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 Quere IMA Workshop January, 2001

Electrospinning is complicated Electrohydrodynamics, evaporation, rheology, air drag, electrostatics wetting, 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 that something 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(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 Nayyar and Murty, 1960 Saville (1970) Saville (1972) Saville (1970) Nayyar and Murty, 1960 Saville (1972) Nayyar and Murty, 1960 Experiments: Must Include Surface Charge Experiments

Electrostatics E Line Dipole + Line Charge  P(z) (z) dielectric free charge  D free charge 

Long wavelength Instabilities varicose whipping h h<<

Whipping Mode: the electrostatics Field from a line charge Field from a line dipole  P determined by matching outside field to field inside the jet. (and using Gauss’ Law) E.G: dielectric polarization dipolar free charge density

Whipping Mode: the fluid mechanics External Forces: Surface Tension+ Electrical Stress acceleration Force Balance Torque Balance Local Couple: Electrical Stresses Bending Moment: viscous (Maha) dielectric

Perfect Conductor: Waves spring

Finite K:Tangential Stresses Drive Whipping Instability torque-producing instability

 Comparison with Saville (1972) inviscid K=0.7 no charge density 10 0 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 water E=2 kV/cm log 10 (  / (esu / cm 2 )) log 10 (h / cm) Whipping Varicose

log 10 (  / (esu / cm 2 )) log 10 (h / cm) Whipping Varicose Phase Diagrams 2% solution of PEO in water E=2 kV/cm

Phase Diagrams Varicose Whipping 2% solution of PEO in water E=2 kV/cm

Phase Diagrams 2% solution of PEO in water Varicose Whipping

Calculating the Jet Shape Momentum Balance field from jet + images external field ( h(z),  z),E(z)) = F(,Q)

What sets the current? Mathematical Fact: There is unique solution of equations given surface charge density at nozzle:  Our procedure: Iterate calculations for jet shape w/ experiments. Result: Can only find theoretical steady state profiles at the ( , Q) observed experimentally only if  

E = 5kV/cm; Solid Line: Image processed experiment Q= 1ml/minDashed Line: Calculation. A Comparison: A Disaster...

Q E Agreement much improved when Including Nozzle Fringe Fields Shape of nozzle is important for quantitative thresholds

Lessons (1)  (0)=0 gives best theoretical description of experiments. Why? (2) Nozzle fringe fields strongly affect (h(z),  z), I). (3) Whipping versus Dripping depends strongly on (h(z),  z)). “Dirty Details” are very significant in determining properties of Spinning (and hence the fabric)

Whipping Mode

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 other Problems.