequencies o z+ 1, eqn. (14) simplifies to eqn. (15) [5-71: z-‘(s)
= [(1+ a)&]
-l +s(c,,
+ CDL)
(15)
TABLE 1 Kinetic data of the polycrystalline Ag/Ag+ System
nss/mV
i,/mA
0.05 M cath.
- 1.4 - 10.4 - 18.4 6.5 10.6 13.5 - 2.5 - 6.5 - 9.6 1.9 10.5 13.0 -1.9 - 4.6 - 10.0 1.6 5.3 10.9 -4.2 - 9.3 - 12.6 3.5 5.1 6.3
- 0.4 -0.7 -1.1 0.6 1.3 1.6 -0.1 - 0.6 - 1.1 0.5 1.3 2.1 -0.7 - 2.2 - 4.3 0.2 2.4 1.1 - 5.5 - 17.3 - 22.5 8.3 15.7 23.2
0.05 M anod. 0.1 M cath. 0.1 M anod. 0.2 M cath. 0.2 M anod. 0.5 M cath. 0.5 M anod.
cm-’
electrode
(l+ o)R,/mS1 9.9 9.6 8.4 3.3 3.5 3.9 13.7 22.0 22.5 5.6 6.2 5.9 3.3 2.9 2.6 2.1 2.0 2.2 5.1 5.5 4.2 1.7 1.5 1.5
cell
cm*
0
R , /mQ cm*
i,/A
2.3 2.0 1.8 0.3 0.4 0.3 9.5 10.0 8.0 3.3 3.4 3.2 2.3 1.9 1.4 1.1 1.2 1.2 7.5 5.9 5.0 1.5 1.1 1.2
3.0 3.2 3.0 2.5 2.5 3.0 1.3 2.0 2.5 1.3 1.4 1.4 1.0 1.0 1.1 1.0 0.9 1.0 0.6 0.8 0.7 0.7 0.7 0.7
8.5 8.0 8.6 10.0 10.0 8.5 19.2 12.5 10.0 19.2 18.0 18.0 25.6 25.6 23.3 25.6 28.5 25.6 42.8 32.1 36.1 36.7 36.1 36.7
cm-*
419
tog (&t-i
Fig. 16. Concentration
1-l)
dependence
of the exchange
current
density.
which corresponds to a semicircle in the Nyquist plot with diameter (1 + a)R,. The experimental impedance spectra shown in Figs. 12-14 were fitted by the transfer function of eqn. (14), giving the optimum fit data summarized in Table 1. These data can be interpreted as follows: (i) The surface 2-D inhomogeneity expressed in the parameter u depends strongly on the pretreatment of the electrode surface [5-7,16-201. Under the given conditions (cf. Appendix), u is nearly independent at low overvoltages, but higher under cathodic than under anodic polarization. (ii) For a given silver ion bulk concentration, the charge-transfer resistance, R, and the exchange current density, i, = RT/( FR,), are nearly constant within the over-potential range studied 1v 1 Q 15 mV. This agrees well with the linearized Butler-Volmer equation of the charge-transfer process. (iii) The exchange current densities related to the active surface area 1 - 13= l/(1 + a) are about half an order of magnitude higher than the corresponding data reported in the literature [1,8-231. (iv) From the concentration dependence of the exchange current density, an electrochemical charge-transfer coefficient of (Y= 0.7 f 0.05 can be estimated, as demonstrated in Fig. 16. (v) Double-layer capacitances of about Co, = 30 f 10 I.LF cmp2 were obtained in all cases from the optimum fits indicating a relatively low 3-D surface roughness. (VI) CONCLUSIONS
The developed high-frequency impedance spectroscopy (ICEP) technique allows the exact quantitative determination of fast electrode kinetics. The experimental results obtained in the Ag/Ag+ system show the capability of this method. The data are measured with a high accuracy of better than 0.05% in the frequency range lo3 s-i
420
For monocrystalline substrates, two separated working electrodes must be used. In this case, the electrodynamic field distribution changes. Investigations on the fast charge-transfer kinetics of electrode processes using defined single-crystal faces as substrates are in progress. It can be expected, that the dynamic analysis of electrochemical systems at high frequencies will lead to a better understanding of many fundamental and practical processes.
ACKNOWLEDGEMENTS
The authors thank the “Deutsche Forschungsgemeinschaft” (DFG) and “Arbeitsgemeinschaft Industrieller Forschungsvereinigungen” (AIF) for financial support. Mrs. M. Stoll, Mr. M. Metzner and Mr. M. Weinhold are gratefully acknowledged for performing theoretical and experimental work.
