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Induced-Charge Electrokinetic Phenomena

Paris-Sciences Chair Lecture Series 2008, ESPCI. Induced-Charge Electrokinetic Phenomena. 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

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Induced-Charge Electrokinetic Phenomena

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  1. Paris-Sciences Chair Lecture Series 2008, ESPCI Induced-Charge Electrokinetic Phenomena 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

  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

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

  4. 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

  5. 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)

  6. 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

  7. 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…

  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 around10 mM • Decreases with ion size, a • Decreases with ion valence, z Not predicted by the theory

  9. Induced-charge electrophoresisof 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 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

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

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

  17. 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

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

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

  20. “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

  21. 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…

  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 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

  24. 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…

  25. Outline • Experimental puzzles • Strongly nonlinear dynamics • 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. Crucial new physics: Ion crowding at large voltages

  28. 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

  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) 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

  30. 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.

  31. 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

  32. steric effects 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)

  33. 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

  34. 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

  35. Generic effect: Saturation of induced zeta

  36. 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-)

  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

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