Relaxation impedance equivalent circuit

Under potentiostatic conditions, the photocurrent dynamics is not only determined by faradaic elements, but also by double layer relaxation. A simplified equivalent circuit for the liquid-liquid junction under illumination at a constant DC potential is shown in Fig. 18. The difference between this case and the one shown in Fig. 7 arises from the type of perturbation introduced to the interface. For impedance measurements, a modulated potential is superimposed on the DC polarization, which induces periodic responses in connection with the ET reaction as well as transfer of the supporting electrolyte. In principle, periodic light intensity perturbations at constant potential do not affect the transfer behavior of the supporting electrolyte, therefore this element does not contribute to the frequency-dependent photocurrent. As further clarified later, the photoinduced ET... 

In studies of these and other items, the impedance method is often invoked because of the diagnostic value of complex impedance or admittance plots, determined in an extremely wide frequency range (typically from 104 Hz down to 10 2 or 10 3 Hz). The data contained in these plots are analyzed by fitting them to equivalent circuits constructed of simple elements like resistances, capacitors, Warburg impedances or transmission line networks [101, 102]. Frequently, the complete equivalent circuit is a network made of sub-circuits, each with its own characteristic relaxation time or its own frequency spectrum. 

About 30 individual grain boundaries, i.e. 30 different electrode configurations were investigated by microcontact impedance spectroscopy. The resulting histograms of the resistance, capacitance and relaxation frequency obtained from an equivalent circuit fit of the low-frequency arc are shown in Fig. 38. For comparison, a conventional (macroscopic) impedance measurement was performed on an identically prepared sample. The relaxation frequency of the grain boundary semicircle is indicated in Fig. 38c by a solid line. 

Alternatively, an equally powerful visualization of impedance data involves Bode analysis. In this case, the magnitude of the impedance and the phase shift are plotted separately as functions of the frequency of the perturbation. This approach was developed to analyze electric circuits in terms of critical resistive and capacitive elements. A similar approach is taken in impedance spectroscopy, and impedance responses of materials are interpreted in terms of equivalent electric circuits. The individual components of the equivalent circuit are further interpreted in terms of phemonenological responses such as ionic conductivity, dielectric behavior, relaxation times, mobility, and diffusion. 

It has to be mentioned that such equivalent circuits as circuits (Cl) or (C2) above, which can represent the kinetic behavior of electrode reactions in terms of the electrical response to a modulation or discontinuity of potential or current, do not necessarily uniquely represent this behavior that is other equivalent circuits with different arrangements and different values of the components can also represent the frequency-response behavior, especially for the cases of more complex multistep reactions, for example, as represented above in circuit (C2). In such cases, it is preferable to make a mathematical or numerical analysis of the frequency response, based on a supposed mechanism of the reaction and its kinetic equations. This was the basis of the important paper of Armstrong and Henderson (108) and later developments by Bai and Conway (113), and by McDonald (114) and MacDonald (115). In these cases, the real (Z ) and imaginary (Z") components of the overall impedance vector (Z) can be evaluated as a function of frequency and are often plotted against one another in a so-called complex-plane or Argand diagram (110). The procedures follow closely those developed earlier for the representation of dielectric relaxation and dielectric loss in dielectric materials and solutions [e.g., the Cole and Cole plots (116) ]. 

The internal cell resistance is approximately the sum of / , Rc i, and Rah- The values of the resistance were determined by using the complex non-linear least squares (CNLS) fitting of the impedance spectra to the equivalent circuit of Figure 10(b). As a matter of fact, whatever models one selects for the impedance spectra in Figure 10(a), the values of total internal cell resistance and relaxation time necessary for charge/discharge of all the capacitive elements remain constant (see below). 

The direct result of the measurement and calculations are the time constants and the relaxation strength. Furthermore, if one seeks the impedance of the elements of the equivalent circuit, it can be calculated for the condition shown in Figure 8.31 using ... 

The impedance plot shown in Figure 2.1a vs. or the Nyquist plot) corresponds to an electrochemical cell (electrode/NaCl solution/electrode) and the equivalent circuit consists of a resistance (R) in parallel with a capacitor (C), which is represented as RQ, while Figure 2.1b shows the variation of the phase angle 4) = arc tan(Zi g/Z,e ) with frequency (4) vs./), but other typical impedance representations correspond to the variation of Z, and -Zj g with frequency (Bode plots), as indicated in Figure 2.1c and 2.1d. This latter representation allows the determination of the interval of frequency associated with a given relaxation process, between KT and 10 Hz, with a maximum frequency around 2 x 10 Hz, for the NaCl solution... 

Impedance plots (Nyquist and Bode plots) for the PS/PA-PEG5 and PS/PA-PEG25 membranes are shown in Figure 2.5b and 2.5c, where the effect of the asymmetric structure on the impedance (Nyquist) plot is indicated, but the differences depending on the PEG concentration are also evident. The equivalent circuit for the total membrane system, (R C ) - (RmQmX is also indicated in Figure 2.5b the depressed semicircle, attributed to the nonconstant phase circuit element (Q, is due to the porous structure of these membranes and the mixture of the relaxation times associated with their electrical response (polymeric matrix and solution). 

Figure 2.11 shows a comparison of the impedance plots obtained for clean ZIOOS and protein-fouled ZIOOSh-BSA manbranes in contact with the same NaCl solution. As can be observed, a unique semicircle was obtained for the ZIOOS membrane with a maximum frequency at 2.5 MHz, which indicates that only one relaxation process for the whole membrane/electrolyte system exists. However, when fitted to a parallel RC equivalent circuit (/ smCsJ, which includes the contribution of both membrane and electrolyte solution, two different contributions (semicircles) can be observed for the ZIOOS+BSA samples, one associated with the fouled membrane and the other associated with the electrolyte between the electrodes and the membrane surface the equivalent circuit for the whole system is (R C ) - (Ryj. 