APPENDIX
Conventional and high-frequency impedance measurements were carried out in the systems Ag(polycrystalline *)/x M AgNO, + 1 M HClO,, with x = 0.05, 0.1, 0.2 or 0.5, and at T= 298 K [5-71. Prior to each experiment, the WEs were polished mechanically, etched chemically in a chromate bath, and prepolarized by cyclic voltammetry with ]d E/dt 1 = 0.1 mV s-l in the range 3 mV d 111 =G15 mV in order to obtain reproducible and active surface states. A platinum plate and a Hg/Hg,SO,, SO,‘- electrode were used as counter and reference electrode, respectively. For conventional measurements, a platinum probe coupled by a 10 PF capacitor to the RE (cf. Fig. 1) served as the reference electrode. The electrolytes were prepared from p.a. AgNO, (Merck), suprapure HClO, (Merck) and fourfold-distilled water and were deaerated by purified nitrogen. The conventional impedance measurements were carried out at rotating disc electrodes and using a Solartron Equipment (1286 and 1250). The high-frequency experiments were done at fixed WEs with the ICEP system, a Rhode and Schwarz network analyser (ZPV) and a HP signal generator (3314 A). All measurements were digitally controlled and analyzed by a HP computer system (9845 B). The perturbation amplitude was I AE I < 1 mV. Conventional and high-frequency impedance measurements were performed in the frequency ranges 10e3 s-l
* Marzgrade,
Ag 2 99.995!%, Materials
Research.
421
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7 8 9 10 11 12 13 14 15
16 17 18 19 20 21 22 23
K.J. Vetter, Electrochemical Kinetics, Academic Press, New York, London, 1967, p. 424 ff. H. Gerischer and M. Krause, Z. Phys. Chem. N.F., 10 (1957) 264. H. Schweickert, A.A. Mihgy, A. Melendez and W.J. Lorenz, J. Electroanal. Chem., 68 (1976) 19. D.D. Maedonald, Transient Techniques in Electrochemistry, Plenum Press, New York, London, 1977. G. Schone, PhD Thesis, University of Karlsruhe, 1986. G. &hone, W. Wiesbeck, M. Stall and W.J. Lorenz, Paper presented at the international Conference on Structure and Dynamics of Solid/Electrolyte Interphases, Fritz-Haber-Institut der Max-PlanckGeseflschaft, Berlin, 2-5 September 1986. M. Stall, Diplomarbeit, University of Karlsruhe, 1986. H. Get&her and R.P. Tischer, Z. Elektrochem., 61 (1957) 1159. H. Gerischer, Z. Elektrochem., 62 (1958) 256. H. Gerischer in N. Ibl, K.M. Oesterle and A.L. Saboz @Is.), Proceedings of “Surface 66”, Forster-Verlag, Ztirich, 1967, p. 11. T. Vitanov, E. Sevastianov, Z. Stoynov and E. Budevski, Elektrokhimiya, 5 (1969) 238. T. Vitanov, E. Sevastianov, V. Bostanov and E. Budevski, Elektrokhimiya, 5 (1969) 451. T. Vitanov, A. Popov and E. Budevski, J. Electrochem. Sot., 121 (1974) 207. E. Budevski, V. Bostanov and G. Staikov, Annu. Rev. Mater. Sci., 10 (1980) 85. E. Budevski in B.E. Conway, J.G’M. Bockris. E. Yeager, S.U.M. Kahn and R.E. White (Eds.), Comprehensive Treatise of El~tr~he~st~, Vol. 7, Plenum Press, New York, London, 1983, p. 399 ff. J. Hitzig and J. Titz, Paper No. 501, 34th ISE-Meeting, Erlangen. 1983. Ext. Abstr. 0501. J. Hitzig, PhD Thesis, University of Karlsruhe, 1984. J. H&zig, J. Titz, K. Jiittner, W.J. Lorenz and E. Schmidt, Electrochim. Acta, 29 (1984) 287. E. Schmidt, J. Hitzig, J. Titz, K. Jttttner and W.J. Lorenz, Electrochim. Acta, 31 (1986) 1041. G. Schone, W. Wiesbeck, M. Stoll and W.J. Lorenz, Ber. Bunsenges. Phys. Chem., 91 (1987) 469. M.G. Natrella, Handbook 91, Experimental Statistics, National Bureau of Standards, Washington, DC, 1963, reprinted 1966. J. Newman, J. Electrochem. Sot., 117 (1970) 198; P. Pierini and J. Newman, J. Electrochem. Sot., 126 (1979) 1348; E. McCafferty, J. Electrochem. Sot., 124 (1977) 1869. M. Boheim, PhD Thesis, University of Karlsruhe, 1985.
