Effects of large-scale geometry on wettingdynamics: a story of sliding drops

L. Limat, A. Daerr (MSC, CNRS & Univ. Paris Diderot)I. Peters, K. Winkels, J. Snoeijer (Univ. Twente) B. Andreotti (PMMH,Univ. Paris Diderot & ESPCI)

T. Podgorski (UJF Grenoble), J.-M. Flesselles (Saint-Gobain),N. Le Grand, E. Rio (Univ. Orsay)

coll: H.-A. Stone (Princeton), J. Eggers (Bristol)

GDR films, 2012

Partial wetting

Static drop

[Young](real world: hysteresisΘa < Θ < Θr )

Moving drop

advancing front steeper (Θ1 > Θa),receding gentler (Θ2 < Θr )

-10-2-5 10-305 10-3

0.884

0.882

0.880

0.878

0.876

0.874

0.872

0.870

10-9 10-8 10-7 10-6 10-5

60

40

20

0

Rolley et al, Delon et al.

Dynamics of wetting: multi-scale problem

drop

m

θap

θm

R

U=Rθap

θ

mλ ∼nm

apθ

.

3CaR/

from: Bonn et al2009

molecular scale: unknownout-of-equilibrium physics(evaporation, diffuse interfaces,. . . )

intermediate scale: viscosity vs.capillary pressure

large scale: outer boundarycondition

Viscous-capillary range

Huh & Scriven’s paradox

U

−∇px

z

σ = ηU/(θx)F = η(U/θ)

∫dx/x

ε = 12η(U2/θ)

∫dx/x

→∞Divergence both at small and large scales

Cox-Voinov relationViscous stress balanced by surface curvature gradient, alsodiverging:

3ηU/h = h∇(γ∆h)

θ3 = θ3micro ± 9Ca log x/a, Ca = ηU/γ

(+ for advancing CL; a: microscopic cut-off)

Wetting transitions

Wetting transitions (2)

Delon et al2008

60

50

40

30

20

10

0

1050–5–10

θapp

Ca(×10–3)

Ca ∗

CacCa >

Ca d

Contact linePlate

Liquidbath

Overhangingridge

Landau–LevichfilmCapillary jump

2 film thicknesses, 3 velocities (Ca∗ film entrainment, Ca> obliquecontact line speed; Cad is the ‘corner’ contact line speed in drops) flow must match different boundary conditions

Sliding dropsexperimental set-up

α

miroir

verre avecrevêtementfluoro-polymères 3M

huile de silicone 10 ... 1000 cP

source de lumière

camera

Liquid: silicon oil

η = 20.8 cPγ = 20.1 mN/mρ = 0.95 g/cm3

Fluor-polymer coating

Θa=54° Θr=48°

A L

Θa

A L

Θr

Sliding dropsobserved regimes

by increasing speed:

roundblunt cornercornercuspperling (small)perling (big)

singularity at the rear

Transition to corner shape

round

dessus côté reflet

corner

conical geometry (non-zero rear contactangle) !

Contact line normal speed

a)

b) c)

Ca sin Φ ≈ cst in corner regime(green solid line →)

1mma)

b)

c)

0

20

40

60

80

100

0

0,2

0,4

0,6

0,8

1

0 0,001 0,002 0,003 0,004 0,005 0,006 0,007

10 cP

θ (°) sinφ

Ca

θa

θr

Oval

Roundedcorner

Corner

Cusp

Pearling

sinΦ

)2Φ 1/3((3/2).Ca.tan

Drop shape

0

20

40

60

80

100

0

0,2

0,4

0,6

0,8

1

0 0,001 0,002 0,003 0,004 0,005 0,006 0,007

10 cP

θ (°) sinφ

Ca

θa

θr

Oval

Roundedcorner

Corner

Cusp

Pearling

sinΦ

)2Φ 1/3((3/2).Ca.tan

θa

θr gouttes ovales

Ω gouttes pointues

2Φ

2θa

2Ω

Cox-Voinov fit of receding contact angle reaches zero atCa = 0.004. Corner transition occurs slightly before.

