A cation-exchange membrane between two salt solutions

A cation-exchange membrane between 0.1 M NaCl and a more dilute solution, at open circuit: the voltage between Ag/AgCl electrodes in the two solutions, against the Teorell–Meyer–Sievers theory, and each ion's level through the membrane. Vary the dilute side, the membrane's fixed charge and the unstirred layers of water beside it. (More about the model below.)

dilute side fixed charge unstirred layers

About this simulation

Each solution is stirred, with an unstirred layer of water between it and the membrane. The membrane's fixed negative charge keeps chloride out (Donnan exclusion), so the salt that leaks through is carried almost entirely by Na⁺. Na⁺ then stays nearly level across the membrane, in equilibrium between the solutions, and the whole salt difference lands on Cl⁻, which the electrodes sense: E = 2t₊V_T ln(c₁/c₂), with t₊ → 1 for an ideal membrane. (The often-quoted 59 mV per decade "membrane potential" is this minus the electrodes' own Nernst terms, a conventional split.) With less fixed charge, chloride gets in and t₊ falls toward its free-solution value, D₊/(D₊ + D₋) = 0.40.

The dashed curve is the Teorell–Meyer–Sievers theory: a Donnan equilibrium at each face and diffusion between. It's the limit of thin unstirred layers. Thicker ones polarise, since the salt leaking through piles up on the dilute side and is drawn down on the other, so the membrane sees less of the difference.

The level diagram uses the species voltage V_i = μ̄_i/(z_iF): a species' electrochemical potential per unit charge, in volts, so electrons and ions share one axis. Each ion moves down its own μ̄_i, so its current flows down slopes in its own V_i (cations move downhill, anions uphill), and it's in equilibrium where its V_i is level. Each ion's V_i (thick) is continuous through each face, while its standard level V°_i (thin, where V_i would sit at the reference concentration) steps there, following φ. Those steps are the Donnan potentials, bookkeeping that cancels from anything measured. Na⁺ is drawn 4.4 V higher (⌇) to share the plot.

The ions, drawn as dots

An illustration of the solution above, not a separate simulation: particles() from the kit samples its concentrations as dots, each a fixed amount of ions, hopping between cells at rates taken from the solution, so that on average they fill the space as the concentrations do and cross each place at the fluxes. Where ions are too plentiful to draw they're a shaded sea, darker where denser: the 0.1 M solution, and the Na⁺ that balances the membrane's fixed charge. Chloride is the one to watch. A strongly charged membrane keeps it out, far thinner inside than in either solution and thinnest at the dilute side, and the few that wander across limit the salt leaking through. Na⁺, plentiful inside, comes and goes across the dilute face in near balance.

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