Jens Eggers Hydrodynamic theory of coating …with Bruno Andreotti, Marc Fermigier, Jacco Snoeijer,...

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