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### Induced-Charge Electrokinetic Phenomena

### Induced-charge electrokinetics: Theory

Paris-Sciences Chair Lecture Series 2008, ESPCI

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

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

Outline

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

Induced-Charge Electro-osmosis

= 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

Low-voltage “weakly nonlinear” theory

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)

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

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…

Flow depends on solution chemistry

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

Induced-charge electrophoresisof metallo-dielectric Janus particles

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

Similar concentration dependence for velocity of Janus particles in NaCl

Apparent scaling for C > 0.1 mM

(or perhaps power-law decay;

need more experiments…)

AC electro-osmotic pumps: Theory

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

Low-voltage experimental data

- 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

High-voltage data

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

More puzzling high-voltage data

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?

Faradaic reactions

- 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

Outline

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

The simplest problem of diffuse-charge dynamics

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

Electrokinetics in a dilute electrolyte

point-like ions

Poisson-Nernst-Planck equations

Singular perturbation

Navier-Stokes equations with electrostatic stress

“Weakly Nonlinear” Charging Dynamics

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

Weakly nonlinear AC electro-osmosis

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…

“Strongly Nonlinear” Charging Dynamics

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

The simplest problem in d>1

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

Strongly nonlinear electrokinetics

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…

Outline

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

Breakdown of Poisson-Boltzmann 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!

Steric effects in equilibrium

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

Steric effects on electrolyte dynamics

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

Steric effects on diffuse-layer relaxation

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.

Example 1:Field-dependent mobility of charged metal particles

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

Example 2:Reversal of planar ACEO pumps

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)

Towards better models

- 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

Crowding effects on electro-osmotic slipBazant, Kilic, Storey, Ajdari (2007, 2008), arXiv:cond-mat/0703035v2

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

Example: Ion-specific electrophoretic mobility

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-)

Electrokinetics at large voltages

- 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

Conclusion

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

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