Gouy-Chapman diffuse-charge model

The Ionic Cloud The Gouy-Chapman Diffuse-Charge Model of the Double Layer... 

Gouy—Chapman Diffuse-Charge Model Jec0kT % ze y 2n Smh 2kT 2 nkT J kT V, = e" Pol 0 er s tial x-> It predicts that differential capacities have the shape of inverted parabolas. Ions are considered as pointcharges. Ion-ion interactions are not considered. The dielectric constant is taken as a constant. 

The Gouy Chapman diffuse layer model has been shown to describe adequately the electrostatic potential produced by charges at the surface of the membrane [137]. For a symmetrical background electrolyte, a and i// are related by ... 

The earliest models used to describe the distribution of charges in the edl are, besides the Helmholz model, the Gouy-Chapman diffuse layer model and the Stem-Graham model. Details of these models are given in Westall and Hohl (1980), Schindler (1981, 1984) and Schindler and Stumm (1987). 

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 layer model, Figure 9.19). A balance between electrostatic and thermal forces is attained. 

Gouy and Chapman later suggested that the thermal energy of the ions would result in a diffuse layer on the solution side of the interface with the concentration of excess charges at a maximum close to the electrode surface and gradually decreasing with distance into the electrolyte (on the order of nanometers). However, the Gouy-Chapman diffuse layer model also does not match well with aU experimental data. 

The Stern model (1924) may be regarded as a synthesis of the Helmholz model of a layer of ions in contact with the electrode (Fig. 20.2) and the Gouy-Chapman diffuse model (Fig. 20.10), and it follows that the net charge density on the solution side of the interphase is now given by... 

Some emphasis is given in the first two chapters to show that complex formation equilibria permit to predict quantitatively the extent of adsorption of H+, OH , of metal ions and ligands as a function of pH, solution variables and of surface characteristics. Although the surface chemistry of hydrous oxides is somewhat similar to that of reversible electrodes the charge development and sorption mechanism for oxides and other mineral surfaces are different. Charge development on hydrous oxides often results from coordinative interactions at the oxide surface. The surface coordinative model describes quantitatively how surface charge develops, and permits to incorporate the central features of the Electric Double Layer theory, above all the Gouy-Chapman diffuse double layer model. 

The interfacial capacitance increases with the DDTC concentration added. The relationship among potential difference t/ of diffusion layer, the electric charge density q on the surface of an electrode and the concentration c of a solution according to Gouy, Chapman and Stem model theory is as follows. 

The next step is to determine the electrical charge and potential distribution in this diffuse region. This is done by using relevant electrostatic and statistical mechanical theories. For a charged planar surface, this problem was solved by Gouy (in 1910) and Chapman (in 1913) by solving the Poisson-Boltzmann equation, the so called Gouy-Chapman (G-C) model. 

Thus, the corresponding field or gradient of potential at a distance. v from the electrode according to the diffuse-charge model of Gouy and Chapman is given by the expression ... 

Fig. 1.10 Schematic view of the electrical double layer in agreement with the Gouy-Chapman-Stem-Grahame models. The metallic electrode has a negative net charge and the solvated cations define the inner limit of the diffuse later at the Helmholtz outer plane (OHP). There are anions adsorbed at the electrode which are located at the inner Helmholtz plane (IHP). The presence of such anions is stabilized by the corresponding images at the electrode in such a way that each adsorbed ion establishes the presence of a surface dipole at the interface...

The Poisson-Boltzman (P-B) equation commonly serves as the basis from which electrostatic interactions between suspended clay particles in solution are described ([23], see Sec.II. A. 2). In aqueous environments, both inner and outer-sphere complexes may form, and these complexes along with the intrinsic surface charge density are included in the net particle surface charge density (crp, 4). When clay mineral particles are suspended in water, a diffuse double layer (DDL) of ion charge is structured with an associated volumetric charge density (p ) if av 0. Given that the entire system must remain electrically neutral, ap then must equal — f p dx. In its simplest form, the DDL may be described, with the help of the P-B equation, by the traditional Gouy-Chapman [23-27] model, which describes the inner potential variation as a function of distance from the particle surface [23]. 

Many more-sophisticated models have been put forth to describe electrokinetic phenomena at surfaces. Considerations have included distance of closest approach of counterions, conduction behind the shear plane, specific adsorption of electrolyte ions, variability of permittivity and viscosity in the electrical double layer, discreteness of charge on the surface, surface roughness, surface porosity, and surface-bound water [7], Perhaps the most commonly used model has been the Gouy-Chapman-Stem-Grahame model 8]. This model separates the counterion region into a compact, surface-bound Stern" layer, wherein potential decays linearly, and a diffuse region that obeys the Poisson-Boltzmann relation. 

Fig. 10.14 Schematic diagram of the double layer according to the Gouy-Chapman-Stern-Grahame model. The metal electrode has a net negative charge and solvated monatomic cations define the inner boundary of the diffuse layer at the outer Helmholtz plane (oHp).

A commonly used model for describing counterion distribution at a charged surface is based on the Gouy-Chapman diffuse double-layer (DDL) theory. This model assumes that the surface can be visualized as a structurally featureless plane with evenly distributed charge, while the counterions are considered point charges in a uniform liquid continuum. In this simplified picture, the equilibrium distribution of counterions is described by the Boltzmann equation ... 

In AOT microemulsions, where the aqueous core of the droplets also contains counterions, a considerable part of the dielectric response to the applied fields originates from the redistribution of the counterions. As mentioned in Sec. II, the counterions near th charged surface can be distributed between the Stem layer and the Gouy-Chapman diffuse double layer (28-31). The distribution of counterions is essentially determined by their concentration and the geometry of the water core. Thus, for very large droplets the diffuse double layer peters out and the polarization can be described by the Schwarz model (32). However, as already mentioned, this approach is more relevant to the dielectric behavior of emulsions than to that of micro emulsions. 

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