Asymptotic matching problem

Visco-capillary solution (valid near contact line)must match

gravito-capillary solution (valid away from contact line):

limx→0

hgc(x) = limx→b

hvc(x)

where the matching scale for the inner solution is the scale atwhich gravity starts dominating viscous shear:

ρgh(b)P ≈ 3ηU/h(b), P = sinα− h′ cosα

viscosity molecule9 log

x

a

x/a Ca∗(x) Cao→c Cao→c

(cP) size (µm/nm) meas. theo.10 3 nm 129.0 200 / 0.1 4.0e-3 3.7e-3 3.5e-3

104 20 nm 80.0 200 / 28 5.1e-3 4.4e-3 4.4e-31040 100 nm 70.5 200 / 84 7.6e-3 6.2e-3 6.4e-3

20 n.c. 80 50 / 7 6.0e-3 4.0e-3 4.5e-3

How is the wetting transition avoided ?

x

y

z

R

2d

3d

2d

U

−∇px

z

viscous stress ⇔ longitudinalcurvature variation

3d

x

y

zaugmentationcourbure &pression

viscous stress ⇔ transversecurvature variation

Dewetting in conical geometry

x

y

zΩ

Φθ

Self-similar solutions h(x , y) = Ca1/3xH(y/x) by Stone & Limat(2003), Snoeijer et al(2005) for 3d lubrication model

3Cahx = ∇[h3∇(∆h)

]∧ ht = −U hx

link cone opening angles Ω and Φ:

tan Ω =35

16Ca tan2 Φ

Dewetting in conical geometry (2)

Imposing a Cox-Voinov-type law on the cone boundary selects theopening angle:

Ca =2Φ

35 + 18(log ba )Φ2

What about the tip itself ?

Corner formation a way to avoid wetting transition: normalcontact line speed reduced to U0 sin Φ < U0.

x

y

z

R

2d

3d

But: always small region near tip where contact line speed is U0.

Tip curvature increases

. . . but remains finite.

Tip matching

Idea: match near 2d solution at tip to 3d cone solution at scale R.

x

y

z

R

2d

3d

θ(R)3 = θ3e − 9Ca logR

a∧ θ(R) = Ω

⇒ R ≈ a exp[(θ3e − Omega3)/9Ca] ≈ a exp[θ3e/9Ca]

Tip matching advanced

More precise matching procedure with hyperbolic Ansatzh(x , y) = h0(x)[1− y2/w(x)2],w(x)2 ≈ 2Rx + Φ2x2 yields

Ω3 = θ3e − 9Ca logβR

Φ2awith Ω3 ≈ 35

16Ca tan2 Φ, β = O(1)

1

R≈ f (Φ)

aexp[− θ3e

9Ca]

tip size comes from microscopic scale !

Tip measurements in two geometries: sliding drop &immersion set-up

turntable setup: Riepen et al. 2008 (Twente)

wafer velocity

Axi-symmetric

side view

camera

bott

om

vie

w

cam

era

water

supply

water

supply

motor

wafer

extraction

flow

a) waferside view

bottom view

b)

Microscopic origin

Conclusion

• Drops sliding down an incline develop a conical pointsingularity, on the already singular contact line.

• The regularisation of this point singularity involves an ‘inner’length scale, not the ‘outer’ one.

• Corner is a truly three-dimensional, self-similar solution (bothcontact line and bulk dissipation select the opening angle).

• Classical hydrodynamics with a molecular cut-off describesthese experiments satisfactorily. Need lower speeds to seemicroscopic effects ?

Contact angle varies along drop perimeter

• How to go smoothly from Θa at front toΘr at rear ?

• On substrate with hysteresis, whathappens at transition from advancing(red) to receding (blue) part ?

Contact angle measurementthrough light refraction

laser sheet

glassplate

translucidscreen

CCDcamera

α

translucentscreen

Φ

laserdiode

sliding puddle

Φ∆

a) b)

Measurement scale:∼ 10µm

Results on contact angleMeasurement around a single drop

Θas

Θrs

Ca sin Φ: capillary number based on contact line speed

Results on contact angleMeasurement around a single drop

Θas

Θrs

Ca sin Φ: capillary number based on contact line speed

Results on contact angleMeasurement around a single drop

Ca sin Φ: capillary number based on contact line speed

Results on contact angleComparison to measurements at front and rear of multiple drops

et

black discs: front/rear of drops empty circles: around one drop

Velocity field of sliding drop

1mma)

b)

c)

Velocity ⊥ contact line

Velocity field of sliding drop

1mma)

b)

c)

Velocity ⊥ contact line

Velocity normal to contact linemicroscopic interpretation

Near contact line,

• viscous stresses and balancing forces(capillarity, disjoining pressure, . . . ) diverge

• properties vary much slower tangent to CLthan normal to CL

tangential velocity component quickly dampedby viscosity⇒ ~u ⊥ contact line

situation essentially symmetry wrt planenormal to contact line

Axis of

symmetry