as?!”

VISCOSITY OF MOLTEN LITHIUM-6 AND LITHIUM—7

by

Nguyen Tu Ban

AN ABSTRACT

Submitted to the School for Advanced Graduate Studies of
Michigan State University of.Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy
1960

Approved "’{>/1(3'7"ZJ"E’7~1’;¢[ A]: [[47Maai

 

men:

Nguyen Tu Ban
1

ABSTRACT

This study is an extension of a program exploiting
isotOpic mass as a probe for liquid-state and solid-state
investigations to non-equilibrium phenomena of liquids.

The viscosity of molten lithium-6 and lithium-7 was
measured by observing the damping of a freely oscillating
sphere filled with molten lithium. The viscosity coeffi-
cient was found by treating the damping of a slowly oscil-
lating sphere containing a viscous fluid as a problem.in
the laminar flow of incompressible fluid.

The temperature dependence of the viscosity of lithium-6
‘was found to be in general agreement with a law stating
that the viscosity varies eXponentially in the reciprocal
of the absolute temperature. This law is also predicted by
some existing theories of viscosity. The extrapolated
value for the absolute viscosity of lithium~6 at the melting
point (180.4 °C) was found to be 4.50 mp.

The effect of isotOpic mass on viscosity was studied
by measuring the viscosity coefficient of lithium~7 at a
given temperature and compare it with that for lithiumpé
at the same temperature. The ratio of the viscosity of
lithium»? to that of lithium~6 at 236.8 °C was found to be
(4.186:0.080/3.83110.016) = 1.09330.025, in general agree-

ment with the theoretical prediction that it is equal to the

Nguyen Tu Ban
2

square root ratio of the atomic mass of lithium-7 to that
of lithium-6, that is to (7.018/6.017)% = 1.080.

On the assumption that the viscosity coefficient for a
mixture of isotopes of an element, consisting predominantly
of one component, is a linear function of the number compo-
sition, the absolute viscosity of lithiumpnatural (92.5%
lithium-7, 7.5% lithium~6) at 236.8 00 is estimated to be
4.16 mp. This value is 24% lower than the corresponding
value obtained by Andrade and Dobbs using lithiumpnatural.
A.detailed examination of their paper led to the conclusion
that their values are significantly in error. This fact is

of technical interest.

VISCOSITY OF MOLTEN LITHIUM-6 AND LITHIUM—'7

by

Nguyen Tu Ban

A THESIS

Submitted to the School for Advanced Graduate Studies of
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy
1960

To Quy, Bi’ch
and MS; Hudng

ACKNOWLEDGMENTS

This study would not have been possible without the
assistance of many individuals:

To Professor D. J. Mentgomery for the suggestion of the
problem, for his guidance and encouragement,

To Dr. P. S. Baker and Mr. E. B. Olszewski, of the Stable
IsotOpes Research and Production Division of the Oak Ridge
National Laboratory for their invaluable help in filling
the spheres,

To Professor S. K. Haynes for assuming the Chainmmnship
of the Guidance Committee while Professor Mcntgomery is on
sabbatical leave,

To Professor R. D. Spence for his stimulating discussions,

To Professor R. Schlegel for his constructive criticisms
during the writing of the manuscript,

To Mr. J. u. Boyd for his help in programming the
calculations on NISTIC,

To Mr. C. L. Kingston and Mr. P. Solonika for their help
in the design and construction of the apparatus,

To the above-mentioned and to many others, the author
humbly takes this opportunity to eXpress his most sincere
thanks.

The author also deeply appreciates the financial support
provided by a research contract granted to Michigan State
University by the Metallurgy and Materials Branch of the

Division of Research of the Atomic Energy Commission.

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

.A. Choice of TOpic . . . . . . . . . . . . . .

B. Choice of Material . . . . . . . . . . . . .

C. Choice of Mbthod of Measurement . . . . . .

D. Statement of Problem . . . . . . . . . . . .
THEORETICAL CONSIDERATIONS ON THE VISCOSITY OF LIQUIDS

THEORY OF THE EJQDERIIIEI‘IT . . . . . . . . . . . . . .

APPARATUS AND I-IETHOD . . . . . . . . . . . . . . . .

A. Description of the Apparatus . . . . . . . .

l. Oscillating System . . . . . . . . . . .

2. Vscuum.Enclosure . . . . . . . . . . .

3. Furnace and Temperature Control . . . .

4. Sample Holder . . . . . . . . . . . . .

B. Methods of Measurement . . . . . . . . . . .

1. Preparation of the Samples . . . . . . .

2. Determination of the Mbment of Inertia

of the Suspended System . . . . . . . .

3. Mbasurement of the Amplitude and Period

of Oscillation . . . . . . . . . . . . .

4. Temperature Measurement of the Lithium.

filled Sphere . . . . . . . . . . . . .

C. Calculation of the Viscosity Coefficient . .

1. Logarithmic Decrement 6 . . . . . . . .

2. .

TABLE OF CONTENTS

Density'13 . . . . . . . . . . . . . .

PAGE

H

mOiUN

21
24
24
24
28

31
34
34

38

42

48

48

49
49

TABLE OF CONTENTS (001112.)

3. Inner Radius of Sphere R . . . . . . .
4. ‘MOment of Inertia I . . . . . . . . .

5. Method of Calculation . . . . . . . .

m m MAL RES LTLTS . C C O O O O O O O O O O O O

A.

B.

Verification of Method . . . . . . . . . .
Effect of Temperature on Viscosity of
Lithiump6 . . . . . . . . . . . . . . . .
Effect of IsotOpic Mass on Viscosity of
Lithimm . . . . . . . . . . . . . . . . .
Estimation of the Absolute Value for

Natural Lithium. . . . . . . . . . . . . .

DISCUSSIONOFRESULTS ..............

A.
B.
C.

D.

Verification of Method . . . . . . . . . .
Effect of Temperature on Viscosity . . . .
Effect of IsotOpic Mass on Viscosity . . .
Accuracy of the Estimated Absolute Values

for the Viscosity of Lithium . . . . . . .

APPENDIXA- INNER DIAMETER OF SPHERE . . . . . .

APPENDIX B - DIMENSIONS OF INERTIA DISKS . . . . .

APPENDIX C - DATA FOR VISCOSITY OF WATER . . . . .

DATA FOR VISCOSITY OF LITHIUM-6 AND
LImIID‘L7 o o o o o o o o g o o o o 0

APPENDIX D - LOGARITHMIC DECREMENT DATA. . . . . .

RESIDUAL DAMPING FOR LITHIUM-6 SPHERE

APPENDIX E - SPHERE TEI-iPERATURE AS A FUNCTION OF

PAGE
50
50

51
51

52

55

56
6O
6O
6O
61

62
66
67
68

69
7O
71

TABLE OF CONTENTS (Cont.)

FURNACE TEMPERATURE . . . . . . . . .
APPENDIX F - DEPENDENCE OF V\ON R . . . . . . . .
APPENDIX G - SPECTROGRAPHIC ANALYSIS OF LITHIUM-6
AFTER VACUUM DISTILLATION . . . . . .

REFERENCES CITED 0 o o 0 O o o o o o O O o O o o 0

PAGE
72
73

74
75

FIGURE

1.

D bl N
o

11.

12.

13.

LIST OF FIGURES

General View of Apparatus . . . . .

SUSpension'Wire Clamps . . . . . . .
Details of Connecting Shaft E . . . .
General View of EXperimental Setup .
Li Sample Holder . . . . . . . . . .
Apparatus for Transferring Lithium to
Sample Holder . . . . . . . . . . . .
Adjustable Mement of Inertia System .
Gauges for Setting Moment of Inertia
Circuit for Amplitude Measurement . .

Thyratron Phase-control Circuit . . .

Sphere Temperature as a Function of Furnace

Temperature . . . . . . . . . . . . .

Viscosity of Lithium-6 as a Function of

Temperature . . . . . . . . . . . . .

Plot of 10310 \10:5 WL) against 103/T . . .

PAGE

58

59

65

LIST OF TABLES

TABLE PAGE

I. Viscosity of Water . . . . . . . . . . . . . . 53
II. meent of Inertia for Li-6 Sphere . . . . . . 53
III. Residual Damping for Li-6 Sphere . . . . . . . 53

IV. Viscosity of Lithium-6 . . . . . . . . . . . . 57

V. Calculated Viscosity of Lithiumpnatural . . . 64

INTRODUCTION

It was realized by the earliest investigators of isotopes
that isotOpic mass is of potential value as a means for
investigating the behavior of matter in the aggregatea.
In studies of isotopes of the same element, the atomic mass
can be changed while the atomic force field is kept the
same. The dependence of physical prOperties on mass is
usually simple, whereas the dependence on force field is
very complicated. In the early days, however, low enrichments
of isotopes obtainable precluded exploitation of isotopic
‘mass in bulk studies. It was not until the establishment
of atomic energy projects that an array of elements became

available in substantial amountsls.

‘A. Choice ot Tepic

This study grew out of a program, supported by the
Atomic Energy Commission, eXploiting isotOpic mass as a
probe for liquid-state and solid-state investigations.
Earlier studies in this program had shown that such studies
were feasible, but most of the tOpics investigated depended
only on higher order effects of isotOpic massS6. Moreover,
these topics had been for the most part deliberately chosen
in fields where the phenomena were relatively well under-

stood. It was decided to extend the eXploitation of isotopic

mass in bulk studies to imperfectly understood prOperties,

particularly properties involving nonpequilibrium phenomena
of liquids. Here viscosity suggests itself as a likely
candidate.

Technically, viscosity is important because most time
dependent processes in liquids - fluid flow, heat transfer,
mixing, etc... - depend on it; scientifically, because its
study may point the way to a better understanding of the
structure and behavior of liquids. Moreover, the existing
theories predict that viscosity depends on mass in She first

order.

B. Choice of Material

For greatest clarity in interpreting the results, a
monatomic substance, that is an element, should be studied
first. It is true that the liquids for which the theory
is simplest are those for which the alterations in electronic
distribution are least when the isolated molecules condense
to form a liquid, i.e., the inert gases. Indeed, the pro-
perties of helium~3 and helium~4 have been studied extensively.
Little attention has been given to neon-20 and neon-22,
argon-36 and argon-40, and the others, because of their
unavailability in bulk, let alone their small relative mass
difference. But there are other reasons for extending the
study beyond this type of substance. The eXperflmental
difficulties at the low temperatures required for liquefaction,
and the narrow temperature range in which the liquid is

stable at ordinary pressures, are two shortcdmings of the

inert gases. More telling perhaps, is the restricted
nature of the interactions in these materials. With metals,
one can attain interactions of more general interest, and
at the same time retain simplicity, particularly if a simple
metal is chosen. Atomically speaking, the simplest metal,
at least at ordinary pressures, is lithium, with its low
atomic number (2:3), and its simple structure (bee) in the
solid state. Fortunately, there are two stable isotOpes,
lithium~6 (7.5% abundance ratio) and lithium-7 (92.5%),
occurring in nature with a large relative mass difference
(14%). Finally, both isotOpes are available in quantity

at high enrichment (up to 99.9%) at moderate costla.

