Porous electrode equivalent circuit

Fig. 19 Equivalent circuit models for caibon-based porous electrodes RC circuits for a series and b parallel connections, representing an equivalent circuit (simplest) of a capacitor. R resistor, C capacitor. Equivalent circuits of only one capacitor (Cdl or CP) in parallel to a resistor R and in series to resistor RS (c) and considering both Cdl (in parallel to RE) and CP (in parallel to RE ) in series with RS (d) are also shown. The ac responses to the latter two circuits are shown in (e, f) [33] (Reprinted with permission from Ref. [33] Copyright (2012) by John Wiley and Sons)...

Figure 2. An equivalent circuit of a porous carbon electrode in SC.

Cf are the resistance and capacitance due to the particulate semiconductor film R m and are the resistance and capacitance of the parts of the BLM which remained unaltered by the incorporation of the semiconductor particles Rsc and Csc are the space charge resistance and capacitance at the semiconductor particle-BLM interface and Rss and Css are the resistance and capacitance due to surface-state on the semiconductor particles in the BLM. Electrolytes short circuit the porous semiconductor particles (Rf = Rsol = 1.4 kO) such that their contribution, along with that due to the Helmholtz layer, can be neglected. This allows the simplification of the equivalent circuit to that shown in Fig. 108c. As seen, the working electrode is connected (via ions) to the semiconductor particulate film. 

Why such a difference between the real EDLC and the model proposed here In other words, what are the specific features the equivalent circuit does not take into account The oversimplified equivalent circuit presented in Figure 1.22 considers two planar electrodes face to face, with a constant thickness of dielectric material between them. The reality is much more complex since EDLCs use three-dimensional porous electrodes, and the porous electrodes are responsible for the particular shape of the Nyquist plot presented in Figure 1.23. 

Basically, the impedance behavior of a porous electrode cannot be described by using only one RC circuit, corresponding to a single time constant RC. In fact, a porous electrode can be described as a succession of series/parallel RC components, when starting from the outer interface in contact with the bulk electrolyte solution, toward the inner distribution of pore channels and pore surfaces [4], This series of RC components leads to different time constant RC that can be seen as the electrical response of the double layer charging in the depth of the electrode. Armed with this evidence, De Levie [27] proposed in 1963 a (simplified) schematic model of a porous electrode (Figure 1.24a) and its related equivalent circuit deduced from the model (Figure 1.24b). 

Figure 18 Various models proposed for the surface films that cover Li electrodes in nonaqueous solutions. The relevant equivalent circuit analog and the expected (theoretical) impedance spectrum (presented as a Nyquist plot) are also shown [77]. (a) A simple, single layer, solid electrolyte interphase (SEI) (b) solid polymer interphase (SPI). Different types of insoluble Li salt products of solution reduction processes are embedded in a polymeric matrix (c) polymeric electrolyte interphase (PEI). The polymer matrix is porous and also contains solution. Note that the PEI and the SPI may be described by a similar equivalent analog. However, the time constants related to SPI film are expected to be poorly separated (compared with a film that behaves like a PEI) [77]. (With copyrights from The Electrochemical Society Inc., 1998.)...

To construct an equivalent circuit of a complicated electrode process (e.g., a porous electrode) and calculate its impedance, more knowledge about the network circuit may be necessary. In this section, we will spend some time discussing network circuit analysis. 

In the study of impedance plots, we may observe the depression of semicircles. This is the so-called semicircle rotation of the impedance. This phenomenon is associated with electrode/electrolyte interface double-layer properties. For example, the rough surface of the electrodes or porous electrodes can result in an uneven distribution of the double-layer electric field. This semicircle rotation can be explained using the equivalent circuit presented in Figure 3.10, where R is inversely proportional to the frequency CO (and b is a constant). 

For a porous electrode such as is found in a fuel cell, since the capacitance caused by double-layer charging is distributed along the length of the pores, the conventional double-layer capacitance is often replaced by a CPE. Then the equivalent circuit in Figure 4.15 can be modified to that shown in Figure 4.16a. 

Equivalent circuits for the catalyst layer are similar to those for porous electrodes, where charge-transfer resistance, capacitance, and Warburg resistance should be considered. The catalyst layer can be conceived of as a whole uniform unit or as a non-uniform circuit. In the case of a uniform unit, the equivalent circuits are similar to the modified ones discussed in Section 4.2.2 2, and the equations in that section apply. In many cases, such as in the presence of adsorbents, the surface is covered by the adsorbed species. For example, in direct methanol fuel cells and in H2/air fuel cells, CO adsorption should be considered. One example is illustrated in Ciureanu s work [7], as shown in Figure 4.31. 

When discussing the ionic conductivity of catalyst layers, one must mention the finite transmission-line equivalent circuit, which is widely used to model porous electrodes and was shown as Figure 4.33 in Chapter 4. For ease of discussion, the circuit is re-plotted here as Figure 6.23. 

To elucidate methanol crossover at the DMFC cathode, the active electrode surface of the cathode was divided into two separate parts one for oxygen reduction and the other for oxidation of crossover methanol. In this model, the methanol oxidation and oxygen reduction occur in parallel at different sites or pores because of the porous structure of the catalyst layer. The equivalent circuit for this model is presented in Figure 6.69. 

