Also Double layer interaction constant surface charge

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. 

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 solvent also acts as a dielectric medium, which determines the field diji/dx and the energy of Interaction between charges. Now, the dielectric constant e depends on the inherent properties of the molecules (mainly their permanent dipole moment and polarizability) and on the structure of the solvent as a whole. Water is unique in this sense. It is highly associated in the liquid phase and so has a dielectric constant of 78 (at 25 C), which is much higher than that expected from the properties of the individual molecules. When it is adsorbed on the surface of an electrode, inside the compact double layer, the structure of bulk water is destroyed and the molecules are essentially immobilized... 

It must also be recognized that, because of the usually much finer subdivision of natural ion exchangers in soils as compared with commercial synthetic resins, surface phenomena may play a more prominent role. Among these effects are water is more structured at the interface than in the bulk liquid ions and water molecules are less mobile at the interface because of stronger interactions the dielectric constant of water is lower than in the bulk solution and surface charges produce an electric double layer (Horst, 1990). 

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