C. Choice of Method of Measurement

 

In the theory of hydrodynamics when one goes from ideal
fluids to real fluids, it is necessary to introduce a
dependence of stresses on relative motion of portions of the
fluid. The simplest useful assumption is that the "drag"
between adjacent portions of the fluid is prOportional to
the velocity gradient at the region of interest, that is,

for a specific component

21-? .. 3a
Lim'as " "y‘yl 3,,
where the viscosity Yk is taken to be independent of the
gradient. Although in principle one needs only to produce
and measure a velocity gradient and a viscous stress (that

is, a force and an area), and thereby use the defining

equation directly, in practice it is almost always necessary

to use an integrated form of the equation resulting from.the
application of the principles of hydrodynamics.

In choosing a method of measurement, it is important to
find an experimental system which can be represented
adequately by a theoretical model simple enough to be analyzed
mathematically. Beyond this, the method must be adapted to
the magnitude of the viscosity, the chemical reactivity of
the substance, the amount available, the temperature range
contemplated, and the precision desired. This last depends
on the precision possible in the measurements of dimensions
and time, and if it is to be an absolute measurement, on the
adequacy of the theoretical representation and solution.

The well-known methods which are susceptible to mathematical
treatment so that absolute viscosity may be obtained, may
be grouped as follows:

1. The measurement of the resistance offered to a
moving body in contact with the viscous fluid. Popular
methods are the falling sphere method (Stokes*, 1845), the
rising bubble of gas method (Marshall, 1917).

2. The measurement of the rate of flow of a viscous
fluid. The single most important representative is the
transpiration or Poiseuille method (1842).

3. Methods in which neither the flow nor the resistance

 

* Only the name of the investigator who first used the mgthod
eXperimfintally is citeg The reader is referred to Barr ,
Bingham , and Hatschek 5 for detailed development of these
methods and of others not mentioned, and for references to
the original papers.

to flow are measured. To cite a few:

a. Concentric cylinders; the outside cylinder is
rotated at constant angular velocity, and the torque exerted
upon the inner coaxial cylinder which is inmersed in the
viscous fluid is measured (Stokes, 1845).

b. A horizontal disk supported at its center by a
wire and oscillating around the wire as an axis (Coulomb,
1801).

c. An oscillating solid sphere imersed in the
viscous fluid and supported by a bifilar suspension (KBnig,
1885).

d. A hollow sphere filled with liquid and oscillating
around its vertical axis (Helmholtz and Piotrowski, 1860).

Due to the extreme corrosion of lithium in its molten

28'35, its readiness to react with

state on most materials
non-inert atmospheres, and the small amounts of the isotopes
available, it is necessary to confine a small volume of the
sample in contact with non-reacting materials. These
requirements eliminate practically all the methods mentioned
except those involving observation of the damping forces of
a confined liquid. Furthermore, the experimental system
must be of a design such that the confined liquid can be
maintained at constant temperature with relative ease over
the temperature range contemplated.

The method which seems to fulfill all the requirements
is that of Helmholtz and Piotrowskize, to which Barr5 refers

as being "at least of historical interest". The original

paper is certainly not such as to encourage others to use
the method*. The calculations given there are very compli-
cated and troublesome, the results obtained, very unsatis-
factory. However, subsequent investigators (for example,
Ladenburg32, 1908, Verschaffelt41, 1915) have modified and
refined this method, and shown that it can be a satisfac-
tory one. More recently, Andrade and his group adapted the
method to measure successfully the viscosities of liquid
sodium and potassiuml, liquid natural lithium, rubidium and

caesiumz, liquid indium and tin16

l7

, and liquid magnesium and
calcium Andrade and Chiong1 also presented a method of
calculation, adapted from that used by Verschaffelt41 for
calculating the viscosity of liquefied gases from the damping
of the oscillation of a sphere immersed in the liquid, which
simplified the calculation greatly. In view of all these
considerations, it was decided to adapt the method of
observing the damping of free oscillation of a hollow sphere

filled with lithium.

D. Statement of Problem
It is proposed to measure the absolute viscosity of
molten lithiumpo and lithium»? as a function of temperature,

using a freely oscillating sphere filled with molten lithium,

 

. "On voit combien ces relations sont compliquees et quelle
incertitude le calcul des experiences laissera planner sur
la valeur de / , qu'slles sont destinees a donner. "
Brillouin. Viscosité des liguides, part 1, p. 102,1907.

 

and to compare the eXperimmntal results with the prediction

of current theories.

THEORETICAL.CONSIDERATIONS ON THE VISCOSITY OF LIQUIDS

Although the theory of the viscosity of dilute gases is
highly satisfactory, that of liquids remains confused. This
state of affairs is understandable in.view of the continuous
interactions between the molecules of liquids, as contrasted
with the momentary interactions between the molecules of
gases. A critical survey of existing theories of liquid
viscosity has just been given by Bondig. Other recent papers
which include a treatment of viscosity as one of the impor-
tant transport processes in liquids have been written by
Eisenschitzla, by Longuet-Higginsx’, and by Colels. Much
of the following discussion is based on these papers.

From dimensional analysis alone, it is possible to draw
useful conclusions about the dependence of the viscosity
coefficient V\ of a hard-sphere fluid on temperature T and
molecular mass m, the variables of primary interest in the
present study. If [m] , [XI] , [t], and ['1‘], represent the
dimensions of mass, length, time and temperature respec-
tively, then the viscosity coefficient has the dimensions

lvu = [m] [13" It)“ <1)
Since the temperature may be taken to have the dimensions
[k]'1 [m] [1.12 [t] '2, where k is Boltzmann's constant, it
is possible to eliminate [t] and thereby to obtain

[v0 = [a]: [11-2 [14* [TR

It follows then that

Vl = constant . (kafk/LZ (2)

and that if /8 does not depend strongly on m.or T, the
viscosity coefficient increases as the square root of‘m
and T.

This prediction is indeed verified very well experimenp
tally, and moreover appears in the first approximation as
a result of the molecular theory of transport properties of
dilute gases.

For liquids, in contrast, this happy state does not
obtain. Let us consider first the effect of temperature.
It is found eXperimentally that almost always the viscosity

coefficient decreases and moreover decreases very rapidly

 

'with increasing temperature. It is apparent then that the
characteristic length ,C must be changing with temperature.
Now measurements of liquid viscosity are normally made at
constant pressure, and hence the mean intermolecular sepa-
ration is indeed changing. It might be thought then that
measurements at constant volume would give agreement for the
so-predicted dependence of VL on T, with l considered as
constant. Indeed, as has been demonstrated by Bridgman12
with non-polar liquids, the dependence of viscosity on
temperature is much less at constant volume than at constant
pressure. Nevertheless, the simple law for dilute gases

is still far from.being fulfilled. In the theories as well
as the experiments, no simple and satisfactory relation.has

been derived for the temperature dependence. Some of the

expressions that have been proposed will be shown later in

10

this section.

We will now turn to the effect of isotopic mass. When
substances that differ in chemical composition as well as
in molecular mass are studied, the difference in molecular
force fields precludes a study of the effect of mass inde-
pendent of simultaneous effects of changes in molecular
diameter and intermolecular spacing. However, substances
that differ only in isotOpic mass usually have only very
slight differences in their atomic geometry, and with these
substances it would be possible to study the effect of mass
in isolation from the effect of length. The eXperimental
data are scanty - in the literature there seem to be reported38
measurements on only the pairs H2O'DZO' HZ-DZ, CH4-CDA,

4

(apart from HeB-He , where completely new phenomena come

into play). Only for light and heavy methane does the ratio
of viscosities follow the square root of the mass dependence37.
Since the shape of the molecules studied is not simple, it

is perhaps not surprising that the simple law is not followed.
Accordingly, to minimize the complications arising from
shape, spherical molecules are desirable, and the present
study was undertaken with the isotOpes of lithium, lithium-6
and lithium-7. The experimental results will be given in

a later section of the thesis. So far as the predictions

of molecular theory are concerned, there is insufficient
agreement among the different workers to allow a brief

statement to be made.

we shall now summarize the various approaches to the

ll

theory of viscosity of liquids:

l. The modern point of view anchors the theory of
viscosity of liquids solidly in the statistical theory of
transport processes in liquids. The central problem here
is the calculation of the molecular distribution functions,
and particularly the pair distribution function. When a
pair distribution function, apprOpriate to a steady non-
uniform state has been found, the flow of momentum or
energy necessary to establish uniform conditions in a liquid
can be calculated, and the transport coefficients are derived
by averaging the contributions made by a representative pair
to these fluxes.

10’ and independently Kirkwood30 have given

Born and Green
methods for doing this. Their theories are very much similar
in Spirit, but differ in the approximations used to reduce
distribution functions of higher order to those of lower
order, thus enabling the circumvention of the nearly impos-
sible task of evaluating the complete partition function.

Owing to the proximity of the molecules to each other
in a liquid, the distribution function is primarily deter-
mined by the magnitude of the intermolecular forces. Born
and Green's theory relates viscosity to the intermolecular
potential ¢(') by considering the following physical pictureszs:

The exchange of momentum.in liquids depends on the
continuous action of the intermolecular forces, in contrast

with the transport of momentum in gases which depends largely

on the motion of the molecules themselves. Hence, if two

12

adjacent layers in the liquid move with different velocities,
each will tend to drag the other in such a manner as to
dissipate the state of relative motion, in the absence of
sustaining external stresses. As a result of this drag, the
molecular structure of the fluid is deformed. A study of
the deformation of the molecular structure will enable one
to establish the quantitative relations which determine the
coefficient of viscosity.

11

Born and Green obtained for the viscosity coefficient

the eXpression
_ i ' 3 T -1. v‘d?

'1 - 3'6 1’43“)»: dr iSmjw ,3)
where A? is a measure of the deformation of the molecular
distribution from its equilibrium configuration, and a: a
function of the magnitude of the velocity v, n, and T. By
assuming that the second term in equation (3), due to thermal
motion, can be neglected for the liquid state, and using for
quO the Lennard-Jones 6-12 potential, they derived as a
"preliminary" formula for viscosity:

vi ~ (31%;) (428% (“HS fl 2'¢m/£T <4)

. C
where ro is the equilibrium distance in the molecular field,
r1, distance of nearest neighbors, and “fl , the frequency
of vibration of the molecules about the equilibrium point r0.

Kirkwood, Buff and Green31

, by introducing a principle
of irreversibility which plays for liquids a role equivalent

to the hypothesis of molecular chaos in the theory of dilute

l3

gases, obtained for the coefficient of viscosity an exceed-
ingly complex relation which lends itself well to numerical
calculation. It would serve little purpose to present it
here, since the mass and temperature dependence of this coef-
ficient is not at all apparent. On the other hand, Kirkwood
obtained for the first time an expression for the volume
viscosity* of liquids, a result hardly accessible to elemen-
tary theories.

