( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−Δ=−+ π

μϑ6

1 3,

eff

cB

AL

Tky

System and experimental setupSystem 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 MC-rich liquid wets the liquid/Si interface and PFMC isPFMC is favored at the liquid/vapor interfacefavored at the liquid/vapor interface.

• Intrinsic chemical potential Δμ of the film relative to bulk liquid/vapor coexistence was controlled by temperature offset ΔT between the substrate and liquid reservoir [10].

Observation of Critical Casimir Effect in a Binary Wetting Film: Observation of Critical Casimir Effect in a Binary Wetting Film: An X-ray Reflectivity StudyAn X-ray Reflectivity Study

Masafumi Fukuto, Yohko F. Yano, and Peter S. PershanMasafumi Fukuto, Yohko F. Yano, and Peter S. PershanDepartment of Physics and DEAS, Harvard University, Cambridge, MA Department of Physics and DEAS, Harvard University, Cambridge, MA

What is a Casimir force?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

Casimir forces in adsorbed fluid Casimir forces in adsorbed fluid films near bulk critical pointsfilms near bulk critical points(i) Fluctuations: Local order parameter (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 = 0 t ~ 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.

47.7 °C

46.2 °C

45.6 °C

From: Heady & Cahn, 1973 [9], Tc = 46.13 0.01 °C xc = 0.361 0.002

x (PFMC mole fraction)

Tem

pera

ture

[C

]

PFMC rich

MC rich

Methylcyclohexane (MC)

Perfluoro-methylcyclohexane

(PFMC)

Inner cell(0.001C)

Outer cell(0.03C)

SaturatedMC + 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 + ΔT.

z

Incident X-rays = 1.54 Å (Cu K)

qz = (4π/)sin()

Si (100)

MC + PFMCL

SpecularReflection

ΔT = 0.50 °C

ΔT =

0.10 °C

ΔT =

0.020 °C

Tfilm [°C]

Tot

al f

ilm

thic

knes

s L

[Å

]

y = (L/)1/ = t (L/0)1/

ϑ +,

= (

k BT

c)1

[Δμ

L3 –

Aef

f/6π]

MFT

2Δ+, (RG)

y = (L/)1/ = t (L/0)1/

ϑ+,

(+,)

(+,+)

2Δ+,

2Δ+,+

MFT scaling functions for Casimir pressure, where the ordinate has been rescaled so that

½ϑ+,±(0) = Δ+,±(RG) at y = 0. (Based on [3])

qz [Å1]

Nor

mal

ized

Ref

lect

ivit

y R

/RF

ΔT = 0.50 °C

ΔT =

0.10 °C

ΔT =

0.020 °C

At Tfilm = 46.2 °C ~ Tc

Comparison with theoryComparison with theory• Film thickness L is determined by

Δμ = (L) + pc(L, t) i.e., a balance between:

(i) Chemical potential (per volume) of film relative to bulk liquid/vapor coexistence:

Δμ > 0 tends to reduce film thickness. Can be calculated from ΔT and known latent heat of MC and PFMC.

(ii) Non-critical (van der Waals) disjoining pressure: = Aeff/[6πL3]

Effective Hamaker constant Aeff > 0 for the MC/PFMC wetting films (T > Twet). 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]ϑ+,(y)

ϑ+, > 0 pc tends to increase film thickness.

Scaling variable: y = (L/)1/ = t(L/0)1/,

where = 0.632 and 0+/0

= 1.96 for 3D Ising systems [11], and 0

+ = 2.79 Å (T > Tc) for MC/PFMC [12].

• Scaling function can be extracted experimentally from the measured L, using:

Theoretical backgroundTheoretical 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 ϑ are universal in the critical regime (t 0, , 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:

( ) ⎟⎠

⎞⎜⎝

⎛=⎟⎠

⎞⎜⎝

⎛=

ωω L

L

TkLtL cB

cc 2,

( ) ⎟⎠

⎞⎜⎝

⎛=⎟⎠

⎞⎜⎝

⎛∂∂−=

ϑ

ω L

L

TkLL

tLp cBcc 3

,

( ) ( ) ( )xdxd

xxx θθϑ −= 2 where

( ) ( )00 21ϑθ =≡Δ Recent observations of Casimir effect in Recent observations of Casimir effect in

critical fluid filmscritical 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])

Method Δ+, Δ+,+

RG: Migdal-Kadanoff procedure [4] 0.279 0

RG: = 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

Δμ ΔT = Tfilm – Trsv

( ) Tss

TTT

lv

vl

vl

Δ−=

Δ⎟⎠⎞

⎜⎝⎛

∂

∂−

∂

∂≈

−=Δ

μμ

μμμ

Thickness measurements by Thickness measurements by x-ray reflectivity x-ray reflectivity

References: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.

ΔT = Tfilm – Trsv [K]

Δ +,

= ½

ϑ+

, (y

= 0

) Δ+, (RG)

At Tfilm = 46.2 °C ~ Tc

• Thickness enhancement near Tc for small ΔT, with a maximum slightly below Tc.

Qualitatively consistent with theoretically expected repulsive Casimir forces for (+,).

Symbols are based on the measured L, Δμ = (2.2 1022 J/Å3)ΔT/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 ΔT = 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.

Summary:Summary:• Both the extracted Casimir amplitude Δ+, and scaling function ϑ+,(y) appear to converge with decreasing ΔT (or increasing L). This is consistent with the theoretical expectation of a universal behavior in the critical regime [2].

• The Casimir amplitude Δ+, extracted at Tc and small ΔT agrees well with Δ+, ~ 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].