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1 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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2 Relaxation of 8CB on Water/Air Interface Why Don’t Tethers Snap?

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3 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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4 Rayleigh Instability [1878] Pure, cylindrical 3D fluid Varicose mode fluctuations Decrease area/surface energy Break into droplets

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5 Hele-Shaw Cell Height of gap constrains

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6 Evolution of a long, narrow bubble Ref:Glasner, Karl A diffuse interface approach to Hele-Shaw flow NONLINEARITY 16 (1): JAN 2003

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7 A dynamic model of molecular monolayers Z = 0 Ω Subphase fluid Z Fundamental Hydrodynamic Equations Stokes Equation Continuity Equation

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8 Assumptions on the subphase fluid Horizontal flow Boundary condition Bulk viscosity η bulk [Ref] Ref:Elizabeth K. Mann Hydrodynamics of Domain Relaxation in a Polymer Monolayer PRE 51 (6): JUN 1995

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9 Assumptions on the surface 2D Fluid (η and K G ) One component [Ref1] : –Elasticity K G [Ref1] : –Surface pressure Π Surface Viscosities [Ref2] : Electrostatic forces Ref1:H. A. Stone; H. M. McConnell; Proc. R. Soc. Lond. A 448: Ω gas liquid Ref2:Elizabeth K. Mann; PRE 51 (6): JUN 1995

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10 Result on Small Distortion Limit For 2D Ref:H. A. Stone; H. M. McConnell Hydrodynamics of quantized shape transitions of lipid domains Proc. R. Soc. Lond. A 448: (n=2) w L

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11 Lubrication Theory X H(x, t) Ref:L. Zhornitskaya; A. L. Bertozzi Positivity-preserving numerical schemes for lubrication-type equations SIAM J. NUMER. ANAL. 37(2):

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12 Simulation result Initial state:

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13 Discussion on the Simulation Periodic Boundary condition No ends What constrains should be applied at the ends of the tether?

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

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15 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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16 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

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17 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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18 Brewster Angle Microscope (set-up) CCD Water Surface L2L2 L1L1 A P BB EiEi

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19 Hole Closing Linearly

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