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; Rayleigh-Brillouin Light Scattering. i .
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ABSTRACT

RAYLEIGH-BRILLOUIN LIGHT SCATTERING
IN LIQUID SILICONES

by
ANIL KUMAR

Rayleigh and Brillouin light scattering was used as an experimental
technique to determine light scattering and acoustical properties of
.chain and ring dimethylsiloxanes. The properties determined were the
Brillouin shift, sound velocity, adiabatic compressibility, Brillouin
linewidth, sonic absorption co-efficient, Landau-Placzek ratio and de-
polarization ratio. Fluctuation theory and the linearized hydrodynamical
equations of Mountain were used to arrive at numerical values. Refractive
indices were also measured where required. Nine silicones with molecular-
weights ranging from l60 to 7,000, were examined spectrometrically at ten
different temperatures falling between 20° and 800C.

It was observed that, as molecular chains lengthened, sound velocity
increased while adiabatic compressibility decreased. Sound velocities
had a tendency to level off in the region of higher molecular weights,
and were in good agreement with ultrasonic measurements in the MHz range.

Propagating hypersonic waves died out in a distance of about 50,000 A ,

roughly ten
Values of tr.
tones.

The Lam
molecular we
Va terperat

|
close to the
1molds. T!
55 was inve
rm sound vel

DEtermi

deoolarized

 

moreISOtm;

 

Anil Kumar
roughly ten times the wavelength of the incident green light (5145 3).
Values of the adiabatic compressibility were unusually high for all sili-
cones.

The Landau-Placzek ratio was found always to increase with increasing
molecular weight, but in every case to decrease monotonically with increas-
ing temperature. These values were strikingly low (0.2 to 2.0) and were
close to the values obtained for water and other low molecular weight
liquids. The spectral distribution of the scattered light from MD7M and

9 Hz to 5 x 109 Hz, but

05 was investigated in the frequency range 2 x l0

no sound velocity dispersion was detected.
Determination of the depolarization ratio, pv, from polarized and

depolarized measurements established that linear silicone samples were

more isotropic than cyclic silicone samples at the same molecular weight.

RAYLEIGH-BRILLOUIN LIGHT SCATTERING

IN LIQUID SILICONES

by

Anil Kumar

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Metallurgy, Mechanics and Materials Science

l973

TO MY PARENTS

Shri and Smt. H. Prasad

I want
Ms counsel,
acting as ng
Mon to Prol
atHichlgan
ESPecially '
the eqlipre

ADDre:

valuable So
at mmlgar

NS. Bey/9'1.)

ACKNOWLEDGEMENT

I want to express my deep appreciation to Professor T. Triffet for
his counsel, patience and active interest in this work as well as for
acting as my major Professor. I also want to extend my sincere apprecia-
tion to Professor J. B. Kinsinger, Chairman of the Chemistry Department
at Michigan State University, for his guidance and encouragement,and
especially for letting me make use of departmental facilities, including
the equipment without which this work would not have been possible.

Appreciation is also extended to Professor 0. J. Montgomery for
valuable suggestions, Mrs. Shukla Sinha of the Department of PhilOSOphy
at Michigan State University for help in numerical calculations and
Ms. Beverly Anderson for typing the thesis.

The financial support of the Department of Metallurgy, Mechanics and

Materials Science, Michigan State University is greatly appreciated.

U "H EC'P'C‘.
r‘.',.l.mnt_L;,_z._

LIST OF F19W

l INTROD'

 

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ....................... iii
LIST OF TABLES ........................ vi
LIST OF FIGURES ........................ x
I INTRODUCTION ....................... l
l. General Background ................ l

2. Purpose of Research ................ 3

II THEORY .......................... ‘ 5
l. General Concepts ............... 5. . 5

2. Thermodynamic Theory ............... 6

3. Fluctuation Theory ................ l0

4. Continuum Theory ................. l5

III EXPERIMENTATION ..................... 24
1. Sample Preparation ................ 24

2. The Brillouin Spectrometer ............ 26

a. Laser ..................... 26

b. Interferometer ................ 28

c. Optics .................... 29

d. Alignment ................... 30

e. Detection and Recording ............ 30

3. The Temperature Control Cell ........... 31

4. The Scattering Cell and the Holder ........ 36

iv

N RES’JL .

HYDI a

l I‘PK K
‘U‘IUJRAV
I.

h|PD:H ,

 

5. Refractive-Index Measurements ...........
IV RESULTS AND DISCUSSION ..................
l. Variation of the Refractive-Index with
Temperature and Molecular Weight . . .......
2. Variation of the Brillouin-shift with
Temperature and Molecular Weight . . .......
3. Dispersion Measurements ..............
4. Velocity of Sound and Adiabatic Compressibility. .
5. Variation of the Brillouin Line Width and the
Sonic Absorption Co-efficient with Temperature
and Molecular Weight ...............
6. Variation of the Ratio of the Intensities of
the Central and Shifted Peaks with Temperature . .
7. Depolarization Measurements. . . .........
V CONCLUSIONS .......................
BIBLIOGRAPHY .........................
APPENDIX A Calculation of the Refractive-Index at 5l45 A. . .
APPENDIX B Variation of JV with Temperature .........
APPENDIX C Laser—Brillouin-Velocimeter ...........
The Experimental Set-up ............
Discussion ....................

Page
37
40

42

49
70

87

95
105
108
III
115
I31
I37
138
I39

lahe

L

MEOSUT:
Brillo.

Cyclic

 

 

 

Refract
Linearl
Refrac1
Linear
Refrac
Cyclic
Observ
TemPOr
and S;
Funct‘

ObSEri

LIST OF TABLES

Table

1.

Measured and Corrected Temperature Obtained during

Brillouin Scattering Measurements for Linear and

Cyclic Polydimethylsiloxanes ................

Refractive-index as a Function of Temperature for

Linear Silicones (MMMO3

Refractive-index as a Function of Temperature for

Linear Silicones (o.c. 200 fluids. loo cts. and 105 cts.. .

Refractive-index as a Function of Temperature for

Cyclic Silicones ......................

Observed Brillouin Frequency Shifts (v3), Velocities (VS)’
Temporal Attenuation Co-efficients (PB), Life times (l/rB)

and Spatial Attenuation Co-efficients (a) of MN as a

Function of Temperature ..................

Observed Brillouin Frequency Shifts (v8), Velocities (VS),
Temporal Attenuation Co-efficient (PB), Life times (l/rB)

and Spatial Attenuation Co-efficient (a) of MD3M as a

Function of Temperature ..................

Observed Brillouin Frequency Shifts (VB), Velocities (VS),
Temporal Attenuation Co-efficient (TB), Life times (I/FB)
and Spatial Attenuation Co-efficient (a) of MD7M as a

Function of Temperature ..................

Observed Brillouin Frequency Shifts (VB), Velocities (VS),

Temporal Attenuation Co-efficient (r8), Life times (I/FB)

vi

M and MD7M) .............

Page

34

43

44

45

54

55

56

lame

l3.

and S;
fluid,
Observ
Temper
and S;
fluid,
Obseri

Temper
and S;

Funct'

OOSErl

Table Page
and Spatial Attenuation Co-efficient (a) of D.C. 200
fluid, lOO cts. as a Function of Temperature. . ...... 57
9. Observed Brillouin Frequency Shifts (VB), Velocities (VS)’
Temporal Attenuation Co-efficient (PB), Life times (l/IB)
and Spatial Attenuation Co-efficient (a) of D.C. 200
fluid, 105 cts. as a Function of Temperature ...... . . 58
10. Observed Brillouin Frequency Shifts (v8), Velocities (VS)’

Temporal Attenuation Co-efficient (PB), Life times (l/rB)
and Spatial Attenuation Co-efficient (a) of 05 as a

Function of Temperature .................. 59
ll. Observed Brillouin Frequency Shifts (VB), Velocities (VS)’

Temporal Attenuation Co-efficient (PB), Life times (l/rB)

and Spatial Attenuation Co-efficient (a) of 03 as a

Function of Temperature .................. 59
12. Observed Brillouin Frequency Shifts (VB), Velocities (VS)’

Temporal Attenuation Co-efficient (r , Life times (l/FB)

B)
and Spatial Attenuation Co—efficient (a) of 09 as a
Function of Temperature .................. 60
l3. Observed Brillouin Frequency Shifts (VB), Velocities (VS)’
Temporal Attenuation Co-efficient (PB), Life times (I/FB)
and Spatial Attenuation Co—efficient (a) of 015 as a
Function of Temperature .................. 6l
l4. Intercept and Slope for the Brillouin Half-width (PB) -
Temperature Relationship .................. 62

15. Intercept and Slope for the Absorption Co-efficient -

Temperature Relationship ........... . . . . . . . 62

vii

l8.

20.

21.

22.

23.

24.
25.

26.
27.

Table Page
16. Intercept and Slope for the Brillouin-shift-

Temperature Relationship .................. 63
17. Intercept and Slope for the Sound Velocity -

Temperature Relationship .................. 63
18. Intercept and Slope for the Brillouin-shift -

Temperature Relationship .................. 64
19. Intercept and Slope for the Sonic Velocity -

Temperature Relationship .................. 64
20. Intercept and Slope for the Brillouin Half-width -

Temperature Relationship .................. 65
21. Intercept and Slope for the Absorption Co-efficient -

Temperature Relationship .................. 65
22. Variations in Brillouin Frequency-shifts (v8), Velocities

(VS)’ Temporal Attenuation Co-efficients (TB), Life-times

(l/FB), Spatial Attenuation Co-efficient (a) and Landau-

Placzek Ratio (JV) of MD7M with Scattering angle 6 ..... 71
23. Variations in Brillouin Frequency-shifts (v8), Velocities

(VS), Temporal Attenuation Co-efficients (PB), Life-times

(l/rB), Spatial Attenuation Co-efficient (a) and Landau-

Placzek Ratio (JV) of 05 with Scattering angle a ...... 72
24. Ultrasonic Data ...................... 74
25. Adiabatic Compressibility of MM, MD3M, MD7M, D.C.ZOO fluid.

100 cts., 05, and 015 at 22°C ............... 77

26. Adiabatic Compressibility as a Function of Temperature. . . 78
27. Observed Landau—Placzek Ratio (JV) of MM as a Function

of Temperature ....................... 96

viii

laMe

 

28. Obser
Funct
29. Cbser
100 c
33. ODSEr
of Te
31. Obser
105 c
32. Obser
Funct
33. Obser
Funct
34. Obser
Funct
35' Obser
Funct
56' Inter
Scat:
37' OPSEr
D.C.
E.

ObSEr

 

Table

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

Observed Landau-Placzek Ratio (JV) of
Function of Temperature .......
Observed Landau-Placzek Ratio (JV) of
100 cts. as a Function of Temperature
Observed Landau-Placzek Ratio of MD7M
of Temperature ............
Observed Landau-Placzek Ratio (JV) of
105 cts. as a Function of Temperature
Observed Landau-Placzek Ratio (JV) of
Function of Temperature .......
Observed Landau-Placzek Ratio (JV) of
Function of Temperature .......
Observed Landau-Placzek Ratio (JV) of
Function of Temperature .......
Observed Landau-Placzek Ratio (JV) of
Function of Temperature .......

Intercept and Slope for the Intensity
Scattered Light (Jv)—Temperature (TC)

Observed depolarization Ratios for MM,

5

D.C. 200 fluids, lOO cts. and 10 cts.

Observed Depolarization Ratios for O3,

ix

MD3M as a

D.C. 200 fluid,

as a Function

D.C. 200 fluid,
03 as a
05 as a

09 as a

015 as a

Ratio of the

Relationship ......
MD3M, MD7M,

D5, 09 and D15 . . .

Page

96

97

97

98

99

99

100

100

101

106
. -107

Figures
L
2.

10.

11.

Class
Spec:
F11tr

Rayle

 

Picto
Cross
ca1i:
Cylir
Light
Refr.
D.C.

RBfr

LIST OF FIGURES

Figures
1. Classical Description of Brillouin Spectra .........
2. Spectrum for MM at 40.0°c, e = 90° and X0 = 5145 K .....
3. Filtration Apparatus ....................
4. Rayleigh-Brillouin Spectrometer ..............
5. Pictorial View of the Temperature Control Cell .......
6. Cross-sectional View of the Temperature Control Cell. . . .
7. Calibration Data for the Temperature Control Cell .....
8. Cylindrical and Brewster Light Scattering Cell .......
9. Light Scattering Cell Holder. ...............
'10. Refractive—Index Versus Temperature for MM, MD3M, MD7M,
D.C. 200 fluids, loo cts. and lo5 cts ............
11. Refractive-Index Versus Temperature for D3, 05, D9 and 015. .
12. Refractive-Index Versus Temperature for 09, MD7M, D5 and
MD3M ............................
13. Spectrum for MM at 70°C, a = 90° and x0 = 5145 K .....
14. Spectrum for MD7M at 40°C, a = 90° and X0 = 5145 A .....
15. Spectrum for MD7M at 50°C, a = 90° and X0 = 5145 K .....
16. Spectrum for 05 at 45°C, 6 = 90° and X0 = 5145 A .....
17. Brillouin-shift Versus Temperature for MM, MD3M, MD7M,
D.C. 200 f1uid,1o5 cts ...................
18. Brillouin-shift Versus Temperature for D3, 05, D9 and D15 . .
19. Brillouin-shift Versus Temperature for 09, 05, MD7M and
MD3M ............................

Page
12
21
25
27
32
33
35
38
39

46
47

48
so
51
52
53

66
67

68

Figure

22.

23.

24.

25.

26.

27.

1"»)
O

29.

