Double layer, diffuse charge distribution

This is not immediately applicable to electrophoresis, however. The solid particle with its fixed double layer (net charge Q) is moving relative to a solution in which the diffuse double layer is distributed (see electrical double layer). The latter is equivalent to a charge -Q spread out on a concentric sphere of radius where this is the thickness of the ionic atmosphere. The presence of this atmosphere reduces the mobility, and the potential at the surface of the particle, by the factor 1/(1+ Kr) so that in place of equation (E.16) the zeta-potential (see electrokinetic effects) is given by... 

So far, we have used the Maxwell equations of electrostatics to determine the distribution of ions in solution around an isolated, charged, flat surface. This distribution must be the equilibrium one. Hence, when a second snrface, also similarly charged, is brought close, the two surfaces will see each other as soon as their diffuse double-layers overlap. The ion densities aronnd each surface will then be altered from their equilibrinm valne and this will lead to an increase in energy and a repulsive force between the snrfaces. This situation is illustrated schematically in Fignre 6.12 for non-interacting and interacting flat snrfaces. 

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. 

Equation (48) is identical in form to Equation (13) for a parallel plate capacitor, with pQ replacing A p and k 1 replacing 8. This result shows that a diffuse double layer at low potentials behaves like a parallel plate capacitor in which the separation between the plates is given by k . This explains why k 1 is called the double-layer thickness. It is important to remember, however, that the actual distribution of counterions in the vicinity of a charged wall is diffuse and approaches the unperturbed bulk value only at large distances from the surface. 

The electric field which actually affects the charge transfer kinetics is that between the electrode and the plane of closest approach of the solvated electroactive species ( outer Helmholtz plane ), as shown in Fig. 2.2. While the potential drop across this region generally corresponds to the major component of the polarization voltage, a further potential fall occurs in the diffuse double layer which extends from the outer Hemlholtz plane into the bulk of the solution. In addition, when ions are specifically absorbed at the electrode surface (Fig. 2.2c), the potential distribution in the inner part of the double layer is no longer a simple function of the polarization voltage. Under these circumstances, serious deviations from Tafel-like behaviour are common. 

The diffuse double layer is generated by the anions trapped in the potential well. The surface charge density , due to their distribution between 0 and dE, is given by ... 

Electrokinetic effects — A number of effects caused by the asymmetric distribution of charged particles in the electrochemical - double layer and subsequent charge separation during relative motion of liquid and solid phase. They can occur when the diffuse double layer is thicker than the hydrodynamic boundary layer. 

The chemical system at the surface will be quite complex. There will be strongly bound species at various sites on the metal surface. Where these are ionic, the electric field established at the surface will tend to attract ions of opposite charge from the solution. The first layer has been termed the electrode double layer, while the gegenion distribution in the solution is called the diffuse double layer. A theoretical analysis of the double layer has been made by Gouy and Chapman and adapted to kinetic analysis by Stern. For references, and discussion see paper by D. C. Grahame, J. Chem, Phys, 21, 1054 (1953). 

Figure 9.19. The diffuse double layer, (a) Diffuseness results from thermal motion in solution, (b) Schematic representation of ion binding on an oxide surface on the basis of the surface complexation model, s is the specific surface area (m kg ). Braces refer to concentrations in mol kg . (c) The electric surface potential, falls off (simplified model) with distance from the surface. The decrease with distance is exponential when l/ < 25 mV. At a distance k the potential has dropped by a factor of 1/c. This distance can be used as a measure of the extension (thickness) of l e double layer (see equation 40c). At the plane of shear (moving particle) a zeta potential can be established with the help of electrophoretic mobility measurements, (d) Variation of charge distribution (concentration of positive and negative ions) with distance from the surface (Z is the charge of the ion), (e) The net excess charge.

The preceding method (Section 6.3.1) relies completely on the limited equilibration of the ion distribution in solution, due to the hindered diffusion. Additionally, the charging time of the double layer can be exploited for the local machining of surfaces. As mentioned in Section 6.2.2, the time constant for the double-layer charging is given by the product of solution resistance and double-layer capacity. In the above experiments employing the tip of an STM, which is a few nms distance to the surface, as local counter electrode this time constant is well below nanoseconds, even for diluted electrolytes. Nonetheless, for electrode separations in the... 

Adamson (51) proposed a model for W/0 microemulsion formation in terms of a balance between Laplace pressure associated with the interfacial tension at the oil/water interface and the Donnan Osmotic pressure due to the total higher ionic concentration in the interior of aqueous droplets in oil phase. The microemulsion phase can exist in equilibrium with an essentially non-colloidal aqueous second phase provided there is an added electrolyte distributed between droplet s aqueous interior and the external aqueous medium. Both aqueous media contain some alcohol and the total ionic concentration inside the aqueous droplet exceeds that in the external aqueous phase. This model was further modified (52) for W/0 microemulsions to allow for the diffuse double layer in the interior of aqueous droplets. Levine and Robinson (52) proposed a relation governing the equilibrium of the droplet for 1-1 electrolyte, which was based on a balance between the surface tension of the film at the boundary in its charged state and the Maxwell electrostatic stress associated with the electric field in the internal diffuse double layer. 

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 ... 

Figure 3.24, Diffuse double-layer model of cation and anion distribution near a permanent-charge clay surface. (Adapted from H. van Olphen, 1977. An Introduction to Clay Colloid Chemistry 2nd ed. New York Wiley.)...

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