1. Induced-Charge Electrokinetic Phenomena Martin Z. Bazant Department 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)

2. 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 Acknowledgments

3. Outline 1.Experimental puzzles 2.Strongly nonlinear dynamics 3.Beyond dilute solution theory

4. Induced-Charge Electro-osmosis Gamayunov, Murtsovkin, Dukhin, Colloid J. USSR (1986) - flow around a metal sphere Bazant & 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

5. Low-voltage “ weakly nonlinear ” theory 1. Equivalent-circuit model for the induced zeta potential 2. Stokes flow driven by ICEO slip 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)  Gamayunov et al. (1986); Ramos et al. (1999); Ajdari (2000); Squires & Bazant (2004). AC linear response

6. 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” frequency In-phase E field (insulator) Out-of-phase E (negligible) Induced dipole Levitan,... Y. Ben,… Colloids and Surfaces (2005). Time-averaged velocity Normal current

7. 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 slice 10  m above the wire Data collapse when scaled to characteristic ICEO velocity Levitan et al (2005)

8. 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 around 10 mM Decreases with ion size, a Decreases with ion valence, z Not predicted by the theory

9. Induced-charge electrophoresis of metallo-dielectric Janus particles S. Gangwal, O. Cayre, MZB, O.Velev, Phys Rev Lett (2008)

10. 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…)

11. 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

12. 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

13. 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

14. 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) Double peaks? Reversal at high frequency? Concentration decay?

15. 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

16. Outline 1.Experimental puzzles 2.Strongly nonlinear dynamics 3.Beyond dilute solution theory

17. What is the time to charge thin double layers 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

18. Equivalent Circuit Approximation Answer: What about nonlinear response? Few models…

19. Electrokinetics in a dilute electrolyte Poisson-Nernst-Planck equations Navier-Stokes equations with electrostatic stress Singular perturbation point-like ions

20. “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)

21. Weakly nonlinear AC electro-osmosis Nonlinear DL capacitance shifts 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):

22. “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

23. 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

24. 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)

25. Outline 1.Experimental puzzles 2.Strongly nonlinear dynamics 3.Beyond dilute solution theory

26. 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!

27. Ion crowding at large voltages Crucial new physics:

28. 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

29. 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) 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)

30. Steric effects on diffuse-layer relaxation Kilic, 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:

31. Example 1: Field-dependent mobility of charged metal particles AS Dukhin (1993) predicted the effect for small E. PB predicts no motion in large E: Bazant, Kilic, Storey, Ajdari, in preparation (2008) Opposite trend for steric models

32. 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, Bazant Phys. Rev. E to appear (2008) log V steric effects log(frequency)

33. Towards better models Bazant, Kilic, Storey, Ajdari (2007, 2008) Model using Carnahan-Starling entropy 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)

34. 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

35. Generic effect: Saturation of induced zeta

36. 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-)

37. 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

38. 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