HIGH-FREQUENCY FtEACl-IONS l
G. SCH6NE
407 in The Netherlands
IMPEDANCE
SPECTROSCOPY
OF FAST ELECTRODE
and W. WIESBECK
Institute of Microwave
Techniques, University of Karlsruhe, Karlsruhe (F.RG.)
W.J. LORENZ Institute of Physical Chemistry and Electrochemistry, (Received
23rd January
Unrversity of Karlsruhe, Karlsruhe (F R. G.)
1987)
ABSTRACT A new technique of high-frequency impedance spectroscopy working in the frequency range between 1 kHz and 5 MHz has been developed. The method is characterized by a special electrochemical cell with two working electrodes, a newly designed dc control being completely ac-decoupled, a highly precise impedance measurement technique (ICEP = Impedance Conversion and Error Processing) and quantitative calculation of the electrodynamic field distribution in the cell. The accuracy of the measurements is better than 0.05%. The method was tested on the fast Ag/Ag+ system. The results obtained are in quantitative agreement with a theoretical transfer function based on a 2-D inhomogeneous surface model.
(I) INTRODUCTION
The charge-transfer kinetics of fast electrode reactions are characterized by a high exchange current density, i,, corresponding to a low charge-transfer resistance, R,.Until now, fast electrode reactions have been investigated mainly in the time domain using large-signal pulse perturbation [l-4]. However, this transient technique is relatively inaccurate because of low reproducibility and restricted resolution of the system response at high frequencies. Moreover, electrochemical systems have a non-linear signal transfer and, therefore, large-signal perturbation gives a system response representing the dynamic system behaviour only qualitatively. On the other hand, conventional electrochemical impedance spectroscopy (EIS) cannot be applied to analyze the kinetics of fast systems in the high-frequency domain due to the following reasons:
In honour Institute.
l
of Professor
0022-0728/87/$03.50
H. Gerischer
on the occasion
0 1987 Elsevier Sequoia
S.A.
of his retirement
as Director
of the Fritz-Haber
Fig. 1. Block diagram of Conventional Electrochemical Impedance Spectroscopy (EIS). (G) Generator, (PC) personal computer, (WE) working electrode, (CE) counter-electrode, (RE) reference electrode, (S) high-frequency probe, (FRA) frequency response analyzer, Cp) potentiostat.
(i) Impedance measurements including a usual three-electrode cell coupled with a commercial potentiostat and a frequency response analyzer (FRA) as shown in Fig. 1 are restricted to a maximum frequency of f = 50 kHz. At higher frequencies, the potentiostat causes phase shifts and amplitude deformations in order to establish stability. Moreover, the accuracy of the FRA is too low to separate exactly R, from the electrolyte resistance R,, when R, e R,. (ii) The resonance frequency 2nfr = w, = (R,C,,)-’ of fast charge transfer reactions with an exchange current density of i, 2 1 A cme2 is in the range f, a 250 kHz, assuming a double-layer capacitance of Co, = 25 X lop6 F cmp2. (iii) The electrolyte resistance, R,, is considered to be real and constant even at high frequencies. In this paper, the development of a high-frequency ‘impedance spectroscopy HF-EIS) is described which allows the exact determination of the kinetics of fast electrochemical reactions. Here, the following problems had to be solved [5-71: (i) Design of a new electrochemical cell and a suitable potentiostat for dc control. (ii) Extension of measurements to higher frequencies covering the range lo3 S -I
s-‘.
