Double-layer capacitance systems

Figure 5.32. Double layer capacitance as a function of overpotential of the system a) Pt/YSZ, b) Au/YSZ, c) Ni/YSZ and d) Au/YSZ before ( ) and after (O) prolonged anodic overpotential application.55 Reprinted with permission from the National Institute of Chemistry, Ljubljana, Slovenia.

Under this electrochemical configuration, it is commonly accepted that the system can be expressed by the Randles-type equivalent circuit (Fig. 6, inset) [23]. For reactions on the bare Au electrode, mathematical simsulations based on the equivalent circuit satisfactorily reproduced the experimental data. The parameters used for the simulation are as follows solution resistance, = 40 kS2 cm double-layer capacitance, C = 28 /xF cm equivalent resistance of Warburg element, W — R = 1.1 x 10 cm equivalent capacitance of Warburg element, IF—7 =l.lxl0 F cm (

charge-transfer resistance, R = 80 kf2 cm. Note that these equivalent parameters are normalized to the electrode geometrical area. On the other hand, results of the mathematical simulation were unsatisfactory due to the nonideal impedance behavior of the DNA adlayer. This should... 

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.

AC Impedance measurements enable the determination of charge transfer resistance and double layer capacitance and other parameters related to coated systems. 

We have seen that the instantaneous faradaic current at an electrode is related to surface concentrations and charge transfer rate constants, and exponentially to the difference of the electrode potential from the E° of the electrochemical couple. This is represented in Figure 5.1c by Zf. With very few exceptions, this leads to intractable nonlinear differential equations. These systems have no closed form solutions and are treatable only by numerical integrations or numerical simulations (e.g., cyclic voltammetry). In addition, the double-layer capacitance itself is also nonlinear with respect to potential. 

Rs (Figure 1.22a). The double layer capacitance is represented by the capacitance C, and Rs is the series resistance of the EDLC, also named the equivalent series resistance (ESR). This series resistance shows the nonideal behavior of the system. This resistance is the sum of various ohmic contributions that can be found in the system, such as the electrolyte resistance (ionic contribution), the contact resistance (between the carbon particles, at the current collector/carbon film interface), and the intrinsic resistance of the components (current collectors and carbon). Since the resistivity of the current collectors is low when A1 foils or grids are used, it is generally admitted that the main important contribution to the ESR is the electrolyte resistance (in the bulk and in the porosity of the electrode) and to a smaller extent the current collector/active film contact impedance [25,26], The Nyquist plot related to this simple RC circuit presented in Figure 1.22b shows a vertical line parallel to the imaginary axis. 

Finally, the basic equivalence of the two measuring techniques should be appreciated. Although there are many ways to approach such a comparison, the following simplified explanation will, we hope, give a more intuitive feeling for the relationship between EIS and PR measurements. As stated above, both techniques rely on the frequency dependence of the impedance of the double-layer capacitance in order to determine the polarization resistance. EIS uses low frequencies to force the capacitor to act like an open circuit. PR measurements use a slow scan rate to do the same thing. To make comparisons, the idea of equivalent scan rate is useful. Suppose that a particular electrochemical system requires EIS measurements to be made down to 1 mHz in order to force 99% of the current through Rp. What would the equivalent scan rate be for PR measurements A frequency of 1 mHz corresponds to a period of 1000 s. If the sine wave is... 

Fig. 11. Schematic representation of an equivalent circuit for a mixed potential system. Q double layer capacitance, R i solution resistance, R anodic charge transfer resistance, R cathodic charge transfer resistance.

The existence of a current hump near Tc is confirmed by several additional facts. In the first place, these are deduced from the results of the quantitative treatment of the impedance spectra of the HTSC/solid electrolyte system [147]. This approach consists of calculating from the experimental complex-plane impedance diagrams the parameters characterizing the solid electrolyte, the polarization resistance of the reaction with the participation of silver, and the double-layer capacitance (Cdi) for each rvalue (measured with an accuracy of up to 0.05°). Temperature dependence of the conductance and capacitance of the solid electrolyte (considered as control parameters) were found to be monotonic, while the similar dependences of two other parameters exhibited anomalies near Tc- The existence of a weakly pronounced minimum of Cji near Tc, which is of great interest in itself, was interpreted by the authors as the result of sharp reconstruction of the interface in the course of superconducting transition [145]. 

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