System and experimental setup

1
Observation of Critical Casimir Effect in a
Binary Wetting Film An X-ray Reflectivity
Study Masafumi Fukuto, Yohko F. Yano, and Peter
S. Pershan Department of Physics and DEAS,
Harvard University, Cambridge, MA

• System and experimental setup

• What is a Casimir force?
• A long-range force between two macroscopic
  bodies
• induced by some form of fluctuations between
  them.
• Two necessary conditions
• (i) Fluctuating field
• (ii) Boundary conditions (B.C.) at the walls

• Comparison with theory
• Film thickness L is determined by
• Dm P(L) pc(L, t)
• i.e., a balance between
• (i) Chemical potential (per volume) of film
  relative to bulk liquid/vapor coexistence
• ? Dm gt 0 tends to reduce film thickness.
• ? Can be calculated from DT and known latent
  heat of MC and PFMC.
• (ii) Non-critical (van der Waals) disjoining
  pressure P Aeff/6pL3
• ? Effective Hamaker constant Aeff gt 0 for the
  MC/PFMC wetting films (T gt Twet).
• ? P tends to increase film thickness.
• ? Aeff for mixed films can be estimated from
  densities in mixture and
• constants Aij estimated previously
  for pairs of pure materials 10.
• (iii) Critical Casimir pressure pc
  kBTc/L3?,-(y)

Thickness measurements by x-ray reflectivity

• Casimir forces in adsorbed fluid films near bulk
  critical points
• (i) Fluctuations Local order parameter f(r,z)
• e.g., mole fraction x - xc in binary mixture
• (ii) B.C. Surface fields, i.e., affinity of one
  component
• over the other at wall/fluid and fluid/vapor
  interfaces.
• As T ? Tc, critical adsorption at each wall.

Symbols are based on the measured L, Dm (2.2 ?
10-22 J/Å3)DT/T, and Aeff 1.2 ? 10-19 J
estimated for a homogeneous MC/PFMC film at bulk
critical concentration xc 0.36. The red line
() is for DT 0.020 C. The dashed red line
(---) for T lt Tc is based on Aeff estimated for
the case in which the film is divided in half
into MC-rich and PFMC-rich layers at
concentrations given by bulk miscibility gap.

• Theoretical background
• Finite-size scaling and universal scaling
  functions
• (Fisher de Gennes, 1978 1)
• Casimir energy/area
• Casimir pressure
• For each B.C., scaling functions ? and ? are
  universal
• in the critical regime (t ? 0, x ? ?, and L ?
  ?) 2.
• Scaling functions have been calculated using
  mean field
• theory (MFT) (Krech, 1997 3).

• Summary
• Both the extracted Casimir amplitude D,- and
  scaling function ?,-(y) appear to converge with
  decreasing DT (or increasing L). This is
  consistent with the theoretical expectation of a
  universal behavior in the critical regime 2.

• The Casimir amplitude D,- extracted at Tc and
  small DT agrees well with D,- 2.4 based on the
  renormalization group (RG) and Monte Carlo
  calculations by Krech 3.
• The range over which the Casimir effect (or the
  thickness enhancement) is observed is narrower
  than the prediction based on mean field theory
  3.

• Thickness enhancement near Tc for small
• DT, with a maximum slightly below Tc.
• ? Qualitatively consistent with theoretically
• expected repulsive Casimir forces for (,-).

• Recent observations of Casimir effect in critical
  fluid films
• Thickening of films of binary alcohol/alkane
  mixtures on Si near the
• consolute point. (Mukhopadhyay Law, 1999 6)
• Thinning of 4He films on Cu, near the superfluid
  transition.
• (Garcia Chan, 1999 7)
• Thickening of binary 3He/4He films on Cu, near
  the triple point.
• (Garcia Chan, 2002 8)

References 1 M. E. Fisher and P.-G. de
Gennes, C. R. Acad. Sci. Paris, Ser. B 287, 209
(1978). 2 M. Krech and S. Dietrich, Phys. Rev.
Lett. 66, 345 (1991) Phys. Rev. A 46, 1922
(1992) Phys Rev. A 46, 1886 (1992). 3 M.
Krech, Phys. Rev. E 56, 1642 (1997). 4 J. O.
Indekeu, M. P. Nightingale, and W. V. Wang, Phys.
Rev. B 34, 330 (1986). 5 Z. Borjan and P. J.
Upton, Phys. Rev. Lett. 81, 4911 (1998).
6 A. Mukhopadhyay and B. M. Law, Phys. Rev.
Lett. 83, 772 (1999) Phys. Rev. E 62, 5201
(2000). 7 R. Garcia and M. H. W. Chan, Phys.
Rev. Lett. 83, 1187 (1999). 8 R. Garcia and M.
H. W. Chan, Phys. Rev. Lett. 88, 086101
(2002). 9 R. B. Heady and J. W. Cahn, J. Chem.
Phys. 58, 896 (1973). 10 R. K. Heilmann, M.
Fukuto, and P. S. Pershan, Phys. Rev. B 63,
205405 (2001). 11 A. J. Liu and M. E. Fisher,
Physica A 156, 35 (1989). 12 J. W. Schmidt,
Phys. Rev. A 41, 885 (1990). Work supported by
Grant No. NSF-DMR-01-24936.