FR17 · A metal post moves the liquid

Induced-Charge Electro-osmosis

Why does reversing the field leave the flow unchanged? The field creates the surface charge that it then pushes. Reversing the field reverses that induced charge too, so the liquid keeps flowing from the cylinder’s poles towards its equator. In the thin, weakly charged double-layer limit, the flow grows with the square of the field.

A fixed metal cylinder polarizes in an electrolyte. Follow the liquid past its charged surface, reverse the electric field and turn the four-lobed flow.

Model assumptions

The steady analytic solution describes an isolated, neutral, ideally polarizable cylinder in a dilute symmetric electrolyte. The double layer is thin and its potential small; inertia, walls, Faradaic reactions and surface conduction are omitted. The diffuse layer is enlarged in the drawing. Tracers follow the calculated velocity at eight times physical time.

Steady flow · enlarged diffuse layer · tracer time runs eight times faster

Maximum surface slip

2.81\,\mu\mathrm{m/s}

Maximum diffuse-layer potential

9\,\mathrm{mV}

Maximum diffuse charge density

62.5\,\mu\mathrm{C/m^2}

Charging time scale

1\,\mathrm{ms}

Charge and surface slip

Gold shows normalized diffuse charge; cyan shows normalized tangential slip. Angles start from the applied field axis.

Field reversal and quadratic response

The dot marks the current field. Reversing its sign preserves the slip magnitude; doubling it multiplies the speed by four.

Model, units and sources

A neutral metal cylinder is fixed in an unbounded liquid. The electric field lies in the section plane; angles are counterclockwise from it. Water permittivity is 0.695 nanofarads per metre, viscosity 1 millipascal second and ion diffusivity one billionth of a square metre per second. The four-lobed flow has zero net pumping.

\zeta=2E_0a\cos\theta,\quad E_\theta=-2E_0\sin\theta,\quad u_s=-\frac{\epsilon\zeta E_\theta}{\eta}

The analytic velocity assumes a thin, weakly charged double layer and negligible surface conduction. A warning appears when the maximum potential exceeds the thermal voltage or the layer ceases to be thin.

Squires & Bazant (2004), §3.1, Eqs. 3.5–3.9 · Continue the electrochemistry course

Start with the essentials

Focus question: Why does reversing the field leave the flow unchanged?
One-sentence intuition: The induced charge and the driving field both reverse, leaving their product and the liquid flow unchanged.

Core mathematical model

The field induces a dipolar double layer

\zeta(\theta)=2E_0a\cos\theta,\qquad q_d=-\epsilon\zeta/\lambda_D

The angle starts from the applied field axis. The diffuse ionic charge has the opposite sign to the surface potential; its integral around the cylinder is zero.

The same charge drives the surface slip

u_s=-\frac{\epsilon\zeta E_\theta}{\eta}=2U_0\sin(2\theta),\qquad U_0=\frac{\epsilon a E_0^2}{\eta}

The tangential electric field acts on the diffuse charge. The resulting slip carries liquid from the field-axis poles towards the equator.

Charging occurs before steady flow

\tau_c=\frac{\lambda_D a}{D}

The steady flow shown here follows the charging transient. Increasing the Debye length increases this time scale but does not change the leading steady thin-layer velocity.

Common difficulties

A fixed post and a moving particle

Typical misconceptionEvery electro-osmotic flow means the cylinder should drift.

Better mental modelThe cylinder here is held fixed and has zero total charge. Its symmetric quadrupolar slip has no mean translation; freely moving, asymmetrically charged particles require a separate force-free calculation.

Run the experiment

01
Predict the reversal
Watch Four lobes of flow, then choose Reverse the field.
What to observe: Gold and cyan charge patches exchange places; the tracer paths keep their directions.
02
Rotate the symmetry axis
Drag the bright handle around the post, or choose Turn the flow.
What to observe: The inflow axis follows the electric field. Turning the field by a right angle swaps the inflow and outflow directions.
03
Test the square law
Compare Four lobes of flow with Twice the field and read the maximum slip.
What to observe: The velocity grows by four while the induced potential grows by two. Raising the field eventually leaves the small-potential approximation.

Induced-Charge Electro-osmosis

Charge and surface slip

Field reversal and quadratic response