Parallel-capacitor model

The above relationships were derived for low electrode coverages by the adsorbed substance, where a linear adsorption isotherm could be used. Higher electrode coverages are connected with a marked change in the surface charge. The two-parallel capacitor model proposed by Frumkin and described by the equation... 

The parallel-capacitor model suggested by Frumkin [8] is an attempt to turn this into a quantitative argument we discuss a simplified ver-... 

Using a parallel capacitor model, Peled and Straze calculated the apparent thickness of the SEI for a series of active metal electrodes, including lithium, calcium, and magnesium, with the equation ... 

The adsorption of some neutral organic compounds on the individual faces of zinc single-crystal electrodes was described by the Frumkin-Damaskin theory with the use of two parallel capacitors model [14]. For pc-Zn electrode, a departure from the two parallel capacitors model due to the energetic inhomogeneity of the surface was observed. 

As stated in Sect. 6.4.1, in the theoretical treatment of the electrochemical responses of surface-bound molecules, it has been assumed that the measured experimental currents and converted charges when a potential Ep is applied can be considered as the sum of a pure faradaic contribution, given by Eqs. (6.130) and (6.131), and a non-faradaic one, Qp nf and fpnt (given by Eqs. (6.150) and (6.157)). The correction of this non-faradaic component of the response can be done simply when subtractive electrochemical techniques are used). We assume the parallel capacitors model introduced by Damaskin [78], for which CAnf can be written as... 

For cyclohexanol adsorption on the three faces of lowest indices of zinc (in 100 mM KCl + 0.1 mM H2SO4), C(E) curves are given in Fig. 50. Obviously, only one of the adsorption-desorption peaks is observable in the dl region. These results were shown to fit the two-parallel-capacitors model (dashed lines). For a given concentration of adsorbate, the potentials of the peak and of the maximum of adsorption shift in the same order as the pzc s in base electrolyte. From the complete analysis of the curves, all adsorption parameters were found to be co dependent.f... 

Evidence for two-dimensional condensation at the water-Hg interface is reviewed by de Levie [135]. Adsorption may also be studied via differential capacity data where the interface is modeled as parallel capacitors, one for the Hg-solvent interface and another for the Hg-adsorbate interface [136, 137]. 

Here not only does the resistive portion of the capacitor model cause problems, but if the PCB is laid out asymmetrically between paralleled capacitors, the trace inductance causes unbalanced heating within the capacitors, thus shortening the life of the hottest capacitor. 

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 electrified interface is generally referred to as the electric double layer (EDL). This name originates from the simple parallel plate capacitor model of the interface attributed to Helmholtz.1,9 In this model, the charge on the surface of the electrode is balanced by a plane of charge (in the form of nonspecifically adsorbed ions) equal in magnitude, but opposite in sign, in the solution. These ions have only a coulombic interaction with the electrode surface, and the plane they form is called the outer Helmholtz plane (OHP). Helmholtz s model assumes a linear variation of potential from the electrode to the OHP. The bulk solution begins immediately beyond the OHP and is constant in potential (see Fig. 1). 

FIG. 11.2 The variation of electrochemical potential in the vicinity of the interface between two phases, a and / (a) according to a schematic profile and (b) according to the parallel plate capacitor model. 

The Stern layer resembles the parallel plate capacitor model for the double layer. Therefore Equation (13) may be applied to this region ... 

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. 

We have failed to discuss so far the numerical value of the capacitance of the compact layer and its dependence on potential (or charge), both of which are in disagreement with the simple parallel-plate capacitor model proposed originally by Helmholtz. These issues, and the important effect of the solvent in the interphase, are discussed in Section 16.5. 

In the case of adsorption of more different adsorbable solutes, the capacitive behavior of the Hg-water interface can be described by the model of parallel capacitors (Jehring, 1974). For two substances, it follows... 

The first model of this approach is the two parallel condenser model (TPC), which visualizes the adsorbed layer as two capacitors connected in parallel. Only water (solvent) molecules are present between the plates of one of these capacitors and only solute molecules between the plates of the other. The model has been progressively generahzed and extended to take account of the reorientation of the solute and solvent molecules, the heterogeneity of the surface of solid electrodes as well as variations in the adsorbed layer thickness upon adsorption. " ... 

Constant capacitance model (CCM) was proposed in 1972 by Schindler and Stumm (Schindler, R. W. et at, 1976 Stumm, W. et at, 1980) mostly for the surface of oxides. It is based on the very first model of the dual electric layer developed by Helmholtz. Its core concept is an assumption that only inner-sphere ion complexes form, which are positioned as an individual layer at some distance from the surface, and the diffusion layer is absent. It is believed that Na+, K+, Cb and NO ", as well as inert, do not form bond with the surface and affect only the ion force of the solution. For this reason the model is viewed as two parallel capacitor plates surface of the mineral with charge a, on the one hand, and adsorbed H+, OH and other ions (Figure 2.18, A) with charge + a. on the other. At that, the electric potential value on the surface of the mineral is equal to... 

In electrical or dielechic measurements, the material to be characterized is usually placed between two conducting electrodes, where an electric field can be created within it by application of a voltage. Ihe extent to which a material responds to an applied electric field can be discussed using a parallel-plate capacitor model (see Figure 1). This section is intended to provide a brief introduction to tiie general dielectric properties of materials and more and detailed information can be found in [1-4]) for example. 

Complex conductivity a is used when the material is considered as a conductor with capacitive properties. Complex conductivity is according to the basic capacitor model shown in Figure 3.1, where capacitance and conductance physically are in parallel ... 

In Figure 3.9(a), the case becomes quite different and much more complicated (see Section 12.2.3). The total immittance, and also the voltage at the interface between the two dielectrics, will be determined by the resistors at low frequencies, but by the capacitors at high frequencies. The analysis of Figure 3.9(a) will be very different dependent on whether a series (impedance) or parallel (admittance) model is used. 

By inspection of the equations for the two-capacitor models found in Section 12.2, we find the parallel version best adapted. According to Eq. 12.24, it can be characterized by one single time constant. The equivalent circuit is shown on Eigure 9.19. 

Figure 11.4 shows the OLED model used in this Chapter. The model consists of a series resistance and a diode parallel with a capacitor. The capacitor models the total capacitance of the layers, the series resistance models the total resistance of the device and the diode models the rectifying nature of the OLED, the model is based on the... 

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