Topological impact of nanopore electrodes on the structure of the electrical double layer and the differential capacitance

Abstract

The electrical double layer for three different topologies of nanopore electrodes is studied, i.e., the interior and exterior electrical double layers of planar, cylindrical and spherical nanopores immersed into a point-ions electrolyte, and not connected to a power source, are analytically attained through the linearized Poisson-Boltzmann equation. Thus, analytical formulas for the mean electrostatic potential, electrolyte's reduced concentration, and electrical field profiles, are exhibited. Their corresponding analytical expressions for their differential capacitances are presented. All the nanopores are treated as permeable, so the electrolyte outside and inside the electrodes are at the same chemical potential. Analogous analytical formulas for solid nano-electrodes are obtained as a corollary of those for nanopores. In particular, their analytical expressions for the differential capacitance here derived are shown to be consistent with the capacitive compactness proposed in the past by one of us. Numerical results of all of the above functions are analyzed as a function of the nanopores geometrical parameters and the electrolyte's temperature and molar concentration. It is found that the spherical topology, at lower temperatures, has the higher differential capacitance. It is demonstrated that for the three nanopore topologies here considered their capacitances reduce to that of a single planar electrode, in the limit of infinitely wide nanopores. The electrical double layer and mean electrostatic potential of the three topologies are in qualitatively agreement with those from the non-linearized Poisson-Boltzmann, hypernetted chain/mean-spherical approximation (HNC/MSA) equations and computer simulations results presented in the past, within the low mean electrostatic potential assumption.