2. The liquid is considered to behave like a dense gas.
The elementary kinetic theory is accordingly adapted without
any appreciable change in the fundamental concepts or comp
bined with the elementary notions concerning the Brownian
movement. The physical fact here is that in a gas, energy
and momentum are transported by diffusion, that is, by
bodily movement of individual molecules through relatively
large distances, and that the exchange of energy and momentum
occurs almost entirely through bimolecular collisions, which
are relatively infrequent events for any given molecule.
In liquids, on the other hand, a given molecule spends the
greater part of the time interacting strongly with the other
molecules, and the very idea of a collision loses its signi-
ficance. A.more serious criticism to this approach is per-

haps the fact that it is doomed to failure in explaining the

 

* O O O O O O O

The definition of volume Viscosity in terms of hydrodynamics
is given in Lamb, hydrodynamics, (Dover Publications, 1945),
Attlcle 326s

14

temperature effect on viscosity of liquids. It may be recalled
that it is empirically established that for all except the
superfluids, liquid viscosity decreases with increasing temp
perature in contrast to the viscosity of gases which increases
as the temperature rises.

3. The elementary kinetic theory is applied to the

interstices between the liquid molecules. This is the hole

theory of viscous flow of liquids, first pr0posed by Ffirthzz.

He stated:

"The holes are considered as the counterparts of the
clusters in a dense gas or vapour. They are formed
by the action of the irregular thermal movement (or
by the statistical fluctuations), and they are
destroyed again by the same process; they interact
with each other, and they perform.a kind of Brownian
motion.... Since in our theory the holes in a liquid
are the counterparts of the clusters in a gas, we may
expect that the phenomenon of viscous flow in a
liquid can be eXplained in a similar way, namely by
assuming that the irregular "Brownian” motion of the
holes produces a transfer of momentum between adjacent
layers in a moving liquid."

From this crude model*, Ffirth obtained for the viscosity

coefficient:

RT 1 .-
eA/xT

V\ = 0.915— (m /a)‘5 (5)
v

where A is the so-called work function, (r , the surface

tension of the liquid, and R, the universal gas constant.

 

"This inner inconsistency of the theory makes it clear
from the beginning that it can provide only a rough picture
of ths real behaviour of matter in the liquid state, ..."
Fiirt 2.

15

4. Self diffusion is considered to be a simple phenom,
enon and accessible to an elementary treatment, from which
conclusions regarding viscous flow are drawn:

a. by using Einstein's formula for the diffusion
coefficient
D = kT / 6 17’ a q (6)

Frenkel2O assumes that the equilibrium position of the
atoms in a liquid is not absolutely permanent. After
performing a more or less large number of oscillations
about the same equilibrium position during a certain time
‘t’, each atom can jump into a new equilibrium.position,
the jump 6 being of the same order of magnitude as the mean
distance between adjacent atoms. This step by step wandering
or self diffusion can be described as a sequence of two
processes: the "evaporation" of the atom from its initial
equilibrium position into an intermediate one, and the
"condensation" from this intermediate position, into a new
equilibrium position. Frenkel gives for the diffusion

coefficient

2
D ,__ 6 e-W/kT

(7)
at;

but gives no theory from.which to calculate I; or w.
Following his earlier argumentzl, 6/ to would be the thermal
velocity (kT/ZTTm)% giving for the viscosity
1
E
= (2m1TkT) eW/kT
1Taa 6

where a is the particle radius, and w, the activation energy.

(8)

16

b. by regarding the movement of a molecule in
viscous flow as a diffusion under external constraint.

Eyring19 pictured the process of diffusion as continu-
ally taking place in the liquid, with or without an applied
force tending to cause viscous flow. The application of
such a force simply tends to make a preferred direction in
which the molecules move.

"Viscous flow is assumed to take place by the activated
jumping of an aggregate composed of one or more molecules
from an initial normal configuration to a second normal
configuration29."

Eyring does not specify the mechanism of momentum
transfer, he simply assumes that the mechanism is such that
the theory of absolute reaction rates is applicable. Then
the coefficients of viscosity and diffusion are related by
the equation

kT / D‘ = a1. a2 / a3 (9)
'where a1, a2, and a3 are lengths of molecular dimensions.

2

Kincaid, Eyring and Stearn 9 give for the viscosity coef-

ficient the expression

A
. MZT
V\ = 7.71 x 10'4--§-- eEv/nRT e'Es/R (10)
V E

v
where Ev,'measured in calories per mole, is the energy of
activation for viscous flow. The n appearing in the
exponential term was chosen to give the correct temperature
coefficient of viscosity by computing values for V\ neglecting

the entrOpy of activation E8 which is small by comparison

with experiment .

17

For this reason, Eyring's theory does not lend itself
to fully absolute calculations, and in Onsager's words, the
method "lends itself to a facile interchange of deduction
.and induction."

5. A mechanism of momentum transfer between contiguous
layers is postulated.

It is assumed that the arrangement of molecules in the
liquid state is very similar to that in the solid state, and
that consequently the average intermolecular force which acts
on a given molecule is, on the whole, very little different
from the average intermolecular force in the solid state.

The essential difference between the two states is not the
magnitude of the intermolecular force, under which the mol-
ecule vibrates, but the amplitude of motion.

To derive an expression for the coefficient of viscosity
of simple liquids in the neighborhood of the freezing point,
Andrade appealed to the following physical pictures:

a. There is for a liquid molecule, a vibration about
a slowly displaced equilibrium position, with a frequency
which is the same as that of the molecule in the solid state:

b. At every extreme libration, the molecules of one
layer come into contact with those of an adjacent layer.
At this contact they will, in general, enter into a tempo-
rary union, the duration of which does not exceed the very
brief time necessary for the molecules to acquire a common
velocity of translation. This sharing of velocities between

molecules in adjacent layers introduces viscous forces.

18

Andrade then postulated that the rate of transfer of
momentum between contiguous layers in laminar flow was pro-
portional to the difference in velocity between them, hence

4 l dv dv

___ym_°r

35 0'2 dz Wk dz

 

or
4 Vfin
V1 = —— (11)
3 6’
where 0' is the average distance between the centers of
molecules, and ‘Y , the frequency of vibration of the mole-

cules about their slowly diffusing equilibrium.position.

Using Lindemann's theory of melting, 1f can be written as

2
’Y = K (Tm / szfi‘ (12)

012

where K is about 3.1 x 1 , which gives for the viscosity

coefficient the relation

-4

q = 5.7 x 10 (Armfi v-% «13)

where A.is the atomic weight, V, the volume of a gram-atom
at the melting point Tm, and V = Hf .
By the assumptions made in its deduction, formula (13)
can only be expected to apply where
(1) the molecules are very simple and symmetrical
and do not themselves consist of a vibrating
system of atoms, that is, to monomolecular
liquids;

(2) the atomic arrangement is not much influenced

19

by melting, that is, to substances that have
a simple cubic structure in the solid crystal
and which do not experience a large change of
volume on melting.
To account for the observed temperature behavior of
viscosity, Andrade took into account the effect of eXpansion
on the average atomic distances and the effect on the poten-

tial energy, and derived

where v is the specific volume, and B and 0, two arbitrary

constants.

In so far as the above account is a fair summary of the
status of the theory of liquid viscosity, one is led to the
conclusion that, at present, there is no statistical theory
of transport processes in liquids Which is at the same time
absolute, practically useful, and free from mathematical
uncertainty. On the other hand, while the elementary theories
give the temperature effect as observed, the various inter-
pretations given to this effect can hardly be reconciled,
and it is difficult to decide between them. However, many
of these theories have also the feature that the viscosity
coefficient varies as the square root of the isotOpic mass,
'when the other factors are nearly constant as they certainly
must be for isotopes of the same element at higher temper-

atures. Hence measurement of the viscosity of separated

isotopes would further provide a critical check on their

adequacy.

2O

21

THEORY OF THE EXPERIMENT

The damping of a slowly oscillating Sphere containing
a viscous fluid can be treated as a problem in the laminar
flow of incompressible fluids. A very brief summary of the
method, originally treated by Helmholtz26, later modified
by Verschaffelt41, Andradel, and others, will be given.
The general equation of motion for a viscous fluid may
be written in vector notation as
3??
3t:

where 3'is the velocity, G the body forces (gravitational

 

- 1
+ RV? = G -—f-Vp - G’vax'x'x’ (15)

force in the present case), 5> the density, p the pressure,
and ’Y 5 V‘/? the kinematic viscosity, where V\ is the
viscosity coefficient.

We make the following assumptions:

1. the liquid moves in concentric Spherical shells
about the axis of rotation Z, each infinitesimally-thick
shell behaving as if it were solid;

2. the components of velocity in the x-Y plane are

of the form:

vX = -y VV(r,t) vy = x 1V(r,t)

where r is the distance from the center of the Sphere.
Furthermore assume
Y’(r,t) = ¢kr) e“:
where 0‘ E - /3 + if

22

with /3 E (S/T and T '=' 277/T

where 6 is the logarithmic decrement and T is the period
of oscillation of the sphere filled with the sample in its
molten state.

Then equation (15) reduces to

2

d ¢ 4 d¢

+— = b2¢ (16)
dr2 r dr

 

 

‘where b a afl/fi”.

The solution of this equation is
(p (r) = r":5 [Akbr - 1) ebr + Btbr + 1) e'br] (17)

where A and B are constants to be determined by the boundary
conditions.
On the other hand, the equation of motion of the oscil-

lating sphere is given by

c129
I 2 =-M6-P-P' \18)
dt

 

where 9 is the angular displacement, M the restoring force
of the SUSpension per unit angle, P the viscous force due
to the sample in its molten state, and P' the external
damping, due mainly to the internal friction of the suSpen-
sion.wires.
On the further assumptions that:

l. the external damping is responsible for only
a small contribution to the total damping;

2. there is no slip between the liquid and the

container wall, in accordance with all modern eXperiments;

equations (17)

x2 - q)V\ -

where q

w

and I

and (18) combine to give:

_,_ l 316 T2
WIT/T)2 aRvk2 - m -' +1 = 0 £19)

gR-l

 

(gR - 1)2 + \hRJ2

2
1 _ 6/417’ + (5/3277'2, neglecting higher terms,

(Trf/TV1)% . a,
(fl’f/Tvlfi‘ . u + 5/417' + 62/327”),

= the period of oscillation of the spherv
filled with the sample in its solid state,

= the inner radius of the Sphere,

= the moment of inertia of the suspended

system.

The viscosity coefficient is then approximated Very

closely by

YL.
where /fifi

 

1T 2
4 (2 2 —L{1 - (1 70%} «20)
’ Q)

T
3(2-q)1<§[’1‘ ]
2“1

2 1r2a25> R5 T o

N

 

the measured quantities being I, R, To’ T, £5, and‘? .

