Shear-Induced Electrophoretic Migration Perpendicular to the Electric Field

This study proposes a theoretical mechanism explaining shear-induced lateral migration of dielectric particles perpendicular to an electric field, demonstrating that shear flow breaks ionic concentration symmetry to drive migration via coupled electrophoretic and diffusiophoretic effects dependent on the zeta potential and Dukhin number.

The Gist

Imagine you are watching a tiny, invisible particle floating in a drop of water inside a microscopic glass tube. Normally, if you turn on an electric current, this particle would simply swim straight along the current, like a boat following a river's flow. If you push the water with a pump, the particle would just drift with the current.

But scientists recently noticed something strange: when they turned on both the electric current and the water flow at the same time, the particle didn't just go straight or drift with the water. Instead, it started swimming sideways, perpendicular to both forces, moving toward the center of the tube or toward the walls depending on the conditions.

For a long time, people thought this sideways movement was caused by the "heaviness" or inertia of the water pushing the particle (like a heavy truck swerving in a strong wind). However, the math showed that the water was moving too slowly for that to be the cause. The mystery remained: Why was the particle moving sideways?

The New Explanation: A "Crowded" Party

The authors of this paper propose a new explanation based on how ions (tiny charged particles) behave around the main particle.

1. The Setup: The Electric Field and the "Crowd"
Think of the main particle as a celebrity at a party. The "ions" in the water are the fans.

• When you apply an electric field, the fans (ions) start moving. Because the celebrity (the particle) has a surface that conducts electricity, the fans crowd up on one side and leave the other side empty. This creates a "concentration polarization"—a lopsided crowd around the celebrity.
• In a still room (no water flow), this crowd is perfectly symmetrical left-to-right. The celebrity feels a push straight ahead, but no push to the side.

2. The Twist: The Shear Flow
Now, imagine the room starts to rotate or the air starts blowing in a specific way (this is the shear flow).

• As the air blows, it sweeps the "fans" (ions) around the celebrity.
• Because the air is blowing, the crowd on one side gets pushed differently than the crowd on the other. The perfect symmetry is broken. The "fans" are no longer evenly distributed around the celebrity.

3. The Result: The Sideways Drift
Because the crowd is now uneven, the pressure isn't balanced anymore.

• The imbalance creates a new force that pushes the celebrity sideways.
• The paper explains that this sideways speed comes from two sources working together:
  • The Electric Push: The uneven crowd creates a tiny, local electric field that pushes the particle sideways.
  • The Diffusion Push: The particle also tries to move away from the crowded side toward the empty side (like a person trying to find space in a room), which also pushes it sideways.

The Surprising Reversal

The most interesting part of their discovery is that the direction of this sideways swim depends on a specific number they call the Dukhin number. You can think of this number as a measure of how "sticky" or conductive the particle's surface is compared to the water.

• Low Conductivity (Low Dukhin Number): The particle swims one way (e.g., toward the center of the tube).
• High Conductivity (High Dukhin Number): As the particle's surface becomes more conductive, the direction flips. The particle suddenly swims the opposite way (e.g., toward the walls).

It's like a car that drives forward when the road is dry, but as the road gets wetter, it suddenly starts driving backward.

How Fast Does It Go?

The scientists calculated that for a particle about the size of a grain of sand (1 micrometer), this sideways speed is roughly micrometers per second. While that sounds slow, in the microscopic world of a tiny channel, it's fast enough to move a particle across the entire width of the channel in just a few seconds.

Summary

The paper solves a mystery by showing that when you mix an electric field with a flowing liquid, the flow distorts the cloud of charged ions around a particle. This distortion breaks the balance, creating a "sideways wind" that pushes the particle across the stream. The direction of this push depends on how conductive the particle's surface is, and it can even flip direction as that conductivity changes. This explains why particles in micro-channels behave in ways that simple physics couldn't previously predict.

