Variational approach to the electrostatic free energy in charged colloidal suspensions: general theory for open systems

Abstract

A general variational technique for calculating the electrostatic free energy of a system of interacting double layers is presented. A stationarity principle for a large class of electrical potential- and position-dependent ion distribution functions and boundary conditions is derived and shown to be unique. The functional satisfying this principle for an open system is identified with the corresponding grand canonical excess surface free energy Ω. The special case of a functional quadratic in the potential ψ(i.e. the Debye–Hückel approximation for the ion distributions together with generalized linear boundary conditions) is shown to permit reduction of the free energy calculation to the evaluation of surface integrals. Illustrations of the approach are presented in the Debye–Hückel approximation for two interacting planar double layers and in the non-linear Poisson–Boltzmann approximation for an isolated planar interface under both constant surface charge density and constant surface potential conditions.

The variational approach is straightforward to implement for a variety of double-layer geometries and surface charge conditions. Moreover, it directly and unambiguously identifies the appropriate electrostatic free energy. We believe the variational technique to be powerful and versatile method for calculating interactions between colloidal particles.