Constant surface charge density model

FIGURE 9.4 Reduced potential energy V (h) — Kl64nkT)V h) as a function of scaled plate separation Kh for the constant surface charge density model calculated with Eq. (9.125) for a = 0, 0.1, and 1 in comparison with V h) = (K/64nkT)V h) for the constant surface potential model calculated with Eq. (9.141). 

Equation (9.136) (or Eq. (9.137)) is a transcendental equation for which can be solved numerically. By substituting the obtained value into Eq. (9.86), which holds irrespective of the type of double-layer interaction (constant surface potential or constant surface charge density models), we can calculate the interaction force P h). 

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. 

Comparison is made with the results for the two conventional models for hard plates given by Honig and Mul [11]. We see that the values of the interaction energy calculated on the basis of the Donnan potential regulation model lie between those calculated from the conventional interaction models (i.e., the constant surface potential model and the constant surface charge density model) and are close to the results obtained the linear superposition approximation. 

At constant surface potential, the force provided by the new model is smaller than predicted by the Gouy-Chapman theory. In contrast, at constant surface charge density, the dielectric constant and the thickness of region I have effects opposite to those at constant surface potential. 

One of the most fruitful applications of the PB equation is in the description of long cylindrical polyelectrolytes, of which DNA is the prototypical example. The same simplifications used in the planar case with a uniform and constant surface charge density and a restricted primitive model of the electrolyte are assumed here. 

Consider a plane metal electrode situated at z = 0, with the metal occupying the half-space z < 0, the solution the region z > 0. In a simple model the excess surface charge density a in the metal is balanced by a space charge density p(z) in the solution, which takes the form p(z) = Aexp(—kz), where k depends on the properties of the solution. Determine the constant A from the charge balance condition. Calculate the interfacial capacity assuming that k is independent of a. 

Alternatively, in the literature, the constant capacitance model and the Stern model were used to describe the dependence of the surface charge density on the surface potential. In the constant capacitance model, the surface charge is defined as ... 

The constant capacitance model was originally introduced by Helmholtz for the surface charging of mercury. In this section the model is used in eombination with the 1-pK model (reaction (5.23) or (5.25)). In the constant capacitance model the surface potential (used in expressions for the equilibrium eonstants, cf. Eq. (5.24)) is proportional to the surface charge density... 

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