Transient finite element analysis of electric double layer using Nernst-Planck-Poisson equations with a modified stern layer

Abstract

Finite element analysis of electric double layer capacitors using a transient nonlinear Nernst-Planck-Poisson (NPP) model and Nernst-Planck-Poisson-modified Stern layer (NPPMS) model are presented in 1D and 2D. The NPP model provided unrealistic ion concentrations for high electrode surface potential. The NPPMS model uses a modified Stern layer to account for finite ion size, resulting in realistic ion concentrations even at high surface potential. The finite element solution algorithm uses the Newton-Raphson method to solve the nonlinear problem and the alpha family approximation for time integration to solve the NPP and NPPMS models for transient cases. Cubic Hermite elements are used for interfacing the modified Stern and diffuse layers in 1D while serendipity elements are used for the same in 2D. Effects of the surface potential and bulk molarity on the electric potential and ion concentrations are studied. The ability of the models to predict energy storage capacity is investigated and the predicted solutions from the 1D NPP and NPPMS models are compared for various cases. It is observed that NPPMS model provided realistic and correct results for low and high values of surface potential. Furthermore, the 1D NPPMS model is extended into 2D. The pore structure on the electrode surface, the electrode surface area and its geometry are important factors in determining the performance of the electric double layer capacitor. Thus 2D models containing a porous electrode are modeled and analyzed for understanding of the behavior of the electric double layer capacitor. The effect of pore radius and pore depth on the predicted electric potential, ion concentrations, surface charge density, surface energy density, and charging time are discussed using the 2D Nernst-Planck-Poissonmodified Stern layer (NPPMS) model.