Time-Constant Dispersion

The impedance models developed in Chapters 9,10,11, and 12 are based on the assumption that the electrode behaves as a uniformly active surface where each physical phenomenon or reaction has a single-valued time constant. The assumption of a uniformly active electrode is generally not valid. Time-constant dispersion can be observed due to variation along the electrode surface of reactivity or of current and potential. Such a variation is described in Section 13.1.1 as resulting in a 2-dimensional distribution. Time-constant dispersion can also be caused by a distribution of time constants that reflect a local property of the electrode, resulting in a 3-dimensional distribution. 

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Some issues pertaining to mass transfer to electrodes are described in Section 5.6, and the associated issues for cell design are considered further in Section 8.1.2. In many cases, a uniformly accessible electrode cannot be used. The time-constant dispersion that can arise as a result of nonuniform mass transfer is discussed in Section 13.2. 

The presence of time-constant (or frequency) distribution is frequently modeled by use of a constant-phase-element (CPE), discussed in Section 13.1. As discussed in Section 13.1.3, use of a CPE eissumes a specific distribution of time constants that may apply only approximately to a given system. The objective of this chapter is to describe specific situations for which time-constant dispersion can be predicted based on fundamental phenomena such as are Eissociated with distributions of mass-transfer rates and Ohmic currents. 

While assumption that the time constant is distributed can be better than assuming that the time constant has a single value, the physical system may not follow the specific distribution implied in equation (13.7). The examples presented in the subsequent sections illustrate systems for which a time-constant dispersion results that resembles that of a CPE, but with different distributions of time constants. 

The geometry-induced current and potential distributions cause a frequency or time-constant dispersion that distorts the impedance response of a disk elec-trode. Huang et ai, 7,i02.,205 j ye shown that current and potential distributions induce a high-frequency pseudo-CPE behavior in the global impedance response of a disk electrode with a local ideally capacitive behavior, a blocking disk electrode exhibiting a local CPE behavior, and a disk electrode exhibiting Faradaic behavior. 

Huang et al. ° ° demonstrated for blocking disk electrodes that, while the local interfacial impedance represents the behavior of the system unaffected by the current and potential distributions along the surface of the electrode, the local impedance shows significant time-constant dispersion. The local and global Ohmic impedances were shown to contain the influence of the current and potential distributions. 

The frequency K = 1 at which the current distribution influences the impedance response is shown in Figure 13.7 with k/Co as a parameter. As demonstrated in Example 13.2, the influence of high-frequency geometry-induced time-constant dispersion can be avoided for reactions that do not involve adsorbed intermediates by conducting experiments below the characteristic frequency given in equation (13.57). The characteristic frequency can be well within the range of experimental measurements. The value k/Cq = 10 cm/s, for example, can be obtained for a capacitance Co = 10 (corresponding to the value expected for the dou-... 

Example 13.2 Characteristic Frequency Consider an experimental system involving a Pt disk in 0.1 M NaCl solution at room temperature for which impedance measurements are desired to a maximum frequency of 10 kHz. Estimate the maximum radius for a disk electrode that xoill avoid the influence of high-frequency geometry-induced time-constant dispersion. 

Consider a 0.25 cm radius steel disk covered with a native oxide layer. The electrolyte is a 0.1 M NaCl solution at room temperature. Estimate the frequency above which geometry-induced time-constant dispersion will influence the impedance response. 

This part demonstrates how deterministic models of impedance response can be developed from physical and kinetic descriptions. When possible, correspondence is drawn between hypothesized models and electrical circuit analogues. The treatment includes electrode kinetics, mass transfer, solid-state systems, time-constant dispersion, models accounting for two- and three-dimensional interfaces, generalized transfer functions, and a more specific example of a transfer-function tech-nique.in which the rotation speed of a disk electrode is modulated. 

Typical conducting polymer films exhibit low-frequency time constant dispersions that vary as a function of applied potential (affecting the redox state of the polymer). We have investigated the characteristics and response of protein-coated and untreated conducting polymer films from 0.5 Hz to 10 kHz with an EG G PAR 5210 lock-in amplifier. 

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