Electrochemistry · A selective interface runs short of salt

Concentration Polarization and Space Charge

What changes when a membrane draws ions faster than diffusion can replace them? A selective membrane removes one ion species and leaves a depleted layer in front of it. The electroneutral approximation approaches a diffusion-limited current. When depletion becomes strong, net space charge can develop and the full transport model departs from that limit.

Pull cations toward an ideal selective membrane. Watch the depleted region grow, compare its final profile with the electroneutral approximation, and increase the voltage until space charge carries the current beyond the classical limit.

Time after voltage step $0.120\,\mathrm{ms}$

Current magnitude $0.0200\,j_0$

Minimum salt concentration $0.120\,c_0$

Current / classical limit $0.500$

One voltage step, one evolving solution

Let the membrane draw ions

Applied voltage

V/(k_{\rm B}T/e)

4

Screening length / gap

\lambda_{\rm D}/L

0.06

Drag the glowing marker below the cell to inspect time. The colored particles show the computed ion density.

A voltage step rearranges the ions.

Units and numerical accuracy

L=2\,\mu\mathrm m

D=10^{-9}\,\mathrm{m^2\,s^{-1}}

T=298\,\mathrm K

t_0=L^2/D=4.00\,\mathrm{ms}

V_T=k_{\rm B}T/e

V_T=25.7\,\mathrm{mV}

Changing screening length changes the initial salt concentration in water with relative permittivity 78.5. The finite-volume solution uses 112 cells, positive implicit ionic fluxes and a self-consistent Poisson solve at each time step.

\Delta j_{\rm tot}<10^{-5}\,j_0

c_0=6.44\,\mu\mathrm{mol\,L^{-1}}

N_+/N_{+,0}=1.00

N_-/N_{-,0}=1.00

The saved file contains dimensional scales, every profile, boundary fluxes, electrode charge, ionic amounts and the spatial spread of total current. The instantaneous ideal switching impulse is excluded.

Model and approximation limits

The reservoir fixes both ion concentrations. The membrane fixes cation concentration at the reservoir value, blocks anions, and is held at the negative applied potential. The electroneutral line is a late-time outer reference; it omits the membrane charge layer.

Dilute point-ion transport omits molecular packing and correlations. Larger voltages may require steric, reaction or concentrated-solution models. Fluid motion is not computed here.

Model assumptions

This is a one-dimensional dilute ideal electrolyte with equal ion diffusivities, a fixed-concentration reservoir and a cation-selective interface. The interface fixes cation concentration and voltage and blocks anions. Applied voltage includes the interfacial Donnan drop. The two-micrometre gap and enlarged screening length are teaching scales. The calculation has no fluid flow, electroconvective instability, surface conduction, water splitting or real membrane microstructure.

Simulated transport record

Follow the ions and the current together

Ion concentration across the cell

Potential across the cell

Current after the voltage step

CationsAnionsPotential / total currentDashed: electroneutral steady limit

The neutral reference approaches twice the current-density scale. The full model includes the charged interface layer and its transient. Both are simulated or analytic model results.

Equations and primary sources

J_\pm=-\partial_xc_\pm\mp c_\pm\partial_x\phi,\quad -2\delta^2\partial_x^2\phi=c_+-c_-,\quad j_{\rm tot}=J_+-J_-+2\delta^2\partial_tE

Bazant, Thornton and Ajdari (2004), blocking-cell charge dynamics

Rubinstein and Zaltzman (2016), concentration polarization and electroconvection

Learn how diffusion, charge and flow connect

Physics tutorial

Concentration Polarization and Space Charge

BackgroundThe reservoir can supply both ions, but the membrane takes only cations. Anions cannot pass through it. Their zero steady flux links the local concentration to the potential, and salt becomes scarce just in front of the membrane.

Why it mattersThe classical limiting current follows from a nearly neutral liquid. Strong depletion is precisely where this approximation becomes fragile, so it is useful to keep Poisson’s equation and inspect the local charge.

Start with the essentials

Focus question: What changes when a membrane draws ions faster than diffusion can replace them?
One-sentence intuition: The depleted concentration and the space-charge field must be solved together. A one-dimensional current above the neutral diffusion limit does not predict the onset of fluid motion.

Core mathematical model

Diffusion and electrical drift

J_\pm=-\partial_xc_\pm\mp c_\pm\partial_x\phi,\qquad \partial_tc_\pm=-\partial_xJ_\pm

Distance is scaled with the full gap, concentration with the reservoir concentration, potential with the thermal voltage, and time with gap squared divided by diffusivity. Positive flux points from left to right. Both ions have the same diffusion coefficient.

Charge changes the potential

-2\delta^2\partial_x^2\phi=c_+-c_-,\qquad \delta=\lambda_{\rm D}/L

The concentration difference is the net charge in these units. A small screening ratio lets appreciable charge collect in narrow layers. The numerical mesh resolves these layers, with its convergence checked separately.

The electroneutral steady comparison

c(x)=1-\frac{j x}{2},\quad \phi(x)=\ln c(x),\quad j=2(1-e^{-V/2})

The left reservoir has unit concentration and zero potential. The right membrane fixes cation concentration at unity and potential at minus the applied voltage while blocking anions. The voltage includes the Donnan drop; the neutral outer profile approaches a limiting current of two.

Common difficulties

A neutral limit is an approximation

Typical misconceptionThe classical limiting current is an absolute ceiling for every electrolyte system.

Better mental modelThe neutral diffusion limit assumes local electroneutrality and no other transport mechanisms. Space charge changes the electrical migration; fluid flow and surface transport are separate extensions.

Run the experiment

01
Start where diffusion keeps up
Choose Diffusion keeps up and let the evolution settle. Predict where the two concentration curves should almost coincide, then compare them with the dashed neutral reference.
What to observe: Away from the membrane charge layer, the two ion concentrations nearly coincide and decline toward the interface. The reference is the steady outer solution, so early transient curves need not match it.
02
Follow the depleted region
Choose A depleted layer. Drag the time marker in the cell to compare an early frame with a late one. Then increase Applied voltage.
What to observe: Salt is depleted near the membrane and the potential drop becomes concentrated there. The ionic flux and current are computed from these same concentration and potential arrays.
03
Inspect space charge
Choose Space charge spreads. Compare the full current with the classical limit and inspect the separation of the positive and negative concentration curves.
What to observe: The full one-dimensional model can carry current above the neutral limit through a charged region. Flow, surface conduction and water splitting would require additional physics.

Concentration Polarization and Space Charge

Follow the ions and the current together

Ion concentration across the cell

Potential across the cell

Current after the voltage step

Concentration Polarization and Space Charge

Core mathematical model

Diffusion and electrical drift

Charge changes the potential

The electroneutral steady comparison

Common difficulties

A neutral limit is an approximation

Run the experiment

Start where diffusion keeps up

Follow the depleted region

Inspect space charge

Related experiments and articles

From ion diffusion to charge, flow and impedance