24

APPARATUS AND METHOD

.A. Description of the Apparatus

1. Oscillating System

The principal part of the apparatus is the oscillating
system shown in Figure 1. The suspension head consisted of
a brass cylinder A fitting into a cylindrical brass block
B rigidly fastened to a levelling table C, the three level-
ling screws being of the type used in the standard transit
theodolite. Two rubber O-rings were used for sealing the
cylinder A to the cylindrical block B for'vacuum tightness
while still allowing relative rotation. The suspended sys-
tem was set in oscillation by turning the lever L through
a small angle. In order to keep this angle within suitable
limits, a stop 8, on which was attached a vernier scale with
one-tenth degree least count, was mounted on the block 8.
The stop 8 could be adjusted to bring the suspended system
to any desired equilibrium position.

While the suspension head was being machined in the
lathe, the end carrying the lever L was faced so that the
circular table D to which the lever L was attached was accu-
rately perpendicular to its axis. Thus adjustment of the
circular table D to be truly horizontal, by means of a pre-
cision Wyler level with sensitivity 0.013 m/m per 1000 m/m,

automatically set the suspension head with its axis vertical.

25

 

 

 

 

 

 

 

 

 

Scale: M!»
L
fig

A
/
, C 1 C

WALL J
” H H V , _
A ’ ____.-To dlifusmn pump.
A F

 

 

 

 

 

 

 

 

 

 

 

 

 

{rIIIIIlII

 

 

 

7! 111111} [(1111 n
‘ i

r

 

 

  

fl

2

-Il'

ll

 

 

 

FIGURE I
GENERAL VIEW OF APPARATUS

26

.A threaded hole was chased into the other end of the suSpen-
sion, to which a special clamp holding the suSpension wires
could be screwed. When the suspension wire clamp was in
place, the two wires which are held fixed at opposite ends
of a diameter, were to be equidistant from the axis of the
suSpension head. To ensure this, each half of the clamp shown
in Figure 2, was machined separately from a length of cylin-
drical brass stock. Two holes of diameter slightly less
than 0.0048 inch were bored in the brass stock, the holes
being equidistant from the axis of the cylindrical brass
stock, which was then milled to obtain exactly a half-cylin-
der. In this way, it was possible to obtain two perfectly-

matched halves for the cylindrical plud It may be mentioned

that the wires were only clamped tightly along a length of
one-eighth inch, thus insuring that they were held rigidly
up to the point where they leave the clamp. The necessity
for clamping a suspension wire correctly has been emphasized
by Beardené.

The bifilar suspension was of 40 s.w.g. tungsten wire*.
The virtues of tungsten have been stressed by Heyl and
Chrzanowski27. To obtain pure rotation of the Sphere, it
is critical that the center of mass of the suspended system

be on the truly vertical axis of the suspension head.

 

* The tungsten wire was obtained from the General Electric
Company through the fine cooperation of Hr. Dallas T. Hurd
of the Advanced Deve10pment Engineering Section. It was
kept straight during shipping, in length of 2.5 meters,
through special arrangement with Mr. Hurd.

IIN.

     

UPPER CLAM P

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L : MIRROR
I

,— _. L.

1 j LOWER CLAMP ——1
I

up =‘
r =
q a.

 

SUSPENSION WIRE CLAMPS
FIGURE 2

28

To ensure this, the bifilar suspension was made of a single
'wire looped around a pulley. The half loop of the wire
carried a small wheel W provided with a fine circumferential
groove, the wheel itself being part of the lower clamp shown
in Figure 2. This clamp was very similar to the one described
above. The wheel was added to allow the suspended system
to settle with its center of mass in the vertical axis of
the suSpenSion head, and to equalize the tension in the
bifilar suspension. After the suspension was in position,
and the center of mass of the whole suspended system had
settled, the two wires were clamped tightly at the lower end
by screwing the two half-cylinders together. In this way,
the suspension wires were held rigidly by the lower clamp
jaws without the wheel having to bear any weight.

The inertia rod I was connected to the lower clamp through
a brass shaft E. The sphere* itself was connected to this
shaft through a long thindwalled stainless steel tubing G.
Details of the brass shaft E and its connection to the stain-

less steel tubing G are shown in Figure 3.

2. Vacuum.Enclosure

The vacuum enclosure consisted of a brass tube H, fur-
nished with two glass windows, one Shown at P, the other
parallel to the plane of the paper, a cylindrical brass box

Q housing the inertia rod 1, and another brass tube, the

 

* The term "sphere" is used throughout this thesis to denote
the sample holder.

| IN.

JL

I To lower wire clamp

 

\i

 

\

 

 

 

“

 

 

 

 

 

 

 

 

 

,5,ij In rflo , ,0
,“l /’I R 0d

 

 

 

 

 

 

To sample holder

FIGURE 3
DETAILS OF CONNECTING SHAFT E

30

lower end of which carried a stainless steel tube, to which
was soldered a c0pper cylinder. The stainless steel tube
minimized heat conduction to the upper part of the apparatus.
The capper cylinder, machined from a length of capper stock,
had one-half inch thick wall and base. The upper brass tube
H was actually two tubes connected by a flexible cOpper
bellows J. This arrangement allowed adjustment of the lower
part of the vacuum enclosure so that the sphere hung cen-
trally in the cepper cylinder. Evacuation was made through
the side tube V. The vacuum was maintained at 0.01 micron

or less during an experiment.

3. Furnace and Temperature Control

For maintaining the Sphere at constant temperature, a
non-inductively wound electric furnace F, deep enough to
heat the whole length.ofthe copper cylinder, was used. An
iron shield was placed between the c0pper cylinder and the
wall of the furnace to minimize Spurious damping of the:
oscillating liquid metal due to stray magnetic field from
the heating coils. A.water-cooling jacket U, separated from
the top of the furnace by a thick insulating board, was used
to keep the inertia rod and the suspension wires from getting
warm.

The temperature of the furnace as indicated by a c0pper-
constantan thermocouple T Shown in Figure 1 was measured on
a 10-mv Speedomax recorder, and controlled by a Series 60,

2-Action Duration Adjusting Type, Leeds and Northrup con-

31

trol unit. To maintain the thermocouple voltage within the
range of the recorder at high temperature, a simple poten-
tiometer was constructed. The settings on the potentiometer
were calibrated against a K93 Leeds and Northrup Universal
Potentiometer (stated accuracy 0.02% of voltage). The cone
trol unit held the temperature at any setting very constant,
as indicated by a perfect straight line on the recorder.
'The whole apparatus, except the furnace, was supported
by the three levelling screws R, resting on small brass cups
which were placed on a thick wooden platform. The wooden
platform was in turn bolted to the platform of a scaffold,
a welded construction of 2 x 2 inches steel angles. The
scaffold, shown in Figure 4, measuring nine feet, was bolted

solidly to the wall of the building.

4. Sample Holder

The sample holder or Sphere was made in two halves, from
lengths of stainless steel type 446 1 1/4-inch rod. Each
half of the Sphere was first roughed out using a.one-inch
ball end mill, then hand ground to a hemisphere with a one-
inch steel ball and jeweler's rouge to obtain a satinpsmooth
finish. The upper hemisphere was provided with a.neck through
which the molten lithium.could be poured. The two hemispheres
were fastened together inside a tight-fitting cylindrical
sleeve, and hard silver soldered as shown in Figure 5. It
‘was critical that the two hemispheres did not overlap where
they joined.

Aumatching cap was made for each sphere using gold 0-ring

aspen Hausmafiaoexm mo 3oH> Hmumcoo

: mesmam

 

l INCH

7
/ 45° BEVEL .039 x .039 IN.

FOR GOLD O- RING

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
   

 

 

 

 

 

 

 

 

 

 

 

 

 

J/ k / I
/
é ? g /
é / / /
dl/ / /
é / /
9
5 3
/ / GOLD O'RING
j S
VJ» V .4 w
2/ ed?
/
x
/
/
x
/
\ \/
\\\
HARD \
SILVER SOLDER I
T6- IN. THICK SLEEVE

FIGURE 5
LI SAMPLE HOLDER

MATERIAL: STAINLESS STEEL TYPE 446

34

for vacuum.eeal. A stainless steel type 446 vane, in the
form.of a cross, was used to prevent rotation of the liquid
in the neck of the sphere during an experiment.

The inner radius was calculated from the volume of dis-
tilled water needed to fill the sphere, the volume of the
neck being known accurately from measurements with a depth
‘micrometer. The calibration was done using a certified 10-m1
burette. The error on volume measurement was estimated to
be 1 part in 1000, with a resulting error in the radius of

1 part in 3000*. The data are shown in Appendix A.

B. thhods of Measurement

1. Preparation of the Samples

The apparatus shown in Figure 6 was used for transferring
lithium to the Sphere without contamination#. A lump of
lithium, larger than necessary to fill the sphere, was put
in the stainless steel funnel with the freshly cut surface

down. The stem of the funnel, slightly flared at the end

 

* It was to the credit of the machinist, Mr. Solonika that
when the first sphere made was cut in half to check, the
line where the halves joined could hardly be detected visup
ally. In matter of fact, the fitting was so good that only
careful inspection showed that the Sphere was constructed
from two hemiSpheres. The Sphericity of the cavity, as
checked with a one-inch steel ball covered with jeweler's
rouge, was judged excellent.

# This'work‘was conducted at the Oak Ridge National Labo-
ratory. The author assisted in the transferring of lithiumpéz
the filling with lithium~7 was done by Mr. E. B. Olszewski,
chemist at the laboratory.

35

TO VACUUM PUMP

 
     

 

‘ III/IIIIII\\I:"II
.‘\\§

     

 

 

 

I ALUMINUM
O-RING
MATERIAL'
STAINLESS STEEL '
FIGURE 6
I
INDUCTION

 
   

COILS

 

 

000000

0
O
O
O
O

 

 

 

 

I
APPARATUS FOR TRANSFERRING LITHIUM TO THE
SAMPLE HOLDER

36

to prevent overflow, extended into the neck of the sphere
to an appropriate depth. The chamber was then evacuated
and the lithium melted by induction heating. The oxide-
nitride coating on the lithium rested intact on the funnel
and pure lithium ran down into the sphere. The lithium-
filled Sphere was allowed to cool under vacuum, then the
chamber flooded with argon gas before it was opened. Upon
cooling, the lithium in the stem of the funnel shrank and
pulled up, breaking away from the solidified lithium inside
the sphere. A jet of argon gas was directed on the Open-
ning of the Sphere while the vane was inserted and the cap
attached.

One of the filled spheres was immersed in mineral oil
and the lithium.was melted out to check.visually whether
any contamination had been introduced by this method of
filling. Hardly any oxide or nitride of lithium was detected,
the molten lithium floating to the top of the oil with a
Silvery brilliance.

