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Electrohydrodynamic (EHD) Instabilities Lectures 3-4: Electrokinetic Flow Instabilities

Chuan-Hua Chen

Dept. Mechanical Engineering and Materials ScienceDuke University,

Durham, NC 27708-0300, USAchuanhua.chen@duke.edu

CISM Advanced School on Electrokinetics and Electrohydrodynamics in Microsystems, Udine, Italy, June 22-26, 2009

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Outline

� Leaky-dielectric model» Ohmic model derivations» Maxwell stresses» Jump conditions» Applications in microsystems: the high-conductivity, small-scale limit

� Electrokinetic flow instabilities» Bulk-coupled model» Temporal, convective and absolute instabilities» EHD instabilities with electroosmotic convection» Applications in electrokinetic assays and micromixing

� Electrohydrodynamic cone-jets» Surface-coupled model» Choking: supercritical flow and pulsating jet» Varicose and whipping instabilities» Applications in droplet microfluidics and electrospinning

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Electrohydrodynamics in Microsystems

� Charge relaxation is instantaneous » Typically aqueous electrolyte (high-conductivity limit)

� Diffusive process becomes important» Bulk-coupled model for miscible interfaces (small length scale)

� Electromechanical coupling

( ) 0 ( ) 0fDDt�

� �� � �� � �� �EE

2( ) 0 ( ) e f fDtt� �� �� � �� � �� �

� �

�v v

* 2 212fED p

Dt� � � � � �� �� � �v v E

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Polarization Force

bf

af

a < b , � = 0

Melcher 1974, IEEE T. Educ. E-17, 100.

20( )

2( )b a

b a

Ehg

� ��

��

Polarization force driving the rise ofdielectric liquid

E

f

The interface of a polarizable fluid in a tangential electric field is drawn toward the region of lesser polarizability (to minimize energy).

a

b

Medium

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X-junction (Micralyne, 50x20 µm2)under same voltage scheme

0.1 10c1/c2 = 1

FlowDirection

c2 c2

c1

50 µm

Electrokinetic Flow Instability

� Electrokinetic instability is undesirable for robustness» “Catastrophic leakage” with ionic

strength mismatch [1]

» “Further increase (in ionic mismatch) … resulting in a surprising decrease in separation efficiency” [2]

� Gradients in EK systems» Designed (sample stacking)

» Unavoidable (2D assay)

� What is the instability mechanism?

[1] Shultz-Lockyear et al. (1999), Electrophoresis 20, 529.[2] Dang et al. (2003) Anal. Chem. 75, 2439.

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Electrokinetic Instability Mixing

[1] Oddy, Santiago and Mikkelsen, Anal. Chem. 2001, 73, 5822.[2] Lin, Storey, Oddy, Chen and Santiago 2004, Phys. Fluids, 16, 1922.

EK instability mixingat 5Hz AC-field [1]

EK instability mixingat DC field [2]

� What is the role of alternative electric field (instability time scale)?� What is the role of electrokinetics (electroosmotic flow)?

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Cross section: 150x10 µm2

150 µm

EK Instability in High Aspect-Ratio T-junction

Real time

Instability originates at the intersection (in the bulk)

1 mM BorateV1

V210 mM Borate

GND

t

V1 = V2

t = 0

Vcr

Chen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.

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Gradient Induced Convective Instability

� Gradient induced» Stable without gradients up to

1.5 kV/cm

» Instability originates at intersection (highest gradient)

� Threshold electric field» Nominal Ea,cr = 0.5 0.1 kV/cm

� Convective instability» Disturbances grows downstream

» Propagation speed proportionalto electric field

Chen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.

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Scalar Perturbation Energy

Downstream location

Sca

lar p

ertu

rbat

ion

ener

gy 1.5 kV/cm1.25 kV/cm

1.0 kV/cm

1.5 kV/cm

1.25 kV/cm

1.0 kV/cm

RMS Intensity showing perturbation boundary

Chen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.

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Hypothesis of Instability Mechanism

� Electrohydrodynamic instability» Instability originates at intersection; not at double layer

� For various aspect ratios, electrolyte concentrations, wall materials

� Induced by conductivity gradients» Unstable only with concentration (conductivity) gradients

� Coupled with electroosmotic convection» Disturbance grows downstream; propagates at speed � E-field

� More hints from E-field dependence» Threshold E-field: instability waves set in (onset of instability)» Moderate E-field: exponential spatial growth (convective instability)» High field: constant perturbation energy (absolute instability)

11Huerre & Rossi (1998), In Hydrodynamics and Nonlinear Instabilities, pp. 81-294. Cambridge Univ. Press.

