Transcript Critical Casimir effect and wetting by helium mixtures

Anomalous wetting
and
critical 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
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.
3He-rich
liquid
q
4He-rich
liquid
sapphire
window
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
the phase diagram
of helium mixtures
normal
T
superfluid
0.87 K
tri-critical
point
a tri-critical point:
superfluidity + phase separation
at Tt = 0.87 K
0
0.675
3He concentration
1
optical interferometry
mixing chamber
copper
10 mm
He-Ne laser
vapor
3He-rich
4He-rich
liquid
liquid
optical
interferometric
cavity
(sapphire treated for 15% reflection)
Images at 0.852 K
the empty cell:
stress on windows
fringe bending
vapor
liquid-gas interface
3He-rich
"c-phase"
3He- 4He
4He-rich
interface
"d-phase
zone to be analyzed
gas
c-phase
11 mm
d-phase
the contact angle q
and the interfacial tension si
c-phase
sapphire
c-phase
q
d-phase
d-phase
zoom at 0.841 K
the interface profile at 0.841K
fringe pattern --> profile of the méniscus --> q and si
typical resolution : 5 mm
capillary length: from 33 mm (at 0.86K) to 84 mm (at 0.81K)
a zoom of the fringe pattern
c-phase
d-phase
experimental difficulties and error bars
the sapphire wall is
tilted by 8.8°
its position is
determined within
± 5 mm
zone to be
analyzed
fits at 0.852 K
given the adjusted wall
position,
q = 38°
at 0.852 K, the typical
error bar is ± 15° on
the contact angle
data at 852mK
fit of the wall
and of the profile
fit with a wall
at - 5 mm
540
vertical position ( mm)
if we displaced the wall
position by -5 mm, we would
find
q = 24°
550
530
520
510
500
490
480
-100
-80
-60
-40
horizontal position ( mm)
-20
0
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
"critical point wetting "
2
s2
Tc
s12
1
q s1
substrate
1
cos q = (s2 - s1)/s12
2
X1
Xc
Young - Dupré :
X2
Moldover and Cahn (1980) :
near the critical point at Tc
s12 --> 0 as T --> Tc
(s2 - s1) --> 0 also , but usually with
a smaller critical exponent,
especially if
(s2 - s1) ~ X2 - X1
--> cos q increases with T up to Tw
where cos q = 1 and q = 0
the contact angle usually decreases
to zero at Tw < Tc
cos q
q
1
Tw
Tc
Tw
Tc
Moldover and Cahn 1980:
a wetting transition takes place at Tw < Tc
P.G. de Gennes (1981):
not necessarily true in the presence of long range forces
a 4He-rich superfluid film
Romagnan, Laheurte and Sornette (1978 - 86):
normal
T
van der Waals attraction
a 4He-rich film grows on the substrate
4He-rich
superfluid
Teq
superfluid film
tri-critical
point
leq
substrate
0
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)
1
3He
concentration
superfluid film
4He-rich
bulk phase
q
substrate
leq
the "critical Casimir effect"
2 walls
confined fluctuations
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
the critical Casimir effect in helium
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
leq
superfluid film
3He-rich
q
4He-rich
substrate
the fluctuations of superfluidity are confined inside a
film of thickness leq , between the substrate and the 3Herich phase
an effective attraction of the film surface by the
substrate (symmetric boundary conditions for
superfluidity)
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/ is the
function" of this force
an approximate calculation
the contact angle q is obtained from the "disjoining pressure"  (l) (see D.Ross,
D.Bonn and J.Meunier, Nature 1999):

cosq = 1 
l eq

(l)dl
si
3 contributions to  (l) from long range forces:
van der Waals (repulsive)
 vdW (l ) =
Casimir (attractive)
Q (l/x) < 0 is the scaling function
1000  (1/ Vd  1/ Vc )
l 3 (1  l / 193)
in
K. A3
Tt
Cas (l ) = 3  q(l / x)
l
which can be estimated from the measurements of Garcia and Chan
2

3
s
2
s
l

i
i
the entropic or "Helfrich" repulsion
 H (l ) =
e xp
2l
 3kBT 
originates in the limitation of the fluctuations of the film surface
the disjoining pressure at 0.86K (i.e. t = 10-2)
Helfrich
where  (l) = 0
-9
van der Waals
20
total pressure
(l) (10
leq = 400 Å
~ about 4x ,
disjoining pressure
the equilibrium
thickness of the
superfluid film:
K.A -3)
40
0
-20
van der Waals + Casimir
-40
Casimir
0
200
400
600
800
film thickness l (Angström)
1000
1200
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
Comparison with the experiment
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 (T=Tc) ~ - 0.3
BUT, for Dirichlet boundary conditions (i.e. if the order parameter vanishes at the
boundary), Krech and Dietrich predict that q (T=Tc) ~ - 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
Complementary theories:
M. 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 (T=Tc ) ~ - 0.3
BUT, for Dirichlet boundary
conditions (i.e. if the order
parameter vanishes at the
boundary),
Krech and Dietrich predict that
q (T=Tc ) ~ - 0.03
(10 times less !)
As for Garcia and Chan, they found a large minimum of order -1.5 below Tc ...
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)