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Induced-Charge Electrokinetic Phenomena

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Paris-Sciences Chair Lecture Series 2008, ESPCI

Induced-Charge Electrokinetic Phenomena

Introduction (7/1)

Induced-charge electrophoresis in colloids (10/1)

AC electro-osmosis in microfluidics (17/1)

Theory at large applied voltages (14/2)

Martin Z. Bazant

Department of Mathematics, MIT

ESPCI-PCT & CNRS Gulliver

Acknowledgments

Induced-charge electrokinetics: Theory

CURRENT

Students:Sabri Kilic, Damian Burch,

JP Urbanski (Thorsen)

Postdoc: Chien-Chih Huang

Faculty: Todd Thorsen (Mech Eng)

Collaborators: Armand Ajdari (St. Gobain)

Brian Storey (Olin College)

Orlin Velev (NC State), Henrik Bruus (DTU)

Maarten Biesheuvel (Twente),

Antonio Ramos (Sevilla)

FORMER

PhD: Jeremy Levitan, Kevin Chu (2005),

Postodocs: Yuxing Ben, Hongwei Sun (2004-06)

Interns: Kapil Subramanian, Andrew Jones,

Brian Wheeler, Matt Fishburn

Collaborators: Todd Squires (UCSB),

Vincent Studer (ESPCI), Martin Schmidt (MIT),

Shankar Devasenathipathy (Stanford)

- Funding:
- Army Research Office
- National Science Foundation
- MIT-France Program
- MIT-Spain Program

- Experimental puzzles
- Strongly nonlinear dynamics
- Beyond dilute solution theory

= nonlinear electro-osmotic slip at a polarizable surface

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

Gamayunov, Murtsovkin, Dukhin, Colloid J. USSR (1986) - flow around a metal sphere

Bazant & Squires, Phys, Rev. Lett. (2004) - theory, broken symmetries, microfluidics

Gamayunov et al. (1986); Ramos et al. (1999); Ajdari (2000); Squires & Bazant (2004).

1. Equivalent-circuit modelfor the induced zeta potential

Bulk resistor (Ohm’s law):

Double-layer BC:

Gouy-Chapman

Stern model

Constant-phase-angle impedance

2. Stokes flow driven by ICEO slip

b=0.6-0.8

AC linear response

Green et al, Phys Rev E (2002)

Levitan et al. Colloids & Surf. (2005)

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

Low frequency DC limit

At the “RC” frequency

In-phase E field (insulator)

Normal current

Out-of-phase E (negligible)

Induced dipole

Time-averaged velocity

Theory vs experiment at low salt concentration

Levitan et al (2005)

Horiz. velocity from a slice

10 mm above the wire

Data collapse when scaled to

characteristic ICEO velocity

- Scaling and flow profile consistent with theory
- Velocity is 3 times smaller than expected (no fitting)
- BUT this is only for dilute 0.1 mM KCl…

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

- ICEO flow around a gold post
- in “large fields” (Ea = 1 Volt)
- Flow vanishes around10 mM
- Decreases with ion size, a
- Decreases with ion valence, z

Not predicted by the theory

S. Gangwal, O. Cayre, MZB, O.Velev, Phys Rev Lett (2008)

Apparent scaling for C > 0.1 mM

(or perhaps power-law decay;

need more experiments…)

Bazant & Ben (2006)

Planar electrode array. Brown, Smith & Rennie (2001).

Same geometry with raised steps

Low-voltage theory always predicts

a single peak of “forward” pumping

Stepped electrodes, symmetric footprint

- Brown et al (2001), water
- straight channel
- planar electrode array
- similar to theory (0.2-1.2 Vrms)

Reproduced in < 1 mM KCl

Studer 2004

Urbanski et al 2006

V. Studer et al. Analyst (2004)

- Dilute KCl
- Planar electrodes, unequal sizes & gaps
- Flow reverses at high frequency
- Flow effectively vanishes > 10 mM.

C = 10 mM

C = 1 mM

C = 0.1 mM

Bazant et al, MicroTAS (2007) Urbanski et al, Appl Phys Lett (2006)

KCl, 3 Vpp, planar pump

De-ionized water (pH = 6)

Reversal at high frequency?

Concentration decay?

Double peaks?

- Ajdari (2000) predicted weak low-frequency flow reversal
in planar ACEO pumps due to Faradaic (redox) reactions

- Observed by Gregersen et al (2007)
- Lastochkin et al (2004) attributed high frequency ACEO reversal
to reactions, but gave no theory

- Olesen, Bruus, Ajdari (2006) could not predict realistic
ACEO flows with linearized Butler-Volmer model of reactions

- Wu et al (2005) used DC bias + AC to reverse ACEO flow
- Still no mathematical theory

Wu (2006) ACEO trapping e Coli bacteria with DC bias

- Experimental puzzles
- Strongly nonlinear dynamics
- Beyond dilute solution theory

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

A sudden voltage across parallel-plate blocking electrodes.

What is the time

to charge thin double

layers of width

= 1-100nm << L?

2

2

Debye time, / D ?

Diffusion time, L / D ?

