Casimir forces, surface
fluctuations, and thinning of
superfluid films
Mehran Kardar (MIT)
Roya Zandi
Joseph Rudnick (UCLA)
Phys. Rev. Lett. 93, 155302 (2004)
4He
thin film experiments
R. Garcia and M.H.W. Chan, Phys. Rev. Lett. 83, 1187 (1999)
Superfluid
transition
Superfluid
Normal
fluid
The film is thinner at the transition, and in the superfluid phase
Question: Why are films thinner in the
superfluid state?
Casimir effect
Proc. K. Ned. Akad. Wet. 51, 793 (1948)
Quantum fluctuations of the EM field between conducting
plates in vacuum results in long-ranged forces
• Normal modes of Electro-Magnetic field between plates:
H
• The ground state energy of quantized normal modes:

•An attractive force between plates:
Finite-size effects at Criticality
•
Analogs in Statistical Mechanics
•
Phase diagrams:
C
•
Free energy of the long-wavelength fluctuations:
A
F(H)  kB T  2  c
H
M.E. Fisher and P.-G. de Gennes, C.R. Acad. Sci. Ser. B 287, 207
(1978)
Superfluid Helium
Pressure
fluid
C
superfluid
gas
temp
•
Superfluid He has “massless” Goldstone modes (phonons)
associated with the phase of the quantum condenstae.
H   
•
K
2
 d x()
3
2
The interaction resulting from (thermal) fluctuations of these
modes 
is
A  (3)
F(H)  kB T  2 
H 16
H. Li and M. Kardar, PRL 67, 3275 (1991); PRA 46, 6490 (1992)
Wetting by a Superfluid film
•
R. Garcia and M.H.W. Chan, Phys. Rev. Lett. 83, 1187 (1999).
d
He (vapor)
He (liquid)
•
Question: What determines the thickness
of the wetting layer?
Thinning of a Superfluid film
d
h
He (vapor)
He (liquid)
•
•
Thickness of the wetting film is obtained by minimizing
1/ 3




C C
2(C  CCas )
E(d)  Aghd  vdw 2 Cas , d   vdw



d
gh


The film is thinner at the transition, and in the superfluid
phase
Ccas  0
Ccas 
Goldstone
modes + ?

•

Critical Ccas
The observed thinning of the film is larger than can be
accounted by the Casimir forces associated with Goldstone
modes.

Surface fluctuations
•
•
•
Normal fluid is clamped due to viscosity
Superfluid films have a velocity field
associated with the superfluid phase 
Kinetic energy

1
2 k tanh kd
2
 d r 2 sv  Pk 2
s
k
3
•
Free energy associated with superfluid flow

• Dzyaloshinskii, Lifshitz, Pitaevskii (1961)
• Mahale and Cole (1986)
• Not a Helfrich interaction which is repulsive

Thinning of a Superfluid film
•
Casimir force due to surface fluctuations
7 kB T  (3) 7
Fsurface  
 FGoldstone
3
4 8 d
4
•
Total Casimir force:
0.15
Fcasimir  Fsurface  FGoldstone   3 kB T
d

M. Krech
Ueno & Balibar (2004)
Summary
• Bulk Goldstone modes + surface
fluctuations suffice to explain the excess
thinning of the film in the superfluid
region.
Future work
• Effect of surface fluctuations at--and
especially immediately below--the
superfluid transition