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: Postdoc:

Sabri Kilic

, Damian Burch, JP Urbanski (Thorsen) 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 Antonio Ramos (Twente), (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 AC linear response (a) Gouy-Chapman (b) Stern model (c) Constant-phase-angle impedance

Z DL

 (

i



A

/  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).

Low frequency DC limit In-phase E field (insulator) Re(   ) At the “RC” frequency 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 m m 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 • Decreases with ion size, a ) • Flow vanishes around 10 mM • Decreases with ion valence, z Not predicted by the theory QuickTime™ and a DV/DVCPRO - NTSC decompressor are needed to see this picture.

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 Stepped electrodes, symmetric footprint Low-voltage theory always predicts a single peak of “forward” pumping

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

C = 10 mM C = 1 mM

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 = 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 Reversal at high frequency?

Concentration decay?

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 layers of width  2 = 1-100nm << L?

Debye time,  2 / 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). Simulations of U vs log(V) and log(freq): Nonlinear DL capacitance shifts flow to low frequency 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 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

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). 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

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.

Related physics: Carbon nanotube ultracapacitor (Schindall/Signorelli, MIT) Induced-charge electro-osmosis Papers, slides: http://math.mit.edu/~bazant