In corrosion systems, a salt film may cover an electrode that is itself covered by a porous oxide layer. If two different layers are superimposed, the geometrical analysis shows that the equivalent circuit corresponds to that described in Section 9.3.1 with an additional series Rti a. circuit to take into account the effect of the second porous layer. The circuit shown in Figure 9.5 is approximate because it assumes that the botmdary between the inner and outer layers can be considered to be an equipotential plane. This plane will, however, be influenced by the presence of pores. The circuit shown in Figure 9.5 will provide a good representation for systems with an outer layer that is much thicker than the inner layer and with an inner layer that has relatively few pores. 

When the polymer flhn is oxidized, its electronic conductivity can exceed the ionic conductivity due to mobile counterions. Then, the film behaves as a porous metal with pores of limited diameter and depth. This can be represented by an equivalent circuit via modified Randles circuits such as those shown in Figure 8.4. One Warburg element, representative of linear finite restricted diffusion of dopants across the film, is also included. The model circuit includes a charge transfer resistance, associated with the electrode/fllm interface, and a constant phase element representing the charge accumulation that forms the interfacial double... 

An equivalent circuit can be derived for the surface-bound membrane formed in this work similar in a manner to the approach taken for porous anodic films and porous electrodes (41-46). An equivalent circuit network, proposed in Figure 8a, corresponds to the model in Figure 7. This network has three RC subnetworks that represent the oxide layer, the surface-bound membrane layer, and the double layer. Cox and Rox are the capacitance and resistance of oxide. and Rdl are the double-layer capacitance and the polarization resistance, known as the charge transfer resistance at the membrane-water interface. For the subnetwork of the surface-bound membrane layer, one branch represents a tightly packed alkylsilane and lipid bilayer in series, and the other branch represents the pores and defects through the bilayer. Calk, Clip and Ralk, Rhp are the capacitances and resistances of... 

A very important issue to consider when working with porous electrodes is that the capacitance is only accessible through a distribution of ohmic resistances, due to the finite resistance of flie supporting electrolyte inside the pores. These situations can be roughly represented by an equivalent circuit, as shown in Fig. 11, where the porous electrode is described by a truncated RC transmission line of R and C elements representing the double Ityer capadtance and the electrolyte resistance in a particular pore size. 

Fig. 14 Scheme of a porous electrode with the equivalent general circuit model according to the theory explained earher. Note that in this picture Rs is the electrolyte resistance, which in the previous section was... 

The frequency dispersion of porous electrodes can be described based on the finding that a transmission line equivalent circuit can simulate the frequency response in a pore. The assumptions of de Levi s model (transmission line model) include cylindrical pore shape, equal radius and length for all pores, electrolyte conductivity, and interfacial impedance, which are not the function of the location in a pore, and no curvature of the equipotential surface in a pore is considered to exist. The latter assumption is not applicable to a rough surface with shallow pores. It has been shown that the impedance of a porous electrode in the absence of faradaic reactions follows the linear line with the phase angle of 45° at high frequency and then... 

The film grown on an electrode surface has a duplex structure with a thin, compact first layer that is directly on the electrode surface and a porous second layer contacting the electrolyte. An equivalent circuit can be used to represent the electrical properties of this film. The components of an equivalent circuit can be determined by impedance spectroscopy. Therefore, this method has become one of the key methods for the characterization of conducting polymers. 

Figure 11.17 Equivalent circuit of the conducting polymer fUm on an inert electrode surface. The model assumes a compact film on the metal surface with semiconducting properties in the neutral state (/ 5( and C q) and a porous film towards the electrolyte with electron resistance and ion resistance and a Faraday capacitance Cp. is the electrolyte resistance.

Figure 27.11. (a) Equivalent electric circuit and (b) impedance complex plane plot for an ideally polarizable porous electrode. 

The so-called ladder equivalent circuit shown in Figure 27.13 is characteristic of many ACs. It represents a set of several R-C parallel circuits and also Warburg diffusion impedance. Herewith, apart from the proper distributed line related to a porous structure of the studied object, one or several circuits in the ladder characterize parallel faradaic redox reactions of surface groups on the electrode. It was shown theoretically that phase angle (p = 45° independent of frequency co is observed... 

For porous electrodes, an additional frequency dispersion appears. First, it can be induced by a non-local effect when a dimension of a system (for example, pore length) is shorter than a characteristic length (for example, diffusion length), i.e. for diffusion in finite space. Second, the distribution characteristic may refer to various heterogeneities such as roughness, distribution of pores, surface disorder and anisotropic surface structures. De Levie used a transmission-line-equivalent circuit to simulate the frequency response in a pore where cylindrical pore shape, equal radius and length for all pores were assumed [14]. 

Figure 8.1 Schematic representation of the charged state of a symmetric electrical doublelayer capacitor using porous electrodes and its simplified equivalent circuit. (From Ref. [26].)...

Recently, a new equivalent circuit was proposed for porous carbon electrodes (Figure 12.5). Naively, one might suppose that this would involve multiple ladder networks in parallel, in order to model the response of multiple pores in parallel. However, the somewhat surprising result is that the circuit in Figure 12.5 is able to capture the complete multipore behavior [37]. 

---