Linearly Stable Temporally unstable

Instability Basics

� Temporal & convective instabilities share the same threshold

• In convective framework, instability classified as� Convective (amplifier)� Absolute (oscillator)

Linearly Stable Convectively unstable Absolutely unstable

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Instability Examples

Temporal InstabilityKelvin-Helmholtz Instability

in closed flow

Convective InstabilityKelvin-Helmholtz Instability

in open flow

Absolute InstabilityVon Karman Vortex Street(wake behind a cylinder)

www.youtube.com

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Glass wall

Glass wall

AqueousSolution

Electroosmosis (EO)

Ea

�D � c-1/210 nm @ 1 mM

Electroosmoticvelocity profile

U � �EaU ~ 1 mm/sBulk:

electro-neutral

Electricdoublelayer

Electricdoublelayer

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Electrokinetics (EK) vs. Electrohydrodynamics (EHD)

� Electrokinetics» Electroosmosis & electrophoresis

» Electric field acts on interfacialcharges in electric double layer

» Velocity scaling: VEK � E

� Electrohydrodynamics» System originally neutral

» Electric field acts on induced (orinjected) charges in bulk fluid

» Velocity scaling: VEHD � E2

+ -Electrophoresis

Electrokinetics

Electrohydrodynamics

EO

+ -

M1

M2

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Temporal Instability Grows as Electro-viscous Time

Lin, Storey, Oddy, Chen and Santiago 2004, Phys. Fluids, 16, 1922.

2 22 ~ ~ ~~evf

LU

EL E

U EL

��� �

�� � �v E

Maximum transverse velocity perturbation

• Electroviscous time by balancing electric and viscous forces

• Typically �ev ~ 10 ms in electrokinetic instabilities- Low frequency AC can be approximated as steady field

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Permittivity Gradient

� For aqueous solution with moderate ionic concentrations (~10 mM), the permittivity is approximately the same as pure water [1].

� Permittivity gradient (w/o conductivity gradient) orthogonal to external field is stabilizing in the surface-coupled model [2]» Perturbation tends to shield electric field out of the fluid of high permittivity

» Perturbation polarization forces tends to restore the perturbed interface

[1] Pottel 1973, Dielectric properties. In Water: a Comprehensive Treatise (ed. F. Franks), vol. 3, ch. 8, Plenum.[2] Melcher and Schwarz 1968, Phys. Fluids, 11, 2604.

F: polarization force density

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Surface Coupled Model

� For surface coupled model: conductivity gradient only does not yield any unstable mode in the linear stability limit [1,2].» No perpendicular perturbation force exists because the free-charge force is

in the direction of applied E-field

� Relaxation of interfacial free charge leads to overstability (oscillatory marginal stability) [1].» Hexane doped with ethyl alcohol

� = 2 x 10-11 C/(V�m); � = 3 x 10-8 S/m; � = 3.2 x 10-4 Pa�s; E = 3 x 105 V/m

» Charge relaxation time ~ 1 ms» Electroviscous time ~ 0.2 ms

� Instantaneous charge relaxation for aqueous solution » Deionized water � ~ 1x10-4 S/m

� Sharp interface is unrealistic at small length scales» Diffusivity of ions D ~ 10-9 m2/s � diffusion length = 1 �m in 1 ms

[1] Melcher and Schwarz 1968, Phys. Fluids, 11, 2604.[2] Hoburg and Melcher 1976, J. Fluid Mech. 73, 333.

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Lecture 3 Summary

� Electrokinetic effects» Electroosmotic flow may be just a convecting medium.» Electro-diffusion equations are too stiff to solve; Stiffness relieved if

leaky dielectric model is adopted to account for bulk properties.

� AC field (inconsequential at low frequency)� Charge relaxation effects (negligible)� Permittivity gradient (negligible)� Conductivity gradient induced instability?

f f

e

DDt� � � �

� �

� �� �� � � �� �

� �E

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Electrolytic Ohmic Model

• Approx. neutrality:

Equivalent diffusivity:

• Avoids stiffness of species conservation (Nernst-Planck) eqns

Navier-Stokes Eqns

0�� �v

2( ) fpt�� � � � �� � �� � � �

v v v v E

2( ) e f ftD �� � � �� � �

v

( ) 0��� �E

“Conservation” of conductivity

Current continuity

1f c cc c

��

� �

� �

��

� �

Gauss’ Law

Levich (1962), Physicochemical hydrodynamics.Lin et al. (2004), Physics of Fluids, 16, 1922.

Ef��

��

� � �

Hoburg & Melcher (1977), Phys. Fluids, 20, 903.Chen et al. (2005), J. Fluid Mech. 524, 263.

Governing Equations for Bulk Fluid

2e f f

D DDD D

� �

� �

��

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Linearized Thin-Layer Model

� Linear stability analysis» Uniform diffusion layer

» Spatial framework (mostly)

� Thin-layer limit» Depth-averaging through

asymptotic expansion� Hele-shaw type parabolic

velocity profile in z-direction

» Electro-viscous velocity

� Analytical potential-streamfunction relation

» Without electroosmosis

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d

dqdz

d ��

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0 0H fdp� �� � � �v E

� �232 0H a yd E�� �� � �y� ��

( , , ) ( )x y t g z� �� ��� �

Chen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.

