04/20/23 1

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

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

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