Double layer at a blocking electrode: dilute vs crowded ions

A salt solution against a blocking electrode held ψ above the bulk: the double layer's charge against Gouy–Chapman theory for ideal ions and Kilic–Bazant–Ajdari theory for crowded ones, and the crowded layer's levels and concentrations. Vary ψ, the ion size and the salt. (More about the model below.)

About this simulation

Ideal (Poisson–Boltzmann) ions pile up without limit: the Gouy–Chapman charge grows as sinh(Fψ/2RT), so by half a volt the counter-ions at the wall would be packed 10⁸ times denser than in the bulk. On a lattice of sites of size a (Bikerman / Kilic–Bazant–Ajdari statistics) the counter-ions saturate at one per a³ instead, the layer thickens, and the charge grows only as √ψ. Crowding starts once the wall concentration nears the lattice's, so it starts sooner for big ions and strong solutions. Both simulations (points) are drawn against their closed-form theory (lines).

The level diagram is the crowded layer, ion by ion, using 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. An ion is in equilibrium where its V_i is level, and here each ion's V_i (thick) is level from the bulk right up to the wall, while its standard level V°_i (thin, where V_i would sit at the reference concentration) bends with φ. The gap between them is the ion's concentration term, and its activity term on a crowded lattice. Each ion is drawn with an offset (⌇), Na⁺ above and Cl⁻ below.