1. Observation of Critical Casimir Effect in a Binary Wetting Film:
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
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
System and experimental setup
Studied a wetting film of binary mixture MC/PFMC on Si(100),
in equilibrium with the binary vapor and bulk liquid mixture at
critical concentration.
Anti-symmetric (+,-) B.C.: Previous study at 30°C [10] showed
that MC-rich liquid wets the liquid/Si interface and PFMC is
favored at the liquid/vapor interface.
Intrinsic chemical potential Dm of the film relative to bulk
liquid/vapor coexistence was controlled by temperature offset
DT between the substrate and liquid reservoir [10].
Thickness measurements by x-ray reflectivity
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 > 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 > 0 for the MC/PFMC wetting films (T > 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)
 +,- > 0  pc tends to increase film thickness.
 Scaling variable: y = (L/x)1/n = t(L/x0)1/n,
where n = and x0+/x0- = 1.96 for 3D Ising systems [11], and
x0+ = 2.79 Å (T > Tc) for MC/PFMC [12].
Scaling function can be extracted experimentally from the measured L, using:
47.7 °C
46.2 °C
45.6 °C
From: Heady & Cahn, 1973 [9],
Tc =  0.01 °C
xc =  0.002
x (PFMC mole fraction)
Temperature [C]
PFMC
rich MC
Methylcyclohexane
(MC)
Perfluoro-
methylcyclohexane
(PFMC)
Incident X-rays
l = 1.54 Å
(Cu Ka)
qz = (4p/l)sin(a)
Si (100)
MC + PFMC L a
Specular
Reflection
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.
For sufficiently small t = (T – Tc)/Tc,
correlation length x = x0 t-n ~ film thickness L
 Each wall starts to “feel” the presence of the other wall.
 “Casimir effect”: film thinning (attractive) for (+,+)
and film thickening (repulsive) for (+,-) when t ~ 0.
qz [Å-1]
Normalized Reflectivity R/RF
DT = 0.50 °C
DT = 0.10 °C
DT = °C
At Tfilm = 46.2 °C ~ Tc
y = (L/x)1/n = t (L/x0)1/n
+,- = (kBTc)-1 [Dm L3 – Aeff/6p]
MFT
2D+,- (RG)
Symbols are based on the measured L, Dm = (2.2  J/Å3)DT/T, and Aeff = 1.2  J estimated for a homogeneous MC/PFMC film at bulk critical concentration xc = The red line (—) is for DT = °C. The dashed red line (---) for T < 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.
Inner cell
(0.001C)
Outer cell
(0.03C)
Saturated
MC + PFMC
vapor
Bulk reservoir:
Critical MC + PFMC
mixture (x ~ xc = 0.36)
at T = Trsv.
MC + PFMC wetting film on Si(100) at
T = Trsv + DT. z
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]).
“Casimir amplitudes” at bulk Tc (t = 0),
for 3D Ising systems:
DT = 0.50 °C
DT = 0.10 °C
DT = °C
Tfilm [°C]
Total film thickness L [Å]
DT = Tfilm – Trsv [K]
D+,- = ½ +,- (y = 0)
D+,- (RG)
At Tfilm = 46.2 °C ~ Tc
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].
Dm  DT = Tfilm – Trsv
y = (L/x)1/n = t (L/x0)1/n
+,
(+,-)
(+,+)
2D+,-
2D+,+
MFT scaling functions for Casimir pressure, where the ordinate has been rescaled so that
½+,±(0) = D+,±(RG)
at y = 0. (Based on [3])
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, (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, (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
Method
D+,-
D+,+
RG: Migdal-Kadanoff procedure [4]
0.279
RG: e = 4 – d expansion [3]
2.39
-0.326
Monte Carlo simulations [3]
2.450
-0.345
“Local free-energy functional” [5]
3.1
-0.42