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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
- - - - - - - -
++
++++
++++
++
++
++
++++
++++
++++++
++++
++++
++
++
++
++
++
++
-
-
-
++
- - - - - - -
++
++++
++++
++
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++++++
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++
++
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++
++
++
-
-
-
++
++
++ -
-
++
++ -
-
++
++ -
--
-
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