Double-layer capacitors circuits

Fig. 18b.5. (a) The capacitor-like metal solution interface, the double layer, (b) The equivalent circuit with solution resistance and overall double-layer capacitor, (c) Charging current transient resulting from a step-potential at... 

In a simple case, the electrochemical reaction at the electrode-electrolyte interface of one of the electrodes of the battery can be represented by the so-called Randles circuit (Figure 8.19), which is composed of [129] a double layer capacitor formed by the charge separation at the electrodeelectrolyte interface, in parallel to a polarization resistor and the Warburg impedance connected in series with a resistor, which represents the resistance of the electrolyte. 

The first step in developing an equivalent electrical circuit for an electrochemical system is to analyze the nature of the overall current and potential. For example, in the simple case of the uniformly accessible electrode shovm in Figure 9.1(a), the overall potential is the sum of the interfacial potential V plus the Ohmic drop Rgi. Accordingly, the overall impedance is the sum of the interfacial impedance Zo plus the electrolyte resistance Re- At the interface itself, shown in Figure 9.1(b), the overall current is the sum of the Faradaic current if plus the charging current I c through the double layer capacitor C. Thus, the interfacial impedance results from the double-layer capacity in parallel with the Faradaic impedance Zf. 

With a dripping mercury electrode the surface is ideal and the double layer is modeled as a pure, frequency independent capacitor, somewhat voltage-dependent. The capacitance values are very high because of the small double-layer thickness, Cdi is about 20 pF/cm. With solid electrode materials, the surface is of a more fractal nature, with a distribution of capacitive and resistive properties. The actual values are dependent on the type of metal, the surface conditions, the type of electrolyte, and the applied voltage. The capacitance increases with higher electrolyte concentration. The double-layer capacitor is inevitable it is there as long as the metal is wetted. Cdi may dominate the circuit if there are no sorption or electrode reaction processes, or if the frequency is high. 

However, in various papers by Miller [1972], he has been able to model the impedance spectrum and power-density (related to the distribution of RC time-constant values) of practical carbon double-layer capacitor devices by a five-element ladder circuit of C and R elements as illustrated in Figure 4.5.27. The resulting impedance behavior is exemplified in Figure 4.5.28. 

The equivalent circuit fitting for these experimental data is a Randles circuit. The Randles equivalent circuit is one of the simplest and most common circuit models of electrochemical impedance. It includes a solution resistance, Rs in series to a parallel combination of resistor. Ret, representing the charge transfer (corrosion) resistance and a double layer capacitor, Cai, representing the electrode capacitance (Badawy et al, 1999). In this case, the value of Rs can be neglected because the value is too small as compared to that of the value of Ret. The equivalent circuit for the Randles cell is shown in Figure 2. ... 

Figure 14.1 A simple electrified interface, in which the vertical dotted lines in (a) are represented by the electronic components in (b). (a) The oxidants (red) with a positive charge diffuse toward the negatively charged electrode, accept electrons from the electrode at the interface, become the reductants (blue), and diffuse to the bulk of the solution. The oxidant is also a counterion to the electrode. No specific adsorption is considered at the interface. IHPs and OHPs are the inner and outer Helmholtz planes, respectively, (b) An equivalent circuit representing each component at the interface and in the solution during an electrochemical reaction is shown for comparison with the physical components. double-layer capacitor ...

Figure 24. Hypothetic equivalent circuit for a cell of one hybrid electrode where the insertion materials and electrochemical double layer capacitor (EDLC) materials are conceptually combined.

In all real systems, some deviation from ideal behavior can be observed. If a potential is applied to a macroscopic system, the total current is the sum of a large number of microscopic current filaments, which originate and end at the electrodes. If the electrode surfaces are rough or if one or more of the dielectric materials in the system is inhomogeneous, many of these microscopic current filaments would be different. In a response to a small-amplitude excitation signal this would lead to frequency-dependent effects, which can often be modeled with simple distributed circuit elements. For example, many capacitors in EIS experiments, most prominently the double-layer capacitor often do not behave ideally due to the distribution of currents and electroactive species. Instead, these capacitors often act like a constant phase element (CPE), an element that has found widespread use in impedance data modeling. 

