Electrochemical Laplace-plane impedance analysis of coated electrodes containing redox systems

29

J. Electroanal. Chem., 212 (1989) 29-36

Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

Electrochemical Laplace-plane impedance analysis of coated electrodes containing redox systems Jianhui Ye * and Karl Doblhofer Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6,

D-1000 Berlin 33 (F.R.G.)

(Received 2 March 1989; in revised form 3 July 1989)

ABSTRACT The impedance function in the real-axis Laplace plane (the “operational impedance” or “transient impedance”) of a thin-layer electrochemical cell containing a quasi-reversible redox system is derived. It allows the analysis of low-level perturbations from equilibrium conditions. To demonstrate the method, the real-axis impedance of an electrode coated with a redox-active polymer film is determined by small-amplitude potential step experiments, and analysed.

INTRODUCTION

Among the techniques for investigating electrochemical systems, impedance measurements have a central place [l--7]. Such measurements can be conducted frequency-by-frequency with a steady-state sinusoidal ac-perturbation of the system. Alternatively, the investigated system can be subjected to a perturbation containing a range of frequencies, such as a potential step or white noise. To obtain, in this second case, the impedance as a function of the frequency, the perturbation and the system’s response, F(t), are both transformed [6-91 into the frequency domain, &s(s). This is done by the Laplace transformation:

F(s)=lmf(r)eWS’dt+

F(t + cc) s

(1)

where f(t) = F(t) - F(t + CQ), i.e. the part of F(t) which varies with time. The frequency s is a complex number: S=U+jw

(2)

* Present address: Department of Chemistry, Xiamen University, Xiamen, People’s Republic of China. 0022-0728/89,‘$03.50

6 1989 Blsevier Sequoia S.A.

30

\/--r.

where j = Thus, the “real axis transformation” and the “imaginary axis transformation” can be performed [6-91. Consequently, from an electrode-potential/time function, E(r), and the corresponding current/time function, I(t), the real axis impedance, Z(a), and the imaginary axis impedance, Z(jw), can be derived:

The real-axis impedance, Z(a), has been termed “operational impedance” [6,7] or “transient impedance” [8,9]. It permits experimental results to be related with impedance functions in the Laplace space. Z(jw) is a complex function having both real, Re[Z(jw)], and imaginary, Im[Z(iw)], parts. It describes the system behaviour in the same manner as if the perturbation were a steady-state sinusoidal function. It has been shown that valuable information can also be derived from the analysis of the electrochemical impedance of electrodes coated with redox-active thin films [5,10-171. The analyses were usually based on the complex-plane impedance, determined from measurements of sinusoidal ac signals. Only in one case [16,17] has the real-axis impedance, Z(a), of an electroactive thin-film system been analysed. This is the approach that will also be used in this work. It will be demonstrated that the mathematical basis of the real-axis impedance analysis is straightforward and the experimental implementation is uncomplicated; no special equipment is required. THE REAL-AXIS FARADAIC IMPEDANCE OF A THIN-LAYER ELECTROCHEMICAL CELL

To derive the faradaic impedance function of the coated electrode in the real-axis Laplace plane we assume that, initially, the redox system is incorporated homogeneously in the coating, i.e. the equilibrium concentrations of the redox ions, ct and cg prevail at time t = 0 at all positions x in the film, up to the value of x = 8, where 6 is the film thickness: co(x, 0) = c;

+(x,

0) = c;

(5)

Secondly, we assume that no redox species cross the coating/electrolyte interface. The redox system is thus confined in the thin film as in a thin-layer cell [16,18]. For such systems the boundary conditions

are valid. The other concepts for deriving the impedance function are the conventional ones. The equation for the kinetics of quasi-reversible electrode reactions is employed in the linearized form [19-211. Only diffusion (governed by Fick’s first

31

and second laws) is regarded as the mechanism for the redox species’ transport towards and from the electrode surface where the electrochemical charge-transfer reaction proceeds. Electron exchange may also take place between redox sites in the film; however, this does not alter the description because charge transport via the homogeneous electron exchange process follows the diffusion laws [22]. On this basis, the faradaic impedance in the Laplace plane can be derived for the thin-film situation considered as for the semi-infinite linear diffusion case (cf., for example, refs. 7 and 8 or p. 67 of ref. 3). One obtains

where Do and D, are the diffusion coefficients of the oxidized and reduced forms of the redox system in the coating, and R, T, n and F have their usual meaning. R,, is the heterogeneous charge-transfer resistance which is related to the exchange current density [i,, = RT/(nFR.,)]. The “complex-plane” impedance function, which may be derived from eqn. (7) (cf. eqn. 2), has been discussed by other authors (see the references given above). In this work we consider the real-axis impedance:

Equation (8) is the basis for the following electrochemical impedance analyses. Note that for x << 1 (x2 -=xx), coth x + l/x, and for x B 1, coth x + 1. Thus, at “low” frequencies where the condition

is valid, the faradaic impedance is described by the equation z&J)

