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### Induced-Charge Electro-osmosis and Electrophoresis

Nonlinear Electrokinetics @ MIT

Students:Jeremy Levitan (ME PhD’05),

Kevin Chu (Math PhD’05), JP Urbanski (ME),

Mustafa Sabri Kilic, Sergiy Sidenko (Math)

Postdocs: Yuxing Ben, Hongwei Sun (Math)

Faculty: Todd Thorsen (ME), Martin Schmidt (EE)

Visitors: Armand Ajdari, Vincent Studer (ESPCI)

Collaborators: Todd Squires (UCSB),

Shankar Devasenathipathy (Stanford)

Howard Stone (Harvard)

Martin Z. Bazant

Department of Mathematics & Institute for Soldier Nanotechnologies, MIT

Funding: US Army Research Office

(Contract DAAD-19-02-002) and

MIT-France Program

ICEO in a microfluidic device.

The Electrochemical Double Layer

+

+

+

neutral

bulk

electrolyte

solid

Electrostatic potential

Ion concentrations

0

continuum region

Electrokinetic Phenomena

Helmholtz-Smoluchowski fluid “slip” formula:

Electro-osmosis

Electrophoresis

The classical theory assumes that the “zeta potential” z (or charge density q) is a constant material property, but what happens at a polarizable (e.g. electrode) surface?

Diffuse-Charge Dynamics

Bazant, Thornton, Ajdari, Phys. Rev. E. (2004).

Analysis of the Poisson-Nernst-Planck equations

by time-dependent matched asymptotic expansions.

Model Problem

Classical “equivalent circuit” in

the thin-double-layer approximation

Time scales

AC Electro-osmosis

Ramos et al., JCIS (1999); Ajdari, Phys. Rev. E (2000)

Steady flow for

AC period =

How general is this phenomenon?

Need electrode arrays? Need “AC”?

“Induced-Charge Electro-osmosis”

= nonlinear electro-osmotic slip at a polarizable surface

Bazant & Squires, Phys, Rev. Lett. 92, 0066101 (2004).

Example: An uncharged metal cylinder in a suddenly applied DC field

Same effect for metals & dielectrics, DC & AC fields…

Double-layer polarization and ICEO flow

A conducting cylinder in a suddenly applied uniform E field.

Electric field ICEO velocity

FEMLAB simulation by Yuxing Ben

Poisson-Nernst-Planck/Navier-Stokes eqns

l/a=0.005

Experimental Observation of ICEO

J. A. Levitan, S. Devasenathipathy, V. Studer, Y. Ben, T. Thorsen, T. M. Squires, & M. Z. Bazant,

Colloids and Surfaces (2005)

100 mm Pt wire

on channel wall

Viewing plane

PDMS

polymer

microchannel

Bottom view

of optical slice

Inverted optics

microscope

Micro-particle image

velocimetry (mPIV) to

map the velocity profile

“Induced-Charge Electrokinetic Phenomena”

1. Prior examples of “ICEO”

- Electro-osmotic flows around metal particles
- Dielectrophoresis of spheres in electrolytes (“dipolophoresis”)
- AC electro-osmosis & colloidal aggregation at electrodes
- DC “electrokinetic jet” at a microchannel corner

Gamayunov, Murtsovkin, Dukhin, Colloid J. USSR (1986); Levich (1960)

Simonova, Shilov, Colloid J. USSR (1981, 1998)

Ramos et al. (1998); Ajdari (2000); “EHD” Ristenpart, Saville (2004)…

Thamida & Chang (2002)

2. Some new examples - breaking symmetries

- ICEO pumps and mixers in microfluidics
- “Fixed-potential ICEO”
- “Induced-charge electrophoresis” (ICEP) particle motion

Bazant & Squires, PRL (2004); Levitan et al. Colloids & Surfaces (2005).

Squires & Bazant, JFM (2004); Levitan, PhD thesis MIT (2005).

Bazant & Squires, PRL (2004); Yariv, Phys. Fluids (2005);

Squires & Bazant, JFM (2006); Saintillon, Darve & Shaqfeh, preprint.

“Fixed-Potential ICEO”

Squires & Bazant, J. Fluid Mech. (2004)

Idea: Vary the induced

total charge in phase

with the local field.

Generalizes “Flow FET” of

Ghowsi & Gale, J. Chromatogr. (1991)

Example: metal cylinder grounded to an electrode supplying an AC field.