CL)
-—.4

(vs)
{\i
a

w
LA,
0

 

Cycli

. Brill

Yeloc

Si1ic

 

Figure

20.

21.
22.

23.
24.
25.
26.
527.
28.

29.
30.

31.
32.

33.

avB/aTC Versus Molecular Weight for Linear and

Cyclic Silicones .......................

Brillouin-shift Versus sin(e/2) for MD M and D

7
Velocity of Sound Versus Molecular Weight for Linear

Silicones at 29.32°C, 33.88°C, 48.60°C and 58.25°C ......

Velocity of Sound Versus Molecular Weight for Cyclic

Silicones at 24.75°C, 34.21°C, 38.88°C and 52.83°C ......

Variations of the Velocity of Sound with Temperature for

MM, MD M, MD

3 7
Variation of the Velocity of Sound with Temperature for

D D D and D

3’
Variation of the Velocity of Sound with Temperature for
D MD

M, MD M and D

9’ 7 3
Temperature Co-efficient of the Velocity of Sound Versus

Molecular Weight for Linear and Cyclic Silicones .......

Adiabatic Compressibility as a Function of Molecular

Weight at Constant Temperature of 22°C ............

Adiabatic Compressibility Versus Temperature for D9 and D15 .

Half of Brillouin Line Width Versus Temperature for MM,

MD3M, M07

Half of Brillouin Line Width Versus Temperature for

05. 09 and D

aI‘B/aTC Versus Molecular Weight for Linear and Cyclic

Silicones ..........................

Sonic Absorption Co-efficient Versus Temperature for

D MD M and MD

5’ ”9' 3 7

xi

5 .......

M and D.C. zoo fluid, 105 cts ..........

5, 9 15 oooooooooooooooooooooo

5 ....................

M and D.C. 200 f1uid,100 cts .............

15 ........................

M ....................

Page

69
73

79

80

81

82

84

85
86

9O

91

92

93

Figure Page
34. act/8TC Versus Molecular Weight for Linear and
Cyclic Silicones ................ . . . . . . . 94

35. JV Versus Temperature for 05, 09, MD3M and MD7M . . . . . . . 102

xii

1. Historic
The S1

ever since t

 

and the redn
”90W in 18'
His theory d
Isotropic pa
of the lncid
N9 EXpl
Portia“.

555°] 1ioui

distributhF

535‘ Scatte

deI'moileéd a

CHAPTER I
INTRODUCTION

1. Historical Background.

 

The scattering of light has challenged the understanding of men
ever since the first attempt to explain the blueness of the daytime sky
and the redness of the sunset. Lord Rayleigh [1,2] developed the basic
theory in 1871 and 1881, identifying the process as one of diffraction.
His theory dealt only with non-interacting, transparent, optically-
isotropic particles with dimensions small compared with the wavelength
of the incident radiation.

5 No explanation was provided for effects in liquids with interdependent
particles. Smoluchowski [3] suggested that the density is not uniform in
a real liquid; fluctuations in the density occur over a region of the
order of the wavelengths of light. At each instant a different density
distribution exists and light is scattered from the fluctuations of the
density about its average value. Einstein [4] calculated the amount of
light scattered from these density fluctuations, and in 1908 Mie [5]
developed a general electromagnetic theory of light scattering for non-
interacting, spherical particles of arbitrary size. His theory is par-
ticularly useful for analyzing scattering by particles width dimensions
that are large compared with the wavelengths of the incident radiation,

as in the "Laser Doppler Velocimeter" application [6,7,8].

Before '
with the elal
light. Fret.
in 1922 inve
tile, and pr?
acoustical h:

Such shl
velocity of .
difficult. i
neat was poml
were not ava,
scattering c:
11.-ch aitentig
Extended ela:
mleclllar we1

Landau
the Central ‘1
irinmn “3%
Miller I15] 1

555555 "91.:

 

Cant only 1n

The dEVe
555 it Possi

5'55 the 1as

:31-
5mm th
533%

“l9. H?

Before 1922 the theoretical work on light scattering dealt only
with the elastic case and emphasized the intensity of the scattered
light. Frequency dependence was completely neglected. Brillouin [9]
in 1922 investigated the problem of frequency dependence for the first
time, and predicted that light would be inelastically scattered by
acoustical waves in a liquid and should have two Doppler-shifted peaks.

Such shifts in frequency are small and directly related to the
velocity of thermal waves in the liquid; hence spectral resolution was
difficult. Gross [10] did verify them, but the accuracy of his experi-
ment was poor owing to the fact that intense monochromatic light sources
were not available. However, Raman's [11] discovery that inelastic
scattering could also be caused by non-linear polarizability received
much attention from scientists working in related areas. Debye [12,13]
extended elastic scattering theory and used his results to predict the
molecular weight of polymers.

Landau and Placzek [l4] calculated the ratio of the intensity of
the central Rayleigh peak (1c) to the intensity of the Doppler-shifted
Brillouin peaks (218) from classical density-fluctuation theory, and
Miller [15] later developed the same equation using strictly thermo-
dynamic arguments. Miller also noticed that macromolecules significantly
increase the amount of scattered light, but that the change was signifi-
cant only in the intensity of the central peak [16,17].

The development of the laser by Maiman [18] and Javan [19] in 1961
made it possible to study the fine structure of the scattering spectrum.
Since the laser beam is monochromatic, coherent, intense, and well-
collimated the separation of the Rayleigh and Brillouin peaks became

possible. High-resolution interferometers [20,21] could then be used to

study the SD".

 

Mountai'
linearized h,
liquids.
colleagues [2

wattering Si

 

 

 

atthe incidq

byimavily-dg

2. t
m
The 1
556““;11l511(lyre

5559395 Of i

study the spectra of light scattered by density fluctuations.

Mountain [22-27], followed by Bhatia and Tong [28], introduced
linearized hydrodynamic theory for describing the Brillouin spectra of
liquids. Recently Volterra, Stoicheff, Stegeman, Cummins and their
colleagues [29-33] have made high-resolution studies of depolarized light-
scattering spectra of many molecular liquids, revealing a doublet centered
at the incident frequency. This must be due to orientational motion caused

by heavily-damped shear waves.

2. Purpose of Research.

The low intermolecular forces of silicones (linear and cyclic poly-
methylsiloxanes) lead to unusually high compressibilities and very small
changes of viscosity with temperature [34,35]. The general structural
changes with temperature and molecular weight that form the basis for
these remarkable properties are the main features investigated in the
present work. Some insight into such matters has previously been gained
by ultrasonic measurements [36,37].

The purpose of the present Brillouin-scattering study of silicones
is to elucidate physical and structural changes as the temeprature is
varied from 20°C to 80°C for polymers of molecular weight 160 to 12,500.
It is intended:

1. To measure the index of refraction of silicones for wavelengths
of 5145 A and 5890 A at varying temperatures. A more detailed
discussion is given in Chapter IV.

2. To show that the dependence of certain light-scattering para-

meters (the Landau-Placzek ratio and the soundwave attenuation

. 10 r

rela

give

terj
latal
res:
the
$111

the

. 10 c

 

COD

data

'TOC

Velc

4

co-efficient) on molecular weight and temperature reflect well-
defined structural changes. The Landau-Placzek ratio (JV) is
related to the ratio of specific heats; a brief derivation is
given in the Section 2 of Chapter II.

To measure the velocity of sound in silicones as a function of
temperature and molecular weight. This quantity can be calcu-
lated from the spectral positions of the Brillouin peaks with
respect to the central Rayleigh peak. A brief discussion of
the velocities, frequencies and wave lengths of sound waves in
silicones is given in Section 3 of Chapter II. The widths of
the Brillouin component give the life times of the sound waves,
as is explained in Section 4 of Chapter II.

To calculate the adiabatic compressibilities of certain sili-
cones from the above sound-velocity measurements and density
data. Calculations are presented in Chapter IV.

To consider the feasibility of developing a Laser-Brillouin-

Velocimeter. Conclusions are presented in Appendix C.

 

1. General
Ligt
hmOmogene'
Mom inclu:
nations 1
In the lat
tolecular
Hhene

the visib'
dipole mo:
secondary
95-3t0n at

If the so

frequency

CHAPTER II
THEORY

1. General Concepts.

 

Light passing through any medium is partially scattered by optical
inhomogeneities in the medium. Such inhomogeneities may result either
from inclusions of foreign particles, or from small-scale density fluc-
tuations in the medium - even though the medium contains no inclusions.
In the latter case the scattered light will carry information on the
molecular structure of the medium.

Whenever electromagnetic radiation of a given incident frequency in
'the visible region interacts with a scattering molecule, it induces a
dipole moment in the molecule, which will then oscillate and act as a
secondary source of energy. If this secondary source scatters a light
photon at the incident frequency, this process is said to be elastic.

If the source scatters a photon at a frequency different from the incident
frequency, this process is said to be inelastic.

Rayleigh scattering is an elastic process in which the maximum
linear dimension of the scattering particles is small compared to the
wavelength of the incident radiation. Where the corresponding dimension
of the scattering particles is not small compared with the wavelength of
the incident radiation, the process is called Mie scattering.

The induced dipole moment pi of the particle is connected with the

amplitude of the incident electromagnetic wave Ej by a polarizability

tensor 5i j

where ,

 

 

The lnc

Particl e.

Eq.(2) are
eouation f
to. (l) in 1
used for di
A genera] e

Aflimica] E

tensor 013 ;

pi = alj ° Ej[38] 3 T.j = 1,2,3 (1)
where,
“11 “12 “13
“ij 5 “21 “22 “23 (2)
“31 “32 “33

The induced dipole moment p1 is dependent upon the shape of the
particle. In the case of a homogeneous medium the off-diagonal terms of
Eq. (2) are zero and all diagonal elements are equal. The scattering
equation for an independent particle can be obtained by substituting
Eq. (1) in Maxwell's equations. But the resulting relation can only be
used for dilute gases. because in liquids the particles are interdependent.
A general equation for the intensity of light scattered from a liquid can
be obtained by treating the liquid as a continuum and utilizing hydro-
dynamical equations. [22-28]

2. Thermodynamic Theory.

 

In the classical theory of light the sample volume V is divided
into small elements of VOTUME\?. The latter is large enough to contain
many molecules but its linear dimensions are small compared with the wave-
length of light. The incident light wave induces a dipole moment in the
volume elementls which in turn, acts as a secondary source. If the

medium is homogeneous, the induced polarization will be constant and the

scattered 1
scattering
ever, this I
local diele

1

An eq.

stant thrc

from Maxwe‘.

each elenegl

there 1 is
is goo, I
.554ttering
light, and

E about ltfi

 

Einstfi

l. "rlting

He assumed

. d from tn

scattered radiation will act destructively in all directions producing no
scattering except in the forward direction. Even in pure liquids, how-
ever, this does not occur. There is a small random fluctuation in the
local dielectric constant (A6) and the induced polarization is not con-
stant throughout the medium.

An equation for the intensity of scattered light can be developed
from Maxwell's equations [39,40]. The intensity of light scattered by

each elemental volume u-is given by:

2 2

lit-4- <(Ag)2> (3)

I = I
o sz

0

Where I is the intensity of the scattered light when the scattering angle
a is 90°, Io the intensity of incident beam, R the distance between the
, scattering volume and the detector, X0 the wavelength of the incident
light, and <(Ae)2> the mean-square fluctuation of the dielectric constant
a about its mean value in the elemental volume.

Einstein [4] expressed e as a function of density p and temperature

T, writing

Ae = (3%) up + (3%) AT . (4)
T 0
He assumed that
92. 22. 2E.
AT (3p)T >> (3T)p

and from thermodynamic fluctuation theory

where K 1i

Eq.

and substi

If od

Pressure 1

01‘

 

 

2 = KTBT

<Ap>

 

. (5)

D

where K is Boltzmann's constant and 8T is the isothermal compressibility.

Eq. (4) reduces to

 

<(Ae)2> = (1934130123 = (o 9:— 2

and substituting Equation (6) in (3) gives

2
11 KTBTU 3t: 2

I-Io—é-Z—j—(o-i) . (7)
x0 T

If on the other hand e is chosen as a function of entropy (s) and

-pressure (p) it follows that

9;“,
AP = (3; A5 + (%5) Ap (8)
p S
or,
2 BE 2 at 2 2
<(4e) > =(-,-; <<as1 + (,3 <<up1> . (91
P S

where the cross term vanishes since fluctuations in s and p are independent.

Landau and Placzek [41,42] identified the first term of Equation (9) as
entropy fluctuations at constant pressure which do not propagate and
therefore give rise to the Rayleigh peak. The second term they identified
as pressure fluctuation at constant entropy, which is the sound wave re-

sponsible for Brillouin doublets.

 

By K

where c
P

the adi ab

 

lead to:

EqL
Equatioi

yield

01‘

By making use of thermodynamic theory [33] Eq. (9) can be reduced to

 

2 2
T 3T 5
<(4€)2 > = (fi -5-' + o -—- \, (10)
OT p Cppo 8p 5

where cp is the specific heat per unit mass at constant volume and Bs is
the adiabatic compressibility (BT/Bs = cp/cv). Equations (10) and (3)

lead to:

M2 2KT2 3&2
1=I o ;2‘13c1-g—T —) p37,; + (p37) Kiss] . (11)
(Rayleigh) (Brillouin)
Equation (7) gives the total intensity I and the first term of

'Equation (11) gives the intensity of the Rayleigh peak, Ic. The second
term gives the intensity of the Brillouin peaks, 2IB. Hence (7) and (11)

yield
.I._ . _____IC+ZIB .21 (12)
21B 21B as
or
I a
555:5:51' and (13)
S
IC
'2"','3’=RLp=l"1 (l4)

where Y =
and specif
This
liquids be
like CV.
calculate:
frequency.

above men:

1M

flUCtuatiQ.

