Integral capacitance of diffusion layer for rectangular structures

Electrical-double-layer capacitors are one of the important devices and systems used in energy storage. In the heart of the electrical-double-layer capacitors is the accumulation of ions/charges on the surfaces of “active” materials. Porous materials, such as carbon nanotubes and activated carbon, have been widely used in the electrical-double-layer capacitors due to large surface area. However, there are limited analytical solutions available for the calculation of the capacitance of porous materials. In this work, we use spectral method to solve linearized Poisson-Boltzmann equation under small electric potential for three different geometrical configurations and obtain analytical expressions of integral capacitances for slab-like structure, rectangular pore and 3D-rectangular box. The numerical results reveal that slit-like structures, such as multilayer graphene, are preferable for the energy storage in the electrical-double-layer capacitors. We also obtain analytical expressions of the integral capacitances for porous materials with parallel, cylindrical and rectangular-like pores of the same cross-sectional area, respectively. The results obtained in this work can also be applied to the design of electrodes of mesosizes and microsizes used in supercapacitors.