DEPENDENCE OF THE HELIUM II FILM
TRANSFER RATE 0N PRESSURE HEAD,
FILM HEIGHT, AND SUBSTRATE

Thesis far '?:he Degree of Ph, D.
MICHIGAN STATE UNIVERSITY
CARL JAMES DUTHLER '
A 1970

   
     

  

   

 

r'rlfi”435! ‘IE‘Jljzallfit‘tél'zzl 1:!
Michigan State
University
I“
This is to certify that the
thesis entitled
Dependence of the Helium II Film Transfer Rate
on Pressure Head, Film Height, and Substrate
presented by
Carl James Duthler
has been accepted towards fulfillment
of the requirements for
Eh.D. degree in_Eh¥S.i_QS
Major professor
Date August 28 , 1970
0-169 ’

 
 
 
   
     

  

3" muomc av "TWA”
IIIIAE & WW I ,,
590K BRIBERY INC. I ,

‘ L'BRA B <D£R$ '

ABSTRACT

DEPENDENCE or ran mum 11 mm mm mm on
pussm m1). rm HEIGHT, m 303st

37
Carl Jenee Duthler

Heliue film trenefer retee. a. have been neaeured ae a function
of level difference, 2, and height, h to the beaker tin, for filling
clean glue and neon coated beekere. Heaeured trenefer ratee ae
a function of level difference or preeenre head are deecribed by:
0(a) - l/[A - B tn:(cn)]. For 1' - 1.65 I and h - 7 cm, the valuee
A - (1.80 a 0.09) x 10‘ eec/cnz and n - (3.9 a 1.9) x 102 eec/cnz
are found. For '1' - 1.28 K and h - 7 can, the valuee A - (1.35 3 0.02)

4 lee/t:-2 and B - (8.4 * 1.4) x 102 eec/cIII2 are found. The

x 10
dependence of the tranefer rate on height in

a(h)' - 10(p./p)h"m'26 * 0’05) 2 105 cn3/eec-cn. Theee obeervatione
give the dependence, P. a: exp[f(h,'r) /v.], of the frictional preeeure.
P. which anyone the flow, on euperfluid velocity, v.. n. depend-

ence, vc c «1.1”.

, of the critical velocity, 'c’ on film thickneee,
d, ie eleo determined fron theee reeulte.

A 161 decreaee ie obeerved in a going from a clean glaee enh-
etrete to a neon coeted enbetrate in the ease epoeretne. lhie re-
enlt ie in reeeoneble agree-ent with the calculated helium film

1/3 -6

thickneee for a neon eubetrate of 2.3h' x 10 cm.

DEPENDENCE OF THE HELIUH II FILM TRANSFER RATE 0N

PRESSURE HEAD, FILM HEIGHT, AND SUBSTRATE

By
Carl Jenee Duthler

A 138818

Submitted to

Michigan State Univereity

in partial fulfillment of the requirenente

for the degree of

Dacron OF PHILOSOPHY

Depertnent of Phyeice

1970

v

i; \

Qt-)

\/\l
a; e

To Karen

11

mmmrs

l'nie enerinent wee enggeeted by Profeeeor G. L. Pollack.
I an very Ilch indebted to him for hie guidance and encouragenent.
Thank- ere eleo due to Mr. Gerald hit: and Mr. Charlee Lening for
many intereeting diecueeione and for their aeeietence in the labo-
ratory. I would eleo like to acknowledge ly wife. Karen, for her
aeeietance in date acquieition and for typing thie theeie. Finally
I would like to acknowledge the financial eupport of the U. 8. Atomic

Energy Callie eion.

iii

TABLE OF CONTENTS

Chapter

I.

II.

III.

V.

INTRODUCTION............

.A. Propertiee of Liquid Heliun

B. Propertiee of the Helium Film

C. Purpoee
THEORETICAL

A. 'lheoriee of the Critical Velocity
B. ‘lheoriee of the Re File michneee
C. Dynanice of the Film ‘rranefer

00000000000

nPBRm'rAI‘0000000000000

A. Clean Glee: thetrate Apparatne

Be Nnon Substrate Apparatus e e e e
Ce Experimental Procedure e e e e e
D. Data Reduction and Experimental Accuracy

RESULTS AND DISCUSSION e e e e

A. Preeeure Head Dependence

3. Height Dependence

3mm AND COMLUS IONS

LISTOFRBFSRENCES......

Appendicee

A. COMPUTER PROGRAM FOR DATA RBWCTION

TABULARTRAIBFERRATBDATA...o.

Ce Subetrete “pun“ e e

iv

Page

0‘ “WI"

‘19

22

22
29

62
47
47
54
60
66

68

72
78

Table

A1

A3

31

32

B3

LIST OF TABLES

Helium film thickness calculated by Schiff.

Values 2f the Hie-Lennard-Jones potential parameters
e and r and nunber density n for rare-gas solids
and liquid helium.

Values the inflamed-Jones potential parameters
e and r for the interaction of helium with rare
8‘. ltOIle

Calculated thickness of Be film on rare-gas solid
substrate.

Fortran listing of the computer program, Sigma-a,
used in the 0(2) analysis.

Typical input data to conputer program for one
beaker filling.

Computer output for data of Table 2.

Measured transfer rate parameters for a clean
glass substrate and '1' - 1.65 K.

Measured transfer rate parameters for a clean
glass substrate and '1' - 1.28 K.

Measured transfer rate data for a neon substrate
0“ T - 1e65 Re

Page
13

15

15

16

72

75

76
78

80

81

LIST OF FIGURES

Figure Page

1 Calculated thickness of the helium film, d with 18
h - l cm,'versus thickness of he costing, f, on
a Xe substrate.

2 Schematic drawing of clean glass substrate 24
apparatus.

3 Schematic drawing of Re substrate apparatus. 31

4 Gas handling system for Ne. 37

5 Level difference, a, as a function of time for 41

three beaker fillings at three different heights,
h, to the beaker rim.

6 Transfer rates, 0(2), from the s(t) curve of 44
a typical run.

7 Transfer rate as a function of level difference 48
for the three beaker fillings of Figure 5.

8 Log—log plot of the critical transfer rate, ac S6
with T - 1.65 K, versus height, h, to the
beaker rim.

9 Log-log plot of the critical transfer rate, 0c 59
'with T - 1.65 K, versus corrected height,
hc orr - h +tl 1/2 cm.
10 Semi-log plot of the critical transfer rate, 0° 62
with h - 7 cm and '1' - 1.65 K, versus thicknessc
of He coatins. 5.

vi

I . INTRODUCTION

A. Properties of Liquid Helium

Helium was first liquefied by H. Kamcrlingh-Onnes in 1908.
Because of weak interatomic forces and large zero point motions,
helium, under its saturated vapor, remain liquid even at absolute
zero where a pressure of at least 25 atmospheres is necessary for
solidification. The normal boiling point of helium is 4.2 1.1'7

Early experimenters did not expect anything unusual from the
liquid phase of these chemically inert and spherically symetric
atom. However, there is a phase change in the liquid at 2.172 K,
the lambda point. Below the lanbda point, Tl’ liquid helium has
many unmual properties. lhe name lanbda point comes from the shape
of the specific heat curve, which resembles the Greek letter A,
in the vicinity of 2.172 K. Reason and Reasons discovered this
transition in 1936 from measurements of the specific heat. Above
TA liquid helium behaves as an ordinary liquid and was called helium I

by Keesom. Liquid helium below 1: is called helium II.

A

Usual methods of assuring the viscosity of He 11 gave a very
low viscosity and different experiments appeared to be contradictory.1"7
In fact, under certain conditions He II can flow frictionlessly
through very narrow channels. Also, the thermal conductivity greatly
increased below Tl' Helium I has a thermal conductivity comparable
with room tesperature air while He II has an effective thermal

2

conductivity about 1000 times greater than room temperature copper.

These and other unusual properties of He II are interpreted
as a macroscopic mnifestation of quantum effects. Because the weakly
interacting Hel' atoms consist of an even number of nucleons and elec-
trons having no net spin, they are somewhat analogous to a non-inter-
acting Bose-Einstein gas.10 Non-interacting bosons can, below a cer-
tain critical temperature, condense into the quantum mechanical ground
state with all the atoms in the ground state at absolute zero. The
unusual properties of He II are interpreted as a result of the con-
densation of atoms into the quantum mechanical ground state.
As a result of the theoretical work of Tiers,9 and of London10
and Landau,u the two fluid model was developed. The two fluid model
is a phenomenological model that has been very successful in explain-
ing experimental properties of He II. Helium II is regarded as con-
sisting of two interpenetrating fluids: the normal fluid and the
superfluid. Each constituent fluid has its own particle density
and to a first approximation its own velocity field. The density
of He II, p, is the sum of the normal fluid density, on, and the
superfluid density, 9.:

p - on + 9.. ’ (1)

me superfluid fraction, p./p, is associated with the fraction of
the stone in the ground state and is a function of tamerature. At
T - Tl’ there is only noml fluid, 11.6., p - on. At T - 0 K, there
is only superfluid, 13.0., p - 9..

The hydrodynamics of He II is governed by the two fluid equations
of motion:

Dali/dc - -(p./p)VP + pas-5T (2)

and

pndvn/dt - -(pn/p)VP - pasVT - "n3 x (U x 3n). (3)

Here 3. is the velocity of the euperfluid, 3n the velocity of the
normal fluid, "n the coefficient of viscosity of the normal fluid,
P the pressure, a the entropy per gram, and T the temerature.
We notice that the superfluid equation has no viscosity term. This
thesis is concerned with the conditions under which the flow of the
superfluid in the helium film remains frictionless.

B. Properties of the Helium Film

In 1932 Kan-erlingh-(hnes noticed that the levels in two con-
tainers, connected by the He II film, tended to equalize. This was
the first observation of transport in the helium film but the trans-
port was unfortunately interpreted by Kammerlingh-mnes as being
due to evaporation and recondensation. In 1937. Rollin and Simon12
calculated that evaporation was not able to explain the observed
transport. They correctly postulated that mass flow was taking place
in a relatively thick, mobile film of helium. As a result, the helium
film is often referred to as the "Rollin film".

Physical adsorption is a familiar phenonanon with the usual
adsorbates being thin and immobile films, except for small, random
atomic motions. The adsorbed He II film is, on the other hand,
approximately 80 layers thick and very mobile due to frictionless
flow of the superfluid. Also, ordinary film are easily evaporated,
but He II files are readily replenished by superfluid flow.

The transport of the He II film provides a convenient means

of studying superfluid flow in very small geometries. Typically,

(4
flow into or out of a beaker is measured and the superfluid velocity,
vi, is related to the transfer rate per unit circumference, o, by:

a - (p./p)v.d. (4)
where d is the thickness of the film. The factor (pa/p) expresses
the fact that only the superfluid component moves while normal fluid,
as an ordinary adsorbed film, is immobile.

In 1938, Daunt and Mendelssohn13 reported the first systematic
observations of transfer in the film. In a very thorough and elegant
series of experiments they observed that the transport rate depended
primarily on the temerature. They also observed that, for constricted
beakers, the total transfer rate was proportional to the smallest
circumference separating the levels. In addition, the flow was
nearly independent of the level difference and the length of the
film. These observations led them to postulate that the flow was

taking place at a characteristic or critical velocity, vc.

