Instabilities of Electrically Forced Jets

Moses Hohman (Univ. of Chicago Thoughtworks)

Michael Shin (Materials Science, MIT)

Greg Rutledge (Chemical Engineering, MIT)

I hate Computers

David QuereIMA WorkshopJanuary, 2001

Electrospinning is complicated

Electrohydrodynamics, evaporation, rheology, air drag, electrostaticswetting, solid-liquid charge transfer, temperature gradients, etc.

Which factors influence the final product?

The Product:

The Physics:

Approaches:

(1) Experimental: Try to control various processes, in hope thatsomething jumps out.

(2) Numerical simulations. Include all physical factors and try to understand which dominate.

(3) “Theory”. Understand a single effect quantitatively. Do not “curve fit” results to experiments but instead try to assess how much of the physics stems from this effect.

Caveats:Free parameters are absolutely unacceptable.

Numerical simulations of parts of the system always necessary.

A principle advantage of “theory” as opposed to numerical simulations and experiments is that

one also studies what does not happen.

I. Procedure for calculating instability thresholds (flavor)

II. Difficulties

III. Applications. Electrospinning, etc.

“Strange” effects in Fluid Conductors:

(1) Surface Charge Density + Tangential Electric field

Tangential Electrical Stress.

In a fluid, this must be balanced by viscous stress (flow).

Both viscosity and conductivity are singular parameters.

(2) A Non-Ohmic mechanism for conduction:

G.I. Taylor, 1964 (78 years old)

h

QEKhI 22

h(z)K

Stability of a thinning Jet:.

(1) Locally jet is a cylinder (constant radius h, surface charge

Find h

(2) Find global shape:

(h(z),z),E(z))

(3) Piece together stability properties along the jet

Previous Work on Linear Stability of uncharged cylinders

(Mestel JFM 1994,1996)

K 0 1

0

1

Nayyar andMurty, 1960

Saville (1970)

Saville (1972)Saville (1972) Saville (1970)

Nayyar andMurty, 1960

Saville (1972)

Nayyar andMurty, 1960

Experiments: Must Include Surface Charge

Experiments

Electrostatics

+

+

-

-

E

Line Dipole + Line Charge

P(z) (z)dielectric dielectric

free charge D free charge

Long wavelength Instabilities

varicose whipping

h

h<<

Whipping Mode: the electrostatics

3|)(|

))(()(

|)(|

)()()(

srx

srxsPds

srx

sdsxx

Field from aline charge

Field from aline dipole

E

P

determined by matching outside field to field inside the jet. (and using Gauss’ Law)

R

hhnE

hP D

022

2

2ˆ

)(2

)(

E.G:

dielectric polarization

dipolar free charge density

Whipping Mode: the fluid mechanics

centerlinetzr ),(

Kds

dTrh 2 External Forces:

Surface Tension+Electrical Stress

acceleration

Force Balance

Torque Balance

0ˆ NTtds

Md

Local Couple:Electrical Stresses

Bending Moment: viscous (Maha)

dielectric M

E

Perfect Conductor: Waves

0

)4

)(( 2

222

XEh

hXh s

++

++--

--

spring

E

++

++--

--

Finite K:Tangential Stresses Drive Whipping Instability

....4

)4

)((

222

222

2

XK

Eh

XEh

hXh

st

s

torque-producing instability

Comparison with Saville (1972)

• inviscid• K=0.7

• no charge density

10-2 10-1 100 10110-6

10-5

10-4

10-3

10-2

10-1

100

Re

k

E

There is also an unstable varicose mode.

The mechanism is not the Rayleigh instability, but is electrically driven.

Varicose

Have 2 Unstable (Electrically Driven) Modes:

Who wins at high field?

Phase Diagrams

2% solution of PEO in waterE=2 kV/cm

AI

mlQPcmSK

10

min/15,7.16,/120

-0.3

-0.2

-0.1

0

0.1

0.2

-1 -0.5 0 0.5 1 1.5 2 2.5-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

log10

( / (esu / cm 2))

log 10

(h /

cm)

Whipping

Varicose

-0.3

-0.2

-0.1

0

0.1

0.2

-1 -0.5 0 0.5 1 1.5 2 2.5-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

log10

( / (esu / cm 2))

log 10

(h /

cm)

Whipping

Varicose

Phase Diagrams

AI

mlQPcmSK

10

min/15,7.16,/120

2% solution of PEO in waterE=2 kV/cm

Phase Diagrams

Varicose

Whipping

2% solution of PEO in waterE=2 kV/cm

AI

mlQPcmSK

10

min/15,7.16,/120

I

Qh

2

Phase Diagrams

2% solution of PEO in water AI

mlQPcmSK

10

min/15,7.16,/120

Varicose

Whipping

Whipping Mode

jet

2),,(),(

Q

hEhQE

2 4 6 8 10 12 14 16 18

0.5

1

1.5

2

2.5

3

3.5

Q (ml/min)

E (

kV

/cm

)

BENDING

BENDING + VARICOSE

VARICOSE STEADY JET

Viscosity Viscosity/10

Conclusions

The procedure quantitatively capture aspects of electrospinning.

Honest comparisons with experiments allow us to hone in on subtle details.

The Ideas are fairly general. Should have applicibility to many otherProblems.