The amount of lithium.needed for each isotope to fill
the sphere while still leaving ample room for thenmal expan-
sion in the temperature range contemplated was calculated,
using the densities given by Snyder and Montgomeryzg. As a
check the completed sphere was weighed. If the proper amount
of lithium was not present, the transferring process was
repeated. TWO trials were necessary for lithium-6.

The lithium samples used were purchased from the Stable

Isotopes Division of the Oak Ridge National Laboratory.

37

The following analyses were given by the supplier along
with these clarifications:

1. The Spectrographic results reported are semi-
quantitative estimates and should not be interpreted or
construed to be precise quantitative determinations;

2. [(-No Spectrum line visible. Probably absent,
definitely less than value given;

3. (T -Present, but less than value given.

Lithiumpé Lot No. SS 5kb)

 

Isotopic Analysis:
Lithium-6, 99.3 t 0.2%; Lithiump7, 0.7 t.0.1%, (atomic percent)

Spectrographic Analysis (element and weight percent):

A1 (.01T Fe .05 Pb (.01
Ba .01 K (.01 Sn <.o1
Be (.001 Mg .01 Sr .01
Ca .25 1&1 <.01 v (.01
Cr (.OlFT Na .02 Zn (.25
Cu .02 Ni (.01

The lithium~6 sample with the above analysis was vacuum
distilled for further purification of the metallic lithium,
at the suggestion of Dr. P. S. Baker of ORNL, before being
used to fill the Sphere. The analysis of the final product

is not available from ORNL at this time. (See Appendix G).

Lithium~7 Lot No. SS 7kb)

IsotOpic Analysis:

Lithiumh7, 99.99%; Lithiumpo, 0.01%, (atomic percent)
Spectrograpnic Analysis (element and weight percent)

Al (.05 K .01 Rb (.01

,. s,
*.

Ba <.02 Mg <.01T Si (.05

Ca .02 'Mn (.02 Sn (.05
Cr (.05 Mo (.05 Sr (.2
Cu (.OlT Na .01 V (.02
Fe (. 05 Ni (.05 ‘

2. Determination of the Moment of Inertia of the
Suspended System

The moment of inertia of the suSpended system.I was made
up of a fixed part Io due to the sphere, the inertia rod,
and various connecting rods, and a variable part In’ due to
the two movable disks K in position (In which can be calcu-
lated. This arrangement, Shown in Figure 7, was adopted so
that the total moment of inertia I could be changed while
keeping the load constant. Thus, the tension in the sus-
pension wires, and hence the restoring force, would remain
constant. It may be mentioned that this load was kept the
same for the whole period during which the eXperiments were
conducted, a Sphere taken off the suspended system being
immediately replaced by a dummy.

To ensure uniformity in density and true geometrical
form, the disks K Were cut from the same stock of brass and
carefully turned in the lathe. The inertia rod was mounted
on the suspended system in the manner shown in Figure 3,
'with a close sliding fit between the disks and the inertia
rod.

.A Sheffield Master Gagemakers Comparator, sensitive to
variation of twenty millionth inch (20 x 10'6 inch), and a

set of Johansson Gage Blocks were used to check the uni-

I INCH
MOVABLE DISCS

    

I
II I
5.5I23 7

 

FIGURE 7

ADJUSTABLE MOMENT OF INERTIA SYSTEM

formity of the disks and the inertia rod. The maximum
variation in the thickness of either disk was 1 part in
1500, and only 1 part in 6000 for the diameter. The mean
thickness of the two disks did not differ by more than 1
part in 25,000, and the mean diameter, by more than 1 part
in 40,000. The excellent match in geometrical dimensions
was exceeded by that for the weights of the two disks, which
differed only by 1 part in 100,000. The results are tabu-
lated in Appendix B. The inertia rod showed a maximum
variation of 1 part in 4,000.

A Special gauge, Shown in Figure 8, was made for setting
the disks on the inertia rod to obtain different values for
the moment of inertia In. It consisted of a brass cylinder
of diameter slightly larger than.that of the inertia rod,
and two "standard” cylindrical blocks Y and Z. The dimenp
Sions shown on Figure 8, measured with the Sheffield compa-
rator, were chosen to obtain large variation in the period
of oscillation. Three different moments In could be set
with the gauge. A groove was cut in the cylinder X to give
an outlet for the trapped air when the gauge was pushed over
the inertia rod to displace the disks K.

The moment of inertia I is related to the period of
oscillation by
'1‘n = 21T(I/M)'i‘
where I = Io + In, and M being the restoring couple.
Although in principle M may be calculated from a knowledge

of dimensions and elastic constant, it is simpler and more

 

I/////////////////x///W

 

 

 

 

 

 

I’///////////////////W

 

3. 9949

I

 

 

‘ 2.54-—____a4

 

 

 

0.40

 

 

 

(f

 

I

Y

 

 

 

 

 

GAUGES FOR

W... M

\ I

LOO I96

 

 

/ I

 

m

2

 

 

 

 

 

FIGURE 8

SETTING

MOMENT 0F INERTIA

42

accurate to eliminate it by measuring the period for different
moments of inertia I. Thus, any two values of n enable Io,
hence I to be computed, that is,

2 2

 

I =
0 2 2
T2 ‘ T1

Furthermore, the moment of inertia In is given by:
= I + I + (m + m ) d2
In 12' 22' 1 2 n

where 110,. and 122, are the moment of inertia about the

vertical axis passing through the center of mass of the two

disks rSSpectively; d the distance between the center of

n,

mass of each disk and the center of the inertia rod; and m

 

 

l
and m2, tRe mass of the two disks. Hence,
dng - diTfi
Io ‘ ' (112' + 122') + (”1 + “2) T2 s2 ‘21)
- 1
2 l
and the total moment of inertia I is given by the measured
quantities n1, m2, dn’ d1, d2, T1' and T2 through:
2.2 2m2
I = I + = (n + m ) 2 + dzfl dlLZ I22)
0 In ‘1 2 dn “2 n9
*2 ' li

Three different values of In were used, giving two inde-
pendent values of 10' The period of oscillation varied from
about 8 seconds to about 15 seconds in the different cases.

The results will be given in a later section of the thesis.

3. Measurement of Amplitude and Period of Oscillation

 

Since the logarithmic decrement was not very large, of

the order of 500 x 10-6, it would have been cumbersome to

43

record the amplitude photographically. Instead, the ampli-
tude decrease was detected electrically using a Kodak Ektron
cell, a lead-sulfide photoconductor, measuring 0.02 x 2 mm.
A beam.of light of suitable strength from a zirconium-arc
lamp, admitted at window P of Figure 1, was reflected by a
front-Silvered mirror mounted on the lower wire clamp of the
suspended system, and emerged horizontally from.a second
window, the plane of which was perpendicular to that of the
first. The oscillating system was adjusted so that at rest
the mirror was at 45° to both light beams.

The Kodak Ektron detector, provided with a shield against
stray illumination, was mounted on a Spindler and Hoyer pre-
cision cross carriage with micrometer screw permitting a
lateral movement of 25 millimeters. The screw allowed pre-
cise positioning of the detector to 0.01 millimeter. The
screw carriage was in turn mounted on a sliding carriage
provided with a vernier scale with a least count of 0.01
millimeter. The sliding carriage was used for coarse dis-
placement of the detector in excess of 25 millimeters, and
the screw carriage for fine adjustment. The sliding car-
riage was initially positioned so that the light beam with
the mirror at rest lay along the perpendicular bisector of
the carriage axis.

The electrical signal from the Ektron detector was re-
corded on a Millivac microvolt recorder, model MR-65A. The
circuit used is shown diagrammatically in Figure 9. For

any one run, the detector was initially positioned so that

44

_ \

* MIRROR

 

 

Millivac
M‘Icrovolt
Recorder

 

 

 

 

I20 K I.5 volts
mercury battery

SIGNAL FROM OSCILLATION COUNTER—7
,/

, . f/I »
AWL. (Sale FuLpLgufei‘e PIE

I l I I I

I: ‘ IE

SIGNAL FROM EKTRON DETECTOR

SAMPLE STRIP CHART RECORD

FIGURE 9
CIRCUIT FOR AMPLITUDE MEASUREMENT

45

the light beam Slightly swung past it at the extremity of

a swing, the signal then appearing as a double pulse. As
the amplitude decreased the overshoot of the light beam
also decreased, bringing the two peaks of the signal closer
to each other. Finally, when the light beam just reached
the detector at the extremity of its swing, the signal be-
came a single pulse. A sample strip chart record is shown
in Figure 9.

The same light source used for recording the amplitude
decrease was also used for counting the oscillations. A
cadmium-selenide photoconductive cell, Clairex-3, placed as
shown in Figure 4, was used in connection.with a thyratron
phase-control circuit to activate a counter. The circuit
used is shown in Figure 10. The signal from this circuit
was also recorded on the'Millivac recorder at the same time
that the amplitude decrease was being measured, and thus
acted as a time marker. The Clairex cell was placed at the
center of and immediately behind the sliding carriage to give
two counts for each complete oscillation of the system. The
period of oscillation was measured by timing 500 counts
(250 complete oscillations) with a st0p watch reading to
0.01 second. The watch was calibrated repeatedly for the
same amount of time taken to measure 500 counts using the
time signal broadcasted by the National Bureau of Standards.
In this way the period was measured with an absolute accu-
racy of one millisecond. The period was measured at least

three times during an eXperiment, and it was found to be

 

 

 

 

I25 v.
..... “I I-..___ 2.___.._____-
I l
I i "5 N030?
l
.. .L
"3—: GHEPO '7I’L'r5
o 0- I00 Ins

 

 

220 330 H RELAY 20.
n. a
21! 2,” I0 ms

 

3m

 

_-—_—— ——-—————-—— —— —‘
——————_—-_—-——_———- _ ————-———-

L_.

 

 

 

 

 

 

 

   
   
   

RELAY
CONTACS

22K
5!.

22‘ OJ '1‘

 

 

FIGURE I0
THYRATRON PHASE - CONTROL CIRCUIT

CL-3 L I
DETECTORI I 2:
I25 a JI-‘J '
'L—J- - r _ ------ 3 L————-——-I:::::-:: ZZZ-.1: _ j

l
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
l
I
I
I
I
I
I
I

"Os
8 VI

D C
COUNTER

 

 

47

constant to within 1' 0.5 millisecond.

For any one experiment, the Sphere was set oscillating
about the position of equilibrium by turning the lever arm
1 1/2 degree away from the position of equilibrium.and back,
and the counting circuit energized. The screw carriage was
then moved on the sliding carriage to obtain a slight over-
shoot of the light beam past the Ektron detector, and the
Signal was recorded. When the signal changed as discussed
in the above, the detector was advanced a fixed amount using
the micrometer screw. This process was repeated fifteen
times for each run. The lateral diSplacement of the Ektron
cell for each advance was 0.50 millimeter during the lithium
runs, and 0.20 millimeter during the runs to find the resi-
dual damping with the lithium solid. Each time the Sphere
is set into oscillation from rest, at least three runs were
taken over a period of about four hours, giving three values
for the logarithmic decrement A.