14 15

Subsequently, Atkins, and independently Esel'son and Lazarev,
observed that the transfer rate did depend on the level difference
and the height of the beaker rim above the bulk liquid. The depend-
ence of the transfer rate on height arises from the dependence of
the film thickness on height which is usually of the form:

a - m1”, (5)
Here d is the thickness of the film at a height b above the bulk
liquid and k is a constant, for a given substrate material, which

is equal to 3.0 x 10.5 elf/3 1'6

for clean glass. However, the actual
dependence of o on h is less than the dependence of d on h, because
vc is itself a function of d.

be parameter k in Equation 5 for the film thickness depends

5
on the substrate material and results in a dependence of the transfer

17 did a series of careful

rate on substrates. Smith and Boorse
measurements for several glass and metal substrates. They concluded,
however, that the typical 102 background variations in their experi—
ments prevented the resolution of a dependence of a on substrate

for the materials they investigated.

The above general properties of helium are closely related to
this thesis. There are several excellent reviews of these and other
properties of liquid helium from which more detailed information
ten be obtained. 1'7

C. Purpose

In the present investigation the dependence of the transfer
rate on pressure head and film height was examined in more detail.
In particular the functional form of these dependences was of in-
terest for comparison with theory. To keep background variations
to a minimum, the geometry of the apparatus is kept simple and a
very clean, and vibration free, envith is provided for the
beaker.

The dependence of the transfer rate on substrate has also been
examined for neon coated glass beakers. by coating the beakers with
neon, the substrate could be changed from clean glass to neon in
the same apparatus. The thickness of the adsorbed helium film on

the effective neon substrate was then determined from the transfer

rate measurements and compared with the calculated value.

II. THEORETICAL

A. Theories of the Critical Velocity

Theories of the critical velocity of He II flowing in narrow
channels are examined in this section. A narrow channel is defined
to be a channel that is snll enough that the normal fluid is immobile.
The He II film is considered to be a special case of narrow channels.

The earliest theory of the critical velocity is due to Landau.1'1
Although the physical mechanism proposed for the critical velocity
is probably not valid, the general approach used by Landau is still
used in modern theories. This theory and possible modifications
are outlined below.

Lendau's theory of He II emphasises the elementary excitations
which are associated with the normal fluid rather than the conden-
sation of the superfluid into the Bose-Einstein ground state. Landau
separated the elementary excitations into two regions. At low momenta
the excitations are phonone, as in a solid, which have a linear
dispersion relation:

d?) " c111- (6)
Here a is the energy of the phonon with momentum p and c1 is a con-
stant equal to the velocity of sound. At higher momenta, Landau
proposed another excitation which is unique to this fluid and is
called a roton. A roton is a co-operativa excitation of many He

stor- which is visualized as being similar to a quantized smoke
6

7
ring. Rotons have the parabolic dispersion relation:
6(p) - A +'(p-po)2/2u. (7)
Here e is the energy of a roton of momentum.p. The minimum roton
energy A occurs at momentumpO with A/k - 9 K.and pofh - 2.0 A71.
The effective mass of the roton is one quarter the helium atomic
mass, £.¢., u - m/4.5

Below the critical velocity, the superfluid flows'without dis-
sipation. At the critical velocity dissipation sets in, which Landau
regarded as being due to the creation of elementary excitations.

If an excitation of energy e and momentum p is created in superfluid,

flowing with uniform velocity v”, the energy change of the super-

fluid system is:5
“3) + 30;.<0. (3)

This energy change must be less than zero and the momentum, 3, of

the excitation is opposite to the superfluid velocity because of

the energy and momentum transfer to the wall of the channel.

The critical velocity, vc, is the minimum velocity at which
this occurs. Solving Equation 8 for '5' we get the Landau criterion
for the critical velocity:

vc - I¢(p)/pI-1n<v.. (9)

For creation of phonons, the resulting critical velocity is, from

the dispersion relation,

I I - 4
vs,ph c1 velocity of sound 2 x 10 cm/sec. (10)

For creation of rotone, the resulting critical velocity is, from

the dispersion relatim
+(_p___9_2w "’ P )2 m - 6 x 103 cm/sec. (ll)
v'Is,rot

8

These critical velocities, v§.ph and vs,rot’ are orders of
magnitude larger than.the observed critical velocities. The largest
observed critical velocity occurs in the He II film and is approxi-
mately equal to 50 cm/sec. However, the Landau criterion may still
be valid for a higher momentum "excitation". These higher momentum
"excitations" are thought to be Feynman-Onsager vortices.

Landau regarded the frictionless character of the superfluid
to be a result of curl free behavior in the superfluid, i.e. 3.x 3;- 0.
An ideal fluid can, however, have vorticity without curl in the case

of multiply connected geometries. Onsager18

pointed this out in

a paper on vorticity where he also mentions, in a footnote, that

the circulation should be quantized in multiples of h/q,
Subsequently, Feynman19 derived this independently from consid-

erations of the wave function of He II. For He II in uniform motion,

at absolute zero the wave function is:

+3
¢ . mo exp(ik g fig) (12)

where is is the position of the jth atom and $0 is the wave function
for the fluid at rest. The total momentum of the N helium atoms
is NH: where I: - urge/K.

For slightly non-uniform motion the wave function is:

w - we explimmg3j<§y~§jl - wo exptmfin. (13)

Feynman points out that the quantum mechanical phase or order para-
meter, ¢(§), is a slowly varying function of position and time and
is the potential for EIOW.

Using the normalization hpl2 - p!5 and taking the quantum mechani-

cal mementum density equal to the superfluid momentum.density,

T- 3...:-
‘3 ps 3’ 8 '

33 - 38/98 = -(ifi/2m8)(w*W - WV) - (ii/110%. (14)

The condition 3.x 3; - 0 is automatically satisfied. The quan-
tized circulation, K, is obtained by integrating the velocity around
a hole or obstruction in the superfluid:

K s 538.39. - 65/11087de - nh/m. (15)

Quantization is a result of the requirement that w be single valued.
For one revolution around the obstruction the phase, ¢, must change
by 2n. Feynman also points out that a physical obstacle is not
necessary. All that is needed is a non—superfluid core of atomic
dimensions.

we can now apply the Landau criterion to the case of vortices

O In this case we consider the critical

in a channel of diameter d.2
velocity to be a result of the production of quantized vortex rings
having a diameter approximately equal to the channel diameter.
A free vortex ring has the following energy t, and momentum, p:

e - (Osz/4)[£n(4d/ao) - «1 (16)
and

p = npKdzlh. (17)
Here a0 is the diameter of the vortex core and K is a constant of
order unity. Applying the Landau criterion we get:

vc =- (alp)mm - (K/ud)[£n(4d/ao) - K]. (18)

This critical velocity has the correct magnitude but the depend-

ence of Vb on channel width does not agree with experiment. From

the examination of many experimental critical velocities spanning

several decades of channel width, Van Alphen et alglhave proposed

10
the empirical relation vc-d-lfl'.

Vortex production is, however, considered to be the correct
methanism responsible for the critical velocity. the disagreements
with experiment are most likely a result of a lack of knowledge of
how the vortices are nucleated and how they interact with the walls
of the channel. Some ideas of how the vortices are nucleated are
presented below.

Vinen22 proposed that there is always vorticity present in
He II in the form of small pieces of vortex lines. Glaberson and
Dnnnelly23 extended this idea in their vortex-mill model of nucleation.
These pieces of vortex lines are considered to be attached or pinned
to irregularities on the channel wall. When the superfluid velocity
exceeds the critical velocity, these primordial vortex lines grow,
generating new lines at the expense of the kinetic energy of the
flow.

Peshkov24 suggested that the energy to create a vortex comes
from the kinetic energy of a certain volume of the superfluid in
a short time 1 - 10-4 sec. In this case the length of the volume
increases as the channel width decreases resulting in a critical
velocity that increases with decreasing channel width. Craig25
has extended Peshkov's ideas and finds

v3cd - Inna + C/vc). (19)
This functional form does fit the data. However, the constants
B and C were obtained from experimental data rather than from basic
principles.

Langer and Fisher26 have proposed that vortices can be nucleated

by thermal activation. They regard the flowing superfluid as a

11
metastable state. Thermal fluctuations cause the superfluid density
to vanish at isolated points in the superfluid. A vortex can then
be created and result in a decrease in the velocity of the superfluid.
Langer and Fisher find that the vortices are created at the rate
fo exp(-AFB/kT) per unit volume in the channel. Here AFB is a funda-
mental fluctuation frequency. The resulting decrease in velocity
is Avh - h/mi where i is the length of the channel.

This agrees with the observations by Clow and Reppy27 of the
decay of superfluid persistent currents when the temperature is
near the lambda point.

Notarys28 extended this theory to pressure driven superfluid
flow. He derives the frictional pressure along a channel as a func-
tion of superfluid velocity and temperature:

p a (h/mpro)exp(-AFB/kT), (20)
where V is the volume of the channel. Using vortex rings for the

dissipation mechanism gives:28

p
-. 1‘. 5...?—
p - m pro exv(- k pvsT)’ (21)

where 8 is nearly constant.
Notarys Finds that this theory agrees with his observations
for superfluid flowing through narrow pores with temperatures down
to 1 K. In this case there is no "critical" velocity. For any
velocity there is a pressure gradient which, however, is very small
at low velocities and increases very rapidly at larger velocities.
Although the dynamics of the nucleation of vortices is not known,
the rate of production is known from the Josephson frequency. Recent
theories have extended Feynman's original wave function and Beliaev29

finds that the time dependence of the quantum mechanical phase, ¢.

12
is determined by the chemical potential u:
{do/dt 8 u (22)
Anderson30 and Donnelly31 have extended this idea to studies of the
critical velocity. If we have two reservoirs at different chemical
potentials connected by a channel, the relative quantum mechanical
phase slips at the rate

fidwl - ¢2)/dt = 1:1 - 112. (23)

Anderson points out that each time the relative phase slips by 2H
a vortex is produced. Hence vortices are produced at the rate:

dn/dt = (ul - u2)/h. (24)

This is the well known Josephson frequency. This says that the
channel or He II film is analogous to a Josephson junction, producing
vortices at the above rate.
B. Theories of the He Film Thickness

The transfer rate per unit circumference, a, measured in beaker
filling experiments, is proportional to the product of the film
thickness, d, and superfluid velocity, v8:

0 *3 (pa/owed. (25)

If the film thickness is known, prcperties of the critical velocity
can be determined from measurements of the transfer rate. The thicke
ness of the He film, covering a solid surface in contact with a
reservoir of bulk helium, changes with the height of the film above
the reservoir. By changing the height, h, of the beaker rim above
the reservoir, we change the thickness of the film at the beaker
rim. This is the region where dissipation takes place. In this
way the dependence of the critical velocity on film thickness can

be determined from measurements of the dependence of a on h.

13
The thickness of the helium film is determined by the
van der Waals attraction between the solid surface and the helium
film. Schiff32 has calculated the thicknesses of the film for some
substrates and finds that the thickness, d, of the film at a height,
h, above the reservoir is given by:
d = k/h1/3 (26)

Here k is a constant for a given substrate material. Schiff's cal-

culated values of k are given in Table 1.

