HEAT CONDUCTION BY RIPPLONS ON THE SURFACE OF 4He

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HEAT CONDUCTION BY RIPPLONS ON THE

SURFACE OF 4He

D. Edwards, I. Mantz, V. Nayak

To cite this version:

JOURNAL DE PHYSIQUE _{Colloque C6, supplPment au no 8, Tome 39, aolit IY 18, page C6-300}

HEAT

CONDUCTION

BY

RIPPLONS ON

THE

SURFACE OF

4 ~ e f ,

D.O. ~dwardsf I.B. Mantz and V.S. Nayak

The Ohio S t a t e University, CoZwnbus, Ohio 43210, U.S.A.

RBsum6.- La conductivit6 thermique associde aux "ripplons" sur la surface d'un film de 'He a 6t6

mesurse. Le transport de chaleur est attribu6 5 un dcoulement hydrodynamique du gaz de ripplons,

limit6 par le transfert de l'impulsion au substrat du film. Ce courant est fortement r6duit par

la prgsence de 3 ~ e sur la surface.

Abstract.- The heat conductivity due to ripplons on the surface of the 'He film has been measured. The transport of heat is attributed to a hydrodynamic flow of the ripplon gas, limited by inomentum transfer to the substrate beneath the film. The ripplon current is greatly reduced by a smallamount

of 3 ~ e on the surface.

This paper describes an experiment on the surface transport of heat by quantized capillary waves or "ripplons" /I/. The effect studied is ana-

logous to the two-fluid convection process in bulk

4 ~ e , in which the heat is carriedby a hydrodynamic

flow of normal fluid excitations, and there is a corresponding flow of superfluid in the opposite

direction. In bulk "e, the heat transport is

limited by the normal viscosity rl because of the

-f n

boundary condition v = 0 at the walls. In our

n

surface experiment, which is performed on a He film to minimize heat flow through the bulk, there are no boundaries to the surface, and the flow islimi- ted by momentum transfer from the surfaceexcita- tions to the substrate of the film.

The measurements were carried out in a cell

containing a mylar strip 42 mm wide by 120 mm high

and 0.025 mm in thickness. The mylar has a heater

at the upper end and two thermometers 30 mm apart

near its center. Both thermometer,^ and heater are

thin layers of "Dag" connected to extremely thin, superconducting aluminium strips for electrical connections. The lower end of the mylar is therma- lly attached to the bottom of the copper cell.

'He containing less than a few parts in 101° of 3 ~ e was gradually added to the cell so to vary the

thickness of the film on the mylar, up to satura- tion. The cell contained some sintered copper

i' This work was supported by the US National

Science Foundation, grant number DMR-19546-A01.

*

Ecole Normale Supbrieure, Groupe de Physique des

Solides, 24, rue Lhomond75231 Paris Cedex05 FRANCE

which increased the total area for He film toabout

10 m2. This reduced the effect of 3 ~ e impurity but,

since some of the pores in the sinter gradually fill with liquid as 'He is added, the relation between the amount of %e and the thickness of the film on the mylar has to be determined empirically.

A measure of the film thickness d was obtai- ned from the heat required to quickly evaporate the helium convering a graphite film resistor in the cell. The completion of evaporation was marked by a sudden rise in the temperature of the resistor af- ter 50 to 200 ys of steady heating. This was too short a period for the film to be appreciably re- plenished, as was verified by varying the length of the heating period. We did not devise this techni- que until the thermal conductivity of a number of samples had been measured, but data were taken for the two thickest samples in figure 1.

The thermal conductance of the mylar subs-

trate without 'He was found to be % ( 1 2 ~ ~ ) erg s-'

deg-'. This is typical for a glassy substance. To

obtain the two-dimensional conductivity K of the helium, the mylar conductance was substractedfrom the total conductance, which was then multiplied by the distance between the thermometers and divi- ded by twice the width of the strip. The factor of two takes into account the He film on the back of

the strip. As shown by figure 1 , the temperature

dependence of K varies from about for the

thinnest sample, to % T"' for the saturated film

and for the next thickest sample. For these two, the ratio of the conductivities and the ratio of

film thickness are both 2.1 so that we deducethat, in this case, K is proportional to the filmthick- ness d.

Fig. 1 : The surface heat conductivity K of pure 'He

for various film thicknesses d. The upper curve is

for the saturated film (d

*

200 A), the curvebelow

for d 'L 100

i.

The lower three curves are for

monotonically decreasing but unknown values of d.

A very brief analysis of the data is as

follows : we first eliminate the possibility that

the heat is carried by the phonon modes of the film

since, to obtain a conductivity com~srable to the

1

data using a formula of the type

-

CVX, requiresan

3

implausibly long phonon free path of % 1 cm. A

similar calculation indicates that, for diffusive heat conduction by ripplons, the ripplon free path

would have to be % 1 mm at 0.1 K. This is much lon-

ger than the free path for internal equilibrium in the ripplon gas since, according to Saam 121, even

at 0 K the free path for a ripplon of energy 0.1 K

to decay into two others is

*

0.03 mm. At finite

temperatures, the free path is further reduced due to stimulated emission, four-ripplon processes,etc. We deduce that we are observing a hydrodynamicflow of the ripplons. In this picture the normal fluid

+

velocitg v is related to the heat current. per unit

n ₊

lengh qS by q' = BrTvn

-

-K grad T, where Sr is the

ripplon entro y per unit area, equal to

g

(15.2x10-3T4' ) erg cm-'deg-' [I /. The driving force

is provided by the gradient of the surface tension,

grad a =

-

S grad T, so that the quantity dlgrad

/vn = K~/(s'T), which has the dimensions of a vis-

cosity, is a measure of the momentum transfer to

-2 2.6

the substrate. Empirically, it is

*

3x10 T

poise for our two thickest samples.

The mechanism for momentum transfer to the substrate is not yet known, but one possibility is via low frequency ripplons with wave vectors in the

range kd $, 1. These have energies << kgT, so that

they do not contribute much to the thermalenergy, yet they are continually emitted and absorbed by

ripplons with energies Q, k T due to the 3-ripplon

B

process 121. They are presumably quite strongly coupled to the substrate because, in contrast with

ripplons with kd >> 1, they produce a substancial

pressure and velocity field at the liquid-substrate interface. For small d or very low frequenciesthere is also the direct Van der Waals interactionbetween the substrate and the liquid surface. We speculate that the important process for momentum transfer is

one in which two thermal, kd $ 1 ripplonsannihilate

to form a phonon in the substrate. Such a process would have a rate which should rise quite rapidly

with temperature, and decrease with filmthickness

,

and it might explain the ~ " ~ / d dependence observed

for the momentum transfer in our two thickest samples.

We have also measured the effect of adding

a small amount of 3 ~ e

-

equivalent to 0.25 layers

on the surface of the saturated film. The heat conductance was reduced to that of the mylar so that, within the error, the ripplon heat conduc- tivity of the helium was reduced to zero. From the

known binding energy of 3 ~ e on the surface, we

calculate that the 3 ~ e remained on the surface and

did not dissolve in the bulk of the film. Such a drastic effect on the ripplon current is not sur- prising. It is entirely analogous to the effect

that dissolved 3 ~ e has on the bulk phonon conduc-

tivity.

References

/I/ Atkins, K.R., Can. J. Phys. (1953) 1165.