Low Potential Case

For the low potential case, simple analytic expressions for the force and potential energy of the double-layer interaction between two plates can be derived. In this case Eq. (9.26) for the interaction force P h) per unit area between the plates at separation h reduces to 

DOUBLE-LAYER INTERACTION BETWEEN TWO PARALLEL SIMILAR PLATES 

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Equation (62) describes the variation in potential with distance from the surface for a diffuse double layer without the simplifying assumption of low potentials. It is obviously far less easy to gain a feeling for this relationship than for the low-potential case. Anticipation of this fact is why so much attention was devoted to the Debye-Hiickel approximation in the first place. Note that Equation (62) may be written... 

The nonlinear equations (4.76) and (4.77) have not been solved analytically except in the low potential case (i.e., the low fixed-charge density case). In this case, the solution to Eqs. (4.76) and (4.77) is... 

In this chapter, we discuss two models for the electrostatic interaction between two parallel dissimilar hard plates, that is, the constant surface charge density model and the surface potential model. We start with the low potential case and then we treat with the case of arbitrary potential. 

Note that for the low potential case, the Donnan potential regulation model agrees exactly with the LSA results. 

The interaction force P(h) driving the two membranes apart per unit area in the low potential case is given by Eq. (8.25), namely,... 

In the limit a O, the particle core vanishes and the particle becomes a cylindrical polyelectrolyte (a porous charged cylinder) of radius b. For the low potential case, Eq.(21.79) gives... 

As seen in equation 23 for the low-potential case, the concentrations of excess ions in the diffuse layer are related to the surface potential, the bulk solution concentrations of the individual ions, and the overall composition of the bulk solution phase. Equation 23 may be rewritten as... 

Cio parameter describing relation between [Pg.707]

In the literature, a large number of various formulae for the electrostatic interaction energies can be found, derived under various approximations. Ohshima (33) has derived an equation for the interaction energy between two spherical particles in the low-potential case. However, the resulting equation is rather complicated. Efforts have been made to derive simple but yet accurate analytical equations for the interaction energy. In this respect, substantial progress has been made by Camie and co-workers (36, 37). For spherical particles at an arbitrary surface potential, relatively simple equations have been derived. The approximations in the Pois-son-Boltzmann equation are, of course, still included. 

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