Ion Permeation in Nanoscale Films: Fundamental Limitation and Evaluation of Dielectric Properties

Abstract

Nanoscale films play a central role in biology and osmotic separations. Their water/salt selectivity is often regarded as intrinsic property, favoring thinner membranes for faster permeation. Here we highlight and quantify a fundamental limitation arising from the dependence of ion self-energy on film thickness, governed by its ratio to Bjerrum length. The resulting relation factors out this dependence from intrinsic ion permeability, which agrees well with available data and enables evaluation of dielectric properties of ultrathin films, advancing understanding of ion transport in membranes.

Figures (3)

• Figure 1: Sketch of an ion crossing a planar film (membrane) of thickness $d$. $z$ is the thickness coordinates, and $\lambda_m$ represent the characteristic thickness of regions adjacent to the interfaces where ion exclusion weakens substantially.
• Figure 2: (a) Dependence of ion permeability, scaled with $P_{\infty}$, on the film thickness, scaled with the Bjerrum length $\lambda_m$. The solid and dashed lines represent the full numerical solution and the analytical approximation for different values of $\lambda_m$ (see eqs. (S10) and (S15) in Supplemental Material). (b) Profiles of the thickness-dependent part of ion self-energy $U(z)$ across the film, presented as $e^{U(z)}$, for films much larger ($d = 40\lambda_m$) and equal ($d = \lambda_m$) to the Bjerrum length, and much smaller than the Bjerrum length ($d = 0.1\lambda_m$). The Bjerrum length is 10 nm.
• Figure 3: Thickness dependence of ion permeability. (a) Proton permeation through lipid membranes, reproduced from paula1996permeation. (b) Potassium and chloride permeation through lipid membranes from paula1996permeationpaula1998permeation. (c) NaCl permeation through polyamide films, reproduced from mulhearn2021thickness. In all panels, symbols show experimental measurements, and solid lines are fits to Eq. \ref{['P approx']}. The reported permeances ($P/d$) were scaled by inverse thickness to obtain the permeability dependence $P(d)$.