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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