Free energies of macroionic suspensions and the equation of state for osmotic pressure

Abstract

A generating functional which, together with an extremal condition, results in the Poisson-Boltzmann equation and its boundary conditions for the average electric potential of a macroionic suspension is constructed within mean field theory. We demonstrate that the extremum of the generating functional is identical to the Helmholtz free energy of the system, which has an integral representation in terms of the average electric potential. From the Helmholtz free energy, the chemical potentials of small ions and the chemical potentials of the effective charge numbers of macroions are calculated, and, in the form of their sum, an integral representation of the Gibbs free energy of the system is derived. The difference between the two free energies leads to an equation of state for the osmotic pressure of the small ion gas in the environment of the macroionic suspension.