Impedance spectroscopy (IS) is a non-destructive technique which has been widely used to determine the electrical properties of materials. Since a TBCs has a multilayered structure, with each layer having different electrical properties, it can be simplified to an Equivalent Circuit, consisting of parallel resistor (R) and capacitor (C) units that are connected in series (Fig. 19). Consequently, the electrical properties (R or C) of the TCK) and the YSZ layer can be obtained by fitting the impedance spectra with an equivalent circuit model. Fig. 19 shows typical Nyquist plots and Bode plots for APS TBCs oxidised for different times. The typical relaxation frequencies of constituent layers of TBCs are given. It was noted, especially in the Bode plot, that with increased heat-treatment the response from the TGO increased. 

Electrochemical impedance spectroscopy is extensively employed for the investigation of SAMs because the broad range of frequencies covered by this technique (usually from 10 to 10 Hz) may allow processes with different relaxation times taking place within the electrified interphase to be detected and sorted out. Unfortunately, the various relaxation times often differ by less than 2 orders of magnitude, thus requiring a certain amount of arbitrariness and of physical intuition for their separation. In fact, it is well known that the same impedance spectrum can often be equally well fitted to different equivalent circuits, which are consequently ascribed to different relaxation processes. Impedance spectra are frequently reported on a Y /co versus Y"/co plot, where Y and Y" are the in-phase and quadrature components of the electrochemical admittance and co is the angular frequency. This plot is particularly suitable for representing a series RC network. Thus, a series connection of R and C yields... 

Figure 4.5.46. Equivalent circuit with relaxation impedance (Zt).

On the basis of this model and the equivalent circuit shown in Figure 4.5.67, the changes and differences, depending on the used anode in the fuel cell (Pt/C or PtRu/C) in the impedance spectra during the experiment, are dominated by the changes of the charge transfer resistance of the anode (Raj), the surface relaxation impedance (Rg, tg) and the finite diffusion impedance (Z ). 

Oftentimes, it is difficult to develop a first approximation of an equivalent circuit upon which to test various experimental conditions. This is because more than one process can have characteristic frequencies so close to each other that it is almost impossible to distinguish the two in Nyquist or Bode plots. To overcome this limitation, researchers in the field of solid oxide fuel cells have developed alternative techniques that allow for a better estimation of equivalent circuit to use to fit the data. Two examples of these are the distribution of relaxation times (DRT) method and the analysis of difference in impedance spectrum (ADIS). While these have never been applied to MXC studies, we believe that these could provide new information of processes that researchers may not have been able to identify previously. We direct the readers to references where these methods are discussed in more detail [40-42]. In addition to these two methods, Dominguez-Benetton and Sevda recently reviewed other alternative methods of data analysis that could be applied to MXCs that we strongly suggest the readers investigate also [37]. 

As can be observed, two semicircles were obtained for dense membrane/NaCl solution and the equivalent circuit is (ReCe)-(RmCm), that is, a series association of two resistance-capacitor subcircuits, one for the electrolyte solution and other for the membrane while data for the ultrafiltration membrane corresponds to a depressed semicircle (associated with a CPE) and the equivalent circuit for the total membrane system is (ReCe)-(RmQm)- However, for the porous membrane/NaCI solution and charged membrane/KCl solution systems, a unique relaxation process (a semicircle) was obtained, which makes it impossible to evaluate the separate contributions associated with the membrane and the electrolyte solution in both cases the equivalent circuit is given by a parallel association of resistance and capacitor representing the total membrane system (RsmQm)-Nevertheless, these two latter systems (shown in Figure 9.3e, f) represent two completely different situations, as can be seem when the corresponding impedance plots for 0.002 M NaCl and 0.01 M KCl solutions are also considered (measurements carried out without membranes in the test cell) ... 

Moreover, when two-layer membranes (asymmetric or composite) are studied, the total system is electrode/solution (c)/membrane (dense/porous layers)/solution (c)/ electrode and the impedance plots can then present three relaxation processes (two at the lowest frequencies which are associated with the membrane itself plus the contribution of the electrolyte solution at high frequencies with fmax 10 Hz), as can observed in Figure 9.4. Note, equivalent circuit (ReCe)-(RiQi)-(R2C2). 

A common approach to model the dielectric response, typically used for impedance spectroscopy, is based on equivalent circuits consisting of a number of resistors, capacitors, constant phase elements, and others. Alternatively, the dielectric response can be modeled by a set of model relaxation functions like the Debye function or more generalized (semiempirical) Cole-Cole, Cole-Davidson, or Dissado-Hill equation (Kremer and Schonhals 2002). 

Figure 7.25 Equivalent circuit models proposed for the interpretation of EIS results measured in corroding systems (a) simplest representation of an electrochemical interface (6) one relaxation time constant with extended diffusion (c) two relaxation time constants and (d) the impedance of pitting processes of Al-based materials.

Figures. The generalized equivalent circuit of a single interface. Rpisthe resistance associated with the Faradaic current flow, is a generalized impedance associated with disorder either in the structure or in the dynamics (diffusion), C, and R, are associated with parallel charge accumulation modes with different relaxation times than the majority carriers such as surface states or minority carriers, is the space charge...

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