Double parallel-plate model

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). 

Fig. 6.62. The Helmholtz-Perrin parallel-plate model, (a) A layer of ions on the OHP constitutes the entire excess charge in the solution. (b) The electrical equivalent of such a double layer is a parallel-plate condenser, (c) The corresponding variation of potential is a linear one. (Note The solvation sheaths of the ions and electrode are not shown in this diagram nor in subsequent ones.)...

Going a step further, what does the parallel-plate model of the double layer have to say regarding the capacity of the interface Rearranging Eq. (6.119) in the form of the definition of differential capacity [Eq. (6.97)],... 

Parallel-plate model (for adsorption isotherm), 332 Parallel-plate model (of the double layer), 188 Partially blocked electrode, 450 Partial surface coverage, 131 Passivation, 506... 

The double-layer capacitance is taken into account by assuming a simplified Helmholtz parallel plate model (1). On opening the circuit, the potential difference, V, across the double layer must be reduced by diminution of the charge on each plate. For a cathodic reaction, each electron being transferred from the metal to the solution side of the interface effects an elementary act of reaction and reduces the charge, q, on each plate. Consequently the rate of reduction of this charge is equal to the faradaic current, and Eq. (55) follows, y is assumed to differ from rj simply by the value of the reversible potential ... 

A hint may be found by observing the extreme negative end of the plots in Figures 8.2 and 8.3. Over about 1V, from - 0.8 to - 1.8 V vs. SCE, the double-layer capacitance in this region is almost independent of concentration and only slightly dependent on potential, as expected from the simple parallel-plate model. The numerical value is offby more than an order of magnitude, but this can be remedied by a better choice of the values of e and d in Eq. (8.3), as seen below. 

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). 

A simple parallel plate condenser model (Fig. 5-12) gives the electric capacity Ch of the compact double layer as shown in Eqn. 5-8 ... 

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... 

It appears that an electrified interface does not behave like a simple double layer. The parallel-plate condenser model is too naive an approach. Evidently some crucial secrets about electrified interfaces are contained in those asymmetric electrocapillaiy curves and the differential capacities that vary with potential. One has to think again. 

FIG. 11.4 Two models for the double layer (a) a diffuse double layer and (b) charge neutralization due partly to a parallel plate charge distribution and partly to a diffuse layer. 

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. 

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). 

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).

In Eq. (15), the electrostatic potential, iJ/, is for the overlapping electric double layer of the interacting particles. Numerous models have been created to predict the overlapping field electrostatic potential between parallel plates. However, calculation of the EDL interaction for the common geometry of two spheres has not been satisfactorily resolved, due mainly to the nonlinear partial differential terms in Eq. (13) arising because of the three-dimensional geometry of the system. As a consequence, a number of approximate and numerical models have been developed for the calculation of the EDL interaction between two spheres. These models are briefly described below. 

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... 

The earliest theoretical studies of the behavior of an electrified interface were made by Helmholtz (1879). He discussed the adsorption of ions at a fixed double layer and he believed that this double layer formed the equivalent of a parallel-plate condenser. But this double layer model is an inadequate description of particles in electrolyte-containing systems. 

Analytical models of double layer structures originated roughly a century ago, based on the theoretical work of Helmholtz, Gouy, Chapman, and Stem. In Figure 26, these idealized double-layer models are compared. The Helmholtz model (Fig. 26a) treats the interfacial region as equivalent to a parallel-plate capacitor, with one plate containing the... 

Because we have the model of parallel charged layers of different potential we have a direct analogy with a parallel plate condenser and it is therefore perhaps no surprise that a capacitance exists across the double layer. 

Capacitive effects. The presence of a protective layer on the surface of a metallic cluster decreases the capacity of the cluster. For a planar metal electrode, the electrical double layer comprised of the charge on the surface of the electrode and the ions of opposite charge in the solution (or the solvent dipoles) can be modeled as a parallel plate capacitor with capacity Cpp in Farads (F) given by... 

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