A Randles circuit is one of the simplest and most conunon cell models used for many aqueous, conductive, and ionic solutions [4]. It includes only solution resistance a parallel combination of a double-layer capacitor... 

The following analysis will be shown for a realistic system equivalent circuit model (Figure 6-1), further simplified by replacing CPE with as shown in Figure 2-6A. The resistance of the material dominates the lower cutoff frequency/j. At lower frequencies the double-layer capacitance and other interfacial processes will cause the impedance to decrease with increasing frequency. This will continue until the impedance from the double-layer capacitor becomes lower than the impedance representing the bulk-material resistance RguLK/ which occurs at the frequency ... 

The simplest test-cell that can be envisaged consists of two blocking electrodes in contact with the polymer whose conductance is to be measured. The three-component equivalent circuit is shown in Fig. 1.4(a). The polymer acts as a resistor, R, which is in series with the double layer capacitor, Cdi, at the interface, and in parallel with the geometric capacitance, Cg, as discussed in Section 1.4.2. This simple network is dealt with by evaluating the parallel term first to give an impedance Zp, which is then added to the double-layer impedance. 

To evaluate the magnitude of capacitive currents in an electrochemical experiment, one can consider the equivalent circuit of an electrochemical cell. As illustrated in Figure 24, in a simple description this is composed by a capacitor of capacitance C, representing the electrode/solution double layer, placed in series with a resistance R, representing the solution resistance. 

Figure 3 The simplest equivalent circuit of an electrochemical cell. Cd = capacitor emulating the double layer Rn + R c = solution resistance...

A more realistic picture of the double-layer has an RC element (that is, a capacitor and resistor in parallel) itself in series with a second resistor Rs (see Figure 8.11(d)). This circuit yields a similar Nyquist plot to that of an RC element... 

To a first approximation, the BLM can be considered to behave like a parallel plate capacitor immersed in a conducting electrolyte solution. In reality, even such a thin insulator as the modified BLM (designated by and R, in Fig. 108) could block the specific adsorption of some species from solution and/or modify the electrochemical behavior of the system. Similarly, System C may turn out to be a semiconductor(l)-insulator-semiconductor(2) (SIS ) rather than a semiconductor(l)-semiconductor(2) (SS ) junction. The obtained data, however, did not allow for an unambiguous distinction between these two alternative junctions we have chosen the simpler of the two [652], The equivalent circuit describing the working (Ew), the reference (Eg), and the counter (Ec) electrodes the resistance (Rm) and the capacitance (C of the BLM the resistance (R ) and capacitance (Ch) of the Helmholtz electrical double layer surrounding the BLM as well as the resistance of the electrolyte solution (RSO ) is shown in Fig. 108a [652],... 

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... 

The second meaning of the word circuit is related to electrochemical impedance spectroscopy. A key point in this spectroscopy is the fact that any -> electrochemical cell can be represented by an equivalent electrical circuit that consists of electronic (resistances, capacitances, and inductances) and mathematical components. The equivalent circuit is a model that more or less correctly reflects the reality of the cell examined. At minimum, the equivalent circuit should contain a capacitor of - capacity Ca representing the -> double layer, the - impedance of the faradaic process Zf, and the uncompensated - resistance Ru (see -> IRU potential drop). The electronic components in the equivalent circuit can be arranged in series (series circuit) and parallel (parallel circuit). An equivalent circuit representing an electrochemical - half-cell or an -> electrode and an uncomplicated electrode process (-> Randles circuit) is shown below. Ic and If in the figure are the -> capacitive current and the -+ faradaic current, respectively. 

Double-layer capacity (Capacitance) — The excess charge stored on both sides of the double layer depends on the -> electrode potential, therefore the double layer can be represented by a capacitor in equivalent circuits. In general, this capacitor in nonlinear, i.e., the stored charge is not proportional to the potential. Hence two different capacities can be defined ... 

Immittance — In alternating current (AC) measurements, the term immittance denotes the electric -> impedance and/or the electric admittance of any network of passive and active elements such as the resistors, capacitors, inductors, constant phase elements, transistors, etc. In electrochemical impedance spectroscopy, which utilizes equivalent electrical circuits to simulate the frequency dependence of a given elec-trodic process or electrical double-layer charging, the immittance analysis is applied. 

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