Randles impedance

Figure II.5.3 represents the Nyquist plot of the Randles impedance with the semicircle at higher frequencies and the straight line at an angle of 45° to the real axis at lower frequencies.

The cell consists of two electrodes and an electrolyte consequently the simplest analogue is that of two Randles impedances[3] in series with the electrolye resistance as shown in Figure 20. The impedances of two-terminal cells have been discussed elsewhere[4] for... 

The impedance data have been usually interpreted in terms of the Randles-type equivalent circuit, which consists of the parallel combination of the capacitance Zq of the ITIES and the faradaic impedances of the charge transfer reactions, with the solution resistance in series [15], cf. Fig. 6. While this is a convenient model in many cases, its limitations have to be always considered. First, it is necessary to justify the validity of the basic model assumption that the charging and faradaic currents are additive. Second, the conditions have to be analyzed, under which the measured impedance of the electrochemical cell can represent the impedance of the ITIES. 

In particular, the coupling between the ion transfer and ion adsorption process has serious consequences for the evaluation of the differential capacity or the kinetic parameters from the impedance data [55]. This is the case, e.g., of the interface between two immiscible electrolyte solutions each containing a transferable ion, which adsorbs specifically on both sides of the interface. In general, the separation of the real and the imaginary terms in the complex impedance of such an ITIES is not straightforward, and the interpretation of the impedance in terms of the Randles-type equivalent circuit is not appropriate [54]. More transparent expressions are obtained when the effect of either the potential difference or the ion concentration on the specific ion adsorption is negli-... 

FIG. 6 Randles equivalent circuit for the ITIES Zq is the interfacial capacitance, Zy)v are the faradaic impedances of the charge transfer reactions, and is the solution resistance. 

Since the ion transfer is a rather fast process, the faradaic impedance Zj can be replaced by the Warburg impedance Zfy corresponding to the diffusion-controlled process. Provided that the Randles equivalent circuit represents the plausible model, the real Z and the imaginary Z" components of the complex impedance Z = Z —jZ " [/ = (—1) ] are given by [60]... 

Under this electrochemical configuration, it is commonly accepted that the system can be expressed by the Randles-type equivalent circuit (Fig. 6, inset) [23]. For reactions on the bare Au electrode, mathematical simsulations based on the equivalent circuit satisfactorily reproduced the experimental data. The parameters used for the simulation are as follows solution resistance, = 40 kS2 cm double-layer capacitance, C = 28 /xF cm equivalent resistance of Warburg element, W — R = 1.1 x 10 cm equivalent capacitance of Warburg element, IF—7 =l.lxl0 F cm (

charge-transfer resistance, R = 80 kf2 cm. Note that these equivalent parameters are normalized to the electrode geometrical area. On the other hand, results of the mathematical simulation were unsatisfactory due to the nonideal impedance behavior of the DNA adlayer. This should... 

This circuit is usually referred to as the Randles circuit and its analysis has been a major feature of AC impedance studies in the last fifty years. In principle, we can measure the impedance of our cell as a function of frequency and then obtain the best values of the parameters Rct,<7,C4i and Rso by a least squares algorithm. The advent of fast micro-computers makes this the normal method nowadays but it is often extremely helpful to represent the AC data graphically since the suitability of a simple model, such as the Randles model, can usually be immediately assessed. The most common graphical representation is the impedance plot in which the real part of the measured impedance (i.e. that in phase with the impressed cell voltage) is plotted against the 90° out-of-phase quadrature or imaginary part of the impedance. 

Further information on this subject can be obtained by frequency response analysis and this technique has proved to be very valuable for studying the kinetics of polymer electrodes. Initially, it has been shown that the overall impedance response of polymer electrodes generally resembles that of intercalation electrodes, such as TiS2 and WO3 (Ho, Raistrick and Huggins, 1980 Naoi, Ueyama, Osaka and Smyrl, 1990). On the other hand this was to be expected since polymer and intercalation electrodes both undergo somewhat similar electrochemical redox reactions, which include the diffusion of ions in the bulk of the host structures. One aspect of this conclusion is that the impedance response of polymer electrodes may be interpreted on the basis of electrical circuits which are representative of the intercalation electrodes, such as the Randles circuit illustrated in Fig. 9.13. The figure also illustrates the idealised response of this circuit in the complex impedance jZ"-Z ) plane. 

Fig. 9.13 Randles-type circuit (top) and its idealised response in the impedance plane. / = frequency of the ac signal...

A significant step forward in our understanding of Pt was taken by Verkerk and Burgraff, who in 1983 analyzed the impedance of porous sputtered Pt (and Pt gauze) electrodes on YSZ and gadolinia-doped ceria (GDC). As shown in Figure 11, they used a Randles circuit to model the interfacial contributions to the impedance, allowing them to subtract from the data the contributions of uncompensated iR and... 

This circuit is usually referred to as the Randles circuit and its analysis has been a major Feature of AC impedance studies in the last fifty years. In principle, we can measure the impedance of our cell as a Function of Frequency and then obtain the best values of the parameters and by a... 

