Porous electrodes, frequency dispersion

In recent years it has been demonstrated by many researchers16,172 173 that the frequency dispersion or capacitance dispersion is intimately related to PSD or pore length distribution (PLD). In this case, the frequency dispersion is not called CPE behavior since the phase angle of the impedance spectra did not show a constant value over the whole frequency range. The phase angle of the impedance spectra measured on the porous electrode with broad PSD or PLD is larger than 45° in value at high frequencies and smaller than 90° in value at low frequencies. 

Dispersion — Frequency dispersion results from different frequencies propagating at different speeds through a material. For example, in the electrochemical impedance spectroscopy (EIS) of a crevice (or porous) electrode, the solution resistance, the charge transfer resistance, and the capacitance of the electric double layer often vary with position in the crevice (or pore). The impedance displays frequency dispersion in the high frequency range due to variations in the current distribution within the crevice (pore). Additionally, EIS measurements in thin layer cells (such as electro chromic... 

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... 

Perez, J, E.R, Gonzalez, and E.A. Ticianelli, Oxygen electrocatalysis on thin porous coating rotating platinum electrodes. Electrochimica Acta, 1998, 44 pp. 1329-1339 Song, H,-K, H,-Y, Hwang, K.-H. Lee, and L.H, Dao, The effect of pore size distribution on the frequency dispersion of porous electrodes. Electrochimica Acta, 2000. 45 pp. 2241-2257 Srikumar, A, T.G. Stanford, and J.W, Weidner, Linear sweep voltammetry in flooded porous electrodes at low sweep rates. Journal of Electroanalytical Chemistry, 1998. 458 pp. 161-173... 

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 4.5.35. Illustrating dispersion of phase angle of impedance of a porous electrode over a wide range of frequencies, w, covering various dispersion mechanisms, as indicated, with increasing (0 (from de Levie [1963, 1964]).

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 ... 

Figure 4.538. Broad range of dispersion of pseudocapacitance of a porous RUO2 electrode with frequency (from Wojtowicz and Conway, unpublished see Conway [1999]).

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