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Tail wags the dog: Macroscopic signature of nanoscale interactions at the contact line

Tail wags the dog: Macroscopic signature of nanoscale interactions at the contact line. Len Pismen Technion, Haifa, Israel. Outline. Nanoscale phenomena near the contact line Perturbation theory based on scale separation Droplets driven by surface forces Self-propelled droplets.

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Tail wags the dog: Macroscopic signature of nanoscale interactions at the contact line

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  1. Tail wags the dog:Macroscopic signature of nanoscale interactions at the contact line Len Pismen Technion, Haifa, Israel Outline • Nanoscale phenomena near the contact line • Perturbation theory based on scale separation • Droplets driven by surface forces • Self-propelled droplets

  2. Hydrodynamic problems involving moving contact lines • (a) spreading of a droplet on a horizontal surface • (b) pull-down of a meniscus on a moving wall • (c) advancement of the leading edge of a film down an inclined plane • (d) condensation or evaporation on a partially wetted surface • (e) climbing of a film under the action of Marangoni force (a) (b) (c) (e) (d)

  3. Contact line paradox:Fluid-dynamical perspective normal stress balance: determine the shape Dynamic contact angle differs from the static one. Stokes equation no slip multivalued velocity field: stress singularity Use slip condition to relieve stress singularity. molecular-scale slip length

  4. Physico-chemicalperspective Diffuse interface thermodynamic balance: determines the shape variable contact angle Stokes equation + intermolecular forces precursor (nm layer) Kinetic slip in 1st molecular layer interaction with substrate  disjoining potential

  5. Kavehpour et al, PRL (2003) bulk precursor MD simulations, PRL (2006)

  6. Film evolution – lubrication approximation involves expansion in scale ratio eq. contact angle technically easier but retains essential physics Mass conservation: Pressure: Generalized Cahn–Hilliard equation • surface disjoining gravity • tension pressure • disjoining pressure is defined by the molecular interaction model • mobility coefficient k(h) is defined by hydrodynamic model and b.c.

  7. Disjoining potential (computed by integrating interaction with substrate across the film) partial wetting vdW/nonlocal theory 0 precursor thickness complete wetting polar/nonlocal theory

  8. Mobility coefficient (computed by integrating the Stokes equation across the film) ln k diffuse interface sharp interface k=h3/3 h

  9. Configurations: a multiscale system length scales differ by many orders of magnitude! h droplet meniscus bulk precursor hm precursor R precursor horizontal slip region bulk

  10. Multiscale perturbation theory dynamic equation dimensionless quasistationary equation small parameter – Capillary number expand Inner equation Outer equation precursor: zero order: static solution macroprofile gives profile near contact line V = 0: parabolic cap dry substrate: assign

  11. Passive Interacting Active chemically reacting Moving droplets

  12. Numerical slip:NS computations(O. Weinstein & L.P.) grid refinement ln (cR/)

  13. Larger drops change shape upon refinementNS computations(O. Weinstein & L.P.) higher refinement

  14. Solvability condition: general quasistationary equation expand 1st order equation inhomogeneity linear operator adjoint operator translational Goldstone mode solvability condition solvability condition in a bounded region

  15. Solvability condition – dry substrate area integral friction factor bulk force contour integral contour force F solvability condition defines velocity

  16. Friction factor (regularized by slip) contact line bulk add up bulk R slip region log of a large scale ratio (can be replaced byhm)

  17. Motion due to variable wettability driving force variable part of contact angle velocity

  18. T>Tml Time LV LV SVa SLa a SLb SVa b Surface freezing experiment, Lazar & Riegler, PRL (‘05)

  19. Surface freezing experiment, Lazar & Riegler, PRL (‘05) simulation, Yochelis & LP, PRE (‘05)

  20. Surface freezing stable at obtuse angle

  21. Self-propelled droplets(Sumino et al, 2005) Chemical self-propulsion(Schenk et al , 1997)

  22. Adsorption / Desorption rescaled length rescaled velocity H = 1 H = 0 H = 1 dimensionless eqn in comoving frame concentration on the droplet contour

  23. Self-propulsion velocity traveling bifurcation a=4 a=2 a=1

  24. Traveling threshold from expansion at : a mobility interval a immobile when diffusion is fast a

  25. Non-diffusive limit

  26. Size dependence (no diffusion) saturated nonsaturated experiment capillary number vs. dimensionless radius

  27. Scattering far field standing moving scattering angle

  28. Solvability condition – precursor translational Goldstone mode area integral perturbation of contact angle related to perturbation of disjoining pressure transform area integral to contour integral

  29. Inner solution – precursor integrated form: 1d: zero-order: static scaled by precursor thickness hm =1; fit to =1 e.g. boundary conditions: “phase plane” solution (n=3) h

  30. Friction factor (2D) (regularized by precursor) • contact line region: use here static contact line solution • droplet bulk: use spherical cap solution • add up: NB: logarithmic factor bulk and contact line contributions cannot be separated in a unique way

  31. Friction factor (3D) (regularized by precursor) • contact line region: multiply local contribution by cos and integrate • (is the angle between local radius and direction of motion) • droplet bulk (spherical cap) • add up: NB: logarithmic factor bulk and contact line contributions cannot be separated in a unique way

  32. Interactions through precursor film flux larger drop in equilibrium with thinner precursor flux smaller droplet is sucked in by the big one larger droplet is repelled in by the small one ripening flux smaller droplet catches up

  33. Mass transport in precursor film • negligible curvature • almost constant thickness • quasistationary motion Spherical cap in equilibrium with precursor: film thickness distribution created by well separated droplets:

  34. Migration on precursor layer driving force on a droplet due to local thickness gradient droplet velocity: flux

  35. Migration & ripening

  36. Conclusions • Interface is where macroscopic meets microscopic; this is the source of complexity; this is why no easy answers exist • Motion of a contact line is a physico-chemical problem dependent on molecular interaction between the fluid and the substrate • Near the contact line the physical properties of the fluid and its interface are not the same as elsewhere • The influence of microscale interactions extends to macroscopic distances • Interactions between droplets and their instabilities are mediated by a precursor layer • There is enormous scale separation between molecular and hydro dynamic scales, which makes computation difficult but facilitates analytical theory

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