TABLE 1
Helium film thickness calculated by Schiff.a
-6 4/3 Cu Ag Class Rocksalt
1910.0 cm ) (5.2 407 :4 2'2
aRef. 32.

In this section the dependence of thickness on height is cal-
culated for pure rare-gas solid substrates. Also for neon coated
beakers, the dependence of the He film.thickness on neon coverage
is calculated.

The Ha film thickness is determined by a balance between
van der Waals attraction to the substrate and gravitational poten-

tial energies.7’33 34’35

The Mic-Lennard-Jones potential , ¢(r),
is used for van der Heals potential energy between an atom in the
film and an atom in the substrate:

0(r) - e(r*/r)12 - 2e(r*/r)6. (27)
Here r is the separation between the atoms and e is the depth of
the potential well which is at an interatomic separation r*.

Fe will consider atoms at the surface of the filmwwhere the

separation, r, is large enough that we need only consider the

14

*
attractive part of the interaction: i.e.. ¢ - - 26(r lr)6.

attr
A helium atom a distance y from the surface of the substrate has
the attractive potential energy

Usttr ' upattrdv’ (28)

where n is the number density of substrate atoms and the integral
is taken over the substrate. The cylindrical co—ordinate r', z',
and 6' are used in a sendpinfinite substrate where z' is the dis-

tsnce from the surface of the substrate to a plane containing r'

and 6'. Substituting for ¢attr and using r . [r'2 + (z' + y)2 1,2.
we get:
- *6 2w . . . .2 . 2 -3 (29)
"attr Zenr ’0 d6 IZdr [Zdz [r ‘+ (z + y) l
or,
*6 3
Uattr " 8r “Tl/3y e (30)

A.helium atom at the surface of a film of thickness d at a

height h above the reservoir, has the additional potential energy:

u - mgh - er*6m/3d3. (31)
In order for the film to be in equilibrium, the energy of a He atom
must not change when it is virtually displaced from the surface of
the reservoir to the surface of the film. Therefore Equation 31
can be set equal to zero and we can solve for the equilibrium fihm
thickness as a function of height:

d - k/h1/3, (32)
where

k ' (er*6rn/3ms)1/3. (33)

The thickness of the helium film on a rare-gas solid substrate

a
can now be calculated. Potential parameters a and r are determined

15

from the pure rare-gas solid parameters using the coupling rules:36

* * *

rHe-Ne " (rNe-He + the-He)” (3")

and
eHe-Ne ' (ENe-NeeHe-He)1/2’ (35)

Values for the pure rare-gas parameters are given in Table 2
with the number density, n, evaluated at absolute zero. Values of
the parameters for the interaction of heliumwwith rare gas atoms
are given in Table 3. The parameters in Table 3 are calculated using
Equations 34 and 35 and Table 2.

TABLE 2

Value! of the Mie"1ennard-Jones potential parameters a
and r and number density n for rare-gas solids and liquid

helium.

a e/km 3(3) n(1022/cm3)
He 10.2 2.89 2.18
Nab 36.3 3.16 4.54
Arb 119 3.87 2.66
Krb 159 4.04 2.22
Xeb 228 4.46 1.73
:Ref. 5.

Ref. 34.

TABLE 3 ,

Values of the Mic-Lennard-Jones potential parameters 6 and r
for the interaction of helium‘with rare gas atoms.

He-Ne He-Ar He-Kr He-Xe

s/k(K) 19.2 34.9 40.3 48.3
*0

r (A) 3.02 3.38 3.46 3.68

One additional correction is made before values are calculated

for the thickness. Correction is made for the helium displaced by

16
the substrate since we are interested in the additional attraction
of the substrate as compared to pure helium:

- U - U

Uattr He-Ne He-He' (36)

Values of k, obtained when this corrected Uattr is substituted in

Equation 31, are given in Table 4.

TABLE 4
Calculated thickness of He film.on rare-gas solid substrate.
Ne Ar Kr Xe
kao'6 cm4/3) 2.3 3.0 3.1 3.5

In the experiments, neon coated glass beakers are used rather
than pure neon beakers. The neon coverage necessary to have an ef-
fective neon beaker is now calculated. Because the interaction be-
tween helium and glass is not known, the calculation is done for
a neon coated xenon beaker. It is assumed that these calculated
results are qualitatively valid for glass as well as xenon.

To calculate the He film thickness for a neon coated xenon
beaker, the attractive potential is integrated over the Ne layer
and over the semi-infinite Xe substrate. Correcting again for the

displaced helium, the attractive potential energy is

Uattr - UHe-X’e +UHe-Ne - UHe—He' (37)
For a neon layer having thickness E, the potential energy terms,
in Equation 37, are for a He atom a distance y from the surface:

0 - * + )3 (38)

He—Xe Elie-XerHe—XemXe ’3 (y E .

* 3 3

"us-Ne ' eHe-Nere-Ne'nNeIl/3y - 1/3(y + t) l. (39)
and * 3

UHe-He - cue-HerHe-He'nHe/3y ° (40)

17

The potential energy, U is, as before, equated to the

attr’
gravitational potential energy giving the thickness of the He film.

However, when this U r is substituted in Equation 31, it can not

att
be easily solved for d as a function of height. Since this is a
small correction, we solve for the thickness at h - 1 cm. (This
thickness is taken to be numerically equal to k and the functional
form of Equation 26 is assumed.

The resulting thickness with h - 1 cm, as a function of neon
coverage, 5, is shown in Figure 1. For E < 10 Z, the thickness
of the He film is equal to the thickness over a pure Xe substrate.
For E > 1000 X, the thickness of the He film is equal to the thickp
ness over a pure Ne substrate so that we have an effective Ne beakp
er. One-half saturation, where the He film has the average of these
thicknesses, occurs at E - 100 X.

The thickness as a function of Na coverage, as shown in Figure l,
is assumed to be qualitatively valid for a glass substrate instead
of Xe substrate. That is, we assume that we need only put a 1000 X
neon layer on a glass beaker to have an effective Ne beaker.

Dzyaloshinskii at al.37 have objected to the use of the
Hie-Lennard-Jones potential for a solid interacting with a liquid.
They point out that the Mic-Lennard-Jones potential is strictly
valid only for isolated atoms and that corrections must be made
for the screening of neighboring atoms. This requires knowing the
entire electromagnetic spectrum of both the helium and neon. This
is beyond the scope of this thesis. However, the correction should
be smallest for the case of neon substrate.

The simple calculation above is for a static film interacting

360

340

0

He Film Thickness d(A)

N N 01 (,4
(D (D O N
O O O O

N
43
O

220

0

Figure 1: Calculated thickness of the helium film, d with h - 1 cm,
ssssss thickness of Ne coating, 5, on a Xe substrate.

18

 

 

 

 

 

l l i l
.1
I I I l

 

 

IO
Ne Thickness {(A)

no2

I03

oIo“

IOS

19

with an infinite wall. Kontorovich38 predicted that the thickness
of a moving Film would be less than the thickness of a static F11m.
However, Keller39 has found experimentally that the thicknesses
are the same. Goodstein and Safr'ifmanl.0 have considered the theory
of a moving film in more detail and find that both moving and static
films have the same thickness as is determined by Equation 31 above.

Various methods have been used to measure the thickness of the
Film experimentally.‘ Bowers41 has weighed the Film adsorbed on
a metal foil. Atkins42 has calculated the thickness of the film
from measurements of oscillations of the film. Keller39 has also
measured the thickness, using a capacitance technique which is sens
sitive to changes in the thickness.

The most direct and accurate measurements of the helium film
thickness are by L. C. Jackson and co-workers.‘ They measure the
ellipticity of light reflected from a surface that is covered by

the He film. Ham and Jacksonl'3

k, is equal to 3.0 x 10.6 cm4/3. This is the basic measurement

find that the thickness parameter,

that is used for the thickness of the He film on a glass substrate
in this thesis.
C. Dynamics of the Film Transfer
Dynamical limitations to measurements of the properties of
critical velocity in beaker filling experiments are examined in
this section. The flow is considered to take place in a channel

h8V1n2 the vertical cross—section d - k/hl'l3

, as discussed before.
In the experiment, the rate at which the beaker Fills is measured.
The transfer rate per unit circumference, o, is related to the ve-

locity of the inner level, dz/dt, for a beaker of radius r, by:

20
o - (r/2)dz/dt. (41)
These measurements of o are related to the superfluid velocity by

0 ' (pa/p)v.d. (42)

The superfluid obeys the two fluid equation, Equation 2:

dv’s/dc - 4m)??? + air. (43)

In the case of isothermal, irrotational flow, as we have for film
transfer at subcritical velocities, this becomes”2

dvs/dt - ave/at + (383)38 - -(1/p)'v’1>. (44)

For a co-ordinate 1 parallel to the surface of the beaker and par-

allel to 3;, this becomes:

3 3 1 2 l 3?
at V3 + a“? v3 - 73- !, (45)

Integrating along the path 1 from the outside level of the beaker

to the inside level, we get:

 

inside
[Give/add). + 6- v3) =- -(1/p)AP - -gz, (46)
outside

where z is the level difference. Because of the nearly zero velocity
inside and outside of the beaker, the second term on the left hand
side of Equation 46 drops out. Using the continuity equation,

vhd - constant along path, (47)

and substituting for v8 in terms of dz/dt from Equation 41, we get
a
M dzz/dt2 - -pgz, (48)

a
where M - (pz/Zps)f(r/d)dl. This relates the acceleration of the
measured level difference to the driving pressure, age, for sub-
critical flows

To account for the critical velocity, we add a frictional

21

pressure, Ps’ to the right hand side of Equation 48:20

t
n dzz/dtz - p. - pgz. (49)

We are primarily interested in the frictional pressure as a
function of superfluid velocity, v8, or transfer rate, a. This
can be measured directly for low acceleration and large a. In this
case M? dzz/dt2 is small compared to pg: so that Equation 49 says
that the frictional pressure is equal to the observed driving pres-
sure, i.e., P. - pgz.

For small 2 and large acceleration, the acceleration term he-
comes important so that the frictional pressure cannot be measured
directly. In this case Pa is small and the levels oscillate with
small clamping.1l"l'2 The apparatus used in this thesis, however,

is not suitable for studying these oscillations.

III. EXPERIMENTAL

A. Clean Glass Substrate Apparatus.

Atkins44 in 1948, and van den Berg and de Haasl'5 independently
in 1949, reported transfer rates as much as twenty times larger
than those originally reported by Daunt and Mendelssohn.13 Bowers
and Mendelssohnl‘6 showed that these enhanced transfer rates were
a result of impurities, such as air, condensed on the beaker sur-
face.

These enhanced transfer rates for contaminated beakers can
arise either from an increased film thickness or an increased micro-
perimeter of the beaker. In the case of a granular impurity, the
microperimeter is larger than the apparent macroperimeter and, in
addition, a thicker film can result from liquid helium.he1d between
the grains by surface tension. McCrunb and Eisenstein“7 pointed
out that polar impurities, such as water, can also result in a thick-
er film. The thick film, in this case, is a result of a stronger
attractive force, of the polar substrate to the helium, than the
usual van der Weals forces.

Smith and Boorse17 investigated these problems in more detail.
They found that, by being careful to prevent impurities, the enhanced
transfer rates could be eliminated but a, typically 10%, background

variation in the transfer rate remained. These variations prevented

them from seeing a dependence of the transfer rate on substrate

22

23
material for the meterials they investigated.

