Triple line dynamics in superfluid helium

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Triple line dynamics in superfluid helium

Françoise Brochard-Wyart

To cite this version:

Classification

Physics

Abstracts

fi7.70-68.10

Short Communication

lkiple

line

dynamics

in

superfluid

helium

Frangoise

Brochard-ivj'art

Institut Curie, 11 _rue Pierre et Alarie Curie, 75231 Paris Cedex 05, France

(Received

14

SeiJtember1992,

accepted

in final form 28

October1992)

Abstract.

Superfluid

~IIe

fails to ivet _any

substrate,

as has been shown

theoretically

and

experimentally.

~Ve

study

_some

liydrodynainic

features

expected

in

partial

wetting

(in

conditions where

exchanges

with the _{vapor are}

negligible):

I)

the

dynamics

of

dewetting:

thin

films

deposited

on non wettable surfaces are metastable below a critical thickness and evolve

by

nucleation and

growth

of

dry

holes. The boles _are

predicted

here to open at constant

velocity.

ii)

The collective triodes of a contact

line,

"triplons",

which are

propagative.

1 Tut io duct ion.

It has been sho,vn _very

recently

that

superfluid

~IIe

fails to wet all substrates.

Theoretically,

Cheng

et al.

[II

predicted

that,

at T ₌

0,

~ _lie

(leposited

on weak

binding

substrates such as the

alkali metals could

e,;hibit,

depending

on the substrate

potential,

either non

wetting

or

wetting:

finite contact

angles

are

predicted

for ~ lie

deposit.ed

on

potassium,

rubidium and

cesium,

while

complete

wetting

is

predicted

for

lithium,

sodiuiii and

hydrogen.

In a more recent paper,

they

explore

the

temperature

dependence

of

the,vetting-non

wetting

phase

diagram.

Observations

by

Naclier and

Dupont-Roc

[2] of heat. flow in

~lle

films in a

glass

tube,

with and without

an interior cesium

ring

show that

~lle

does not ,vet cesium. However

wetting

is observed for

Rb and Ii

rings. Using

_{vapor pressure} isotherm

measurements,

&Iukherjee,

Druist and Chan

[3] indicate a

complete

absence of

~IIe

adsorption

on cesium coated

graphite

for T < 2 K.

Ketola,

Wang

and lIallock _[4]

(using

third

sound)

have observed anomalous

wetting

of

partly

Cs-coated

glass.

Increasing

the _{vapor pressure,}

they

monitor

a transition between

dry

cesium

and cesium covered

by

_a thin

film,

,i=hereas Naclier observed _no

wetting

for

~He

in coexistence

with bulk

liquid.

This

discrepency

has been attributed to

roughness and/or

surface

impurities

(e.g.:

_oxygen

promoting wetting),

,vhicli _are different in the two

experiments.

In _a more recent

work,

Rutledge

and Taborek _{[5] ,vere} able to prepare very smooth cesium

evaporated

films onto

the surface of a quartz niicrolJalance an(I slio,v that

~lle

film

undergoes

a

wetting

transition

22 JOURNAL DE

PHYSIQUE

II N°1

Our aim in the

present

note is to

study

the

dynamics

of

wetting by

superfluid

helium,

as-suming:

a)

_a finite contact

angle

SE,

(or

a

negative spreading

coefficient S = 7so-

(7sL+7),

where _{7;j are} the bare

solid,

solid/liquid

and

liquid/vapor

interracial

tensions,

and

b)

negli-gible exchanges

between the

liqui(I

and the _vapor

(thus

restricting

our attention to very low

temperatures).

If _a helium film is

deposited

on a non-wettable

substrate,

the film is unstable below a critical

thickness _[6]

ec =

2K~~

sin~~

₍₁₎

2 where Jc~~ is the

capillary

length.

~-l

~ 7

(~~

~

with ₇

= 0.35

niN.ni~~,

p = 0.14

g.cni~~,

one

expects

K~~

+~ 0.5 mm. "Thick" films are metastable:

they

dewet

I)y

nucleation and

growth

of

dry

patches

[6, 7].

Microscopic

films

are unstable and dewet

by

a meclianisiu of

spinodal

decomposition,

where

capillary

waves are

amplified [8-10].

The

dewetting

of viscous

licjuids

has been studied

recently

both

experimentally

and

theoretically:

dry

holes in the film _open and _a

liquid

rim is formed at the

periphery

of the

dry

region.

