Resistive-capacitive behavior

Figure 7. (A, top) Simple battery circuit diagram, where Cdl represents the capacitance of the electrical double layer at the electrode—solution interface (cf. discussion of supercapacitors below), W depicts the Warburg impedance for diffusion processes, Rj is the internal resistance, and Zanode and Zcathode are the impedances of the electrode reactions. These are sometimes represented as a series resistance capacitance network with values derived from the Argand diagram. This reaction capacitance can be 10 times the size of the double-layer capacitance. The reaction resistance component of Z is related to the exchange current for the kinetics of the reaction. (B, bottom) Corresponding Argand diagram of the behavior of impedance with frequency, f, for an idealized battery system, where the characteristic behaviors of ohmic, activation, and diffusion or concentration polarizations are depicted.

Keiser et al.164 first showed that the more occluded the shape of the pore, the more distorted the impedance locus from the ideal capacitive behavior. However, the pore shapes in real system turn out to be much complicated and thus a straightforward analytical calculation is not usually possible of the overall impedance for those complicated pores. In connection with this problem, the fractal geometry has given a powerful tool for the analysis of the CPE behavior of the porous electrode. A number of theoretical papers166,179 191 have devoted to investigate the relationship between the fractal geometry of the electrode and the CPE impedance on the basis of the electrolytic resistive distribution due to the surface irregularity. 

The capacitance behavior is strongly dependent on the contributing surface position in the electrode [50], If the considered surface lays deep into carbon micropores, the access time for the ions to this area will be longer and more resistive than for external carbon surfaces. In other words, it means that the deep internal areas are relatively inaccessible for short-electric impulses or for high-frequency currents. These deep surfaces contribute to the capacitance with a high RC time constant. As a result, the capacitance amplitude drops with increasing signal frequency. 

High r factors are, however, not without some other complications since they imply porosity of materials. Porosity can lead to the following difficulties (a) impediment to disengagement of evolved gases or of diffusion of elec-trochemically consumable gases (as in fuel-cell electrodes 7i2) (b) expulsion of electrolyte from pores on gas evolution and (c) internal current distribution effects associated with pore resistance or interparticle resistance effects that can lead to anomalously high Tafel slopes (132, 477) and (d) difficulties in the use of impedance measurements for characterizing adsorption and the double-layer capacitance behavior of such materials. On the other hand, it is possible that finely porous materials, such as Raney nickels, can develop special catalytic properties associated with small atomic metal cluster structures, as known from the unusual catalytic activities of such synthetically produced polyatomic metal clusters (133). 

The fractal concept has proved to the helpful in describing such systems. In this connection our attention has been focused on using the fractal concept to make predictions concerning the physical properties of inhomogeneous media with a random structure. We remark that numerous other examples of fractal behavior than those treated here appear in the literature. We should mention resistance capacitance transmission lines [240] and fractal models for the alternating current response at a rough interface between materials of very dissimilar conductivities [241,242] and how a resistance capacitance line may be used as a semiintegrator [243]. 

Table 39.2 lists the pressure-volume relationships for various geometries the fluidic capacitance is found simply by differentiating with respect to pressure. For small deformations, volume varies linearly with applied pressure, such that the capacitance is not a function of the pressure it merely defines the proportionality between increases in pressure and increases in stored mass. For such cases, the fluid circuit analysis is linear, because flow rate and pressure drops are related via linear expressions. For large deformations (i.e., the membrane limit), the fluidic capacitance is a function of the pressure this implies that the fluidic circuit behavior will be nonlinear. Obviously, once the fluidic resistance, capacitance and inductance have been identified via geometry (and material properties), complicated networks can be analyzed using commercially available circuit analysis software such as SPICE [42]. 

Finally, electrochemical impedance spectroscopy (EIS) can be used to quantify the impedance characteristics of a CP. The vast majority of literature has shown that CP coating of conventional electrode materials can reduce the interface impedance by several orders of magnitude [12,52,79,139]. This has widely been attributed to the increase in surface area available for charge transfer when CPs are employed. Importantly, EIS is also used to model the electrode interface. CPs reportedly alter the capacitive behavior of conventional electrode materials to produce a predominantly resistive interface with significantly smaller phase angles at low frequencies [12,51,139,140]. 

The use of an alternating current will let us avoid the problems we had to face with a direct current. In order to avoid those inconveniences, we will have to work at high frequencies, which is to say at a few Hertz. Working with high frequencies will also make it possible for us to characterize the resistive and capacitive behavior of the set up we are studying. 

Placement of electrodes directly on the skin will show a large electrical resistance, often in the megohm (10 ) region. This is due mostly to the surface layer of dead and dehydrated skin cells on the epidermis, known as the corneum. Below the epidermis, the next sublayer is the living part of the skin, known as the dermis. It exhibits a complex electrical behavior having a resistance, capacitance, and potential generators similar in some respects to that of electrodes. 

To understand the impedance of electrochemical objects, it is necessary to understand the behavior of simple electrical circuits, first in steady state, then in transient conditions. Such circuits contain simple linear electrical elements resistance, capacitance, and inductance. Then the cmicept of electrical impedance will be introduced. It demands an understanding of the Laplace and Fourier transforms, which will also be presented. To understand impedance, it is necessary to thoroughly understand the complex plane and Bode plots, which will be presented for a few typical connections of the electrical elements. They can be computed using Excel, Maple, Mathematica, and specialized programs such as ZView. Several examples and exercises will be included. 

Equation (9.19) differs from de Levie s equation, Eq. (9.7), by the presence of one additional term. It should be noted that when r = 0, Eq. (9.19) reduces to Eq. (9.6) or (9.7). When the frequency m oo, A -> oo, the second term in Eq. (9.19) goes to zero and the first term becomes Wa,p, which corresponds to the average harmonic resistance of the solution and the electrode. When w 0, the real part of the first term in parentheses goes to 2/3 and the first term becomes 2/ n,p/3, while the real part of the second term goes to R o,p/3. This means that the low-frequency impedance is real and equal to 2R o,p/3 + R o,p/3. At low frequencies the imaginary parts of both terms go to infinity and the electrode displays its capacitive behavior. The complex plane plots of the total impedance, as well as those of the first and second terms, are compared in Fig. 9.15. The second term in Eq. (9.19), Fig. 9.15c, shows a complex plane plot that is similar to that in the absence of the electrode resistance (de Levie s solution). The first term shows a... 

Figure 2.1.12. A resistive-capacitive transmission line which describes the behavior of a semiinfinite diffusion process.

The deviations from ideal capacitative behavior of the double layer at solid electrodes also become apparent in the impedance spectroscopy behavior of electrode reactions at such electrodes where a Faradaic resistance is coupled with the capacitance then, instead of regular semicircular complex-plane plots, the Nyquist plots are arcs of semicircles, i.e. with centers depressed below the Z axis. 

Pt surface, as studied by Pajkossy [1994], almost ideally capacitative behavior can, in fact, be observed this is obviously a critical result indicating that it is not inseparable coupling between solution resistance and capacitance at a roughened (Pt) electrode surface (Pajkossy [1994]) that is the origin of dispersion effects. This led (Pajkossy [1994]) to the conclusion that it is ion (anion) adsorption that plays a crucial role in capacitance dispersion, because of frequency-dependent adsorption pseudocapacitance associated with anion chemisorption and associated kinetics of that process (Pajkossy [1994], Pajkossy et al. [1996]). 

This is the first-order system discussed previously, which is the analog of a resistance-capacitance circuit in electronics. It is mainly apphcable to responses of instruments (sensors and valves) but usually is inadequate to describe dynamic behavior of actual chemical processes. 

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