At the end of each run, the position of equilibrium was
detenmined by measuring the left and right excursions of the
beam, thus allowing the computation of the amplitude of
swing. The equilibrium position was found in this way to
stay constant to within 0.05 millimeter.

Both the temperature of the furnace and the pressure in
the vacuum enclosure were monitored during the entire period
taken for each run. The pressure did not exceed 0.01 micron,
ensuring that the residual damping due to air was negli-

gible. The temperature recorded on the Speedomax remained

48

constant as discussed earlier.

4. Temperature Measurement of the Lithiumpfilled
Sphere

After the eXperiments with lithiumpo were completed, a
c0pper-constantan thermocouple, insulated by a thin Sheet
of mica, was directly attached to the outside of the sphere
on the horizontal equator. The Sphere temperature thus meas—
ured was determined as a function of the furnace setting as
indicated by the thermocouple T of Figure 1. During this
calibration, the conditions under which the SXperiments were
run were strictly observed. The copper-constantan thermo-
couple attached to the sphere were taken from the same spools
used for making the thermocouple T. The temperature meas-
urement was carried out using a Leeds and Northrup Univer-
sal potentiometer, type K-3.

The Sphere itself was found to reach an equilibrium
temperature about ten hours after the furnace was started
at room temperature, and about six hours after a change in
controller setting. For the eXperimentS, the apparatus was
generally left for twenty-four hours at a particular tem-

perature before the decrement was measured.

C. Calculation of the Viscosity Coefficient

 

The viscosity coefficient V\ was calculated by the

relation
2 2

_ a R Trf i 2
VI — T {- - (1 -/M) } I23)

4 I2 - q)2

 

49

the various symbols having been defined earlier. The quan-
tities determined experimentally were 8 , j) , R, I, To'
and T. The temperature dependence of these quantities will

be now discussed.

1. Logarithmic Decrement 6

The plot of the logarithm of the amplitude as a function
of the cumulative number of oscillations at any given temp
perature yielded a straight line, the slope of which was
the logarithmic decrement 5. The logarithmic decrement was
Icalculated by the least-squares method, the variables being

the logarithm of the amplitude and the number of oscillations.

2. Dens ity f

Precise determinations of the density of liquid natural
lithium apparently have not been reported, let alone those
of the separated isotopes. However the densities of the
solid isotopes have been measured by Snyder and Mentgomerysg.
The coefficients of thermal expansion in the solid and liquid
state, as well as the volume change upon melting of the above
isotOpes, are still lacking, but they Should vary little
with isotOpic mass. Consequently, the values given by
Bernini and Cantoni7 for the coefficients of thermal expan-
sion, and that of Losana34 for the volume change upon melting

for natural lithium have been adopted.

We may then write:

Li6
L16 i) °m

 

th 1 + 174 x 10'6 (T - 180.4) + 106 x 10'"9 (T - 180.4)2

.7
f2.

L17 =
f t -6 -9 2
1 + 174 x 10 (T - 180.7) + 106 x 10 (T - 180.7)

.6 7
Li Li . . .
where Pom and Pom are the dens ities of liquid lithium-6
and lithium-7 respectively at 180 C-deg., and
.6 0.460 [1 + 153.5 x 10"6 x 20 + 92 x 10'9 12°F]

 

L1
Pm“ 1.0157 [1 + 153.5 x 10"5 x 130 + 92 x 10'9 (180)2J
-5 -9 2
1.17 0.537 [1 + 153.5 x 10 x 20 + 92 x 10 120) ]

 

13mm 1.0157 [1 + 153.5 x 10"6 x 180 + 92 x 10"9 (180)21

3. Inner Radius of the Sphere R
The change in the inner radius of the sphere with temp
perature was calculated, using as the linear coefficient of

thermal expansion for stainless steel, type 446, the value
(K

9.9 x 10‘6 / C-deg.

4. Mbment of Inertia I

No correction was made for the change in the moment of
inertia I due to the expansion of the sphere, Since this
change amounted to only 5 parts in 10,000 when the temp

perature rose from 20 to 300 C-deg.

5. Method of Calculation

The actual calculation for the viscosity coefficient
‘was carried out by programming equation (23) for MISTIC,
a high speed digital computer. The degree of approxi-
mation used and the results obtained will be given in the

following section of the thesis.

EXPERIMENTAL RESULTS

.A. Verification of Method

 

The earlier history of the oscillating-Sphere method
for the determination of viscosity would give one little
confidence in it. Later investigators have said that it
is reliable, but certain details in their presentation lead
to the suspicion that the precision may not be as high as
claimed. Hence it was deemed essential to verify the va-
lidity of the method with use of some standard substance.
Water seems suitable, since its viscosity is extremely well
established, and is of the same order of magnitude as that

expected for the lithium isotOpes.

51

The sample of water was maintained at a fixed.temperature

by a rudimentary system consisting of a water bath in Which
was immersed a c0pper tube helix carrying tap water as a

refrigerant. The experiment was carried out in accordance

with the procedure described earlier. The results are Shown

in Table I. The values found eXperimentally are a few per
cent lower than the accepted one324. The SXplanation may
lie in the fact that the temperature given in the table is
the temperature of the copper jacket forming the bottom part
of the chamber, whereas the temperature of the Sphere may
be a little larger, say a degree or two. It is possible to

get better agreement by adjusting the temperature, but such

procedure seems unwarranted. The chief point to be estab—

52

lished was the absence of gross effects that might induce

error.

B. Effect of Temperature on Viscosity of Lithium-6

For the study of the effect of temperature on viscosity,
only the lithium-6 sample proved to be useful. It was h0ped
to determine the effect of temperature with both isotopes,
but with lithium-7 difficulties in the sealing of the Sphere
prevented an extensive study.

To make sure that the Spherical portion of the sample
holder was completely filled, the amount of lithium~6 that
it contained was determined by weighing the sample holder
before and after pouring the sample. The weight was 4.074
grams, corresponding to a volume large enough to fill the
Spherical cavity and a portion of the neck, while leaving
ample room for thermal expansion over the temperature range
used.

The moment of inertia of the suSpended system, with the
lithiump6 in the solid state, was then.measured, with the
results shown in Table II. The error is believed not to
exceed 1 part in 6,000.

The residual damping, with the sample in the solid state,
was measured according to the procedure outlined earlier,
with the results shown in Table III. It is satisfying that
the decrement is very small, and that it does not increase
with temperature, a finding which shows that any damping due

to the magnetic field of the furnace current is negligible.

Table I

Viscosity of Water

 

 

 

 

 

 

 

 

Temp. , V\Iexpt.) V\\8td.) per cent

00 . mp. deviation

16.6 10.48 10.94 - 4.2

17.1 10.42 10.80 - 3.5

18.6 9.87 10.40 - 5.1
Table II

Moment of Inertia for Li-6 Sphere

 

 

 

 

 

 

T10 = 8.795 sec. 8% = 5.13204 cm?
T20 = 15.048 " ' 83 = 27.71601 "
T30 = 12.787 " 6% = 18.18255 "
With 81 and d2 I = 4,443.79 gmpcmz
with 81 and d3 I = 4,443.75 "
with 82 and d3 I = 4,444.01 "
Average value I = 4,443.86 "

Table III

Residual Damping for Li-6 Sphere

 

 

 

Temp. Residual
°C logarithmic decrement ‘So
22 41.79 x 10"6
26 41.90 "
152 41.21 "
178 38.63 "

 

 

 

53

54

The value of the residual logarithmic decrement adopted was
the one obtained at the temperature closest to the melting
point of lithium96.

The viscosity was then determined by observing the vis-
cous damping at various temperatures. The first measurement
was made at the highest temperature contemplated (266.8 0C).
At the completion of this measurement, the temperature was
decreased in steps until the lowest temperature used k188.3
0C) was reached, then brought up stepwise to the maximum
again, and finally decreased once more to the lowest temp
perature. No effect due to a possible thermal hysteresis
was noticed. The detailed sequence of measurements is given
in Appendix C. At each temperature setting, the experiments
were run at least once on each of two consecutive days.
During each day of data-taking, three measurements of the
logarithmic decrement were made as the amplitude of swing
decreased. The decrements measured during any one day were
found in most cases to be constant within.1/2 per cent, the
largest deviation between extreme values never exceeding 1
per cent. The decrements obtained for each day were aver-
aged, and the resulting value was used in the calculation
after the residual decrement had been subtracted from it.

The temperature of the Sphere could not be measured
directly while a determination of damping was being carried
out. It was determined indirectly by measuring the furnace
temperature, which.was found in a subsidiary investigation

to be a linear function of Sphere temperature, as Shown in

55

9
I

Figure 11. The Sphere temperature was measured according
to the procedure described earlier. The effect of temper-
ature on the radius R, the density, and the periods TO and
T is given in Appendix C.

It was little trouble to make successive approximations
to an exceedingly high accuracy in the calculation of the
viscosity coefficient with‘MISTIC. The calculation was
continued until the relative difference between successive
results was less than 10'7. (For example, for lithiumpo
at 188.3 C-deg., the final approximation was taken as
4.389 740 32 mp., the preceding one being 4.389 740 28 mp.;
physically, the value is meaningful only to three or four
significant figures).

The values found for the viscosity coefficient of lith-
ium-6 as a function of temperature are summarized in Table
IV and Shown (together with one value for lithium-7) in
Figure 12. The detailed results are tabulated in Appendixes

C, D, and E.

C. Effect of Isotopic Mass on Viscosity of Lithium

 

To study the effect of isotopic mass on viscosity, the
value for the viscosity coefficient of lithium~7 was meas-
ured at a given temperature and compared with that for
lithium~6 at the same temperature. A defect in the construc-
tion of the lithium-7 sphere resulted in a leak which devel-
Oped after the Sphere had been held at high temperature for

a time, and consequently the comparison can be made at only

 

 

56

one temperature.

The weight of the sample was 4.660 grams, correSponding
to the proper volume of material. Because of the destruc-
tion of the sample, the moment of inertia and the residual
damping could not be determined directly. However, it
appears that use of the corresponding values for the lithi-
um~6 sphere can not introduce appreciable error, in as much
as the sample holders for lithium-6 and for lithium-7 are
nearly identical, and the difference in mass and in moment
of inertia for the actual isotope samples are negligible
in comparison with the corresponding values for the whole
system. The residual damping, therefore, should be very
nearly the same, since the tension on the suspension wires
and the geometry of the system are almost unchanged.

The viscosity coefficient of lithium-7 at 236.8 C-deg.
was found then to be

“Li-7 = 4.19 :t 0.08 mp.

'his value is also included in Figure 12.

0. Estimation of the Absolute Value for Natural Lithium

On the assumption that the viscosity coefficient for a
mixture of isotopes of the same element is a linear function
of the number composition, the absolute viscosity of natural
lithium (92.5% lithium-7, 7.5% lithium-6) at 236.8 C-deg.
is estimated to be

\Li-nat. = 4°16 mp.

