Double layer, capacitance/capacitor models

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). 

The relationship between the fractional surface coverage 9 and the double-layer capacitance C may be better understood in terms of the following model. The doublelayer capacitance at the electrode in the presence of adsorption can be viewed as consisting of two capacitors connected in parallel. One capacitor corresponds to the electrode areas that are unoccupied (free) and the other to the electrode areas that are occupied (covered) with adsorbate (13-15). These two condenser-capacitors have different dielectrics and thus different capacitances. The capacitance of a parallel combination of capacitors is equal to the sum of the individual capacitances. 

An ideally polarizable electrode behaves as an ideal capacitor because there is no charge transfer across the solution/electrode boundary. In this case, the equivalent electrical model consists of the solution resistance, R, in series with the double-layer capacitance, Cdi. An analysis of such a circuit was presented in Section I.2(i). 

It is often found that the double-layer capacitance or a coating capacitance does not behave like an ideal capacitor, experimentally manifested in the complex plane plot by a depressed semicircle whose center lies below the real axis. This behavior is usually attributed to some distribution (or dispersion) in some physical property of the system (e.g., the porous surface of the metal or the varying thickness or composition of a coating) and is modeled by the use of a constant phase element (CPE) [30]. 

CPE is used in a model in place of a capacitor to compensate for non-homogeneity in the system. A rough or porous surface can cause double-layer capacitance to appear as CPE and Warburg element [116, 117]. 

EIS data are analysed by fitting them to an equivalent electrical circuit model consisting of resistors, capacitors, and inductors. The working electrode interface undergoing an electrochemical reaction is analogous to an electronic circuit and can be characterised as an electrochemical system in terms of equivalent circuit. Typical circuits are shown in Figs. 1.10, 1.11, 1.12 and 1.13 where is admittance (ohm-cm ),Cf is double-layer capacitance and a is the exponents [114]. (/ .E Reference Electrode and W Working Electrode)... 

The first two terms on the right-hand side of Eq. (21) represent the static part of the charging current, reflecting the change in the double-layer capacitance before and after the desorption, whereas the third term represents the dynamic part or pseudocapacitance part involving d0/dE. In the Erumkin s two-parallel-plate capacitor model, j c is then given simply by... 

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 ... 

Here Cdi is the experimentally measmed double-layer capacitance. Equation (8.13) has the usual form for two capacitors coimected in series, in agreement with the model postulated by Stern, in which the two parts of the double layer are consecutive in space. 

This chapter is devoted to the behavior of double layers and inclusion-free membranes. Section II treats two simple models, the elastic dimer and the elastic capacitor. They help to demonstrate the origin of electroelastic instabilities. Section III considers electrochemical interfaces. We discuss theoretical predictions of negative capacitance and how they may be related to reality. For this purpose we introduce three sorts of electrical control and show that this anomaly is most likely to arise in models which assume that the charge density on the electrode is uniform and can be controlled. This real applications only the total charge or the applied voltage can be fixed. We then show that predictions of C < 0 under a-control may indicate that in reality the symmetry breaks. Such interfaces undergo a transition to a nonuniform state the initial uniformity assumption is erroneous. Most... 

Solution-phase DPV of Au144-C6S dispersed in 10 mM [bis(triphenylpho-sphoranylidene)-ammoniumtetrakis-(pentafluorophenyl)-borate (BTPPATPFB)/ toluene] [acetonitrile] 2 1 revealed well-behaved, equally spaced and symmetric quantized double-layer charging peaks with AE - 0.270 0.010 V. Applying the classical concentric spheres capacitor model (8) reveals an individual cluster capacitance of 0.6 aF [334, 335]. 

Thus, according to this model, the interphase consists of two equal and opposite layers of charges, one on the metal ( m) the other in solution (q ). This pair of charged layers, called the double layer, is equivalent to a parallel-plate capacitor (Fig. 4.5). The variation of potential in the double layer with distance from the electrode is linear (Fig. 4.4). A parallel-plate condenser has capacitance per unit area given by the equation... 

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. 

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