(iv) Calculation of the electrodynamic field distribution in the electrochemical cell to separate the frequency-dependent electrolyte impedance go(w) from the total measured impedance z,(w). This non-conventional impedance measurement technique (called ICEP, Impedance Conversion and Error Processing) was checked on the fast system Ag/Ag+, which has an extremely high exchange current density depending on the surface structure of the substrate and the silver ion concentration in the electrolyte [8-151. Recently, conventional impedance measurements showed that this system is purely transport-controlled up to frequencies of about 5 X lo4 [5-7,16-201. S -’ taking into account 2-D surface inhomogeneity (II) CELL DESIGN
AND
POTENTIOSTATIC
DC CONTROL
In contrast to conventional impedance measurement techniques, two identical working electrodes, WE1 and WE2, with equivalent surface preparation and thus
I
1
/
Cell
Fig. 2. Concept of High-Frequency working electrodes. Fig. 3. Coaxial
ring-disc
configuration
Impedance
Spectroscopy
(HF-EIS)
with
a configuration
of two
of both electrodes.
the same interphase impedances, are used (Fig. 2). The frequency response analyser is coupled directly to the working electrodes, measuring the sum of the interphase impedances at WEl, WE2, and the impedance of the bulk solution in between. A slow ac-decoupled potential-control system regulates the identical dc potential of both WEs vs. the reference electrode, RE. This concept eliminates the above-mentioned potentiostat errors in the frequency range above 1 kHz. The inner and outer conductor of a coaxial line are used as ring and disc working electrodes WE1 and WE2 (Fig. 3). A great advantage of this configuration is the well-defined contact between both electrodes and the FRA, allowing an exact external error correction (cf. Section 111.2). Moreover, the surface preparation of the WEs can be carried out reproducibly and nearly identically. The new dc potential-control system is based on two current sources decoupled up to 10 MHz (Fig. 4). These sources have a high output impedance of more than 100 k!G!in the whole frequency range from 1 kHz to 10 MHz. The current of source 2 is controlled by a slow comparator levelling the dc voltage between WE1 and WE2 to zero. The current of source 1 is controlled by a slow PI regulator (Proportional Integral regulator) to adjust a defined potential of the WEs vs. RE.
Fig. 4. New concept of dc potential control of the working electrodes using voltage-controlled sources. (LP) Low-pass filter, (S,, S,) current sources, (D) differential amplifier.
current
410 (III) IMPEDANCE
MEASUREMENT
The measurement of the unknown impedance _Z, between the working electrode WE and the bulk solution, usually called the electrode impedance, faces three major difficulties: (i) _Z, cannot be measured separately as the bulk resistance & is always in series. (ii) _Z, is small compared to _Z,. (iii) Systematic and statistical errors are nearly of the same order of magnitude as _Z In the following, ways are shown to overcome these problems. The prime ideas are (1) to compensate the bulk impedance &, resulting in an impedance conversion and (2) to eliminate errors by calibration and integration. The procedure developed according to these ideas is called Impedance Conversion and Error Processing (ICEP). (III. 1) Impedance conversion The idea behind impedance conversion is to increase the accuracy of a given commercial Network Analyzer (NWA) by a factor of 5.. .25 inserting an interface between LZ, and the NWA. According to Fig. 5, this interface, the so-called “Impedance Converter”, subtracts a precisely known, real impedance _Z, from the cell impedance z,, compensating nearly the entire real part of &. The resulting residual impedance _Z, can be determined by the NWA with the analyzer’s standard accuracy. Adding the values of the small, not very precisely measured impedance _Z, to the large, precisely known compensation impedance _Zk, leads to a very accurate result for the interesting cell impedance z,. Figure 6 shows the block diagram of the impedance converter. It is based on a bridge, but has an additional circuit to subtract that part of the voltage at z, which is in phase with the current. This circuit consists of a power divider, deriving an in-phase internal reference signal from the generator’s source voltage, a variable attenuator reducing the reference signal by one of 128 available steps, and a differential amplifier which subtracts the residual output voltage of the attenuator from the voltage at z,.
Fig. 5. Real-part correction applied to &. residual measuring value.
(gx)
uncompensated, (Zk) compensated part of &,
(z,)
411
Ref.
Int
‘.‘. \
.
r
Variable
A(tenuation
k I
K
I.
-I-
Fig. 6. Impedance
Conversion
and Error Processing (ICEP) method.
In series to the impedance converter a commercial network analyser (NWA) is doing the measurement. A personal computer, HP 9845, controls the generator amplitude and frequency, the degree of attenuation and accepts the displayed voltages at the NWA. By variation of the compensation characterized by the step number k at the beginning of an impedance measurement, the computer system minimizes the voltage _U, at the input of the NWA. Knowing the step number k and the result &, = _V,/U_ of the network analyzer’s measurement, the personal computer is able to calculate the compensation rate & defined as the ratio ( 1Zk 1- Re@,))/Re(&), which is an indicator for the increase of accuracy. Applying an easy matrix algorithm to 46 and & leads to the wanted electrode impedance z,. The accuracy of the impedance measurement of & is limited directly by (i) the inaccuracy of stepwise attenuation, the power divider, the bridge, and the differential amplifier,
Relative A Error (*/d
Fig. 7. Relative measurement
error as a function of the real-part compensation
factor &.
