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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