Gouy-Chapman diffusion-double-layer theory

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. 

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 ... 

The Gouy-Chapman theory provides a better approximation of reality than does the Helmholtz theory, but it still has limited quantitative application. It assumes that ions behave as point charges, which they cannot, and it assumes that there is no physical limit for the ions in their approach to the TPB, which is not true. Stem, therefore, modified the Gouy-Chapman diffuse double layer. His theory states that ions do have finite size, so they cannot approach the TPB closer than a few nm [54, 60], The first ions of the Gouy-Chapman diffuse double layer are in the gas phase but not at the TPB. They are at some distance 8 away from the zirconia-metal-gas interface. This distance will usually be taken as the radius of the ion. As a result, the potential and concentration of the diffuse part of the layer are low enough to justify treating the ions as point charges. Stem also assumed that it is possible that some of the ions are specifically adsorbed by the TPB in the plane 8, and this layer has become known as the Stem layer. Therefore, the potential will drop by T o - Pg over the molecular condenser (i.e., the Helmholtz plane) and by T g over the diffuse layer. Pg has become known as the zeta (Q potential. 

This model is based on the Gouy-Chapman theory (diffuse double-layer theory). The theory states that in the area of the boundary layer between solid and aqueous phase, independently of the surface charge, increased concentrations of cations and anions within a diffuse layer exists because of electrostatic forces. In contrast to the constant-capacitance model, the electrical potential does not change up to a certain distance from the phase boundaries and is not immediately declining in a linear manner (Fig. 14 a). Diffusion counteracts these forces, leading to dilution with increasing distance from the boundary. This relation can be described physically by the Poisson-Boltzmann equation. 

The Diffuse-Double-Layer Theory of Gouy and Chapman... 

It is evident now why the Helmholtz and Gouy-Chapman models were retained. While each alone fails completely when compared with experiment, a simple combination of the two yields good agreement. There is room for improvement and refinement of the theory, but we shall not deal with that here. The model of Stem brings theory and experiment close enough for us to believe that it does describe the real situation at the interface. Moreover, the work of Grahame shows that the diffuse-double-layer theory, used in the proper context (i.e., assuming that the two capacitors are effectively connected in series), yields consistent results and can be considered to be correct, within the limits of the approximations used to derive it. 

Measurements based on the Gouy-Chapman-Stem theory to determine the diffuse double-layer capacitance 10, 24,72, 74... 

To evaluate the contribution of the SHG active oriented cation complexes to the ISE potential, the SHG responses were analyzed on the basis of a space-charge model [30,31]. This model, which was proposed to explain the permselectivity behavior of electrically neutral ionophore-based liquid membranes, assumes that a space charge region exists at the membrane boundary the primary function of lipophilic ionophores is to solubilize cations in the boundary region of the membrane, whereas hydrophilic counteranions are excluded from the membrane phase. Theoretical treatments of this model reported so far were essentially based on the assumption of a double-diffuse layer at the organic-aqueous solution interface and used a description of the diffuse double layer based on the classical Gouy-Chapman theory [31,34]. 

For present purposes, the electrical double-layer is represented in terms of Stem s model (Figure 5.8) wherein the double-layer is divided into two parts separated by a plane (Stem plane) located at a distance of about one hydrated-ion radius from the surface. The potential changes from xj/o (surface) to x/s8 (Stem potential) in the Stem layer and decays to zero in the diffuse double-layer quantitative treatment of the diffuse double-layer follows the Gouy-Chapman theory(16,17 ... 

The Helmholtz model was found not to be able to give a satisfactory analysis of measured data. Later, another theory of the diffuse double layer was proposed by Gouy and Chapman. The interfacial region for a system with charged lipid, R-Na+, with NaCl, is shown in Figure 4.10. 

However, it is interesting to note that the theory of the diffuse double layer was presented independently by Gouy and Chapman (1910) 13 years before the Debye-Hiickel theory of ion ion interactions (1923). The Debye-Hiickel theory was immediately discussed and applied to the diffuse charge around an ion, doubtless owing to the preoccupation of the majority of scientists in the 1920s with bulk properties rather than those at surfaces. 

This same relationship is the starting point of the Debye-Hiickel theory of electrolyte nonideality, except that the Debye-Hiickel theory uses the value of V2 p required for spherical symmetry. It is interesting to note that Gouy (in 1910) and Chapman (in 1913) applied this relationship to the diffuse double layer a decade before the Debye-Hiickel theory appeared. 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

While the linear adsorption isotherms of Figure 4 are illustrative only, they are not inconsistent with reality. The simplest theory of the electrical double layer, the Gouy-Chapman approximation, predicts that if the pH is not far from the isoelectric point, the charge represented by counter ions in the diffuse double layer is related to the surface potential as follows (4, 52, 86) ... 

Diffuse-double-layer model. The DDLM is similar to the CCM but the charge potential is formulated from the Gouy-Chapman theory ... 

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