Fixed-potential ICEO mixer

ICEO Microfluidic Elements

J. A. Levitan, Ph.D. Thesis (2005).

Fixed-potential ICEO “pump”

(u = 3 mm/sec)

ICEO “mixer” or “trap”

(u = 0.2 mm/sec)

E = 100V/cm (< 10 Volt), 300 Hz AC, 0.1 mM KCl, 0.5 mm fluorescent tracers

50-250 mm electroplated gold posts, PDMS polymer microchannels

A promising platform for portable microfluidics…

“Induced-Charge Electrophoresis”= ICEO swimming via broken symmetries

Bazant & Squires, Phys. Rev. Lett. (2004); Yariv, Phys. Fluids (2005).

I. Heterogeneous Surfaces

Squires & Bazant, J. Fluid Mech. (2006).

A metal sphere with a partial dielectric

coating swims toward its coated end,

which rotates to align perpendicular to E.

An “ICEO pinwheel” rotates to align and

spins continuously in a uniform AC field!

Stable

Unstable

ICEP II. Asymmetric Shapes

Squires & Bazant, J. Fluid Mech. (2006).

ICEP can separate polarizable colloids by shape

and size in a uniform DC or AC electric field,

while normal (linear) electrophoresis cannot.

- long axis rotates to align with E
- a “thin arrow” swims parallel to E,
- towards its “blunt” end
- a “fat arrow” swims transverse to E
- towards its “pointed” end

Perturbation analysis

E u

An asymmetric metal post

can pump fluid in any direction

in a uniform DC or AC field, but

ICEO flow has quadrupolar rolls,

very different from normal EOF.

FEMLAB finite-element simulation (Yuxing Ben)

ICEP III. Non-uniform Fields

Shilov & Simonova, Colloid J. USSR (1981, 2001). Metal sphere “dipolophoresis”

Squires & Bazant, J. Fluid Mech. (2006). General problem of DEP + ICEP

- Must include electrostatic force and torque (Maxwell stress tensor)
- Dielectrophoresis (DEP) + ICEP
- For metals, ICEP points up, and DEP down, an electric field gradient
- ICEP cancels DEP for a metal sphere (but not a cylinder or other shapes)

Electric Field

Fluid Streamlines

General solution for any 2d shape in any non-uniform E field bycomplex analysis…

Electric Field

Fluid Streamlines

“Weakly Nonlinear” Theory of ICEO

Gamayunov et al. (1986); Ramos et al. (1998); Ajdari (2000); Squires & Bazant (2004).

1. Equivalent-circuit modelfor the induced zeta potential

Bulk resistor (Ohm’s law):

Double-layer BC:

Double-layer circuit elements:

Gouy-Chapman capacitor

Stern model

Constant-phase-angle impedance

2. Stokes flow driven by ICEO slip

b=0.6-0.8

Dimensionless BC for AC forcing

Green et al, Phys Rev E (2002)

Levitan et al. Colloids & Surf. (2005)

FEMLAB simulation of our first experiment:ICEO around a 100 micron platinum wire in 0.1 mM KCl

Levitan, ... Y. Ben,… Colloids and Surfaces (2005).

Low frequency DC limit

At the “RC” frequency

Electric field lines:

Electric Field lines

Electric field lines

Electric field lines

Velocity fields

Velocity fields

Comparision of Simulation and PIV Data:Velocity Profiles

Raw data from a slice

0-10 mm above the wire

Data collapse when scaled to

characteristic ICEO velocity

- Scaling and flow profile consistent with ICEO theory
- Flow magnitude roughly 2 times smaller than in simple theory
- Need better theories for large voltages and varying solution chemistry…

Theory of “strongly nonlinear” electrokinetics?

Use the basic methods of applied mathematics:

(Analysis) Solve the existing equations in a new regime.

This leads to some interesting new effects, but does not explain all

the experimental data (e.g. decrease in ICEO flow for C > 10 mM).

More importantly, the solutions contain physical nonsense!

(Modeling) Postulate new equations, solve & compare to experiments.

This is now the only choice, and progress is underway.

Classical Equations of “Dilute Solution Theory”

Poisson-Nernst-Planck ion transport equations

Singular perturbation

Navier-Stokes fluid equations with electrostatic stresses

Strongly Nonlinear Solutions to the Classical Equations

1. Breakdown of circuit models: Surface adsorption and bulk diffusion

Bazant, Thornton, Ajdari, PRE (2004).

2. Tangential transport of ions in the double layer

Bikerman (1933), SS Dukhin & Deryaguin (1969, 1974)

Linear theory for small E, highly charged surfaces

Kevin Chu, Ph.D. thesis (2005).

Nonlinear theory for large E, uncharged conductors

3. Diffusio-osmosis (= flow due to gradients

in bulk salt concentration)

Deryaguin (1964)

Bulk diffusion around an

uncharged metal sphere

in a uniform E field.

