Induced-Charge Electro-osmosis and Electrophoresis Martin Z. Bazant Department of Mathematics & Institute for Soldier Nanotechnologies, MIT Nonlinear Electrokinetics @ MIT Students: Jeremy Levitan (ME PhD’05), Kevin Chu (Math PhD’05), JP Urbanski (ME), Mustafa Sabri Kilic (Math) Postdocs: Yuxing Ben, Hongwei Sun (Math) Faculty: Todd Thorsen (ME), Martin Schmidt (EE) Visitors: Armand Ajdari, Vincent Studer (ESPCI) Collaborators: Todd Squires (UCSB), Shankar Devasenathipathy (Stanford) Howard Stone (Harvard) Funding: US Army Research Office (Contract DAAD-19-02-002) and MIT-France Program ICEO in a microfluidic device.

The Electrochemical Double Layer + neutral bulk electrolyte solid Electrostatic potential Ion concentrations continuum region

Electrokinetic Phenomena Helmholtz-Smoluchowski fluid “slip” formula: Electro-osmosis Electrophoresis The classical theory assumes that the “zeta potential” z (or charge density q) is a constant material property, but what happens at a polarizable (e.g. electrode) surface?

AC Electro-osmosis How general is this phenomenon? Ramos et al., JCIS (1999); Ajdari, Phys. Rev. E (2000) Steady flow for AC period = How general is this phenomenon? Need electrode arrays? Need “AC”?

“Induced-Charge Electro-osmosis” = nonlinear electro-osmotic slip at a polarizable surface Bazant & Squires, Phys, Rev. Lett. 92, 0066101 (2004). Example: An uncharged metal cylinder in a suddenly applied DC field Integral boundary conditions for surface charge -- Q = 0 --> phi not specified Same effect for metals & dielectrics, DC & AC fields…

Double-layer polarization and ICEO flow A conducting cylinder in a suddenly applied uniform E field. Electric field ICEO velocity FEMLAB simulation by Yuxing Ben Poisson-Nernst-Planck/Navier-Stokes eqns l/a=0.005

Experimental Observation of ICEO J. A. Levitan, S. Devasenathipathy, V. Studer, Y. Ben, T. Thorsen, T. M. Squires, & M. Z. Bazant, Colloids and Surfaces (2005) 100 mm Pt wire on channel wall Viewing plane PDMS polymer microchannel Bottom view of optical slice Inverted optics microscope Micro-particle image velocimetry (mPIV) to map the velocity profile

Movie: Optical slice sweeping through the 100 mm Pt wire

“Induced-Charge Electrokinetic Phenomena” 1. Prior examples of “ICEO” Electro-osmotic flows around metal particles Dielectrophoresis of spheres in electrolytes (“dipolophoresis”) AC electro-osmosis & colloidal aggregation at electrodes DC “electrokinetic jet” at a microchannel corner Gamayunov, Murtsovkin, Dukhin, Colloid J. USSR (1986); Levich (1960) Simonova, Shilov, Colloid J. USSR (1981, 1998) Ramos et al. (1998); Ajdari (2000); “EHD” Ristenpart, Saville (2004)… Thamida & Chang (2002) 2. Some new examples - breaking symmetries ICEO pumps and mixers in microfluidics “Fixed-potential ICEO” “Induced-charge electrophoresis” (ICEP) particle motion Bazant & Squires, PRL (2004); Levitan et al. Colloids & Surfaces (2005). Squires & Bazant, JFM (2004); Levitan, PhD thesis MIT (2005). Bazant & Squires, PRL (2004); Yariv, Phys. Fluids (2005); Squires & Bazant, JFM (2006); Saintillon, Darve & Shaqfeh JFM (2006); Rose & Santiago (2006).

“Fixed-Potential ICEO” Squires & Bazant, J. Fluid Mech. (2004) Idea: Vary the induced total charge in phase with the local field. Generalizes “Flow FET” of Ghowsi & Gale, J. Chromatogr. (1991) Example: metal cylinder grounded to an electrode supplying an AC field. Fixed-potential ICEO mixer

ICEO Microfluidic Elements J. A. Levitan, Ph.D. Thesis (2005). ICEO “mixer” or “trap” (u = 0.2 mm/sec) Fixed-potential ICEO “pump” (u = 3 mm/sec) E = 100V/cm (< 10 Volt), 300 Hz AC, 0.1 mM KCl, 0.5 mm fluorescent tracers 50-250 mm electroplated gold posts, PDMS polymer microchannels A promising platform for portable microfluidics…

