Porous electrode transmission line model

Figure 16.18 Modeling scheme for a porous two-phase Ni/8YSZ composite electrode (transmission line model), rion and CNi describe the resistance per unit length (f2m ) of the effective ionic and electronic...

Eloot etal. suggested a new general matrix method for calculations involving noncylindrical pores, in which the pore is divided into sections and for each section a transmission line model with constant impedances is used. Direct simulations of the impedances for porous electrodes were also carried out using a random walk method. ... 

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... 

Figure 4.4.41. Disaetized form of the transmission line model. e and e, are the potentials in the magnetite and solution phases, respectively. Here i and i, are the currents in the magnetite and solution phases, respectively I and / are the total current and the current flowing across the metal-solution interface and base of the pore, respectively RE and M designate the reference electrode and metal (working electrode) locations, respectively. (Reprinted with permission from J. R. Park and D. D. Macdonald, Impedance Studies of the Growth of Porous Magnetite Films on Carbon Steel in High Temperature Aqueous Systems, Corros. Sci. 23, 295 [1983]. Copyright 1983, Pergamon Journals Ltd.)...

Figure 4.53. Discretization of a transmission-line model of a porous electrode or diffusion process.

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. 

It is well known that for optimal performance of electrochemical energy storage and conversion devices, it is necessary to have a nonplanar electrode to increase reaction area. One requires a porous electrode with multiple phases that can transport the reactant and products in the electrode while also undergoing reaction [1] an analogy in heterogeneous catalysis is reaction through a catalyst particle [2], For traditional devices, porous electrodes are often comprised of an electrolyte (which can be solid or liquid) that carries the ions or ionic current and a solid phase that carries the electrons or electronic current. In addition, there may be other phases such as a gas phase (e.g., fuel cells). Schematically one can consider the porous electrode as a transmission-line model as shown in Fig. 1. 

Hence we note that the simple semiinfinite transient current response obtained from analyzing a porous electrode model is similar in form to the Cottrell equation derived via a diffusion-based approach in both cases i varies as In the transmission line model, we examine... 

For supercapacitor development, special ECs have been proposed to fit the experimental results. For example. Figure 7.13 shows three proposed ECs cited in the literature. The first (a) is similar to that in Figure 7.10a, except that the parallel leakage resistance is connected in parallel to the pseudocapacitance Cp rather than to - Cp [10]. The second EC (b) and the third (c) are transmission line models to take care of the porous electrode layer [8,10,11]. Note that the third EC model uses the constant phase elements (Q,) rather than pure capacitances that mainly deal with the inclined Nyquist line at the low frequency range. In Figure 7.13b and c, the magnitudes of R, Q, and can be the same or different, depending on the real situation. The constant phase element (Qj) can be expressed as... 

Fig. 15 (a) Transmission line model for the generalized diffusion impedance, (b) Transmission line model for a porous electrode, (c) Transmission line impedance model for diffusion-recombination in a mesoporous Ti02 electrode, including also interfacial impedances and mass transport impedance in electrolyte... 

A simple but intuitive way to illustrate potential distributions and current fluxes in a porous electrode is the transmission line model (TLM) that was developed by R. De Levie in the 1960s (de Levie, 1964 Levie, 1963). Figure 1.10 shows the transmission-line equivalent circuit for the CCL under steady-state current flux. Resistances due to electron transport in the metal phase, Rm, proton transport in... 

Both blocks, when considered separately, are analogous to transmission line models of porous electrodes that were introduced by De Levie in the 1960s (de Levie, 1963,1967). Two types of coupling exist between them. An explicit form of coupling is due to the electrochemical source term, Rreac, that appears in both sets. An implicit coupling is due to the dependence of the solution on the spatially varying liquid saturation, s x). The relations p s and D s(x)), D" s x)), k s(x)), f s x)),... 

The simplest approach to understanding interfacial charging processes in porous eiectrodes is through the use of uniform transmission line models (26), such as that shown in Fig. 5. Here, the electrode is supposed to consist of a set of uniform cylindrical pores, each of length I. The electrolyte resistance per unit length is r (ohm/cm), and the capacitance per unit length Is c (F/cm). The differential equations describing the current and potential variation with distance are... 

As the first approximation, impedance of a porous electrode can always be considered as a series combination of two processes—a mass-transport resistance inside the pores and impedance of electrochemical reactions inside the pores. De Levie was the first to develop a transmission line model to describe the frequency dispersion in porous electrodes in the absence of internal diffusion limitations [66]. De Levie s model is based on the assumption that the pores are cylindrical, of uniform diameter 2r and semi-infinite length /, not intercoimected, and homogeneously filled with electrolyte. The electrode material is assumed to have no resistance. Under these conditions, a pore behaves like a imiform RC transmission line. If a sinusoidal excitation is applied, the transmission line behavior causes the amplitude of the signal to decrease with the distance from the opening of the pore, and concentration and potential gradients may develop inside the pore. These assumptions imply that only a fraction of the pore is effectively taking part in the double-layer charging process. The RpQi i- [ohm] resistance to current in a porous electrode structure with number of pores n, filled with solution with resistivity p, is ... 

Numerical simulations - The double-layer charging process for a porous electrode consisting of cylindrical pores can be simulated with the use of the transmission line model [24-26]. If the cylindrical pores are characterized by radius r, length 1 and number of pores n, the mathematical form for the transmission line model is... 

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. 

The model to describe the electrochemical behavior of the porous electrode was first treated by De Levie.160-162 He represented a pore surface by a transmission line as shown in Figure 8, and derived the following expression for the impedance of the pore, Z0... 

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. 

The models that consider this approach are largely based on the assumption of effectively homogeneous local relaxation processes related to transport in each of the phases and electrical charge exchange between them. Thus, the complex problem of an uneven distribution of electrical current and potential inside the electrode can be described analytically, and impedances can be calculated. Furthermore the models may be conveniently pictured as a double-channel transmission line (Fig. 3.5). In several papers, the theory of the impedance of porous electrodes has been extended to cover those cases in which a complex frequency response arises in the transport processes [100] or at the inner surface [194,203]. 

There are several other, more complicated elements available to describe the various processes that can occur in a photoelectrochemical cell, such as the Warburg element (to model diffusion), the Constant Phase Element (CPE, used to describe processes that have a distribution of time constants or activation energies), and transmission lines (to model porous electrodes [47]). Porous electrodes and CPE elements that represent nonideal capacitive elements are briefly discussed below. For more detailed information, the reader is referred to the literature [48, 49]. 

The porous electrode model described in Eq. (9.7) cannot be represented by a simple connection of R, L, and C elements. However, it can be represented by a semi-infinite series of R-C elements called a transmission fine [410,411], shown in Fig. 9.4. Of course, this representation is equivalent to Eq. (9.7). Some authors tried to use a transmission line to approximate experimental data using a sufficient number of RC elements and verifying whether the number of these parameters was sufficient. This procedure can approximate, then, experimental impedances, but the use of Eq. (9.7) is more appropriate because it allows for the direct estimation of certain parameters and their standard deviations. This model is included in the recent version of the ZView program. 

---