The apparatus used in this thesis was designed to provide a
very clean, and vibration free, environment for the beaker in order
to minimize these problems. To do this the beaker was enclosed in
an experimental chamber rather than exposing it to the possibly dirty
environment of the liquid helium bath. In addition, rather than
condensing possibly impure helium gas into the experimental chamber,
the chamber was filled by admitting helium in liquid form from the
bath. Another advantage of the chamber was that the helium level
of the chamber remained constant, while the beaker was filling,
whereas the helium level of the bath decreased slowly with time.

The experimental chamber and beaker are shown in Figure 2.

A 2 in. Kovar to glass seal was used to make the chamber. The champ
her was demountable at the top by means of brass flanges that were
sealed with a lead O-ring. Five ampere fuse wire was used to make
the O-ring whose ends were soldered together to form a closed loop.
Liquid helium was admitted to the chamber through the needle valve
at the top of the apparatus. This valve was made from a Hoke

#3242 M 43 needle valve which was modified to reduce its mass.

Connections from outside the dewar to the experimental chamber
were made with a 3/8 in. o.d. stainless steel tube. Two other 3/8
in. stainless steel tubes served as supports for the chamber. Con-
nections were made from the experimental chamber to mercury and
oil nanometers, a mercury bubbler, and a diffusion pump system.

The nanometers were used to measure the vapor pressure of the liq-
uid helium in the chamber in order to determine its temperature.

The bubbler served as a vacuum tight, safety valve which would allow

24

To
Monomeier

”a

 

 

W He

Inlet Bath

Valve ’

 

 

//////

 

 

 

 

\
Pb
O-ring

 

 

 

 

___ .——_-—_4
__ -—_._
.__. —4 ___.___.
__ h———
—— -< .—————4
_— ~———
_——-— .1 v—————--q
——— l————
_— p—____‘

___ _._—_4
_.__. _‘
,__._ ._ ..
__.__ ___

 

 

 

Figure 2: Schematic drawing of clean glass substrate apparatus.

25
helium gas to escape from the chamber in case of pump failure.

The diffusion pump system consisted of a CVC model 21, air
cooled diffusion pump which was backed by a Welch model 1402 B,
140 liter per minute mechanical pump. An, approximately 2 liter
capacity, liquid nitrogen cold trap placed between the pumps and
apparatus, served as e cryopump and in addition prevented pump oil
from diffusing to the beaker. Before each run the diffusion pump
system was used to degas the beaker. After each run the mechani-
cal pump, without the diffusion pump, was used to remove the liquid
helium from the experimental chamber.

3

Pressures above 10- Torr, in the diffusion pump system, were

treasured with a CVC model GP-140 pirani gauge. Pressures below

10"3

Torr were measured with an H. S. Martin cold cathode ioniza-
tion gauge. The highest vacuum attainable with this system was a-
bout 10-5 Torr.

Beakers used in these experiments, shown in Figure 2, were
made of 3 mm i.d., 5mm o.d., and 10 cm height Corning #7740 Pyrex
tube. The beaker was mounted on a 1 1/2 in. Pyrex base which had
small Pyrex hooks for removing the beaker from the demounted cham-
ber. Care was taken to specially select glass tube, from the stock
of glass at the M. S. U. Glass Fabrication Laboratory, that was
clean and scratch free. The rim of the beaker was formed by care-
fully cutting the Pyrex tube with a Carborundum wheel. The rim was
then lightly fire-polished to remove scratches and to smooth the
edges.

The exact radius of the beaker was determined volumetrically.

The mass of room temperature, distilled water in the beaker was

26
measured as a function of height with a mettler analytical balance.
Using the density of water, the resulting volume versus height curve
was plotted and was fit by a least squares straight line. The slope
of this curve gave the cross-sectional area of the inside of the
beaker from which the inner radius was determined. Twa different,
but otherwise identical, beakers were used in these experiments.
Beaker 1 has an inside radius of 0.162 cm and beaker 2 had an in-
side radius of 0.156 cm.

When the Pyrex beaker'was removed from the demounted chamber,
it could be cleaned in solvents that were not compatible with the
metal parts of the apparatus. The following cleaning procedure
gave reproducible transfer rates and was used before each run during
the clean glass substrate study:

The beaker was first rinsed with distilled water and then with
methanol. This was followed by cleaning with detergent in an ultra-
sonic bath and rinsing ultrasonically with distilled water. The
beaker was then rinsed with nitric acid and rerinsed with distilled
water. It was then heated to 200 °C in air with a heat gun to re-
move adsorbed water. Finally, it was mounted in the experimental
chamber and heated to roughly 75 °C with an infrared heat lamp while
pumping to about 10'6 Torr with the diffusion pump system. The
pumping*was continued for one or two days prior to cooldown.

The entire cleaning procedure above may not be necessary be-

fore each run. Allen and Armitagel'8

have used a similar cleaning
procedure and report that, after the initial thorough cleaning, only
heating and pumping between runs is necessary. With the neon sub-

strate apparatus, described in the next section, it was also found

27
that it was only necessary to prevent impurities from entering the
chamber and to diffusion pump between runs after the initial thor-
ough cleaning.

Impurity problems were encountered in the preliminary runs be-
fore the actual thesis data were taken. It appeared that the beaker
in the original non-demountable apparatus got dirtier each time
it was cleaned, £.e., the transfer rate increased after each cleaning.
This was most likely a result of water, used in the cleaning pro-
cedure, being adsorbed on the beaker. These problems led to the
design of the demountable apparatus so that the beaker could be re-
moved to be cleaned in stronger solvents and to be heated to higher
temperatures. These observations with the nonedemountable, demount-
able, and neon substrate apparatuses indicate that the crucial step
in the cleaning procedure is heating to 200 'C to remove adsorbed
water.

Although these observations indicate that the beaker may have
to be cleaned thoroughly only once, the cleaning procedure, never-
theless, was used with slight variations before each run as a test
of the reproducibility of the cleaning procedure. This also ensured
that the maximum initial cleanliness was reached.

The experimental chamber'was immersed in a liquid helium.bath
in an H. S. Martin, glass, liquid helium dewar with a removable
liquid nitrogen dewar. Both the nitrogen and helium dewars were
strip silvered with 3/4 in. viewing slits. Liquid helium bath temp-
eratures were lowered by evacuating the helium dewar with a
Heraeus-Engelhard #8225, air-cooled, 147 cfm vacuum.pump. The temp-

erature of the helium bath was regulated to better than *0.01 K

28
for temperatures above 1.5 K.with a Cryonetics Mark II mechanical

pressure regulator. Comparable temperature regulation was achieved
for temperatures near 1.3 K‘by regulating the pumping speed with

a 2 in. gate valve in the pumping line. The lowest temperature
attainable with this apparatus was about 1.2 K.

Vibrations of the apparatus were minimized by placing flexible
bellows along the pumping line. A 4 in. x 24 in. copper bellows
was located in the main 4 in. pumping line near the pump which was
about 15 ft. from the apparatus. Also an asymmetric arrangement
of five automobile radiator hoses connecting the helium dewar to
the 4 in. pumping line further reduced vibration. Three 2 in. x
15 in. hoses, one 1 1/2 in. x 12 in., and one 1 1/2 in. x S in. hose
were used in this arrangement.

Two different means of illuminating the beakers during these
experiments gave identical results. This agrees with the results

49 who found that the trans-

of Bowers and Mendelssohn46 and Pious,
fer rate was independent of the intensity of the illuminating radia-
tion, as long as it was not too intense. For this experiment, radia-
tion, other than that used for illumination, into the dewar was mini,
mized by placing radiation shields, made of aluminum foil, in the
helium bath. These shields were placed above the apparatus and around
it, except for vertical viewing slits. Additional shielding was
provided by covering the slits on the outside of the nitrogen dewar
with aluminum.foil except for 6 in. near the bottom which was left
uncovered for viewing.

The first means of illumination made use of an 8 watt floures—

cent light. This light was shielded by a plastic diffuser which,

except for a narrow slit, was covered with aluminum foil. In

29
addition, a green glass filter was placed in front of this slit.
Infrared radiation from this light source was filtered by a water
cell placed in front the dewar slit. using this illumination the
experiment was done with the laboratory completely dark except for
other low intensity lights which illuminated the laboratory note-
book and instruments.

A second, more convenient method of illumination used no spe-
cial light source. In this case illumination was provided by room
light with the laboratory slightly darkened. Radiation shields on
the nitrogen dewar and in the helium bath were again used in this
arrangement.

B. Neon Substrate Apparatus

Neon beakers were made by coating clean glass beakers with
neon. To have an effective neon beaker, as was discussed in section
II—B, a 1000 Z or greater thickness neon coating on the glass beaker
is required to saturate the forces between the helium and substrate
material. However, the coating must be smooth and uniform to avoid
the rough substrate problems that were discussed in the previous
section.

The neon substrate apparatus is similar to the clean glass
substrate apparatus except that, in order to raise the temperature
of the chamber above 6 K, it was provided with a vacuum space and
heating coil. If neon is condensed directly onto a surface having
a temperature of A K, enhanced transfer rates are seen presumably
due to rough substrate conditions. This was observed in preliminary
experiments. The rough substrate in this case was most likely a

result of the fact that neon atoms stick, as soon as they strike,

30

a 4 K surface.50 At higher surface teuperatures, near 20 K for neon,
the adsorbed neon atoms are distributed more uniformly because they
have a lower probability of sticking when they strike the surface.
Also because of their thermal energy, the adsorbed neon layers, at
these temperatures, are somewhat mobile and as a result are able
to smooth themselves.51 Therefore the apparatus was designed so
that the neon could be condensed with the temperature of the glass
beaker in the range 20K to 25 K.

The neon substrate apparatus is shown in Figure 3. The inner
chamber, containing the beaker, was made from a 1 l/2 in. Kovar
to glass seal. The outer chamber, used to form a vacuum space a-
round the inner chamber, was made from another Kovar to glass seal
of 2 in. diameter. The tops of both chambers were soldered to brass
flanges.

Two 1/2 in. stainless steel tubes supported the apparatus.
One of these tubes served both as an.inlet to the vacuum, or exchange
gas, space and as an outlet for electrical leads from the heater
and thermometer. The other 1/2 in. tube formed a vacuum space around
a 1/4 in. stainless steel tube which served as an outlet from the
inner chamber. The tube from the inner chamber, as in the clean
glass substrate apparatus, was connected to mercury and oil mano-
meters, the mercury bubbler, and the diffusion pump system.

Liquid heliumnwas admitted to the inner chamber from the bath
through a 1/8 in. copper tube from the needle valve at the top of
the outer chamber. The needle valve used in this apparatus was
constructed by the ML 8. 0. Physics Department Machine Shop.

These chambers were not easily demountable so that the beaker,

31

Outer tube from inner chamber
Exchange gas and wires
Needle valve

Pt thermometer

Vacuum space

Outer chamber

Inner chamber

Beaker
Heater

32

 

 

 

 

 

 

 

 

TII'TIUOCD

 

 

 

 

 

' __.[[fl]]]

Figure 3: Schematic drawing of Ne substrate apparatus.

