S. Balibar, T.Ueno*, T. Mizusaki**,

1
Anomalous wetting andcritical Casimir forces

• S. Balibar, T.Ueno, T. Mizusaki,
• F. Caupin and E. Rolley
• Laboratoire de Physique Statistique de l ENS,
  Paris, France
• now at MIT, Cambridge, USA
• Dept of Physics, Kyoto University, Japan

Kyoto, 22 oct 2003
2
abstract

• an optical measurement wetting by a 3He - 4He
  liquid mixture near its tri-critical point
• a remarkable exception to "critical point
  wetting"
• interpretation a consequence ot the "critical
  Casimir effect", i.e. the confinement of critical
  fluctuations in a film of finite thickness.

3
Ueno et al. (Kyoto 2000)magnetic resonance
imaging (MRI)
3He-4He mixtures near Tt the capillary length l
(s???/Dr?g)1/2 tends to zero the contrast
between the 2 phases also MRI no data very close
to the wall large uncertainty near Tt
4
the phase diagram of helium mixtures
a tri-critical point superfluidity phase
separation at Tt 0.87 K
5
optical interferometry
copper
6
Images at 0.852 K
the empty cell stress on windows fringe bending
7
gas
c-phase
11 mm
d-phase
8
the contact angle q and the interfacial tension
si
c-phase
c-phase
sapphire
d-phase
d-phase
zoom at 0.841 K
the interface profile at 0.841K
fringe pattern --gt profile of the méniscus --gt q
and si typical resolution 5 mm capillary
length from 33 mm (at 0.86K) to 84 mm (at 0.81K)
9
a zoom of the fringe pattern
c-phase
d-phase
10
experimental difficulties and error bars
the sapphire wall is tilted by 8.8 its position
is determined within 5 mm
zone to be analyzed
11
fits at 0.852 K

• given the adjusted wall position,
• 38
• if we displaced the wall position by -5 mm, we
  would find
• 24
• at 0.852 K, the typical error bar is 15 on the
  contact angle

12
experimental results
the interfacial tension agreement with Leiderer
et al. (J. Low Temp. Phys. 28, 167, 1977) si
0.076 t2 where t 1 - T/Tt and Tt 0.87 K
the contact angle q is non-zero it increases
with T
13
"critical point wetting "
Young - Dupré cos q (s2 - s1)/s12
Moldover and Cahn (1980) near the critical
point at Tc s12 --gt 0 as T --gt Tc (s2 - s1) --gt
0 also , but usually with a smaller critical
exponent, especially if (s2 - s1) X2 - X1 --gt
cos q? increases with T up to Tw where cos q 1
and q 0
14
the contact angle usually decreasesto zero at Tw
lt Tc
Moldover and Cahn 1980 a wetting transition
takes place at Tw lt Tc P.G. de Gennes
(1981) not necessarily true in the presence of
long range forces
15
a 4He-rich superfluid film
Romagnan, Laheurte and Sornette (1978 - 86)
van der Waals attraction a 4He-rich film grows on
the substrate
leq (T - Teq)-1/3 up to 60 Angstöms
two possibilities - leq tends to a macroscopic
value complete wetting (q 0) - leq saturates
at some mesoscopic value partial wetting (q ? 0)
q
16
the "critical Casimir effect"
the original Casimir effect confinement of the
fluctuations of the electromagnetic field the
two electrodes attract each other the critical
Casimir effect (P.G. de Gennes and M. Fisher,
1978) near a critical point, confinement of the
fluctuations of the order parameter a force of
order kBT q?(d/x)/d3 where the correlation
length x t -? and the universal "scaling
function" q?(d/x) 1 at Tc - the sign of
the"critical Casimir effect" (P. Nightingale and
J. Indekeu 1985, M.Krech and S. Dietrich 1991-92)
depends on the symmetry of the boundary
conditions on each side attractive if symmetric,
repulsive if anti-symmetric
17
the critical Casimir effect in helium
the fluctuations of superfluidity are confined
inside a film of thickness leq , between the
substrate and the 3He-rich phase an
effective attraction of the film surface by the
substrate (symmetric boundary conditions for
superfluidity) the full calculation of this
effect has not yet been done an experimental
measurement by R. Garcia and M. Chan (1999 -
2001) in a similar situation
18
the experiment by R. Garcia and M. Chan
a non-saturated film of pure 4He (200 à 500
angströms) in the vicinity of the superfluid
transition (a critical point at 2.17 K), the
film ges thinner evidence for long range
attractive forces
agreement with predictions by M.Krech and S.
Dietrich ? a critical Casimir force q?(x)/l 3
where q?(x t l1/n??is the?????????
function"?of this force
19
an approximate calculation
the contact angle q is obtained from the
"disjoining pressure" P?(l) (see D.Ross, D.Bonn
and J.Meunier, Nature 1999) 3 contributions
to P?(l) from long range forces van der Waals
(repulsive) Casimir (attractive) Q?(l/x) lt 0 is
the scaling function which can be estimated from
the measurements of Garcia and Chan the
entropic or "Helfrich" repulsion originates in
the limitation of the fluctuations of the film
surface
20
the disjoining pressure at 0.86K (i.e. t 10-2)
the equilibrium thickness of the superfluid
film leq 400 Å about 4x , where P?(l) 0
21
a theoretical estimate of the contact angle q
at T 0.86 K, i.e. t 1 - T/Tt 10 -2 leq
400 Å , 4 times the correlation length x? By
integrating the disjoining pressure from leq to
infinity, we find q 45 near a tri-critical
point, the casimir amplitude should be larger by
a factor 2 this would lead to q 66 , in even
better agreement with our experiment At lower
temperature (away from Tt ) si is larger, van
der Waals also, while Casimir is smaller, so
that q is also smaller the contact angle
increases with T, as found experimentally
22
Comparison with the experiment
23
future experiments
better accuracy near Tt (MRI Kyoto, in progress)
lower T is q different from zero ? if yes
(Ueno (Kyoto, 2000), Ishiguro (Paris) in
progress) a new explanation is needed Goldstone
modes (phase fluctuations in the superfluid film)
? M.Kardar and R. Golestanian (Rev. Mod. Phys.
71, 1233, 1999)
The amplitude is too small, but , in fact, the
theoretical estimates for the usual Casimir
amplitude seem to be also too small
Krech Dietrich (above Tc) agree with G.
Williams (below Tc) for periodic boundary
conditions. In this case, the Casimir amplitude q
(TTc) - 0.3 BUT, for Dirichlet boundary
conditions (i.e. if the order parameter vanishes
at the boundary), Krech and Dietrich predict that
q (TTc) - 0.03 (10 times less !) As for Garcia
and Chan, they found a large minimum of order
-1.5 ...
the van der Waals force is about 4/l3 K/A3, much
larger in principle
24
Complementary theoriesM. Krech S.Dietrich
(Stuttgart) and G.Williams (UCLA)
Krech Dietrich ( above Tc ) agree
with G. Williams ( below Tc ) for
periodic boundary conditions. In this case, the
Casimir amplitude q (TTc ) - 0.3
BUT, for Dirichlet boundary conditions (i.e. if
the order parameter vanishes at the boundary),
Krech and Dietrich predict that q (TTc ) -
0.03 (10 times less !)
As for Garcia and Chan, they found a large
minimum of order -1.5 below Tc ...
25
future theoretical work
a rigorous calculation of the scaling function
in the critical case with realistic boundary
conditions compare with Garcia and Chan not only
above Tc understand the magnitude and the
displacement of the minimum with respect to Tc
(bulk) and in the tri-critical case (coupled
fluctuations of concentration and superfluidity)