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

1
Tail wags the dogMacroscopic 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)
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Contact line paradoxFluid-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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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
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 CahnHilliard 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 kh3/3
h
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Configurations a multiscale system
length scales differ by many orders of magnitude!
h
droplet
meniscus
bulk
precursor
precursor
hm
R
precursor
horizontal
slip region
bulk
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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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Moving droplets
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Numerical slip NS computations (O. Weinstein
L.P.)
grid refinement
ln (cR/?)
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Larger drops change shape upon refinement NS
computations (O. Weinstein L.P.)
higher refinement
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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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Solvability condition dry substrate
area integral
friction factor
bulk force
contour integral
contour force F?
solvability condition defines velocity
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Friction factor (regularized by slip)
contact line
bulk
add up
bulk
R
log of a large scale ratio (? can be replaced by
hm)
slip region
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Motion due to variable wettability
driving force
variable part of contact angle
velocity
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?LV
?LV
?SVa
?SLa
?a
?SLb
?SVa
?b
Surface freezing
experiment, Lazar Riegler, PRL (05)
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Surface freezing
experiment, Lazar Riegler, PRL (05)
simulation, Yochelis LP, PRE (05)
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Surface freezing
stable at obtuse angle
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Self-propelled droplets (Sumino et al, 2005)
Chemical self-propulsion (Schenk et al , 1997)
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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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Self-propulsion velocity
traveling bifurcation
a4
a2
a1
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Traveling threshold
from expansion at
a
mobility interval
a
immobile when diffusion is fast
a
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Non-diffusive limit
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Size dependence (no diffusion)
saturated
nonsaturated
experiment
capillary number vs. dimensionless radius
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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
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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 (n3)
h
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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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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
larger droplet is repelled in by the small one
smaller droplet is sucked in by the big 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
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Migration on precursor layer
driving force on a droplet due to local thickness
gradient
droplet velocity
flux
35
Migration ripening
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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