412
(ii) the possible compensation rate fik, (iii) the non-linearity and noise of the differential amplifier, and (iv) the residual errors of the network analyser NWA. The influences of (i) and (ii} can be di~nish~ by a ~mputer-~ded error correction described below and by an optimized compensation state characterized by k, respectively. The influence of (iii) limits the voltage level U_ at z, to the range 0.1 mV G IQ ( G 5 mV. The influence of (iv) on the measuring results is low, as demonstrated in Fig. 7. All experiments were carried out under the condition 0 d I& 1 G 28, giving a small relative error. The absolute error of the ICEP measurement technique without error processing to reduce the influence of (i) amounts to 0.5%. (III. 2) Error processing The computer-aided error processing involves the preceding calibration of systematical errors, i.e. converter and external errors, as well as the elimination of random errors during the measurement by a tracking mechanism, the so-called windowing technique. All correction parameters are stored in the computer system and applied automatically to correct each measured value. To calculate and eliminate internal errors of the impedance converter, the error model shown in Fig. 8 was supplied. Each of the five blocks in Fig. 8 represents a possible error source. The transfer properties of these blocks were assumed to be linear but frequency-dependent due to the applied small signal perturbation. The error equation (1) can be derived from the error model in Fig. 8 by combination of all linear-dependent block parameters
z=__L3Kn+l _L?_L+L,
_x
where &, denote linear independent correction parameters and &
Fig. 8. Error model of the impedance converter. sources. (4) Voltages, (Ampl.) amplitude.
The five blocks indicated
the meas~ement
may be influenced
by error
413
Impedance
Fig. 9. Error model of the external circuit.
value. Both &, and L;, depend on the frequency and the degree of the real-part compensation k. The correction parameters L,, can be calculated according to the matrix algorithm
(2) by measuring three frequency-independent calibration standards _ZX1,L_zX,,, &. One obtains more than 4 X lo4 sets of _Li using 340 discrete frequencies at 128 different k values. Potentiostat, cables and electrode configuration cause external errors which can be represented by an error-f-port model shown in Fig. 9 with a linear transfer behaviour. The corresponding error equation
%=S,t+
S:*rL 1 -
S22IL
contains the reflection factors rL and rE and the scattering parameters giJ of the external error sources. The s,, values can be dete~ed according to eqns. (4) and 15): S;; = \iSzzK
(4)
I
by measuring the three calibration standards mentioned above. During calibration, _r,, are measured at different frequencies, whereas the _T~,~ values are the well-known reflection factors corresponding to the calibration standards z,,. In eqns. (4) and (5) &. and 4” are matrix-normalization factors. Using corrections to both converter and external errors, the overall systematic system error could be reduced to less than 0.05% in the whole frequency range lo3 s-i
414
Region:
m1
0 c] 0
11 III
0
Fig. 10. Windowing technique. (P,) Measured value, (/,) radius of an error region.
peaks or electromagnetic field influences, two different correction procedures were applied. Experimental values with high standard deviations are cancelled by a statistical averaging method [21] using at least ten measurements at each frequency. A so-called windowing technique was used additionally in the case of calibration measurements. It is based on the principle of maximum likelihood illustrated in Fig. 10. The reliability of a measurement point can be estimated on the basis of three previous points in a sequence of impedance measurements made at various consecutive frequencies. Windowing builds up three regions of probability around an estimated point in the impedance domain. Depending on the position of the next experimental value, the measurement is either continued or repeated up to 5 times. Thus, the most reasonable result will be taken as the real one. Windowing detects and eliminates more than 99% of the random errors due to the repetition of measurements caused by a disagreement between estimation and measured result. (Iv) ELECTRODYNAMIC FIELD DISTRIBUTION To calculate the true impedance of the electrode interphase Z?(u), it is necessary to determine exactly the electrolyte impedance &n(w), which must be subtracted from the measured and error-corrected cell impedance z,(w). However, z*(o) can be calculated only if the electrodynamic current distribution in the electrochemical cell is known. Previous calculations of the field distribution in electrochemical cells dealt with the assumption of static or inductive fields [22]. In this paper, a real electrodynamic field distribution is presumed, i.e. the field strength depends on time and place. The following assumptions are made to limit the mathematical formalism and the computation time in the calculation of the electrodynamic field distribution: (i) All metal surfaces are considered to be electric walls. (ii) All bulk-isolator surfaces are assumed to be magnetic walls. (iii) Small-signal perturbation is applied. (iv) The bulk electrolyte is considered as homogeneous, lossy and isotropic.
