When a depleted layer starts to move

Give a disturbance a transverse direction

Extend the channel sideways. Coordinate $X$ still runs from reservoir to membrane; $Y$ runs along the membrane, both in metres. The stationary background varies only with $X$ . Imagine a small patch just in front of the membrane becoming slightly saltier than its neighbours. It introduces variation along the interface.

The saltier patch conducts more readily and redistributes potential, potentially creating a tangential field. That field acts on charge and drives a small liquid motion. If the motion brings still more salt into the original patch, the disturbance grows. If it levels the concentration difference, the disturbance decays. Stability depends on the sign of this feedback.

A one-dimensional space-charge solution supplies the stationary conducting background, including a strong field in the depleted region. It has no transverse coordinate in which a circulation can form. The next step couples transverse disturbances and liquid transport to ion conservation.

Reading: Rubinstein & Zaltzman

Let the liquid carry ions

Velocity $\mathbf u=(u_X,u_Y)$ is in metres per second and number density $n_\pm$ in inverse cubic metres. Ions move with the liquid in addition to diffusion and electrical drift, adding density times velocity to their flux. The operator $\nabla$ takes spatial changes in both directions; a divergence adds the net outflow from them.

At low inertia, pressure $p$ , viscosity and electrical force per volume balance. Viscosity $\eta$ is in pascal seconds and charge density is $\rho_e=e(n_+-n_-)$ . Pressure includes equilibrium contributions that can be absorbed into a gradient. It can balance part of the electric force, so charge in a field alone does not specify a circulation.

\partial_\tau n_\pm=-\nabla\cdot\mathbf N_\pm,\qquad\mathbf N_\pm=n_\pm\mathbf u-D\left(\nabla n_\pm\pm\frac{en_\pm}{k_BT}\nabla\Phi\right)

The dilute ideal monovalent ions share a diffusivity. Keeping dimensional quantities first makes it possible to check that every term is a number flux.

-\nabla p+\eta\nabla^2\mathbf u+\rho_e\mathbf E=0,\quad\nabla\cdot\mathbf u=0,\quad\mathbf E=-\nabla\Phi

The second condition is incompressibility. Ion distributions remain coupled to Poisson’s equation. No slip or effective thin-layer slip at a solid must match which charge layers the calculation actually resolves.

Reading: Rubinstein & Zaltzman · Dydek et al.

Estimate how quickly salt can be transported

Take a depleted-region thickness $\ell=100\;\mathrm{\mu m}$ and diffusivity $D=10^{-9}\;\mathrm{m^2\,s^{-1}}$ . Suppose a circulation with observed speed $U=20\;\mathrm{\mu m\,s^{-1}}$ already exists. Advection crosses this length in five seconds; diffusion takes ten. Their ratio defines the Péclet number, measuring the ability of a specified flow to transport concentration.

Here $\mathrm{Pe}=2$ and advection can substantially alter depletion. At $U=5\;\mathrm{\mu m\,s^{-1}}$ , the advective time becomes twenty seconds and $\mathrm{Pe}=0.5$ . These are teaching-scale estimates with a prescribed observed speed. The speed has not been inferred from a one-dimensional current.

Equal advective and diffusive times do not establish instability. The Péclet number uses an existing velocity. Predicting whether stationary liquid starts moving requires the additional feedback calculation: does the flow induced by a small disturbance reinforce that disturbance?

\tau_D=\frac{\ell^2}{D}=10.0\;\mathrm s,\qquad\tau_A=\frac\ell U=5.00\;\mathrm s,\qquad\mathrm{Pe}=\frac{U\ell}{D}=\frac{\tau_D}{\tau_A}

Use the same length to compare advection and diffusion. Locating onset requires a disturbance growth rate.

Teaching time-scale estimates at prescribed speed
$U/(\mathrm{\mu m\,s^{-1}})$	$\tau_A/\mathrm s$	$\mathrm{Pe}$
5.00	20.0	0.500
10.0	10.0	1.00
20.0	5.00	2.00

Read stability from a disturbance growth rate

Let a quantity initially have stationary profile $f_0(X)$ . Add a small wave along the membrane, separating its changing amplitude from its repeating spatial form. A complex exponential packages sine and cosine together; taking the real part returns a physical real-valued disturbance. The real part of growth rate $s$ determines whether amplitude increases. Wavenumber $k$ sets its repetition distance and has units of inverse metres.

Substitute this form into transport and flow equations and keep only terms linear in disturbance amplitude. For each voltage and wavenumber, solve for the growth rate. The stationary state loses stability when its fastest mode changes from decay to growth. Subsequent nonlinear dynamics determine whether the mature motion is steady, oscillatory or more complicated.

Open Fig. 2 of Rubinstein and Zaltzman, arXiv:1602.00137v1. The horizontal axis is interfacial wavenumber and the vertical axis is voltage, both in the paper’s own dimensionless scales. The region above each neutral curve is unstable. Solid curves retain space charge and dashed curves use the electroneutral limit. The figure fixes the paper’s layer-width ratio $L=1$ and Debye parameter $\varepsilon=10^{-4}$ , and varies membrane fixed-charge parameter $N$ . Read geometry and boundaries before comparing curve minima.

f=f_0(X)+\Re\!\left[\widehat f(X)e^{s\tau+ikY}\right],\qquad\lambda_Y=\frac{2\pi}{k},\qquad\Re(s)=0

The final equality defines neutral stability. A wavelength

\lambda_Y=50.0\;\mathrm{\mu m}

gives

k=1.26\times10^5\;\mathrm{m^{-1}}

. A teaching example with growth rate

\Re(s)=0.200\;\mathrm{s^{-1}}

increases amplitude by

e^1=2.72

in five seconds. Actual rates require a linear stability calculation with specified boundaries.

