Transcript Document
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
• A long-range force between two macroscopic bodies
induced by some form of fluctuations between them.
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.
Specular
Reflection
a
47.7 °C
L
(ii) Non-critical (van der Waals) disjoining pressure:
At Tfilm = 46.2 °C ~ Tc
(iii) Critical Casimir pressure:
• Intrinsic chemical potential of the film relative to bulk
liquid/vapor coexistence was controlled by temperature offset
T between the substrate and liquid reservoir [10].
L
k T
L
L k BTc L
Casimir pressure: pc L, t c 3
L x
L
x
T = Trsv + T.
z
Saturated
MC + PFMC
vapor
Bulk reservoir:
Critical MC + PFMC
mixture (x ~ xc = 0.36)
at
T = Trsv.
d
x
dx
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]).
T =
0.50 °C
MC + PFMC wetting
film on Si(100) at
Inner cell
(0.001C)
B c
Casimir energy/area: c L, t c 2
L
x
x
T =
0.10 °C
Total film thickness L [Å]
• Finite-size scaling and universal scaling functions
(Fisher & de Gennes, 1978 [1])
T =
0.020 °C
qz [Å1]
Outer cell
(0.03C)
Theoretical background
where x 2 x x
• 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.
2+,
T =
0.020 °C
T =
0.10 °C
T =
0.50 °C
T = Tfilm – Trsv
+,
(+,+)
2+,+
MFT scaling functions
for Casimir pressure,
where the ordinate has
been rescaled so that
½+,±(0) = +,±(RG)
at y = 0. (Based on [3])
l v T
T T
sv sl T
• “Casimir amplitudes” at bulk Tc (t = 0), 0
for 3D Ising systems:
1
2
Qualitatively consistent with theoretically
expected repulsive Casimir forces for (+,).
0
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])
+,
+,+
RG: Migdal-Kadanoff procedure [4]
0.279
0
RG: e = 4 – d expansion [3]
2.39
0.326
• Thinning of 4He films on Cu, near the superfluid transition.
(Garcia & Chan, 1999 [7])
Monte Carlo simulations [3]
2.450
3.1
0.345
0.42
• Thickening of binary 3He/4He films on Cu, near the triple point.
(Garcia & Chan, 2002 [8])
Method
“Local free-energy functional” [5]
y = (L/x)1/n = t(L/x0)1/n,
where n = 0.632 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:
1
3 Aeff
L
, y
k BTc
6
MFT
2+, (RG)
Symbols are based on the measured L,
= (2.2 1022 J/Å3)T/T, and Aeff = 1.2
1019 J estimated for a homogeneous
MC/PFMC film at bulk critical
concentration xc = 0.36. The red line (—)
is for T = 0.020 °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.
At Tfilm = 46.2 °C ~ Tc
Summary:
• Both the extracted Casimir
amplitude +, and scaling function
+,(y) appear to converge with
decreasing T (or increasing L).
This is consistent with the
theoretical expectation of a
universal behavior in the critical
regime [2].
Tfilm [°C]
• Thickness enhancement near Tc for small
T, with a maximum slightly below Tc.
y = (L/x)1/n = t (L/x0)1/n
Scaling variable:
y = (L/x)1/n = t (L/x0)1/n
l v
(+,)
pc = [kBTc/L3]+,(y)
+, > 0 pc tends to increase film thickness.
Normalized Reflectivity R/RF
45.6 °C
P = Aeff/[6L3]
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].
46.2 °C
• 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.
correlation length x = x0 tn ~ 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.
> 0 tends to reduce film thickness.
Can be calculated from T and known latent heat of MC and PFMC.
PFMC
rich
From: Heady & Cahn, 1973 [9],
Tc = 46.13 0.01 °C
xc = 0.361 0.002
• For sufficiently small t = (T – Tc)/Tc,
i.e., a balance between:
(i) Chemical potential (per volume) of film relative to bulk liquid/vapor coexistence:
MC + PFMC
x (PFMC mole fraction)
• As T Tc, critical adsorption at each wall.
• Film thickness L is determined by
a
Si (100)
MC
rich
Comparison with theory
= P(L) + pc(L, t)
qz = (4/l)sin(a)
Incident X-rays
l = 1.54 Å
(Cu Ka)
Temperature [C]
Two necessary conditions:
(i) Fluctuating field
(ii) Boundary conditions (B.C.) at the walls
Methylcyclohexane
(MC)
Perfluoromethylcyclohexane
(PFMC)
Thickness measurements by
x-ray reflectivity
+, = ½ +, (y = 0)
System and experimental setup
+, = (kBTc)1 [ L3 – Aeff/6]
What is a Casimir force?
+, (RG)
T = Tfilm – Trsv [K]
• The Casimir amplitude +, extracted at Tc and small T agrees well with +, ~
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].
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.