Capacitance behavior

The value a = 1 corresponds to ideal capacitive behavior. The fractal dimension D introduced by Mandelbrot275 is a formal quantity that attains a value between 2 and 3 for a fractal structure and reduces to 2 when the surface is flat. D is related to a by... 

Although the conductivity change Aa [relation (8)] of microwave conductivity measurements and the Ac of electrochemical measurements [relation (1)] are typically not identical (owing to the theoretically accessible frequency dependence of the quantities involved), the analogy between relations (1) and (8) shows that similar parameters are addressed by (photo)electrochemical and photoinduced microwave conductivity measurements. This includes the dynamics of charge carriers and dipoles, photoeffects, flat band and capacitive behavior, and the effect of surface states. 

F/g, whereas the composite in the optimal proportions can supply quite interesting capacitance behavior explained only by a positive effect of nitrogen. 

Figure 5 presents the capacitance-frequency dependence from impedance spectroscopy measurements for CS48 and CS15 in acidic and organic medium. In the low frequency region (from ImHz to lOOmHz) nearly a complete penetration of the ions into the pores is allowed and the quite stable values indicate the domination of the capacitive behavior at the electro 1 ytc/carbon interface. All the curves show a typical drop of... 

The N-doped carbons with a nanotube backbone combine a moderate presence of micropores with the extraordinary effect of nitrogen that gives pseudocapacitance phenomena. The capacitance of the PAN/CNts composite (ca. 100 F/g) definitively exceeds the capacitance of the single components (5-20 F/g). The nitrogen functionalities, with electron donor properties, incorporated into the graphene rings have a great importance in the exceptional capacitance behavior. 

For the first time, a-Mn02nH20 based composites have been studied in real two electrode capacitors. The a-Mn02nH20/CNTs electrodes demonstrate an ideal capacitive behavior and high values of capacitance. Compared to the conventional carbon black, multi-walled CNTs are a very promising conductivity additive for capacitor or battery electrodes. 

Sometimes molecules are used as layers in other devices. For example, molecules can act as capacitors, and this relatively new field is promising for applications in energy storage as well as providing typical capacitance behavior in thin film devices [121, 122]. 

A constant phase element (CPE) rather than the ideal capacitance is normally observed in the impedance of electrodes. In the absence of Faradaic reactions, the impedance spectrum deviates from the purely capacitive behavior of the blocking electrode, whereas in the presence of Faradaic reactions, the shape of the impedance spectrum is a depressed arc. The CPE shows... 

Rao, Jayalakshmi and coworkers have prepared NPs of SnO, SnS and ZnS in order to study their capacitance behavior [88-90]. The metal oxide and metal sulfide NPs were prepared using hydrothermal methods. After immobilization in PIGE electrodes, their electrochemical properties were examined. Capacitance values in the 4—15 Fg were reported for SnS. Comparable values were reported for ZnS. 

A method of avoiding the effect of potential differences arising at the electrodesolution interface is to take advantage of the capacitive behavior of the double layer at the electrode surface to make ac (alternating current) contact with the solution. To understand how this may be accomplished, it is necessary to consider a basic model of a conductance cell and examine its behavior under the influence of ac excitation. A review of ac circuit principles at a level sufficient for understanding the behavior of conductance cells and the instrumentation for conductance measurement is presented. The reader who desires a more thorough study of this topic is directed to material contained in the references [4-7]. 

Frequency as an experimental variable offers additional design flexibility. This approach has several advantages. The most important one is the lack of polarization of the contacts. The second one is the fact that equivalent electrical circuit analysis can be used that aids in elucidation of the transduction mechanisms. Perhaps the most important distinguishing feature of this class of conductometric sensors is the fact that their impedance is measured in the direction normal to their surface. In fact, there may be no requirement on their DC conductivity and their response can be obtained from their capacitive behavior. In the following section, we examine so-called impedance sensors (or impedimetric sensors see Fig. 8.1b). 

The chapter begins with a presentation of available evidence of cell and system transient behavior. Based on this evidence, parameters that have been proposed to account for various capacitive behaviors within the cell and fuel cell system are presented. A brief discussion for how such transient behavior affects the safe operation of a fuel cell and system is then given. Models are then developed for predicting the behavior of the cell and system, and, in the final section of this chapter, examples of their application are given. 

As already noted, the present study of dynamic fuel cell behavior involves the analysis of systems with capacitive elements. These elements control the rate at which process parameters change due to net forces imposed by other coupled process parameters. A general dynamic equation showing capacitance behavior is ... 

Fig. 9.21 Simulink electrochemical model with R-C model capacitance behavior. The BV Fen block is the quasi-steady Butler-Volmer overpotential equation giving current through the Ret charge transfer resistor as a function of charge transfer overpotential, r).

The Coupled Lumped Simulink model presented in Section 9.5.1 was extended to analyze for the double layer capacitance behavior using the model given in Section 9.3.3. The Simulink block for calculating the capacitance behavior is shown in Figure 9.21. Results for a load decrease and increase are shown in Figure 9.22. Figure 9.23 shows the same data for the load decrease case but over an expanded timebase to more clearly show the transient behavior due to the electrochemical capacitance. Similar results are found for the load increase case. 

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. 

For a perfectly smooth surface with c/F ss = 2 at all scales, Eq. (33) predicts a = 1, i.e., purely capacitive behavior. In other limit as c/F ss = 3, a = 0.5 which is de Levie s well-known result for the electrode with cylindrical pore. Eq. (33) also implies that the surfaces with different morphologies but with same surface fractal dimension are equivalent as far as the impedance is concerned. The relationship between dFss and a is dependent upon the model used in the fractal characterization of the electrode. 

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. 

Finite-space diffusion takes place during the charging of insertion electrodes at moderate frequencies, transforming into mainly capacitive behavior within the limit of very low frequencies, in contrast to the semi-infinite diffusion for solution redox-species (except for thin-layer solution electrochemistry) electrochemical impedance spectroscopy becomes a very useful diagnostic tool for the characterization of insertion mechanisms ... 

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