Also Double layer interaction density

Equation (9.197) as combined with Eq. (9.201) is the required expression for the potential energy of the double-layer interaction per unit area between two parallel similar plates with constant surface charge density cr. From the nature of this linearization, the obtained potential distribution (Eq. (9.187)) is only accurate near the plate surface and thus the interaction energy expression (9.197) is also only accurate for small plate separations h. Indeed, as /z —> 0, Eq. (9.197) gives the following correct limiting expression [15, 16] ... 

In Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

The electrical double layer has been dealt with in countless papers and in a number of reviews, including those published in previous volumes of the Modem Aspects of Electrochemistry series/ The experimental double layer data have been reported and commented on in several important works in which various theories of the structure of the double layer have been postulated. Nevertheless, many double layer-related problems have not been solved yet, mainly because certain important parameters describing the interface cannot be measured. This applies to the electric permittivity, dipole moments, surface density, and other physical quantities that are influenced by the electric field at the interface. It is also often difficult to separate the electrostatic and specific interactions of the solvent and the adsorbate with the electrode. To acquire necessary knowledge about the metal/solution interface, different metals, solvents, and adsorbates have been studied. 

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. 

The effect of electrolyte concentration on the transition from common to Newton black films and the stability of both types of films are explained using a model in which the interaction energy for films with planar interfaces is obtained by adding to the classical DLVO forces the hydration force. The theory takes into account the reassociation of the charges of the interface with the counterions as the electrolyte concentration increases and their replacements by ion pairs. This affects both the double layer repulsion, because the charge on the interface is decreased, and the hydration repulsion, because the ion pair density is increased by increasing the ionic strength. The theory also accounts for the thermal fluctuations of the two interfaces. Each of the two interfaces is considered as formed of small planar surfaces with a Boltzmannian distribution of the interdistances across the liquid film. The area of the small planar surfaces is calculated on the basis of a harmonic approximation of the interaction potential. It is shown that the fluctuations decrease the stability of both kinds of black films. 

Even with these useful results from statistical mechanics, it is difficult to specify straightforward criteria delineating when the Poisson-Boltzmann or linear Poisson-Boltzmann equations can be expected to yield quantitatively accurate results for particle-wall interactions. As we have seen, such criteria vary greatly with different types of boundary conditions, what type of electrolyte is present, the electrolyte concentration and the surface-to-surface gap and double layer dimensions. However, most of the evidence supports the notion that the nonlinear Poisson-Boltzmann equation is accurate for surface potentials less than 100 mV and salt concentrations less than 0.1 M, as stated in the Introduction. Of course, such a statement might not hold when, for example, the surface-to-surface separation is only a few ion diameters. We have also seen that the linear Poisson-Boltzmann equation can yield results virtually identical with the nonlinear equation, particularly for constant potential boundary conditions and with surface potentials less than about 50 mV. Even for constant surface charge density conditions the linear equation can be useful, particularly when Ka < 1 or Kh > 1, or when the particle and wall surfaces have comparable charge densities with opposite signs. 

We also mentioned earlier the oscillatory nature of the density as a function of metal-surface distance. If the electron density is high the oscillation decreases by the electron-electron interactions. We will revert to this later. Because of the electron spillover at z > 0, negative charges will not be balanced by the positive background. As a result a surface-dipole layer develops. The potential due to this double layer can be calculated with Poisson s equation ... 

The electrostatic stability of a colloidal system depends not only on the magnitude of the electrical surface charge density but also on the dielectric properties of the medium, on its ionic strength, on the valence of the ions in the double layer, on the size of the particles, and on the temperature of the system (only slightly). The total interaction potential between two spherical particles charged by a single type of ions at the surface can be determined using the DLVO equation ... 

Electrostatic interactions in air are very different compared to the electrostatic interactions in aqueous systems, as there is no ion solubility and no double layer is formed. The source of the surface charges is also different. In air, there are no possibilities for acid-base interactions or for ion disassociation. Charging in air is caused by static electricity and is stable for non-conducting particles. This makes the interaction coulombic in nature, with a range comparable to the radius of the particle. The electrostatic interactions easily dominate when the particle size is large, the density is low, and there is a low water content (low conductivity). Typical examples could be during the drying of milk powder or the transport of coffee powder. 

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