Kirchhoff's Laws Based on Electrochemical Potential of Charge Dictate Double-Layer Charging in Porous Media

3D-printed electrode structures are a promising avenue for advancing next-generation miniaturized energy storage devices. However, a rational approach to designing such electrodes remains elusive since direct numerical simulations of transport in these complex geometries are prohibitively expensive. We address these limitations by developing a Debye-Hückel model to describe the electric double-layer charging of a binary electrolyte in an arbitrary network of pores. We propose novel boundary conditions and interpret the model using a transmission line circuit approach, specifically for the electrochemical potential of charge – the valence-weighted sum of ionic electrochemical potentials. This innovative methodology unveils the effective Kirchhoff's laws in the network and demonstrates excellent agreement with direct numerical simulations while offering computational speeds that are significantly faster. Leveraging the derived model, we investigate the effects of pore arrangement and polydispersity on given pore-size distributions. Our analysis reveals valuable insights into how different pore configurations can influence the performance of energy storage devices.