Sébastien Balibar and Ryosuke IshiguroSébastien Balibar and Ryosuke IshiguroLaboratoire de Physique Statistique de l ’Ecole Normale Supérieure,Laboratoire de Physique Statistique de l ’Ecole Normale Supérieure,

associé au CNRS et aux Universités Paris 6 & 7associé au CNRS et aux Universités Paris 6 & 7

Paris, FranceParis, France

critical Casimir forces critical Casimir forces andand

anomalous wetting anomalous wetting

StatPhys Bangalore, july 2004StatPhys Bangalore, july 2004

for references and files, go to for references and files, go to http://www.lps.ens.fr/~balibar/

abstract

a critical introduction to a critical introduction to and discussion ofand discussion of the "critical Casimir effect"the "critical Casimir effect" "critical point wetting", i.e. wetting near a critical point"critical point wetting", i.e. wetting near a critical point 4 experiments:4 experiments:

Garcia and Chan (Cornell, 1999)Garcia and Chan (Cornell, 1999) Ueno et al. (Kyoto, 2000)Ueno et al. (Kyoto, 2000) Ueno et al. (Paris, 2003)Ueno et al. (Paris, 2003) Ishiguro and Balibar (Paris, 2004)Ishiguro and Balibar (Paris, 2004)

the standard Casimir effect : the standard Casimir effect : confinement of the fluctuations of the confinement of the fluctuations of the electromagnetic field electromagnetic field the two electrodes attract each otherthe two electrodes attract each other

the "critical Casimir effect"the "critical Casimir effect"

confined fluctuations2 plates2 plates

the critical Casimir effect (Fisher and de Gennes, 1978):the critical Casimir effect (Fisher and de Gennes, 1978):near a critical point, near a critical point, confinement of the fluctuations of the order parameterconfinement of the fluctuations of the order parameter

a singular contribution to the free energy E ~ ka singular contribution to the free energy E ~ kBBT /LT /L2 2

a force between the two plates Fa force between the two plates FCasCas = - dE/dL ~ 2 k = - dE/dL ~ 2 kBBT /LT /L33

LL

the universal scaling functions the universal scaling functions and and

Further work Further work ((Nightingale and J. Indekeu 1985, M.Krech Nightingale and J. Indekeu 1985, M.Krech and S. Dietrich 1991-92) shows that and S. Dietrich 1991-92) shows that

E = kE = kBBT/LT/L22 (L/ (L/))

where where the "universal scaling function"the "universal scaling function" depends on depends on the bulk correlation function the bulk correlation function ~ t ~ t -- which diverges near the critical temperature Twhich diverges near the critical temperature Tc c . .

At TAt Tc c , i.e. t = 0, , i.e. t = 0, , the "Casimir amplitude". , the "Casimir amplitude".

a similar scaling function is introduced for the forcea similar scaling function is introduced for the force

FFCasCas = k = kBBT/LT/L33 (L/ (L/))

UniversalityUniversality

the scaling functions only depend onthe scaling functions only depend on- the dimension of spacethe dimension of space- the dimension of the order parameterthe dimension of the order parameter- the type of boundary conditions :the type of boundary conditions :

- periodic or antiperiodicperiodic or antiperiodic- DirichletDirichlet- von Neumann von Neumann

the 5 different values the 5 different values = = (T(Tcc) have been calculated, ) have been calculated,

but not but not at any T nor with any boundary conditions at any T nor with any boundary conditionsfor example Dirichlet-Dirichlet below Tfor example Dirichlet-Dirichlet below Tcc

the sign of the forcethe sign of the force

attractiveattractive if symmetric boundary conditions ( if symmetric boundary conditions ( < 0) < 0)repulsiverepulsive if antisymmetric ( if antisymmetric ( > 0) > 0)

the Casimir amplitude the Casimir amplitude = = (L/ (L/ = 0) = 0)

~ 0.2 to 0.3 for periodic boundary conditions~ 0.2 to 0.3 for periodic boundary conditionsproportional to the dimension N of the order parameterproportional to the dimension N of the order parameter10 times smaller if the order parameter vanishes at the 10 times smaller if the order parameter vanishes at the wall (Dirichlet-Dirichlet)wall (Dirichlet-Dirichlet) twice as large if tri-critical instead of criticaltwice as large if tri-critical instead of critical