=

‘+r JE n2F2d [ Co*

+Rct Cg 1

The proportionality between Z,(a) and l/a demonstrates capacitive behaviour of the system (in the corresponding complex-plane impedance plots at these frequencies the vertical section is produced [5,15]). A “low frequency capacitance”, C,,, of the system can thus be defined:

At constant temperature, CLF is a function only of the amount of electroactive material present in the coating (per unit electrode surface area). At “high” frequencies, at which the condition

32

is valid, eqn. (8) assumes the form

1

+Rct

03)

Equation (13) is identical to the corresponding semi-infinite linear diffusion case [8,9,19-211. In terms of equivalent circuits, eqn. (13) corresponds to the “Warburg” diffusion impedance, Z,(a), in series with the resistance R,,. INFLUENCE OF DOUBLE-LAYER CAPACITANCE AND UNCOMPENSATED

RESISTANCE

The measurable impedance, Z(a), of the coated electrode in the electrochemical cell can, in the absence of complications [4], be described by the “Randles” equivalent circuit [8,9,19-21,231. It contains the double-layer capacitance, Cd,, in parallel, and the uncompensated resistance, R,, in series with the faradaic impedance: Z(a)=R,+

1

Z&J)

1 + UC,,

To determine the system parameters one will normally evaluate Z(u) over a wide frequency range. At high frequencies, Z(u) will be determined largely by R,, which can thus be evaluated. If, at accessible frequencies, the condition l/Z,(u) K UC,, can be achieved, then, according to eqn. (14), C,, can be determined from the slope of a plot of these high-frequency Z(u) data vs. l/u. For the analysis of the faradaic impedance it is convenient to subtract the R, and UC,, values at each frequency, according to eqn. (14). The diffusion- and charge-transfer kinetic parameters can then be derived from plots of Z,(u) vs. l/G (eqn. 13). This procedure is, of course, identical to the more conventional semi-infinite linear diffusion case. A PRACTICAL EXAMPLE

To demonstrate the above concept, the real-axis impedance of a glassy carbon electrode coated with the perfluorinated cation-exchange polymer “Nafion” (cf. ref. 24), in a sulphuric acid electrolyte containing the redox system Ru(bpy):+/3+ (bpy stands for bipyridyl), was determined and analysed. The electrochemical complexplane impedance of this system has been studied before [13,25]. The redox cations are partitioned strongly into the coating, and are extracted only very slowly by aqueous electrolytes. The experiments can thus be conducted in the absence of redox ions in the electrolyte (the thin-layer cell assumption is justified). The experiments were conducted by applying small-amplitude potential steps (4-5 mV), starting at equilibrium potentials. The formal potential of the redox system is 1.06 V (vs. SCE), as estimated from the positions of the cyclic voltammet-

33

0

2

4 6 lo5 0-l / Iradls)”

Fig. 1. High-frequency (a) real-axis impedance, Z(a), of a glassy-carbon electrode coated with a 0.3 pm Nafion/Ru(bpy):+/2+ film in 0.2 M H,SO, electrolyte, at the indicated equilibrium potentials, E (V vs. a saturated calomel electrode, SCE).

ric peaks. The potentiostatic technique was used, with conventional electrochemical laboratory equipment. A two-compartment glass cell with a Luggin capillary was employed. The potentiostat was built in this institute. A 20 MHz, 8 bit, two-channel transient recorder allowed the overvoltage-and the current-time functions, including the rising portions, to be recorded simultaneously. Each transient for the faradaic reaction considered was recorded several times, with identical experimental parameters, but at different time intervals. From these data the final array of values, f(ti) = 3000 points, characterizing the voltage-and the current-time functions, was composed. They were transformed into the real-axis Laplace plane by the numerical integration of eqn. (1). This was done by the algorithm proposed recently [26]:

F(u) =

-F$ [f(ti+l)

+f(t,)] [e-“‘i+’ - evuri] +

F(tz O”)

(15)

Typically, 10 or 20 frequency values of Z(a) were calculated per decade. The frequency range lo6 > u > 10m2 rad s-r was routinely evaluated. The procedure was, in principle, that described previously [26]. Figure 1 is a representation of high-frequency Z( a)/(l/o) plots. The double-layer capacitance values derived from the slopes are included in Table 1. They are in good agreement with the values obtained in the absence of the redox system. As expected from eqn. (14), at all the potentials considered, the extrapolation l/o + 0 leads to the uncompensated resistance (R, = 0.95 !J cm2). The thickness of the film used for the reported experiment was estimated, from the volume of the Nafion solution evaporated on the electrode surface [26], to be S = 0.3 pm. The diffusion coefficient in the film probably describes electron hopping between the redox sites [27]. It is of the order of D = 4 x 10-l’ cm2 s-l [27]. Thus, with S/n = 1.5 s"~, eqn. (10) should be applicable to the Z,(u) values of the lowest available frequencies. Such impedance values at the equilibrium potentials 0.98, 1.02 and 1.06 V are reported in Fig. 2 as a function of l/u. The faradaic current at 0.94 V was so small that the faradaic impedance at this electrode

