Jens Eggers

Hydrodynamic theory of coating

…with Bruno Andreotti, Marc Fermigier,

Jacco Snoeijer, Johanna Ziegler

Dynamical wetting transition

crU

2 cL

Deraguin+Levi 1964

crU

2 cL

continuous transition discontinuous transition

de Gennes 1986

apparent contact angle at the transition?

Small scale physics

moving contact lineHuh and Scriven’s paradox: no motion for 0!micro

stationary

U

h

23 ln( / )micro

L

diss micro

dzU L

h

Lubrication equations

liquid bath

gh(x)

x

y

plate

/slip e /plate e viscous s.t. gravity

(0) 0, (0) 1, ( )h h h

cin units of /( ) :g 33 /( )eCa U

( ,0) slip

uu x U

y

eqnm

(0) eqh

2 3

Cah h

h h

note additional parameter

outer meniscus

( ) ( ) xout apph x e

Matching: Pull!

U<Uc

liquid bath

gmatch!

U>Uc

liquid bath

g

no overlap!

inh outh

compare

0

curvature

matching impossible!

: microscopic length

2

3

3inin in

Cah

h h

1(3 )( ) 3in advh x H x

ap

Critical speed

0

curvatureouth

app

1/ 3

2/3 1/(3 )1

1/3 21 1

2 ( )

( ) 3 2 ( )

CaAi s e

Ca Ai s Ai s

1solve for !s

2/3 1/31

1

2 ( )

( )ap

Ca Ai s

Ai s

Ca

crCa

inh

1/3 1/(3 )

1/3 212 ( )

CaCa e

Ai s

1/3 1/(3 )

1/3 2max2 ( )

Ca

cr

Ca e

Ai s

1/3 1/(3 )

1/32 0.287

Ca

cr

Ca e

1

1/3 2max

1ln

3 54 ( )cr

cr

CaCa

Ai s

“Cox-Voinov”

3 2( 1)1 3 ln

3ap Ca

matching condition

Wetting transition

Delon et al., JFM 2007

transient experiment:

pl crU U

0app at transition

•Derjaguin+Levi 1964:

2cl

c

x

Ca

0.3plate eq

410

simtheory

Beyond the transition

h

x

before

contact line

apparent

contact line

h

x

after

Another transition

Ca

plate angle

Ziegler et al.,

EPJ `09

Wetting transition

Delon et al., JFM 2007

*Ca Ca0q

new film solution!

The Landau Levich Derjaguin film

fhCa

2

31 1

q

h

Ca

h

I: bath

2

2

31Q

HH H

LL-equation

II

2/31/3

CaCa

xh H

2/3q QCa

2/30.95f Cah

2H ( ) 0H

The dimple!

fh

Ca

I: bath

2 II

2/31/3

CaCa

xh H

2/3q QCa

2H ( ) 0H III

1/ 21/ 6

xh Ca H

Ca

balance

includes

gravity!

2

31 1

q

h

Ca

h

A range of new solutions

2/31/3

CaCa

xh H

2/3q QCa

2H

( ) 0H

0Q

2

31Q

HH H

max 1.376Q Q

Phase diagram

Ca

fh0Q

max 1.376Q Q

2

31 1

q

h

Ca

h

1/ 2 2/332f

QCh a Ca

new thick film

solutions!

2/30.95f Cah partial wettingcomplete wetting

Experiment:

The film profile

fh

Snoeijer et al., PRL `08

bath

39.3 10Ca 38.1 10Ca

new thick

film

theory

LLD

film

theory

the dimple!

The big picture

log( )Ca1

log( )fh

2/3

1/2

so far!

1/2

sinfp

Cah

Derjaguin

A fun movie made in Bristol

…work withPhilippe Brunet,Robert Deegan

Bristol Evening Post