Transmission Line Circuit and Equation for an Electrolyte-Filled Pore of Finite Length

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 τ , an expression for which consistently follows from the TL circuit, TL equation, and from the pore's known impedance. An approximation to τ explains the numerically determined relaxation time of the stack-electrode model of Lian et al. [Phys. Rev. Lett. 124, 076001 (2020)].

Charging dynamics of electrical double layers inside a cylindrical pore: predicting the effects of arbitrary pore size

“…This assumption is valid for non-interacting pores where the radial currents are identically zero, consistent with the treatment of SDL in the literature. 2,29,36 We solve the set of eqn ( 2)-( 8) numerically using OpenFOAM. 47,48 The details of geometry, mesh, and algorithm have been described in ref.…”

“…( 1, enabling us to assume electroneutrality inside the SDL. In contrast, the smallest length scale over which the charge gradients are present inside the pore is a p ' p , which implies r C D ¼ Oð1Þ; 36,44,[49][50][51] see also eqn (29). Next, in the transition region, r varies from zero to the value inside the pore.…”

“…Several studies on electrode charging invoke transmission line representations, theoretically developed for thin double layers, [27][28][29] to address electric potential and capacitance in these systems. [22][23][24][25] In this section, we demonstrate that a similar transmission line model can be constructed for arbitrary pore sizes.…”

“…We employ the definition of effective volumetric capacitance C eff as the ratio of total charge stored per applied voltage and unit volume. Going back to dimensionless variables and performing a radial integration of the steady-state charge density profile of eqn (29), it is straightforward to obtain…”

“…27,28 Later, Biesheuvel and Bazant 2 extended the circuit to high potentials for capacitive deionization applications. Recent papers 26,29 further discuss the relationship between the equivalent circuit and the corresponding transmission line continuous equation for pores with finite lengths. Throughout most pore charging models, common assumptions are either of thin double layers 2,27,28,30,63 within the pores, i.e., such that the length of the charged region is much smaller than the pore size, or overlapping double layers, 31,32 where the charged regions extending from the opposite sides of the surface meet.…”