Electrokinetic flow in microfluidics: problems at high voltage Brian D. Storey Olin College of Engineering

People and funding Collaborators – Martin Bazant (MIT) – Sabri Kilic (former PhD student MIT) – Armand Ajdari (ESPCI) UG students – Jacqui Baca – Lee Edwards Funding – NSF

Today Classic linear electrokinetics Induced charge and nonlinear electrokinetics Classical theory and its breakdown What can we do?

What’s electrokinetics? Interaction of ion transport, fluid flow, and electric fields. – Electrophoresis – Electroosmosis – Sedimentation potential – Streaming potential Discovered in 1809, theory is over 100 yrs old. Today we are only concerned with transport in simple aqueous, dilute electrolytes.

What’s an electrolyte? A material in which the mobile species are ions and free movement of electrons is blocked. (Newman, Electrochemical Systems) Na + Cl - Na + Cl - Na + Cl - Na + 1 mM of salt water is a 3 mm salt cube in 1 liter 1 ion per 10,000 waters

The electric double layer counter-ions co-ions Glass + water Glass Salt water

Electroosmosis (200 th anniversary) Electric field

Electroosmosis in a channel (the simplest pump?) Charge density Velocity Y Y Electric field Electroneutral in bulk

Double layers are typically thin ~10 nm Helmholtz-Smolochowski

Pressure-drivenElectrokinetic Molho and Santiago, 2002 Electroosmosis-experiments

Near a wall, steady state, 1D: Chemical potential of dilute ions: Wall voltage =.025 V Classical electrokinetics double layer structure Poisson’s eqn for electric potential: n

“Classical” microfluidic application Sustarich, Storey, and Pennathur, 2010

Linear EK devices 1 Problem: High voltage, restricted to the lab 1 Solution: High fields can be generated at low voltage if electrodes are placed very close to each other.

Applied voltage via electrodes 1D transient problem Bazant, Thorton, Ajdari PRE 2004

Applied voltage via electrodes 1D problem Position Concentration Electric Potential C=1 Φ=+V Φ=-V

Applied voltage via electrodes 1D problem

Induced charge electromosis (ICEO) Bazant & Squires PRL & JFM2004 Flow is proportional to the square of the electric field, nonlinear.

Ramos, Morgan, Green, Castellenos 1998 Flat electrodes and pumps

ICEP Gangwal, Cayre, Bazant, Velev PRL 2008

And don’t think this is all new…

The “standard model” for ICEO Trivial to implement and solve in a commercial finite element package

Some problems with the standard model

Flow reversal Ajdari, PRE 2000 Storey, Edwards, Kilic, Bazant, PRE 2008

Unexplained freq response Huang, Bazant, Thorsen, LOC 2010

Universal flow decay with concentration Urbanski et al Studer et al, 2004

Flow decay with concentration Bazant, Kilic, Storey, Ajdari ACIS 2009

ICEO microfluidics For engineers, ICEO operates at low voltage. For theory, ICEO operates at high voltage ~100 kT/e Classical theory is great for some features, a number of phenomena have been predicted before observation. Classical theory misses some important trends and cannot get quantitative agreement. Would like a better theory, but one simple enough to be practical for device design.

The ICEO standard model Poisson-Nernst-Planck Navier Stokes Do some math (asymptotics) Is this OK? ICEO Standard model, Linear PDEs Flow and electrical problems are decoupled. Trivial. Fundamental. Non-linear PDEs Flow and electrical problems are coupled. Very thin boundary layers. A bit nasty.

Near a wall, steady state, 1D: Chemical potential of dilute point ions: Applied voltage =.025 V Applied voltage =0.75 V Would need ions to be 0.01 angstrom Classical theory – one problem

Stern layer (1924) Zembala, Diffuse layer Diffuse +Stern layer Solid Bulk fluid

Steric effects – continuum theory Bare Hard sphere Hydrated Borukhov and Andelman 1997 Iglic and Kralj-Iglic 1994 Strating and Wiegel 1993 Wicke and Eigen 1951 Dutta and Bagchi 1950 Grimley and Mott 1947 Bikerman 1942 Stern 1924 Classic

Stern 1924 On the other hand, it is easy, instead of introducing the gas laws for osmotic pressure, to introduce the laws of the ideal concentrated solutions. Under this assumption, which simplifies to (2a) when the second addend in the square brackets is small compared to 1. (as translated by a German student in my class, Johannes Santen)

Bikerman model Kilic, Bazant, Ajdari – PRE 2007 n, dimensionless, ν, volume fraction in equilibrium ν

Bikerman model Bazant, Kilic, Storey, Ajdari ACIS 2009 KPF 6 on silver, no adsorption Potassium Hexafluorophosphate

Linearized, DH Non-linear, GCS Bikerman model Model applied to ICEO pump Storey, Edwards, Kilic, Bazant PRE 2008

Theory and experiment Ion is 4 nm to best fit data. Bazant, Kilic, Storey, Ajdari, ACIS 2009 Exp. from Studer, Pepin, Chen, 2004

Carnahan-Starling - hard spheres “volume effects can be underestimated significantly” using Bikerman’s model. (Biesheuvel & van Soestbergen, JCIS 2007). 1-2 nm ion needed to fit the flow data – but capacitance data look more like Bikerman!

Flow halts at high concentration Why?

Continuum model of the slip plane Stern, 1924 (picture from Zembala, 2004)

A simple continuum model Electroosmotic mobility Valid for any continuum model Simplest model of thickening effect Bazant, Kilic, Storey, Ajdari ACIS 2009 Other power laws explored

Charge induced thickening Jamming against a surface (MD simulations, colloidal systems/granular ) Electrostatic correlations (ion pulled back to correlation “hole”) Dielectric saturation, permittivity thought to be ~5 near surface. Alignment of solvent dipoles can increase viscosity (MD). Viscosity in bulk known to increase with ion density (solubility limits usually don’t let us see this effect)

Charge induced thickening Applied voltage Apparent induced voltage Helmholtz-Smolochowski

Model applied to an ICEO pump Need an ion size of ~4 nm to fit flow data 1 μM 10 mM

What’s still missing? Electrostatic correlations– initial work indicates this may help correct the ion size issue. Faradaic reactions Surface roughness Ion-surface correlations Specific adsorption Perhaps a continuum model is just doomed from the start.

Conclusions ICEO applications has opened new avenues for study in theoretical electrokinetics. Crowding of ions, increased viscosity, and decreased permittivity are not new ideas (Bikerman, 1970). Accounting for steric effects can effect qualitative and quantitative predictions in ICEO. More work is needed for a truly useful theory. Goal: A simple continuum model that can be solved or implemented as simple boundary conditions in simulations. “Surfaces are the work of the devil”

Some recent experiments, do work Pascall & Squire, PRL 2010 No dielectric assumed Thin dielectric coating nm Thin dielectric coating and accounting for chemistry

Carnahan Starling 1-2 nm ion needed to fit the data