Cylindrical porous electrode model

Cylindrical Porous Electrode Model. In electrocatalysis, and in catalysis generally, there is a great interest in increasing the real surface area, especially the electrochemical active area of electrodes. In such cases, porous electrodes are used. First investigations of porous electrodes with EIS were applied by de Levie (for details see Section 2.1.6—Rough and Porous Electrodes), presenting a model describing... 

Figure 4. 45. One-dimensional network distributed homogeneous cylindrical porous electrode model after Gbhr [1997].

The random structure of the porous electrode, illustrated in Figure 13.11(a), leads to a distribution of pore diameters and lengths. Nevertheless, the porous electrode is usually represented by the simplified single-pore model shown in Figure 13.11(b) in which pores are assumed to have a cylindrical shape with a length i and a radius r. The impedance of the pore can be represented by the transmission... 

In electrocatalysis there is great interest in increasing the real surface area of electrodes. In such cases porous electrodes are used. Because modehng of real electrodes is difficult, a simpler model is usually used in which it is assumed that pores have a cylindrical shape with a length / and a radius... 

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 cylindrical pore model is an idealization of a real porous electrode. Other pore geometries were also studied. De Levie [413] obtained an analytical solution for the impedance of V-grooved pores. Such pores might be obtained, for example, by scratching the electrode surface. A cross section of such a groove is displayed in Fig. 9.8. Its impedance per unit of groove length is... 

Simple models of porous structures involve straight cylindrical pores. An understanding of the processes which determine the operation of a single pore under certain conditions is important. The discussion of the properties of single pores represents the first step. It is followed by the consideration of more elaborate models of porous structures. A comprehensive review of the progress made in the theory of porous electrodes in the last three decades was given by Chismadzhev [1] recently. Transient responses of porous electrodes are not discussed in this chapter since steady-state conditions are chiefly of interest for the operation of fuel cells. The reader is referred to the review by de Levie [2]. 

Several important technological applications, such as battery devices and electrocatalysis, need a very large effective surface of contact between the electrode and the electrolyte. This expanded surface can be developed on porous electrode surfaces. The complexity of the random structure of the porous electrode and various experimental situations related to mass-transport impedance in the pores, coupled with interfacial kinetics inside the pores, led investigators initially to investigate simple single-pore models. Of the possible shapes modeled, the cylindrical pore with a length I and a radius r has been... 

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... 

In order to see how the electrode thickness might be optimized in order to provide the lowest electrode resistivity, we have developed a theoretical model to describe the charge/discharge processes in porous carbon electrodes. As a first approximation, let us consider an electrode having two sets of cylindrical pores, namely, nanopores (NP) of less than 3 nm in diameter and transport channels (TC) of more than 20 nm in diameter, with each nanopore having an exit to only one TC. ... 

The idea that porous Ebonex might be used as a flow-through permeable electrode, originally suggested by Chen et al. [16], was exploited by Zaky and Chaplin [17], who used aporous, cylindrical Ebonex anode as a reactive electrochemical membrane for the oxidation of several model compounds. These included oxalic acid and p-.methoxyphenol that might have been expected to be inert to oxidation based on the... 

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... 

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