 

 

 

 

33
which was fastened to the inner chainer, could not be removed for
cleaning before each run. The beaker was initially cleaned and heated
using the procedure given in the previous section. The inner cham-
ber was then assembled taking care not to get the beaker dirty.
Transfer rates obtained with the asseabled apparatus were the same
as those obtained during the clean glass study. This indicated that
the beaker was not contaminated during the assembly of the inner
chamber.

As was discussed in the previous section, the beaker was kept
clean between runs by pumping. Occasional runs were made without
a neon coating during the course of the neon substrate study. The
transfer rates observed in these runs were again the same as the
transfer rates obtained during the clean glass study. This demon-
strated that the glass beaker in the neon substrate apparatus remained
clean during the neon substrate study.

The exact inner radius of the beaker was again determined volu-
metrically. Because the beaker was fastened to the massive inner
chamber it could not be calibrated directly. Another beaker, made
from the same section of glass tube that was used in the construc-
tion of the experimental beaker, was used for calibration. The inner
radius of this beaker was found to be 0.160 cm.

Manganin resistance wire having a resistance of 10 ohms per
foot was used for the heating coil. Approximately 100 feet of this
wire was wound longitudinally around the inner charter, and the
base of inlet tubes, leaving vertical slits for viewing the beaker.
Nylon thread and masking tape were used to fasten the heater wire

to the inner chafler. The ends of the heater wire were connected

34
to a small terminal strip at the top of the inner chamber in the
vacuum space. he leads from the terminal strip were routed through
the exchange gas tube and were connected to 2 pins of a vactnm: tight,
9 pin connector at room temperature.

A miniature platinum resistance thermometer was used to meas-
ure the temperature of the inner chamber. The thermometer used in
this apparatus was an Artronix #PS-l which had the serial nunber
X-159. The thermometer had a square cross-section, 0.2 in. on a
side, and was 0.05 in. thick. Thermal contact between the brass
top of the inner chadaer and the thermometer was made using Apiezon
N vacuum grease. The bottom of the terminal strip, used for connect-
ing electrical leads, held the thermometer in place.

The resistance of the platinum resistor as a function of tempera-
ture was calibrated from basic measurements of the resistance at
4.2 K and 273 K. These twa measurements of the resistance were
made with the apparatus immersed in a liquid helium bath, at atmos-
pheric pressure and. an ice and water bath‘ respectively. The re-
sistance, R, as a function of tenperature, T, is then given by

Rm " ”1.2 x "' 201001273 K ' Rmz x ' (50)

Here MT) is a tabulated function given by Whites;z for Pt.

The miniature Pt resistance thermometer had only two short leads.
In order to eliminate lead resistance, it was converted to a four-lead
arrangement by connecting it to four inlet leads at the terminal
strip. The of these leads supplied a 1 ma current to the resistor
and the other two leads served as zero current, potential leads.
has four leads also were routed up the exchange gas tube and were

connected to four other pins of the nine pin connector.

35

The resistance of the thermometer was measured by connecting
the potential leads from.the nine pin connector to a Leeds and
Northrup model n+3 potentiometer. The auxiliary emfi termmnals of
this potentiometer were used to monitor the l .A current through
the platinum resistance thermometer. This was done by measuring
the potential difference across a one ohm standard resistor which
was in series with the resistance thermometer.

The heater leads from the nine pin connector were connected
to a Variac variable ac voltage source. A.1000 ohm resistor was
placed in series with the heater and Viriac to reduce the heat imp
put. Temperature regulation. for temperatures above 6 K, was achieved
by manually regulating the voltage output of the Variac. This was
done by adjusting the voltage while variations of the resistance
of the thermometer, from the value set on the potentiometer, were
monitored on the dc potentiometer null detector. .

While the neon was being condeued, the chasbsr tenperature
was above 4 K and a vacuum was maintained in the vacuum, or exchange
gas, space. After the neon had been condensed,the inner chamber
'was brought to a temperature of 4 K. At this time helium gas was
admitted to the vacuum.space before admitting liquid helium to the
inner chamber.

The helim exchange gas was stored at atmospheric pressure
in a one-half liter, glass flask outside the dewar. A glass, vacuum
stopcock‘was opened to admit the exchange gas to the evacuated vacuum
space. This procedure ensured that the exchange gas pressure‘would
never exceed atmospheric pressure even when the apparatus was warmed

to room teqerature. In addition, the stopcock served as a safety

36
valve in case liquid helium should leak into the vacuum space from
the bath.

The amount of neon needed to coat the beaker was determined
by assuming that the neon uniformly covered all the surfaces of the
beaker and the inside surfaces of the inner chadaer. Using the area
of these surfaces, the amount of, room temperature, neon gas required
to form a desired thickness was easily calculated.

The neon gas required to form a desired thickness neon coating
was stored in a one liter glass flask outside the dewar. Typically
a 1000 Z coating was used which required a room temperature, neon
pressure of 5.3 Torr in the storage flask. The gas handling system
for the neon is shown schematically in Figure lo. Mercury and oil
nanometers were used to measure the pressure of the neon. After
the desired partial pressure of neon was admitted to the flask,
helium gas was added filling the flask to a pressure of one atmos—
phere.

Matheson Research grade neon having a purity of 99.9991 was
used in these experiments. The helium gas having an estimated puri-
ty of at least 99.99% was obtained from the M. S. 0. Physics Depart-
ment. The purity of the helium gas was tested by admitting only
helium gas, without neon, to the experimental chamber. It was then
observed that the transfer rate was the same as for clean glass.
This demonstrated that no impurities from the helium gas that was
mixed with the neon gas were condensed on the beaker surface.

During a run, the helium and neon gas mixture was admitted to
the experimental chatter with the initial chamber temperature at

25 K. The helium gas served bath to distribute the neon and to

37

 

[“°‘”?_.L

 

1* H

L Pumps] 1- ; IL Monometer l

 

 

 

 

 

 

Expt.
Chamber

 

 

 

Figure 4: Gas handling system for He.

38
maintain thermal equilibrium within the chamber. After the gases
were admitted to the chamber, the chamber temperature was than low-
ered at a rate of 0.2 K/min from 25 K.to A K condensing the neon
on the beaker. Transfer rate data indicate that this procedure re-
sulted in a smooth surface for coverages of about 1000 A, although
no independent test of the surface‘was made.

For the condensation of the neon,the liquid helium.level in
the dewar‘was set about half way up the outside of the experimental
chamber. The tube from.the inner chamber to the needle valve pro-
vided thermal contact between the inner chamber and the helium bath.
The heat from the inner chamber was dissipated primarily in the cold
helium vapor surrounding the top of the outer chamber. Heat was
also dissipated in the liquid helium.by heat conduction down the
glass walls of the outer chamber.

The liquid helium transfer tube from the storage dewar to the
experimental dewar remained connected during the neon condensation.
A slight overpressure on the storage dewar maintained a slight flow
of cold helium gas through the transfer tube to the experimental
dewar. This cold gas served two purposes. It, first of all, main?
tained a stream of cold gas on the top of the apparatus to provide
cooling to the inner chamber. In addition, this cold gas kept the
transfer tube cold so that more liquid helium.could be transferred
after the condensation of the neon without warming the apparatus.

0. Experimental Procedure

The measurements made during an experimental run to determine

the He film.transfer rate are described in this section. As was

described in the previous two sections, the surface of the beaker

39
was prepared for running before liquid helium was admitted to the
experimental chamber. This preparation involved, in the clean glass
substrate study, thoroughly cleaning the glass beaker one or two
days prior to cooldown. In addition, during the neon substrate study,
a layer of neon was carefully condensed on the surface of the beaker

before admitting liquid helium to the experimental chamber.

During an experimental run, approximately four liters of liquid
helium were transferred into the experimental dewar and cooled by
pumping to the desired temperature. Liquid helium was then admitted
to the experimental chamber from the bath, by means of the needle
valve, filling the chamber outside the beaker to a depth of about
2 cm.

'me position of helium level inside the beaker was then amas—
ured as a function of time until the inner and outer beaker levels
reached equilibrium. Usually the level difference, 2, was monitored
over a distance of 1 1/2 on which took a period of time of about
30 minutes. While the beaker was filling, the outer level remained
at a nearly constant distance h below the beaker rim. The position
of the inner and outer levels with respect to the beaker along with
the distances h and s are shown in Figure 2.

The positions of the helium levels were measured with an ele-
gant Wild-Hesrbrug model 104326 cathetometer. During a beaker fill-
ing, the position of the inner beaker level was measured and recorded
as a function of time with an accuracy of 0.001 cm taking height
versus time readings usually every 30 sec. The time during the
beaker filling was measured with a stopwatch. The stopwatch was

started at the time of the first height reading and subsequent

60
readings were made with an accuracy of *1 see without stopping the
stopwatch.

Figure 5 shows the level difference, 2, as a function of time
for three of five beaker fillings at different h's during the same
run. The transfer rate is proportional to the derivative of the
x(t) curve. we notice that these curves are nearly straight and
have only slightly different slopes. This agrees with the original
observations of Daunt and Mendelssohn;3 that o is nearly independent
of pressure head, 2, and height, h, of the beaker rim.above the
reservoir.

However this thesis is concerned with the slight departures
of o from this simple behavior. These curves in Figure 5 are slightly
concave downwards reflecting the dependence of a on 2. Also the
height dependence of o is manifested in the different slopes of

the curves taken at different h's.

41

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42
D. Data Reduction and Experimental Accuracy

The transfer rate per unit circumference, o, is proportional
to the derivative of the 2(t) curves shown in Figure 5 of the pre-
vious section, £.e.,

o - (r/2) dz/dt. (51)

Here r is the inside radius of the beaker. In this section the
computer method that is used to evaluate these derivatives or trans-
fer rates is discussed. In addition, the random error in o from
the measurements and the differentiation procedure is estimated.

The data reduction was done using the computer terminal in
the Physics - Astronomy Building. This terminal was connected to
the Michigan State University, Control Data Corporation 6500 com-
puter which was located in the Computer Center.

The following procedure is used in the computer program to

53

calculate the derivatives. A quadratic equation,

z(t) - A + Bt + Ctz, (52)
is fit to the first eleven data points of a beaker filling. The
derivative,

dz/dt - B + 2Ct, (53)
is then evaluated at datum.point number 6, the midpoint of the eleven
data point segment. Next, data points 2-12 are fit in the same way
and the derivative is evaluated at the midpoint of this segment.

By taking overlapping segments of the z(t) curve in this manner,
the derivative of the curve can be determined either as a function
of z or t.

Transfer rates, a, are then obtained by multiplying the deriva-

tives by c - (r/Z) x 10’s. This gives oin units of 10'5 cm3/sec-cm

63
so that o is numerically between 1 and 10. Figure 6 shows a as
a function of x for a typical run. Each point on this curve was
obtained by evaluating the derivative, which was then multiplied
by G, at the midpoint of one of the overlapping eleven data point
segments. The solid curve is a fitted curve which shows the trend
of data. This curve will be discussed in the next section.

Details of the computer program, Sigma-s, are given in Appendix A.
A Fortran listing of the program Sigma-s is given in Table A1.
Table A3 gives the computer output for these data. Although the
data for only one beaker filling are shown in Appendix A, the pro-
gram was designed so that data from several beaker fillings could
be submitted and analyzed at the same time.

