Induced-Charge Electrokinetic Phenomena

Martin Z. BazantDepartment of Mathematics, MIT

ESPCI-PCT & CNRS Gulliver

Paris-Sciences Chair Lecture Series 2008, ESPCI

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)

Induced-charge electrokinetics: Theory

CURRENTStudents: 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)

FORMERPhD: Jeremy Levitan, Kevin Chu (2005),Postodocs: Yuxing Ben, Hongwei Sun (2004-06)Interns: Kapil Subramanian, Andrew Jones, Brian Wheeler, Matt FishburnCollaborators: 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

Acknowledgments

Outline

1. Experimental puzzles

2. Strongly nonlinear dynamics

3. Beyond dilute solution theory

Induced-Charge Electro-osmosis

Gamayunov, Murtsovkin, Dukhin, Colloid J. USSR (1986) - flow around a metal sphereBazant & Squires, Phys, Rev. Lett. (2004) - theory, broken symmetries, microfluidics

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

= nonlinear electro-osmotic slip at a polarizable surface

Low-voltage “weakly nonlinear” theory

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:

(a) Gouy-Chapman(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. (1999); Ajdari (2000); Squires & Bazant (2004).

AC linear response

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” frequencyIn-phase E field (insulator)

Out-of-phase E (negligible) Induced dipole

)Re( Φ∇−

)Im( Φ∇−

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

Time-averaged velocity

Normal current

Theory vs experiment at low salt concentration

• 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…

Horiz. velocity from a slice10 m above the wire

Data collapse when scaled tocharacteristic ICEO velocity

Levitan et al (2005)

Flow depends on solution chemistry

QuickTime™ and aDV/DVCPRO - NTSC decompressor

are needed to see this picture.

J. A. Levitan, Ph.D. Thesis (2005). ICEO flow around a gold postin “large fields” (Ea = 1 Volt)• Flow vanishes around 10 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

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

Same geometry with raised steps

Stepped electrodes, symmetric footprint

Bazant & Ben (2006)

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 2004Urbanski et al 2006

High-voltage dataV. 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 dataBazant et al, MicroTAS (2007) Urbanski et al, Appl Phys Lett (2006)

KCl, 3 Vpp, planar pumpDe-ionized water (pH = 6)

Double peaks?Reversal at high frequency?Concentration decay?

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

What is the timeto charge thin doublelayers of width = 1-100nm << L?

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

Debye time, / D ?

Diffusion time, L / D ?

2

2

The simplest problem of diffuse-charge dynamics

A sudden voltage across parallel-plate blocking electrodes.

2

Equivalent Circuit Approximation

Answer:

What about nonlinear response? Few models…

Electrokinetics in a dilute electrolyte

Poisson-Nernst-Planck equations

Navier-Stokes equations with electrostatic stress

Singular perturbation

point-like ions

“Weakly Nonlinear” Charging Dynamics

Ohm’s Law in the neutral bulk

Effective “RC” boundary condition

Derive by boundary-layer analysis(matched asymptotic expansions)

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

Weakly nonlinear AC electro-osmosis

Nonlinear DL capacitanceshifts flow to low frequency

Faradaic reactions “short circuit” the flow

Classical models fail…

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

“Strongly Nonlinear” Charging DynamicsBazant, 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

A metal cylinder/sphere in a sudden uniform E field

Chu & Bazant, Phys Rev E (2006).

• Surface conduction through double layers sets in at same time as bulk salt adsorption• yields recirculating current

Dukhin (Bikerman) number

Strongly nonlinear electrokinetics

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

Some new effects

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…

Laurits Olesen, PhD Thesis, DTU (2006)

Outline

1. Experimental puzzles

2. Strongly nonlinear dynamics

3. 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!

Ion crowding at large voltages

Crucial new physics:

Steric effects in equilibrium

Modified Poisson-Boltzmann equation a = minimum ion spacing

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)

Borukhov et al. (1997)Large ions, high concentration

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

“Fermi-Dirac” statistics

Steric effects on electrolyte dynamicsKilic, Bazant, Ajdari, Phys. Rev. E (2007). Sudden DC voltageOlesen, Bazant, Bruus, in preparation (2008). Large AC voltage (steady response)

Modified Poisson-Nernst-Planck equations

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

1d blocking cell, sudden V

dilute solution theory + entropy of solvent (excluded volume)

Steric effects on diffuse-layer relaxationKilic, Bazant, Ajdari, Phys. Rev. E (2007).

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

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:

Example 1: Field-dependent

mobility of charged metal particles

AS Dukhin (1993) predicted theeffect for small E.

PB predicts no motion in large E:

Bazant, Kilic, Storey, Ajdari,in preparation (2008)

Opposite trend for steric models

Example 2: Reversal of planar

ACEO pumps

A. Large electrode wins(since it has time to charge)

B. Small electrode wins(since it charges faster at high V)

Storey, Edwards, Kilic, BazantPhys. Rev. E to appear (2008)

log V

steric effects

log(frequency)

Towards better models

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

Model using Carnahan-Starlingentropy for hard-sphere liquid

• 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…

Biesheuvel, van Soestbergen (2007)

Storey, Edwards, Kilic, Bazant (2008)

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

Larger cations

Divalent cations

Mobility in large DC fields:

ICEP of a polarizable uncharged sphere in asymmetric electrolyte

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

ConclusionNonlinear electrokinetics is a frontier of theoretical physics and applied mathematicswith many possible applications in engineering.

Related physics: Carbon nanotube ultracapacitor (Schindall/Signorelli, MIT)Induced-charge electro-osmosis

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