Layer Capacitance

FIGURE 4.15. Four possible cases, A, B, C, and D, for a charge transfer resistance and double-layer capacitance connected to a transmission line. The movements of + and X show charging the double-layer capacitance at high frequencies or short time periods. 

At high frequencies the double-layer capacitance shunts both resistances, while at low frequencies the transmission line dominates. The behavior of Eqn. 49 is further discussed in the following paragraphs. 

We now consider the behavior of the combined circuit in Cases A and B. The impedance of the Randles element is written 

Three cases are identified. In Case 1 the transmission line is dominant p is greater than unity, and at all frequencies terms from either Eqn. 51 or 52 are larger than those from Eqn. 50. In Case 2 there is a high-frequency loop in addition to the transmission line. Most of the loop is given by the Randles semicircle in Eqn. 50, but we have found that for the high-frequency part of the loop, the impedance is a mixture, with the real part given by the transmission line and the imaginary part by the double-layer capacitance 

The subscript M stands for mixed. This interesting result arises because at high frequencies 

---

Figure Bl.28.8. Equivalent circuit for a tliree-electrode electrochemical cell. WE, CE and RE represent the working, counter and reference electrodes is the solution resistance, the uncompensated resistance, R the charge-transfer resistance, R the resistance of the reference electrode, the double-layer capacitance and the parasitic loss to tire ground.

Electrically, the electrical double layer may be viewed as a capacitor with the charges separated by a distance of the order of molecular dimensions. The measured capacitance ranges from about two to several hundred microfarads per square centimeter depending on the stmcture of the double layer, the potential, and the composition of the electrode materials. Figure 4 illustrates the behavior of the capacitance and potential for a mercury electrode where the double layer capacitance is about 16 p.F/cm when cations occupy the OHP and about 38 p.F/cm when anions occupy the IHP. The behavior of other electrode materials is judged to be similar. 

From an electroanalytical point of view, the double-layer capacitance is a nuisance resulting in the charging current, which has no analytical value. 

Active electrochemical techniques are not confined to pulse and linear sweep waveforms, which are considered large ampHtude methods. A-C voltammetry, considered a small ampHtude method because an alternating voltage <10 mV is appHed to actively couple through the double-layer capacitance, can also be used (15). An excellent source of additional information concerning active electroanalytical techniques can be found in References 16—18. Reference 18, although directed toward clinical chemistry and medicine, also contains an excellent review of electroanalytical techniques (see also... 

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. 

Instead of an exact calculation, Gouy and Chapman have assumed that (4) can be approximated by combining the Poisson equation with a Boltzmann factor which contains the mean electrical potential existing in the interface. (This approximation will be rederived below). From this approach the distribution of the potential across the interface can be calculated as the function of a and from (2) we get a differential capacitance Cqc- It has been shown by Grahame that Cqc fits very well the measurements in the case of low ionic concentrations [11]. For higher concentrations another capacitance in series, Q, had to be introduced. It is called the inner layer capacitance and it was first considered by Stern [1,2]. Then the experimental capacitance Cexp is analyzed according to ... 

Figure 1-13 displays the experimental dependence of the double-layer capacitance upon the applied potential and electrolyte concentration. As expected for the parallel-plate model, the capacitance is nearly independent of the potential or concentration over several hundreds of millivolts. Nevertheless, a sharp dip in the capacitance is observed (around —0.5 V i.e., the Ep/C) with dilute solutions, reflecting the contribution of the diffuse layer. Comparison of the double layer witii die parallel-plate capacitor is dius most appropriate at high electrolyte concentrations (i.e., when C CH). 

FIGURE 1-13 Double-layer capacitance of a mercury drop electrode in NaF solutions of different concentrations. (Reproduced with permission from reference 5.)... 

Measurements of the double-layer capacitance provide valuable insights into adsorption and desorption processes, as well as into the structure of film-modified electrodes (6). 

The heterogeneity of a metal surface is responsible for the curvature of the Parsons-Zobel (PZ) plot (1/Cvs. 1/Q, where Cis the experimental capacitance and Qthe diffuse layer capacitance calculated on the basis of... 

For (ideally) polarizable metals with a sufficiently broad double-layer region, such as Hg, Ag, Au, Bi, Sn, Pb, Cd, H, and others, Ea=to can be obtained from measurements of the double-layer capacitance in dilute... 

Measurements based on the Gouy-Chapman-Stem theory to determine the diffuse double-layer capacitance 10, 24,72, 74... 

Frumkin was the first to give a qualitative consideration of the electrochemical properties of pc electrodes.10,20 70 He noted that the charge fixed value of the potential E and this may change the form of the capacitance curve near the diffuse-layer capacitance minimum. Important results were obtained in a pioneering paper by Valette and Hamelin.67 They compared experimental capacitance curves for a pc-Ag electrode and its three basic faces. They found that the capacitance of a pc-Ag electrode can be obtained by the superposition of the corresponding Cj, E curves for individual faces exposed at the pc surface, i.e. 

A weighted sum of C,E curves for the faces was found to be similar to the C,E curve for a pc electrode. According to Valette and Hamelin,67 all main Ag faces [(111), (100), and (110)] are exposed on the surface, their fractions 0j on the surface being 0.31, 0.23, and 0.46, respectively. These authors demonstrated that the diffuse-layer capacitance minimum potential E a of a pc-Ag electrode was only slightly less negative (30 mV) than the pzc of the Ag(110) face, i.e., for the face with the more negative value of EamQ. The diffuse-layer capacitance minimum for pc-Ag was wider and less deep than for the Ag faces. 

Mathematical simulation of C, E curves shows that the shape of the diffuse-layer capacitance minimum depends on the difference of Eamo in individual faces and their fractions, as well as on the shape of partial Cj, E curves (Fig. 9). 

The results of experimental capacitance studies at two plane model pc-Bi electrodes were in agreement with these conclusions.2 266 Thus it has been shown that the potential of the diffuse-layer capacitance minimum for a pc electrode does not correspond to the zero charge potential of the whole surface, i.e., Zfipj Oat E n-... 

The idea in these papers67,223,224 was to identify the potential of the capacitance minimum in dilute electrolyte solutions with the actual value of Ea=o (i.e., <7ge0m( min) = Ofor the whole surface) and to obtain the value of R as the inverse slope of the Parsons-Zobel plot at min.72 Extrapolation of Cwom vs- to Cgg0m = 0 provides the inner-layer capacitance in the / C geom, and not C ea as assumed in several papers.67,68,223,224 In the absence of ion-specific adsorption and for ideally smooth surfaces, these plots are expected to be linear with unit slope. However, data for Hg and single-crystal face electrodes have shown that the test is somewhat more complicated.63,74,219,247-249 More specifically,247,248 PZ plots for Hg/... 

A new approach to the double-layer capacitance of rough electrodes has been given by Daikhin et al.m m The concept of a Debye length-dependent roughness factor [i.e., a roughness function R LD) that deter-... 

The inner-layer capacitance of Cd faces increases as the atomic density decreases. It has been suggested that hydrophilicity increases in the order Cd(0001) < Cd(10T0) < Cd(llZO). The same order has been proposed on the basis of data on organic compound adsorption.153... 

Figure 24. Linear dependence of the reciprocal of the inner-layer capacitance...

Daikhin, double layer capacitance of solid at rough electrodes, 52 of the double layer, of non-aqueous solutions, 61... 

---