S

1
critical Casimir forces and anomalous wetting

• Sébastien Balibar and Ryosuke Ishiguro
• Laboratoire de Physique Statistique de l Ecole
  Normale Supérieure,
• associé au CNRS et aux Universités Paris 6 7
• Paris, France

for references and files, go to
http//www.lps.ens.fr/balibar/
StatPhys Bangalore, july 2004
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abstract
a critical introduction to and discussion of
the "critical Casimir effect" "critical point
wetting", i.e. wetting near a critical point 4
experiments Garcia and Chan (Cornell, 1999)
Ueno et al. (Kyoto, 2000) Ueno et al. (Paris,
2003) Ishiguro and Balibar (Paris, 2004)
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the "critical Casimir effect"
L
the standard Casimir effect confinement of the
fluctuations of the electromagnetic field
the two electrodes attract each other
the critical Casimir effect (Fisher and de
Gennes, 1978) near a critical point,
confinement of the fluctuations of the order
parameter a singular contribution to the free
energy E kBT /L2 a force between the two
plates FCas - dE/dL 2 kBT /L3
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the universal scaling functions Q and q
Further work (Nightingale and J. Indekeu 1985,
M.Krech and S. Dietrich 1991-92) shows that E
kBT/L2 Q (L/x) where the "universal scaling
function" Q depends on the bulk correlation
function x t -n which diverges near the
critical temperature Tc . At Tc , i.e. t 0,
Q(0) D , the "Casimir amplitude". a similar
scaling function is introduced for the
force FCas kBT/L3 q (L/x)
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Universality

• the scaling functions only depend on
• the dimension of space
• the dimension of the order parameter
• the type of boundary conditions
• periodic or antiperiodic
• Dirichlet
• von Neumann

the 5 different values D Q (Tc) have been
calculated, but not Q at any T nor with any
boundary conditions for example
Dirichlet-Dirichlet below Tc
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the sign of the force
attractive if symmetric boundary conditions (q lt
0) repulsive if antisymmetric (q gt 0)
the Casimir amplitude D Q (L/x 0)
0.2 to 0.3 for periodic boundary
conditions proportional to the dimension N of the
order parameter 10 times smaller if the order
parameter vanishes at the wall (Dirichlet-Dirichle
t) twice as large if tri-critical instead of
critical
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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 gets thinner evidence for long range
attractive forces
comparison with predictions assume a critical
Casimir force q?(x)/l 3 measure q?x
(L/x)1/n?, the????????? function"?of this force
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comparison with theory
above Tc agreement with Krech and Dietrich
Phys. Rev. A 46, 1886 (1992)
below Tc no theory the magnitude of the
experimental q depends on L (not universal ??) it
is also surprisingly large (1.5 to 2, no
theoretical result larger than 0.5)
far below Tc a finite value of q ? confinement
of Godstone modes (Ajdari et al. 1991, Ziherl et
al. 2000, Kardar et al. 1991-2004, Dantchev and
Krech 2004)
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Phys. Rev. E 2004
periodic boundary conditions the Casimir
amplitude is larger by a factor 2 for the XY
model (N 2) the scaling function does not
vanish as T tends to 0 for the XY model
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the magnitude of the effect of Godstone modes
for Dirichlet-Dirichlet boundary conditions,
Kardar and Golestanian (Rev. Mod. Phys. 1999)
predict a very small amplitude q -
0.05 Garcia's measurement q - 0.3 in
agreement with Dantchev (but with periodic
boundary conditions) at the 2004 APS march
meeting, R.Zandi, J. Rudnick and M. Kardar invoke
the surface fluctuations of the film which would
enhance the Goldstone mode contribution, but the
sign of this last effect is somewhat
controversial. In fact the situation is not
settled better experiments, and calculations
with the right boundary conditions are needed
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"critical point wetting "wetting near a
critical point
Young - Dupré cos q (s2 - s1)/s12
Moldover and Cahn (1980) near the critical
point at Tc s12 ? 0 as T --gt 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
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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)
Nightingale and Indekeu (1985) not necessarily
true in the presence of long range forces
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a possible exception to critical point wetting
the example of helium 3 - helium 4 liquid
mixtures
a tri-critical point superfluidity phase
separation at Tt 0.87 K
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a 4He-rich superfluid film
Romagnan, Laheurte and Sornette (1978 - 86)
van der Waals attractive field a 4He-rich film
grows on the substrate
leq (T - Teq)-1/3 up to 60 Angstöms

• two possibilities
• van der Waals only,
• leq tends to a macroscopic value
• complete wetting (q 0)
• vdW an attractive force (Casimir),
• leq saturates at some mesoscopic value
• partial wetting (q ? 0)

q
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an approximate calculation
the contact angle q is obtained from the
"disjoining pressure" P?(l) (see Ueno, Balibar et
al. PRL 90, 116102, 2003 and Ross, Bonn and
Meunier, Nature 1999) 3 contributions to
P?(l) from long range forces van der Waals
(repulsive on the film surface) Casimir
(attractive) Q?(l/x) lt 0 is the scaling function
which can be estimated from Garcia and Chan the
entropic or "Helfrich" repulsion due to the
limitation of the fluctuations of the film
surface
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optical interferometry
copper
copper
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Images at 0.852 KT. Ueno et al. 2003
the empty cell stress on windows fringe bending
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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 meniscus --gt q
and si typical resolution 5 mm capillary
length from 33 mm (at 0.86K) to 84 mm (at 0.81K)
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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
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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
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the calculated 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 and van der
Waals are larger, Casimir is smaller, so that q
should also be smaller the contact angle
increases with T, as found experimentally
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In 2003, our exp. results (Ueno et al. , JLTP
130, 543, 2003) agreed with our approximate
calculation (Ueno et al. PRL 60, 116102, 2003)
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new setup for experiments at lower T(R. Ishiguro
and S. Balibar, in progress)
closer to normal incidence less distortion due to
refraction effects, better control of the fringe
pattern measurements at lower T is the contact
angle ? 0 ? Goldstone modes ? amplitude ?
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Ishiguro's profiles
the contact angle is zero at low T (237 mK) and
near Tt (840 mK)
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Ishiguro's results
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the interfacial tension near Tt
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the contact angle
Ishiguro and Balibar (2004) find q 0 in
contradiction with previous measurements
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could the Casimir force be 5 times smaller than
measured by Garcia and Chan ?
the disjoining pressure would be dominated by the
van der Waals field, always positive, implying
complete wetting (q 0)
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summary
the exception found by Ueno et al. to "critical
point wetting" is not confirmed by our more
careful, and more recent experiment still
possible if the substrate exerted a weaker van
der waals field ? the amplitude of the critical
Casimir force measured by Garcia and Chan is not
really universal and its amplitude looks
large but there is no available calculation with
the right boundary conditions below Tc where it
is large. more work ...