34 TABLE 1 Analysis of the real-axis impedance electrolyte: 0.2 M H,SO,)

measurements

0.94 0.02 2.4 13.8

E/V vs. SCE lo4 c;/mol cme3 lo4 cg/mol cmP3 RJQ cm’ i,/mA cm-’ lo4 k “/cm s-l 10” D/cm2 s-l cdl/pF cm -2 C,,/mF cm-’

(electrode:

0.98 0.1 2.3 9 2.8 6 7.9 19 1.1 (1.1)

glassy carbon/Nafion/Ru(bipy):+/3+;

1.02 0.4 2.0 5 5 6 3.9 27 2.9 (3.9)

1.06 1.2 1.2 3 8.5 7 2.9 34 4.3 (6.7)

potential could not be determined with an acceptable precision. From the slopes of the low-frequency plots of Fig. 2, the values of CL, reported in Table 1 were derived. The values included in parentheses were calculated with eqn. (11). The required equilibrium concentrations of the redox ions (cf. Table 1) were calculated from the Nernst equation, the total redox-ion concentration derived from a linearsweep voltammogram, and the above value of 8. “High-frequency” faradaic impedance results (cf. eqn. 12) are summarized in Fig. 3 on the l/G scale, relating to eqn. (13). The extrapolations l/h + 0 yield the R,, values collected in Table 1. From the corresponding exchange-current densities, i,, the apparent standard rate constants, k” ‘, were evaluated according to i, =

“Fk”‘(c;;)a(c~)l-a

(16)

The charge-transfer coefficient, LT,was assumed to be 0.5. Practically identical rate constants (Table 1) were obtained, as expected.

0 0

20

LO

60

60

100

041/lmdlsI-'

Fig. 2. Low-frequency values of the real-axis impedance, Z(o), of the system characterized in Fig. 1, at the indicated equilibrium potentials, E (in V vs. SCE). At these low frequencies, Z,(a) is practically identical to Z(a).

35

“E U 60 G \

‘;;lo

r-4

20 0 r

I

0

I

0.l

I

0.2

I

0.3

,

I Oh

(I-"2/ (rad/s)-“’

Fig. 3. Real-axis faradaic impedance, Z,(o), of the coated electrode characterized in Fig. 1, at the indicated equilibrium electrode potentials, E (in V vs. SCE).

Finally, from the slopes of the Z,(a)/(1/\/;7) plots (Fig. 3), the diffusion coefficients for charge transport across the film were derived. Assuming Do = D, = D, one obtains the diffusion coefficients included in Table 1. Variations of D as a function of the system parameters have been observed before [13]. Their discussion is not the subject of this work. REFERENCES 1 M. Sluyters-Rehbach and J.H. Sluyters in A.J. Bard (Ed.), Electroanalytical Chemistry, Vol. 4, Marcel Dekker, New York, 1970, pp. 1-128. 2 D.D. Macdonald, Transient Techniques in Electrochemistry, Plenum Press, New York, 1977. 3 D.D. Macdonald and M.C.H. McKubre in J.O’M. Bockris, B.E. Conway and R.E. White @is.), Modem Aspects of Electrochemistry, Vol. 14, Plenum Press, New York, 1982, pp. 61-150. 4 H. Giihr, Ber. Bunsenges. Phys. Chem., 85 (1981) 274. 5 C. Gabrielli, 0. Haas and H. Takeuouti, J. Appl. Electrochem., 17 (1987) 82. 6 E. Poirier d’Ange d’orsay, CR. Acad. Sci., 260 (1965) 5266. 7 E. Levart and E. Poirier d’Ange d’Orsay, J. Electroanal. Chem., 12 (1966) 277. 8 A.A. Pilla, J. Electrochem. Sot., 117 (1970) 467. 9 A.A. Pilla in J.S. Mattson, H.B. Mark, Jr. and H.C. MacDonald, Jr. (Eds.), Computers in Chemistry and Instrumentation, Vol. 2, Marcel Dekker, New York, 1972, p. 139. 10 R.D. Armstrong, J. Electroanal. Chem., 198 (1986) 177. 11 R.D. Armstrong, B. Lindholm and M. Sharp, J. Electroanal. Chem., 202 (1986) 69. 12 C. Ho, I.D. Raistrick and R.A. Huggins, J. Electrochem. Sot., 127 (1980) 343. 13 I. Rubmstein, J. Rishpon and S. Gottesfeld, J. Electrochem. Sot., 133 (1986) 729. 14 D.R. Franceschetti and J.R. MacDonald, J. Electrochem. Sot., 129 (1982) 1754. 15 K. Doblhofer and R.D. Armstrong, Electrocbim. Acta, 33 (1988) 453. 16 0. Contamin, E. Levart, G. Magner, R. Parsons and M. Savy, J. Electroanal. Chem., 179 (1984) 41. 17 0. Contamin, E. Levart, G. Magner, M. Savy and G. Scarbeck, J. Electroanal. Chem., 237 (1987) 39. 18 D.M. Ogle-sby, S.H. Omang and C.N. Reilley, Anal. Chem., 37 (1965) 1312. 19 J.E.B. Randles, Faraday Sot. Discuss., 1 (1947) 11. 20 H. Get&her, Z. Phys. Chem., 198 (1951) 286.

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