Reading: Rubinstein & Zaltzman

Several mechanisms can carry overlimiting current

Begin with a narrow channel whose side walls carry negative charge. Excess cations screen that charge and can conduct along the walls even when the central liquid is depleted. Narrowing the channel increases wall area per liquid volume. This surface contribution can raise current without vortices.

In a wider channel, electro-osmosis on charged side walls can supply salt near the membrane and drive return flow near the closed end. A still wider geometry can permit transverse disturbances of the membrane’s depleted layer to develop into electro-osmotic instability. Fig. 1 of Dydek and colleagues, arXiv:1108.2325v1, separates surface conduction, side-wall flow and membrane instability into three physical pictures. Each belongs to specified geometry and wall conditions.

A useful experiment records current and tracer velocity together while changing channel height, wall charge or reservoir concentration. Suspected water splitting calls for product or acid–base measurements as well. Comparing these observations with the current–voltage curve tests each proposed source of extra transport.

Reading: Dydek et al.

Turn a frontier question into a testable calculation

Rubinstein and Zaltzman’s study was published in Physical Review Fluids in 2016; we read its fixed preprint version here. It compares driving terms through linear stability calculations for prescribed membrane selectivity and boundaries. Dydek and colleagues’ microchannel work appeared in Physical Review Letters in 2011 and combines reduced models with two-dimensional numerical results to study geometrical selection of transport mechanisms.

A next calculation can start from a verified one-dimensional conduction state. Add small disturbances at separate transverse wavelengths and measure their early growth rates. Refine the charge-layer mesh and time step independently. Reducing initial amplitude should preserve the linear growth rate; changing lateral boundaries also changes the allowed wavelengths. Stationary conduction, instability onset and mature flow can then be tested as separate steps.

Strong driving may also crowd ions near an interface. Kilic, Bazant and Ajdari’s finite-ion-volume models show how point-ion extrapolations of capacitance and salt adsorption can change. Questions controlled by crowding, concentrated-solution activity or distributed porous-electrode reactions require their own constitutive models. The present one-dimensional transport and weak-field cylinder flow supply independently reproducible starting points for that research sequence.

Reading: Rubinstein & Zaltzman · Dydek et al. · Kilic

Try the prediction

Identify what a stationary conduction calculation answers

Predict how voltage changes concentration, electric field and transverse velocity. State which quantities a one-dimensional model can determine.
Compare Diffusion keeps up and Space charge spreads. Save trajectories with their boundary conditions and record late current and minimum salt concentration.
Specify the additions needed for a two-dimensional calculation: transverse wavelength, velocity and lateral and membrane flow boundaries. Explain how you would measure a growth rate.

Distinguish imposed electro-osmotic flow from instability

Observe the steady cylinder flow and predict what field reversal will change.
Reverse the field, compare flow direction, then rotate the field to reorient the four-lobed pattern.
Explain how this analytic flow fixed by surface geometry and applied field differs from spontaneous growth of a depleted-layer disturbance.

Exercises and explanations

Current exceeds the classical diffusion limit, but microscopy resolves no vortices. What else should be investigated?
Read the explanation
Consider one-dimensional space-charge transport, conduction along charged walls and chemical production of ions. Unresolved motion can also reflect limited imaging sensitivity. Compare geometry and wall-charge dependence, improve velocity sensitivity and, where relevant, measure acid–base changes or reaction products.
Halve transport length at fixed velocity and diffusivity. How do the two transport times and Péclet number change? Does this locate the instability voltage?
Read the explanation
Diffusion time falls to one-quarter, advective time to one-half and Péclet number to one-half. Diffusion has more opportunity to smooth concentration relative to advection. Instability voltage also depends on how disturbances generate flow, membrane selectivity and fluid boundaries; a prescribed-speed estimate cannot replace that calculation.
A disturbance doubles in amplitude in four seconds during linear exponential growth. What is its growth rate, and why repeat with a smaller starting amplitude?
Read the explanation
The rate is $\Re(s)=\ln2/(4\;\mathrm s)=0.173\;\mathrm{s^{-1}}$ . A smaller starting amplitude should give the same early rate. Strong amplitude dependence can indicate nonlinear evolution or unresolved numerical error.

Sources and calculation scope

Rubinstein & Zaltzman, The driving factors of electro-convective instability in concentration polarization, arXiv:1602.00137v1 (2016)
Sections I–II, equations (3)–(9) and Figures 1–3 distinguish selective conduction, fluid coupling and a transverse stability calculation. The course reads this published research; its one-dimensional PNP experiment does not reproduce the neutral-stability curves.
Dydek et al., Overlimiting current in a microchannel, arXiv:1108.2325v1 (2011)
Figures 1–3 and the separate surface-conduction, side-wall-flow and instability mechanisms. Their channel geometry and voltage convention differ from the course half-cell; our PNP curve is not a reproduction of those figures.
Kilic, Bazant & Ajdari, Steric effects in the dynamics of electrolytes at large applied voltages I, arXiv:physics/0611030v1 (2006)
Sections I–II explain why point-ion dilute theory can overpredict near-surface concentrations and introduce finite-size alternatives. These modifications are reading material, not active equations in the ideal PNP browser solver.