Equilibria for Interacting Macroions

Until we discovered the constancy of the surface potential from the uniaxial stress results, like most other people, I had been more interested in constant surface charge models. If you do not know how the valency of a macroion varies with the external conditions, it is reasonable to assume it to be constant unless given evidence to the contrary. Given the evidence that y/0 70 mV is roughly constant for the n-butylammonium vermiculite system, what other consequences follow from this In particular, what happens if we apply the coulombic attraction theory with the constant surface potential boundary condition  

FIGURE 4.1 Schematic illustration of the phenomenon (a) a swollen n-butylammonium vermiculite gel and (b) the components present in the two phases, the gel (I) and supernatant fluid (II). The dotted line represents an effective membrane enclosing the plate macroions Pn. The symbols M+, X , and S stand for univalent counterions (n-butylammonium ions), univalent co-ions (chloride ions) and solvent (water) molecules, respectively. 

As elsewhere, we will confine our attention to a uni-univalent electrolyte solution for the sake of simplicity. The inverse Debye screening length k is then given by 

We are going to make some brutal approximations to bring out the leading features of the behavior. The first one, which we shall reconsider later in the discussion, is 

As discussed previously, this approximation is certainly valid in dilute (n 1019 cm 3) electrolyte solutions, and we use it for all salt concentrations as a first approximation. It is also straightforward to correct for the term containing a, as we shall see later in the chapter. 

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