Although the above assumption might be debatable in the

general case, it appears safe for a mixture consisting

predominantly of one component, such as natural lithium in.

Table IV

Viscosity of lithium~6

 

 

 

Temperature Densi y Viscosity Lo 103 T OK
188.3 461.5 0.44019 4.39 0.642 2.17
198.0 471.2 .43944 4.28 .631 2.12
217.4 490.6 .43792 4.00 .602 2.04
236.2 509.4 .43642 3.85 .585 1.96
236.8 510.0 .43637 3.83 .583 1.96
266.6 539.8 .43395 3.69 .567 1.85
266.8 540.0 .43390 3.67 .565 1.85

 

 

TEMP. (”c)

SPHERE

280

240

 

 

 

 

220 _
200 p ’
'80 1 L i 1
200 220 240 260 280 300

FURNACE TEMP. (°C)

FIGURE ll

SPHERE TEMPERATURE AS A FUNCTION OF
FURNACE TEMPERATURE

59

CON

 

N. 550.“.

8.. umakdmuezmk
ecu can cos

0. $.00. KL...

 

a 2...»:

00 ¢.OO_ .0 u—J

.a 2:22

mmahgmfim... no 20:023.; 4 m4 0! 3.1:... no >tmoom;

O!

 

p-525... x

c-6354 .

ALISOOSM

(dun)

DIS 31 ‘SS ION OF RESULTS

A. Verification of Method

The results with water give confidence that the method
is basically sound. The eXperiments reported, which were
performed at the beginning of the eXperimental study, were
not carried out carefully enough to allow a thorough eval-
uation of the linits of precision of the method. Neverthe-
less, it appears that the precision claimed by earlier
workers may be illusory, since the final value calculated
for viscosity is far more sensitive to inaccuracies in
Sphere diameter than had apparently been realized*. More-
over, tFe measurement of the temperature of the actual
sample appears to be more difficult than had been considered
by earlier investigators. With the present apparatus and
technique, it is likely that the absolute value obtained
for a viscosity coefficient can not be claimed to be accu-
rate to within less than 3-5 per cent. The relative accu-
racy is considerably better, probably within 2 per cent or

better.

B. Effect of Temperature on Viscosity
Previous experimental work for a variety of liquids has

established a law that the logarithm of the viscosity is a

 

‘3‘

* .
See Appendix L

61

linear function of the reciprocal of the absolute tempera-
ture. Moreover, some theories claim to give this sort of
dependence. The present data, when plotted in the form
log.qk against 1/T, shows a general agreement with this law,
as may be seen in Figure 13. The only serious deviation
occurs at 266.8 C-deg., which may be due to either the tem-
perature measurement or a slight increase in residual damp-
ing at high furnace current or to both. Preposed variants
of this law, with some dependence on specific volume, do
not give significantly different fits. The temperature range
is sufficiently limited that a severe test of the preposed
law is not possible.

The curve of Figure 13, plotted from Table IV, gives for
the absolute viscosity of lithium-6 at the melting point

(180.4 0C) the value

YL = 4.50 mp.

C. Effect of Isotopic fiass on Viscosity

 

According to the data previously cited, the ratio of the
viscosity of lithium-7 to that of lithium-6 at 236.8 C-deg.
is approximately (4.1910.08)/(3.83:0.02) = 1.09:0.02, or
when more exact figures are used to avoid rounding errors,
1.09310.025. The square root ratio of the atomic mass of
lithium-7 to that of lithium-6 is (7.018/6.017)% = 1.080.
Thus it appears that the eXperimental results are consistent
with the theoretical predictions discussed earlier. Data
at other temperatures are desirable in order to reduce the

uncertainty in the above figure, if indeed the dependence

62

of viscosity coefficient on isotOpic mass is independent of
temperature. Data on "isotOpic alloys" would be valuable,
but caution must be exercised in the interpretation of the
results, as the phenomena.for mixtures of isotOpes of.a
given average atomic weight are intrinsically more compli-
cated than would be phenomena for an isotOpically pure

substance of the same atomic weight.

D. Accuracy of the Estimated Absolute Values for the

Viscosity of Lithimm

 

The absolute value for the viscosity coefficient of
lithium-6, lithium-7, and lithium-natural are of theoretical
interest with respect to the verification of the molecular
theory of liquids, and of practical interest because of
potential applications of lithium in nuclear-reactor technol-
ogy. Consequently it is important to assess the accuracy
of the absolute values reported herein, and to search for
eXplanations of discrepancies with previous work.

The only work that need be considered is Andrade and
Dobbsz. Their value for lithiumpnatural at 236.8 C-deg. is
about 24 per cent higher than the present result. In order
to carry out the comparison on a more extensive scale, we
assumed that the viscosity coefficient varies as the square
root of isotopic mass, and calculated the viscosities for
lithiumpnatural from the results obtained for lithiump6.

It is very significant to note then that the per cent
deviation between Andrade and Dobbs results and the present

results is very constant over the whole range of temperature

63

used, as shown in Table V. The only exception occurs at
266.3 C-dcg., for the reasons given in a previous section.
This constancy strongly suggests that the difference in the
results may be due to a systematic error in Andrade and Dobbs
technique. A detailed examination of their paper led to the
following observations:

1. The sphericity of their sample holder, which
influenced the radius calculation, is very questionable,
especially after a close study of their calibration data for
the Sphere; we have demonstrated earlier the extreme sensi-
tivity of.the viscosity coefficient VL with respect to inac-
curacies in sphere diameter.

2. No data were given either for the logarithmic
decrements or for the residual damping, even though they
constitute one of the most significant results directly
obtained fron the exneriuents.

3. These authors claim that there exists no temper-
ature differential between the jacket and the sample holder,
this is very questionable, especially since we have found
a temperature difference of about 30 C-deg. in our work.
However this alone is not sufficient to exalain the differ-
ence in the results.

Even though we only claim that our absolute values are
accurate to within 5 per cent, and our relative values to
2 per cent or better, we have to conclude from the above
observations that Andrade and Dobbs results seem to be

significantly in error. This fact is of technological

interest.

Table V
Calculated Viscosity of Lithium-natural

 

 

 

Temp. w“ 3 .“(mp) per cent
°C mp Ban Andrade difference
188.3 4.39 4.71 5.85 24
198.0 4.28 4.60 5.71 24
217.4 4.00 4.30 5.40 25
236.8 3.83 4.11 5.12 25
266.8 3.67 3.94 4.73 20
180.4 4.50 4.33 6.02 25

 

 

 

 

 

 

* The viscosity coefficient of lithium-natural was calculated
as follows:

Tod. 1' ykw'[(1.080 x 0.925) + 0.075] = who x 1.074

0.6 5

 

 

Lithium- 6

 

 

 

 

'2' r
'43 oso
:: F
g L Melting
U
.9. I point
_ : I80.4°C
I
F l
e I
L- I
I
055 . 1 l l
|.80 I.90 2.00 2.!0 2.20 2.30
IDs/T (II’K)
266.8 236.8 2|7.4 l98.0 I883 l80.4
l l L 1 L 1
FIGURE I3
PLOT or (10311) AGAINST l03/T

log]Io

(’C)

66

APPENDIX A
INNEfl DIAMETER OF SPHERE

Volume of Spherical cavity V8 = Total volume of
water used Vt - Volume of neck Vn’
1
v8 = -—1103
6

Volume of neck (Cf. Figure 5):

vn = 1[(o.350)2 x 0.060] + -—1T—[(o.3125)2 x 0.752] in}
4 4
= 6.345 x 10"2 in3 = 1.04 cm:5

The base of the cylindrical neck joining the tOp of the

Spherical cavity is not exactly flat, the error introduced

by assuming that it is flat was computed to be 7.69 x 10"4

cm}, i.e., completely negligible.

 

 

 

Measurement No. Sphere No. 2 Sphere No. 3
(Lithium-6) (Lithium-7)
Vt cm:5 Vt cm3

1 9.88 9.91

2 9.88 9.92

3 9.89 9.91

4 9.88 9.91

5 9.88 9.90

6 9.87 9.91
Average Vt 9.88 9.91
Average Vs 8.84 8.37
Diameter D 2.5654 cm 2.5683 cm

 

 

 

 

 

Note:

APPENDIX B

unless otherwise indicated.

DB-ENSIONS 0F INERTIA DISKS

All dimensions given are measured in inch,

 

Disk No. 1

Disk No. 2

 

Thickness

Average

.5828 .5329 .5830
.5830 .5330 .5823
.5830 .5830 .5831
.5830 .5823 .5827

.58292

.5827 .5829 .5830
.5331 .5830 .5830
.5328 .5830 .5830
.5830 .5329 .5329

.58294

 

Diameter

Average

1.7718 1.7717
1.7718 1.7719
1.7717

1.77178

1.7719 1.7717
1.7717 1.7720
1.7718

1.77132

 

 

jeight

 

189.6274 grams

 

189.6290 grams

 

 

67

APPENDIX C

DATA FOR VISCOSITY OF WATER

 

Temperature (00)

 

 

R (cm)
To (see)
T (sec)

Densit
(gm/cm )

I (gm-cmz)

Viscosity
(mp) YR

 

 

16.5 17.1 18.6
1.2793 1.2793 .2827
12.317 8.812 3.812
12.758 8.773 8.768
0.998843 0.998757 0.998482
9,396.91167 4,447.42603 4,447.42603
44.6562 45.4292 45.4292
580.7983 1,052.1470 1,045.6000
536.1421 1,006.7173 1,000.1703
10.4799 10.4177 9.8666

 

 

 

 

68

APPENDIX C (Cont.)

