Capacitance, potential dependence

The capacitance-potential dependences of Cd(OOOl) in dilute solutions of Cl04, N02, and NOs" were also studied [6]. A weak specific adsorption of anions increasing in the order Cl04 < N02 < N03 was observed. The adsorption of halides on the Cd(OOOl) single crystal electrode was studied [7], and was found to increase in the sequence Cl < Br < 1 [8]. Analysis of the impedance data does not point to the specific adsorption of Cl ions, and shows that the surface excess (T) of halide ions changes with potential and increases from Br to 1 (Fig. 1) [7]... 

Hence, we suggest a simple, serial equivalent circuit analog that describes the impedance behavior of carbon electrodes as seen in Figure 12. It contains a Voight-type analog in series with R-C, which reflects the charge transfer, a potential-dependent Warburg -type element (solid state diffusion of Li-ions), and, finally, a capacitive potential-dependent element that reflects the accumulation of lithium. This relatively simple model has already been discussed in depth [105-107]. 

Due to the small amplitude of the superimposed voltage or current, the current-voltage relationship is linear and thus even charge-transfer reactions, which normally give rise to an exponential current-potential dependence (Chapter 4), appear as resistances, usually coupled with a capacitance. Thus any real ohmic resistance associated with the electrode will appear as a single point, while a charge transfer reaction (e.g. taking place at the tpb) will appear ideally as a semicircle, i.e. a combination of a resistor and capacitor connected in parallel (Fig. 5.29). 

Following the concepts of H. Helmholtz (1853), the EDL has a rigid structnre, and all excess charges on the solntion side are packed against the interface. Thus, the EDL is likened to a capacitor with plates separated by a distance 5, which is that of the closest approach of an ion s center to the surface. The EDL capacitance depends on 5 and on the value of the dielectric constant s for the medium between the plates. Adopting a value of 5 of 10 to 20 nm and a value of s = 4.5 (the water molecules in the layer between the plates are oriented, and the value of e is much lower than that in the bulk solution), we obtain C = 20 to 40 jjE/cm, which corresponds to the values observed. However, this model has a defect, in that the values of capacitance calculated depend neither on concentration nor on potential, which is at variance with experience (the model disregards thermal motion of the ions). 

FIGURE 10.2 Potential dependence of differential capacitance calculated from Gouy-Chapman theory for z+ = = 1 and various concentrations (1) 10 , (2) 10 , (3) 10 M. 

Figure 4.8 Potential-dependent reaction energies for water dissociation to form OH, O, and H over Pt(l 11). (a) Energy curves based on the full charge model the nonlinearity of these plots expresses the capacitance of the interface, (b) Differences of the curves indicate the reaction energies. The nonlinear terms cancel almost completely. The dashed lines indicate predictions made from the linear model, whereas the solid lines are predictions made from a fuU solvation/ charge-based model [Rossmeisl et al., 2006].

Hiratsuka et al102 used water-soluble tetrasulfonated Co and Ni phthalocyanines (M-TSP) as homogeneous catalysts for C02 reduction to formic acid at an amalgamated platinum electrode. The current-potential and capacitance-potential curves showed that the reduction potential of C02 was reduced by ca. 0.2 to 0.4 V at 1 mA/cm2 in Clark-Lubs buffer solutions in the presence of catalysts compared to catalyst-free solutions. The authors suggested that a two-step mechanism for C02 reduction in which a C02-M-TSP complex was formed at ca. —0.8 V versus SCE, the first reduction wave of M-TSP, and then the reduction of C02-M-TSP took place at ca. -1.2 V versus SCE, the second reduction wave. Recently, metal phthalocyanines deposited on carbon electrodes have been used127 for electroreduction of C02 in aqueous solutions. The catalytic activity of the catalysts depended on the central metal ions and the relative order Co2+ > Ni2+ Fe2+ = Cu2+ > Cr3+, Sn2+ was obtained. On electrolysis at a potential between -1.2 and -1.4V (versus SCE), formic acid was the product with a current efficiency of ca. 60% in solutions of pH greater than 5, while at lower pH... 

Figure 3 also contains an example of an ISER-flrel plot for a simple specifically adsorbed species, bromide on silver (solid curve). This plot was extracted from bromide coverage-potential data, obtained from differential capacitance measurements, along with the corresponding potential-dependent intensity of the SERS bromide-surface stretching mode at ca. 160 cm"1 (19.). In this case, the maximum (i.e. unity) value of 0r>1 corresponds to a close-packed bromide monolayer, ca. 1.4 x 10"9 mol cm 2. Again, the ISER-0t 1... 

The Gouy-Chapman theory was tested experimentally on the basis of the doublelayer capacity measurements. This theory predicts a parabolic capacitance-potential relationship and a square-root dependence on concentration at constant e and T ... 

The general method for the determination of the flat band potential is based on the Mott-Schottky linear plot based on ca-pacitance/voltage relation. Starting from Eq. (9) the space charge distribution was calculated, and its potential dependence lead to the derivation of a model equivalent to a capacitance, given by ... 

The Gouy-Chapman Model Provides a Potential Dependence of the Capacitance, but at What Cost ... 

The interphase between an electrolyte solution and an electrode has become known as the electrical double layer. It was recognized early that the interphase behaves like a capacitor in its ability to store charge. Helmholtz therefore proposed a simple electrostatic model of the interphase based on charge separation across a constant distance as illustrated in Figure 2.12. This parallel-plate capacitor model survives principally in the use of the term double layer to describe a situation that is quite obviously far more complex. Helmholtz was unable to account for the experimentally observed potential dependence and ionic strength dependence of the capacitance. For an ideal capacitor, Q = CV, and the capacitance C is not a function of V. 

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. 

Noting these uncertainties, we have evaluated the equivalent circuit and present the results in Fig. 3. The potential dependence of the three capacitive elements is shown in Figs. 

We have extended the technique of Relaxation Spectrum Analysis to cover the seven orders of magnitude of the experimentally available frequency range. This frequency range is required for a complete description of the equivalent circuit for our CdSe-polysulfide electrolyte cells. The fastest relaxing capacitive element is due to the fully ionized donor states. On the basis of their potential dependence exhibited in the cell data and their indicated absence in the preliminary measurements of the Au Schottky barriers on CdSe single crystals, the slower relaxing capacitive elements are tentatively associated with charge accumulation at the solid-liquid interface. 

The metal-water interaction has also been suggested [373] to play a role in determining the extent and strength of hydrogen adsorption. The metal-water interaction is potential dependent in particular it decreases as the potential is made more cathodic. Thus, pseudo-capacitances were observed at higher overpotential than 0.3 V because of the appearance of a considerable amount of adsorbed hydrogen. 

Occasionally, the impedance spectra of diamond electrodes are well described by the Randles equivalent circuit with a frequency-independent capacitance (in the 1 to 105 Hz range) [66], Shown in Fig. 11 is the potential dependence of the reciprocal of capacitance squared, a well-known Mott-Schottky plot. Physically, the plot reflects the potential dependence of the space charge region thickness in a semiconductor [6], The intercept on the potential axis is the flat-band potential E whereas the slope of the line gives the uncompensated acceptor concentration NA - Nd in what follows, we shall for brevity denote it as Na ... 

Nevertheless, despite the fact that the theory [84] looks convincing, we think it doubtful that the electrode surface roughness alone, without any additional conditions imposed, should not only cause the frequency dispersion of the capacitance, but also provide for its characteristic potential dependence, described by the Schottky theory. Moreover, a well-substantiated conclusion was drawn recently that the roughness of electrode surface does not necessarily cause the emergence of a CPE in the electrode s impedance [88],... 

---