critical Casimir forces and anomalous wetting

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

2. 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)

3. 2 plates confined fluctuations 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

4. 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/L2Q (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/L3q (L/x)

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 D = Q (Tc) have been calculated, but not Q at any T nor with any boundary conditions for example Dirichlet-Dirichlet below Tc

6. the sign of the force attractive if symmetric boundary conditions (q < 0) repulsive if antisymmetric (q > 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-Dirichlet) twice as large if tri-critical instead of critical

7. 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

8. 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)

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 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

11. 2 1 s12 s2 s1 q substrate Tc 1 2 X1 Xc X2 "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 --> 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

12. cos q q 1 Tw Tw Tc Tc the contact angle usually decreasesto zero at Tw < Tc Moldover and Cahn 1980: a wetting transition takes place at Tw< Tc P.G. de Gennes (1981) + Nightingale and Indekeu (1985): not necessarily true in the presence of long range forces

13. normal T superfluid 0.87 K tri-critical point 0 1 0.675 3He concentration 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

14. normal T superfluid Teq 4He-rich superfluid film leq tri-critical point substrate 0 1 3He concentration superfluid film 4He-rich bulk phase leq substrate 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)-1/3up 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

15. 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) < 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

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

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

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

19. 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

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. 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

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. dilution refrigerator copper frame laser beam 3He-rich liquid optical cavity 4He-rich liquid (sapphire treated for 15% reflection) 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 ?

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

25. Ishiguro's results

26. the interfacial tension near Tt

27. the contact angle Ishiguro and Balibar (2004) find q = 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 (q = 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 ...