415
Fig. 11. Mathematical
model of the dual-working-electrode
concept in Fig. 2.
(v) The thickness of the interphase is assumed to be very small compared to the bulk dimensions. (vi) The coaxial cable (electrode configuration plus transmission line) and the cell have infinite length. (vii) Only the TEM mode is able to propagate within the coaxial cable. A hypothetic third medium III, the length of which tends to zero, is inserted between the electrode surface and the bulk in order to solve the boundary-value problems (Fig. 11). So the boundaries I/III and III/II consist of electric and magnetic walls only, respectively. As usual, the mathematical treatment starts with the electrodynamic potentials _T defined in eqn. (6): rot -fH = -E’
rotfE=ti -
(6)
-
where @ and @ denote the electric and magnetic fields, respectively. These trodynamic potentials are adjusted by the Lorentz convention expressed as ArE.H + b2_TE.H = 0
elec-
(7)
with the wave number /c, = o u/g,,, where We and c, are the eigenfrequency and the phase velocity of mode m. Due to the coaxial-cable symmetry (cf. Figs. 3, 11) the potential functions can be described more practically in cylinder coordinates p, rp and z as follows: 2
--
a
P ag [ p
aTE.H
1 a2TE*'
L
ap
The solution
T E,H= B”,“(p)
+_----=---+
1
P*
ad
of this differential
a2TE.H L+k2TE.n=0
az2
equation
. gEJy cp) . _z”.“( z)
--
(8)
is given by eqn. (9):
(9)
where 4, 2 and _Z are functions dependent only in one coordinate, as indicated. The solution of the boundary-value problem in medium I (Fig. 11) results in a universal description of the electrodynamic field distribution in a coaxial cable. The same treatment applied to media II and III (Fig. 11) gives the electrodynamic field distribution in a lossy dielectric cylindric waveguide and in a cylindric waveguide with conducting walls, respectively. In order to couple the different fields I/III and III/II, their development in orthogonal functions is appropriate:
I?r is an independent field vector parallel to the boundary I/III or III/II. 1, denotes a set of orthogonal functions and g, represents the corresponding weighting factor, which can be derived from an integration over the entire boundary area “A ” between two of the media (e.g. media I and II as shown in eqn. 11)
01) The resulting sets of coupled fields I/III and III/II can be combined under elimination of the fields in medium III by a well-known matrix algorithm [23]. The resulting matrix describes the coupling between all modes in media I and II, Using the knowledge about the physics of the problem, one can reduce the described equation (not provided in this text) drastically. First of all, it is taken into account, that only a TEM-mode causes the electrodynamic field distribution in medium I, and no other mode from outside incides into medium II. In a second step, it is assumed that all modes, except the TEM, are highly attenuated in the coaxial cable. So, the residual equation is very small according to
(12) where c(: * describes the amp litude of the incident ( + ) and emitted ( - ) TEM-mode into and from the coaxial cable, and $$, the ~~espon~ng coupling coefficient. Equation (12) describes the reflection of a TEM-mode being incident from the coaxial cable to the boundary I/III. The value s& is the reflection coefficient from which the impedance follows as: (13) &, denotes the well-known characteristic impedance of the coaxial cable. &(w) represents the electrolyte impedance which has to be calculated for each frequency. Now, all the tools necessary to correct the impedance measurements of fast electrode systems have been given. (V) EXPERIMENTAL
RESULTS
AND
DISCUSSION
The newly developed high-frequency impedance spectroscopy described above was tested on the fast electrode system Ag/Ag+ *. Recently, conventional impedance measurements were carried out on poly- and monocrystalline silver electrodes in the frequency range 10e3 s-’
* The experimental
details of the present
measurement
are described
m the Appendix
417
Re(Z-&$/fkm*~lO~*
Re(Z-Z~)/RxmZ~lO-*
Fig. 12. Conventional (0, 0) and high-frequency (A, A) impedance measurements. System: Ag/O.l AgNO, + 1 M HCIO,, qss = - 6.5 mV, R, = 0.17 D cm*. Frequency marks in kHz.
