Overlapping of the Diffuse Double Layer in the Vicinity of the Microporous Electrode
Authors: Kitazumi Yuki , Shirai Osamu , Yamamoto Masahiro , Kano Kenji 
Published: February 2020

Introduction
A microporous electrode with nm-sized micropores on the surface is an attractive subject in electrochemistry. Some experimental results show the physicochemical properties of microporous electrodes are not explained only in terms of the surface area. We have an interest in the electrical double layer in the narrow space. When the size of the micropore on the electrode is smaller than the diffuse part of the electrical double layer, the diffuse double layer show overlap across the pore. This overlapping of the diffuse double layer cannot be estimated on the one dimension model, such as Gouy–Chapman (GC) theory for planar electrodes. We propose a model of the diffuse double layers in the vicinity of micropores on the basis of the Poisson-Boltzmann equation without any complicated assumptions. This calculation shows essential features of the diffuse double layer on the electrode with micropores. The size dependence, the permittivity dependence, the potential dependence, and the electrolyte concentration dependence of the diffuse double layer and also the differential capacitance of the electrode with micropores are discussed.
Model
The model is described with a cylindrical coordinate. The electrode was assumed as a conductor with an equipotential surface. The diameter of a pore on the electrode was varied from 0.2 to 5 nm. To avoid singular points in the calculation, the radius of the corners was assumed to be 50 pm in the geometry of this model. The solution phase contains a 1:1 electrolyte. In the solution phase, the Poisson–Boltzmann equation was applied to obtain the profile of the electric potential. In this calculation, the relative permittivity was set to be from 10 to 78. The nonlinear Poisson–Boltzmann equation was numerically solved by the finite element method. The differential capacitance of the double layer was estimated from the dependence of the total charge on the electrode surface on the surface potential.
Results and discussion
The calculated potential profile showed the overlapping of the diffuse double layer in the micropore. The electric field in the micropore is weaker than that outside of the micropore, that is, the overlapped diffuse double layer provides the mild electric field. Assuming that the thickness of the diffuse double layer is the division of the surface potential by the gradient of the potential at the electrode surface, the degree of the overlapping can express qualitatively. When the size of the micropore on the electrode is smaller than the thickness of the diffuse double layer, the calculated capacitance agrees with the GC capacitance at the planner electrode. On the other hand, when the size of the micropore on the electrode is larger than the thickness of the diffuse double layer, the double layer capacitance converges on the GC capacitance at the microporous electrode. This means that the surface area of micropore becomes effective only at thinner diffuse double layers. The change of the thickness of the diffuse double layer in the micropore causes the strong dependence of the capacitance on the potential and the electrolyte concentration compared with the situation of planar electrode. The characteristic features of microporous electrodes would be originated from the heterogeneous electric field caused by the overlapping of the diffuse double layer.