A renewal of interest in the other rate-controlling processes started in those groups who were developing the impedance method [49, 53] and the a.c. polarographic method [12, 25], probably because it was found that, in many cases, Randles equivalent circuit did not hold and also because the appropriate mathematics are more tractable in the frequency domain. Still, it is recommended that the a.c. studies are combined with the diagnostic results which can be obtained from steady-state techniques and/or cyclic voltammetry. 

Zac represents the ac impedance of the usual Randles equivalent circuit. 

In a simple case, the electrochemical reaction at the electrode-electrolyte interface of one of the electrodes of the battery can be represented by the so-called Randles circuit (Figure 8.19), which is composed of [129] a double layer capacitor formed by the charge separation at the electrodeelectrolyte interface, in parallel to a polarization resistor and the Warburg impedance connected in series with a resistor, which represents the resistance of the electrolyte. 

The geometrical capacitance Cg is of the order of 10 10 F cm 2 for the majority of films studied, whose thickness is 1 pm by order the corresponding impedance would be an order higher than the film resistance Rs (with exception of very low-doped films) and can be neglected. We then obtain a simpler three-element circuit (Fig. 10b) often called the Randles circuit [65], An essential assumption is that all elements of the circuits in Figs. 10a and b are frequency-independent. 

Occasionally, the impedance spectra of diamond electrodes are well described by the Randles equivalent circuit with a frequency-independent capacitance (in the 1 to 105 Hz range) [66], Shown in Fig. 11 is the potential dependence of the reciprocal of capacitance squared, a well-known Mott-Schottky plot. Physically, the plot reflects the potential dependence of the space charge region thickness in a semiconductor [6], The intercept on the potential axis is the flat-band potential E whereas the slope of the line gives the uncompensated acceptor concentration NA - Nd in what follows, we shall for brevity denote it as Na ... 

The experimental impedance is always obtained as if it were the result of a resistance and capacitance in series. We have already seen in (11.20) and (11.21) the relation between an RC series combination and the Rct + zw combination. It can be shown for the full Randles equivalent circuit for this simple charge transfer reaction, see Fig. 11.4, on separating the in-phase and out-of-phase components of the impedance, that... 

The unit of Rct is fl cm2. Rct is also called activation resistance. It follows from Eq. (1) that the higher is ja, the smaller is Rct. Rct can be calculated also at different potentials far from the equilibrium which is a general practice in - electrochemical impedance spectroscopy. It is based on the concept that at small signal perturbation (< 5 mV) the response is essentially linear. Rct values are obtained either from the diameter of the - Randles semicircle or from the angular frequency to) at which Z" exhibits a maximum vs. Z ... 

The second meaning of the word circuit is related to electrochemical impedance spectroscopy. A key point in this spectroscopy is the fact that any -> electrochemical cell can be represented by an equivalent electrical circuit that consists of electronic (resistances, capacitances, and inductances) and mathematical components. The equivalent circuit is a model that more or less correctly reflects the reality of the cell examined. At minimum, the equivalent circuit should contain a capacitor of - capacity Ca representing the -> double layer, the - impedance of the faradaic process Zf, and the uncompensated - resistance Ru (see -> IRU potential drop). The electronic components in the equivalent circuit can be arranged in series (series circuit) and parallel (parallel circuit). An equivalent circuit representing an electrochemical - half-cell or an -> electrode and an uncomplicated electrode process (-> Randles circuit) is shown below. Ic and If in the figure are the -> capacitive current and the -+ faradaic current, respectively. 

The Nyquist plot of a Randles cell is always a semicircle. At high frequencies the impedance of Cdl is very low, so the measured impedance tends to Reh At very low frequencies the impedance of Cm becomes extremely high, and thus, the measured impedance tends to Rct + Rd. Accordingly, at intermediate frequencies, the impedance falls between Rd and Rct + Rd. Therefore, the high-frequency intercept is associated with the electrolyte resistance, while the low-frequency intercept corresponds to the sum of the charge-transfer resistance and the electrolyte resistance. The diameter of the semicircle is equal to the charge-transfer resistance. 

The Nyquist plot is presented in Figure 4.9b. At high frequencies (real axis at a value of R. At low frequencies ( 0), it intercepts the real axis at a value of R+R0. Note that the bounded Warburg impedance is easily recognized from its Nyquist plot. At high frequencies, this circuit element looks like a traditional Warburg impedance and shows a 45° line on the Nyquist plot. At low frequencies, it looks like the semicircle of a Randles cell,... 

If a resistor is added in series with the parallel RC circuit, the overall circuit becomes the well-known Randles cell, as shown in Figure 4.11a. This is a model representing a polarizable electrode (or an irreversible electrode process), based on the assumptions that a diffusion limitation does not exist, and that a simple single-step electrochemical reaction takes place on the electrode surface. Thus, the Faradaic impedance can be simplified to a resistance, called the charge-transfer resistance. The single-step electrochemical reaction is described as... 

Figure 4.16b shows the simulated Nyquist plot of the modified Randles cell shown in Figure 4.16a. Impedance diagrams with variations in the CPE parameters are included in Appendix D (Model D15).

Figure 4.176 shows the simulated Nyquist plot of a modified adsorption model with an extra CPE in the Randles circuit. This modification of the adsorption model strongly influences the low-fiequency shape of the complex plane impedance diagram. More examples with variations on the CPE exponent are presented in Appendix D (Model D16). 

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