“Induced-Charge Electrophoresis” = ICEO swimming via broken symmetries Bazant & Squires, Phys. Rev. Lett. (2004); Yariv, Phys. Fluids (2005). I. Heterogeneous Surfaces Squires & Bazant, J. Fluid Mech. (2006). A metal sphere with a partial dielectric coating swims toward its coated end, which rotates to align perpendicular to E. An “ICEO pinwheel” rotates to align and spins continuously in a uniform AC field! Stable Unstable

ICEP II. Asymmetric Shapes Squires & Bazant, J. Fluid Mech. (2006). ICEP can separate polarizable colloids by shape and size in a uniform DC or AC electric field, while normal (linear) electrophoresis cannot. long axis rotates to align with E a “thin arrow” swims parallel to E, towards its “blunt” end a “fat arrow” swims transverse to E towards its “pointed” end Perturbation analysis E u An asymmetric metal post can pump fluid in any direction in a uniform DC or AC field, but ICEO flow has quadrupolar rolls, very different from normal EOF. FEMLAB finite-element simulation (Yuxing Ben)

ICEP III. Non-uniform Fields Shilov & Simonova, Colloid J. USSR (1981, 2001). Metal sphere “dipolophoresis” Squires & Bazant, J. Fluid Mech. (2006). General problem of DEP + ICEP Must include electrostatic force and torque (Maxwell stress tensor) Dielectrophoresis (DEP) + ICEP For metals, ICEP points up, and DEP down, an electric field gradient ICEP cancels DEP for a metal sphere (but not a cylinder or other shapes) Electric Field Fluid Streamlines

General solution for any 2d shape in any non-uniform E field by complex analysis… Electric Field Fluid Streamlines

“Weakly Nonlinear” Theory of ICEO Gamayunov et al. (1986); Ramos et al. (1998); Ajdari (2000); Squires & Bazant (2004). 1. Equivalent-circuit model for the induced zeta potential Bulk resistor (Ohm’s law): Double-layer BC: Double-layer circuit elements: Gouy-Chapman capacitor Stern model Constant-phase-angle impedance 2. Stokes flow driven by ICEO slip b=0.6-0.8 Dimensionless BC for AC forcing 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 At the “RC” frequency Electric field lines: Electric Field lines Electric field lines Electric field lines Velocity fields Velocity fields

Comparision of Simulation and PIV Data: Velocity Profiles Raw data from a slice 0-10 mm above the wire Data collapse when scaled to characteristic ICEO velocity Scaling and flow profile consistent with ICEO theory Flow magnitude roughly 2 times smaller than in simple theory Need better theories for large voltages and varying solution chemistry…

Theory of “strongly nonlinear” electrokinetics? Use the basic methods of applied mathematics: (Analysis) Solve the existing equations in a new regime. This leads to some interesting new effects, but does not explain all the experimental data (e.g. decrease in ICEO flow for C > 10 mM). More importantly, the solutions contain physical nonsense! (Modeling) Postulate new equations, solve & compare to experiments. This is now the only choice, and progress is underway.

Classical Equations of “Dilute Solution Theory” Poisson-Nernst-Planck ion transport equations Singular perturbation Navier-Stokes fluid equations with electrostatic stresses

Strongly Nonlinear Solutions to the Classical Equations 1. Breakdown of circuit models: Surface adsorption and bulk diffusion Bazant, Thornton, Ajdari, PRE (2004). 2. Tangential transport of ions in the double layer Bikerman (1933), SS Dukhin & Deryaguin (1969, 1974) Linear theory for small E, highly charged surfaces Kevin Chu & MZB (2006). Nonlinear theory for large E, uncharged conductors, Matched asymptotic expansions…. 3. Diffusio-osmosis (= flow due to gradients in bulk salt concentration) Deryaguin (1964) Bulk diffusion around an uncharged metal sphere in a uniform E field.

Modified Theory of Electrokinetics Sabri Kilic, Bazant, Ajdari (2006). Steric effects (ion size = a) in an equilibrium double layer: Borukhov et al. (1997). 2. Steric effects on dynamics: Modified Nerst-Planck Eqns Zeta Steric & viscoelectric effects: Modified Smoluchowski slip formula DL Voltage (kT/ze) New prediction: “Entropophoresis” of an uncharged metal in asymmetric electrolyte.

