The Exact Mean Field Theory Solution for Plate Macroions

Before setting out on the exact mean field theory solution to the one-dimensional colloid problem, I wish to emphasize that the existence of the reversible phase transition in the n-butylammonium vermiculite system provides decisive evidence in favor of our model. The calculations presented in this chapter are deeply rooted in their agreement with the experimental facts on the best-studied system of plate macroions, the n-butylammonium vermiculite system [3], We now proceed to construct the exact mean field theory solution to the problem in terms of adiabatic pah-potentials of both the Helmholtz and Gibbs free energies. It is the one-dimensional nature of the problem that renders the exact solution possible. 

FIGURE 6.1 The geometry of the problem. Regions R, RJ and RJ are assumed to be in thermodynamic equilibrium. 

As a basic postulate, we assume that thermal equilibrium is achieved throughout all regions of the solution and, accordingly, the number density of the small ions with valency z- (z+ = 1), which are described as point particles, is determined by the common Boltzmann distribution 

In the mean field description, the potential 0(x) is assumed to obey the PB equation 

The geometrical symmetry of the plate configuration requires the derivative of the potential to vanish at the midpoint x = x,° of the region R i.e., 

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The Exact Mean Field Theory Solution for Plate Macroions... 

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