Transcript Document

Instability of electro-osmotic channel flow
with streamwise conductivity gradients
J. Jobim Santos
Brian D. Storey
Franklin W. Olin College of Engineering
Needham MA
PHYSICAL REVIEW E 78, 046316 2008
NSF CTS-0521845 (RUI)
EHD instability in microfluidics
…building on Hoburg and Melcher (JFM 1976)
Lin, Storey, Oddy, Chen Santaigo PoF2004
Baygents, Baldessari PoF1998
Chen, Lin, Lele, Santiago JFM 2005
ElMochtar, Aubry, Batton, LoC 2003
Posner, Santiago, JFM 2006
Computation
Experiment
Lin, Storey, Santaigo JFM 2008
Problem statement
Electric field
~50 micron channel
E
High cond.
buffer, 1
Low cond.
sample, 2
+
High cond.
buffer, 1
-
V
Question: Is this flow stable?
Example of axial conductivity gradients in EK
Field Amplified Sample Stacking (FASS)
t=0
- -- - - - - -
+
High Conductivity buffer
Low Conductivity Sample
High Conductivity buffer
ES
E
t>0
-
EB
+
UB
-
US
--- -- - --
-
EB
-
Stacked Analyte
Burgi & Chein 1991, Analytical Chem.
Electrokinetic dispersion
ueof, 1
High conductivity, E1
E
ueof, 1
ueof, 2
Low conductivity, E
2
High conductivity, E1
•Electroosmotic velocity depends upon the electric field
•Electric field is high when conductivity is low
•Low conductivity = high electroosmotic velocity
•No applied pressure gradient – pressure is generatedEO mismatch
Red; cond =10
Blue; cond =1
Ghosal, EP 2004
Baradawaj & Santiago JFM 2005
Ren & Li JCIS 2006
Sounart & Baygents JFM 2007
Model summary
•
•
Incompressible Navier-Stokes plus electric body force
Ion transport based on Nernst-Planck for binary, symmetric electrolyte; simplified by
assuming fluid bulk is electro-neutral.


Dv
2
Re
 P   v   e E
Dt

v  0
D
Ra
  2
Dt

  E  0
 hi
( is conductivity,  
)
 lo
Boundaryconditionsat the channel wall :


v  Rv E
(Helmholtz - Smolochowski slip)
 
E n  0
(Electrica ly insulating)

  n  0 (no loss of ions)
Ls/L
H/L
Hoburg & Melcher, JFM 1976
Lin, Storey, Oddy, Chen Santaigo PoF2004
Unstable flow
E=25,000 V/m, Conductivity ratio=10
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 measure
100
E=10000 V/m
200
300
400
100
E=25000 V/m
200
300
400
500
Maximum vertical vel.
along the centerline
600
Stability measure as function of
applied field
Unstable E field
Phase diagram
E
Typical exp. range
 1 
 hi   lo
 lo
E 2 H 2
Ra 
D
 2
Rv Ra 
D
2
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.
• Future work could include; role of instability on stacking
efficiency, role of analyte on stability, single interface
FASS, and experimental validation.
Dimensionless parameters
U ev H
Rae 
D
Re 
U ev H

 high

 low
U eo
Rv 
U ev

H
L
Lsample
H
Electric Rayleigh number

E 2 H 
U ev 

 

Reynolds number
Electrical conductivity ratio
Ratio of electro-osmotic to electroviscous velocity
Channel aspect ratio
Ratio of sample length to channel height
Role of electric body force
Phase diagram
Baygents, Baldessari PoF1998
Generalized governing equations
two symmetric species, dilute
Convective diffusion
(+) and (-)
c
  c v   z Fc E  D c   0
t
Convection
Charge Density and
Gauss Law
Navier-Stokes
Equations
Note (c+-c-)/(c++c-)~10-5
Electromigration
Diffusion
 E  F ( z c  z c ) ~ (c  c )
 ( r 0 E )   E
 v  0
 v

 (v )v   p  2v   E E
 t


Electro-neutral bulk assumption
Thin double layer approx.
c   c   c (Electroneutral)



c

   v c  FcE  Dc  0
t



c
    v c  Fc E  Dc   0
t


c


    v c  Dc   0
t



Sub :   cE  0
Add :
 
Final eqns & mechanism for flow
v
1
 v  v 
( p   2v   E E )
t
Re
 v  0
 ( E )  0


  E    E  0

E    E / 

1 2
 v  
 ,
t
Rae
HS electro-osmotic slip boundary conditions
uslip  E
No electro-osmotic slip (zeta=0)
E=10,000 V/m (much lower field than with EO)