Fast AC Electrokinetic Pumps Bazant, Ben (2005) The “conveyor belt principle”: Raised pumping surfaces, recess reverse rolls. Apply to symmetric array of electrodes in existing ACEO pumps Ramos et al (1999), Ajdari (2000) Raise half of each electrode to make a fast pump

Optimization of ICEO/ACEO pumps Bazant, Yuxing Ben (2005) Fastest existing ACEO pump Green et al. (2003) theory; Bornw & Rennie (2001); Studer et al. (2004) expt. New design: 10 times faster!

Engineering of Electrokinetic Pumps JP Urbanski, Levitan, Bazant, Thorsen (2006) Exploit fixed-potential ICEO, and standard ACEO Electroplated interdigitated & recessed gold electrodes on glass PDMS soft lithography for microchannels Microfluidic loop for testing pumps (Studer et al. 2004)

Experimental Results Raised pumps are at least 3-5 times faster than existing planar pumps 10 micron electrodes can pump at mm/sec using only 1 Volt, kHz AC. Demonstration of fast flows for voltage steps 1,2,3,4 V (far from pump). Tour of the 20mm microfluidic loop in steady ACEO flow. http://web.mit.edu/urbanski/Public/Microfluidics/

ICEO: a platform for portable microfluidics? State-of-the-art “table-top microfluidics” Pressure-driven microfluidics (e.g. K. Jensen) Capillary electro-osmosis (e.g. J. Santiago) Soft microfluidic networks (e.g S. Quake) Possible advantages of ICEO: Low voltage (< 10 Volt), low power (< 1 mW) AC (< kHz) reduces unwanted reactions / bubbles in linear EOF Time-dependent local flow control for mixing, trapping, switching,… Excellent scaling with miniaturization Standard “hard” microfabrication methods Possible disadvantages: Requires low ionic strength (< 10 mM) Sensitive to solution chemistry, surface contamination our “micro” experiment

Commercial Applications Engineering Applications of ICEO Commercial Applications 1. Battery-powered microfluidics Portable/implantable devices for medical or chemical monitoring Localized drug delivery Pressure control (e.g. glaucoma) Cooling portable electronics Example: on-field detection of exposure to biowarfare agents for the dismounted soldier by monitoring nanoliters of blood. (T. Thorsen @ MIT Mech Eng) 2. Polarizable colloids ICEO flows in dielectrophoresis ICEO manipulation of nanobarcodes (Santiago, Shaqfeh @ Stanford Mech Eng) www.studybusiness.com

ICEO & ICEP From mathematical theory…. to scientific experiments and engineering applications. http://math.mit.edu/~bazant/ICEO

Diffuse-Charge Dynamics Bazant, Thornton, Ajdari, Phys. Rev. E. (2004). Analysis of the Poisson-Nernst-Planck equations by time-dependent matched asymptotic expansions. Model Problem Classical “equivalent circuit” in the thin-double-layer approximation Time scales

“Strongly Nonlinear” Solutions (as required by the experimental parameters) Breakdown of circuit models at “large” voltages when V > 2 kT/e = 0.05 V (z=V) “Transient Dukhin number” Bazant, Thornton & Ajdari, Phys. Rev. E 70, 021506 (2004). 1d model problem (PNP equations) BREAK INTO TWO SLIDES: 1d problem + Kevin V = 4 kT/e potential charge density salt concentration Neutral salt adsorption by the diffuse charge layer and bulk diffusion

Strip resist; cap with PDMS ICEO microfluidic pumps without moving parts Jeremy Levitan, Ph.D. thesis, Mechanical Engineering MIT (2005) Experimental fabrication: soft lithography for micro-channels (50-200 mm) and electroplating for gold structures (25-200 mm wide, 5-50 mm tall) on glass Deposit and pattern gold on glass wafer Electroplate gold Deposit and pattern thick resist mold Strip resist; cap with PDMS to form micro-channel

Comparision of Simulation and PIV Data: Scaling with Voltage and Frequency Similar ”ICEO flow” observed around mercury drops (without any quantitative analysis): Gamayunov, Mantrov, Murtsovkin, Colloid J. USSR (1992)

Towards a new mathematical model… 1. Anolmalous “constant phase angle” double-layer impedance Data suggests BC for power-law “fractional relaxation”: Hypothesis: long waiting times for Stern-layer adsorption (not fractal surface roughness) KCl/Au expt By J. Levitan 2. Strong dependence on surface and solution chemistry ICEO flow decreases with concentration and depends on ion valence, size,… Hypothesis: steric effects + variable viscosity in the Stern layer Borukhov et al Phys Rev Lett (1997)