22 2

2

( 1)( 1)

aev

EdU �� ��

��

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Electrohydrodynamic Instability (W/O Electroosmosis)

Ea

�L

�H

y xv �� Analytical solution:

E-fieldPerturbationv

v

1eeff

evU hRaD

� !2

effev

h hU D

"

Electroviscous time < Diffusion time

Unstable if

Chen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.

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Convective Electrokinetic Instability

�L

�H

Convective Instability Conductivity Perturbation

Potential Perturbation Stream Function Perturbation

� Electroosmotic flow as convective medium» Instability onset still governed by Rae

� Zeta dependence on concentration (or conductivity) not essential for instability

0( , , ) ( ) ( , )iwtq x y t eq y q x y ��� �

( )( , , ) ( ) I Ri k x wtk xq x y t q y e e� �� � �

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Convective Stability Diagram

Wave #

Growthrate

Frequency

Ele

ctric

Ray

leig

h #

Rae,cr = 11

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Conductivity Potential

Charge density

Stream function

Most Unstable Eigen Mode

Chen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.

Ey-fieldy� ��

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Zeta Dependence on Conductivity

� Empirical power law» n =1/4 for borate buffer

� Non-uniform base flow� Electroosmotic velocity

perturbation due to» Electric field perturbation» Conductivity perturbation

n

r r

� �� �

�� �

� � �� �

* * *eo xn�� �v eE

eo

��

�E

v

• Zeta dependence on conductivity not essential

Chen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.

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Absolute Electrokinetic Instability

• Upstream not perturbedConvectively unstable

• Upstream perturbedAbsolutely unstable

Absolute if 1v eveo

URU

� !ev eo

h hU U

"

Electroviscous time < Convection time

Huerre & Rossi (1998) Hydrodynamics and Nonlinear Instabilities (ed. C. Godreche & P. Manneville), pp. 81, Cambridge.

V+V- > 0 V+V- < 0

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� Selection of physical modes» Briggs test based on the principle of causality

� Detection of absolute instability» Briggs-Bers zero group velocity criteria» Saddle point detection vs. cusp point detection

Convective instability Absolute instability

Briggs-Bers Criteria

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Briggs-Bers Criteria

Absolute Stability: Cusp Point vs. Saddle Point

Frequency

Spa

tial g

row

th ra

te VelocityRatio

Rv,cr = 4.9

Chen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.

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Base state

Perturbed state

Worst case scenario

��: conductivity ratio)

0

0

~f��

��

� �E

02 ~ fevUd� � E

2

2

22( 1)( 1)

aev

dU E��

�

��

�

Viscous ~ electric stresses

Electroviscous velocity

11 aE�

��� 1

11�� ��

Gauss’ Law

Chen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.Posner and Santiago 2006, J. Fluid Mech. 555, 1. [the handling of conductivity ratio is slightly different]

Electroviscous Velocity Scaling

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Stability Phase Diagram

Nine parameters collapsed into two groups» Rae governing onset of convective instability » Rv governing onset of absolute instability

Rae,crRv,cr

Rv

Ra e

2 22

2

( 1)( 1)e e ff eff

aev EU h d hRaD D

�� �

�� �

�2 2

2

( 1)( 1)

av

ev

eo

U dRU

E �� �

�� �

�

ConvectiveA

bsol

ute

Convective stability

Absolute stabilityChen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.

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Theory vs. Experiments

� Theoretical assumptions» Linearized, depth averaged» Uniform diffusion length (h/4)

� Theory vs. experiments» Trends captured at increasing E-field

� Quantitatively within a factor of three

» Convective instability threshold� Exper.: Ea,cr = 0.5 kV/cm � Theory: Ecr = 0.14 KV/cm

» Absolute instability� Exper.: Ea = 1.5 kV/cm (indication)� Theory: Ecr = 0.65 KV/cm

� Controlling parameters: Rae, Rv» Verified by follow-up numerical and

experimental studies

0.5 kV/cm

1.0 kV/cm

1.5 kV/cmChen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.

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ExperimentSimulation

0.0 s

0.5 s

1.5 s

2.0 s

2.5 s

3.0 s

4.0 s

5.0 s

1.0 s

Lin, Storey, Oddy, Chen and Santiago 2004, Phys. Fluids, 16, 1922.

Nonlinear Simulation vs. Experiment

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Instability in Electrokinetic Flow Focusing

Center streamless conducting

Center streammore conducting

Rayleigh # (Rae) predicts the onset for the case where the center stream is more conducting, but fails for the opposite configuration.

Posner and Santiago 2006, J. Fluid Mech. 555, 1.

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Summary

� Electrokinetic instability mechanism» Conductivity gradient induced » Electrohydrodynamic instability» Coupled with electroosmotic convection

� Controlling parameters collapsing nine physical variables» Rae: onset of instability

� Robustness of electrokinetic analysis

» Rv: onset of absolute instability� Promotion of microfluidic mixing