Free energy of electrical double layers: Entropy of adsorbed ions and the binding polynomial

We adapt the method of binding polynomials to general problems of binding equilibria of ions to polybases, polyacids, and mixed polyelectrolytes, such as proteins and other colloids. For spherical particles with a smeared charge the interaction effects are taken into account using the Poisson-Boltzmann equation, which is shown to differ little from the Debye-Hückel approximation under conditions met in most protein solutions. Examples are given of the salt dependence of pH titration equilibria. Binding polynomials produce an extra term in the free energy of the electrical double layer, which arises from the entropy of the adsorbed ions. The maximum term method applied to the binding polynomial yields an expression which is similar to that derived by the charging process of Chan and Mitchell. Applications to monolayers and to polyelectrolyte gels are also discussed.