Casimir forces, surface fluctuations, and thinning of

superfluid films

Mehran Kardar (MIT)

Roya Zandi

Joseph Rudnick (UCLA)

Phys. Rev. Lett. 93, 155302 (2004)

Superfluid transition

Question: Why are films thinner in the superfluid state?

Normalfluid

Superfluid

The film is thinner at the transition, and in the superfluid phase

4He thin film experimentsR. Garcia and M.H.W. Chan, Phys. Rev. Lett. 83, 1187 (1999)

Casimir effect

Quantum fluctuations of the EM field between conducting plates in vacuum results in long-ranged forces

Proc. K. Ned. Akad. Wet. 51, 793 (1948)

• 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:

• Free energy of the long-wavelength fluctuations:

M.E. Fisher and P.-G. de Gennes, C.R. Acad. Sci. Ser. B 287, 207 (1978)

M.E. Fisher + P.-G. de Gennes, C. R. Acad. Sci.Ser. B 287, 207 (1978)

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δF(H) = −kBT ×A

H 2× c

C

Superfluid Helium

• Superfluid He has “massless” Goldstone modes (phonons) associated with the phase of the quantum condenstae.

• The interaction resulting from (thermal) fluctuations of these modes is

H. Li and M. Kardar, PRL 67, 3275 (1991); PRA 46, 6490 (1992)

temp

Pressure

superfluid

fluid

gas

C

€

H φ[ ] =K

2d3x(∇φ)2∫

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δF(H) = −kBT ×A

H 2×

ζ (3)

16π

Wetting by a Superfluid film• R. Garcia and M.H.W. Chan, Phys. Rev. Lett. 83, 1187 (1999).

• Question: What determines the thickness of the wetting layer?

dHe (vapor)

He (liquid)

Thinning of a Superfluid film

• Thickness of the wetting film is obtained by minimizing

• The film is thinner at the transition, and in the superfluid phase

• The observed thinning of the film is larger than can be accounted by the Casimir forces associated with Goldstone modes.

d

h He (vapor)

He (liquid)

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E(d) = A ghd +Cvdw − CCas

d2

⎡ ⎣ ⎢

⎤ ⎦ ⎥,⇒ d =

2(Cvdw − CCas)

gh

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 3

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Ccas = 0

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CcasCritical€

Ccas =

Goldstone

modes + ?

• Normal fluid is clamped due to viscosity

• Superfluid films have a velocity field associated with the superfluid phase

• Kinetic energy

• Free energy associated with superfluid flow

Surface fluctuations

€

d3r1

2ρ sv

2 =∫ Pk

2 k tanhkd

2ρ sr k

∑

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φ

• 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

• Total Casimir force:€

Fsurface = −7

4

kBT

8π

ξ (3)

d3=

7

4FGoldstone

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Fcasimir = Fsurface + FGoldstone ≅ −0.15

d3kBT

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