Paris-Sciences Chair Lecture Series 2008, ESPCI
Induced-Charge Electrokinetic Phenomena
Martin Z. Bazant
Department of Mathematics, MIT
ESPCI-PCT & CNRS Gulliver
1. Introduction (7/1)
2. Induced-charge electrophoresis in colloids (10/1)
3. AC electro-osmosis in microfluidics (17/1)
4. Theory at large applied voltages (14/2)
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
Outline
1. Experimental puzzles
2. Strongly nonlinear dynamics
3. 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 model for the induced zeta potential
Bulk resistor (Ohm’s law):
Double-layer BC:
2. Stokes flow driven by ICEO slip
(a) Gouy-Chapman
(b) Stern model
(c) Constant-phase-angle impedance
Z DL 
AC linear response
A
(i / 0 ) 
0.6-0.8
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).
At the “RC” frequency
Low frequency DC limit
In-phase E field (insulator)
- Re()
Normal current
Out-of-phase E (negligible)
- Im()
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).
Not predicted by the theory
QuickTime™ and a
DV/DVCPRO - NTSC decompressor
are needed to see this picture.
ICEO flow around a gold post
in “large fields” (Ea = 1 Volt)
• Flow vanishes around 10 mM
• Decreases with ion size, a
• Decreases with ion valence, z
Induced-charge electrophoresis
of 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)
KCl, 3 Vpp, planar pump
Reversal at high frequency?
Concentration decay?
Urbanski et al, Appl Phys Lett (2006)
De-ionized water (pH = 6)
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
1. Experimental puzzles
2. Strongly nonlinear dynamics
3. 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
2
layers of width
 = 1-100nm << L?
2
Debye time,  / D ?
2
Diffusion time, L / D ?
Equivalent Circuit Approximation
Answer:
What about nonlinear response? Few models…
Electrokinetics in a dilute electrolyte
Poisson-Nernst-Planck equations
point-like ions
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).
Nonlinear DL capacitance
shifts flow to low frequency
Simulations of U vs log(V) and log(freq):
Faradaic reactions
“short circuit” the flow
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 doublelayer 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
1. Experimental puzzles
2. Strongly nonlinear dynamics
3. Beyond dilute solution theory
Breakdown of Poisson-Boltzmann theory
• At high voltage, Boltzmann statistics predict unphysical
surface concentrations, even in very dilute bulk solutions:
Packing limit
Impossible!
• 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.
Crucial new physics:
Ion crowding at large voltages
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).
Olesen, Bazant, Bruus, in preparation (2008).
Sudden DC voltage
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
steric effects
Example 2:
Reversal of planar
ACEO pumps
log V
Storey, Edwards, Kilic, Bazant
Phys. Rev. E to appear (2008)
log(frequency)
A. Large electrode wins
(since it has time to charge)
B. Small electrode wins
(since it charges faster at high V)
Towards better models
Bazant, Kilic, Storey, Ajdari (2007, 2008)
• Bikerman’s lattice-based
MPB model under-estimates
steric effects in a liquid
• Can use better models
for ion chemical potentials
Biesheuvel, van Soestbergen (2007)
• Still need a>1nm to fit
experimental flow reversal
Storey, Edwards, Kilic, Bazant (2008)
Model using Carnahan-Starling
entropy for hard-sphere liquid
• Steric effects alone cannot
predict strong decay of
flow at high concentration…
Crowding effects on electro-osmotic slip
Bazant, 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
Generic effect: Saturation of induced zeta
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.
Induced-charge electro-osmosis
Related physics: Carbon nanotube
ultracapacitor (Schindall/Signorelli, MIT)
Papers, slides: http://math.mit.edu/~bazant