Revisiting the access conductance of a nanopore in a charged membrane

TL;DR

This work develops a unified theory for electric-field-driven ionic transport through circular pores in ultrathin charged membranes, deriving a Debye–Hückel analytic expression and a semi-analytical formula for arbitrary surface potentials that predict fractional-power scaling with pore radius $a$ and Debye length $\\lambda_D$. By validating against finite-element simulations, the authors generalize Hall’s access-conductance framework to regimes with small pores, moderate-to-high surface charge, and varying electrolyte concentrations, and they reveal that fractional scaling with concentration is an intrinsic property of entrance effects, extending to thicker, ion-selective membranes where the Dukhin and Gouy–Chapman lengths govern transport. A central result is the surface-conductance expression $\\kappa_s \\approx \\frac{\\epsilon\\epsilon_0 (D_+ + D_-)}{l_{GC}} \\[ - \\frac{l_{GC}}{\\lambda_D} + \\sqrt{ (\\frac{l_{GC}}{\\lambda_D})^2 + \\frac{2}{3}(\\frac{a}{\\lambda_D})^{1/2}} \\]$, which, together with the bulk term, yields a generalized conductance $G$ that captures fractional scaling in both thin and thick regimes. These insights clarify experimental observations of fractional scaling in 2D nanopores and inform design of osmotic power, sensing, and iontronic devices by highlighting entrance effects as a fundamental mechanism rather than an anomaly.

Abstract

Electric-field-driven electrolyte transport through nanoporous membranes is important for applications including osmotic power generation, sensing and iontronics. We derive an analytical equation in the Debye--Hückel regime and a semi-analytical equation for arbitrary surface potentials for the electric-field-driven electric current through a pore in an ultrathin membrane, which predict scaling with fractional powers of the pore size and Debye length. We show that our theory for arbitrary electric potentials accurately quantifies the ionic conductance through an ultrathin membrane in finite-element method numerical simulations for a wide range of parameters, and generalizes a widely used theory for the access electrical conductance of a membrane nanopore to a broader range of conditions. Our theory predicts that fractional scaling of the ionic conductance with electrolyte concentration at low concentrations is an intrinsic property of charged ultrathin membranes and also occurs for thicker membranes for which the access contribution to the conductance dominates, which could help to explain experimental observations of this widely debated phenomenon.

Figures (5)

• Figure 1: Schematic of flow of an electrolyte solution through a circular aperture of radius $a$ in an infinitesimally thin planar wall with surface charge density $\sigma$. $c_{\mathrm{H}}^{(i)}$ is the concentration of species $i$, $p_{\mathrm{H}}$ is the pressure, and $\psi_{\mathrm{H}}$ is electric potential far from the membrane in the upper half plane. $c_{\mathrm{L}}^{(i)}$ is the concentration of species $i$, $p_{\mathrm{L}}$ is the pressure, and $\psi_{\mathrm{L}}$ is electric potential far from the membrane in the lower half plane. $Q$, $J$ and $I$ are the flow rate, solute flux and electric current, respectively. $\zeta$ and $\nu$ are oblate–-spheroidal coordinates, while $r = a \sqrt{(1 + \nu^2)(1 - \zeta^2 )}$ and $z = a\nu\zeta$ are cylindrical coordinates. Contours of constant $\zeta$ and $\nu$ are shown as solid red lines and unit vectors are shown at one point in space. The system has rotational symmetry about $z$.
• Figure 2: Access conductance $G_\mathrm{a}$ vs (a) pore radius $a$ for surface charge density $\sigma = -60$ mC m$^{-2}$ and bulk electrolyte concentration $c_{\mathrm{\infty}} = 0.3$ (blue), $1$ (red), $3$ (green), $10$ (orange), and $30$ (purple) mol m$^{-3}$ and (b) bulk electrolyte concentration $c_{\mathrm{\infty}}$ for pore radius $a = 1$ nm and surface charge density $\sigma = -1$ (blue), $-10$ (red), $-30$ (green), $-60$ (orange), and $-100$ (purple) mC m$^{-2}$. Symbols are FEM simulations of an ultrathin membrane, solid lines are the theory in this work (Eqs. \ref{['eq:surf_conduct']} and \ref{['eq:acc_resist']}), dashed lines are the theory in Ref. leeLargeElectricSizeSurfaceCondudction2012 (Eqs. \ref{['eq:surf_conduct-plane']} and \ref{['eq:lee-access_conductance']}) for $\alpha = \beta = 2$, and dotted lines indicate scaling relationships.
• Figure 3: Access conductance $G_\mathrm{a}$ from the theory derived in this work (Eqs. \ref{['eq:surf_conduct']} and \ref{['eq:acc_resist']}) and from the theory in Ref. leeLargeElectricSizeSurfaceCondudction2012 (Eqs. \ref{['eq:surf_conduct-plane']} and \ref{['eq:lee-access_conductance']}) for $\alpha = \beta = 2$, vs corresponding conductance from FEM simulations (symbols). The color map depicts variations in the ratio of the Debye length $\lambda_{\mathrm{D}}$ to the pore radius $a$ and the solid line indicates perfect agreement between the theory and simulations.
• Figure 4: Ratio of access conductance $G_\mathrm{a}$ from theory to that from FEM simulations (symbols) for the theory (a) in this work (Eqs. \ref{['eq:surf_conduct']} and \ref{['eq:acc_resist']}) and (b) Ref. leeLargeElectricSizeSurfaceCondudction2012 (Eqs. \ref{['eq:surf_conduct-plane']} and \ref{['eq:lee-access_conductance']}) for $\alpha = \beta = 2$, vs $l_{\mathrm{Du}}/a$. $l_{\mathrm{Du}}$ is the Dukhin length (of an ultrathin membrane) and $a$ is the pore radius. The color map depicts variations $\lambda_{\mathrm{D}}/a$, where $\lambda_{\mathrm{D}}$ is the Debye length and $a$ is the pore radius. The solid line indicates perfect agreement between the theory and simulations.
• Figure 5: Apparent power-law scaling exponent of the ionic conductance of a membrane pore vs electrolyte concentration from the theory in Eq. \ref{['eq:ion_conduct-finite_len']} (using Eq. \ref{['eq:surf_conduct-plane']} and Eq. \ref{['eq:surf_conduct']} in the pore and access contributions, respectively) as a function of $2a/L$ and $\lambda_{\mathrm{D}}/a$, at $a/l_{\mathrm{GC}} =$$2$ and $20$, where $a$ is the pore radius, $L$ is the membrane thickness, $\lambda_{\mathrm{D}}$ is the Debye length and $l_{\mathrm{GC}}$ is the Gouy--Chapman length.