Circuit elements diffusion-related

For the interpretation of the faradaic impedance in the presence of diffusion-related phenomena, it is convenient to subdivide it into two circuit elements. This can be done in two different ways. Zf can be presented by a resistance Rg in series with a pseudocapacitance Cg according to Fig. 6(a), or it can be subdivided into the charge-transfer resistance and the... 

Diffusion-Related Elements. Although we usually employ ideal resistors, capacitors, and inductances in an equivalent circuit, actual real elements only approximate ideality over a limited frequency range. Thus an actual resistor always exhibits some capacitance and inductance as well and, in fact, acts somewhat like a transmission line, so that its response to an electrical stimulus (output) is always delayed compared to its input. All real elements are actually distributed because they extend over a finite region of space rather than being localized at a point. Nevertheless, for equivalent circuits which are not applied at very high frequencies (say over 10 or 10 Hz), it will usually be an adequate approximation to incorporate some ideal, lumped-constant resistors, capacitors, and possibly inductances. 

When linear electrical elements are used in circuits, the measured impedances are independent of the amplitude of the ac perturbation. However, in electrochemistry, the electrical equivalent parameters related to the kinetics, diffusion, and other aspects are nonlinear because the current is a nonlinear function of the applied potential. In such cases, as was already discussed in Sects. 4.2.1 and 13.2, a small perturbation amplitude must be used to avoid the formation of harmonics. Now let us look in more detail at the formation of nonUnearities and the applicatimis of such methods. 

The most time sensitive impedance elements are the double layer capacity, the Warburg impedance (W) and the charge transfer resistance. The Warburg impedance is related to a diffusion process, as derived in section 2222 (Distributed Circuit... 

In a broad sense a parallel combination of charge transfer resistance and CPE elements, in series with finite diffusion element typically represent the circuit. When potential modulation is introduced, charge-transfer-related impedances decrease with increases in electrochemical potential and capacitance for the metal-polymer interface. The capacitance is usually nonideal due to film or electrode porosity [13] and typically is represented by the CPE element. If the film is formed as a reflective boundary, the angle is sometimes different from -90° because of inhomogeneity of the film and distributed values for diffusion coefficients. If two films are formed on the electrode, two RI CPE semicircles are often observed. 

Figure 11-lOA [75, 76] shows the Nyquist spectra at +1.00 V at TiOj electrodes in solution containing electroactive [Fe(CN)ions (pH 4.7) with and without the addition of HSA into the bulk solution. In the absence of the protein, the equivalent circuit used to fit the experimental data is reduced to a typical Randles circuit containing Warburg diffusion and [Fe(CN) ] charge-transfer resistance elements in parallel with a double-layer capacitance (Figure 11-lOA). In the presence of a full protein layer, additional processes related... 

Features of the impedance spectra of Fig. 3.15a may be modeled by a simple modified Randles-Ershler equivalent circuit shown in Fig. 3.15c. In this model, is the solution resistance, and is the charge-transfer resistance at the electrode/eIectrol e interface. A constant phase element (CPE) was used instead of a doublelayer capacitance to take into account the surface roughness of the particle. Qn is the insertion capacitance, and Zw is the Warbui impedance that corresponds to the solid-state diffusion of the Li-ion into the bulk anode. The Warburg element was used only for impedance data obtained at the tenth charge. The electrical components of the surface film which is likely formed on the electrode were disregarded, because no time constant related to this process could be seen in the electrochemical impedance spectroscopy (EIS) spectra. It was also checked that their inclusion in the model of Fig. 3.15c does not improve the fit. 

Figure 10.6 shows that the overall impedance of the system decreases after addition of plasticizer. The data are in agreement with the increase observed in ionic conductivity. From the parameters obtained by fitting the experimental data shown in Fig. 10.6, the apparent diffusion coefficient can be estimated using equation 10.7,where 4 is the thickness of the electrolyte film and 5 is a parameter related to the element O in the equivalent circuit proposed, which accounts for a finite-length Warburg diffusion (Zd), which represents a kind of resistance to mass transfer. 

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