1 Electrohydrodynamic (EHD) Instabilities Lectures 5-6: Electrohydrodynamic Cone-Jets Chuan-Hua Chen Dept. Mechanical Engineering and Materials Science Duke University, Durham, NC 27708-0300, USA [email protected]CISM Advanced School on Electrokinetics and Electrohydrodynamics in Microsystems, Udine, Italy, June 22-26, 2009
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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
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°”.
» “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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EHD Cone-Jet Stability
Pulsating cone-jet ↔ steady cone-jet» Imbalance of flow (choking in the nozzle)
The mysterious 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)
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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)
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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EHD Cone-Jet Stability
Pulsating cone-jet ↔ steady cone-jet» Imbalance of flow (choking in the nozzle)
The mysterious 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)
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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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EHD Cone-Jet Stability
Pulsating cone-jet ↔ steady cone-jet» Imbalance of flow (choking in the nozzle)
The mysterious 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)
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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∂ + ∂ =
22
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
Hohman, Shin, Rutledge and Brenner 2001, Phys. Fluids. 13, 2201.
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Summary
Hohman, Shin, Rutledge and Brenner 2001, Phys. Fluids. 13, 2221.
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!