M
Fig. 13. High-frequency impedance measurement. System: Ag/0.05 M AgNO, + 1 M HCIO,, vss = - 10.4 mV, R, = 0.16 Q cm*. Frequency marks in kHz. (- 0 -) Optimum fit.
plained quantitatively on the basis of a 2-D inhomogeneous surface model. In the frequency range studied, the kinetics are determined by non-linear mass transport, reversible metal ion adsorption and surface relaxation of the 2-D domain structure. As expected, the fast charge-transfer step of Ag+ ions could not be detected. The aim of the present measurements is to determine quantitatively the chargetransfer resistance, R,, of the polycrystalline Ag/Ag+ electrode depending on the silver ion concentration in the electrolyte and the electrode potential. As shown recently [16-191, the influence of the hydrodynamic conditions on the impedance spectra vanishes at higher frequencies of about f z lo* s- ‘. Therefore, conventional impedance spectroscopy on RDE and high-frequency impedance spectroscopy on fixed working electrodes are comparable in a small frequency window, lo3 s- ’ Gf < 5 x lo4 s-l. Figure 12 shows the good agreement of both techniques within this range and the indication of a capacitive semicircle in the Nyquist representation at higher frequencies, representing the charge-transfer step. Examples of high-frequency impedance spectra on positively and negatively polarized electrodes are shown in Figs. 13 and 14. The theoretical transfer function of the fast Ag/Ag+ system taking into consideration 2-D surface inhomogeneity, charge transfer, non-linear diffusion, and reversible adsorption of Ag+ ions, is described by [S-7,16-20]: z-‘(s)=
[(l+u)R,+&+c~_Z~]-~+.sC~~+
z sC,[(l+~fx,+r~~~]
-1
-1
1
+ SC,,
(14)
where C,, is the double-layer capacity, C,, the adsorption capacity of ad-atoms, & the Nernst diffusion impedance, _Z, the lateral transport contribution, R, the charge-transfer resistance, (I is an index of 2-D surface inhomogeneity (a = B/(1 e)), and s is jo. This transfer function can be represented by the equivalent circuit
‘.
,l.(lK$
u
2
I
6
L
8
t-----i@
Re(Z-Zn)/hm2~10-3 _-
Fig. 14. High-frequency impedance measurement. System: Ag/0.05 M AgNO, + 1 M HCIO,, mV, R, = 0.16 s2 cm’. Frequency marks in kHz. (- 0 -) Optimum fit.
g,, = 6.5
Fig. 15. Equivalent circuit of the theoretical transfer function.
in Fig. 15. Experimentally, z(w) can be determined from the measured impedance z,(w) by subtracting z,(w). At high frequencies o z+ 1, eqn. (14) simplifies to eqn. (15) [5-71: z-‘(s)
= [(1+ a)&]
-l +s(c,,
+ CDL)
(15)
TABLE 1 Kinetic data of the polycrystalline Ag/Ag+ System
nss/mV
i,/mA
0.05 M cath.
- 1.4 - 10.4 - 18.4 6.5 10.6 13.5 - 2.5 - 6.5 - 9.6 1.9 10.5 13.0 -1.9 - 4.6 - 10.0 1.6 5.3 10.9 -4.2 - 9.3 - 12.6 3.5 5.1 6.3
- 0.4 -0.7 -1.1 0.6 1.3 1.6 -0.1 - 0.6 - 1.1 0.5 1.3 2.1 -0.7 - 2.2 - 4.3 0.2 2.4 1.1 - 5.5 - 17.3 - 22.5 8.3 15.7 23.2
0.05 M anod. 0.1 M cath. 0.1 M anod. 0.2 M cath. 0.2 M anod. 0.5 M cath. 0.5 M anod.
cm-’
electrode
(l+ o)R,/mS1 9.9 9.6 8.4 3.3 3.5 3.9 13.7 22.0 22.5 5.6 6.2 5.9 3.3 2.9 2.6 2.1 2.0 2.2 5.1 5.5 4.2 1.7 1.5 1.5
cell
cm*
0
R , /mQ cm*
i,/A
2.3 2.0 1.8 0.3 0.4 0.3 9.5 10.0 8.0 3.3 3.4 3.2 2.3 1.9 1.4 1.1 1.2 1.2 7.5 5.9 5.0 1.5 1.1 1.2
3.0 3.2 3.0 2.5 2.5 3.0 1.3 2.0 2.5 1.3 1.4 1.4 1.0 1.0 1.1 1.0 0.9 1.0 0.6 0.8 0.7 0.7 0.7 0.7
8.5 8.0 8.6 10.0 10.0 8.5 19.2 12.5 10.0 19.2 18.0 18.0 25.6 25.6 23.3 25.6 28.5 25.6 42.8 32.1 36.1 36.7 36.1 36.7
cm-*
419
tog (&t-i
Fig. 16. Concentration
1-l)
dependence
of the exchange
current
density.