The first computer card of the input data in Table A2 gives
an identification number for the data set. The second computer
card of the input gives the height of the heliua level outside the
beaker, no, and the geometrical factor G. Subsequent cards give
the position of the inner level as a function of time. Six height
versus tin data points are recorded on each computer card. The
remaining points, which do not fill one card, are then recorded one

datum point per counter card.

The first line of the computer output .presented in Table A3
gives the identification number of the run; the liquid helium level
outside the beaker, Ho; and the geostrical factor, G. TVo other
numbers given on this line of the output are the parameters A and
B which describe the 0(a) curve. These parameters are discussed
in the next section.

The remaining computer output is presented in eight columns

Mo

 

 

 

 

'E
3 5.8 - .3."-
cu ..
m .
I0\ .0 a.s/
”g 5.6 h .. {4.4. e "‘
Q arr/.5.
b 5 4 — «M. ”‘5’". .3
.2 f
at; 5.2 — 35/. —
5'2 3'
g 5.0 - 1 _
2 s
'—

4.8 l l l l

O 0.5 LO I5 2.0 2.5

Level Difference 2 (cm)

Figure 6: Transfer rates, 0(8), from the x(t) curve of a typical run.

45
in Table A3. The second and third columns of the Table list the
input data, 11.5, the inside beaker level height, E, versus tin.
These data points are numbered in column 1. The fourth column gives
the nearly constant difference in the inside beaker level height
between adjacent data points for proofreading purposes. The trans-
fer rate, a, as a function of s is presented in tabular form in
columns 5 and 6. In the next section the functional form of the
0(a) curve is considered. In that section use is made of the curve
l/o versus in: which is tabulated in colums 7 and 8.

The random errors in the 0(2) points, arising from the measure-
mmlts and the differentiation procedure. are now estimated for the
0(2) points of a particular beaker filling such as these shown in
Figure 6. To do this, the root mean square deviation of the z(t)
data points from the fitted quadratic equation was calculated for
the overlapping segments of a few typical sets of beaker filling
data. Since the z(t) curve is nearly a straight line, the error
in the derivative of the eleven data point segment was taken to
be the difference in the slopes of two straight lines whose end
points differed in s by twice the rms deviation. Using the differ-
ence in time of 300 sec between the first and eleventh data points,
the error in the derivative is estimated to be *2A/300 sec, where
A is the rms deviation. The average percent error for the overlap-
ping segamnts of these s(t) curves was found to be about 0.62.

This agrees with the error that would be estimated for the derivative
using the accuracy *0. 001 cm for the cathetometer readings.

The percent error of o is the sum of the percent errors of

the derivative and the beaker radius. From the volume versus height

46

curve used in calibrating the beaker, random deviations in the radi-
us of the beaker along its length are estimated to be *0.22. This
error has, however, already been taken into account in the deriva-
tive error because the derivative error was estimated from experi-
mental data. Therefore the random error in the 0(2) points for
one beaker filling is estimated to be 0.62.

There are also larger run-to-run differences in the transfer
rate‘which arise from differing runrto-run substrate conditions.
These and other errors will be discussed in the following sections

as the experimental results are presented.

IV. RESULTS AND DISCUSSION

A. Pressure Head Dependence

In this section experimental results of the dependence of the
transfer rate on level difference, or pressure head, are presented
and discussed. This dependence has already been mentioned in the
previous sections regarding the curvature of the 2(t) curves in
Figure 5. The critical velocity is observed to respond to pressure,
causing the transfer rate to decrease by about 102 over a change
in head of 1 1/2 on in these experiments.

As was discussed in Section II—C in reference to Equation 69,
the dependence of the frictional pressure, P., on 0 or vi can be
measured directly when the driving pressure, pgs, is larger than
the acceleration tern,‘M* d22/dt2. Figure 7 shows the dependence
of 0 on 2 for three of five'beaker fillings during the same experi-
mental run. These curves were obtained by differentiating the same
2(t) curves that were shown in Figure 5. Measurements of the accel-

eration, dzz/dt2

, made from experimental 0(2) curves such as those
shown in Figure 7 indicate that for 2 > 0.01 cm the acceleration
is small enough that the frictional pressure, P., can be set equal
to the driving pressure, 032.
For 2 > 0.1 cm, the 0(2) data may be described by 54
0(2) - l/[A - Bm(cm)]. (54)

Curves of this functional formnwith A and B as adjustable parameters

47

a;(l0'5cm3/sec-cm)

Transfer Rate

48

 

8 5 I I I I I I I I I l I l I I

80— zfl'm'm h=0.2lcm—

7.5 _ _.

7.0 -— ,.-
. ,.———'o°"'000'o.".. 0.

”000......

9’
a:
I

6.0 '4’- .4
5.5 -— ”4.4—”;
/" h=8.20cm

5.0 —- .9/

/ T= l.65K

 

l l l

‘}55 I l, 1 1 l .I 1 1 1

ee..e' h=2.l2cm

L

 

 

l
' O 0.5 LO
Level Difference 2 (cm)

Figure 7: Transfer rate as a function of level difference
for the three beaker fillings of Figure 5.

L5

49
are shown ae eolid linee on Figure 7. Thin functional form ie a
good fit to all the 0(2) dete obeerved in 80 beaker fillinge during
the clean gleee eubetrate etudy end aleo for 35 beaker fillinge
during the neon eubetrate etudy.

Ihe curvee in Figure 7 can be interpreted ae giving the I-V
characterietic of the film. Alternatively, Equation 54 cen be in-
verted to give the dependence of the frictional preaeure on a or
v.:

P. " pg: - C WEI/Ba). (55)
where C in a conetant equel to pg exp(A/B). The frictional preeeure
will have the eene functional form ueing v. instead of a. mu 1e

obtained by eubetituting for o in Equetion 55 ueing the relation:

0 - (p./p)v.d. (56)

If 1/0 ie plotted vereue inz(cn) we get a etraight line of elope
-B and intercept A et inr(cn) - 0. me parametere A end B were
obtained by fitting the linearized date with e leaet equered atraight
line in the computer program Sigma-z. Valuee of A end B are given
in the firet line of the computer output euch ea in the output ehown
in 'reble A3.

Graphe were drawn after each run a vereua r end l/o vereue
inz(cm) along with the fitted curvee to vieuelly check the parametere
end goodneee of the fit. 'Ihe data pointe on theee graphe were plot-
ted end the fitted curvee were drorn from the computer output ueing
a Hewlett-Packard 9100A calculator end 9125A Calculetor Plotter.

Valuea of the parametere A end B were obtained at two temere-
turee during the clean gleee eubetrate etudy. At 1.65 K. the values

of theee parametere ere: A - (1.80 * 0.09) x 104 nee/cm2 end

50
n - (8.9 . 1.9) x 1.02 eec/cn2 with h - 7 on. At T - 1.28 x, the
valuee of theee paranetere are: A - (1.35 t 0.02) x 104 eec/cm2

2 eec/cm:z with h - 7 cm. The errore given

and B - (8.3 t 1.4) x 10
here are the root mean equate deviatione of the neaeured A'e and

B'e for h in the neighborhood of 7 on. Then errore will be die-
cueaed in more detail later in thie eection. Theee and other pere-
eetere neeeured during the clean glaee and neon eubetrate etudiee

are preeented in tabular form in Appendix B.

For 2. < 0.1 cm, the acceleration ten: in the equation of motion
beconee inortant no that the frictional preeeure cannot be meeaured
directly. If thie tern wee not included and the fitted curve for
large 1: wee extended to enell r, we would have a + 0 ea 2 + 0.
However, the acceleration tam caueee the inner level to overehoot
the outer level with a 1‘ 0 at r - 0. AI wee diecueeed in Section 11-0,
the inner level then oecillatee about the outer level.

he critical velocity. or critical trenefer rate, ie defined
to occur at the velocity where frictional diaeipation firet occure.
In thin experiment the velocity could not be increaeed from zero
until thie velocity ie reached, no one would like to do. Inetead
the critical velocity wee approached from above with the reeulting
dynamical limitation. To eetinate the zero preeeure head or criti-
cal tranafer rate. a linear extrapolation is made from the fitted
curve for large 2. lhe daehed linee in Figure 7 are euch lineer
extrapolatione. Extrepolating from x - 0.1 on. using do/dz evalu-
ated at x - 0.1 cm. we find the following eatimete of the zero pree-
eure heed tranefer rate:

ac - a(0.1 cu) - B[o(0.l ”>12. (57)

51
Here ac is taken to be the critical transfer rate of the beaker filling
at that h.

The parameter B is observed to be independent of h for h > 4 cm.
For h < 4 cm, B decreases so that o is nearly independent of 2. as
can be seen on the top curve of Figure 7. This decrease of B at
small h may not be a property of the helium but may be a result
of a rapidly changing film thickness when the level difference,

2, is comparable to the height. h. to the beaker rim. However for
h=7 cm, the thickness changes sufficiently slowly with h that pure
pressure head dependence can be seen. ‘Most of the height dependence
of a. which is the subject of the next section, is contained in the
parameter A.

Abrupt changes in the transfer rate from one constant value
to another have been reported by Harris-Lowe et a1.55 and Allen
and Armitagef’8 In this thesis,behavior of the transfer rate has
also been seen which is suggestive of these abrupt changes. As
the level difference decays, it appears that the date shown in Figures
6 and 7 have a stepped structure as reflected in the departure of
the data from the fitted curves.

These deviations of the 0(2) data from the fitted curves are
larger than would be expected from the measurement errors of the
individual 0(a) points. As was calculated in Section III-D, the
average random error in one 0(2) datum point is *O.6Z. The average
rms deviation of the 0(2) points from.the fitted curves is. however,
found to be 1.22. This is twice the deviation that is expected
from the measurements. However. more quantitative information about

these steps cannot be given because of the low signel—to-noise ratio .

52

The errors given earlier in this section with the values of
the parameters A and B are the rms deviations of the measured A's
and B's from their average value for data in the neighborhood of
h - 7 cm. These random errors are primarily a result of the devia-
tions on the 0(2) curves. However, the actual errors in the observed
A's and B's are slightly greater than would be expected from the
2(t) data because of additional run-to-run deviations arising from
differing substrate conditions. The magnitude of o is primarily
determined by the parameter A. The approximately 42 error in A in
this experiment is lower than the 101 run-to-run background devia-

tion in a which was reported by Smith and Boorse. 1'7

Thou 0(2) results are in qualitative agreement with Atkins'su'

original results and with recent results of Martin and Mendelssohn.S6

157 for flow

Similar results were also obtained by Keller and flame
through a narrow slit.

The functional form of the pressure head dependence agreeswith
Notarys's28 observations of superfluid flowing through narrow pores
and with his extension of the hanger—slasher26 thermal activation
theory. However. when this theory is extended to the experiments
of this thesis on He film transfer, there are several disagreements
with the dependence on other quantities.

The most serious disagreement of these experiments with the
Langer-Fisher26 theory is in the temperature dependence of the para-
meters A and B. An extension of the Lenger— Fisher theory gives
A(T), B(T) c T/(o./p)2. The temperature dependences of the experi-
mental parameters are: A(l.65 K)IA(1.28 K) - 1.33 and

B(l.65 K) /B(l.28 K) . 1.1, while the Lsnger-Fisher theory predicts

53
a ratio 1.84 for both A and B. This gives, for the narrow tempera-
ture range investigated, a nearly temperature independent B and
A(T) c 9/990 Hence, within experimental uncertainties we have the
usually accepted temperature dependence of 06:1-7 ac ¢ pulp.

