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
+
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
Salt water
Glass
+
+
+
+
+ +
+
+
+
+
+
+
- + - +
+
+
+
+
+
- +
+
+ - +
-
Glass + water
SiOH  SiO  H3 0
3
2.5
C
2
counter-ions
1.5
1
0.5
0
0
co-ions
1
2
3
X
4
5
Electric field
+
+
+
+
+ +
+
+
+
+
+
- + + +
+
+
- +
+
+
- +
+
+
+
+
+
+
+
+ +
+
+
+
+
+
- + + +
+
+
- +
+
+
- +
+
+
+
-
+
-
+
+
+
+
-
+
-
-
-
+
-
+
-
+
+
+
-
-
+
-
Electroosmosis (200th anniversary)
Electroosmosis in a channel
(the simplest pump?)
1
0.8
0.6
0.4
Y
Electric field
0.2
Y0
-0.2
-0.4
Electroneutral in bulk
-0.6
-0.8
-1
0
0.2
0.4
0.6
Charge density
Charge density
0.8
1
Velocity
1
-0.98
0.8
-0.982
0.6
-0.984
0.4
-0.986
0.2
-0.988
0
-0.99
y
y
Double layers are typically thin
~10 nm
-0.2
-0.992
-0.4
-0.994
-0.6
-0.996
-0.8
-0.998
-1
-1
0
0.2
0.4
0.6
Velocity
0.8
E
U slip 

1
1.2
0
0.2
0.4
Helmholtz-Smolochowski
0.6
Velocity
0.8
1
1.2
Electroosmosis-experiments
Pressure-driven
Electrokinetic
Molho and Santiago, 2002
Classical electrokinetics
double layer structure
Chemical potential of dilute ions:
i  kT ln ni  zi e
Near a wall, steady state, 1D:
zi e
ni  n e kT
Poisson’s eqn for electric potential:
 2     
kT
 25mV
e
 z en
i
i
Wall voltage =.025 V
3
2.5
2
C
n
1.5
1
0.5
0
0
1
2
3
X
4
5
i
“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
C=1
Electric Potential
Concentration
1D problem
Φ=-V
Φ=+V
Position
Applied voltage via electrodes
1D problem
V1
C
R
C
Induced charge electromosis (ICEO)
Flow is proportional to the square of the electric field, nonlinear.
Bazant & Squires PRL & JFM2004
Flat electrodes and pumps
Ramos, Morgan, Green, Castellenos 1998
ICEP
Gangwal, Cayre, Bazant, Velev PRL 2008
And don’t think this is all new…
The “standard model” for ICEO
  (E)   2  0
d
C
 E
dt
u
 P   2u
t
 u  0
u

E

Electronuetral fluid, constantconductivity 
BC : Blockingsurface, acts like a capacitor C, is voltageacross C.
Stokes equation,low Re
Incompressible flow
BC : Helmholtz - Smoluchowski slip boundarycondition
Trivial to implement and solve in a commercial finite element package
Some problems with the standard
model
Ajdari, PRE 2000
Flow reversal
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
Fundamental.
Non-linear PDEs
Flow and electrical
problems are
coupled. Very thin
boundary layers.
A bit nasty.
Poisson-Nernst-Planck
Navier Stokes
Do some math (asymptotics)
Is this OK?
  (E)   2  0
ICEO Standard model,
Linear PDEs
Flow and electrical
problems are
decoupled.
Trivial.
d
C
 E
dt
u
 P   2u
t
u  0
u

E

Is this OK?
Classical theory – one problem
Chemical potential of dilute point ions:
i  kT ln ni  zi e
Near a wall, steady state, 1D:
zi e
ni  ne kT
Applied voltage =.025 V
3
Applied voltage =0.75 V
20
10
2.5
10
10
1.5
C
C
2
0
10
1
-10
10
0.5
0
0
Would need ions to be 0.01 angstrom
-20
1
2
3
X
4
5
10
0
1
2
3
X
4
5
Stern layer (1924)
Solid
CS
CDL
Bulk fluid
20
Diffuse layer
C
15
10
Diffuse +Stern
layer
5
0
-20
-10
0

10
20
Zembala, 2004.
Steric effects – continuum theory
Bare
Hydrated
Hard
sphere
i  kT ln ci  zi e  kT ln(1  )
Classic
•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
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
i  kT ln
ni
 zi e
1 
@ equilibrium
n
1 n
ze
 e kT
n, dimensionless,
ν, volume fraction in bulk
ν
Kilic, Bazant, Ajdari – PRE 2007
Bikerman model
KPF6 on silver, no adsorption
Potassium Hexafluorophosphate
Bazant, Kilic, Storey, Ajdari ACIS 2009
Model applied to ICEO pump
Linearized, DH
Non-linear, GCS
Bikerman model
Storey, Edwards, Kilic, Bazant PRE 2008
Theory
and
Ion is 4 nm to best fit data.
experiment
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
U s  bEt
Valid for any continuum model
b
D

0
b

d 

b
Simplest model of thickening effect
 b
 1 c 
 b
Other power laws explored
Bazant, Kilic, Storey, Ajdari ACIS 2009
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
Helmholtz-Smolochowski
Apparent induced voltage
E
U slip 

Applied voltage
Model applied to an ICEO pump
1 μM
10 mM
Need an ion size of ~4 nm to fit flow data
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
No dielectric assumed
Thin dielectric coating
30-60 nm
Thin dielectric coating and
accounting for chemistry
Pascall & Squire, PRL 2010
Carnahan Starling
1-2 nm ion needed to fit the data