DATA FOR VISCOSITY OF LITHIUM-6 AND LITHIUM-7

69

 

6

 

 

 

Order Temp. R To T
°C cm sec sec g‘uificm3 x (106) zip
Lithium-6
6 188.5 1.23484 8.795 8.794 .44019 441.8940 4.5259
7 " " " 8.795 " 440.2847 4.2768
16 " " " 8.794 " 446.6999 4.4649
17 " " " " " 447.7455 4.4962
Ave. " " " " " 444. 1560 4.5897
5 193.0 1.28496 8.795 8.794 .45944 455.5604 4.1555
14 " " " 8.795 " 442.2466 4.5425
15 " " " " " 442.5005 4.5440
Ave. " ” " " ” 440.0558 4.2787
4 217.4 1.28521 3.795 8.795 .45792 450.1658 4.0251
8 " " " 8.794 " 428.1079 5.9670
9 " " " " " 429.5190 4.0049
Ave. " " " " " 429.2656 5.9980
5 256.2 1.28545 8.795 8.794 .45642 422.5769 5.8574
10 " " " ” " 424.8746 5.8975
11 " " " " " 421.5107 5.8098
Ave. " ” " " " 422.9874 5.8480
18 256.8 1.28545 8.796 8.798 .45657 421.5840 5.8144
19 " ” ” ” " 422.8675 5.8477
Ave. " " " " " 422.2257 5.8511
1 266.6 1.28585 8.795 8.795 .45595 415.4996 5.6852
2 " " " " " 416.4526 5.7067
Ave. " " " ” " 415. 9661 5. 6949
12 266.8 1.28585 8.795 8.795 .45590 414.3041 5.6665
15 " " " " " 415. 5275 5. 6797
Ave. " ” " " " 415.0657 5.6751
Lithium-7

256.3 1.28696 8.796 8.797 .50945 484.0510 4.1864

 

 

 

APPENDL‘C D

LOGARITHHIC DECREI/IENI‘ DATA

7O

 

 

 

Order Temp. Log. degr. Order Temp. Log. degr.
00 6,5: 10 00 6,5: 10
Lithiumpo Lithium-6
6 188.3 482.2725 10 236.2 463.2557
478.3121 465.3280
480.9846 461.9273
7 188.3 478.0755 11 236.2 460.7277
479.6906 460.3303
478.9752 459.3115
16 188.3 486.1153 18 236.8 458.1642
484.5164 462.2620
435.3552 19 236.8 461.5875
17 188.3 485.9834 461.4054
486.7659 1 266.6 455.0963
5 198.0 475.4092 454.6390
475.0178 452.6509
472.1414 2 266.6 456.3262
14 198.0 482.2980 454.9471
482.1750 453.9118
478.1542 12 266.8 453.8711
15 198.0 481.3653 452.1115
479.8177 454.3171
481.6051 13 266.8 455.9060
4 217.4 470.8463 454.7515
467.6874 451.2116
467.8449
8 217.4 467.0125
466.6431
466.5554 Lithium-7
9 217.4 467.3154
470.8317 236.8 523.8293
466.2972 522.3651
3 236.2 460.2000 521.7860
461.5133

461.9048

 

 

 

 

APPENDIX D (Cont.)

Residual damping for lithium-6 sphere

 

 

 

Temperature Logarithmic dgcrement
00 60 x 10
22 41.4377

42.1339
Ave. 41.7858
26 43.0992
40.3912
42.2050
Ave. 41.8985
152 40.6121
41.2272
41.2378
Ave. 41.21422
178 39.0468
38.7209
38.1197
Ave. 38.62913

 

 

SPHERE TEMPERATURE AS A FUNCTION OF FURNACE TEMPERATURE

APPENDIX E

 

 

 

 

Furnace temp.* Sphere temp.# Temperature
Order
mv °C mv °C difference
1 10.17 216.4 8.615 187.3 29.1
9 10.21 217.2 8.659 188.1 29.1
2 10.71 226.4 9.123 196.9 29.5
8 10.74 227.0 9.161 197.6 29.4
3 11.87 247.4 10.212 217.2 30.2
7 11.88 247.6 10.213 217.2 30.4
4 12.95 266.7 11.226 235.2 31.5
6 12.96 266.8 11.234 235.9 30.9
5 14.73 297.7 12.945 266.6 31.1

 

 

 

 

as measured by thermocouple T of Figure 1.

as measured by a thermocouple attached directly to the

outside of the Sphere on its horizontal equator.

72

73

APPENDIX F
DEPENDENCE OF V\ ON R

The following table will clearly demonstrate the extreme
sensitivity of the viscosity coefficient V\ to inaccuracies
in the sphere radius. The data for lithium-6 at 236.8 C-deg.
were used; keeping I, To, T, 6 , and f fixed and varying

only R, the following results were obtained:

R (cm) vk (mp)
1.283 3.922
1.284 3.885
1.285 3.848
1.28545 3.831
1.287 3.775

1.289 3.704

SPECTROGRAPHIC ANALYSIS OF LITHIUML6
AFTER VACUUM DISTILLATION

APPENDIX G

74

The Spectrographic analysis of the lithium-6 sample

which was used in this investigation has been made by ORNL.

Vacuum distillation has in general improved the purity of

the sample with reSpect to impurity contents, with a marked

decrease in the calcium and zinc impurities, the only two

elements present in relatively substantial amount before

distillation.

Lithium-6 Lot No. SS 5(b)

The results are given as follows:

Spectrographic Analysis (element and weight percent):

A8
A1

Ba
Ca
Co
Cr
Cu

<.01T
.01

<.01T
.05

(.05
.01
.05

Fe
K

Mg
Mn
Mo

Na
Ni

.03

.02
(.02
<.01T
(.01

.05
(.OlT

Pb
Si
Sn
Sr
Ti
V

<.OlT
.05
(.02
(.01
(.02
(.01

1.

11.

12.

13.

14.

75

REFERENCES CITED

E. N. da C. Andrade and.Y. S. Chiong, "On the deter-
mination of viscosity by the oscillation of a vessel
enclosing a fluid", Part 1, Proc. Phys. Soc. (London),
22: 247 (1936).

E. N. da C. Andrade and E. R. Dobbs, "The viscosities
of liquid lithium, rubidium and caesium", Proc. Roy.
Soc. (London), A211, 12 (1952).

E. N. da C. Andrade, "A theory of the viscosity of
liquids", Part I and 11, Phil. Ma., 12, 497, 698
(1934).

See e. g., F. W. Aston, Mass Spectra and Isoto s,
(Edward Arnold and Company, London, 1933, I941),

276 pp.

G. Barr, A.Mono a h of Viscometry, (Oxford University
Press, LonHon, 1931), 318 pp.

J. A. Bearden, "A.precision determination ofthe vis-
cosity of air", Phys. Rev. 26, 1023 (1939).

 

A. Bernini and C. Cantoni, "Sulla dilatazione termica
del sodio, del potassio e del litio", Nuovo Cimento
8, 241 (1914).

E. C. Bingham, Fluidit and Plasticit , (hGraw-Hill
Book Company, Inc., New YorE, I922), £40 pp.

A. Bondi, Theories of Viscosity, Rheology $3 321 (1956).
‘M. Born and H. S. Green, "A general kinetic theory of
liquids. Part I: The molecular distribution functions",
Proc. Roy. Soc. (London).A188, 10 (1946).

‘M. Born and H. S. Green, "A general kinetic theory of

liquids. Part III: Dynamical prOperties", Proc. Roy.
Soc. (London) A190, 455 (1947).

P. W. Bridgman, The ghysics of 8%gh gfigssures, (The
Macmillan Company, ew or ,. , . , 398pp.
Catalog_and Price List of Stable Isotgpes, (ORNL, Oak

Ridge, Tenn., April 1954)?

Catalo and Price List of Stable Isotopes, (ORNL, Oak
RIng, Tenn., April I957).

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

- 29.

76

G. H. A. Cole, "The kinetic theory of monatomic liquids
at ordinary temperature", Rep. Progr. Phys. (London:
Physical Society) _1_9_, 1 (1956).

M. F. Culpin, "The viscosity of liquid indium and
liquid tin", Proc. Phys. Soc. (London) B70, 1069 (1957).

M. F. Culpin, "The viscosity of liquid magnesium and
liquid calcium", Proc. Phys. Soc. (London) 870, 1079
(1957).

R. Eisenschitz, "Transport processes in liqui ", Proc.
Roy. Soc. (London) A215, 29 (1952).

H. Eyring, "Viscosity, plasticity and diffusion as
examples of absolute reaction rates", J. Chem. Phys.
3, 283 (1936)..

J. Frenkel, Kinetic Theory of Liqgids, (Dover, New York,
1955), Ch. 1V7 488 pp.

J. Frankel, "fiber die warmebewegung in festen und
flfissigen Kbrpern", Z. Physik §_5_, 652 (1926).

R. Ffirth, "On the theory of the liquid state", Part I,
II and III, Proc. Cambridge Phil. Soc. 31, 252, 276,
281 (1941).

H. S. Green The Molecular Theory of Fluids, (North-
Holland PubiisHIfig Company, Amsterdam, Holland, 1952),
PP. 136-137, 264 pp.

Handbook of Chemist_:_1% and Phfis ics (Chemical Rubber
1s ng mpany, eve an , , 36th edition,

1954-1955). pp. 1962, 2002, 3173 pp.

E. Hatschek, The Viscosit of Liquids, (D. Van Nostrand
Comapny, New YorE, I928), 239 pp.

H. Helmholtz and G. v. Piotrowski, "fiber Reibung
trepfbarer Flfissigkeiten", Sitzungsberichte Akad.
Visa” Wien _4_9_, 607 (1860).

P. R. Heyl and P. Chrzanowski, "A new determination of
the constant of gravitation", J. Research Natl. Bur.
Standards 32, 1 (1942).

E. E. Hoffman and w. D. Manly, "Corrosion resistance
of metals and alloys to sodium and lithium", Advances
in Chemistry Series 12, 82 (1957).

J.l'. Kincaid, H. Eyring and A. E. Steam, "The theory
of absolute reaction rates and its application to

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

41.

. della fus

77

viscosity and diffusion in the liquid state", Chem.
Revs. 28, 301 (1941).

J. G. Kirkwood, "The statistical mechanical theory of
transport processes. Part I: General theory", J. Chem.
Phys. 13, 180 (1946).

J. G. Kirkwood, F. P. Buff, and M. S. Green, "The
statistical mechanical theory of transport processes.
Part III: The coefficients of shear and bulk viscosity
of liquids", J. Chem. Phys. 12:, 988 (1949).

R. Ladenburg, "fiber den Einfluss der Reibung auf die
Schwingungfin einer mit Fliissigkeit geffillten Kugel",
Ann. Phys 21, 157 (1908).

H. C. Longuet-Higgins, "Transport processes in fluids",
Nuovo Cimento Supplement 1, 140 (1955).

L. Losana "Variazione di volume degli elementi all'atto
ione", Gazz. Ghim. Ital. 92, 851 (1935).

R. N. Lyon, Editor-in-Ghief, L1auid-Metals Handbook,
(NAVEXOS -.P - 733 Rev., U. S. t. rint. ce,
Washington, D. C., 1954), 269 pp. '

D. J. Montgomery, "Bulk properties of stable isotopes
as a probe for liquid-state and solid-state investi-
gations", U. N. Peaceful Uses of Atomic Energy, Proc.
Second Intern. Conf. 22, 165 (1958).

J- A- P013193, "The viscosity Of isotopic substances",
Physics 12, 688 (1953).

See e. g., J. S. Rowlinson, "The viscosity of heavy
water vapour", Physics 12, 303 (1953).

D. D. Snyder and D. J. Montgomery, "Bulk density of

separated lithium isotopes", J. Chem. Phys. 31, 1033
(1957).

S erior Tube 0 an Catalo ‘, Section 21, pp. 22,
(Eorrfitown, Penna. E9375.

J. E. Verschaffelt, "The viscosity of liquefied gases",
Gomun. Phys. Lab. Univ. Leiden No. 1483, 17 (1915).

MICHIGAN STATE UNIV. LIsRaRIEs
mWWWI“WWNWWIIHWINNIHII
31293017648370