The

velocit.y

of rim Vd~ #

(0E)~

is constant in

time, inversely

proportional

to the

viscosity

q of the

liquid

and

independent

of the initial thickness

e~.

In section 2 we

study

the

dewetting

velocity

and the

shape

of the

moving

rim for inertial

liquids.

In section

3,

we consider a contact line I)etween a

solid,

liquid

helium and vapor which is

deformed

sinusoidally.

Many

capillary

probleins

involve the deformations of _a contact line: for

instance the

separation

of the line fi.oni _an

anchoring

site _can be

analysed

once undulation

modes

are understood. These modes have been studied for viscous

liquids by Ondarquhu

and

Veyssid

[I

ii.

lvith silicone oils

they

found

single

exponential

relaxations,

suggestive

of

purely

viscous

regimes.

we have calculated these collective mode relaxations for viscous

liquids

[12].

Our aim in section 3 is to discuss t-he _case of inertial

liquids.

Most of _our discussion in section 3 is

only

qualitative,

and

ignores

all nuiuerical

prefactors.

2.

Dewetting:

nucleation and

growth

of

dry

regions,

When _a

liquid

fiIni is

deposited

on a non-wettable

surface,

dry regions

nucleate and grow. We

have

pictured

in

figure

i the

opening

of _a

(lry patch

of _a radius

R(t)

surrounded

by

a

liquid

rim,

which troves at.

velocity

U =

j

_{The native fiIni is immobile. The}

_liquid

_{of the}

_dry

i

region

is collecte(I in the rim of voliiue

flit)

increasing

with time. We can

simply

derive the

motion

equation

of the

inoving

tint front the transfer of surface

energies

into kinetics _energy of the rim.

)pQU~=-SA

(3)

where A

= ~

R~

is the surface of the

dry

surface of radius R and

Q,

(Q

= A

e),

the volume of

the

liquid

which has been collected in the rim. From

equation

(3),

one finds the

velocity

U :

U= ~~

=/T

~~.

(4)

pe

U

my RIM NATIVE FILM

o ~ B

E

(iii)

(

e

Fig.

1.

Opening

of _a

dry patch

of radius

R(t)

in _a helium film of thickness _e. The

liquid

is collected in

a rim of circular

shape.

@E is the

equilibrium

contact

angle.

U is the

dewetting

velocity.

The

velocity

is constant with time! The

driving

force

acting

on the rim is

constant,

but the

mass of the rim increases and the

resulting

motion is uniform.

Equation

(3)

assumes that

e < ec. In the

general

case, one must add a

gravity

energy

contribution,

which leads to:

U =

fi.

(4')

~ e ~

Typically,

with S ~W 7, one

expects

U ~W cm

s~~

for ~~ ~W lo. e

The

velocity

of

dewetting

is similar to the

velocity

of

opening

of a hole in a soap film

[13,

14].

The _case of _soap films is

complicated by

the _presence of the surfactant at the free

surface,

which

gives

rise to a

relatively

complex

elastic behavior. In our case, the interfacial tension 7

is

constant,

alld

equation

(4)

is exact.

The

shape

of rim is deduced front the motion

equation

of a

liquid

sheet. In the lubrication

approximation,

the

velocity

V(z)

of a

liquid

sheet of

height

((~)

is

independent

of z

(plug

flow)

and the Stokes

equation

for a non-viscous

liquid

is:

P(

($

+

V~~)

=

(

7

("'

(5)

The derivative of the moinentuni is balanced

by

the

Laplace

_pressure

gradient.

The

equation

has to be modified at the two contact lines A and B with the solid substrate and the native

film,

respectively.

Because the acceleration term is

negligible

for z -

0,

the static

equilibrium

conditions hold at both ends of the tint

The rim moves at

velocity

U

given

by equation

(4),

except

in a narrow

region

where the

velocity

slows down to zero.

Equat.ion

(5)

then shows

that,

in most of the

rim,

("'

=

0,

I-e- the pressure

is constallt arid the rim

profile

is _a

portion

of _a

perfect circle,

except

near the contact line B

in a thin

region

of extension

f.

The

healing length

f

is deduced from

equation

(5)

by

a

scaling

24 JOURNAL DE

PHYSIQUE

II N°1

Integrating

equation

(5)

between A and B leads to

equation

(4).

We have

pictured

in

figure

i

the

shape

of the rim and the small

region

where 0 falls to zero.

