Continuum and Equivalent Circuit Modelling of Porous Electrode Charging

Abstract

The flow of electrolytes through pores is important in various fields of science and technology. Examples range from axons in the brain and plasmodesmata in plant cells to energy storage in batteries and supercapacitors. In the porous carbon electrodes of supercapacitors, ions move over micrometers through macropores until they arrive at nanopores, forming so-called electrical double layers. In this presentation, I discuss several models for this multi-scale charging process. First, I present analytical [1] and finite-element solutions [2,3] to the Poisson-Nernst-Planck (PNP) equation for the charging of cylindrical electrolyte-filled pores [Figure (a)]. With these methods, I will delineate the validity of the famous “transmission line” circuit model developed by de Levie in the 1960s [4,5] [Figure (b)]. The TL model assumes equipotential lines in a pore to be straight, which is not the case at a pore’s entrance and end, see Figure (a). As a result, the TL model does not accurately describe short-pore charging. Related, the impedance of short pores deviates from the impedance of the TL circuit (the “Warburg open” impedance Wo), especially at high frequencies, see Figure (c).Figure Caption: (a) The electrostatic potential in a cylindrical pore next to an electrolyte reservoir of resistance Rb shortly after applying a potential step. The data represent numerical solutions to the Poisson-Nernst-Planck (PNP) equations, which we solved with finite elements using a Newton solver from the FEniCS library. (b) The transmission line circuit is an equivalent circuit that describes the charging of pores such as the one shown in panel (a). In the circuit, the resistance R and capacitance C of the pore are cut into small pieces r and c. (c) The impedance of the TL circuit is given by Rb +Wo (blue dotted line), where Wo is the Warburg open impedance. Also shown is the impedance of a pore whose length is five times its diameter, as obtained from numerical PNP simulations (black). T. Aslyamov and M. Janssen, Electrochim. Acta 424, (2022)J. Yang, M. Janssen, C. Lian, R. van Roij, J. Chem. Phys. 156, (2022)C. Pedersen, T. Aslyamov, M. Janssen, in preparation R. de Levie, Electrochim. Acta 8, (1963)M. Janssen, Phys. Rev. Lett. 126, (2021) Figure 1