2

Answer:

What about nonlinear response? Few models…

point-like ions

Poisson-Nernst-Planck equations

Singular perturbation

Navier-Stokes equations with electrostatic stress

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

Derive by boundary-layer analysis

(matched asymptotic expansions)

Ohm’s Law in the neutral bulk

Effective “RC” boundary condition

Olesen, Bruus, Ajdari, Phys. Rev. E (2006). Simulations of U vs log(V) and log(freq):

Faradaic reactions

“short circuit” the flow

Nonlinear DL capacitance

shifts flow to low frequency

Classical models fail…

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

New effect: neutral salt adsorption by the double layers

depletes the nearby bulk solution and couples double-layer charging to slow bulk diffusion

Chu & Bazant, Phys Rev E (2006).

A metal cylinder/sphere in a sudden uniform E field

- Surface conduction through
double layers sets in at same

time as bulk salt adsorption

- yields recirculating current

Dukhin (Bikerman) number

Laurits Olesen, PhD Thesis, DTU (2006)

Some new effects

- Surface conduction “short circuits” double-layer charging
- Diffusio-osmosis & bulk electroconvection oppose ACEO
- Space-charge and “2nd kind” electro-osmotic flow

- BUT
- Even fully nonlinear Poisson-Nernst-Planck-Smoluchowski
theory does not agree with experiment

- No high-frequency flow reversal & concentration effects
It seems time to modify the fundamental equations…

- Experimental puzzles
- Strongly nonlinear dynamics
- Beyond dilute solution theory

- Stern (1924) introduced a cutoff distance for closest approach of ions to a charged surface, but this does not fix the problem or describe crowding dynamics.

- At high voltage, Boltzmann statistics predict unphysical
surface concentrations, even in very dilute bulk solutions:

Packing limit

Impossible!

Crucial new physics:

Bikerman (1942); Dutta, Indian J Chem (1949);

Wicke & Eigen, Z. Elektrochem. (1952)

Iglic & Kral-Iglic, Electrotech. Rev. (Slovenia) (1994).

Borukhov, Andelman & Orland, Phys. Rev. Lett. (1997)

Modified Poisson-Boltzmann equation

a = minimum ion spacing

- Minimize free energy, F = E-TS
- Mean-field electrostatics
- Continuum approx. of lattice entropy
- Ignore ion correlations, specific forces, etc.

Borukhov et al. (1997)

Large ions, high concentration

“Fermi-Dirac”

statistics

Kilic, Bazant, Ajdari, Phys. Rev. E (2007). Sudden DC voltage

Olesen, Bazant, Bruus, in preparation (2008). Large AC voltage (steady response)

Chemical potentials, e.g. from a lattice model (or liquid state theory)

dilute solution theory + entropy of solvent (excluded volume)

Modified Poisson-Nernst-Planck equations

1d blocking cell, sudden V

Kilic, Bazant, Ajdari, Phys. Rev. E (2007).

Exact formulae for Bikerman’s MPB model (red) and simpler Condensed Layer Model (blue) are in the paper.

All nonlinear effects are suppressed by steric constraints:

- Capacitance is bounded, and decreases at large potential.
- Salt adsorption (Dukhin number) cannot be as large for thin diffuse layers.

Bazant, Kilic, Storey, Ajdari,

in preparation (2008)

AS Dukhin (1993) predicted the

effect for small E.

PB predicts no motion in large E:

Opposite trend

for steric models

steric effects

log V

Storey, Edwards, Kilic, Bazant

Phys. Rev. E to appear (2008)

log(frequency)

Large electrode wins

(since it has time to charge)

B. Small electrode wins

(since it charges faster at high V)

- Bikerman’s lattice-based
MPB model under-estimates

steric effects in a liquid

- Can use better models
for ion chemical potentials

- Still need a>1nm to fit
experimental flow reversal

- Steric effects alone cannot
predict strong decay of

flow at high concentration…

Bazant, Kilic, Storey, Ajdari (2007, 2008)

Biesheuvel, van Soestbergen (2007)

Storey, Edwards, Kilic, Bazant (2008)

Model using Carnahan-Starling

entropy for hard-sphere liquid

Electro-osmotic mobility for variable viscosity and/or permittivity:

1. Lyklema, Overbeek (1961): viscoelectric effect

2. Instead, assume viscosity diverges at close packing (jamming)

Modified slip formula depends on polarity and composition

Can use with any MPB model;

Easy to integrate for Bikerman

ICEP of a polarizable uncharged sphere in asymmetric electrolyte

Larger

cations

Divalent

cations

Mobility in large DC fields:

Also may explain double peaks in water ACEO (H+, OH-)

- Steric effects (more accurate models, mixtures)
- Induced viscosity increase

- Electrostatic correlations (beyond the mean-field approximation)

- Solvent structure, surface roughness (effect on ion crowding?)
- Faradaic reactions, specific adsorption of ions
- Dielectric breakdown?
- Strongly nonlinear dynamics with modified equations

MORE EXPERIMENTS & SIMULATIONS NEEDED

Nonlinear electrokinetics is a frontier of

theoretical physics and applied mathematics

with many possible applications in engineering.

Related physics: Carbon nanotube

ultracapacitor (Schindall/Signorelli, MIT)

Induced-charge electro-osmosis

Papers, slides: http://math.mit.edu/~bazant