IMECE, November 15 th , 2004, Anaheim, CA

STANFORD MICROFLUIDICS LABORATORY

A DA DEPTHEPTH-A-AVERAGED VERAGED M MODELODEL F FOROR EELECTROKINETIC LECTROKINETIC F FLOWS LOWS I IN N A T A THINHIN

MMICROCHANNEL ICROCHANNEL G GEOMETRYEOMETRY

Hao Lin,1 Brian D. Storey2 and Juan G. Santiago1

1. Mechanical Engineering Department, Stanford University2. Franklin W. Olin College of Engineering

IMECE, November 15th, 2004, Anaheim, CA

STANFORD MICROFLUIDICS LABORATORY

Motivation: Generalized EK flow with conductivity gradients

Field amplified sample stacking (FASS)

Electrokinetic instability (EKI)

Rajiv Bharadwaj

Michael H. Oddy

STANFORD MICROFLUIDICS LABORATORY

Previous WorkLin, Storey, Oddy, Chen & Santiago 2004, Phys. Fluids. 16(6): 1922-1935

– Instability mechanism: induced by bulk charge accumulation; stabilized by diffusion (Taylor-Melcher-Baygents)

– 2D and 3D linear analyses– 2D nonlinear computations

Storey, Tilley, Lin & Santiago 2004 Phys. Fluids, in press.

– Depth-averaged Hele-Shaw analysis (zeroth-order)Chen, Lin, Lele & Santiago 2004 J. Fluid Mech., in press

– Instability mechanism: induced by bulk charge accumulation; stabilized by diffusion (Taylor-Melcher-Baygents)

– Depth-averaged linear analyses– Convective and absolute instability

Experiment

2D Computation

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Thin-Channel ModelPracticality Consideration

– 2D depth-averaged model significantly reduces the cost of 3D computation

– Model well captures the full 3D physics

Develop flow model for generalized electrokinetic channel flows

– Eletrokinetic instability and mixing

– Sample stacking– Other EK flows which involves

conductivity gradients

x

yz

dH

E

1

2

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Full 3D Formulation (Lin et al.)

H. Lin, Storey, B., M. Oddy, Chen, C.-H., and J.G. Santiago, “Instability of Electrokinetic Microchannel Flows with Conductivity Gradients,” Phys. Fluids 16(6), 1922-1935, 2004. C.-H. Chen, H. Lin, S.K. Lele, and J.G. Santiago, “Convective and Absolute Electrokinetic Instabilities with Conductivity Gradients,” J. Fluid Mech., in press, 2004.

eveU HRaD

( ) 0E

21 ,eR

vt a

21 ( )Re E

v v v p v Et

0v

evU HRe

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Depth Averaged ModelAsymptotic Expansion based on the aspect ratio = d/H which is similar to lubrication/Hele-Shaw theory

20 1 2 ...f f f f

Equations are depth-averaged to obtain in-plane (x,y) governing equations

1

1

1( , ) ( , , )2

f x y f x y z dz

Flows in the z-direction are integrated and modeled

2 427( , , ) ( , )

4 30 2eRa zx y z x y U z

x

u z

x

212 2eo

zU

u u

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Depth Averaged Equations

2 221 2[ ( )]

105 e

e

H H H HRat Ra

u U U

( ) 0H H eo U u u

Convective dispersion: Taylor-Aris type

0H u

2 22 23H H H H H HRe pt

u

u u U u

Momentum: Darcy-Brinkman-Forchheimer

H. Lin, Storey, B., and J.G. Santiago, “A depth-averaged model for electrokinetic flows in a thin microchannel geometry,” to be submitted, 2004.

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Field Amplified Sample Stacking (FASS)

+t > 0--

---

--- -

Stacked Analyte

-

t = 0

High Conductivity bufferLow Conductivity SampleHigh Conductivity buffer

---- --

- - - -+

- -UB US Oi E

ESEB

EEB

Rajiv Bharadwaj

STANFORD MICROFLUIDICS LABORATORY

1D Simplification (y-invariant)

( ) ( )I E x x constant

( ) [ ( ), ( )]eo

eo

u U u constant

U x u u E x x

22 2 2

2

1 2( , )

105 e

e

u Ra U x tt x Ra x x x

y

E

x

Dispersion effects include:

•EOF variation in x

•Vertical circulation in z

u ueo, 1 eo, 2

w

z

xHigh Conductivity Low Conductivity

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FASS: Model vs DNS

2 427( , ) ( )

4 30 2eRa zx z x U z

x

DNS

Model

DNS

Model

Model w/o Dispersion

DNS

Model

Model w/o Dispersion

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FASS: Model vs DNS

2 2

2 2 2

2 [ ( )]105

2 | |105

e H H

e

Ra

Ra

U U

Un n

DNS

Model

Model w/o Dispersion

Model RM

S

Time (s)

DNS

Model w/o Dispersion

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Motivation: Electrokinetic Instability (EKI)

No gradient = 10

Stable, conductivity matched condition

50 m

Unstable, fluctuating concentrations in high-conductivity-gradient case

50 m

50 m

1 mm

(Michael H. Oddy)

(C.-H. Chen)(Rajiv Bharadwaj)

STANFORD MICROFLUIDICS LABORATORY

Linear Analysis: 2D vs 3D 3D Linear Analysis

Stable

Ecr,experiment ~ 0.3 kv/cm, Ecr,2D ~ 0.04 kv/cm, Ecr,3D ~ 0.18 kv/cm

H. Lin, Storey, B., M. Oddy, Chen, C.-H., and J.G. Santiago, “Instability of Electrokinetic Microchannel Flows with Conductivity Gradients,” Phys. Fluids 16(6), 1922-1935, 2004.

2D Linear Analysis

Stable

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EKI: Linear Analysis

Model

3D Linear

2 2 23( )H H H E H eo HRe pt

u

u u u u u

zeroth-order momentum1 ( )3eo Ep u u

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EKI: Nonlinear Simulation

ExperimentModel

t = 0.0 s

t = 0.5 s

t = 1.5 s

t = 2.0 s

t = 2.5 st = 3.0 s

t = 4.0 s

t = 5.0 s

t = 1.0 s

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Conclusions and Future Work

Developed depth-averaged model for general EK flows in microchannelsModel validated with DNS and experimentsFuture work:– Modeling and optimization of realistic FASS

applications

– Modeling and optimization of EKI mixing