These observed temperature dependences of A and B give an in-
creasing do/dz with decreasing temperature in qualitative agreement
with the results of Martin and Mendelssohn.56 As suggested by Martin
and Mendelssohn, the increased curvature at lower temperatures may
explain the anomalously increasing o's with decreasing temperature
for T' < l X that were reported by some experimenters.1m7

The functional form of the dependence of the frictional pressure
on superfluid velocity observed in these experiments, and also by
Notarys, 28may be true for lower velocities, and pressures, than the
limited range investigated in these experiments. If this is true
for very small velocities, this would mean that there is no "critical"
velocity. That is, for every superfluid velocity, no matter how
small, there would be a sull frictional pressure. For dynamical
reasons, discussed earlier, this could not be tested in this thesis.

Although the hanger-Fisher26 thermal activation theory does
not appear to apply to this experiment. it may be valid for tempera-
tures near Tl’ As was discussed in Section II-C, there may be two
competing processes for nucleating vortices. It is possible that
a mechanism such as the vortex mill model proposed by Glaberson
and Donnelly23 may dominate at low temperatures while thermal acti-
vation dominates at higher temerstures. In this case, the transi-
tion temperature from one region to the other would depend on the

number of pieces of vortex lines present in the liquid helium.

This could explain why Notarys28 observed the thermal activation

54
temperature dependence for temperatures down to l I while it was
not observed in these experiments.
B. Height Dependence
The critical velocity occurs at the region near the rim of the
beaker where the film has its minimum cross sectional area and the
velocity is a maximum.58 Using the dependence of the film thickness
on height, the dependence of the critical velocity on film thickness
can be determined from measurements of the dependence of a on the
height, h, of the beaker rim above the reservoir. To do this, we
seems that the thickness, d, of the film at the beaker rim is given
by
d - k/h1/3, (58)
as was calculated in Section II—B for liquid helium adsorbed on
a semi-infinite wall.
The critical velocity is expected to have the empirical depend-

c d1“, as was proposed by Van Alphen

at al..21 Using Equation 58, this gives cc 8 5-1/4. However there

ence on film thickness, vc

is a question concerning which height to use in this relation because
the distance to the beaker rim from the bulk liquid is different

for the inner and outer levels. Atkins“

originally suggested that
the proper height to use was the height to the beaker rim from the
source liquid. This is the distance, h which is shown in Figure 2,
from the beaker rim to the outer level for this experiment. This
height was used as a first approximation for the preliminary data
analysis.

The dependence of the critical, or zero pressure heed, trans-

fer rate, oc,on h is determined from a log-log plot of ac versus h.

55
Figure 8 shows a log-log plot of ac versus h for the clean glass
substrate data at 1.65 K. For h > 2 cm, the data are described
by a straight line. A least squares fit to the data for h in the

0.19. 59 For h < 2 cm, the trans-

range 2 cm to 8 cm gives ac c h-
fer rate is nearly independent of h as well as 2.

The observations of Figure 8 indicate that the film thickness
at the beaker rim is not exactly determined by the height, h, from
the outside,or source, level and that further corrections mat be
made. Another possible distance that could be used to determine
the film thickness is the instantaneous distance to the inner level
from.the beaker rim. In this case, the film thickness would increase
with decreasing r as the beaker is filling. As a result, when h
is comparable to z, a should increase with decreasing s as the beaker
fills. As can be seen on the top curve of 0(a) in Figure 7, this
is not observed.

The data suggest a third possibility. It appears that the data
in Figure 8 are displaced by a constant factor in h. This leads
to the plausible assumption that the proper distance to use for

determining the film thickness is a distance:

‘1 - h + h. > be (59)
corr

This correction appears to be a result of the nonzero initial level

difference, 2 , as suggested by the departure of the data from

init

a straight line in Figure 8 at h - 2 cm, £.s., when a is com-

init
parable to h. This might be similar to metastable behavior that
has been seen for beakers that were filled by submersion, i.e.,

a metastable state of thickness,60 or vorticity, is formed appropri-

ate to the distance to the beaker rim from.the initial inner level.

56

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57

Other metastable behavior of the film.is also suggested by the abrupt
changes in the transfer rate, which were mentioned in the previous
section.

The above suggestion wee tested by performing two experimental
runs with a ”init of 0.3 cm instead of the usual 1 1/2 cm to 2 cm.
If the correction is due to 'init’ we would expect that with a smaller
zinit the data‘would both depart from a straight line at a smaller
h and also have a larger magnitude slope in the straight line region.
With zinit - 0.3 cm, the departure of the data from a straight line
on the log-log plot appeared to occur at 0.6 cm instead of 2 cm as
in the original data. However, the slope of the straight line wee

—0.20, essentially the same as the slope for a larger 2 There-

init'
fore this experimental test was inconclusive and, at worst, may not
support the assumption that the correction term h', was equal to
zinit'

The following considerations of the quantum mechanical phase,
¢. make the concept that the thickness at the inside rim of the
beaker is determined by the distance to the outside beaker level,
plus a possible constant displacement, more palatable.31 As was
discussed in Section IIqA, the phase acts as a potential for flow
so that on the outside surface of the beaker, where vi is subcriti-
cal, we have a gradient in ¢,but ¢ is well defined everywhere in
this region. This may again be true along the lower inside surface
of the beaker. The dissipative region, where the critical velocity
occurs, is near the inside rim of the beaker. In the dissipative
region 0 is not well defined but instead slips at the Josephson

frequency. Hence we have a well defined quantum.mechenical phase

58

up the outside surface of the beaker and over the rim up to the dis-
sipative region. This could cause the thickness in the dissipative
region to be coupled to the height from the rim to the outside level.
Ewever, this does not explain the origin of the displacement, h',
unless it is possibly due to the finite thickness of the beaker rim.

Nevertheless, if we make the correction, hcorr - h + 1 1/2 cm,
we not only get a straight line on the log-log plot but also agree
with the empirical relation: ac ¢ hi”. This is shown in Figure 9.

A least squares fit to the corrected 1.65 K data gives:

arr

-5

cc (1. 65 z) - (8. 3 : 0.2)hco1o cm3/sec-cm. (60)

The error given here for the magnitude of (1c is the m scatter

of the data from the fitted curve. The error of the exponent was

estimated by taking the difference in the slopes of “o lines on

the log-log plot whose end points differed by twice the rms scatter.
Although we have less data at 1.28 X, if these data are treated

in the same way we get:

-5

a(1.28 x) - (10):? 1’ ‘ cm3/sec-cm. (61)

corr‘ :10

These two results are now combined by removing the temperature de-

pendence of a :

-1/4 -5

a (h, 1:) - (10) (9 Mb" :10 cm3/sec-cm. (52)

corrx

Using the dependence of the film thickness on height given
in Equation 58, we get the functional dependence of v c on d:
vc c d-IM. This agrees with the empirical relation of Van Alphen
at al..21 This empirical relation was proposed from the examination
of several different experiments spanning several decades of chan-

nel width of which data the helium film critical velocity formed

59

“N00

.5 a: a + a u e .233 332:8 3:3 .e no.“ .. a

u
not. o .3! nauseous 133.8 one no some nonsmom 2.. our»:

2.3 :3; 2.2.: 882.8
m e m a _

 

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60
only one point. In the present experiment, this empirical relation
was verified over the narrow range of thicknesses covered by the
film. This verifies that the proposed dependence is valid for this
narrow range. This indicates that the mechanism responsible for
the critical velocity in the film is probably the same as for the
larger channels.

The dependence, vcd c ind, predicted by many theories that
use Feynman- Onsager vortices is not observed.1"7 This does not
necessarily disprove these theories but may be a result of a lack
of knowledge of how the vortices interact with the channel walls
and how they are nucleated.

This functional form of the dependence of a on h agrees with
the experimental results of Allen and Armitagel'8 for the film.
Their measurements were mde primarily for beaker emptying experi-
ments and were reported at about the same time as these these experi-
ments were begun. The magnitude of ac with h - 2 cm agrees with

1—7

many other experiments although it is larger than the results

of Allen and Armitage."8 Van Alphen at al. predict

l/ 3 -6

ve 8 ld-J’M (c.g.s.).21 If we use the value d - 3.0h- x 10 cm

16 -l/4 «as...»

for the film thickness, Equation 62 gives vc - 1.6d
c. Substrate Dependence
Results are reported and discussed in this section of measure-
ments of the transfer rate on a neon substrate. As was discussed
in Section III-B, an effective neon substrate was prepared by coating
a clean glass beaker with neon taking care to provide smooth and

uniform surface conditions. Using this method, a decrease in a

61
was seen going from a clean glass substrate to an effective neon
substrate in the same apparatus.

Using the results of the previous two sections, the dependen-
ces of a on h and 2 could be separated and the substrate dependence
isolated. During an experimental run with a neon substrate, meas-
urements were made of 2(t) at three h's: typically at h - 5 cm,

7 cm and 9 cm. The 0(a) curves were then determined and the a de-
pendence of the data removed by using the zero pressure head trans-
fer rate, °c° The dependence of a c on b was then removed from ac
by normalizing the data to h II 7 cm using the dependence :

°c ‘ -1/4.

During the neon substrate study, three runs were made using
only the clean glass beaker without a neon coating. The first clean
glass run was made after the final assembly of the apparatus for
comparison with the results of the previous sections. TVo other
clean glass runs were made with the neon substrate apparatus dur-
ing the course of the neon substrate study as a test of the clean-
liness of the beaker. The transfer rates obtained for these runs
agree with the previous clean glass substrate results demonstrating
that the beaker was initially clean, and remained clean during the
course of the substrate study.

All of the data during the neon substrate study were taken at
a temperature of 1.65 K, as were most of the clean glass substrate
data of the previous sections. Transfer rates for a neon substrate
were determined as a function of the thickness of the neon coating,
5. Figure 10 shows ac with h - 7 cm and T - 1.65 K as a function

of E. The three points at the left of this figure are the clean

62

 

 

 

 

5-4 "V I 1 I I
A 5.2 - ° —
s
‘3
§ 5.0 - -
"E O
0
'9 4.8 -
9
b0
0 4.6 o -—
E
a; 4.4 —
“.3
S
s: 4.2 ° e
40 4 I l l J

 

 

 

0 IO :02 I03 . :04 lo5
Ne Thickness of (A)

Figure 10: Semi-log plot of the critical transfer rate, ac with h-7cm
and T - 1.65 K, versus thickness of He coating, 5.

63
glass substrate transfer rates obtained with this apparatus.

The 0(5) results shown on Figure 10 are more meaningful than
seems upon initial inspection. For a neon coating of about 1000 A,
it was possible to see an on-off effect; 11.0., the substrate could
be changed relatively reproducibly from clean glass to neon at will.
Because of the logarithmic neon thickness scale, the larger than
expected transfer rates at very high and very low coverages mks
the 0(5) data look worse than they really are. Although some of
the transfer rates are larger than expected, all of the neon sub-
strate transfer rates for E > 500 A are, however, less than the
clean glass transfer rates.

