Jacco Snoeijer

PHYSICS OF FLUIDS

Hydrodynamics of wetting phenomena

Outline

1. Creeping flow: hydrodynamics at low Reynolds numbers (2 hrs)

2. Thin films and lubrication flows (3 hrs + problem session 1.5 hrs)

A. Bubble entrapment, instabilities, coalescence B. Landau-Levich films C. Problem session (from classic and recent papers)

3. Static and moving contact lines (3 hrs)

4. Wetting on soft substrates (depending on time)hidden theme:

scaling & similarity solutions

Influence of air pressure on impact event

Xu, Zhang, Nagel, PRL (2005)

1 atm 0.2 atm

Earlier indications on role of air: Air bubble entrapment

Van Dam, Le Clerc, Phys. Fluids (2004)

Sketch of impacting drop

pressure buildup!

falling droplet

Dimple-formation!

Thin film interference

• Oil films

• Soap bubbles

• Applications in e.g.anti-reflective coatings

Use the information coded in the

color!

Maximum air bubble

Bouwhuis et al, Phys. Rev. Lett. (2012)

Bubbles: a major nuisance

Immersion Lithography

Bubbles: a major nuisance

before after

Keij, Winkels, Casteleijns, Riepen & Snoeijer, submitted to Phys. Fluids

Lubrication for capillary flows

\frac{\partial h}{\partial t} + \frac{1}{3\eta} \frac{\partial}{\partial x}\left[h^3\left\{\frac{\gamma \partial^3 h}{\partial x^3}\right\}\right] = 0

! =A

12!h2

gravity:

van der Waals:

! = !gh

nonlinear PDE for h(x,t)

!h

!t+

13"

!

!x

!h3

"#

!3h

!x3! !!

!x

#$= 0

merging of steps

polysterene films (30..200 nm)

McGraw, Salez, Baumchen, Raphael, Dalnoki-Veress, Phys. Rev. Lett. 2012

merging of steps

polysterene films

McGraw, Salez, Baumchen, Raphael, Dalnoki-Veress, Phys. Rev. Lett. 2012

- lubrication theory: extremely useful

- similarity solutions can often be used

- next: coalescence phenomena

capillary flows: intermediate conclusion

- lubrication theory: extremely useful

- similarity solutions can often be used

- next: coalescence phenomena

capillary flows: intermediate conclusion

!E ! !R2

coalescence

reduction capillary energy

surface tension

R

coalescence

spherical water drops (timescale ~ millisecond)

Aarts et al. Phys. Rev. Lett. 2005

r(t)

r w

r w

w = r2

R

r w

p ! !w !

!Rr2

w = r2

R

r(t) w

p ! !!

drdt

"2p ! !Rr2

surface tension vs inertia

r(t) ! t1/2

Eggers, Lister & Stone, J. Fluid Mech. 1999Duchemin, Josserand & Eggers, J. Fluid Mech. 2003

2 regimes

Paulsen, Burton & Nagel, Phys. Rev. Lett. 2011Paulsen et al, PNAS 2012

r ~ t

r ~ t1/2

“inertial”

“viscous”

sessile drops

sessile drops

• geometry• solid wall: no slip• moving contact line!

complications:

sessile drops

Ristenpart, McCalla, Roy & Stone, Phys. Rev. Lett 2006

experimentally: r(t) ~ t1/2

r(t)

very viscous silicone oil

Narhe, Beysens & Pomeau, Europhys. Lett. 2008

coalescence of drops on substrate

silicone oil (12.500x water)

1D lubrication model

100 µm

mechanism

silicone oil (12.500x water)

liquid flux Q

low capillary pressure: p ~ - γ/h

flux: Q ~ - dp/dx

mechanism

silicone oil (12.500x water)

liquid flux Q

flux: Q ~ - dp/dx

mass conservation: !h

!t+

!Q

!x= 0

low capillary pressure: p ~ - γ/h

bridge growth

silicone oil (12.500x water)

h0

coalescence dynamics: h0(t) ?

bridge growth

h0 ~ t

bridge shape

silicone oil (12.500x water)

x

shape of bridge: h(x,t) ?

h(x,t)

self-similarity!

x θ/h0

h/h0

bridge shape

silicone oil (12.500x water)

shape of bridge: h(x,t) ?

self-similarity!

x θ/h0

h/h0

Problem session: Hernandez-Sanchez, Lubbers, Eddi & Snoeijer

Phys. Rev. Lett. 2012

!h

!t+

!Q

!x= 0

!h

!t+

"

3#

!

!x

!h3 !3h

!x3

"= 0

lubrication theory

back to topview...

r ~ t1/2 ?

geometry

!0.2 !0.1 0 0.1 0.2 0.30

0.05

0.1

0.15

0.2

0.25

X

H

T = 0T = 0.0234T = 0.0467T = 0.0701T = 0.0935

w ~ h ~ t

w ~ r2/Rr ~ t1/2 !

Narhe, Beysens & Pomeau, Europhys. Lett. 2008Ristenpart, McCalla, Roy & Stone, Phys. Rev. Lett 2006

water drops on substrate (inertial)

- exponent 1/2?

- self-similarity?

Anonin Eddi, Koen Winkels & JHS, submitted

Photron SA1.1

Synchronization and computer

10XLens

Photron APX-RS

water drops

water drops - side view

200.000 frames/second

!4 !3 !2 !1 0 1 2 3 40

1

2

3

4

5

6

7

8

9

X/hb

Y/h b

Rescaled profiles for frames 7,15,25,40,70

self-similar!

x/h0

h/h0

self-similar!

x/h0

h/h0

2D Potential flow: Billigham & King, JFM 2005

exponent: 2/3

2/3

exponent: 2/3

Pcap ! !

w

Piner ! !v2 v ! h0

t

w =h0

tan !

h0 !!! tan "

#

"1/3t2/3

Keller and Miksis (1983)

exponent: 2/3

2/3

h0 = D0

!! tan "#

"1/3t2/3

D0 = 0.89

- lubrication theory: extremely useful

- similarity solutions can often be used

- coalescence phenomena (show movie Marangoni)

capillary flows

Outline

1. Creeping flow: hydrodynamics at low Reynolds numbers (2 hrs)

2. Thin films and lubrication flows (3 hrs + problem session 1.5 hrs)

A. Bubble entrapment, instabilities, coalescence B. Landau-Levich films C. Problem session (from classic and recent papers)

3. Static and moving contact lines (3 hrs)

4. Wetting on soft substrates (depending on time)