Electrohydrodynamic instabilities in microfluidics Brian D. Storey Franklin W. Olin College of...

Electrohydrodynamic instabilities in microfluidics

Brian D. Storey

Franklin W. Olin College of EngineeringNeedham MA

EHD instability in microfluidics

Posner, Santiago, JFM 2006

Chen, Lin, Lele, Santiago JFM 2005ElMochtar, Aubry, Batton, LoC 2003

Lin, Storey, Oddy, Chen Santaigo PoF2004

Lin, Storey, Santaigo JFM 2008

Computation ExperimentSantos & Storey PRE 2008

Hoburg and Melcher (JFM 1976)

Web of science1976-1985 8 citations by the author(s)1982-1994 4 citations2004-today 22 citations

Electrohydrodynamics

• Electrohydrodynamics is the interaction between electric fields and fluid motion.

• Today we will be concerned with EHD of simple, miscible, 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+

Electrolytes and charged surfaces

--------

++

++++

++++

++

++

++

++++ ++

++

++

++++

++

++

++ ++

++

++

++

++

++

++

-

-

-

++

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

X

C

counter-ions

co-ions

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

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

Electrohydrodynamic instabilityExperiments (Mike Oddy of J. Santiago’s group)

1 mm

V

High conductivity fluid

Low conductivity fluid

Miscible interface

Model summary

• Incompressible Navier-Stokes plus electric body force• Poisson-Nernst-Planck for ion transport binary, symmetric

electrolyte; simplified by assuming fluid is nearly electro-neutral.• Helmholtz-Smolochowski electrokinetic slip boundary conditions

0

Ra

0

Re

ty)conductivi is (2

2

E

Dt

D

v

EvPDt

vDe

Lin, Storey, Oddy, Chen Santaigo PoF2004

m a=F

Mass is conserved

Fluid conductivity goes with the flow

Current is conserved, V=iR

Mechanism for charge generation

eE

E

0 /EE

0 EE

High conductivity Low conductivity

Electric field

E

+++++

++++++++

++

positive

positive

e

e

x

Ex

E

Mechanism for flow

/EE

0

Ra

0

Re

2

2

E

Dt

D

v

EvPDt

vDe

E

Dimensionless parameters

Re evU H

eve

U HRa

D

ev

eov U

UR

low

high

Electric Rayleigh number

Reynolds number

Ratio of electro-osmotic to electroviscous velocity

Electrical conductivity ratio

HE

U ev

2

Experiment vs. 2D Computation

Lin, Storey, Oddy, Chen, Santiago, Phys Fluids 2004

Other configurationsHigh conductivity center Low conductivity center

2D Simulation (Storey, Phys D 2005)

Experiment (Ponser & Santiago, JFM 2006) Experiment (Ponser & Santiago, JFM 2006)


Instability at T-junction0.5, 0.75, 1, & 1.25 kV/cm

Chen, Lin, Lele, & Santiago, JFM 2005

Simulations with same basic model provided good agreement

Linear stability results

Ecr,experiment ~ 35,000 V/m,

x

yz

2D Linear Analysis with 1/ 2=10

Stable

Ra e

E (

V/m

)

3D Linear Analysis with 1/ 2=10

Stable

Ra e

E (

V/m

)

Ecrit

Ecrit

Lin, Storey, Oddy, Chen, Santiago, Phys Fluids 2004

So 3D matters3D DNS

Storey, Physica D, 2005

time

As does electroosmosis

Storey, Physica D, 2005

time

Thin channels

• So aspect ratio matters, but can we model flow in thin channels with a 2D model?

x

y

z

d

H

E

1

2

Thin Channel Approx. (Hele-Shaw)

Solid- full 3DDashed – this model

x

yz

d

H

E

1

2

Storey, Tilley, Lin, Santiago, Phys Fluids 2005

Hele-Shaw model works in linear regime, fails in non-linear regime

3D DNS

Depth Ave

Zeroth order

3D DNS

Depth Ave

Zeroth order

Lin, Storey, Santiago JFM 2008

Higher order (includes EK dispersion) works better in NL regime

3D Simulation

Full Depth Ave

Zeroth order

3D Simulation

Full Depth Ave

Zeroth order


Depth-Averaged Model

ExperimentComputation

t = 0.0 s

t = 0.5 s

t = 1.5 s

t = 2.0 s

t = 2.5 s

t = 3.0 s

t = 4.0 s

t = 5.0 s

t = 1.0 s


Computational Results:depth-averaged model


Experiment

Simulation

So…

• Depth averaged, 2D model for electrokinetic flow works.

• Need to include electrokinetic dispersion in the model.

• But what’s electrokinetic dispersion?