The

Constant.
homgeneit
“Electric
Let u
3€Comp086d
l3

8 Cptlca

th

 

10

where Y = Cp/Cv , the ratio of the specific heat at constant pressure
and specific heat at constant volume.

This thermodynamic derivation is not entirely accurate for all
liquids because of the frequency dependence of thermodynamic parameters
like CV“ The measured Landau-Placzek ratio is generally greater than its
calculated value, and the velocity of sound increases with increasing
frequency. A theory is needed which will account for dispersion in the

above mentioned quantities.

3. Fluctuation Theory.

 

Thermal motion of the molecules in a liquid produces density
fluctuations which, in turn, cause fluctuations of the optical dielectric
constant. Light passing through the liquid will be scattered by these in-
Ohomogeneities and its intensity will depend upon the extent to which the
dielectric constant is coupled with the density fluctuations.

Let us assume that the dielectric constant of the liquid e, can be

decomposed into an average dielectric constant, e , and dielectric con-

0
stant Aeij [43] due to the density fluctuations. Thus, Aeij reflects
the optical inhomogeneity in the liquid. We write

a = 6 +238. (15)

where 5ij is Kronecker's delta and i, j = 1,2,3.
The first term on the right-hand side of Eq. (15) gives the value
of the dielectric constant in a homogeneous medium in which there is

no light scattering. The second term can be divided into two terms, a

 

synnetric

as fol l ov.

where

 

 

11

symmetric (or, isotropic) and a skew-symmetric (or, anisotropic) part,

as follows;

Aeij = Ae'dij + Ae'ij » (16)

where
3
921 As ll = 0

The first term on the right hand side of (16) represents isotropic
fluctuations in the liquid and can be determined by the fluctuations in
pressure and entropy, or density and temperature, or any other pair of
independent thermodynamic variables. The light scattered by the
fluctuations represented by As will be completely polarized.

The second term on the right hand side,Ae’ 1. reflects the anisotropy

J

of the medium resulting from thermal motion, but Ac’i also appears to

3
result in part from fluctuations in the orientation of the anistropic

molecules [43]. Light scattered by Ae’ will be depolarized.

13
The frequency distribution of the scattered light from a liquid will
not be identical to that of the incident light since the fluctuations pre-
sent in the liquid are not static. Measurements of the scattered spectrum
make it possible to study the temporal behavior of the thermal fluctuations
in the liquid.
As in a crystal, longitudinal and transverse waves of different fre-

quencies (thermally excited sound waves) may be present in a viscous liquid.

When the viscosity of the liquid is small, transverse waves will not be

present, l
availableJ
needed to

The ‘

 

have in a

 

Nat/e and

The

555t» S1

1»-

i"

12

present, but the longitudinal waves are easily excited. The energy
available as heat (kT) is much more than the energy per phonon (ho)

needed to excite longitudinal waves.

The incident light wave of wave vector 5i is scattered by the sound

wave in a direction E’i satisfying Bragg's Law.

 

 

 

 

 

 

 

Figure 1.

By this, the sum of the vectors for the incident wave, scattered

wave and sound wave must be equal to zero, whence

5’1 = E1. + E1. (17)

The velocity of the sound wave is much smaller than the velocity of

This approxi-

light, so we can make the approximation that Ikil z lk’il .

mation makes the vector triangle isosceles; therefore,

Sin 9 (18)

qi=2k 2

i

where e i

nentatior

where l 1
length of

Vs the so

 

scatterin

FTC»
l3 dePEn.
0f the w

C51CU]at

Shift (

\ .
V

H

We medi

the I,

a o

§5°3Ure

13

where e is the scattering angle which can be controlled during experi-

mentation. Using the relations
k1 = 2n/A and qi = 2n/aS (19)

where A is the wave length of the incident light and AS is the wave
length of the sound wave in the liquid, let ”5 be the sound frequency,
VS the sound velocity and A = Ao/n, with n the refractive index of the

scattering medium. Substituting (19) in (18) we get,

A0
A9. = —___e— ’ (2“)
2n SING?)

 

 

v°l
vS = 5 ° 6 ~ (21)
2n- Sin(§)
. e
V ~2n Sln -
_ _ s 2
VBzvs - A (22)

O

From Eq. (20) it can be seen that the wave-length of the sound wave
is dependent upon the scattering angle, and is of the order of magnitude
of the wave-length of the incident light. Eq. (21) shows that we can
calculate the velocity of sound in the medium by measuring the frequency
shift (vs) of the Brillouin peaks. then inserting known values - the
wavelength of the incident light in vacuum (to), the refractive index of
the medium (n), and the scattering angle (6). If we increase the angle 6
the light scatters from sound waves of increasing frequency; thus we can

measure the dispersion in the velocity of sound by changing the scattering

angle. F

lated, bu
Eq.

measured

the incid

 

for 1 =

o

It f
of the Dr.
PhySical
melecular
frorlcohel
fluctuatjg
prollagate
being reli
time Of: 6r
5 = 5‘5/cc
heat at C:

Central A

re

 

 

14

angle. From Eq. (22) the ultra-high acoustic frequencies can be calcu-
lated, but these can also be measured directly from the spectrum.

Eq. (20) shows that the shortest pressure fluctuation which can be
measured from Brillouin scattering is dependent upon the wave-length of

the incident light,

A = Xo/Zn ’ (23)

5 min

for to = 5145 A and n = 1.40 , A = 1.62 x 10'5 cm .

5 min
It follows that the shortest pressure fluctuation has a wave length

5 cm, and a frequency of the order of 1010 Hz. The

of the order of 10'
physical length of these pressure fluctuations is very long compared with
molecular sizes. Hence, the scattered light can be assumed to emanate
from coherent density fluctuations. In addition to propagating pressure
fluctuations, entropy fluctuations occur in the liquid; but these do not
propagate. They occur only over extremely short time intervals, and
being related to thermal-diffusion processes,are incoherent. The life-
time of entropy fluctuations is determined by the thermal diffusivity

a = A’lpcp , where A’ is the thermal conductivity and cp is the specific

heat at constant pressure per unit mass. Therefore, the width of the

central peak is proportional to the thermal diffusivity a.

4.20331
AI

1
produced
describir
An e

and freq.
developepl
principle
scattered
cm.), cg

hydrodyna

 

higher tli
theory [2
reclProca

This
degrees c
transht,1
Smiller F
55'“ Staci

meaSUred

fluids (c

15

4. Continuum Theory.

 

According to Onsager's hypothesis, the relaxation of thermally-
produced fluctuations can be described by the same equations as those
describing deviations from equilibrium resulting from external causes.

An approximate theory yielding intensity as a function of magnitude
and frequency for the various components of the Brillouin spectrum was
developed by Mountain et al. [22-27] from thermodynamic and hydrodynamic
principles. Eq. (23) establishes that the wave-length 0f the
scattered light is large in relation to molecular sizes (10'5 vs. 10'7
cm.). Consequently the medium can be treated as a continuum and

10 Hz is

hydrodynamic theory can be applied. The frequency of 10
higher than is usually encountered in hydrodynamics but Mountain's
theory [23] should be applicable up to frequencies approximating the
reciprocal of the collision time, 1012 to 1014 Hz [45].
I This theory assumes a model in which the fluid possesses internal
degrees of freedom for the molecules which are weakly coupled to the
translational degrees of freedom of the fluid. Fluctuations in the density
scatter polarized light. and the weak coupling of modes serve to modify
the scattered frequency spectrum. This modification explains why the
measured Landau-Placzek ratio is found to differ from y - l for relaxing
fluids (see Eq. 14).

The intensity of the light scattered from density fluctuations within
a small volume element containing N molecules of the scattering fluid is

given by

1m...) = Io ‘52}; K: sin2 ¢<[Ae(K,w)]2> (24)

1611

There 10

 

vector 1.
iii. )
l is the
angle be
isthe F
and u is

tude of

Where it
Scatteri
Over the
To
1“ the

Erature

Him

("I

.1.)
A
N

I).

16

Where I0 is the intensity of the incident plane-polariZed light of wave
vector Ki, R is the point of observation of the scattered light intensity
I(R, w), and scattering is considered to have taken place at the origin.

R is the vectorial sum of the scattered and incident wave vectors. The
angle between the electric vector of the incident wave and R is o, Ae(R, w)
is the Fourier component of the fluctuation in the dielectric constant,

and w is the shift in angular frequency of the scattered light. The magni-

tude of the wave vector K can be defined as

K = 2nK1. Sin(%) (25)
where n is the index of refraction of the scattering fluid and o is the
scattering angle. The angular brackets < > indicate an ensemble average
over the initial states of the system.

5 To calculate Ae(K, w) in Eq. (24) it is assumed that the fluctuations
in the dielectric constant are due to fluctuations in the density and temp-

erature, e.g., c = €(p, T). Then
_ BE BE (26
A5 — (——-)T Ap + (STlp aT . )

Bo

With

N

 

where :l
the evai
fluctuat

irrevers

 

in terms
adapting
assumed
to the t
the hydr
assumed

be expar

 

‘The

Cor)

1131'

17

where p(K, w) is a Fourier component of the density fluctuation. For

the evaluation of the mean square Fourier component of the density
fluctuation, Mountain utilizes the linearized hydrodynamic equations of
irreversible thermodynamics. These equations must be solved for p(K, w)
in terms of an initial fluctuation p(K), which can be accomplished by
adapting Van HoveS [47] method. In our present analysis it will be
assumed that the transfer of energy from the internal degrees of freedom
to the translational degrees occurs by a single relaxational process. In
the hydrodynamic and energy equations the deviation from equilibrium is
assumed to be small so that the mass density p, and the temperature T can

be expanded about their equilibrium values (po and To, respectively),

p = 90 + 01
(28)
T = To + T1
The linearized hydrodynamic and energy equations are:
Continuity
Bo
l . + =
fi— ‘1’ p0 dTV V 0 (29)

Navier-Stokes:

V

2
089

O

 

3V _ 4 .
po §?-- - grad p1 - grad T1 + (§ Rs + nV) grad dlv V

+ ft n’v(t - t’) grad div V (t’)dt’ ; (30)

Ene

In these
the ther=
is the l.
The bulk

RV. and .

 

 

 

“V (t) I
Accc
CGLLEPEc

18

Energy Transport:

ST] Op] , 2
ovaigf—i ' [CV(Y'1)/B] 3E”" A V T] = 0 . (31)

In these three equations 8 is the thermal expansion co—efficient, X is
the thermal conductivity, y = CP/CV is the ratio of specific heats, V0
is the low-frequency sound velocity, and the shear viscosity *15.
The bulk visocsity consists of two terms, a frequency-independent term
Viv, and a frequency—dependent term which is the Fourier transform of
nv’(t) where t is the time.
According to Komarov and Fisher, [48] the intensity of the light

scattered by N molecules of a fluid with effective molecular polariz-

ability, a, is

 

IN
I(§,u) = ° o2 K14Sin2 4 s(k,u) (32)

where S(K, w) is the generalized structure factor, related to the Fourier

component of the density fluctuations by
S(K,w) = <o(sz) o(-K)> - (33)

The ordinary structure factor S(k) can be calculated by integrating

Eq. (33) over all possible angular frequencies in the liquid. Hence,

I

S(K) = { S(K.w)dw = <o(K) o(-K)> . (34)

N

11 on

The gene‘
tions by

scatterer

where

Here 5 is

Eqs.
fluctuati
the densi

to the fa

 

*[

 

19

The generalized structure factor is related to initial density fluctua-
tions by a function o(K, m), which is the frequency distribution of the

scattered light:

 

S(K.w) = <o(K) o(-K)> o(K.w) (35)
where
0(K,w) = 2 Re [::(E’ Sg(2£)§) ] s = 1w . (36)

Here 5 is a dummy variable utilizing Fourier and Laplace transforms.
Eqs. (29)-(3l) are solved for the time-dependence of the density

fluctuations by eliminating the velocity and obtaining two equations in

the density and the temperature. Approx1mate transformations then lead

to the following result:

 

 

 

 

 

 

2X’ K ZoC/p
C(sz) 5" (1 '1/Y)[ I OP 2 2] +
(A K 2/oOCp)
[(vmz- oZle-(vz/vo 2-1 1)(vO 4/v2 2+ +v02 K2(1 - 1/Y)> 1 x
+
(vo 4/v4 20 + V2K2)
VZ/VZI
X v0 ”/v4 2 2] *
[l-v Wu -1/v)]Iv2K2 +v 2N212] - (v 2-v 2,18
+[ o 4 4 2 2 2 m 0 1 X
vo /v +v K
I' 1"
X I 32 B + B J . (37)

58 is th
phonon Si
is the 11
the theri
internal
In
The first
sents dec

1
half-widi

 

Where

Cn

on the 81 r

 

 

 

20

1‘8 is the half-width at half—height of a Brillouin peak, V is the
phonon speed calculated from the Brillouin shift “B (see Figure 2), V0°
is the infinite-frequency phonon speed, and T is the relaxation time of
the thermal-diffusion process responsible for the weak coupling of the

internal to the translational degrees of freedom.

-1 2).

In Eq. (37) all four terms are Lorentzian in character (I = A + Bw
The first term corresponds to the non-propagating Rayleigh peak, and repre-
sents decay of a density fluctuation by thermal-diffusion processes. Its

half-width at half-height is given by

2
X’K
r = (38)
R pon
where
2 2
k2 = §-"——'—‘— (1 - C05 6) . (39)
A 2
0

Therefore PR for the Rayleigh peak should increase when the scattering
angle 6 increases.

