Faradaic current, double-layer capacitance

The contribution of faradaic and double-layer capacitances in the response of DG-structured V2O5 can be estimated by comparing the CV curves of devices that use RTIL electrolyte with and without lithium salt, shown in Fig. 5.14b. Since V2O5 does not react with either of the ions of pure RTIL, the lithium-free experiment tests the EDLC response. With lithium salt added, the significant increase in capacitive current and the appearance of peak pairs indicates that redox reactions are taking place. These faradaic processes are kinetically facile and thus considered pseudocapacitive, but phase transitions may occur. Although it is difficult to distinguish between redox and intercalation pseudocapacitance, the latter is likely to be present in DG bicontinuous materials. 

Even in the absence of Faradaic current, ie, in the case of an ideally polarizable electrode, changing the potential of the electrode causes a transient current to flow, charging the double layer. The metal may have an excess charge near its surface to balance the charge of the specifically adsorbed ions. These two planes of charge separated by a small distance are analogous to a capacitor. Thus the electrode is analogous to a double-layer capacitance in parallel with a kinetic resistance. 

The Faradaic and capacitive components of the current both increase with the scan rate. The latter increases faster (proportionally to v) than the former (proportionally to y/v), making the extraction of the Faradaic component from the total current less and less precise as the scan rate increases, particularly if the concentration of the molecules under investigation is small. The variations of the capacitive and Faradaic responses are illustrated in Figure 1.7 with typical values of the various parameters. The analysis above assumed implicitly that the double-layer capacitance is independent of the electrode potential. In fact, this is not strictly true. It may, however, be regarded as a good approximation in most cases, especially when care is taken to limit the overall potential variation to values on the order of half-a-volt.10 13... 

The preceding derivation has assumed implicitly that the double-layer charging current is negligible in front of the Faradaic current or that it can be eliminated by a simple subtraction procedure. In cases where these conditions are not fulfilled, the following treatment will take care of the problem under the assumption that the double-layer capacitance is not affected appreciably by the Faradaic reaction but may nevertheless vary in the potential range explored. The first step of the treatment then consists of extracting the Faradaic component from the total current according to (see Section 1.3)... 

The net effect of the presence of the solution resistance on potential excitation methods is that the potential seen by the electrode solution interface is different from the potential applied by the potentiostat. This difference is current-dependent and the current is itself potential-dependent. The resistance also makes it more difficult to separate current components arising from the double-layer capacitance from the faradaic process. Similar complications arise for current excitations. 

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. 

Residual currents, also referred to as background currents, are the sum of faradaic and nonfaradaic currents that arise from the solvent/electrolyte blank. Faradaic processes from impurities may be practically eliminated by the careful experimentalist, but the nonfaradaic currents associated with charging of the electrode double layer (Chap. 2) are inherent to the nature of a potential sweep experiment. Equation 23.5 describes the relationship between this charging current icc, the double-layer capacitance Cdl, the electrode area A, and the scan rate v ... 

Equation 23.6 gives the approximate relationship between the charging current and the peak faradaic current, ip (the target quantity), for a test compound of concentration C° (mM) assuming normal double-layer capacitances [1],... 

Sample C02 Evolved (mmol/g) CO Evolved (mmol/g) Double-Layer Capacitance (F/g) Faradaic Current (mA/g)... 

Here C is the specific differential double layer capacitance. The two terms on the left side of Eq. (4) describe the capacitive and faradaic current densities at a position r at the electrode electrolyte interface. The sum of these two terms is equal to the current density due to all fluxes of charged species that flow into the double layer from the electrolyte side, z ei,z (r, z = WE), where z is the direction perpendicular to the electrode, and z = WE is at the working electrode, more precisely, at the transition from the charged double layer region to the electroneutral electrolyte. 4i,z is composed of diffusion and migration fluxes, which, in the Nernst-Planck approximation, are given by... 

Fig. 10.4. Equivalent circuit for the electrode with hydrodynamic modulation R. denotes the solution resistance, C0/. the double layer capacitance, RF the faradaic resistance and Z the impedance contributed by diffusional transport of the reactants to the surface. Hydrodynamic modulation is represented by the modulated current, JD and the resultant current in the external circuit represented by I,.

The double-layer capacitance is another useful electrochemical method for probing the integrity of molecular assemblies in contact with electrolyte solutions. If a cyclic voltammogram is recorded in a solution of electrolyte only, a non-Faradaic current results, and the double-layer capacitance, Cji (or differential capacitance), is measured as half the width of the voltammogram charging envelope given by [30, 48]... 

The double-layer capacitance is taken into account by assuming a simplified Helmholtz parallel plate model (1). On opening the circuit, the potential difference, V, across the double layer must be reduced by diminution of the charge on each plate. For a cathodic reaction, each electron being transferred from the metal to the solution side of the interface effects an elementary act of reaction and reduces the charge, q, on each plate. Consequently the rate of reduction of this charge is equal to the faradaic current, and Eq. (55) follows, y is assumed to differ from rj simply by the value of the reversible potential ... 

The Faradaic impedance is linked in parallel to the double-layer capacitance Cdi and then to the solution resistance Re as illustrated in Figure 9.1. The double-layer capacitance is in general considered to be constant. It is frequently observed, however, that Cdi changes with the dc current at which the impedance measurements were carried out. TTie capacitance Cdi has been observed, for example, to increase with increasing current density. 

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