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 forcesevidence for long range attractive forcescomparison with predictions :

assume a critical Casimir force assume a critical Casimir force (x)(x)//l l 33 measure [x = (L/)], the function"of this

force

comparison with theory

below Tbelow Tcc : :

no theoryno theorythe magnitude of the experimental the magnitude of the experimental depends on L (not universal ??) depends on L (not universal ??)it is also surprisingly large (1.5 to 2, no theoretical result larger than it is also surprisingly large (1.5 to 2, no theoretical result larger than ~ 0.5)~ 0.5)

above Tabove Tcc : :

agreement with Krech and Dietrich agreement with Krech and Dietrich [Phys. Rev. A 46, 1886 (1992)][Phys. Rev. A 46, 1886 (1992)]

far below Tfar below Tcc::

a finite value of a finite value of ? ?confinement of Godstone modes confinement of Godstone modes (Ajdari et al. 1991, Ziherl et al. 2000, Kardar et al. 1991-2004,(Ajdari et al. 1991, Ziherl et al. 2000, Kardar et al. 1991-2004, Dantchev and Krech 2004)Dantchev and Krech 2004)

Phys. Rev.

E 2004

periodic boundary conditionsperiodic boundary conditionsthe Casimir amplitude is larger by a factor ~2 for the XY model (N = 2)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 the scaling function does not vanish as T tends to 0 for the XY model

the magnitude of the effect of Godstone modes

for Dirichlet-Dirichlet boundary conditions, for Dirichlet-Dirichlet boundary conditions, Kardar and Golestanian (Rev. Mod. Phys. 1999) predict Kardar and Golestanian (Rev. Mod. Phys. 1999) predict a very small amplitude a very small amplitude ~ - 0.05 ~ - 0.05Garcia's measurement : Garcia's measurement : ~ - 0.3 ~ - 0.3 in agreement with Dantchev (but with periodic boundary conditions) in agreement with Dantchev (but with periodic boundary conditions) at the 2004 APS march meeting,at the 2004 APS march meeting,R.Zandi, J. Rudnick and M. Kardar invoke the surface fluctuations R.Zandi, J. Rudnick and M. Kardar invoke the surface fluctuations of the film which would enhance the Goldstone mode contribution, of the film which would enhance the Goldstone mode contribution, but the sign of this last effect is somewhat controversial.but the sign of this last effect is somewhat controversial.In fact the situation is not settled: In fact the situation is not settled: better experiments, better experiments, and calculations with the right boundary conditions and calculations with the right boundary conditions are neededare needed

substrate

12

12

12

"critical point wetting ":"critical point wetting ":wetting near a critical pointwetting near a critical point

Young - Dupré :Young - Dupré :

cos cos = (= (22 - - 11)/)/1212

Tc

Xc

1 2

X1 X2

Moldover and Cahn (1980) :Moldover and Cahn (1980) :near the critical point at Tnear the critical point at Tcc

12 12 0 as T --> T 0 as T --> Tcc

((22 - - 11) ) 0 also , but usually with 0 also , but usually with

a smaller critical exponent, a smaller critical exponent, especially ifespecially if((22 - - 11) ~ X) ~ X22 - X - X11

cos cos increases with T up to Tincreases with T up to Tww

where cos where cos = 1 and = 1 and = 0 = 0

TcTw

cos

Tc

1

Tw

the contact angle usually decreasesthe contact angle usually decreasesto zero at Tto zero at Tww < T < Tcc

Moldover and Cahn 1980:Moldover and Cahn 1980:a wetting transition takes place at a wetting transition takes place at TTww < T< Tcc

P.G. de Gennes (1981) + Nightingale and Indekeu (1985):P.G. de Gennes (1981) + Nightingale and Indekeu (1985):not necessarily true in the presence of long range forcesnot necessarily true in the presence of long range forces

a possible exception to critical point a possible exception to critical point wetting wetting

