Integral Representations for Free Energies of Macroionic Suspensions and Equation of State for Osmotic Pressure

TL;DR

This paper develops an integral representation of free energies for macroionic suspensions using a mean field approach, deriving an equation of state for osmotic pressure based on the Poisson-Boltzmann framework.

Contribution

It introduces a generating functional leading to the Poisson-Boltzmann equation and provides new integral representations for free energies and osmotic pressure in macroionic suspensions.

Findings

Derived integral representation of Helmholtz free energy.

Calculated chemical potentials of ions and macroions.

Established an equation of state for osmotic pressure.

Abstract

A generating functional which results in the Poisson-Boltzmann equation and boundary conditions for an average electric potential of a macroionic suspension through an extremal condition is constructed in a mean field theory. The extremum of the generating functional turns out to be identical with the Helmholtz free energy of the system which has an integral representation in terms of the average electric potential satisfying the Poisson-Boltzmann equation. From the Helmholtz free energy, the chemical potentials of small ions and {\it chemical potentials of effective valencies of macroions} are calculated and, as the total sum of them, an integral representation of the Gibbs free energy of the system is derived. Difference of two free energies leads to an equation of state for osmotic pressure of small ion gas in an environment of macroions in the suspension.