The second term corresponds to a non-propagating density fluctuation
decay and is related to the internal degrees of freedom of the molecules.

This peak is call the "Mountain line", and its half-width at half-height is

O<
N

(40)

N!

1M

<
v-i

Consequently, PM depends on the dispersion of the velocity of sound and

on the single relaxation time (T) of the liquid.

o
.< me_m h e ace com 1 e.g.o.oa ea 2: Lou Eaepoeam .N .mea

O

 

 

 

 

 

m> m9

 

 

 

 

Am_mum ow uocv mvwacwp pmos \
com Fera>p meuomamtcwso__wcm

1.111W1 .

 

 

 

 

 

 

 

T
fluctua

frequen:

where

ln9 scati

Brillouir

 

 

22

The last term corresponds to the decay of the propagating density
fluctuation (Brillouin peaks), and the frequency shift from the incident

frequency wo is given by

Aw = V°K . (41)
where
K = 319— Sin(9) (42)
X0 2 '

Eq. (42) shows that the frequency shift (Aw) decreases with decreas-
ing scattering angle a. The half—width at half-height of the shifted

Brillouin peak is:

2

 

1 4 X’ v0 2
r ={-- -r1 +n +——( -—-)1}1< +
B po 3 s V Cp . v2
2 2
vm-v .2
+( 232m $313516 (43)
1+VIK p0y

TB is strongly dependent upon K2 and, therefore, decreases with a decrease
in the scattering angle 6.

The ratio of the intensities of the central and shifted peaks, corres-
ponding to the Landau-Placzek ratio RLP in the absence of relaxation, is

called JV. From Eq. (37) it follows that

For low

Landau-

and far

 

23

4

 

 

V 2_ 2 V
(will 3 2 + VZKZ) + (Va Vo ) - (V 2 - or 2 2 +v 2K2“ - 1)]
V T V0 V T Y
o 2 2 o 2 2 2
[1 ' 'V‘Z" (1 ' 1/Y)][V K ‘1' E7] " (Vm - V0 ) K ,

For low phonon frequencies (VKT<<1), this equation simplifies to the

Landau-Placzek ratio (13),
J=-—-=R =y-1 ,
and for large phonon frequencies (VKT>>1) it reduces to the simpler form

JV=(V—°°—2)v-1 . (45)
o .

1. San}:

filtered
mounting
Obtain q

Th
With the
and dust
filter 1
milljpor.
149 Cell
and a te‘
PressUre

 

 

d‘Eoend i n;
A VaCUuo

has not 3

 

CHAPTER III

EXPERIMENTATION

1. Sample Preparation.

 

Silicone samples were obtained from Dow Corning C0. and were
filtered by an apparatus, designed primarily by Yuen, [53] before
mounting on the instrument. The liquid must be dust free in order to
obtain quantitative information from the calculations.

The apparatus consists of two filters arranged in series along
with the sample cell, as is shown in Figure 3. Any contact with air
and dust is eliminated while the sample is being filtered. The first
filter is an ultrafine pyrex sintered glass filter with a solvenert
millipore filter with a pore size of 0.25 n. The cylindrical scatter-
ing cell was connected to the apparatus with two Fisher-Porter joints
and a teflon gasket, making a completely sealed system. A hydrogen
pressure of about 20 psi was applied on the t0p of the silicone to
force it through the filters. This was repeated at least four times,
depending upon the silicones, until the sample was completely dust free.
A vacuum pump was connected to the vacuum manifold; the cell was then
vacuum sealed and could be used repeatedly. The Brewster angle cell

was not sealed.

24

if,

Purifie
Nitrogg

Y

10
iaiuum
lfnlfol

9. 3.

 

25

 

 

 

 

 

 

 

 

 

 

     
     

 

         

Purified
Nitrogen
U.F. Sintered
( / Glass Filter
fl ,
._- 0.25 p Solvenert
‘ , Millipore Filter
Valve B
Valve C "'i'"
J— I!
To "" I ’
Vacuum
Manifold o

 

 

11L

 

=4

 

5T\./‘
A

 

Sample Tube

 

 

 

Fig. 3. Filtration Apparatus.

_l

2. [be]
ing was :
Cate Un'
instrume'l
laser, r;
tube, pig
alarge,
ternal V1
keep dusi
tion of

Cussion c

 

 

26

2. The Brillouin Spectrometer.

 

The Brillouin spectrometer for measuring Brillouin light scatter-
ing was designed and constructed in the Chemistry laboratory at Michigan
State University [50]. A diagrammatic sketch and pictorial view of the
instrument is given in Figure 4. The spectrometer consists mainly of a
laser, rotating table, optic housing, collecting lenses, photomultiplier
tube, picoammeter and recorder. The whole optical system is mounted on
a large, flat, acoustically-isolated table in order to eliminate any ex-
ternal vibrations. The incoming room air is also filtered in order to
keep dust particles at a minimum level inside the room. A brief descrip—
tion of the equipment is given in this chapter and a more complete dis-

cussion can be found in Reference 50.

a. L§§g§_
- The source of light is a commercial Spectra Physics argon-ion
laser, Model 165-03. The light coming out of this laser is very intense
(intensity up to 800 mw), monochromatic (single frequency) and fully-
polarized. There are eight different wavelengths which can be used for
light scattering measurements, five of which (5145 K, 4965 K, 4755 A and
4579 A) have sufficient intensity to provide good spectra. In the present
measurements a wavelength of 5145 A was used with the single mode intensity
of 250 mw. This intensity gives better stability of the light output and
hence a good spectrum.

Vertically polarized light was used for most of the measurements;
however,- for the depolarization measurements, the electric vector was

rotated by 90° in order to obtain horizontally polarized light.

27

 

 

MAOIZHQ 1 a
mommHz n z
mzmA l 4
m4um<m t m
ach<mmzmo

az<m m<mzH4
mw<h40> IQHI

prmEoeuumam ew:o_FwLmuzmwm_>mm .v .mwm

 

— mmomoumm

A

 

—

 

mmhmzz<ouma #1

 

 

 

>4am2m

 

 

 

mmzoa

17

 

 

 

 

 

 

.zomwummth 4

 

 

 

 

 

 

 

mmmaa . a aamam .a awe.

Emzzoza a .11 - r
.r 1F yI._L>_ .415: HX>
- _ KIU< _ e

a 11 11 _ -
x. 1.
F _ _
z

 

 

 

 

Fr<

hza

 

 

mmm<4 zowm<

 

 

 

 

 

#11:1.

 

m4m<H 4<uHhao

 

 

 

>uaa=m -
mmzoa

- muNHSm<em1~1
>ozm=ouma _

 

 

 

 

 

 

instrur
0f the ‘
oneter
0 9
f the 1
feromet
surface
d' 55
lstribl
o

f freq

1
11

 

[S
1.56]

 

28

b. Interferometer

The interferometer is the most sensitive component of the entire
instrument. Its alignment and operation is very critical to the quality
of the Brillouin spectrum. A commercial Fabry-Perot scanning interfer—
ometer, Lansing Research Model 30.205, was used for resolving frequencies

9 Hz from the incident frequency of 1014Hz. The inter-

of the order of 10
ferometer consists of two, one-inch diameter mirrors which have inside
surfaces polished to A/lOO flatness. It allows a very narrow frequency
distribution to pass at one time, but can be scanned—over a small range
of frequencies by varying the optical path length between the two mirrors.

The intensity of the light transmitted by the mirrors is given by
[51,56]

1..
I = 1nc1dent (46)

4R . 2 6

Sln -—-
(l-R)2 Ao

 

trans l +

where a = 2nnt Cos o. It can be seen from Eq. (46) that I depends on

trans

the incident intensity, I t’ the reflection co-efficient R(R=98.5%

inciden
for the mirrors used), and the Optical path length 6. In order to obtain

a Brillouin spectrum the optical length a can be changed by varying the
index of refraction n between the mirrors, the mirror separation t, and

the angle of refraction in the material a. For scanning through the inter-
ferometer, the mirror separation t was changed by varying the voltage from
0 to 1600 across two opposite faces of the piezoelectric crystal attached
to one of the mirrors. This voltage was linearly increased with time by a

Lansing Research Model 80.010 power supply. The applied voltage gradually

expanded the crystal, decreasing the mirror separation, and almost five

spectra
were av
Th

the Fre

Th
(1.3 cm
nin : l
in orde
and yet
BrilTOU

Th
and ref
(about
0f the

Quantit

 

29

spectral orders could be obtained. Only the three most accurate orders
were averaged in the final analysis in order to secure better accuracy.
The total frequency range scanned in one spectral order is called

the Free Spectral Range and is defined as

f = (47)

_C._
2nt
This quantity is inversely proportional to the mirrors separation t
(1.3 cm to 1.4 cm for the present measurements) and the index of refractions

n(n = l for air). Care was taken in choosing the correct mirror separation
in order to eliminate the overlap of Brillouin peaks from adjacent orders,
and yet identify the correct central Rayleigh peak associated with each
Brillouin peak.

The instrumental band width VB” is dependent upon the transmissivity
and reflectivity of the mirrors and appears as an instrumental constant
(about 300 MHz for the instrument used). It is the full width at half-height
of the Rayleigh peak when perfectly monochromatic light is passed, a

quantity which depends very much upon the alignment of the instrument.

0992::

There are two achromatic lenses and three pinholes for defining the
direction of the scattered light, as illustrated in Figure 4. These are
mounted on an optical rail inside two light-tight boxes. The scattered
light is collected by a collimating tube near the scattering cell, which has
a variable aperature at each end. The first aperture (1.0 mm in diameter)
determines the acceptance angle of the scattered light, while the second
aperture (3.0 mm in diameter) determines the core angle.

The scattered light is then collected by an achromatic lens of focal

length 50 cms before entering the interferometer. The focal point of this

3O

lens falls within the scattering cell. To reduce the light reflected off
the front surface of the front mirror and prevent it from re-entering the
beam path, a 1.5 cm aperture is placed between the first lens and the front
of the interferometer.

The second lens of focal length 100 cm is placed behind the interfer-
ometer and this focuses on a pinhole in front of the photomultiplier tube,
thus allowing detection of a central spot of the ring and increasing the
fineness by about a factor of two. Fineness is defined as the ratio of the
width at the half-height of the central Rayleigh peak to the separation be-
tween the central Rayleigh peaks of consecutive orders. Hence, the higher

the fineness, the better the alignment.

d. Alignment

A Spectra-Physics Model 125 helium-neon laser was used for initial
alignment; an oscilloscope was then used to improve upon the initial align-
ment. It was essential that the alignment be good in order to obtain re-
producible spectra. Maximum care was taken with these procedures, the
instrument being aligned after every run. A more detailed discussion of

alignment procedures can be found in Reference [62].

e. Dectection and Recording

 

An EMI 9558 B photomultiplier tube was utilized to detect the
scattered light. A regulated high voltage power supply from KEPCO was
used for supplying 1100 volts to the photomultiplier tube. Its cathode
was continuously cooled to -lO°C with a Products for Research Model TE-104TS
refrigerated chamber. The detector Signal was then fed into a preamplifier

with a variable current range (typically 0 - 10.9 amps). The damping

31

control was set at about 25% of the maximum value. The signal was then
fed into a Keithly Model 417 picoamplifier, and finally to a Sargent Model
SRC Strip Chart Recorder for a read out of the final spectrum.

3. The Temperature Control Cell.

 

A pictorial view of the temperature control cell is given in
Figure 5 and a simplified cross-sectional view appears in Figure 6. This
cell consists of a hollow copper cylinder thermostat, insulated on the out*
side, and with a calibrating cell inside. (Cflycerin was used for the cali-
bration of the temperature control cell.) The cylinder is 4 3/4 inches
long and 3 1/4 inches in diameter. The inside cavity which houses the sample
cell is 3 inches long and 2 inches in diameter. The copper base plate (1/4 _
inch thick) is discussed later in Section 3. A cooling coil is bihelically
wound outside the copper cylinder and projects at the top for connection to
an external coolant supply.

An epoxy resin shell 7/8 inch thick fits over the copper cylinder and
is bolted to it at the top, so that the entire temperature control cell can
be conveniently lifted on and off, leaving the sample and the sample-holder
in place. A 180° viewing slot is provided in both the epoxy jacket and the
copper cylinder. The thermometer and thermocouple locations are clearly
shown in the sketch. For more details see Reference [52].

A YSI Model 72 temperature controller was used for applying voltage

to the nichrome wire in order to heat the copper cylinder. A band width of
0.1°C was set on the controller while the temperatures could be set to the

nearest tenth of a degree.

The controller was set at a temperature ranging from 25°C to 80°C

32

Thermometers \C\l‘q

 

 

 

 

 

 

 

 

 

/1 1
x
5%

 

 

 

 

 

 

 

 

3
§ 4%

Thermocouple

Electrical

///// ' Connections

for Heating

-~.Cooling Coils

 

Fig. 5 Pictorial View of the Temperature Control Cell.

 

 

180
Vie
Por

Insd
639'

Flat

F13.

 

 

Epoxy

Jacket ‘\\\\\

Sample

Cell ‘\\\\\\\H

Glycerin

180°

Viewing

Port \\\\\\\\\
Insulating

Base ‘*\~

Plate \\7

33

Mercury or Platinum
Thermometer

5\\\\\\‘q

 

 

\

 

 

Temperature
Control Probe

Copper
”1” Jacket

K

Cooling
r/’/’C011S

 

/
.1
F__J

OGQ‘P

 

 

 

5‘5L\\_L/

 

 

 

06)

 

 

O

 

 

 

 

 

 

 

Heating
L”’ Wires

Sample Cell
we’r”’ Holder

 

 

F/’,,,A

 

 

 

 

[1

L____

 

 

 

 

 

,/’
’/////’

Aluminum Bar on
Rotating Table Top

1 1
Lake:

entering Pin

Fig. 6. Cross-sectional View of Temperature Control Cell

34

TABLE I
Measured and Corrected Temperatures Obtained During
Brillouin Scattering Measurements for Linear

and Cyclic Polydimethylsiloxanes.