which corresponds to a semicircle in the Nyquist plot with diameter (1 + a)R,. The experimental impedance spectra shown in Figs. 12-14 were fitted by the transfer function of eqn. (14), giving the optimum fit data summarized in Table 1. These data can be interpreted as follows: (i) The surface 2-D inhomogeneity expressed in the parameter u depends strongly on the pretreatment of the electrode surface [5-7,16-201. Under the given conditions (cf. Appendix), u is nearly independent at low overvoltages, but higher under cathodic than under anodic polarization. (ii) For a given silver ion bulk concentration, the charge-transfer resistance, R, and the exchange current density, i, = RT/( FR,), are nearly constant within the over-potential range studied 1v 1 Q 15 mV. This agrees well with the linearized Butler-Volmer equation of the charge-transfer process. (iii) The exchange current densities related to the active surface area 1 - 13= l/(1 + a) are about half an order of magnitude higher than the corresponding data reported in the literature [1,8-231. (iv) From the concentration dependence of the exchange current density, an electrochemical charge-transfer coefficient of (Y= 0.7 f 0.05 can be estimated, as demonstrated in Fig. 16. (v) Double-layer capacitances of about Co, = 30 f 10 I.LF cmp2 were obtained in all cases from the optimum fits indicating a relatively low 3-D surface roughness. (VI) CONCLUSIONS
The developed high-frequency impedance spectroscopy (ICEP) technique allows the exact quantitative determination of fast electrode kinetics. The experimental results obtained in the Ag/Ag+ system show the capability of this method. The data are measured with a high accuracy of better than 0.05% in the frequency range lo3 s-i
420
For monocrystalline substrates, two separated working electrodes must be used. In this case, the electrodynamic field distribution changes. Investigations on the fast charge-transfer kinetics of electrode processes using defined single-crystal faces as substrates are in progress. It can be expected, that the dynamic analysis of electrochemical systems at high frequencies will lead to a better understanding of many fundamental and practical processes.
ACKNOWLEDGEMENTS
The authors thank the “Deutsche Forschungsgemeinschaft” (DFG) and “Arbeitsgemeinschaft Industrieller Forschungsvereinigungen” (AIF) for financial support. Mrs. M. Stoll, Mr. M. Metzner and Mr. M. Weinhold are gratefully acknowledged for performing theoretical and experimental work.
APPENDIX
Conventional and high-frequency impedance measurements were carried out in the systems Ag(polycrystalline *)/x M AgNO, + 1 M HClO,, with x = 0.05, 0.1, 0.2 or 0.5, and at T= 298 K [5-71. Prior to each experiment, the WEs were polished mechanically, etched chemically in a chromate bath, and prepolarized by cyclic voltammetry with ]d E/dt 1 = 0.1 mV s-l in the range 3 mV d 111 =G15 mV in order to obtain reproducible and active surface states. A platinum plate and a Hg/Hg,SO,, SO,‘- electrode were used as counter and reference electrode, respectively. For conventional measurements, a platinum probe coupled by a 10 PF capacitor to the RE (cf. Fig. 1) served as the reference electrode. The electrolytes were prepared from p.a. AgNO, (Merck), suprapure HClO, (Merck) and fourfold-distilled water and were deaerated by purified nitrogen. The conventional impedance measurements were carried out at rotating disc electrodes and using a Solartron Equipment (1286 and 1250). The high-frequency experiments were done at fixed WEs with the ICEP system, a Rhode and Schwarz network analyser (ZPV) and a HP signal generator (3314 A). All measurements were digitally controlled and analyzed by a HP computer system (9845 B). The perturbation amplitude was I AE I < 1 mV. Conventional and high-frequency impedance measurements were performed in the frequency ranges 10e3 s-l
* Marzgrade,
Ag 2 99.995!%, Materials
Research.
421
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