Instability of electro-osmotic channel flow with streamwise conductivity gradients

Brian StoreyJose Santos

Franklin W. Olin College of EngineeringNeedham MA

“Electrokinetic instability”2003 Experiments (Mike Oddy of J. Santiago’s group)

1 mm

V

High conductivity fluid

Low conductivity fluid

Model comparison

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

Lin, Storey, Oddy, Chen, Santiago, Phys Fluids 2004Storey, Tilley, Lin. Santiago, Phys Fluids 2005Lin, Storey, Santiago, JFM 2008

Hoburg and Melcher (1976)

Unstable EHD in microfluidics

Posner, Santiago, JFM 2006

Chen, Lin, Lele, Santiago JFM 2005

Baygents, Baldessari PoF1998 ElMochtar, Aubry, Batton, LoC 2003

Storey, PhysD 2005 Boy , Storey, PRE 2007

Field Amplified Sample Stacking (FASS)

+t > 0-

-

---

--- -

Stacked Analyte

-

t = 0

High Conductivity bufferLow Conductivity SampleHigh Conductivity buffer

---- --

- - - -+

- -UB US Oi E

ESEB

EEB

Electrokinetic dispersion

•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 =1

Questions• Can instability and dispersion interact in “stacking”

applications? • Does instability influence stacking efficiency?

Lin, Storey, Santiago, JFM 2008

Generalized governing equations two symmetric species, dilute

Convective diffusion (+) and (-)

0c

c v z Fc E D ct

Convection Electromigration Diffusion

( ) ~ ( )E F z c z c c c Charge Density and Gauss Law

0( )r EE

0v

2( ) E

vv v p v E

t

Navier-Stokes Equations

Note (c+-c-)/(c++c-)~10-5

Electro-neutral bulk assumptionThin double layer approx.

0:Sub

0 :Add

0

0

utral)(Electrone

Ec

cDcvt

c

cDEFccvt

c

cDEFccvt

c

ccc

Final eqns & mechanism for flow

( ) 0E

21,

eRv

t a

21( )

Re E

vv v p v E

t

0v

/EE

0 EE

HS electro-osmotic slip boundary conditions Euslip

Dimensionless parameters

Re evU H

eve

U HRa

D

ev

eov U

UR

L

H

low

high

H

Lsample

Electric Rayleigh number

Reynolds number

Channel aspect ratio

Ratio of electro-osmotic to electroviscous velocity

Electrical conductivity ratio

Ratio of sample length to channel height

HE

U ev

2

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

Posner, 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 V

Stability measure as function of applied field

Unstable E field

Role of electric body force

No electro-osmotic slip (zeta=0)E=10,000 V/m (much lower field than with EO)

Phase diagram

Phase diagram

DRaRv

22

D

HERa

22

lo

lohi

1

Conclusions

• Instability can occur in FASS geometry.

• Simple stability map can be used to predict stability within reason.

• Phenomena seems generic when you drive low conductivity into high conductivity.

• Instability doesn’t impact rate of dispersion that much.

• Preliminary – instability doesn’t seem to impact sample concentration as

much as you might think.