1. 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 Paris, France critical Casimir forces and anomalous wetting StatPhys Bangalore, july 2004 for references and files, go to http://www.lps.ens.fr/~balibar/

2. abstract a critical introduction to and 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)

3. the standard Casimir effect : confinement of the fluctuations of the electromagnetic field the two electrodes attract each other the "critical Casimir effect" confined fluctuations 2 plates 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 ~ k B T /L 2 a force between the two plates F Cas = - dE/dL ~ 2 k B T /L 3 L

4. the universal scaling functions  and  Further work ( Nightingale and J. Indekeu 1985, M.Krech and S. Dietrich 1991-92) shows that E = k B T/L 2  (L/  ) E = k B T/L 2  (L/  ) where the "universal scaling function"  depends on the bulk correlation function  ~ t - depends on the bulk correlation function  ~ t - which diverges near the critical temperature T c. At T c, i.e. t = 0, , the "Casimir amplitude". a similar scaling function is introduced for the force F Cas = k B T/L 3  (L/  )

5. 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  =  (T c ) have been calculated, but not  at any T nor with any boundary conditions for example Dirichlet-Dirichlet below T c

6. the sign of the force attractive if symmetric boundary conditions (  < 0) repulsive if antisymmetric (  > 0) the Casimir amplitude  =  (L/  = 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-Dirichlet) twice as large if tri-critical instead of critical twice as large if tri-critical instead of critical

7. the experiment by R. Garcia and M. Chan a non-saturated film of pure 4 He (200 à 500 angströms) in the vicinity of the superfluid transition (a critical point at 2.17 K), evidence for long range attractive forces the film gets thinner : evidence for long range attractive forces comparison with predictions : assume a critical Casimir force  (x) / l 3 measure  [x = (L/  )   ], the  function"of this force

8. comparison with theory below T c : no theory the magnitude of the experimental  depends on L (not universal ??) it is also surprisingly large (1.5 to 2, no theoretical result larger than ~ 0.5) above T c : agreement with Krech and Dietrich [Phys. Rev. A 46, 1886 (1992)] far below T c : a finite value of  ? confinement of Godstone modes (Ajdari et al. 1991, Ziherl et al. 2000, Kardar et al. 1991-2004, Dantchev and Krech 2004) Dantchev and Krech 2004)

9. 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

10. 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  ~ - 0.05 Garcia's measurement :  ~ - 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

11. substrate 1 2  12 11 22  "critical point wetting ": wetting near a critical point Young - Dupré : cos  = (  2 -  1 )/  12 TcTc XcXc 1 2 X1X1 X2X2 Moldover and Cahn (1980) : near the critical point at T c  12  0 as T --> T c (  2 -  1 )  0 also, but usually with a smaller critical exponent, especially if (  2 -  1 ) ~ X 2 - X 1  cos  increases with T up to T w where cos  = 1 and  = 0

12.  TcTc TwTw cos  TcTc 1 TwTw the contact angle usually decreases to zero at T w < T c Moldover and Cahn 1980: a wetting transition takes place at T w < T c P.G. de Gennes (1981) + Nightingale and Indekeu (1985): not necessarily true in the presence of long range forces

13. a possible exception to critical point wetting T 10 3 He concentration superfluid normal 0.87 K 0.675 tri-critical point a tri-critical point: superfluidity + phase separation at T t = 0.87 K the example of helium 3 - helium 4 liquid mixtures

14. a 4 He-rich superfluid film T 3 He concentration 10 T eq superfluid normal tri-critical point Romagnan, Laheurte and Sornette (1978 - 86): van der Waals attractive field a 4 He-rich film grows on the substrate substrate l eq 4 He-rich superfluid film l eq ~ (T - T eq ) -1/3 up to 60 Angstöms two possibilities: - van der Waals only, l eq tends to a macroscopic value: l eq tends to a macroscopic value: complete wetting (  = 0) - vdW + an attractive force (Casimir), l eq saturates at some mesoscopic value: l eq saturates at some mesoscopic value: partial wetting (  ≠ 0)  substrate superfluid film l eq 4 He-rich bulk phase

15. the contact angle  is obtained from the "disjoining pressure"  (l) (see Ueno, Balibar et al. PRL 90, 116102, 2003 and Ross, Bonn and Meunier, Nature 1999): 3 contributions to  (l) from long range forces: van der Waals (repulsive on the film surface) Casimir (attractive)  (l/  ) < 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 an approximate calculation

16. optical interferometry copper mixing chamber 10 mm He-Ne laser optical interferometric cavity (sapphire treated for 15% reflection) vapor 3 He-rich liquid 4 He-rich liquid copper

17. Images at 0.852 K T. Ueno et al. 2003 the empty cell: stress on windows fringe bending liquid-gas interface vapor 3 He-rich "c-phase" 3 He- 4 He interface 4 He-rich "d-phase zone to be analyzed

18. the contact angle  and the interfacial tension  i fringe pattern --> profile of the meniscus -->  and  i typical resolution : 5  m capillary length: from 33  m (at 0.86K) to 84  m (at 0.81K) zoom at 0.841 K d-phase c-phase the interface profile at 0.841K  c-phase d-phase sapphire

19. experimental results the interfacial tension agreement with Leiderer et al. (J. Low Temp. Phys. 28, 167, 1977):  i = 0.076 t 2 where t = 1 - T/T t and T t = 0.87 K the contact angle  is non-zero it increases with T

20. the disjoining pressure at 0.86K (i.e. t = 10 -2 ) the equilibrium thickness of the superfluid film: l eq = 400 Å l eq = 400 Å ~ about 4 , where  (l) = 0

21. the calculated contact angle  at T = 0.86 K, i.e. t = 1 - T/T t = 10 -2 l eq = 400 Å, 4 times the correlation length  By integrating the disjoining pressure from l eq to infinity, we find  = 45 ° near a tri-critical point, the Casimir amplitude should be larger by a factor 2 this would lead to  = 66 °, in even better agreement with our experiment At lower temperature (away from T t ):  i and van der Waals are larger, Casimir is smaller, so that  should also be smaller the contact angle increases with T, as found experimentally

22. In 2003, our exp. results (Ueno et al., JLTP 130, 543, 2003) agreed with our approximate calculation (Ueno et al. PRL 60, 116102, 2003)

23. new setup for experiments at lower T (R. Ishiguro and S. Balibar, in progress) laser beam 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 ? (sapphire treated for 15% reflection) dilution refrigerator copper frame optical cavity 3 He-rich liquid 4 He-rich liquid

24. Ishiguro's profiles the contact angle is zero at low T (237 mK) and near T t (840 mK)

25. Ishiguro's results

26. the interfacial tension near T t

27. the contact angle Ishiguro and Balibar (2004) find  = 0 in contradiction with previous measurements

28. 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 (  = 0)

29. 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...