Measured Temp. Corrected Temp.
TB(°C) TC(°C)
25.00 24.75
29.60 29.32
34.70 34.21
39.60 38.88
44.60 43.84
49.60 48.60
54.50 53.30
59.30 58.25
64.30 62.90
69.20 67.80

79.20 77.60

35

. F—mo POLpCOU .QEQH mcu. LOW mumo GOTHGLDW F60 .N MLDGwm
AUOV me
ow 0m 00 om ow om ON
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36

using an interval of 5°C for temperature calibration. Two hours were
allowed for the glycerin sample to attain thermal equilibrium after every
change in temperature. In Table 1, TA is the temperature set on the YSI
Model 72, TB the temperature of the copper box, and TC the temperature of
the glycerin sample. The precision of the measured temperature, TB and the
corrected temperature Tc is i O.l°C. A linear relationship between these

variables was established, as illustrated in Figure 7.

4. The Scattering Cell and the Holder.

 

The scattering cell consists of a precision-bore tube one inch in

diameter, sealed to a 15mm Fisher-Porter joint. The total height is about
5 inches with the bottom sealed and flattened for a base, as shown in
Figure 8. A neck was formed about 1 1/2 inches below the Fisher-Porter
joint. This cell was then attached to a similar Fisher-Porter joint on
the filtration apparatus with a teflon gasket, making an air-tight system.

A small amount of the incident beam is reflected back from the glass-
air interface where the incident beam exits from the cell. This reflected-
light causes a very weak Brillouin peak when scattering is measured at some

angle other than 90°,

2V n

- S . 180 - o
vB reflected — A $1” ( ) (48)

To avoid effects due to back-reflection for angular measurements, a
Brewster angle cell was used. At the Brewster angle, vertically-polarized

light will be reflected. The incident light used for Brillouin scattering

neasurer
the Brew
in the s
a 5° ang
the effe
the cyli
A
secured
and can
secure tl

Simple is

 

 

 

37

measurements is vertically-polarized. Hence all light will pass through
the Brewster angle. Some refraction will also occur if there are bends
in the scattering cell. It was found that at the silicone-glass interface
a 5° angle of the exit plane with the vertical was sufficient to cut out
the effects of back reflection. To avoid any undesirable light reflections,
the cylindrical and the Brewster angle cells were painted black.
A sample cell holder is shown in Figure 9. The sample cell is
secured to the holder with an "0" ring which is located between two plates
and can be tightened by the three small screws shown in the figure to
secure the sample in place. The height and vertical alignment of the

sample is adjusted with the three supporting screws indicated.

 

5. Refractive-Index Measurements.

‘ The refractive-indices of the liquid silicones were measured by a
Bausch and Lomb Abbe 3-L Refractometer at a wavelength of 5890 A. Indices
for all nine silicones were measured at five different temperatures (26°C,
35°C, 45°C, 55°C and 70°C). Temperature control for the refractometer
prism was provided by a Haake circulating-bath temperature control unit.

The refractive indices for A = 5145 A were calculated from the infor-
mation given in the dispersion table series 5l6 provided by the Bausch and

Lomb Company. A more detailed discussion is given in Appendix A.

 

 

 

 

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use“, LUuLOQTLozm‘k - IN

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38

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fl .L luv— :8 22.3 $3.655 82am

 

 

 

 

 

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39

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A

 

 

 

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WZ/Z/w/Z33???”52,6—y?éy/4//..,yé////.27/

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ratio Of!

CHAPTER IV
RESULTS AND DISCUSSION

The liquids studied in the present work are linear silicones (linear
polydimethyl-siloxanes)
CH3

MDLM = (CH3)3 Si - 0 -Si - - o - Si(CH3)3

CH3
L

and cyclic dimethylsiloxanes

where 2==0,3,7 and m = 3, 5,9 and TS. Dow Corning 200 fluids, lOO cts. and

105

Cts. were also studied. In the above chemical equations M = (CH3)3Si0]/2
and D = (CH3)25i0.

The siloxane bond flexes and rotates fairly freely about the SiO axis
especially with small substituents, such as methyl on the silicon atom. As
a result of this freedom of motion, intermolecular distances between methyl-
siloxane chains are greater and intermolecular forces are smaller.
The present objective is to calculate the velocity of sound, Vs’ the

adiabatic compressibility 85’ the sonic absorption co-efficient a and the

ratio of the scattered light intensities JV, all as a function of temperature

40

and mall
lhn

where, [
incident
investii
sound wt$
as Showjl

from thq

 

 

 

41

and molecular weight for linear and cyclic silicones.
The velocity of sound V5, is given by Eq. (21), with A0 = %—-,
o

CvB

V = w———————7;
va0 Sin?

5

where, C is the velocity of light in a vacuum, v0 is the frequency of the
incident light wave, n is the index of refraction of the silicone under
investigation, a is the scattering angle and VB is the frequency of the
sound wave - which can be measured directly from the Brillouin spectrum
as shown in Figure 2. The adiabatic compressibility as can be calculated

from the velocity of sound V5 and density data, using the relationship

_ l
85 " V""2' (49)
p s
where p is the density of the liquid.

The sonic absorption co-efficient is defined as

a = I‘B/VS (50)

where r is the half-width at half-height of the Brillouin peak and can be

B
obtained from the spectrum, also as illustrated in Figure 2.

While v3 and PB come from the spectrum, separate measurements and cal-
culations were required for the index of refraction as a function of tem-
perature and molecular weight. These are described in the following section,

prior to the discussion of spectral measurements and related results.

[4
ncnexis
angle
crease

remains

 

42

It should also be noted that velocity dispersion was proved to be
nonexistent at several different frequencies by varying the scattering
angle 6, and that the vertical polarization ratio, pv, was shown to de-
crease with temperature, while the horizontal polarization ratio, ph,

remained constant at all temperatures for every silicone studied.

l. Variation of the Refractive Index n with Temperature TC and Molecular

 

Weight.
Refractive indices of all the silicones studied (MM, MD3M, MD7M,

D.C. 200 fluids of lOO Cts. and 105 Cts., D3, 05’ and 015) were measured
at the wavelength of 5890 A (the sodium D line) for five different tem-
peratures (26°C, 35.l°C, 45°C, 55°C and 70.5°C). Since the light scatter-
ing observations were made using incident light of 5145 K wave-length, the
refractive indices were calculated for A = 5145 A from the A = 5890 A
value (See Eq. A-l). Details of calculations and values of n5145 X at the
different temperatures are presented in the Appendix A.

Data for n5890 K and n5145 K at different temperatures are provided
in Tables 2 - 4. Typical graphs of the calculated refractive index,
n5145 3, versus temperature for all silicones are shown in Figures l0 - 12.
It is obvious from Figures 10 - l2 that there is a linear relationship be-
tween refractive index and temperature for all silicones, and that there is
no anomaly in the refractive index behavior in the temperature range of
25°C to 70°C. It can be observed from Tables 2 - 4 that at constant tem-
perature, refractive index increases as molecular weight is increased. It
is also clear from Figure l2 that, for the same molecular weight, cyclic
compounds (D5 and 09) have a higher refractive index than the linear compounds

(MD M and MD7M).

3

43
TABLE 2
Refractive Index as a Function of Temperature at

A = 5890A and 51453

Liquid T(°C) "5890A n5145K
MM 26.0°C 1.3750 1.3778
35.0°C 1.3700 1.3730

45.0°c 1.3645 1.3673

55.0°c 1.3584 1.3605

70.5°c 1.3496 1.3524

Liqu‘d T(°C) "5890A n51453
MD3M 26.0°C 1.3905 1.3936
35.0°C 1.3860 1.3893

45 0°C 1.3816 1.3847

55.0°c 1.3771 1.3801

70.0°c 1.3705 1.3733
L1°Ui° T(°C) "5890A n151453
MD7M 26.0°C 1.3950 1.3989
35.0°C 1.3924 1.3955

45.0°c 1.3886 1.3915

55.0°c 1.3840 1.3870

70.0°c 1.3778 1.3806

Refractive Index as a Function of Temperature at

A = 5890A and 51453

Liquid

D.C.
fluid.

100 cts.

Liouid

D.C.
f1U1d o

105 cts.

44

TABLE 3

T(°C)

26.0°C
35.1°C
45.0°C
55.0°C
70.5°C

T(°C)

26.0°C
35.1°C
45.0°C
55.0°C
70.5°C
80.0°C

n589011

1.4030
1.3992
1.3955
1.3916
1.3859

"58903

.4036
.4000
.3965
.3926
.3870
.3831

dddddd

n51453

1.4055
1.4025
1.3988
1.3945
1.3888

n514531

.4053
.4033
.3998
.3959
.3901
.3859

dddddd

Refractive Index as a Function of Temperature at

Liquid

Liquid

Liquid

15

45

TABLE 4

T(°C)

70.1°C
80.1°C

T(°C)

26.0°C
35.1°C
45.0°C
55.0°C
70.5°C

T(°C)

26.0°C
35.0°C
45.0°C
55.1°C
70.2°C

T(°C)

26.0°C
35.1°C
45.0°C
55.0°C
70.5°C

A = 58903 and 51453

n58903

1.3602
1.3550

n58903

1.3960
1.3917
1.3873
1.3825
1.3753

n5890A

1.4050
1.4016
1.3980
1.3936
1.3878

n58903

1.4045
1.4015
1.3979
1.3939
1.3878

n51453

1.3633
1.3578

n51453

1.3989
1.3945
1.3905
1.3854
1.3782

n51453

1.4082
1.4047
1.4008
1.3970
1.3819

"51453

1.4080

1.4067
1.4007

1.3971
1.3908

 

 

 

 

i313]

”5145

 

'1JUF9

46

 

 

- 0.0. 200 fluid 105 cts.
] 4]_ U D.C. 200 fluid 100 cts.
° 0 MD7M
- XMM
1.40-
1.39-i
"5145 3
l.38~
1.37%
1.36..
1.354
1.34 1 I .
25 35 45 55 65 75

 

Tc( C)
Figure 10. Refractive Index Versus Temperature for MM , MD3M, MD7M,

5

D.C. 200 fluids, 100 cts. and 10 cts.

 

 

‘ ”5145

 

47

 

 

a. 09
V
D15
£3 05
c> 03
1.41-
1.40-
1.39-1
D9
D15
1.38-i
n5145 3
D5
1.37-4
1.36..
1.35-1
1.34 I
25 35 45 55 65 76

Tera)

Figure 1]. Refractive Index versus Temeprature for D3, 05. 09 and 015.

 

 

514

F1

48

1.41’

  
    

‘09 (M01. wt. 667.0)

1.40— OMD7M (1401. wt. 681.0)

A05 (Mol. wt. 370.5)

01403114 (1401. wt. 384.6)

1.39-

"5145 3

1.38*

1.37

 

 

j l 1 r

l I I | l l
25 30 35 40 45 so 55 60 65 70 75
T(°C)

Figure l2 . Refractive Index Versus Temperature for 09, MD7M, D5 and MD3M.

I I
I
I
I
III
v ,

M

18:
sh

:ed

ll“

UE

1‘61:

1h

at-

Fe?

11'

49

2. Variation of the Brillouin-Shift ”B with Temperature Tc and Molecular

 

Height.

Light waves scattered from the sonic waves in the liquid display'a
three-peaked intensity-frequency distribution, as illustrated in Figures 13 -
16. The two symmetrical Doppler-shifted side peaks are known as the
Brillouin peaks and their shift, vB, from the central Rayleigh peak is given
by the Eq. 22.

The Brillouin-shift can be measured directly from the spectrum as indi-
cated in Figure 2 of Chapter II, and is in the gigahertz (GHz) range. From
Figures l7, l8 and l9 it is clear that VB varies with the temperature,
molecular weight and molecular structure of the silicones. Figures l7 and
l8 Shaw that ”B varies linearly with temperature for the silicones investi-
gated. Figure l9 provides a good comparison of how VB changes with tempera-
ture for linear and cyclic silicones having about the same molecular weights.

The values in Tables 5-13 establish that v increases with increasing molec-

B
ular weight for all silicones.

These tables contain measurements of 03 and PB as well as calculated
values of V5 and 6. Tables l4-2l contain calculations of various slopes

dI‘B

dot . . .
(de/ch, st/ch, Ef“ and a-TZ). All of these results w1ll be utl112ed

as required in the subsequent discussion.
Tables l6 and l8 contain values for the slope and intercept of the

Brillouin-shift-temperature curves. As can be seen from Figure 20, the

dv
relation between aTg-and molecular weight for linear silicones is nonlinear,
C dv
whereas the relation between HTE'and molecular weight for cyclic silicones
c

is linear.

.< mSm n

 

6‘ nee .05 n o .000“ we :2 Lee Esepueam .m_ .533

 

50

 

 

 

 

 

m9

 

 

 

 

 

 

.4 merm n ox eee .08 n o .ooo.oe “5 Ema: toe Eeegeeam .ep .52;

O

 

 

51

 

 

 

 

 

 

 

 

 

 

 

. m. v.
m
a

 

52

 

.< me_m n 6‘ 6:8 .05 n 6 .000m 8e gee: toe Eeeueeem .m_ .323

O

 

 

 

 

m>

 

 

 

 

m>

 

 

53

H

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1:
it

 

 

 

0

Fig. l6. Spectrum for 05 at 45°C, e= 90° and A0 = 5l45 A.

