Asymptotic behavior of the solutions of the poisson-boltzmann equation and its physical consequences

Abstract

Expressions are derived describing the asymptotic behavior of the solutions of the Poisson-Boltzmann equation (PBE) with increase of the surface potential or surface charge. In contrast to the solutions of the linearized PBE, the exact solutions remain finite everywhere in the aqueous medium at infinite surface charge density. This peculiarity gives rise to some notable physical effects: (a) the electrostatic disjoining pressure P el in a liquid film of fixed width cannot exceed a finite limiting value at any surface charge density; (b) at constant pressure, the width of a liquid film tends to a finite value at infinite increase of its surface charge; (c) the ζ-potential of an interface is limited from above and becomes insensitive to the value of the surface charge if the latter is sufficiently high. Numerical estimates show that the specific asymptotic behavior of the exact solutions of PBE manifests itself at surface charge densities which are readily accessible in experimental measurements. A comparison with the data available about foam films and lyotropic lamellar lipid phases provides evidence that P el and the width of a liquid film may become indistinguishably close to their asymptotic values. Thus, the asymptotic effects may be used also as a criterion for the extent of validity of the standard Gouy-Chapman theory.