Technical Summary

Technical Summary: Shear-Induced Electrophoretic Migration Perpendicular to the Electric Field

Problem Statement
Recent experimental observations in microchannels have demonstrated that microparticles subjected to simultaneous electric fields and pressure-driven flows undergo lateral migration perpendicular to the applied electric field. While inertial effects have been proposed as a potential mechanism, theoretical analyses suggest that inertial contributions are negligibly small under typical experimental regimes. Consequently, the physical mechanism driving this cross-stream migration remains an open question, particularly given that the phenomenon emerges only when electric and shear fields act simultaneously, rather than under flow alone.

Methodology
To address this gap, the authors extend previous theoretical work on concentration polarization (CP) to a system involving a dielectric particle with surface conductance subjected to both an external electric field and a linear shear flow. The study employs the macroscale model developed by Schnitzer and Yariv (SY model), which is valid for thin electrical double layers (EDLs) and large zeta potentials ($\zeta_0 \gg 25$ mV).

The theoretical framework involves:

• Governing Equations: The system is described by the convection-diffusion equation for ionic concentration, the continuity equation for electric current, and the Stokes equations for fluid velocity (including electrical body forces).
• Boundary Conditions: The model assumes a thin EDL where co-ions are excluded from the diffuse layer. Boundary conditions at the particle surface account for surface conductance (quantified by the Dukhin number, $Du$) and a slip velocity combining electrophoretic and diffusiophoretic effects.
• Asymptotic Analysis: The governing fields are expanded asymptotically in powers of the dimensionless electric field strength ($\beta$) and dimensionless shear rate ($\Gamma$). The authors focus on the cross-term ($u_{11}$), which is proportional to the product $\Gamma\beta$, as lower-order terms do not induce lateral migration.
• Velocity Calculation: Using the Lorentz reciprocal theorem for Stokes flows, the particle's translational velocity is derived as the surface average of the slip velocity. The calculation utilizes the method of matched asymptotic expansions to handle the concentration field, which fails under regular perturbation schemes.

Key Contributions and Results
The study derives an explicit analytical expression for the cross-stream migration velocity ($U_x$), revealing that the phenomenon is driven by the breaking of symmetry in the ionic concentration field around the particle due to shear advection. The resulting migration velocity comprises two distinct contributions:

1. Electrophoretic Component: Arising from the induced electric field along the direction perpendicular to the applied field.
2. Diffusiophoretic Component: Arising from the asymmetry in the induced ionic concentration gradients.

The derived non-dimensional velocity is expressed as:
$$\frac{U_x}{\Gamma\beta/D} = \frac{Du}{(1 + 2Du)^2} \left[ \frac{|\zeta_0|Du}{4} - (1 + Du) \ln(\cosh(\zeta_0/4)) \right]$$

Key findings include:

• Velocity Magnitude: The model predicts migration speeds on the order of micrometers per second ($\mu$m/s) for particles with radii around 1 $\mu$m, consistent with experimental timescales observed in microchannels.
• Directional Reversal: The theory predicts a reversal in the migration direction as the Dukhin number ($Du$) increases. For low $Du$, particles migrate toward regions of lower fluid velocity (channel walls in aligned fields), while for higher $Du$, they migrate toward regions of higher fluid velocity (channel center).
• Comparison with Inertia: For particles with radii $\approx 1$ $\mu$m, the predicted velocity due to concentration polarization significantly exceeds the theoretical predictions for inertial electrophoretic lift by Khair and Kabarowski.

Significance and Claims
The authors claim that this work identifies concentration polarization, modified by shear flow, as the primary physical mechanism for cross-stream electrophoretic migration, resolving the discrepancy between experimental observations and inertial theories. The model successfully explains the magnitude of observed velocities and qualitatively accounts for the reversal in migration direction reported in experiments by varying electrolyte conductivity (which alters the Dukhin number).

The paper concludes that while the theory aligns well with existing data, future work requires systematic experiments for rigorous quantitative comparison. The authors note that future studies must account for wall effects (specifically concentration-polarization electroosmosis) and extend the analysis to nanocolloids where the thin-EDL approximation is no longer valid.