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

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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

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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)

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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