T

10

3He concentration

superfluid

normal

0.87 K

0.675

tri-criticalpointa tri-critical point:a tri-critical point:

superfluidity + phase separation superfluidity + phase separation

at Tat Tt t = 0.87 K= 0.87 K

the example of helium 3 - helium 4 the example of helium 3 - helium 4 liquid mixturesliquid mixtures

a a 44He-rich superfluid filmHe-rich superfluid film

T

3He concentration

10

Teq

superfluid

normal

tri-criticalpoint

Romagnan, Laheurte and Sornette (1978 - 86):Romagnan, Laheurte and Sornette (1978 - 86): van der Waals attractive fieldvan der Waals attractive field

a a 44He-rich film grows on the substrateHe-rich film grows on the substrate

substrateleq

4He-rich superfluid film

lleqeq ~ (T - T ~ (T - Teqeq))-1/3-1/3 up to 60 Angstömsup to 60 Angstöms

two possibilities:two possibilities:- van der Waals only,van der Waals only, lleqeq tends to a macroscopic value: tends to a macroscopic value:

complete wetting (complete wetting (= 0)= 0)- vdW + an attractive force (Casimir),vdW + an attractive force (Casimir), lleqeq saturates at some mesoscopic value: saturates at some mesoscopic value:

partial wetting (partial wetting ( ≠ 0) ≠ 0)

substrate

superfluid film

leq

4He-richbulk phase

ΠH(l)=3πσi2l

exp−2πσil

2

3kBT

⎛

⎝ ⎜ ⎞

⎠ ⎟

ΠCas(l)=Ttl3 ×θ(l /ξ)

ΠvdW(l)=

1000×(1/Vd−1/Vc)l3(1+l/193)

in K.A−3o

the contact angle the contact angle is obtained from the is obtained from the "disjoining pressure" "disjoining pressure" (l)(l) (see Ueno, (see Ueno, Balibar et al. PRL 90, 116102, 2003 and Ross, Bonn and Meunier, Nature 1999):Balibar et al. PRL 90, 116102, 2003 and Ross, Bonn and Meunier, Nature 1999):

3 contributions to 3 contributions to (l) from long range forces:(l) from long range forces:

van der Waals van der Waals

(repulsive on the film surface) (repulsive on the film surface)

Casimir (attractive)Casimir (attractive)(l/(l/) < 0 is the scaling function ) < 0 is the scaling function

which can be estimated from Garcia and Chanwhich can be estimated from Garcia and Chan

the entropic or "Helfrich" repulsionthe entropic or "Helfrich" repulsion

due to the limitation of the fluctuations of the film surfacedue to the limitation of the fluctuations of the film surface

an approximate calculation an approximate calculation

cosθ=1−Π(l)dl

∞

leq

∫σi

optical interferometryoptical interferometry

copper

mixing chamber

10 mmHe-Ne laser

opticalinterferometric

cavity(sapphire treated for 15% reflection)

vapor

3He-rich liquid

4He-rich liquid

coppercopper

Images at 0.852 KT. Ueno et al. 2003

the empty cell:stress on windows

fringe bending

liquid-gas interface

vapor

3He-rich "c-phase"

3He- 4He interface

4He-rich "d-phase

zone to be analyzed

the contact angle the contact angle and the interfacial tension and the interfacial tension ii

fringe patternfringe pattern --> --> profile of the meniscusprofile of the meniscus --> --> and and ii

typical resolution : 5 typical resolution : 5 mmcapillary length: from 33 capillary length: from 33 m (at 0.86K) to 84 m (at 0.86K) to 84 m (at 0.81K)m (at 0.81K)

zoom at 0.841 K

d-phase

c-phase

the interface profile at 0.841K

c-phase

d-phase

sapp

hire

experimental experimental

resultsresults the interfacial tensionthe interfacial tension

agreement with Leiderer et al. agreement with Leiderer et al. (J. Low Temp. Phys. 28, 167, 1977):(J. Low Temp. Phys. 28, 167, 1977):

ii = 0.076 t = 0.076 t22

where t = 1 - T/Twhere t = 1 - T/Tt t and Tand Tt t = 0.87 K= 0.87 K

the contact anglethe contact angle is non-zero is non-zero

it it increasesincreases with T with T

the disjoining pressure at 0.86K (i.e. t = 10the disjoining pressure at 0.86K (i.e. t = 10 -2-2))

