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
4/20/2017

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)
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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
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4. Physico-chemical perspective
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
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5. Kavehpour et al,
PRL (2003)
bulk
precursor
MD simulations, PRL (2006)
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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.
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7. Disjoining potential
(computed by integrating interaction with substrate across the film)
partial wetting
vdW/nonlocal theory
precursor thickness
complete wetting
polar/nonlocal theory
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8. Mobility coefficient
(computed by integrating the Stokes equation across the film)
ln k
diffuse interface
sharp interface
k=h3/3 h
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9. Configurations: a multiscale system
region
angle scale
length scale
length scales differ by many orders of magnitude!
precursor
 q hm
contact line q
hm or b
bulk
R or  h
droplet
meniscus
bulk hm
precursor
precursor R
precursor
horizontal
slip region
bulk
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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
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11. Moving droplets Passive Interacting Active chemically reacting
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12. Numerical slip: NS computations (O. Weinstein & L.P.)
grid refinement
ln (cR/)
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13. Larger drops change shape upon refinement NS computations (O
Larger drops change shape upon refinement NS computations (O. Weinstein & L.P.)
higher refinement
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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
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15. Solvability condition – dry substrate
area integral
friction factor
bulk force
contour integral
contour force F
solvability condition defines velocity
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16. Friction factor (regularized by slip)
contact line
bulk
add up
bulk R
slip region
log of a large scale ratio
( can be replaced by hm)
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17. Motion due to variable wettability
driving force
variable part of contact angle
velocity
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18. Surface freezing LV LV SVa SLa a SLb SVa b T>Tml Time
experiment, Lazar & Riegler, PRL (‘05)
Time
T>Tml

19. Surface freezing experiment, Lazar & Riegler, PRL (‘05)
simulation, Yochelis & LP, PRE (‘05)
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20. Surface freezing
stable at obtuse angle
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21. Self-propelled droplets (Sumino et al, 2005)
Chemical self-propulsion (Schenk et al , 1997)
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22. Adsorption / Desorption
rescaled length
rescaled velocity
H = 1
H = 0
H = 1
dimensionless eqn in comoving frame
concentration on the droplet contour
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23. Self-propulsion velocity
traveling bifurcation
a=4
a=2
a=1
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24. Traveling threshold from expansion at : a a
mobility interval a
immobile when diffusion is fast a
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25. Non-diffusive limit
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26. Size dependence (no diffusion)
saturated
nonsaturated
experiment
capillary number vs. dimensionless radius
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27. Scattering
far field
standing
moving
scattering angle
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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
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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
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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
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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
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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
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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:
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34. Migration on precursor layer
driving force on a droplet due to local thickness gradient
droplet velocity:
flux
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35. Migration & ripening
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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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