electrohydrodynamic (ehd) instabilities

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Electrohydrodynamic (EHD) InstabilitiesLectures 5-6: Electrohydrodynamic Cone-Jets

Chuan-Hua Chen

Dept. Mechanical Engineering and Materials ScienceDuke University,

Durham, NC 27708-0300, [email protected]

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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Primary ReferencesElectrohydrodynamics

Leaky-Dielectric Model» Melcher and Taylor 1969, Annu. Rev. Fluid Mech. 1, 111.» Melcher 1974, IEEE T. Educ. E-17, 100 (and the corresponding film).» Melcher 1981, Continuum Electromechanics, MIT Press.» Saville 1997, Annu. Rev. Fluid. Mech, 29, 27.» Castellanos 1998, Electrohydrodynamics, Springer.Flow Instabilities» Saad 1993, Compressible Fluid Flow, 2nd Ed. Prentice Hall.» Huerre and Rossi 1998, Ch.2 in Hydrodynamics and Nonlinear Instabilities

(ed. Goreche and Manneville), Cambridge » Eggers and Villermaux 2008, Rep. Prog. Phys. 71, 036601.

Electrokinetic Flow Instabilities» Melcher and Schwarz 1968, Phys. Fluids, 11, 2604.» Hoburg and Melcher 1976, J. Fluid Mech. 73, 333.» Baygents and Baldessari 1998, Phys. Fluids. 10, 301.» Lin, Storey, Oddy, Chen and Santiago 2004, Phys. Fluids, 16, 1922.» Chen, Lin, Lele and Santiago 2005, J. Fluid Mech. 524, 263.» Posner and Santiago 2006, J. Fluid Mech. 555, 1.

Electrohydrodynamic Cone-Jets » Melcher and Warren 1971, J. Fluid Mech. 47, 127.» Cloupeau and Prunet-Foch 1994, J. Aerosol Sci. 25, 1021.» Ganan-Calvo 1997, J. Fluid Mech. 97, 165.» Hohman, Shin, Rutledge and Brenner 2001, Phys. Fluids. 13, 2201; 2221.» Chen, Saville and Aksay 2006, Appl. Phys. Lett. 89, 124103.» Fernandez de la Mora 2007, Annu. Rev. Fluid Mech. 39, 217.

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Electric Stress

2 21 12 2

ef E Eρ ε ε ε⎛ ⎞= − ∇ = ∇⋅ − = ∇ ⋅⎜ ⎟

⎝ ⎠f E EE δ T

( )

( )

( )

2 2 2

2 2 2

2 2 2

12

12

12

x y z x y x z

ex y y z x y z

x z y z z x y

E E E E E E E

E E E E E E E

E E E E E E E

ε

⎡ ⎤− −⎢ ⎥⎢ ⎥⎢ ⎥= − −⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥⎣ ⎦

T

n

E

2

2Ef dSε

=

Helmholtz force density:

Maxwell stress tensor:

Electrical forces on a surface:Assumptions:• Incompressible• Electrically linear

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Jump Conditions for Surface-Coupled Model

0

0m ep

× =

⋅ =

= ⋅ +

n v

n v

n n T T21

2e Eε ε= −T EE δ

( )m Tμ= ∇ +∇T v vwhere,

Melcher and Taylor 1969, Annu. Rev. Fluid Mech. 1, 111.Saville 1997, Annu. Rev. Fluid Mech. 29, 27.

0

( ) ( ) ( ) 0s s

qq qt

ε

σ

× =

⋅ =

∂+ ⋅ ∇ ⋅ +∇ ⋅ + ⋅ =

∂

n E

n E

n v n K n E

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Electrohydrodynamic (EHD) Cone-Jet Transition

Electric stress vs. surface tension (and hydrodynamic forces)» Pendent drop → Taylor cone → Cone-jet

Large draw-down: dn/dj ~ 10-1000» Electrospray mass spectrometry» Electrospinning of nanofiber» Electrohydrodynamic drop production

Image credit: H.F. Poon (2002), Electrohydrodynamic Printing, Ph.D. Thesis, Princeton University.

