A charged nanochannel: selectivity, saturation and rectification

A pore ten nanometres across through a membrane, between two KCl solutions, its walls charged negative (silica or a polymer in water usually is). The charge draws K⁺ in and keeps Cl⁻ out, so the channel conducts almost only K⁺, and goes on conducting when the salt is diluted a thousandfold. Both come from the K⁺ the wall's charge holds inside; the potential steps that keep it there, and Cl⁻ out, sit in layers of space charge at each mouth, and in dilute salt those layers are as long as the channel: the two mouths' layers overlap, and there's no neutral inside. Put the charge on one half, or make the channel bipolar, and it conducts better one way than the other, like a pn junction. Vary the salt on each side, the charge and its pattern, the length and the voltage; long channels are checked against the Teorell–Meyer–Sievers theory. (More about the model below.)

About this simulation

The channel runs along x, between two reservoirs of KCl, and everything is resolved: ε > 0 throughout, so the charge at each mouth, where the wall's fixed charge meets the solution, comes out of Poisson's equation together with every ion's drift and diffusion. The wall's charge is spread through the channel as a density X: a surface charge σ on the walls of a channel of diameter d is X = 4σ/(Fd), so 30 mM is about 7 mC/m² in a 10 nm channel. Spreading it evenly across the channel is fair when the screening length inside the channel is comparable to its radius: with the wall's charge drawing in K⁺, that's the Donnan length below (2.5 nm at 30 mM, against a 5 nm radius), so the screening clouds from opposite walls overlap and the channel is fairly uniform across.

Inside a long channel the bulk is neutral, the wall's charge balanced by excess K⁺: K⁺ = (X + √(X² + 4c²))/2 and Cl⁻ = K⁺ − X, a Donnan equilibrium with the solution at each mouth. With much more wall charge than salt, K⁺ inside is pinned at X and Cl⁻ is shut out, so the conductance stops depending on the salt: the plateau at low concentration, measured in silica nanochannels (Stein, Kruithof and Dekker, 2004). That's the Teorell–Meyer–Sievers theory, the dashed curve: the long-channel limit, where all the space charge sits in thin Donnan layers at the mouths that the theory treats as steps, the physics of an ion-exchange membrane. The dotted line is the same channel uncharged, conducting like the salt. Across a salt gradient at zero current, K⁺'s level is flat through a strongly charged channel, in equilibrium between the two sides, and the voltage between Ag/AgCl electrodes approaches 2V_T ln(c₁/c₂), as for a perfectly selective membrane: V_T ln(c₁/c₂) across the channel, and as much again from the two electrodes, which follow Cl⁻ in solutions of different strength.

A channel not much longer than its layers is all mouth. What sets the scale is the channel's length against the Donnan length √(εRT/F²X) inside (2.5 nm at 30 mM of wall charge) and the Debye length outside. The net charge plot shows each mouth's pair of layers, the wall's charge against its screening cloud, and in a 20 nm channel at 1 mM they run into each other: the channel holds less K⁺ than its wall charge would draw in, and conducts at 0.74 of TMS (0.57 at 10 nm, 0.88 at 50 nm). There Poisson's equation, not electroneutrality, decides what the channel does.

Charge on one half only, or − on one half and + on the other, makes the current depend on its direction: driven one way, ions are carried into the channel faster than they leave, and it fills with salt; the other way, it empties, and conducts poorly. A bipolar channel is an ionic diode (built in conical nanopores by Siwy, Vlassiouk and others), and its physics is a pn junction's: the negatively charged half, where K⁺ is the majority ion, is the p side (fixed acceptors, majority holes), and the positive half, with Cl⁻, the n side. Where they meet, a depletion layer holds the space charge, and the standard levels V°_i bend across it by a built-in potential, just as band edges do. Reverse bias widens it and leaves only the minority ions to carry a small, saturating current; forward bias floods it with salt.

The level diagram draws each ion's species voltage V_i = μ̄_i/(z_iF), with K⁺'s drawn 4.35 V higher to share the plot (⌇: they're that far apart only because each ion's μ° is counted from its elements, and no ion here answers to another's level), and their standard levels V°_i (thin), which follow φ. The steps in V°_i at the mouths are the Donnan potentials; each ion's own level is continuous through them, and an ion flows only where its level slopes.

What a 1D slice leaves out: the reservoirs widen away from the mouth in 3D and add little resistance, so here they're layers whose ions move 10⁴ times faster, standing in for the widening (no equilibrium depends on how fast ions move, so the Donnan layers are unchanged); and the channel's radial structure, which matters once the Debye length is well below the radius (in strong salt). The same equations, Poisson–Nernst–Planck theory, are used for biological ion channels (after Eisenberg and Chen), but there a selectivity filter a couple of nanometres long holds a handful of ions in single file, and a continuum of concentrations and a single permittivity are rough approximations.