 

Ter

241
291

38.
43.

53.

S8.
67.

77,

 

54

TABLE 5
Observed Brillouin Frequency Shifts (93), Velocities (VS),
Temporal Attenuation Coefficients (PB), Life Times (l/PB)

and Spatial Attenuation Coefficients (a) of MM

as a Function of Temperature.

’1

Temp “8 x l09 VS PB x l06 3B x 10'9 a = V:-
Tc (°C) Hz m|sec Hz Hz Cm"1
24.75 3.3438 882.37 95.403 10.482 1081.22
29.32 3.3507 885.92 96.013 10.415 1083.76
34.21 3.3193 879.40 87.785 11.392 998.22
38.88 3.1485 835.70 97.978 10.206 1172.27
43.84 3.0456 810.19 92.919 10.762 1146.88
48.60 2.9640 790.06 88.668 11.278 1122.29
53.30 2.9071 776.46 102.970 9.712- 1326.15
58.25 2.7766 743.14 89.762 11.141 1207.88
67.80 2.62l0 704.440 82.404 12.135 1169.85
77.60 2.5600 690.92 80.628 12.403 1166.96

55

TABLE 6
Observed Brillouin Frequency Shifts (vB), Velocities (VS),
Temporal Attenuation Coefficients (PB), Life Times (l/PB)

and Spatial Attenuation Coefficients of MD

a Function of Temperature.

311 as

Temp 63 x 10 vS r3 %E-x 10‘9 a =‘V3
Tc(°C) Hz m|sec x lOGHz Hz Cm"1

24.75 3.6342 948.40 163.195 6.128 1720.74
29.32 3.5738 934.04 155.747 6.420 1667.46
34.21 3.4704 908.46 156.480 6.391 1722.46
38.88 3.4326 899.99 161.726 6.183 1796.98
43.84 3.4224 898.81 143.424 6.972 1595.71
'48.60 3.2903 865.48 139.262 7.181 1609.08
53.30 3.2445 854.73 140.909 7.097 1648.58
58.25 3.1803 ' 839.23 115.959 8.624 1381.74
67.80 3.0770 814.55 119.022 8.402 1461.19
77.60 2.9343 779.84 95.356 10.487 1222.76

56

TABLE 7
Observed Brillouin Frequency Shifts (vB), Velocities (VS)’ Temporal
Attenuation Co-efficients (PB), Life Times (%—) and Spatial Attenuation
B

Co-efficient (a) of MD7M as a Function of Temperature.

1‘

Temp. “8 x lo.9 VS [‘8 x 10.6 11:3: x lo"9 a = ——2—
(°C) Hz m|sec Hz sec Cm"1
22.35 3.8265 998.6 195.082 5.1260 1953.45
29.60 3.8026 992.4 208.338 4.800 2099.30
40.31 3.6327 948.1 196.455 5.090 2072.18
49.97 3.5867 936.1 174.090 5.744 1854.80

57.35 3.5035 914.43 140.945 7.095 1541.48

 

 

Obse

and S

 

24.75
29.32
34.21
38.88
43.84
348.60
53.30
58.25

 

67.80
77.60

 

57

TABLE 8
Observed Brillouin Frequency Shifts (08), Velocities (VS),
Temporal Attenuation Coefficients (PB), Life Times (l/rB)

and Spatial Attenuation Coefficients (a) of D.C. 200 fluid,l00 cts.

as a Function of Temperature.

Temp ”8 x lo9 VS PB x 106 4E-x l0"9 0 = 1:-
Tc(°C) Hz m|sec Hz Hz Cm'1
24.75 3.9334 1017.6 260.96 3.8321 2564.3
29.32 3.6476 944.9 216.96 4.609 2296.2
34.21 3.6812 954.9 220.26 4.540 2306.7
38.88 3.6992 960.7 192.25 5.201
43.84 3.5638 926.8 196.73 5.083 2122.7
3 48.60 3.5799 932.1 213.16 4.691 2286.7
53.30 3.6466 950.7 209.63 4.770 2204.9
58.25 3.4067 889.4 197.49 5.064 2220.4
67.80 3.3670 881.3 195.79 5.103 2221.4
77.60 3.2554 854.4 198.58 5.035 2324.3

58

TABLE 9
Observed Brillouin Frequency Shifts (03), Velocities (VS)’
Temporal Attenuation Coefficients (PB), Life Times (l/PB)

and Spatial Attenuation Coefficients (a) of D.C. 200 Fluid,

l0S Cts as a Function of Temperature.

Temp 03 x 10"9 VS PB x l06 %E-x 10.9 a =-:E
Tc(°C) Hz m|sec. H2 H2 Cm-1

24.75 4.2614 1102.2 144.32 6.929 1309.35
29.32 4.0813 1056.8 137.68 7.263 1302.75
34.21 4.1456 1074.8 140.69 7.108 1308.99
38.80 3.9852 1034.5 186.07 5.374 1798.65
43.84 3.8796 1008.4 142.87 6.999 1416.89
348.60 3.8311 997.1 161.93 6.176 1624.02
53.30 3.9480 1028.7 154.92 6.455 1505.91
58.25 3.7388 975.5 136.46 7.328 1398.88
62.90 3.6730 959.5 119.39 8.376 1244.26
67.80 3.538 925.3 140.28 7.128 1516.03
77.60 3.489 915.0 133.11 7.513 1454.68

59

TABLE 10
Observed Brillouin Frequency Shifts (vB), Velocities (VS), Temporal
Attenuation Co-efficients (PB), Life Times (l/rB) and Spatial Attenuation

Co-efficients (a) of 05 as a Function of Temperature.
r

1c 68 x 109 vS rB x 10° 1.: x 10’9 Gav:—
(°C) Hz mlsec. Hz Sec. Cm"1
24.75 3.6416 947.8 245.91 4.067 2594.5
29.32 3.4905 908.5 222.39 4.497 2447.9
34.21 3.3999 884.9 202.55 4.937 2289.0
38.88 3.3748 878.4 211.47 4.729 2407.5
43.84 3.3036 859.8 174.85 5.719 2033.5
52.58 3.1891 830.0 162.01 6.173 1951.8
TABLE 11

Observed Brillouin Frequency Shifts (08), Velocities (VS), Temporal
Attenuation Co-efficients (PB), Life Times (%—) and Spatial Attenuation
B

Co-efficients (a) of 03 as a Function of Temperature.

9 6 1 -9 ___8
Temp. VB x lO VS 53 x lO -Fg-x l0 a-VS
Tc(°C) Hz m|sec Hz Hz Cm"1
62.90 2.8279 752.50 118.438 8.443 1573.9
67.80 2.8162 750.86 119.278 8.384 1588.5
72.64 2.7336 730.29 102.283 9.777 1400.6
77.60 2.6346 705.25 120.842 8.275 1713.5
82.28 2.5949 695.89

I‘

60

TABLE 12
Observed Brillouin Frequency Shifts (vB), Velocities (VS), Temporal
Attenuation Co-efficients (r8), Life Times (l/rB) and Spatial Attenuation

Coefficients (a) of 09 as a Funtion of Temperature.

Temp. 03 x 109 vS rB x 106 %—-x 10'9 .=V§-

B s
TC(°C) Hz m|sec. H2 H2 Cm-1
24.75 3.9828 1027.06 266.25 3.756 2592.3
29.32 3.9211 1013.01 239.30 4.179 2362.2
34.21 3.7909 981.34 271.65 3.681 2768.1
38.88 3.5829 929.21 269.68 3.708 2902.2
43.84 3.7233 967.62 275.48 3.630 2847.0
48.60 3.6591 952.76 256.60 3.897 2693.2
53.30 3.5547 927.37 241.67 4.138 2605.9
58.25 3.5222 920 76 233.05 4.291 2531.1
67.80 3.4099 894.94 215.91 4.6316 2412.5

illllullhlu 1| 3.131
I
,. —
. n
0

 

61

TABLE 13
Observed Brillouin Frequency Shifts (vB), Velocities (VS)’ Temporal
Attenuation Co-efficients (PB), Life Times (l/rB) and Spatial Attenuation

(a) of 015 as a Function of Temperature.

"1

Temp. 68 x 109 vS rB x 10° %E-x 10'9 a =°V3
Tc(°C) Hz m|sec. H2 H2 Cm"1
24.75 4.0290 1040.6 246.72 4.053 2370.9
34.21 3.8941 1008.4 189.78 5.269 1882.0
38.80 3.951 1024.5 227.81 4.390 2223.6
53.30 3 659] 952.5 192.79 5.187 2024.0
58.25 3.6026 939.1 192.25 5.202 2047.1
62.90 3.6111 942.5 197.31 5.068 2093.5
77.60 3.3734 884.1 187.87 5.323 2125.0

Intercept and Slope for the Brillouin Half-width (r

Temperature Relationship; r

Liquid

MM

MD3M

MD7M

D.C. 200 fluid.
lOO cts.

D.C. 200 fluid,
105 cts.

Hz

104
199
246
247

163

62

TABLE 14

B

8
3:15—13
C

Hz/°C

-0.284
-1 . 261
-l.574
-7.698

-3.612

TABLE 15

B)-

= A + BTc'

Standard error

of estimate for A

12
15

16

Intercept and Slope for the Absorption Co-efficient-

Temperature Relationship; 0 = A + BTC.

Liquid

MM
MD3M

MD7M

D.C. 200 fluid,
lOO cts

D.C. 200 fluid,
105 cts

A

Cm-1

1017
2016
2372
2429

1386

...dL__

0:
B ’ ‘3Tc)

Cm"/°c

2.365
-0.100
-11.682
-3.005

1.165

Standard error

of estimate for A
46
82

134
105

154

63

TABLE 16
Intercept and Slope for the Brillouin-shift-

Temperature Relationship; VB = A + BTC
2308
Liquid A B = (STr) Standard error
c
Hz Hz/°C of estimate for B
MM 3.812 -0.0170 0.045
MD3M 3.947 -0.0130 0.021
MD7M 4.053 -0.0096 0.022
D.C. 200 fluid,4.079 -0.0105 0.062
100 cts.
D.C. 200 fluid, 4.567 -0.0l42 0.062
105 cts.
TABLE 17

Intercept and Slope for the Sound Velocity-

Temperature Relationship; VS = A + BTc'

3V
Liquid A B = (——§) Standard error
3Tc
m/sec. m/sec/°C of estimate for 8
MM 1010 -4.36 11
MD3M 1021 -3.10
M07M 1058 -2.50
D.C. 200 fluid 1050 -2.49 19
100 cts.
D.C. 2 0 fluid, 1176 -3.44 16

10 cts.

64

TABLE 18

Intercept and Slope for the Brillouin-shift-

 

Temperature Relationship; 08 = A + BTC
1108
Liquid A B = (0T ) Standard error
c
Hz Hz/°C of estimate for B
03 3.691 —0.0133 0.020
05 3.940 -0.0l40 0.022
09 4.240 -0.0126 0.064
015 4.357 -0.0125 0.041
TABLE 19

Intercept and Slope for the Sound Velocity-

Temperature Relationship; VS = A + BTC.

8V

Liquid A B = (r—é- Standard error
aTC

m/sec. m/sec/°C of estimate for A

03 965 -3.272 5

D5 1018 -3.375 6

D9 1045 -2.877 17

D 1118 -2.990 11

65

TABLE 20
Intercept and Slope for the Brillouin Half-width (PB)-
Temperature Relationship; TB = A + BTC.
Liquid A B Standard error
Hz Hz/°C of estimate for A
03 129 -O.194 7
05 230 -0.677 11
09 293 -O.949 14
015 249 -0.884 15

TABLE 21
Intercept and Slope for the Absorption Co-efficient (a) -

Temperature Relationship; 0 = A + BTC.

Liquid A B Standard error
Cm"1 Cm'1/°C of estimate for A
03 1231 4.808 108
D5 2261 1.050 120
D9 2689 -l.947 154
D 2230 -2.405 138

66

 

 

 

5.o~
4.0.
.Bx1o9
Hz
3.04
- D.C. 200 fluid 105 cts.
o MD7M
2.o~ o MD3M
*4 MM
1.5 ‘ 1 l ' T I I 1 I l l
'15 20 25 30 35 40 45 50 55 70 75 8o 85
Tc(°C)

Figure 17. Brillouin Shift Versus Temperature for MM, MD M, MD7M and

0.0. 200 fluid,lO5 cts.

5.00-

9*

4.00-

 

 

wax 10
Hz

‘V 015

3.00-1 A D
A D5
19 D3

2000 I I r 1 l I V I 1 ' I V r I

15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

TT°C)

Figure 18. Brillouin Shift Versus Temperature for D3, 05, D9 and 015.

 

 

‘ 09 (M01. Wt. 667.0)
A 05 (M01. Wt. 370.5)
C MD 714 (M01. Wt. 681. 0)
5 0 .- \(1134013: 384 6)
4.0 ‘ \
VBX'10-9 \\
Hz
3.0 "1
2.0 1 l l 1 1 1 1
15 20 30 40 50 60 70 80
0
Tc( C)

Figure 19. Brillouin Shift Versus Temperature for D9, 05, MD7M and MD3M.

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70

3. Dispersion Measurements.

 

The frequency shift of the Brillouin peaks “8 increases with an
increasing scattering angle 6 and is given by Eq. 22 (See Tables 22 and 23).
Thus by increasing the angle 6 we select pressure fluctuations of higher
frequencies; the velocity of sound can then be calculated at the same
frequencies. A plot of the Brillouin-shift against sin(%), where e is
the scattering angle, shows a linear relationship (Figure 21). If dis-
persion were present in the sound velocity the relation between 03 and
Sing-would be nonlinear; so in the frequency range of interest, there is

no dispersion in the velocity of sound for linear and cyclic silicones.

