Soft Surface with Dissociable Groups

Next we consider the case where the hxed charges in the surface charge layer are due to dissociation of dissociable groups AH in a solution containing ions of 

At thermodynamic equilibrium, p (x) must be equal to the chemical potential p of H in the bulk solution phase, which is 

The total change in the free energy density in the surface charge layer at position X and the surrounding solution is thus given by 

The free energy density of the electrical double layer /(x) =/t(x) —/ (x) is thus given by 

Verwey and J. Th. G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. 

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Consider a cylindrical soft particle, that is, an infinitely long cylindrical hard particle of core radius a covered with an ion-penetrable layer of polyelectrolytes of thickness d in a symmetrical electrolyte solution of valence z and bulk concentration (number density) n. The polymer-coated particle has thus an inner radius a and an outer radius b = a + d. The origin of the cylindrical coordinate system (r, z, cp) is held fixed on the cylinder axis. We consider the case where dissociated groups of valence Z are distributed with a uniform density N in the polyelectrolyte layer so that the density of the fixed charges in the surface layer is given by pgx = ZeN. We assume that the potential i/ (r) satisfies the following cylindrical Poisson-Boltz-mann equations ... 

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