Calculating total electrostatic energies with the nonlinear Poisson-Boltzmann equation

The Poisson-Boltzmann (PB) equation is enjoying a resurgence in popularity and usefulness in biophysics and biochemistry due to numerical advances which allow the equation to be rapidly solved for arbitrary geometries and nonuniform dielectrics. The great simplification of PB models is to use the mean electrostatic potential to give an estimate of the potential of mean force (PMF) governing the distribution of the mobile ions in the solvent. This approximation enables both the mean potential and mean ion distribution to be obtained directly from solutions to the PB equation without performing complex statistical mechanical integrations. The nonlinear form of the PB equation has greater accuracy and range of validity than the linear form, but the approximation of the PMF by the mean potential creates theoretical difficulties in defining the total electrostatic energy for the former. In this paper we use the calculus of variations to provide a unique definition of the total energy and to obtain expressions for the total electrostatic free energy for various forms of the PB equation. These expressions involve energy density integrals over the volume of the system. Various equivalent expressions for the total energy are given and the physical meaning of the different terms that appear is discussed. Numerical calculations are carried out to demonstrate the feasibility of our approach and to assess the magnitude of the various terms that arise in the theory. Both the more familiar charging integral and the energy density integral methods can be applied to the PB equation with equal accuracy, but the latter is much more efficient computationally. The energy density integral involves the integral of the excess osmotic pressure of the ion atmosphere. The various forms of the PB equation which have been most widely discussed to date because of the availability of analytical solutions are shown to be special cases where the osmotic term is absent.