3. Collective uiodes of _a contact line.

We consider the contact line _a non-wettable

solid,

superfluid

~He

and its _vapor

(of negligible

pressure).

The line is deforiued

sinusoidally

with _a

wavelength

~~

(Fig.

2).

We discuss the

q

resulting

oscillation

assuming

a small contact

angle

0E

< 1. In this

limit,

the

capillary

energy

of the

weakly

disturbed

line,

with

amplitude

_uq for _{a wave} vector _q, has been calculated first

in reference

[15]

and is of the form:

fei

q =

)1011ql

lttql~

(8)

y

~

2~

q

6<6~

+ + + + + X

Fig.

2. "Triplon" modes of _a contact line.

Equation

(8)

holds when

gravitational

energies

_are

negligible

(q

>

K).

In this

regime,

the line

elasticity

cannot be described in teriils of a line tension. A tension

(

would

generate

an energy

~W

(

q~

jug

~ _{The distortions of the}

_{liquid-,<apor}

_interface

near the contact line extend

(in

the

direction normal to the

line)

_up to dist.ances of order

q~~

This is the source of the anomalous

factor _(q( in

equation

(8).

In the

opposit.

liiuit

_(q

<

K),

the deformations are screened at a

distance

K~~,

and the _eitergy has a standard foriu:

fAq

_"

(7

01

K (q(~

("q(~

~~~

The

general

expression

interpolating

I)etu~een these two limits has been derived more

recently

[16]:

feiq

"

)7

°I

(fi

)

lug

~

(IQ)

We derive the

dynamical

features of the mode fi.oiu the balance between the elastic

restoring

force deduced froiu

equation

lo)

and the inertial force:

7

0(

(fi

«)

[uq[

= M

w~uq

(ii)

3. I CAPILLARY REGIhfE

(q

>

K).

In tills

limit,

the

perturbation

extends over a

region

of

size

q~~

normal to the direction of the

line,

and the

scaling

form of M is:

hi ₌

p~(.

(12)

q"

From

equation

(ii),

we find the

fi.equency

w(q)

of the

propagative

mode

omitting

an exact

numerical coefficient:

w(q)

₌

q~/~

/~

~

0E.

(13)

Joanny

has calculated the

fi.equency

of the oscillation of

superfluid

droplets

around their

equi-librium

shape

[Iii

and finds _w

=

'

(

in

agreeiuent

with

equation

(13).

~

3.2 GRAVITY REGIhIE _q <

«~~

For

long

waveiengtiis,

the deformation extends on the

capillary

length

K~~

and the _mass fi,I is now:

hi

+~

p0EK~~

(14)

Equation

(10)

leads to standard

propagative

modes of a

vibrating

rope w = cq, where the

velocity

of the wave, c, is:

~

/~

~

' ~~~~ with

K~~

+~ 0.5 runt,

0E

~

10~~,

one finds c +~

ni.s~~

4

Concluding

reiuarks.

We have studied some

dynainical

features of

superfluid

~lle

deposited

on non-wettable

sub-strates at _very low temperatures, where the role of _vapor is

negligible.

I)

Thin films are unstable and dewet

by

nucleation and

growth

of

dry

patches,

which open at _-?

constant

velocity

U

=

U*0E,

where U* = g

~

Contrary

to the _case of viscous

liquids

where

~

the

dewetting

velocity

_i~,as

independent

of the initial film thickness _e, one expects U +~

e~~/~

for inertial

liquids.

By

video

recording

the

opening

of micro or macro

dry

patches,

one can

monitor the filiu thickness

varying

_{over a} broad _range

(100 I-100

pm).

The metastable films

observed

recently by

liutledge

an(I Taborek _[5] do not dewet

spontaneously.

However,

if a

hole in the film is fornied

(I)y

a laser

lJeain),

the film will dewet

probably by

the mechanisms

described here.

ii)

The

"triplons"

iuodes of the contact line are

propagative.

As for

capillary

waves of

usual

liquids

in the inertial

regime,

_,ve find _w

+~

q~/~

for

large

wave vectors q. On the other

hand,

for undulations of siuall wave vectors, the deforiuatioiis are screened and w = cq, where

26 JOURNAL DE

PHYSIQUE

II N°1

Acknowledgements.

We thank P-G- de Gemies and J.F.

Joanny

for

stimulating

discussions _on the

wetting

of

~He,

C. Redon _on

dewetting

and the Referee for his useful comments.

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