Solid electrodes, impedance

Problems similar to those observed on ideally polarizable solid electrodes also arise in the presence of faradaic reactions at these electrodes. In the next section, various models used to explain solid electrode impedance behavior are presented. 

Metal/molten salt interfaces have been studied mainly by electrocapillary833-838 and differential capacitance839-841 methods. Sometimes the estance method has been used.842 Electrocapillary and impedance measurements in molten salts are complicated by nonideal polarizability of metals, as well as wetting of the glass capillary by liquid metals. The capacitance data for liquid and solid electrodes in contact with molten salt show a well-defined minimum in C,E curves and usually have a symmetrical parabolic form.8 10,839-841 Sometimes inflections or steps associated with adsorption processes arise, whose nature, however, is unclear.8,10 A minimum in the C,E curve lies at potentials close to the electrocapillary maximum, but some difference is observed, which is associated with errors in comparing reference electrode (usually Pb/2.5% PbCl2 + LiCl + KC1)840 potential values used in different studies.8,10 It should be noted that any comparison of experimental data in aqueous electrolytes and in molten salts is somewhat questionable. 

The study of metal ion/metal(s) interfaces has been limited because of the excessive adsorption of the reactants and impurities at the electrode surface and due to the inseparability of the faradaic and nonfaradaic impedances. For obtaining reproducible results with solid electrodes, the important factors to be considered are the fabrication, the smoothness of the surface (by polishing), and the pretreatment of the electrodes, the treatment of the solution with activated charcoal, the use of an inert atmosphere, and the constancy of the equilibrium potential for the duration of the experiment. It is appropriate to deal with some of these details from a practical point of view. 

Here j is the imaginary unit, co is the angular frequency, and C is the capacitance. For solid electrodes, however, the impedance response deviates from a purely capacitive one and the empirical equation should be used... 

Fig. 16. Dependence of d(C 2)/dE on the logarithm of frequency, obtained by linear (open circles) and nonlinear (solid circles) impedance techniques. Polycrystalline electrode [83]. Reproduced by permission from Elsevier Science.

Many electrode processes are more complex than those discussed above. Besides this, the impedance of an interface is dependent on its microscopic structure which, in the case of a solid electrode, can have an important influence. Impedance measurements can be used to study complicated corrosion phenomena (Chapter 16), blocked interfaces (i.e. where there is no redox process nor adsorption/desorption), the liquid/liquid interface2425, transport through membranes26, the electrode/solid electrolyte interface etc. Experimental measurements always furnish values of Z and Z" or their equivalents Y and Y", or of the complex permittivities e and e" (e = Y/icoCc, Cc being the capacitance of the empty cell). In this section we attempt to show how to... 

An important example of the system with an ideally permeable external interface is the diffusion of an electroactive species across the boundary layer in solution near the solid electrode surface, described within the framework of the Nernst diffusion layer model. Mathematically, an equivalent problem appears for the diffusion of a solute electroactive species to the electrode surface across a passive membrane layer. The non-stationary distribution of this species inside the layer corresponds to a finite - diffusion problem. Its solution for the film with an ideally permeable external boundary and with the concentration modulation at the electrode film contact in the course of the passage of an alternating current results in one of two expressions for finite-Warburg impedance for the contribution of the layer Ziayer = H(0) tanh(icard)1/2/(iwrd)1/2 containing the characteristic - diffusion time, Td = L2/D (L, layer thickness, D, - diffusion coefficient), and the low-frequency resistance of the layer, R(0) = dE/dl, this derivative corresponding to -> direct current conditions. 

Figure 37. Nyquist plot of the experimental (solid circles) and simulated (solid Une) impedance spectrum of TRPyPz/CuTSPc film modified ITO electrode, from 1 to 100 kHz, at 0.95 and 0.60 V inset). Electrolyte 0.10 Af lithium trifluoromethane sulfonate aqueous solution.

