(399h) Kirchhoff’s Laws Get an Upgrade: Double-Layer Dynamics in Pore Networks Described By a De Levie Circuit for an Effective Electrochemical Potential of Charge

Electric double-layer capacitors (EDLCs) are devices that store energy through the quick electrostatic separation of oppositely charged ions into the inner surfaces of highly porous electrodes subjected to different applied electric potentials. EDLCs are vital to the stability and constant availability of power obtained from intermittent renewable energy sources, such as sunlight and wind. A recent route of optimization of EDLCs consists of the control of electrode topology to simultaneously enhance power and energy densities. While most of the experimental avenues proposed focus on engineering electrode shapes or designing materials with high internal surface area, there is an opportunity to further advance ion transport in EDLCs by controlling their pore network structures. This is a challenging task because of the geometrical complexity of the porous medium.A paradigm in the modeling of double-layer dynamics in EDLCs in de Levie’s transmission line. It describes the evolution of electric potentials in a pore with thin double layers as an equivalent circuit of resistors for Ohmic losses along the axis and capacitive charge storage near the pore surface. However, it hinges on the assumption that double layers are thin relative to pore thickness, invoking electroneutrality in the pore bulk. Therefore, this model does not capture the effects of pore networks of finite double-layer thickness. In this work, we propose a reduced-order model of the Poisson-Nernst-Planck equations in the Debye-Hückel limit to describe EDL charging of symmetric binary electrolytes in networks of long pores for arbitrary Debye lengths.We show that the cross-section averaged (reduced-order) equations can be interpreted as a transmission-line circuit for the effective electrochemical potential of charge density, i.e., the valence-weighted average of the ionic electrochemical potentials. Such a representation is crucial to leverage the continuity of electrochemical potentials across pore junctions, which captures changes in the potential and charge profiles of pores of different radii. Through this framework, we are able to write effective Kirchhoff’s current and voltage laws for pore junctions and loops. The order reduction of the model enables the fast simulation of arbitrary networks, allowing for the simulation of five thousand pores in 6 minutes, and rendering this framework applicable to the design of pore networks and the interpretation of impedance spectroscopy measurements.