Generalized Poisson-Fermi formalism for investigating size correlation effects with multiple ions

We establish a generalized Poisson-Fermi formalism to compute the electrostatic potential next to charged surfaces in the presence of multiple ion species with different sizes. A generalized Fermi-like ion distribution is deduced from the excess free energy, after expansion of the functional entropy of free space in which the ions have all the same size. The ion distribution is expressed in terms of the bulk volume fractions of each ion species rather than their bulk concentrations so as to account for the excluded volumes. We present size correlations effects such as underscreening and ion stratification, which have not been investigated before with such a simple theory. The change of dielectric properties across the space, arising from the finite spatial occupancy of ions, can be solved self-consistently through the Bruggeman model. The generalized Poisson-Fermi formalism is anticipated to be useful for interpreting electrophoretic mobility measurements and for computing the electrostatic potential over the surface of biomolecules in ionic solutions.