Classic Taylor dispersion inpressure driven flow

“Physicochemical Hydrodynamics”Probstein

Electrokinetic dispersion (looking in the thin direction)

•Electroosmotic velocity depends upon the electric field•Electric field is high when conductivity is low•Low conductivity = high EO velocity

High conductivity, E1

ueof, 1 ueof, 2

High conductivity, E Low conductivity, E2

ueof, 1 ueof, 2

1

ueof, 1

High conductivity, E

Red; cond =10 Blue; cond =1Ghosal, EP 2004Baradawaj & Santiago JFM 2005Ren & Li JCIS 2006Sounart & Baygents JFM 2007

Dispersion acts as anisotropic diffusion

3D Simulation

Full Depth Ave

Zeroth order

3D Simulation

Full Depth Ave

Zeroth order


So…

• Is flow stable in the shallow direction?

• How does our shallow model break down?

High conductivity, E1

ueof, 1 ueof, 2

High conductivity, E Low conductivity, E2

ueof, 1 ueof, 2

1

ueof, 1

High conductivity, E

Example of axial conductivity gradients in EKField Amplified Sample Stacking (FASS)

+t > 0-

-

---

--- -

Stacked Analyte

-

t = 0

High Conductivity bufferLow Conductivity SampleHigh Conductivity buffer

---- --

- - - -+

- -UB US

ESEB

EEB

Burgi & Chein 1991, Analytical Chem.

Unstable flowE=25,000 V/m, Conductivity ratio=10

Santos & Storey, PRE 2008

Flow in center similar to other observations

High conductivity center


Experiment (Ponser & Santiago, JFM 2006)

Observations

•“Shock” at the leading edge of the sample.•Vertical velocity at the channel walls pumps fluid toward the centerline.•Unstable flow only inside the sample region.


Stability measureMaximum vertical vel.along the centerline


Stability measure as function of applied field

Unstable E field


A microfluidic EHD mixer

E Field

ElMochtar, Aubry, Batton, LoC 2003 Boy & Storey, PRE 2007

Time periodic forcing for enhanced mixing

Boy & Storey, PRE 2007

Classic problem in electrochemistry

x

y

Binary electrolyte (C+,C-)

•Fixed potential•Fixed concentration of C+•No flux of C-

Solid surfaces are charge selective (electrode or ion exchange membrane).

Current

Steady state V=1

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

V=1

C+,C

-

x

E, flux of C+

Bulk is electro-neutral, linear conc. profile

Double layer, Debye =0.01

Double layer, Debye =0.01

Typical dimensionless Debye =0.0001 or less

Current-voltage relationship

0 5 10 15 200

1

2

3

4

5

6

1D Solution

Observed

Cur

rent

Voltage

Resistor at low voltage

Attributed to instability of double layersZaltzman & Rubinstein, JFM 2007

Different views on bulk stability

•Bulk instability. Grigin (1985, 1992)•Bulk instability, but not sufficient for mixing. Bruinsma & Alexander (1990)•Bulk instability. Rubinstein, Zaltzman, & Zaltzman (1995).•No bulk instability. Buchanan & Saville (1999)•No bulk instability. Highlighted problems with all earlier works reporting instability. Limited parameter space. Lerman, Zaltzman, Rubinstein (2005)

Q: The model equations for bulk instability is the same as ours, why is there no bulk instability? Or is there?

Hoburg-Melcher limit Pe=∞, low

V analysis

D

DD

D

DSS ;

1

1~22

0

01

1

1

2

2

ccD

D

cPe

cvt

c

Summary

•D>1, Real, S2<0, Stable•D<1, Real, S2>0, Unstable•D=1, Imag, Oscillations

Storey, Zaltzman, & Rubinstein, PRE 2007

Bulk electroconvection, finite Pe low V analysis

D

DD

D

DIL ;

1

1

4Pe

2

Current, Imax =4

unstable

L=-68

k=4.74

Summary•D>1, Real, Stable•D<1, Real, Unstable (threshold)•D=1, Stable


BE at finite voltage, D=0.1

Unstable

Pe=9.9


Relationship between BE and microchannel EHD instability

• Bulk instability can exist, in theory. • Threshold is different since conductivity gradient is driven• New bulk instability mechanism found when D+ < D-, that

can occur at low V.• Many previous studies only considered D+=D.• An analysis looking for an application…

Other example of flows driven by concentration polarization

From J. Han, MIT

Device built for bio-molecule preconcentration

Instability observed

From J. Han, MIT

Stuff I didn’t show you..Colloids, Posner Two phase, Zahn & Reddy Two phase, Aubry et al

Electrothermal, Ramos, Gonzalez, Castellanos, et al

Multi-species, Oddy & Santiago

Acknowledgements• Collaborators:

– Hao Lin, Rutgers– Juan Santiago, Stanford– Boris Zaltzman & Isaac Rubinstein, Ben Gurion University of Negev, Israel

• Undergraduate students– David Boy– Jobim Santos– Lee Edwards– Doug Ellwanger– Allison Schmidt– Mark Cavolowsky– Nina Cary– Angela Mao

• Funding– NSF– Olin College

Depth averaged equations

From the DA equations, we can reconstruct the full 3D fields.

Electrohydrodynamic instabilities in microfluidics Brian D. Storey Franklin W. Olin College of...

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