Membrane Electrostatics—A Statistical Mechanical Approach to the Functional Density Theory of Electric Double Layer

Abstract

We describe physical properties of the electric double layer composed of a charged surface in contact with a solution of counter‐ions and coions representing nanoparticles. The electrostatic free energy of the electric double layer is derived using a statistical mechanical approach. The consistently related expressions for the equilibrium ion and solvent distribution functions and the differential equation for the electric potential are derived by minimization of the elctrostatic free energy of the system. The finite size of nanoparticles constituting the solution is taken into account by means of the excluded volume within the lattice model. Different geometries of the electric double layer are considered. We found that an increased size of charged nanoparticles (ions) reduces the number of counter‐ions near the charged surface, leading to an enhancement of the electrostatic surface potential. The linearized Poisson–Boltzmann theory and the influence of the finite size of ions (nanoparticles) on the thickness of the electric double layer are described. Also the intra‐ionic correlations within charged nanoparticles, which have spatially distributed (quadrupolar or dipolar) electric charge, are considered. The interaction between two charged surfaces in the solution composed of charged quadrupolar (divalent) or dipolar rod‐like nanoparticles is calculated. It is shown that for large enough dimensions of charged quadrupolar (divalent) nanoparticles and for large enough surface charge densities of the charged surfaces, two equally charged surfaces experience attractive force owing to spatially distributed charge within the nanoparticles. Also, it is shown that in the vicinity of the charged surface (wall), large rod‐like quadrupolar or dipolar nanoparticles orient in the electric field within the electric double layer. Close to the charged surfaces the orientation of the rod‐like quadrupolar or the dipolar nanoparticles is hindered because of steric restrictions (hard wall).