E↑ E↑

EHD Cone-Jet

dn

dj

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Electrospraying and Electrospinning

Zeleny (1914) “The electric discharge from liquid points”, Phys. Rev. 3, 69.Fenn et al. (1989) “Electrospray ionization for mass spectrometry of large biomolecules”, Science, 246, 64.

ElectrosprayingKebarle et al. (1993), Anal. Chem. 65, 972A.

ElectrospinningHuang et al. (2003), Composites Sci. Tech. 63, 2223.

Formhals (1934), “Processing and apparatus for preparing artificial threads”, US Patent 1975504.Reneker et al. (1996), “Nanometer diameter fibers of polymer, produced by electrospinning”, Nanotechnology, 7, 216.

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Electrohydrodynamic Drop Formation

Link et al. (2006) Angew. Chem. Int. Ed. 45, 2556.Kim et al. (2007) Appl. Phys. Lett. 91, 133106. Yogi et al. (2001), Anal. Chem. 73,1896.

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Electrically Driven Jets: Early Experiments

Gilbert (1600): “When a piece of rubbed amber is held at a suitable distance above it, a spherical drop of water is drawn up into a cone” [1].Zeleny (1914)

[1] Taylor (1969), “Electrically driven jets”, Proc. Roy. Soc. Lond. A. 313, 453.[2] Zeleny (1914), “The electric discharge from liquid points”, Phys. Rev. 3, 69.[3] Zeleny (1917), “Instability of electrified liquid surfaces”, Phys. Rev. 10, 1.

Electrified liquid jets in air [3] Zeleny’s setup [2]

Glycerin Alcohol, 30s exposureAlcohol

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Electrically Driven Jets by G. I. Taylor

Taylor Cone [1]

» “A conical interface between two fluids can exist in equilibrium in an electric field, but only when the cone has a semi-vertical angle 49.3°”.

» Phenomenological approx. assuming Conical surface

Equipotential surface

Jet Stability [2]

» “Jets as small as 20 um in diammeterand 5 cm long were produced which were quite steady, … Attempts to describe them mathematically failed.”

[1] Taylor (1964), Proc. Roy. Soc. Lond., A. 280, 383.[2] Taylor (1969), Proc. Roy. Soc. Lond. A. 313, 453.

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Rayleigh Limit

Rayleigh limit: the max charge an isolated drop can carry» For perfect conductor (typically a good assumption for aqueous

solution), charge will be only distributed on the surface» Surface charge leads to electrostatic repulsion» When the repulsion balances surface tension, the Rayleigh

electrostatic stability limit is reached

+++

+

++

+

+

Rayleigh (1882) Philos. Mag. 14, 184.

2

0 2

30

1 22 4

8

e s

R

Rq a

T T

qa a

π ε γ

γεπ

=

⎛ ⎞

=

=⎜ ⎟⎝ ⎠

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Tangential Shear Stress on the Taylor Cone

Hayati, Bailey and Tadros, 1986, Nature, 319, 41.

Circulation within a Taylor cone

Ef

Circulation pattern exists due to tangential shear stress

Surface charge is driven to the surface by Ohmic conduction within the liquid

A tangential electric field exists because » The liquid is leaky dielectric with

finite resistance; or

» A steady current in the cone-jet case can establish a tangential field within a “perfect” conductor

0( )t n tT E Eε=n

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Electrohydrodynamic Flow Modes

Dripping

PulsatingCone-Jet

Cone-Jet

Instability

Multi-jet

Varicose

Kink

Incr

easi

ng V

olta

ge

Grace and Marijnissen (1994) J. Aerosol Sci., 25, 1005. Cloupeau and Prunet-Foch 1994, J. Aerosol Sci. 25, 1021.

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EHD Cone-Jet Stability

Pulsating cone-jet ↔ steady cone-jet» Imbalance of flow (choking in the nozzle)

What leads to superior stability of EHD jet?» Supercritical flow (choking in the jet)

Steady cone-jet ↔ jet instability» Varicose instabilities (related to electrospraying)» Whipping instabilities (related to electrospinning)

Cautionary notes about lectures 5-6:» Only selected flow stability issues related to the cone-jet mode are

reviewed. (Scaling laws will not be reviewed.) » Most models reviewed are still under development (or even debate).» “Remarkably, despite 50 years and even a Nobel prize, many

aspects of electrospraying remain elusive.” [1]

[1] Basaran and Suryo (2007), Nat. Phys. 3, 679.

