Two charged nanochannel walls (surface potential ζ, half-spacing h) sit in a 1:1 electrolyte of bulk concentration C. Screening length — the Debye length:
λD = sqrt( ε kT / (2 n0 e²) ), n0 = C·NA
Linearized Poisson–Boltzmann gives the potential and ion profiles across the channel (y measured from the centerline):
ψ(y) = ζ · cosh(y/λD) / cosh(h/λD)
c±(y) = C · exp(∓ eψ(y) / kT)
Combining this body charge with Stokes flow under axial field E gives the electroosmotic velocity profile (Rice–Whitehead solution):
u(y) = -(ε ζ E / η) · [1 - cosh(y/λD)/cosh(h/λD)]
- The left panel renders the channel cross-section end-on: a heatmap of net charge density e(c₊(y)−c₋(y)) across the gap, the velocity profile u(y) drawn as a sideways curve, and tracer ions drifting at their local u(y) with vertical position sampled from the Boltzmann-weighted density — a true 2D field slice, not a flattened 3D scene.
- The right panel is a log–log regime map of Debye length λD against channel half-height h: the diagonal κh = 1 line marks full double-layer overlap, κh = 4 marks the isolated-layer limit, and the current (h, λD) state is plotted live as you move the sliders.
- κh ≫ 1 (wide channel / high salt): thin, isolated double layers, plug-like flow, weak selectivity. κh ≲ 1 (narrow channel / dilute salt): the double layers overlap, co-ions are excluded everywhere (Donnan exclusion), and the flow turns parabolic-like with reduced net speed.