The interfacial capacitance may also be measured at solid polarizable electrodes in an impedance experiment using phase-sensitive detection. Most experiments are carried out with single crystal electrodes at which the structure of the solid electrode remains constant from experiment to experiment. Nevertheless, capacity experiments with solid electrodes suffer from the problem of frequency dispersion. This means that the experimentally observed interfacial capacity depends to some extent on the frequency used in the a.c. impedance experiment. This observation is attributed to the fact that even a single crystal electrode is not smooth on the atomic scale but has on its surface atomic level steps and other imperfections. Using the theory of fractals, one can rationalize the frequency dependence of the interfacial properties [9]. The capacitance that one would observe at a perfect single crystal without imperfections is that obtained at infinite frequency. Details regarding the analysis of impedance data obtained at solid electrodes are given in [10]. 

One current-based approach is referred to as impedancemetric sensing [32]. This is based on impedance spectroscopy, in which a cyclic voltage is applied to the electrode and an analysis of the resultant electrical current is used to determine the electrode impedance. As different processes have different characteristic frequencies, impedance spectroscopy can be used to identify and separate contributions from different processes, such as electron transfer at the interface from solid-state electronic conduction. The frequency range ofthe applied voltage in impedancemetric sensors is selected so that the measured impedance is related to the electrode reaction, rather than to transport in the electrode or electrolyte material. Thus, the response is different from that in resistance-based sensors, which are related to changes in the electrical conductivity of a semiconducting material in response to changes in the gas composition. 

III.l [see also Eq. (17) and Fig. 2], and that in the presence of a faradaic reaction [Section III. 2, Fig. 4(a)] are found experimentally on liquid electrodes (e.g., mercury, amalgams, and indium-gallium). On solid electrodes, deviations from the ideal behavior are often observed. On ideally polarizable solid electrodes, the electrically equivalent model usually cannot be represented (with the exception of monocrystalline electrodes in the absence of adsorption) as a smies connection of the solution resistance and double-layer capacitance. However, on solid electrodes a frequency dispersion is observed that is, the observed impedances cannot be represented by the connection of simple R-C-L elements. The impedance of such systems may be approximated by an infinite series of parallel R-C circuits, that is, a transmission line [see Section VI, Fig. 41(b), ladder circuit]. The impedances may often be represented by an equation without simple electrical representation, through distributed elements. The Warburg impedance is an example of a distributed element. 

Electric Double Layer and Fractal Structure of Surface Electrochemical impedance spectroscopy (EIS) in a sufficiently broad frequency range is a method well suited for the determination of equilibrium and kinetic parameters (faradaic or non-faradaic) at a given applied potential. The main difficulty in the analysis of impedance spectra of solid electrodes is the frequency dispersion of the impedance values, referred to the constant phase or fractal behavior and modeled in the equivalent circuit by the so-called constant phase element (CPE) [5,15,16, 22, 35, 36]. The frequency dependence is usually attributed to the geometrical nonuniformity and the roughness of PC surfaces having fractal nature with so-called selfsimilarity or self-affinity of the structure resulting in an unusual fractal dimension... 

Figure 2. Scanning electron micrograph of polymeric (TMHPP)Ni (a). An impedance spectrum for thin-layer polymeric film on a solid electrode (b). An impedance spectrum for polymeric-(TMHPP)Ni film on a glassy carbon electrode (potential 0.52 V, film thickness 0.4 j im) (c).

For an ideally polarized electrode, the impedance consists of the double-layer capacity Cd and the solution resistancein series. In the impedance plane plot, a straight vertical line results intersecting the Z -axis at Z =. At solid electrodes, especially... 

McAdams, E.T., Jossinet, J., 1991a. DC nonlinearity of the solid electrode-electrolyte interface impedance. Innov. Technol. Biol. Med. 12, 330—343. 

The impedance of ideally polarizable liquid electrodes (e.g., mercury, amalgams, indium-gallium) may be modeled by an R-C circuit (Fig. 4.1a). However, most impedance studies are now carried out at solid electrodes. At these electrodes the double-layer capacitance is not purely capacitive and often displays a certain frequency dispersion. Such behavior cannot be modeled by a simple circuit consisting of R, L, and C elements. To explain such behavior, a constant phase element (CPE) is usually used. 

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