The Parallel-Plate Model of Helmholtz

The first attempt to explain the capacitive nature of the interface is credited to Helmholtz (1853). In his model, the interface 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 interface. Here, there can be an excess charge density on the metal, which we denote qM, and an excess charge density, q, on the solution side of the interface. The interface as a whole must be electroneutral. It follows, then, that at any metal/solution interface we can write 

While the Helmholtz model can explain the existence of a capacitance at the interface, it can explain neither its dependence on potential nor its numerical value at any potential. TTie capacitance of a parallel-plate capacitor, per unit surface area, is given by  

If a theory has failed so utterly in describing the experimental results, how could it withstand the test of time, and still merit mentioning a century and a half later 

The Diffltse-Double-Layer Theory of Gouy and Chapman 

A different approach to interpret the behavior of the double-layer capacitance was taken by Gouy (1910) and later by Chapman (1913). The fundamental premise of the diffusC douhle-layer model they proposed is that the ions constituting the charge qs, on 

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What happens when the concentration c0 of ions in solution is very large Equations (6.124) and (6.130) indicate that while CG increases with increasing c0, CH remains constant. Thus, as c0 increases, (1/CG) (1/CH), and for all practical purposes, C CH. That is, in sufficiently concentrated solutions, the capacity of the interface is effectively equal to the capacity of the Helmholtz region, Le., of the parallel-plate model. What this means is that most of the solution charge is squeezed onto the Helmholtz plane, or confined in a region vety near this plane. In other words, little charge is scattered diffusely into the solution in the Gouy-Chapman disarray. 

Figure 8.5b shows the structure of the interface at positive potentials, in the presence of specifically adsorbed anions. Here the physical meaning of the inner Helmholtz plane (IHP) is illustrated. The distance of this layer is always smaller than that of the outer Helmhotz plane OHP, since there is no layer of water molecules between the anions and the surface. The absence of such a layer can qualitatively explain why the value of Cm-z is higher on the positive side of Epzc than on the negative side, taking the parallel-plate model one step closer to the experimental observations. 

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.)...

If we lake median values of e = 7 and d = 0.52 nm, we find from the Helmholtz parallel-plate model C =11.9 uF/cm in reasonable agreement with experiment. A correct calculation will be much more complicated, taking into account, among other things, the possibility that the value of e at the outer Helmholtz plane may already be higher... 

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 ... 

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... 

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 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). 

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. 

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... 

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... 

In 1879, von Helmholtz proposed that all of the counterions are lined up parallel to the charged surface at a distance of about one molecular diameter (Figure 10.5). The electrical potential decreases rapidly to zero within a very short distance from the charged surface in this model. Such a model treated the electrical doublelayer as a parallel-plate condenser, and the calculations of potential decay were based on simple capacitor equations. However, thermal motion leads to the ions being diffused in the vicinity of the surface, and this was not taken into account in the Helmholtz model. 

Thus, the SEI can be modeled as two parallel plate capacitors in series. Because the value of the space charge region capacitor is generally smaller than that of the Helmholtz layer, it dominates the overall capacitance of the SEI. 

Finally, as an illustrative exercise, let us calculate the variation of the Helmholtz free energy of the familiar parallel plate condenser model for two interacting colloidal particles. We choose coordinate x o sl to the surface of the condens6r plates, with x and x therefore lying in the plane of the plates at right angles to each other and normal to Xj. 

The first attempt to describe the electrode-solution interphase in electrostatic terms was made by Helmholtz in 1879. His model, which is shown in Fig. 23 is essentially that of a simple parallel plate capacitor the charge on... 

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