Porous layers circuit model

Figure 31 Time-dependent changes in equivalent circuit model element values due to interrupted sealing, (a) Solution resistance, Rs, (b) porous layer resistance, Ra, (c) porous layer capacitance, CH, (d) barrier layer resistance, Rb, (e) barrier layer capacitance, Cb. (From J. L. Dawson, G. E. Thompson, M. B. H. Ahmadun. p. 255, ASTM STP 1188, ASTM, Philadelphia, PA (1993).)...

The development of an adequate equivalent circuit has been controversially discussed in the literature. Gabrielli et al. considered the polymer primarily as a non-porous layer. Transport processes in the polymer matrix dominated the impedance. Vorotyntsev et al. developed a model that took into account the electron transfer at the metal—polymer interface, transport of charge carriers in the film, and ion transfer at the polymer-electrolyte interface (Figure 11.16). 

Figure 9-31. Equivalent circuit model of corrosion of metals covered by a porous surface layer [according to Jiittner et al. (1988), Jiittner (1990), and Felhdsi et al., (1999a)].

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.)...

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. 

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... 

In experiments covering a larger potential region, from the oxidized state until the complete neutral state, a new resonance circuit was found not described by the transmission line model. A new model was suggested by Pickup et al., which was used and modified later by Rammelt and Plieth et This model is corroborated by the duplex film structure (Figure 11.9). A compact layer on the metal/polymer interface with neutral state properties in the neutral state and double-layer properties in the oxidized state describes the compact polymer film the transmission fine model represents the porous part (Figure 11.17). 

The cathode of a modem Ni-Cd battery consists of controlled particle size spherical NiO(OH)2 particles, mixed with a conductive additive (Zn or acetylene black) and binder and pressed onto a Ni-foam current collector. Nickel hydroxide cathode kinetics is determined by a sohd state proton insertion reaction (Huggins et al. [1994]). Its impedance can therefore be treated as that of intercalation material, e.g. considering H+ diffusion toward the center of sohd-state particles and specific conductivity of the porous material itself. The porous nature of the electrode can be accounted for by using the transmission line model (D.D. Macdonald et al. [1990]). The equivalent circuit considering both diffusion within particles and layer porosity is given in Figure 4.5.9. Using the diffusion equations derived for spherical boundary conditions, as in Eq. (30), appears most appropriate. 

Figure 4.12 Equivalent circuit (porous electrode model) for the evaluation of the impedance spectra of silver gas diffusion electrodes (a). Time dependence of the double layer capacity (b), of the...

A more relevant example is related to the charge transfer at an electrode, which is in contact with a liquid solution. This situation is best described by a parallel combination of a resistance and a capacitance (circuit 2), where the resistance conveys the charge transfer kinetics at the electrode/liquid interface and the capacitance reflects the accumulation of charge at the Helmholtz layer (Helmholtz capacitance). A series resistance (denoted here by R2) associated to this RC combination reflects the Ohmic losses. This equivalent circuit is called a Randles circuit. On the other hand, if the relevant capacitive processes identifled in the electrochemical response of the system are not ideal, which is often the case for porous electrodes, CPEs (Q) can replace the capacitances (circuit 3). The parameter accounting for the nonideality of the system (n) must be comprised between 0.8 and 1 (to model a nonideal capacitance). Otherwise, for < 0.8, the CPE can be related to a different process, which is not essentially capacitive. It is this circuit that is most commonly used to describe charge transfer... 

Capacitive humidity sensors commonly contain layers of hydrophilic inorganic oxides which act as a dielectric. Absorption of polar water molecules has a strong effect on the dielectric constant of the material. The magnitude of this effect increases with a large inner surface which can accept large amounts of water. An example of this type of dielectric is porous j8-alumina. Colloidal ferric oxide, certain semiconductors, perowskites and certain polymers have also been used. /1-alumina is characterized by ionic conductance. Materials of this type can be characterized by a complex resistance composed of real (ohmic) as well as capacitive terms. The behaviour of such solids can be symbolized by a model and an associated equivalent circuit as given in Fig. 5.8. 

The electrode layers formed using die physical loading method are usually relatively thicker (more than 10 pm in thickness), and the composite layers are composed of nanoparticles of the electrode material and the ionic polymer. These layers are both electronically and ionically conductive. The impedance for such electrodes is assumed to be similar to diat of porous electrodes. Levie (1963, 1964) was the first to develop a transmission line circuit (TLC) model of the porous electrode consisting of the electrolyte resistance and the double-layer capacitance. Subsequently, a number of authors proposed modified TLC models for the impedance of porous electrodes on the basis of Levie s model. Bisquert (2000) reviewed the various impedance models for porous electrodes. The composite electrode layers prepared by the physical loading method could be successfully represented by the impedance model for porous electrodes, as shown in Fig. 6d this model is composed of the double-layer capacitance, Cj, the Warburg diffusion capacitance, W and the electrolyte resistance, 7 (Liu et al. 2012 Cha and Porfiri 2013). 

The impedance of a porous electrode can be simulated with the transmission line model, and the penetration depth can be evaluated [24]. For the non-porous Pt-modified as-deposited surface, the methanol oxidation reaction can be simulated as a simple Randles equivalent circuit comprising a parallel combination of a double layer capacitance and a semi-infinite Warburg impedance in series with a charge transfer resistance. 

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