71

TABLE 22
Variations in Brillouin Frequency-Shifts (08), Velocities (VS),Temporal

Attenuation Co-efficients (PB), Life-Times (%—), Spatial Attenuation
B

Co-efficient (a) and Landau-Placzek Ratio (JV) of MD7M with Scattering
Angle (e).
r
o . 0 9 6 1 -9 _ _B
0 S1n 2' ”B x 10 VS PB x 10 78.x 10 a - VS
Hz m/sec Hz Sec. Cm']
45° 0.3827 2.3665 1141.16 85.085 11.753 745.611
60° 0.50 2.7251 1005.79 97.144 10.294 965.850
75° 0.6088 3.2960 999.08 149.050 6.709 1491.880
90° 0.707 3.9068 1033.70 200.203 4.995 1936.750.
105° 0.7934 4.3230 1005.50 262.403 3.810 2609.600
120° 0.8660 4.5978 979.78 301.180 3.320 3073.950
135° 0.9239 5.0720 1013.08 356.732 2.803 3521.200
Temp. IC I JV Fineness
(°)
20.44 1.80 3.866 0.2332 41.0
20.40 2.00 4.165 0.2401 39.4
20.28 1.24 2.430 0,255] 31.3
20.40 2.65 4.840 0.2732 44.7
20.40 41.2
20.36 1.31 2.295 0.2843 41.0

 

£51.13 3 IV. ’1'. .n 5.51

 

72

TABLE 23
Variations in Brillouin Frequency - Shifts (08), Velocities, Temporal
Attenuation Coefficient (PB), Life Times (%—), Spatial Attenuation
B

coefficient (a) and Landau-Placzek Ratio (JV) of 05 with Scattering
Angle (e).

- .e. 1. -9
8 Sin 2 VB . VS PB x PB x 10
x 10 9 Hz m/sec 10 ° Hz Cm"

45° 0.3827 2.1077 1013.6 80.462 12.428

60° 0.500 2.7861 1025.5 145.568 6.870

75° 0.6088 3.0829 932.0 180.968 5.526

90° 0.7071 3.582 932.3 200.306 4.992

105° 0.7934 4.189 971.7 261.880 3.819

I 20° 0.8660 4. 580 973.4 314.867 3.176

135° 0.9239 4.885 973.1 392.462 2.548

P8
0 I V;- Tc(°C) Ic IB JV Fineness
Cm'1

793.814 19.70°C 4.000 8.145 0.2456 30
1419.475 19.68°C 3.235 6.717 0.2408 35
1941.806 19.65°C 3.835 7.100 0.2701 39
2148.524 19.7°C 3.900 6.792 0.2871 30
2694.950 19.7°C 2.275 3.760 0.3025 31
3234.730 19.65°C 2.140 3.700 0.2892 34
4033.190 19.65°C 3.020 5.757 0.2623 34

73

5.5-

5.0-

4.5-1

4.0-

03 X 10-

(H2)
33.5-

303

2.5q

 

 

 

2-0 1 1 1 r

0.3 034 035 0.6 0.7 0.8 0.9

Sim-g) (radians)

Figure 21. Brillouin-shift Versus Sin % for M0714 and 05.

74

4. Velocity of Sound VS and Adiabatic Compressibility BS’

 

Ultrasonic data in the megahertz range have been used for monitoring
structural changes in liquid silicones [36’37’54]; hypersonic data in the
gigahertz range obtained from Brillouin scattering can be used to give
analogous information at different frequency intervals. The velocity of
sound in a liquid varies with the compressibility, which in turn is related
to intermolecular forces.

Velocity of sound results calculated by means of the Brillouin scatter-
ing technique and from ultrasonic data are in very good agreement, as can

be seen from Table 24 and Figure 22.

Table 24

Molecular Velocity of Sound*
Wt. mlsec.
162.2 873.2
236.3 901.3
310.4 919.0
384.5 931.3
520.0 942.2
720.0 953.8
1,160.0 966.5

* Ultrasonic measurements by Heissler[36].

If there is no structural change in going from one liquid to another,
then the change in the velocity of sound of the different liquids (i.e.,

of different molecular weights) will be a smooth, continuous function and
dV
the change in a—é- with molecular weight will be a function with the same
C

75

properties.

Tables 5-13 display Brillouin frequency shifts and velocities of
sound as functions of temperature for linear and cyclic silicones. Tables
17 and 19 contain values for g;%— for all of the silicones investigated.
The velocity-temperature curve for each liquid satisfies the linear rela—
tionship VS = A + BTC’

From Figures 22 and 23 it is clear that there is a nonlinear depen-
dence between the velocity of sound and molecular weight for linear and
cyclic compounds. Figures 24 and 25 show that for every liquid investi-
gated the velocity of sound and temperature were linearly related, but
with different slopes. Figure 26 provides a better comparison between
linear and cyclic compounds having about the same molecular weight.

It is clear from all these graphs that if the basic structure of the
silicones is changed, (linear to cyclic or cyclic to linear) then a sharp
<3Iange in their properties will occur. It can be seen in Figure 27 that
the relationship between §¥§'

C
pound is linear, whereas for the linear compound the same relation is

and molecular weight for the cyclic com-

nonlinear.

The adiabatic compressibility 85 can be calculated from velocity of
sound and density data by means of Eq. (49).

Table 25 contains values of 65 for silicones at 22.0°C, while Table 26
contains the corresponding values for D9 and 015.

The adiabatic compressibility of a liquid is inversely proportional
to the force of attraction between its molecules, which in turn depends
directly on the velocity of sound in the liquid. By examining changes of

85. V5 and HTé' with molecular weight and temperature, insight into local
C
structural changes can be gained.

76

It is obvious from Figure 28 that as the molecular weight is increased
the adiabatic compressibility decreases. This indicates that intermolecular
forces are increasing as the molecular weight increases for both linear and
cyclic silicones. It can also be concluded from Figure 28 that the adiabatic
compressibilities of the linear silicones are greater than those of the
cyclic silicones for the same molecular weight. The implication is that in
this case intermolecular forces are stronger in cyclic chains than in linear
chains.

Figure 29 shows that BS increases linearly as the temperature is in-
creased, which suggests that intermolecular forces (attraction between Si.

and O) are decreasing with increasing temperature.

77

TABLE 25

Adiabatic Compressibility of MM, MD M, MD M, D.C. 200 fluid, 100 cts.,

3 7
05, 09 and 015 at 22°C.

Liquid Velocity Density Adiabatic

of Sound Compressibility

mlsec. g/ml BS

Sq. cm/dyne
MM 898 0.7636 162.4 x 10'12
MD3M 953 0.8755 125.8 x 10‘12
MD7M 998 0.9180 109.4 x 10’12
D.C. 200 fluid 100 Cts.1048 0.9579 95.1 x 10"12

05 962 0.9593 112.6 x 10"2
09 1048 0.9756 93.3 x 10"2
015 1055 0.9737 92.3 x 10‘12

78

TABLE 26

Adiabatic Compressibility of 09 as a Function of Temperature.
T¢(°C) Velocity Density* Adiabatic

of Sound g/ml Compressibility
m/sec. 85
Sq. Cm/dyne
24.75 1027.05 0 9756 97.2 x 10"2
38.88 970.00 0.9624 110.4 x 10"2
58.25 920.76 0.9440 124.9 x 10'12

Adiabatic Compressibility of 015 as a Function of Temperature.
Tc(°C) Velocity Density* Adiabatic

of Sound g/ml Compressibility
m/sec. 85
Sq. Cm/dyne
24.75 1044.0 0 9737 94.2 x 10"‘2
38.88 1001.7 0.9510 103.7 x 10'12
58.25 943.8 0.9435 119.0 x 10'12

* Density measurements for D9 and 015 at different temperatures were
obtained from Dr. J. F. Hampton, Dow Corning Corporation, Midland,
Michigan.

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1000 "‘
900 -
Vs
m|sec.
800 ~
0 0.5. Fluid 105 cts.
Q M0714
° MD3M
*x MM
700 '-
1 1 ' '
650 20 40 60 80

TC(°C)

Fl'9llre 24. Variation of the Velocity of Sound with Temperature for
MM, MD3M, MD7M and 0.0. Fluid 105 cts.

 

 

1100'1 82
1000-
900-
V5 '
m|sec.
800-1 ‘ D9
A US
9 D3
700-
675 T T r 1 1 1 1 l T 1
10 20 30 4O 50 6O 70 80 90 100
Tel c)
Figure 25. Variation of the Velocity of Sound with Temperature for D3, D5

D9 and D15

83

    

A 09 (M01. Wt. 667.0)
0 MD7M (M01. Wt. 681.0)
0 MD3M (M01. Wt. 380.0)
1000' A D5 (M01. Wt. 370.5)
900—
vs
_m|sec.
8004
700 , .

 

 

| 1 l I 1 l l
10 20 30 40 50 6O 70 8O 9O

0
Tc( C)
Figure 26. Variation of the Velocity of Sound with Temperature for D9,

MD7M, MD3M and D5.

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86

1.41 -

130-

-12
85 x 10

Sq. Cm/dyne
120‘-

1101

A
1004 015

90"

 

 

80 1 fi T 1 f
20 30 40 50 60 70

T(°C)
Figure 29, Adiabatic Compressibility Versus Temperature for D9 and 015.

87

5. Variation of the Brillouin Line Width F8 and the Sonic Absorption

 

Coefficient a with Temperature and Molecular Weight.

For a nonrelaxing liquid the Brillouin spectrum contains three
Lorentzian peaks [22] but for relaxing liquids each peak may represent a
continuation of simple Lorentzian forms. If the peaks are simple Lorentzian
shapes the true Brillouin line width (2FB) can be obtained from the following

equation:

21‘ =21‘ -

B BO 1‘5 (5‘)

where PRO is the observed line width of the Brillouin peak and PC is the
instrumental line width, which is the width of the central peak at its
half-height. To identify the actual shapes thirty points were picked from
each of the three peaks and compared with the analytical Lorentzian form.
In general, the central portion of the experimental peak (approximately
fifteen points) fitted the Lorentzian equation very well, while deviating
slightly both at the top and at the bottom.

The calculated values of the Brillouin line widths (using Eq. 51) are
tabulated in Tables 5-13 presented earlier. Values for §;E-are tabulated
in Tables 14 and 20. Typical plots of Billouin line width versus temper-
ature for different silicones are shown in Figures 30-31. For linear
silicones the line width is found to decrease linearly with a rise in the
temperature; however, for cyclic silicones the decrease is nonlinear with
temperature. A plot of g;§-versus molecular weight for linear compounds
shows an increasing nonlinear trend as molecular weight increases; for
cyclic compounds the same trend occurs but terminates with an asymptotic

approach to a single value (See Figure 32). The Brillouin line width also

88

has angular dependence, as predicted by Eq. 43. This dependence can
clearly be seen in Tables 22 and 23 of Section 3.

According to hydrodynamic theory [54] the absorption co-efficient a
is given by Eq. (50),

B

r
“'17?

Tables 5-13 list values of the absorption co-efficient for all silicones
studied as a function of the temperature TC'

The absorption of sound is related to the effectiveness of the trans-
fer of internal vibrational energy to translational energy in a liquid. It
is believed that if the collisions between the molecules in a liquid are in-
efficient in producing a transfer of energy, the absorption will increase.
0n the other hand, if the efficiency of transfer of the internal vibrational
energy into the translational energy is higher, then the absorption will de-
crease and sound propagation will be sustained longer. Due to temperature
change there will be some structural change in the liquid, and this change
should be reflected in the variation of the sonic absorption co—efficient
with temperature because of modified molecular interactions.

Tables 15 and 21 give the experimental values for the temperature
deviation of the sonic absorption co-efficient. From Figure 33 it can be
seen that 8 decreases linearly with an increase in temperature, thereby
indicating higher energy transfer. Figure 33 provides a comparative view
of linear and cyclic silicones of about the same molecular weight. As the
temperature is increased, intermolecular interaction increases; hence a
decreases.

Values for the temperature derivative of the sonic absorption

89

co-efficient were also calculated for all of the silicones investigated;
do

a typical graph of aT—-versus molecular weight is shown in Figure 34.
The negative values of %%—-indicate that according to the classification

C
scheme of Herzfeld and Litovitz[54], these silicones are "associated

fluids".

r x10

(H2)

Figure 30.

260 .
250 4
240 4
230 a
220 4
210 4
200 -
190 ‘
180 4
170 4
160 4
150d
140 d
l30~
120-
110-1
100‘-

90"

 

80

90

    

D.C. 200 f1uid,100 cts.

MD7M

MD3M
MM

0 ID D; D'

 

20

30 40 50 50 70 7'80

Term)

Half of Brillouin Line Width Versus Temperatures for MM,

MD3M, MD7M and D.C. 200 fluid, 100 cts.

Figure 31. Half of Brillouin Line Width Versus Temperature for 05.

280 -

270 ‘
260 _

250 -
240 .1
230 -
220 -.
210 ..
200 -i
190 -1
180 -
170 -

160 .1

91

 

 

 

150

20

D andD .

9

15

1
40
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3000 4
2800 J
2500 4
2400 -
(cm-])2200-4
2000 4
1800 4

1600 -i

1400 -

 

 

 

1200 T . r
20 31) 46 5'0 60 70 80

Tc(°C)
Figure 33.Sonic Absorption Co-efficient (0) Versus Temperature (Tc) for

09, 05, MD7M and MD M.