The lowest transfer rates were seen over the decade of neon
thickness from 500 A to 5000 A. Four experimental runs having a
neon coating in this thickness range gave a critical transfer rate
of oc(h-7 cm) - 4.1 x 10'5 cm3/sec-cm. Two other runs at 1000 A

5

thickness neon coating gave ac - 4.14 x 10. cm3/sec-cm. These larger

transfer rates are most likely a result of accidentally rough neon
coatings. For comparison, the average clean glass transfer rate

5

in this apparatus was: ac - 4.86 x 10- cm3/sec-cm.

0
At 100 A thickness neon coating, we expect to have a helium
film thickness equal to the average of the film thicknesses on clean
glass and pure neon substrates. Two measurements were made of ac

at this neon coverage. One of these experiments gave

5 cm3 lsec-cm,approximately as expected. The other

Cc - As’ 3 10
experiment, probably due to rough substrate conditions, gave the
larger transfer rate: ac - 5.2 x 10'"5 cm3/sec-cm.

0
At 10 000 A thickness neon coverage, larger transfer rates

64
than the minimum are again seen. From the calculated helium film
thickness as a function of 5, this is not expected. Two measure-
ments were‘made of ac at this coverage. These transfer rates had

5 cmglsec-cm. .At this large thick-

an average value of ac - 4.7 x 10-
ness. it may not be possible to get a smooth neon coating using the
method described in Section III-B. Because of the long annealing
times used in this procedure to smcoth the neon coating, it may be
that a granular layer is formed as a result of crystallization of
the thick neon coating.

The functional form of the expected dependence of ac on film
thickness can be calculated using the results of the previous sec-

1/4. 3/4

tions: 0 ¢ vcd and vc c d. These results give oc c d .

c
Using this dependence, the expected dependence of ac on the thick-
nose of the neon costing, E, can be determined from the plot of

the helium film thickness versus E shown in Figure 1. This expected
functional dependence is shown as the solid curve on Figure 10.

This curve has been normalized to go through the experimental clean
glass substrate points and the lowest 1000 A thickness neon coating

5

points‘which have average transfer rates of 4.86 x 10' cm3/sec-cm

and 4.1 x 10-5 cm3/sec-cm.respectively.

The ratio of the thickness of the helium film on the effective
neon substrate to the thickness of the helium film on the clean
glass substrate is new calculated from the experimental results.
using the lowest neon substrate transfer rate and the clean glass
substrate transfer rate, given above. we get

)4/3

dN's/dglaes - (cue/aglass - 0'79' (63)

65

1/3 x -6

Assuming the film thickness, d 10 cm, for a

clean glass substratef3ths thickness of the helium film on a neon

81". - 3e0h

substrate can be calculated ming Equation 63. Solving this equa-

tion, '0 86': d“. - 2.4h'1/ 3 x 10'6 cm. This experimental value of

the film thickness agrees remarkably well with the result:

d“. - 2.3h":l'/3 x 10.6 previously ‘calculated in Section II-B.

V. SWY AM) COICLUSIONS

Measurements of helium film transfer rates for filling clean
glass and neon coated beakers have been reported in the previous
sections. The apparatus used in this study was designed to provide
a clean environment for the beaker, to avoid enhanced transfer rates
and unnecessary background variations. With this apparatus, quan-
titative measurements could be made of the slight dependences of
the transfer rate, a. on pressure head, film height, and substrate
anteriel.

The pressure head dependence of a was measured over a change
in head, a, from 2 cm to 0.1 cm. The data in this range are described
by the functional dependence:

0(2) " l/[A - Bin(s)]. (64)
This gives a dependence on superfluid velocity, v.,of the frictional
presets-91", opposing the flow of P. c exp[f(h,T)/v.]. These results
agree with previous, qualitative measIn-smsnts by other experimenters
of the pressure head dependence for both the film and narrow channels.

The dependence of the critical, or sero pressure head, trans-
fer rate, ac, on the height, h, of the beaker rim above the outer
helium level was measured for h ranging from a few millimeters to
'10 cm. After applying corrections, which were discussed in Section

IV-B, these data are described by:

acmcorr) - Golgi: ’ (65)

66

67

where Go is a content for a given temperature and hoorr is the

-1/4
a

corrected height. This results in the dependence, vc c d of

the critical velocity, vc, on channel width, d. This dependence

(‘8 for the

21

agrees with recent measurements by Allen and Armitags
film and the empirical relation proposed by Van Alphen at al..

A 16: decrease was seen in the transfer rate going from a clean
glass substrate to an effective neon substrate in the same appara-
tus. The helium film thickness on a neon substrate, derived from
this change in o, is in reasonable agreement with the thicknesses
calculated for rare-gas solids in Section II-B.

Although the physical mechanism for the critical velocity is
believed to be vortex production, basic theories using this mecha-
nism are not sufficiently well developed to predict these measured
dependences on pressure head and film height. We hope these results

can be used as a guide in the solution of these problems.

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1.

2.

3.
4.
5.

7.

9.

10.

12.

13.

14.

15.

l6.

17.

LIST OF REFEREMES

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APPENDICES

72

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90.0N
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99.0—

APPENDIX B

TABULAR TRANSFER RATE DATA

TABLE 81
Measured transfer rate parameters for a clean glass
.ubUtnt. and T - 1s65 Ks

Run h A B a;
No. (on) (lO‘sec/enz) (lozsec/cnz) (Id-scnzlsec)
571 7.897 1.843 9.87 4.658
578 6.374 1.769 9.21 4.813
578 4.363 1.676 13.29 4.707
57D 2.758 1.684 6.32 5.589
578 0.413 1.485 3.29 6.683
581 8.867 1.788 11.11: 4.627
588 7.233 1.739 9.32 4.875
58C 5.533 1.678 6.57 5.278
588 3.473 1.612 6.64 5.453
588 0.974 1.448 2.25 6.602
591 8.619 1.735 6.32 5.139
598 6.754 1.667 6.39 5.318
59c 4.879 1.597 5.61 5.685
59D 3.184 1.558 3.61 5.959
598 0.582 1.385 2.07 6.879
60A 8.387 2.134 18.88 4.851
688 6.803 2.026 6.36 4.468
688 4.948 1.847 9.86 4.592
60D 2.910 1.676 4.93 5.434
688 0.213 1.376 1.93 6.944
61A 8.705 1.865 9.96 4.548
618 7.893 1.786 6.96 4.954
618 5.092 1.688 7.18 5.187
618 3.050 1.591 5.57 5.628
618 8.565 1.354 0.82 7.248
62A 8.498 1.876 8.93 '4.598
628 6.675 1.778 7.93 4.919
62c 4.996 1.693 10.39 4.897
628 3.838 1.599 8.14 5.343
628 1.021 1.421 3.43 6.514
631 8.646 1.857 18.32 4.539
638 6.687 1.744 18.58 4.778
638 4.682 1.648 11.57 4.988
630 2.615 1.533 7.89 5.564
638 8.381 1.257 5.87 7.811

78

Run

64A
64B
64C
64D
65A
653
650
65D
65!
66A
66B
66C
660
663
67A
67B
67C
67D
68A
688
68C
68D
683

(cm)

7.960
6.008
4.012
1.614
8.197
5.994
4.191
2.122
0.213
8.395
6.427
4.379
2.452
0.389
7.325
5.300
3.289
1.350
8.478
6.423
4.498
2.471
0.294

79

rubl- Bl (cont'd.)

A.

(loaoac/cn?)

1.863
1.734
1.664
1.426
1.829
1.698
1.616
1.475
1.236
1.914
1.794
1.695
1.555
1.332
2.025
1.792
1.617
1.401
1.777
1.696
1.591
1.472
1.229

B

(lozcoc/cmg)

8.79
10.75
5.89
3.14
9.89
10.25
10.64
5.86
1.46
9.14
9.96
6.11
6.79
2.82
9.04
11.00
4.32
9.00
10.93
6.54
6.43
4.68
1.71

ac

(10-56.2/IOC)

4.636
4.773
5.375
6.534
4.628
4.897
5.066
5.985
7.786
4.505
4.699
5.266
5.612
7.014
4.297
4.626
5.679
5.870
4.664
5.224
5.538
6.143
7.778

80

TABLE B2
‘Msssursd transfsr rsts psrsustsrs for a clean glass
substrsts and T‘- 1.28 K.

Run 13 A B ac
N8. (cu) (104ssc/cn2) (1021.6/8-4) (ld-Scmzlsoc)
71A 8.747 1.374 8.14 6.071
713 6.734 1.330 6.89 6.407
716 4.761 1.311 9.14 6.178
710 2.700 1.257 6.71 6.748
718 0.389 1.100 3.18 8.293
72A 8.410 1.388 8.57 5.967
72B 6.493 1.327 6.43 6.484
720 4.483 1.283 10.64 6.089
72D 2.474 1.213 7.57 6.815
722 0.373 1.096 1.46 8.738
73A 8.760 1.399 9.14 5.860
738 6.894 1.362 7.07 6.254
730 4.949 1.329 7.14 6.376
730 2.975 1.274 6.25 6.742
738 0.911 1.130 2.04 8.349
74A 7.579 1.396 7.46 6.075
748 5.676 1.283 10.82 6.066
740 3.681 1.300 9.11 6.224
740 1.675 1.178 7.75 6.951

81

IABLB B3
‘Msssursd trsnsfsr rats data for s nson substrats and T - 1.65 K.
Run E h A. B ac
No . (A) (cu) (104.16] 431112) (1028 cc/ c102) (IO-Sculz/ sec)
77A 0 9.337 1.972 7.56 4.50
77B 0 7.635 1.876 5.72 4.84
77C 0 5.459 1.776 5.39 5.11
78A 1000 9.350 2.344 9.31 3.77
788 1000 7.517 2.210 6.58 4.12
780 1000 5.588 2.051 8.39 4.29
78D 1000 3.397 1.870 5.03 4.91
79A 1000 9.120 2.208 7.19 4.09
798 1000 7.177 2.046 5.94 4.46
80A 1000 9.130 2.373 11.88 3.61
808 1000 7.142 2.186 8.03 4.08
800 1000 5.116 2.022 5.02 4.57
81A 0 0.092 1.969 7.53 4.50
818 0 7.051 1.837 5.60 4.94
82A 10 000 8.908 1.979 5.87 4.60
828 10 000 6.853 1.832 5.10 5.00
83A 3000 8.991 2.265 9.24 3.89
838 3000 7.039 2.104 6.70 4.30
84A 500 6.435 2.068 8.02 4.28
85A 100 9.288 2.060 7.43 4.33
858 100 7.289 1.930 6.54 4.65
850 100 5.284 1.826 4.44 5.07
86A 10 000 9.126 2.111 8.65 4.17
868 10 000 7.212 1.991 5.34 4.61
860 10 000 5.211 1.873 4.49 4.94
87A 100 1.129 1.864 5.22 4.90
878 100 7.150 1.765 3.62 5.30
870 100 5.120 1.675 3.64 5.57
88A 0 9.129 1.969 6.52 4.57
888 0 7.230 1.857 6.26 4.84
880 0 5.206 1.754 6.47 5.08
894 1000 9.130 2.167 7.57 4.13
898 1000 7.089 1.990 6.34 4.54
89C 1000 4.997 1.850 4.79 4.98

 

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