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The mysterious stability of EHD jet» Supercritical flow (choking in the jet)


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University of Taylor Cone Formation Mechanism

Marginean et al. 2004, Anal. Chem. 76, 4202.

a: steady cone-jet supported on a needleb: transient cone-jet at the Rayleigh limitc: charged drop in an external e-fieldd: uncharged drop in an external e-field

Transient cone-jetsupport on a needle

Fernandez de la Mora 2007, Annu. Rev. Fluid Mech. 39, 217.

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Analogy between Steady and Transient Cone-Jets

Exploding dropSupported meniscus

Intrinsic pulsations on a supported needle are analogous totransient cone-jets on a charged drop experiencing Rayleigh fission

if the jet lifetime is much longer than charge relaxation time

Fernandez de la Mora (1996), JCIS, 178, 209.Chen, Saville, Aksay (2006), APL, 89, 124103.

Fernandez de la Mora (2007), Annu. Rev. Fluid Mech. 39, 217.

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Characteristics of Transient Cone-Jets

Transient cone-jet develops when the surface charge accumulates to a level where the charge has to be redistributed to a larger surface area.

For exploding drops:» Charge loss ratio: Δq/q ~ 10-50%

» Mass loss ratio: Δm/m ~ 1% or less

The rate at which surface charge is accumulated and ejected dictates whether the cone jet is transient or steady.

Juraschek et al. (1998), Int, J. Mass Spec., 177, 1.

Intrinsic cone-jet pulsation(due to imbalance of supply & loss rate at cone)

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Electrohydrodynamic (EHD) Drop Generation

Water

Captured at 2500 fps, 394 µs exp.

Drop formation through slender nozzle

100 µmSlowed 500X

Chen, Saville, Aksay (2006), APL, 89, 124103;

100 µmSlowed 1000X

Captured at 5000 fps, 10 µs exp.

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Drop Formation Rate Scaling

y = 360x - 16R2 = 0.991

0

10

20

30

40

50

0.00 0.05 0.10 0.15 0.20

d4E2L-1 [V2m]

Q [n

l/s]

ID =

50 µm

ID =

75 µm

ID =

100 µ

m

4 200 4~

128 2n

cn

Q Ed

dL

Pπ ε γμ

⎛ ⎞− +⎜ ⎟

⎝ ⎠

Flow rate choked by slender nozzle

Chen, Saville, Aksay (2006), APL, 89, 124103; Choi et al. (2008), APL, 92, 123109.

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Intrinsic Pulsation Scaling

0

1

2

3

4

5

0 1 2 3 4V2 (kV2)

f (kH

z)

2 50

2~pj nf dE /

Intrinsic pulsation frequency

( )3~ jpj nd dV

Minimum drop volume

3.2 ms

3.6 ms

100 µm4.0 ms

Choking leads to pulsations

Chen, Saville, Aksay (2006), APL, 89, 124103;

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Summary: Cone-Jet Pulsations

Choking at the nozzle leads to intrinsic pulsationsIntrinsic pulsations bounds speed and precision of EHD drop formationResults are applicable to nanoelectrospray where slender nozzle is used and flow rate is self-regulated

Notes: » For the pulsation frequency, a competing model exists based on the

oscillation frequency of a charged dropMarginean et al. (2006), Appl. Phys. Lett. 89, 064104.Choi et al. (2008), APL, 92, 123109.

» Steady and transient cone-jets can behave quite differently, particularly for high-viscosity fluid.

Fernandez de la Mora (2007), Annu. Rev. Fluid Mech. 39, 217.

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Melcher’s Demonstration

Surface charge on the jet is controlled by Vs on the jet and Vm on the surround wall

Melcher 1974, IEEE T. Educ. E-17, 100 (and the corresponding film).

