Parallel-plate capacitor model

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 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. 

The negative sign on the left-hand side is for the conformity with the convention of the faradaic current described earlier. One of the simplest models for q is the Frumkin s two-parallel-plate capacitor model [99], which has been successfully employed to describe the adsorption of organic molecules on electrodes [100],... 

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... 

If we consider Fig. 2 again, with a chemisorptive attraction between the cations and the electrode added, the distance d may be shortened slightly, but this is not necessary. The most significant change is that the parallel-plate-capacitor model of the inner layer is no longer valid, because q > QA(j) /d. In aqueous systems, one envisions specific adsorption in terms of ions breaching a hydration sheath to move closer to the electrode surface, but this does not seem relevant to fused and solid-electrolyte systems. In these, it seems necessary to define specific adsorption by an excess of q over... 

The situation of Fig. 20.39 is characterized by the fact that the measurable interfacial capacitance C is defined by the parallel-plate capacitor model ... 

A simple model of the e.d.l. was first suggested by Helmholz in which the charges at the interface were regarded as the two plates constituting a parallel plate capacitor, e.g. a plate of metal with excess electrons (the inner Helmholz plane I.H.P.) and a plate of excess positively charged ions (the outer Helmholz plane O.H.P.) in the solution adjacent to the metal the... 

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). 

Whatever the most acceptable model may be and as we need only a rough estimate of the amount of ions discharged, we start from the Helmholtz model of a simple parallel-plate capacitor, whose potential difference is... 

The electric field or ionic term corresponds to an ideal parallel-plate capacitor, with potential drop g (ion) = qMd/4ire. Itincludes a contribution from the polarizability of the electrolyte, since the dielectric constant is included in the expression. The distance d between the layers of charge is often taken to be from the outer Helmholtz plane (distance of closest approach of ions in solution to the metal in the absence of specific adsorption) to the position of the image charge in the metal a model for the metal is required to define this position properly. The capacitance per unit area of the ideal capacitor is a constant, e/Aird, often written as Klon. The contribution to 1/C is 1 /Klon this term is much less important in the sum (larger capacitance) than the other two contributions.2... 

The growth of an anodic alumina film, at a constant current, is characterized by a virtually linear increase of the electrode potential with time, exemplified by Fig. 10, with a more or less notable curvature (or an intercept of the extrapolated straight line) at the beginning of anodization.73 This reflects the constant rate of increase of the film thickness. Indeed, a linear relationship was found experimentally between the potential and the inverse capacitance78 (the latter reflecting the thickness in a model of a parallel-plate capacitor under the assumption of a constant dielectric permittivity). This is foreseen by applying Eq. (38) to Eq. (35). It is a consequence of the need for a constant electric field on the film in order to transport constant ionic current, as required by Eqs. (39)-(43). 

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... 

This model is better than a parallel-plate capacitor for simulating curves such as in Fig. 3.4fo, but only close to Ez in reality, far from Ez, Cd is, to a first approximation, independent of potential. We remember the approximation that ions are considered as point charges and that, consequently, there is no maximum concentration of ions close to the electrode surface ... 

Since the unloaded QCM is an electromechanical transducer, it can be described by the Butterworth-Van Dyke (BVD) equivalent electrical circuit represented in Fig. 12.3 (box) which is formed by a series RLC circuit in parallel with a static capacitance C0. The electrical equivalence to the mechanical model (mass, elastic response and friction losses of the quartz crystal) are represented by the inductance L, the capacitance C and the resistance, R connected in series. The static capacitance in parallel with the series motional RLC arm represents the electrical capacitance of the parallel plate capacitor formed by both metal electrodes that sandwich the thin quartz crystal plus the stray capacitance due to the connectors. However, it is not related with the piezoelectric effect but it influences the QCM resonant frequency. 

The constant-capacitance model assumes that the double layer on the solid-liquid phase boundary can be regarded as a parallel-plate capacitor (Fig. 14b). 

The first attempt to explain the capacitive nature of the interphase is credited to Helmholtz, in the middle of the nineteenth century. In his model, the interphase is viewed as a parallel-plate capacitor - a layer of ions on its solution side and a corresponding excess of charge on the surface of the metal. It should be noted here that electroneutrality must be maintained in the bulk of all phases, but not at the... 

We now turn to the potential dependence of electrosorption of neutral molecules, considering first the model developed by Frumkin. This is a phenomenological model, which depends on considerations of the changes in the electrostatic energy of the interphase caused by adsorption. Assuming that measurements are taken in concentrated solutions of a supporting electrolyte, we can neglect diffuse-double-layer effects and focus our attention on the Helmholtz part of the double layer, considered as a parallel-plate capacitor. In the pure solvent the... 

Figure 10. Electrical double layer models. Top right (a) typical type of potential vs. composition plot for a charged surface compared to (b) constant capacitance model. Top left Two double-layer models, (a) diffuse double layer, (b) part parallel plate capacitor and part diffuse layer.. Bottom left Stem layer model. Incorporation of adsorbed ions to surface. From Hiemenz and Rajagopalan (1997) Bottom right Comparison of Gouy-Chapman and Stem-Grahame models of the electrical double layer. From Davis and Kent (1990).

The development of microscopic models of the double layer began over 100 years ago with work of Helmholtz [20]. He assumed that the charge on the polarizable metal electrode is exactly compensated by a layer of ionic charge in solution located at a constant distance from the geometrical electrode solution interface. The separation distance was assumed to have molecular dimensions. This simple model which gave rise to the term double layer is the equivalent of a parallel-plate capacitor with a capacitance given by... 

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