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Tail wags the dog:Macroscopic signature of nanoscale
interactions at the contact lineLen Pismen
Technion, Haifa, Israel
Nanoscale phenomena near the contact line Perturbation theory based on scale separation
Droplets driven by surface forces Self-propelled droplets
Outline
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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 paradox:Fluid-dynamical perspective
normal stress balance:determine the shape
multivalued velocity field:stress singularity
Stokes equation
no slip
Dynamic contact anglediffers from the static one.
Use slip condition to relieve stress singularity.
molecular-scale slip length
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Physico-chemical perspective
thermodynamic balance:determines the shape
Stokes equation + intermolecular forces
Kinetic slip in 1st molecular layer
precursor (nm layer)
variable contact angle
interaction with substrate
disjoining potential
Diffuse interface
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04/20/23 5MD simulations, PRL (2006)
precursor
Kavehpour et al, PRL (2003)bulk
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Film evolution – lubrication approximation
Mass conservation:
Pressure:
surface disjoining gravity tension pressure
€
V = ρ g ( h − α x )
involves expansion in scale ratio eq. contact angletechnically easier but retains essential physics
Generalized Cahn–Hilliard equation
∂ h ∂ t = η −1∇ ⋅[k(h)∇P]
P =−γ 2∇2h + Π(h) + V(h)
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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Disjoining potential
(computed by integrating interaction with substrate across the film)
precursor thickness
complete wetting
partial wettingΠ(h) =
A
h31−
1
h3+n
⎛⎝⎜
⎞⎠⎟
vdW/nonlocal theory
polar/nonlocal theory
Π(h) = e−h a − e−h( )
0
Π(h)
h
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Mobility coefficient
(computed by integrating the Stokes equation across the film)
h
sharp interface k=h3/3
diffuse interface
ln k
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Configurations: a multiscale system
precursor hm
contact line hm or b
bulk R or
region angle scale length scale
bulkprecursor precursor
R
length scales differ by many
orders of magnitude!
slip regionhorizontal
hm
h
meniscusdroplet
precursor
bulk
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Multiscale perturbation theory
dimensionless quasistationary equation
dynamic equation
Inner equation Outer equation
∂h
∂t= −
η
3∇ ⋅ h2 (h − b)∇ γ 0θ0
2∇2h − Π(h) − V (h, x)⎡⎣ ⎤⎦{ }
small parameter – Capillary number
∇2h0 − V (h0 , x) = 0∇2h0 − Π(h0 ) = 0
h =h +δh1 + ...
zero order: static solution
gives profile near contact linemacroprofile
δ =
3Uη
γ 0θ03
= 1 expand
precursor:
dry substrate: assignV = 0: parabolic cap
h0 (−∞) =hm, h′(∞) =1
h0 =2
1−r2
R2
⎛
⎝⎜⎞
⎠⎟
δ∂h
∂x+∇ ⋅ h2 (h − b)∇ ∇2h − Π(h) −V (h, x)⎡⎣ ⎤⎦{ } = 0
δ∂h
∂x+∇ ⋅ h2 (h − b)∇ ∇2h − Π(h)⎡⎣ ⎤⎦{ } = 0 δ
∂h
∂x+∇ ⋅ h3∇ ∇2h −V (h, x)⎡⎣ ⎤⎦{ } = 0
€
h0′(1) =1
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Moving droplets
Passive
Interacting
Active
T∇
chemically reacting
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Numerical slip: NS computations (O. Weinstein & L.P.)
ln (cR/)
grid refinement
Ca =Uμγ
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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
1st order equation
adjoint operator
translational Goldstone mode
δ∂h
∂x+∇ ⋅ k(h)∇ ∇2h −V0 (h, x) − δ V1(h, x)⎡⎣ ⎤⎦{ }
L =∇⋅ k(h )∇ ∇2 −V′(h)⎡
⎣⎢⎤⎦⎥{ }
Lh1 +Ψ(h ) =
L† = ∇2 −V′(h)⎡
⎣⎢⎤⎦⎥∇⋅k(h )∇
L†ϕ = ϕ (x) = x̂h0 − C
k(h0 )∫ dx
ϕ (x)Ψ(h0 )∫ dx = 0
quasistationary equation
h =h +δh1 + ...expand
linear operator inhomogeneity
solvability condition
ϕ (x)Ψ(h0 )∫ dx+boundary terms = 0solvability condition in a bounded region
Ψ =∂h0
∂x−∇ ⋅ k(h0 )∇V1(h0 ,x)[ ]
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Solvability condition – dry substrate
friction factorarea integral
bulk force
contour integral
contour force F
solvability condition defines velocity
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Friction factor (regularized by slip)
log of a large scale ratio ( can be replaced by hm)
bulk
R
slip region
contact line
bulk
add up
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Motion due to variable wettability
variable part of contact angle
driving force
velocity
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γSVa
γLV
γSLaa γSL
b
γLV
bγSV
a
Time
T>Tml
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
H = 1 H = 0 H = 1 rescaled velocity
rescaled length
dimensionless eqn in comoving frame
concentration on the droplet contour
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Self-propulsion velocity
a=1
a=2
a=4 traveling bifurcation
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Traveling threshold
from expansion at :
a
a mobility interval
aimmobile when diffusion is fast
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Non-diffusive limit
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Size dependence (no diffusion)
capillary number vs. dimensionless radius
experimentnonsaturated
saturated
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Scattering
scattering angle
far field
standing moving
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Solvability condition – precursortranslational Goldstone mode
perturbation of contact angle related to perturbation of disjoining pressure
transform area integral to contour integral
area integral
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Inner solution – precursor
scaled by precursor thickness hm =1; fit to =1
boundary conditions:
zero-order: static
h =, ′h (x) = at x→ −∞, ′′h (x) = at x→ ∞
y(h) = ′h (x)[ ]2
y(1) =, ′y (h) = at h→ ∞
“phase plane” solution (n=3)
y(h) =(h−1)2
h5
23+43
h+ 2h2 +h3⎛⎝⎜
⎞⎠⎟
e.g.
h
y =slope
δdh
dx+
d
dxh3 d
dx
d 2h
dx2− Π(h)
⎡
⎣⎢
⎤
⎦⎥
⎧⎨⎩⎪
⎫⎬⎭⎪
= 0 δh − hm
h3+
d 3h
dx3−
dΠ(h)
dx= 0
d
dx
d 2h
dx2−Π(h)
⎡
⎣⎢
⎤
⎦⎥= dy
dh−2Π(h) =
Π(h) =n −1
n − 3
1
h3−
1
hn
⎛⎝⎜
⎞⎠⎟
1d: integrated form:
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• contact line region: use here static contact line solution
• droplet bulk: use spherical cap solution
• add up:NB: logarithmic factorbulk and contact line contributionscannot be separated in a unique way
Friction factor (2D) (regularized by precursor)
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Friction factor (3D) (regularized by precursor)
• contact line region: multiply local contribution by cos ϕ and
integrate
(ϕ is the angle between local radius and direction of motion)
• droplet bulk (spherical cap)
• add up:NB: logarithmic factorbulk and contact line contributionscannot be separated in a unique way
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flux
flux
larger drop in equilibrium with thinner precursor
Interactions through precursor film
smaller droplet catches up
flux
larger droplet is repelled in by the small one
smaller droplet is sucked in by the big oneripening
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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
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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