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 -

Cl -

Cl -

Cl -

Na+

Cl -

Cl -

Cl -

Cl -

Na+

Na+

Na+

Cl -

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

--------

++

++++

++++

++

++

++

++++ ++

++

++++

++++

++ ++

++

++

++

++

-

-

-

++

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

X

C

counter-ions

co-ions

-

-

-

-

-

-

-Glass + water

0HSiOSiOH 3

Glass Salt water

Electroosmosis (200th anniversary)

Electric field

- - - - - - - -

++

++++

++++

++

++

++

++++

++++

++++++

++++

++++

++

++

++

++

++

++

-

-

-

++

- - - - - - -

++

++++

++++

++

++

++

++++

++++

++++++

++++

++++

++

++

++

++

++

++

-

-

-

++

++

++ -

-

++

++ -

-

++

++ -

--

-

Electroosmosis in a channel(the simplest pump?)

0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Charge densityCharge density Velocity

Y

Y

Electric field

Electroneutral in bulk

Double layers are typically thin ~10 nm

0 0.2 0.4 0.6 0.8 1 1.2-1

-0.998

-0.996

-0.994

-0.992

-0.99

-0.988

-0.986

-0.984

-0.982

-0.98

Velocity

y

0 0.2 0.4 0.6 0.8 1 1.2-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Velocity

y

E

Uslip Helmholtz-Smolochowski

Pressure-driven Electrokinetic

Molho and Santiago, 2002

Electroosmosis-experiments

Near a wall, steady state, 1D:

Chemical potential of dilute ions:

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

X

C

Wall voltage =.025 V

i

i

i

kT

ez

i

iii

nez

e

kTenn

eznkT

i

2

mV25

ln

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

Con

cent

ratio

nE

lect

ric P

oten

tial

C=1

Φ=+V

Φ=-V

Applied voltage via electrodes 1D problem

V 1

R CC

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

conditionboundary slip kiSmoluchows-Helmholtz :BC

flow ibleIncompress 0

Re low equation, Stokes

C. across voltageis C,capacitor a like acts surface, Blocking :BC

ty conductiviconstant fluid, tralElectronue 0)(

2

2

Eu

u

uu

E

E

Pt

dt

dC

Trivial to implement and solve in a commercial finite element package

Some problems with the standard model

Flow reversalAjdari, PRE 2000

Storey, Edwards, Kilic, Bazant, PRE 2008

Unexplained freq response

Huang, Bazant, Thorsen, LOC 2010

Universal flow decay with concentration

Urbanski et al. 2007 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

0

0)(

2

2

Eu

u

uu

E

E

Pt

dt

dC

Poisson-Nernst-Planck Navier Stokes

Do some math (asymptotics)

Is this OK?

Is this OK?

ICEO Standard model, Linear PDEsFlow and electricalproblems are decoupled.Trivial.

Fundamental.Non-linear PDEsFlow and electricalproblems are coupled. Very thin boundary layers.A bit nasty.

ln eznkT iii

Near a wall, steady state, 1D:

Chemical potential of dilute point ions:

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

X

C

0 1 2 3 4 510

-20

10-10

100

1010

1020

X

C

Applied voltage =.025 V Applied voltage =0.75 V

kT

ez

i

i

enn

Would need ions to be 0.01 angstrom

Classical theory – one problem

Stern layer (1924)

Zembala, 2004.

C D L

C S

-20 -10 0 10 200

5

10

15

20

C

Diffuse layer

Diffuse +Stern layer

Solid Bulk fluid

Steric effects – continuum theory

Bare HardsphereHydrated

)1ln(ln kTezckT iii

•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 studentin my class, Johannes Santen)

Bikerman model

Kilic, Bazant, Ajdari – PRE 2007

ezn

kT ii

i

1

lnkT

ze

en

n

1 n, dimensionless, ν, volume fraction in bulk

@ equilibrium

ν

Bikerman model

Bazant, Kilic, Storey, Ajdari ACIS 2009

KPF6 on silver, no adsorptionPotassium 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 2009Exp. 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 concentrationWhy?

Continuum model of the slip plane

Stern, 1924 (picture from Zembala, 2004)

A simple continuum model

ts bEU

b

bdbD

0

cb

b

1

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

App

aren

t in

duce

d vo

ltage

E

Uslip 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 coating30-60 nm

Thin dielectric coating and accounting for chemistry

Carnahan Starling

1-2 nm ion needed to fit the data