Induced-Charge Electro-osmosis and Electrophoresis

Martin Z. BazantDepartment of Mathematics & Institute for Soldier Nanotechnologies, MIT

Nonlinear Electrokinetics @ MITStudents: 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)

ICEO in a microfluidic device. Funding: US Army Research Office(Contract DAAD-19-02-002) andMIT-France Program

The Electrochemical Double Layer

neutralbulkelectrolyte

+

+

+

solid

Ion concentrations

0 continuum region

Electrostatic potential

Electrokinetic Phenomena

Helmholtz-Smoluchowski fluid “slip” formula:

Electro-osmosis Electrophoresis

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

Diffuse-Charge DynamicsBazant, Thornton, Ajdari, Phys. Rev. E. (2004).

Analysis of the Poisson-Nernst-Planck equations by time-dependent matched asymptotic expansions.

Model Problem

Classical “equivalent circuit” inthe thin-double-layer approximation

Time scales

AC Electro-osmosisRamos et al., JCIS (1999); Ajdari, Phys. Rev. E (2000)

Steady flow forAC period =

How general is this phenomenon? Need electrode arrays? Need “AC”?

“Induced-Charge Electro-osmosis”

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

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

= nonlinear electro-osmotic slip at a polarizable surface

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

Double-layer polarization and ICEO flow

Electric field ICEO velocity

FEMLAB simulation by Yuxing BenPoisson-Nernst-Planck/Navier-Stokes eqns/a=0.005

A conducting cylinder in a suddenly applied uniform E field.

Experimental Observation of ICEO

PDMSpolymermicrochannel

100 m Pt wireon channel wall

Inverted opticsmicroscope

Viewing plane

Bottom viewof optical slice

J. A. Levitan, S. Devasenathipathy, V. Studer, Y. Ben, T. Thorsen, T. M. Squires, & M. Z. Bazant,Colloids and Surfaces (2005)

Micro-particle imagevelocimetry (PIV) tomap the velocity profile

Movie: Optical slice sweeping through the 100 m Pt wire

QuickTime™ and aDV/DVCPRO - NTSC decompressor

are needed to see this picture.

“Induced-Charge Electrokinetic Phenomena”

• 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)

1. Prior examples of “ICEO”

Thamida & Chang (2002)

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

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

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”

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

Fixed-potential ICEO mixer

Idea: Vary the induced total charge in phase with the local field.

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

Generalizes “Flow FET” ofGhowsi & Gale, J. Chromatogr. (1991)

ICEO Microfluidic Elements

E = 100V/cm (< 10 Volt), 300 Hz AC, 0.1 mM KCl, 0.5 m fluorescent tracers50-250 m electroplated gold posts, PDMS polymer microchannels

ICEO “mixer” or “trap” (u = 0.2 mm/sec)

Fixed-potential ICEO “pump”(u = 3 mm/sec)

A promising platform for portable microfluidics…

QuickTime™ and aDV/DVCPRO - NTSC decompressor

are needed to see this picture.

QuickTime™ and aDV/DVCPRO - NTSC decompressor

are needed to see this picture.

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

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

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

Stable Unstable

A metal sphere with a partial dielectriccoating swims toward its coated end,which rotates to align perpendicular to E.

An “ICEO pinwheel” rotates to align andspins continuously in a uniform AC field!

I. Heterogeneous Surfaces Squires & Bazant, J. Fluid Mech. (2006).

ICEP II. Asymmetric Shapes

- 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

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

ICEP can separate polarizable colloids by shapeand size in a uniform DC or AC electric field,while normal (linear) electrophoresis cannot.

An asymmetric metal postcan pump fluid in any directionin a uniform DC or AC field, but ICEO flow has quadrupolar rolls,very different from normal EOF.

Perturbation analysisE u

FEMLAB finite-element simulation (Yuxing Ben)

ICEP III. Non-uniform Fields

• 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)

Shilov & Simonova, Colloid J. USSR (1981, 2001). Metal sphere “dipolophoresis”Squires & Bazant, J. Fluid Mech. (2006). General problem of DEP + ICEP

Electric Field Fluid Streamlines

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

Electric Field Fluid Streamlines

“Weakly Nonlinear” Theory of ICEO

1. Equivalent-circuit model for the induced zeta potential

2. Stokes flow driven by ICEO slip

βωω )/( 0i

AZDL =

Bulk resistor (Ohm’s law):

Double-layer BC:

Double-layer circuit elements:(a) Gouy-Chapman capacitor(b) Stern model (c) Constant-phase-angle impedance

Green et al, Phys Rev E (2002)Levitan et al. Colloids & Surf. (2005)

β

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

Dimensionless BC for AC forcing

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

Low frequency DC limit At the “RC” frequencyElectric field lines:

Velocity fields

Electric Field lines

Velocity fields

Electric field lines Electric field lines

)Re( Φ∇−

)Im( Φ∇−

)Re( Φ∇−

)Im( Φ∇−

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

Comparision of Simulation and PIV Data:Velocity Profiles

• 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…

Raw data from a slice0-10 m above the wire

Data collapse when scaled tocharacteristic ICEO velocity

Theory of “strongly nonlinear” electrokinetics?

Use the basic methods of applied mathematics:

1. (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

Navier-Stokes fluid equations with electrostatic stresses

Singular perturbation

Strongly Nonlinear Solutions to the Classical Equations

2. Tangential transport of ions in the double layer

Kevin Chu, Ph.D. thesis (2005).Nonlinear theory for large E, uncharged conductors

Bikerman (1933), SS Dukhin & Deryaguin (1969, 1974)Linear theory for small E, highly charged surfaces

Bulk diffusion around an uncharged metal spherein a uniform E field.

3. Diffusio-osmosis (= flow due to gradients in bulk salt concentration)

Deryaguin (1964)

1. Breakdown of circuit models: Surface adsorption and bulk diffusionBazant, Thornton, Ajdari, PRE (2004).

Modified Equations for Electrokinetics

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

Sabri Kilic, Bazant, Ajdari, in preparation.

3. 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 ICEPin a large uniform field, if the electrolyte is asymmetric.

Engineering of Microfluidic PumpsJP 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 PumpsBazant, Ben (2006)

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

Apply to periodic array of electrodes in existing ACEO pumps

Ramos et al (1999), Ajdari (2000) Raise half of each electrode to make a fast pump

Optimization of ICEO/ACEO pumps

Fastest existing ACEO pumpGreen et al. (2003) theory; Studer et al. (2004) expt.

Bazant, Yuxing Ben (2005)

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/

Commercial Applications1. Battery-powered microfluidics• Portable/implantable devices for

medical or chemical monitoring• Localized drug delivery• Pressure control (e.g. glaucoma)• Cooling portable electronics

Engineering Applications of ICEO

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 & ICEPFrom mathematical theory….

to scientific experiments and engineering applications.

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

Deposit and pattern gold on glass wafer

Deposit and pattern thick resist mold

Electroplate gold

Strip resist; cap with PDMS to form micro-channel

ICEO microfluidic pumps without moving partsJeremy Levitan, Ph.D. thesis, Mechanical Engineering MIT (2005)

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

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)

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

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

1d model problem(PNP equations)

potential charge density salt concentration

V = 4 kT/e

“Transient Dukhin number”

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 timesfor Stern-layer adsorption(not fractal surface roughness)

2. Strong dependence on surface and solution chemistry

ICEO flow decreases with concentrationand depends on ion valence, size,…

Hypothesis: steric effects + variable viscosity in the Stern layer

Borukhov et al Phys Rev Lett (1997)

KCl/Au exptBy J. Levitan