Modified Equations for Electrokinetics

Sabri Kilic, Bazant, Ajdari, in preparation.

1. Steric effects (finite ion size) on equilibrium:

Modified Poisson-Boltzmann equation

PB = Poisson-Boltzmann theory

Borukhov et al. Phys. Rev. Lett. (1997).

2. Steric effects on dynamics:

Modified Nerst-Planck equations

Steric & viscoelectric effects on electro-osmosis:

Modified Helmholtz-Smoluchowski slip formula

4. Steric & viscoelectric effects on ICEO…

New prediction: An uncharged metal sphere will move by ICEP

in a large uniform field, if the electrolyte is asymmetric.

Engineering of Microfluidic Pumps

JP Urbanski, Levitan, Bazant, Thorsen, in preparation

- Exploit fixed-potential ICEO, and standard ACEO
- Electroplated interdigitated & recessed gold electrodes on glass
- PDMS soft lithography for microchannels

Fast AC Electrokinetic Pumps

Bazant, Ben (2006)

The “conveyor belt principle”: Raised pumping surfaces, recess reverse rolls.

Apply to periodic array of electrodes in existing ACEO pumps

Raise half of each electrode to make a fast pump

Ramos et al (1999), Ajdari (2000)

Optimization of ICEO/ACEO pumps

Bazant, Yuxing Ben (2005)

Fastest existing ACEO pump

Green et al. (2003) theory;

Studer et al. (2004) expt.

New design:

10 times faster!

ICEO: a platform for portable microfluidics?

- State-of-the-art “table-top microfluidics”
- Pressure-driven microfluidics (e.g. K. Jensen)
- Capillary electro-osmosis (e.g. J. Santiago)
- Soft microfluidic networks (e.g S. Quake)
- Possible advantages of ICEO:
- Low voltage (< 10 Volt), low power (< 1 mW)
- AC (< kHz) reduces unwanted reactions / bubbles in linear EOF
- Time-dependent local flow control for mixing, trapping, switching,…
- Excellent scaling with miniaturization
- Standard “hard” microfabrication methods
- Possible disadvantages:
- Requires low ionic strength (< 10 mM)
- Sensitive to solution chemistry, surface contamination

http://www.physics.ubc.ca/~chansen/

Engineering Applications of ICEO

Commercial Applications1. Battery-powered microfluidics

- Portable/implantable devices for medical or chemical monitoring
- Localized drug delivery
- Pressure control (e.g. glaucoma)
- Cooling portable electronics

Example: on-field detection of exposure to biowarfare agents for the dismounted soldier by monitoring nanoliters of blood.

(T. Thorsen @ MIT Mech Eng)

- 2. Polarizable colloids
- ICEO flows in dielectrophoresis
- ICEO manipulation of nanobarcodes(Santiago, Shaqfeh @ Stanford Mech Eng)

www.studybusiness.com

ICEO & ICEP

From mathematical theory….

to scientific experiments and engineering applications.

http://math.mit.edu/~bazant/ICEO

ICEO microfluidic pumps without moving parts

Jeremy Levitan, Ph.D. thesis, Mechanical Engineering MIT (2005)

- Experimental fabrication: soft lithography for micro-channels (50-200 mm) and electroplating for gold structures (25-200 mm wide, 5-50 mm tall) on glass

Deposit and pattern gold

on glass wafer

Electroplate gold

Strip resist; cap with PDMS

to form micro-channel

Deposit and pattern

thick resist mold

Comparision of Simulation and PIV Data:Scaling with Voltage and Frequency

Similar ”ICEO flow” observed around mercury drops

(without any quantitative analysis):

Gamayunov, Mantrov, Murtsovkin, Colloid J. USSR (1992)

“Strongly Nonlinear” Solutions(as required by the experimental parameters)

- Breakdown of circuit models at “large” voltages
- when V > 2 kT/e = 0.05 V (z=V)

“Transient Dukhin number”

Bazant, Thornton & Ajdari, Phys. Rev. E 70, 021506 (2004).

1d model problem

(PNP equations)

V = 4 kT/e

potential charge density salt concentration

Neutral salt adsorption by the diffuse charge layer and bulk diffusion

Towards a new mathematical model…

1. Anolmalous “constant phase angle” double-layer impedance

Data suggests BC for power-law

“fractional relaxation”:

Hypothesis: long waiting times

for Stern-layer adsorption

(not fractal surface roughness)

KCl/Au expt

By J. Levitan

2. Strong dependence on surface and solution chemistry

ICEO flow decreases with concentration

and depends on ion valence, size,…

Hypothesis: steric effects +

variable viscosity in the Stern layer

Borukhov et al

Phys Rev Lett (1997)

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