Transmission line circuit and equation for an electrolyte-filled pore of finite length

TL;DR

This paper explores the relationship between transmission line equations and circuit models for electrolyte-filled pores, deriving boundary conditions and relaxation times that unify physical and circuit perspectives.

Contribution

It demonstrates how boundary conditions and relaxation times in electrolyte-filled pores can be derived from both transmission line equations and circuit models, unifying physical and electrical descriptions.

Findings

Robin and Neumann boundary conditions emerge naturally in circuit models.

Relaxation time τ is consistently derived from multiple approaches.

Approximation of τ matches numerical results from previous studies.

Abstract

I discuss the strong link between the transmission line (TL) equation and the TL circuit model for the charging of an electrolyte-filled pore of finite length. In particular, I show how Robin and Neumann boundary conditions to the TL equation, proposed by others on physical grounds, also emerge in the TL circuit subject to a stepwise potential. The pore relaxes with a timescale $\tau$, an expression for which consistently follows from the TL circuit, TL equation, and from the pore's known impedance. An approximation to $\tau$ explains the numerically determined relaxation time of the stack-electrode model of Lian et al. [Phys. Rev. Lett. 124, 076001 (2020), arXiv:1911.09924].