Complex-Capacitance Format

Remember 16.4 Like the admittance representation, the complex-capacitance representation emphasizes values at high frequency and is often used for solid-state and dielectric systemsfor which information is sought regarding system capacitance. 

The characteristic angular frequency for the blocking circuit is o c = 1/ReC, the same as is found for the admittance of the blocking circuit. At the characteristic angular frequency, the real part of the capacitance is equal to half the double-layer capacitance, and the imaginary part is equal to minus one-half of the double-layer capacitance. The complex-capacitance plot for tire blocking circuit traces a semicircle. 

The corresponding development for the reactive system of Table 16.1b yields 

The characteristic angular frequency for which Cr = C/2 is given by equation (16.31), and the imaginary impedance at low frequencies tends toward —oo. 

---

Impedance data are presented in different formats to emphasize specific classes of behavior. The impedance format emphasizes the values at low frequency, which t5rpically are of greatest importance for electrochemical systems that are influenced by mass transfer and reaction kinetics. The admittance format, which emphasizes the capacitive behavior at high frequencies, is often employed for solid-state systems. The complex capacity format is used for dielectric systems in which the capacity is often the feature of greatest interest. 

The characteristic frequency evident as a peak for the imaginary part of the complex-capacitance in Figures 16.12(b) and 16.13(b) has a value corresponding exactly to fc = 27tReC) only for the blocking system. As found for data presentation in admittance format, the presence of a Faradaic process confounds use of graphical techniques to assess this characteristic frequency. Like the admittance format, the complex capacitance is not particularly well suited for analysis of electrochemical and other systems for which identification of Faradaic processes parallel to the capacitance represents the aim of the impedance experiments. It is particularly well suited for analysis of dielectric systems for which the electrolyte resistance can be neglected. 

The electrical activity of a defect is characterized in part by its electrical level position, which can be determined by capacitance transient methods. When the capacitance transient spectra are monitored before and after exposure to atomic hydrogen, it is found in many systems that these levels disappear. This phenomenon has been associated with the formation of electrically inactive hydrogen-impurity complexes as summarized by Pear-ton et al. (1987) and in Chapter 5 of this volume. 

Since charged particles involve all these processes, including the formation of edge charges (Equations 2.3-2.5), first, the electric properties of interfaces have to be determined. A simple way to do so is the application of a support electrolyte in high concentration. The electric double layer, in this case, behaves as a plane and, as a first approach, the Helmholtz model, that is, the constant capacitance model, can be used (Chapter 1, Section 1.3.2.1.1, Table 1.7). It is important to note that the support electrolyte has to be inert. A suitable support electrolyte (such as sodium perchlorate) does not form complexes (e.g., with chloride ions, Section 2.3) and does not cause the degradation of montmorillonite (e.g., potassium fixation in the crystal cavities). In this case, however, cations of the support electrolyte, usually sodium ions, can also neutralize the layer charges ... 

The complex plane plots in Fig. 18 illustrate the characteristic components of the impedance response for p-type silicon and heavily doped n-type silicon in the absence of illumination. In the region of pore formation where dt//dlog(/) = 60 mV, the impedance response is characterized by an inductive loop at low frequencies and a capacitive loop at higher frequencies, as shown in Fig. 18 a. In the transition region, a second capacitive loop is observed related to oxide formation at the surface (Fig. 18 b). At more positive potentials in the electropolishing domain (Fig. 18 c) only the two capacitive loops are seen. 

In principle, electrochemical transducers can be used to detect the formation of a surface-bound affinity complex when the affinity-binding reaction is associated with a change in electrical properties (e.g., ion permeability or capacitance) of the layer immobilized onto the electrode surface. For example, the so-called ion-chemnel sensors detect permeabilily changes of a film immobilized on an electrode surface to an electroactive molecule, which is used as a redox marker. The formation of a surface-bound affinity complex results in a permeability change, which can be monitored by the change of cyclic voltammetric response of the redox marker. 

The real and imaginary parts of the complexfrequency limit can be interpreted in terms of an effective double-layer capacitance. This relationship is exact for the blocking circuit of Table 16.1(a). The corresponding plots in logarithmic format are presented as Figures 16.13(a) and (b), respectively. 

VIBRATIONAL SPECTROSCOPY Infrared and Raman spectroscopies have proven to be useful techniques for studying the interactions of ions with surfaces. Direct evidence for inner-sphere surface complex formation of metal and metalloid anions has come from vibrational spectroscopic characterization. Both Raman and Fourier transform infrared (FTIR) spectroscopies are capable of examining ion adsorption in wet systems. Chromate (Hsia et al., 1993) and arsenate (Hsia et al., 1994) were found to adsorb specifically on hydrous iron oxide using FTIR spectroscopy. Raman and FTIR spectroscopic studies of arsenic adsorption indicated inner-sphere surface complexes for arsenate and arsenite on amorphous iron oxide, inner-sphere and outer-sphere surface complexes for arsenite on amorphous iron oxide, and outer-sphere surface complexes for arsenite on amorphous aluminum oxide (Goldberg and Johnston, 2001). These surface configurations were used to constrain the surface complexes in application of the constant capacitance and triple layer models (Goldberg and Johnston, 2001). 

Experiments carried out on monocrystalline Au(lll) and Au(lOO) electrodes in the absence of specific adsorption did not show any fre-quency dispersion. Dispersion was observed, however, in the presence of specific adsorption of halide ions. It was attributed to slow adsorption and diffusion of these ions and phase transitions (reconstructions). In their analysis these authors expressed the electrode impedance as = R, + (jco iJ- where is a complex electrode capacitance. In the case of a simple CPE circuit, this parameter is = T(Jcaif. However, an analysis of the ac impedance spectra in the presence of specific adsorption revealed that the complex plane capacitance plots (C t vs. Cjnt) show the formation of deformed semicircles. Consequently, Pajkossy et al. proposed the electrical equivalent model shown in Fig. 29, in which instead of the CPE there is a double-layer capacitance in parallel with a series connection of the adsorption resistance and capacitance, / ad and Cad, and the semi-infinite Warburg impedance coimected with the diffusion of the adsorbing species. A comparison of the measured and calculated capacitances (using the model in Fig. 29) for Au(lll) in 0.1 M HCIO4 in ths presence of 0.15 mM NaBr is shown in Fig. 30. 

---