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Supercritical vs. Subcritical Flow

Jet flow is approximated by a quasi-1D surface-coupled model with mass, momentum and charge conservation and jump conditions for mechanical/electric stresses.

Supercritical: the local jet velocity exceeds the wave velocity» Analogous to supersonic flow in which the speed of flow exceeds the

speed of sound.» Disturbances can not propagate upstream in supercritical flow.

Subcritical flow: local jet velocity is less than wave velocity» Disturbances can propagate upstream in subcritical flow

In our lectures, supercritical is referring to the condition that the velocity ratio is larger than 1. It is not referring to supercritical bifurcation. Similar notes apply to subcritical.

Melcher and Warren 1971, J. Fluid Mech. 47, 127; Ganan-Calvo 1997, J. Fluid Mech. 97, 165.

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Summary of Melcher and Ganan-Calvo’s Results

Supercritical region exists under the right set of conditionsSupercritical region of the jet supplies an impenetrable shield for the fragile conical capillary equilibrium meniscus

» Electric current is choked at the critical point and insensitive to downstream locations

» May explain the remarkable stability of electrohydrodynamic jets

Convective instability sets in at the “point of instability”.If point of instability is too close to critical point, global (absolute) instability results

Notes:» The supercritical concept has not gained

widespread acceptance yet.

Supercritical

Subcritical

ConvectiveInstability

TaylorCone-Jet

Critical Point

Point of Instability

Melcher and Warren 1971, J. Fluid Mech. 47, 127; Ganan-Calvo 1997, J. Fluid Mech. 97, 165.

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Jet Stability Enhanced by Polarization Forces

Current continuity → concentration of longitudinal field (Ez) in the constricted regions» e is perturbation electric field

Increased Ez produces increased outward-directed polarization surface force density T, which tends to restore the equilibrium radius» Interface of polarizable fluid in a

tangential e-field is dawn toward the region of lesser polarizability. Electric forces competing

with surface tension

Melcher and Warren 1971, J. Fluid Mech. 47, 127

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Quasi-1D, Surface-Coupled Model

Experimental setup

Hohman, Shin, Rutledge and Brenner 2001, Phys. Fluids. 13, 2201.

Theoretical model

2(2 ) (2 ) 0t zh h v h KEπ σ π σ π∂ + ∂ + =

2 2( ) ( ) 0t zh h vπ π∂ + ∂ =

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1 3 ( )2t z z tot z z

Eh

v v v p g h vhμσ

ρρ ρ∂ + ∂ = − ∂ + + + ∂ ∂

2 20

1 2 0

( )1 12 2tot

EpR R

ε ε σγε

⎛ ⎞ −= + − −⎜ ⎟

⎝ ⎠

1,2

: ( ), axial electric field Eqn derived for slender jet

: Surface charge density: Surface tension: Conductivity

: Principal radii of curvature

E E Z

KR

σγ

polarization force

electrostatic repulsion

tangential electric force

Troutonviscosity

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Model for Whipping Jet

Varicose mode

Whipping mode


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Varicose vs. Whipping Modes

1 kV/cm 5 kV/cm

(1) Axisymmetric Rayleigh mode (solid line)(2) Axisymmetric conducting mode (solid line)(3) Whipping conducting mode (dashed line)


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Summary


Varicose

Whipping

Solid line: numerical predictionDots: experimental data on glycerol jets

Three modes are identified;:» Rayleigh mode» Varicose conducting mode» Whipping conducting mode

1D, surface-coupled model reproduced much of prior literature with 2D modelsWhipping modes dominate at high field (experimental confirmed)

Notes:» Non-newtonian effects are

not considered; seeReneker et al. 2000, J. Appl. Phys., 87, 4531.

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Concluding Remarks

EHD instabilities in electrokinetic microsystems and electrohydrodynamic cone-jets» Leaky-dielectric model» Surface- and bulk- coupled» Hydrodynamic instabilities

Acknowledgements» Duke Pratt IT (Marc, John)» Duke μPHYL Lab (Chris, Jean, Jonathan, Shenren, Yuejun)» Juan Santiago» Antonio Ramos» All of you for your patience! See you!

electrohydrodynamic (ehd) instabilities

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