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A Dynamical Model of Molecular Monolayers:

Why Tethers Don’t Snap?

Lu Zou,* Violeta Beleva,* Andrew J. Bernoff,# James C. Alexander,+ J. Adin Mann Jr.! Elizabeth K. Mann*

*Dept. of Physics, Kent State University

# Dept. of Mathematics, Harvey Mudd College

+ Dept of Mathematics, Case Western Reserve University

! Dept of Chemical Engineering, Case Western Reserve University

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Relaxation of 8CB on Water/Air Interface

Why Don’t Tethers Snap?

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• Introduction on Rayleigh instability (3D) and Hele-Shaw flow (2D)

• A dynamic model of molecular monolayers (2D)

• Simulation and experimental results

• Conclusion and prospects

OVERVIEW

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Rayleigh Instability [1878]

• Pure, cylindrical 3D fluid

• Varicose mode fluctuations

• Decrease area/surface energy

• Break into droplets

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Hele-Shaw Cell

Height of gapconstrains

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Evolution of a long, narrow bubble

Ref: Glasner, KarlA diffuse interface approach to Hele-Shaw flowNONLINEARITY 16 (1): 49-66 JAN 2003

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A dynamic model of molecular monolayers

Z = 0

Ω

Subphase fluid

Z

Fundamental Hydrodynamic Equations

• Stokes Equation

• Continuity Equation

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Assumptions on the subphase fluid

• Horizontal flow

• Boundary condition

• Bulk viscosity ηbulk [Ref]

Ref: Elizabeth K. MannHydrodynamics of Domain Relaxation in a Polymer MonolayerPRE 51 (6): 5708-5720 JUN 1995

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Assumptions on the surface

• 2D Fluid (η and KG)

• One component [Ref1]:

– Elasticity KG [Ref1]:

– Surface pressure Π

• Surface Viscosities [Ref2]:• Electrostatic forces

Ref1: H. A. Stone; H. M. McConnell; Proc. R. Soc. Lond. A 448: 97-111 1995

Ωgas

liquid

Ref2: Elizabeth K. Mann; PRE 51 (6): 5708-5720 JUN 1995

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Result on Small Distortion Limit For 2D

Ref: H. A. Stone; H. M. McConnellHydrodynamics of quantized shape transitions of lipid domainsProc. R. Soc. Lond. A 448: 97-111 1995

(n=2)

wL

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Lubrication Theory

X

H(x, t)

Ref: L. Zhornitskaya; A. L. BertozziPositivity-preserving numerical schemes for lubrication-type equationsSIAM J. NUMER. ANAL. 37(2): 523-555 2000

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Simulation result

Initial state:

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Discussion on the Simulation

• Periodic Boundary condition

• No ends

What constrains should be applied at the ends of the tether?

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Hole Closing

Poly(dimethyl)siloxane (PDMS) monolayer on water/air interface

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Conclusion

• A simplified model with assumptions close to the real experimental conditions

Prospect

• Line tension determination• Entire range of the relaxation behavior

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Acknowledgement

• Dr. Elizabeth K. Mann (Kent State University)

• Dr. Andrew J. Bernoff (Harvey Mudd College)

• Dr. James C. Alexander (Case Western Reserve University)

• Dr. J. Adin Mann Jr. (Case Western Reserve University)

• Ms. Violeta Beleva (Kent State University)

• Ms. Ji Wang (Kent State University)

• Supported by National Science Foundation under Grant No. 9984304

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Frequent Questions

• Brewster Angle Microscope (set-up)• Green Function Hele Shaw• F(n=2)=5PI/16 (Stone); F(n=2)=5PI/12

• Hole closing, linearly

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Brewster Angle Microscope (set-up)

CCD

Water Surface

L2L1

AP

B

Ei

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Hole Closing Linearly