-40

-20

0

20

40

0 200 400 600 800 1000 1200

disjoining pressure

( )(l

-9

.K A

-3)

( )film thickness l Angström

Helfrich

van der Waals

total pressure

+van der Waals Casimir

Casimir

the equilibrium the equilibrium thickness of the thickness of the superfluid film:superfluid film:

lleqeq = 400 = 400 ÅÅ~ about 4~ about 4,,

where where (l) = 0(l) = 0

the calculated contact angle the calculated contact angle

at T = 0.86 K, i.e. t = 1 - T/Tat T = 0.86 K, i.e. t = 1 - T/Tt t = 10= 10 -2 -2

lleqeq = 400 Å = 400 Å , 4 times the correlation length , 4 times the correlation length

By integrating the disjoining pressure from lBy integrating the disjoining pressure from leqeq to infinity, to infinity,

we find we find = 45 ° = 45 °

near a tri-critical point, the Casimir amplitude should be near a tri-critical point, the Casimir amplitude should be larger by a factor 2larger by a factor 2

this would lead to this would lead to = 66 °, in even better agreement with our = 66 °, in even better agreement with our experimentexperiment

At lower temperature (away from TAt lower temperature (away from Tt t ):):

ii and van der Waals are larger, Casimir is smaller, and van der Waals are larger, Casimir is smaller,

so that so that should also be smaller should also be smaller

the contact angle increases with T, the contact angle increases with T,

as found experimentallyas found experimentally

In 2003, our exp. resultsIn 2003, our exp. results (Ueno et al. , JLTP 130, 543, (Ueno et al. , JLTP 130, 543,

2003)2003) agreed with our approximate calculation agreed with our approximate calculation (Ueno et al. PRL 60, 116102, 2003)(Ueno et al. PRL 60, 116102, 2003)

new setup for new setup for experiments at lower Texperiments at lower T

(R. Ishiguro and S. (R. Ishiguro and S. Balibar, in progress)Balibar, in progress)

laser beamlaser beam

closer to normal incidencecloser to normal incidenceless distortion due to refraction less distortion due to refraction effects, better control of the effects, better control of the fringe patternfringe patternmeasurements at lower T:measurements at lower T:is the contact angle ≠ 0 ?is the contact angle ≠ 0 ?Goldstone modes ? amplitude ?Goldstone modes ? amplitude ?

(sapphire treated for 15% reflection)

dilution refrigeratordilution refrigerator

copper framecopper frame

optical cavity3He-rich liquid

4He-rich liquid

Ishiguro's profiles

the contact angle is zero at low T (237 mK) and near Tthe contact angle is zero at low T (237 mK) and near Ttt (840 mK) (840 mK)

Ishiguro's results

the interfacial

tension near Tt

the contact angle

Ishiguro and Balibar (2004) find Ishiguro and Balibar (2004) find = 0 = 0in contradiction with previous measurementsin contradiction with previous measurements

could the Casimir force be 5 times smaller than measured by Garcia and Chan ?

the disjoining pressure would the disjoining pressure would be dominated by the van der be dominated by the van der Waals field, Waals field, always positive, always positive, implying complete wetting implying complete wetting ((= 0)= 0)

-40

-20

0

20

40

0 200 400 600 800 1000 1200

( )film thickness l Angström

Helfrich

van der Waals

total pressure

+van der Waals Casimir

Casimir

-40

-20

0

20

40

0 200 400 600 800 1000 1200

( )film thickness l Angström

Helfrich

van der Waals

/5total pressure with GC

+ ( /5)van der Waals Casimir GC

( /5)Casimir GC

summary

the exception found by Ueno et al. to "critical point wetting"the exception found by Ueno et al. to "critical point wetting"is not confirmed by our more careful, and more recent experimentis not confirmed by our more careful, and more recent experiment

still possible if the substrate exerted a weaker van der waals field ?still possible if the substrate exerted a weaker van der waals field ?

the amplitude of the critical Casimir force measured bythe amplitude of the critical Casimir force measured byGarcia and Chan is not really universal and its amplitude looks Garcia and Chan is not really universal and its amplitude looks largelargebut there is no available calculation with the right boundary but there is no available calculation with the right boundary conditions below Tc where it is large.conditions below Tc where it is large.

more work ...more work ...