Fixed double layer, model

The Gouy-Chapman theory did not prove entirely satisfactory, and in 1924 a considerable advance was made by the Germ an-American physicist Otto Stern, whose model is shown in Figure 11.18c. Stern combined the fixed double-layer model of Helmholtz with the diffuse double-layer model of Gouy and Chapman, As shown in the figure, there is a fixed layer at the surface, as well as a diffuse layer. On the whole this treatment has proved to be satisfactory, but for certain kinds of investigations it has been found necessary to develop more elaborate models. 

The simplest model of the structure of the metal-solution interphase is the Helmholtz compact double-layer model (1879). According to this model, all the excess charge on the solution side of the interphase, qs. is lined up in the same plane at a fixed distance away from the electrode, the Helmholtz plane (Fig. 4.4). This fixed distance xH is determined by the hydration sphere of the ions. It is defined as the plane of the centers of the hydrated ions. All excess charge on the metal, qM, is located at the metal surface. 

The double layer model of Helmholtz assumes fixed layer of charges on the electrode and the outer Helmholtz plane. This model has been modified by Guoy-Chapman analysis, which assumes that the ions of charge opposite of the charge on the electrode distribute themselves in a diffuse manner as shown in Figure 1.15. 

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. 

Stern combines in his model (1924) the ideas of a diffuse layer and a fixed double layer in the following way ... 

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... 

As we have seen, the electric state of a surface depends on the spatial distribution of free (electronic or ionic) charges in its neighborhood. The distribution is usually idealized as an electric double layer one layer is envisaged as a fixed charge or surface charge attached to the particle or solid surface while the other is distributed more or less diffusively in the liquid in contact (Gouy-Chapman diffuse model, Fig. 3.2). A balance between electrostatic and thermal forces is attained. 

FIG. 6 Comparison between MGC theory (open symbols) and the disk model (filled symbols) for the dependence of Vcx on the electrical double layer thickness [Eq. (14)]. The number of unit layers in a quasicrystal is a fixed parameter for each set of curves [34],... 

Stem s Theory of the Double Layer.—The variations of capacity of the double layer with the conditions, the influence of electrolytes on the zeta-potential, and other considerations led Stern to propose a model for the double layer which combines the essential characteristics of the Helmholtz and the Gouy theories. According to Stern the double layer consists of two parts one, which is approximately of a molecular diame r in thickness, Is supposed to remain fixed to the surface, while ihe other is anlttfuse layer extending for me distance into tlie solution The fall of potential in the fixed layer is sharp while that in the diffuse layer is gradual, the decrease being exponential in nature, as required by equation (5). 

Relatively complete elaborations for the cylinder model have been given by for instance, Anderson and Koh and Levine et al. K In these two theories the solution Is assumed to contain (1-1) electrolytes with =u. Both theories fail to account for conduction behind the slip plane, and both solve the electrokinetic equations, taking double layer overlap into account. Anderson and Koh assume this overlap to take place at fixed surface charge (which, because of the implicit rigid particle model of the cylinder wall, comes down to fixed tr =cT ), whereas Levine et al. do so for constant surface potential (essentially fixed Anderson and Koh also considered capUlaries of other... 

The potential distribution of Fig. 1 is typical of the compact-diffuse layer models (42-44). The potential varies almost linearly within the compact double layer, decaying exponentially within the diffuse layer. The thickness of the latter depends on the electrolyte ionic strength and becomes negligible in strong electrolytic solutions (42). This feature becomes important in electrocatalytic studies, since it is the potential difference (0 — < e) that can be measured or fixed experimentally versus a reference electrode, while reacting ions and molecules experience a potential difference ((j) —... 

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