3

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95

6. Variation of the Ratio of the Intensities of the Central and Shifted

 

Peaks, JV, with Temperature.

 

Tables 27-35 contain values of JV expressed as a function of tem-
perature for all of the silicones under consideration. An ideal way of
comparing theoretical and experimental values would be to compare results
calculated from Eq. 44 with the experimental values of JV listed in these
tables. However, no data exists for several of the quantities that appear
in Eq. 44, so it is not possible to do more than present a semi-quantitative
explanation for the change in JV with temperature and molecular weight.

It is obvious from Figure 35 that Jv decreases with an increase in
temperature. This decrease follows from the decrease in the damping forces
in the liquid with increasing temperature. The magnitude of the pressure
fluctuations is increased because of the decrease in the damping forces.
This in turn is responsible for the change in the Brillouin peaks which
decreases JV.

It can be seen from Figure 36 that as the molecular weight is increased
JV also increases, but that the increase is slow compared with that for
other liquids, like ethylene glycol and l-octyl alcohol[633. Moreover, the
value of Jv is higher for cyclic silicones than for linear silicones at
the same temperature. It is also clear from Figure 37 that the rate of
change of Jv with temperature increases with molecular weight, but tends
toward a constant value of -2 x 10'3 for cyclic silicones beyond a molecular
weight of 600. For the linear silicones, no such definite trend was

observed (Table 36).

96

TABLE 27
Observed Landau-Placzek Ratio JV of MM

as a Function of Temperature

Tc(°C) IC IB JV

24.75 0.915 1.612 0.2837
29.32 0.970 1.858 0.2610
34.21 1.045 1.907 0.2739
38.88 0.915 1.807 0.2531
43.84 1.065 2.095 0.2542
48.60 1.180 2.252 0.2619
53.30 1.255 2.407 0.2606
58.25 1.190 2.227 0.2671
67.80 1.415 2.777 0.2547
77.60 1.555 3.087 0.2518

TABLE 28

Observed Landau-Placzek Ratio JV of MD3M

as a Function of Temperature.

Tc(°C) Ic IB JV

24.75 1.915 3.907 0.2450
29.32 2.015 4.332 0.2325
34.21 1.845 4.252 0.2169
38.88 2.185 4.593 0.2378
43.84 1.935 4.562 0.2121
48.60 2.075 4.497 0.2307
53.30 1.885 4.642 0.2030
58.25 2.425 5.785 0.2096
67.80 0.790 1.935 0.2041
77.60 0.815 2.215 0.1840

97

TABLE 29
Observed Landau-Placzek Ratio JV of D.C. 200 Fluid, 100 cts. as a
Function of Temperature.
Tc(°C) Ic IB JV
24.75 2.885 5.183 0.2783
29.32 2.340 4.275 0.2737
34.21 2.565 4.737 0.2707
38.88 2.310 4.697 0.2459
43.84 2.330 4.600 0.2533
48.60 2.425 5.222 0.2322
53.30 2.585 5.250 0.2462
58.25 3.040 6.415 0.2369
67.80 2.673 6.024 0.2219
77.60 2.523 5.946 0.2122
TABLE 30

Observed Landau-Placzek Ratio of MD7M as

a Function of Temperature .

Temp. Ic IB JV
(0.)

22.35 1.910 3.830 0.25
29.60 0.670 1.423 0.24
40.31 0.705 1.530 0.23
49.97 1.065 2.390 0.22

57.35

1.065

2.588

0.21

98

TABLE 31

Observed Landau-Placzek Ratio JV of D.C. 200 Fluid,

105 Cts. as a Function of Temperature.

Tc(°C) Ic IB JV

24.75 2.830 0.662 2.1358
29.32 4.045 0.660 3.0644
34.21 3.850 0.720 2.6736
38.88 2.435 0.745 1.6342
43.84 3.675 0.777 2.3633
48.60 1.915 0.845 1.1331
53.30 3.456 0.896 1.9290
58.25 2.320 0.957 1.2115
62.90 2.265 0.960 1.1797
67.80 3.685 0.985 1.8706
77.60 1.995 1.205 0.8278

99

TABLE 32
Observed Landau-Placzek Ratio JV of 03 as

a Function of Temperature

Tc( C) Ic IB JV
62.90 1.120 2.880 0.1944
67.80 1.536 1.815 0.4233
72.64 1.060 2.085 0.2542
77.60 2.100 2.445 0.5000
82.80 0.965 2.345 0.2058
TABLE 33

Observed Landau-Placzek Ratio J of D

a Function of Temperature

Tc(°C) Ic IB JV

24.75 2.825 5.092 0.2770
29.32 3.050 5.800 0.2620
34.21 3.260 6.090 0.2676
38.88 3.440 6.860 0.2507
43.84 3.50 7.210 0.2427
52.58 4.235 8.995 0.2352

100

TABLE 34
Observed Landau-Placzek Ratio JV of 09 as

a Function of Temperature.

Tc(°C)

J

c B V
24.75 1.505 2.570 0.2928
29.32 1.145 1.935 0.2957
34.21 2.013 .305 0.3300
38.88 1.385 2.6625 0.2601
43.84 3.055 6.0525 0.2524
48.60 3.145 6.390 0.2461
53.30 3.255 6.6325 0.2454
58.25 3.295 7.035 0.2342
67.80 1.095 2.4475 0.2237
TABLE 35

Observed Landau-Placzek Ratio

a Function of Temperature.

Tc(°C) IC 18 °v
24.75 3.055 2.6975 0.4569
34.21 2.465 3.0625 0.4024
38.80 2.620 3.2825 0.3991
53.30 2.773 3.448 0.4021
58.25 2.895 3.8225 0.3787
52.90 2.975 3.8375 0.3875
77.60 2.905 4.59 0.3154

101

TABLE 36

Intercept and Slope for the Intensity Ratio of the Scattered Light (JV)-

Temperature (TC) Relationship JV = A + BT.

Liquid

MM

MDBM

MD7M
D.C. 200 fluid.
100 Ct.S.

D.C. 200 fluid,

105 Ct.S.

CODE:

15

0.279
0.263
0.274
0.307

3.347

1.163
0.313
0.356
0.493

B:

.632
.612
.102
.245

.113

.102
.515

.073 x 10

.031

dd

_1

dTC

x 10'4
-4
-3

-3

-2
-3
-3
-3

102

 

09 (M01. Wt. 667)
MDyM (M01. Wt. 681)

05 (M01. Wt. 371)

MD3M (M01. Wt. 395)

OODD

0.2‘

 

 

Fig.35. JV Versus Temperature for D9. M07”. 05 and MDBM.

103

.mmCOUTFPm uw_u>u use cmmcwb com pcmwmz cm_:umpoz mzmcm> >a .mm .mwm

 

 

8: 82 8... 8m OS O8 8... 08 8m gm of

r _ _ _ r _ _ _ _ _ .

_

l N.

r. m.

< i 4.
8.8.3 3... 28:: 6
8.8.8 2. 285.. 6
9.8.8 3... 8:98 4
gowmam Z 6:98 4

 

o

 

 

 

 

 

.mmcoUWme uw—oxo use mecPJ Lo» azmwmz cmpzumpoz msmcm>.mwm .Nm .mwm
nu
_
0:98. H.
gamerJ o .1¢1o_
2
28:3 1 o—
Mw.hr111114u11 o. .01 111115111 m1
1 6:98
Tmiop
1_1o_
4 A . w a _ 3 . q q _ q o
oops oopp coop coo com ooN com com oo¢ com com omp

“cave: cmpzumpoz

U
p...
'U

>

at

105

7. Depolarization Measurements.

Identical intensity ratios were obtained using vertically-polarized
incident light with or without a vertical polarizer immediately in front
of the detector, thus confirming the idea that Brillouin spectra result
from vertically-polarized light alone. There was no indication of resolv-
able peaks for a horizontally-oriented polarizer.

The depolarization ratios 6V and ph are defined as
H V

v h
p = ——- and p = ——- , (52)
V Vv h h

where Vv is the intensity of the scattered light whose polarization vector
is perpendicular to the scattering plane before and after scattering, Hv
is the intensity of the scattered light whose polarization vector is vertical
before scattering and horizontal after scattering, Vh defines the case where
the polarization vector is horizontal before and vertical afterward, and Hh
the case where the polarization vector is horizontal both before and after
scattering. Calculated values of pv and ph are tabulated in Tables 37 and
38 for a number of different silicones as a function of temperature.

Evidently for all silicones ph has a constant value of about 1.0 regardless

of temperature while the values of pv decrease with increasing temperature.

106

TABLE 37
Observed Depolarization Ratios for MM.

Tc pv ph
(°C)
24.75 0.0618 1.171
48.60 0.0279 0.982
77.60 0.0084 1.275
Observed Depolarization Ratios for MD3M.
Tc pv ph
(°C)
24.75 0.0183 1.095
48.60 0.0156 1.126
77.60 0.0111 1.169
Observed Depolarization Ratios for MD7M.
Tc pv ph
1°c)
24.75 0.0369 1.035
77.60 0.0227 1.000
Observed Depolarization Ratios for D.C. 200 fluid, 100 cts.
Tc pv ph
<°c1
48.60 0.0154 0.947
77.60 0.0111 1.225
Observed Depolarization Ratios for D.C. 200 fluid, 10 cts.
Tc pv ph
1°C)
24.75 0.0277 0.334
48.60 0.0336 0.343

107

TABLE 38
Observed Depolarizaiton Ratios for 03.
TC 9v ph
(°C)
77.60 0.0415 0.442

Observed Depolarization Ratios for 05.

Tc pv ph
(°C)

48.60 0.0134 1.260
77.60 0.0101 1.108

Observed Depolarization Ratios for 09.

Tc pv ph
(°C)

24.75 0.0131 1.105
48.60 0.0145 1.104
77.60 0.0171 1.300

Observed Depolarization Ratios for 015.

Tc pv ph
(°C)

24.75 0.1704 1.000
48.60 0.1475 0.900

77.60 0.1293 0.956

CHAPTER V

CONCLUSIONS

In linear and cyclic silicones the velocity of sound increases rapidly
with molecular weight initially, but has a tendency to level off in the
region of higher molecular weights (2 7,000 molecular weight). Present
measurements are in good agreement with the only data that exists for such
liquids, results derived from a limited number of ultrasonic measurements[36].
The rate at which the velocity of sound changes with temperature decreases
from -4.4 to -2.5 meters/sec for linear silicones, and from -3.3 to -2.9
meters/sec for cyclic silicones, as one goes from lower to higher molecular
weights. Such slopes are normal for liquids having a low molecular weight
and considerable dependence of viscosity on temperature. However, silicones
exhibit about the same slopes, even though their molecular weight is high
and the dependence of their viscosity on temperature is negligible.

It was found that as molecular chains lengthened, the sound velocity
increased, while the adiabatic compressibility decreased. The latter is
inversely proportional to the magnitude of the intermolecular forces. Thus
an increase in molecular weight increases the strength of intermolecular
forces. Compared to other liquids having similar molecular weights, values
of the adiabatic compressibility were unusually high for all of the sili-

cones studied.

108

109

Increasing the temperature increases the number of molecular inter-
actions, which in turn increases the efficiency of transferring internal
vibrational energy to translational energy. This causes a decrease in the
sonic absorption co-efficient a, as appears in Figure 33. Figure 34

establishes that 1%%—- becomes negative at higher molecular weights, which

C
indicates that most of the silicones studied may be classified as "associated

fluids"[54].i Values of a ranged from 1000 cm'1 to 2500 cm'].

These high
values may be caused by structural absorption - related to the fact that
molecules in an associated liquid can undergo a transition from one type

of structure to another under the influence of a propagating hypersonic wave.

The Mountain theory outlined in Chapter II predicts an extra relaxation
peak centered at the incident frequency for a thermally-relaxing liquid.
This "Mountain line" arises from the exchange of energy between internal
vibrational and transitional modes, which decays ‘with a lifetime of the
order of the relaxation time T. Though.observed in carbon tetrachloride
(CC14) by Cornall [30], et. al., no trace of such a line was found during
the present investigation.

The temporal attenuation co-efficient PB, is strongly dependent upon
the wave number squared (K2) as can be seen from Eq. (43). Tables 22 and
23 confirm that PB decreases rapidly with K in the present case.

The ratio JV of the scattered light intensities decreases monotonically
with temperature for all silicones (Figure 35). Moreover, JV consis-
tently increased with molecular weight. Consequently it is concluded that
the anisotropy of liquid silicones increases with an increase in their
molecular weight. As the chains lengthen more light is scattered at the
incident frequency, increasing the magnitude of the peak. Values of JV

ranged from 0.2 to 2.0 for the silicones studied. Low values like these

110

are commonly observed for water,ethyl alcohol and other low molecular
weight liquids[64], but are scarcely what one would expect for high
molecular weight substances.

Heat capacity depends upon temperature changes, but also upon the
manner of heating. The Specific heat at constant pressure (CD) is expected
to be larger than the specific heat at constant volume (Cv)’ because when
the substance is heated at constant pressure it may expand and do work
against the external pressure, whereas when it is heated at constant vol-
ume, it will do no external work. However the values of .1V obtained for
silicones establish that the ratio Cp/Cv remains fairly close to unity
(1.2 to 3.07),contrary to what might be expected for polymers (2 1000 -
2000).

Brillouin spectra were recorded over a range of scattering angles
from 45° to 135° for D5 and MD7M with a single interferometer mirror
separation (Tables 22 and 23); but no dispersion in the velocity was de-
tected in this range (2 x 109 Hz to 5 x 109 Hz). It should also be noted
that refractive indices were measured for 5890 A wavelength and calculated
for 5145 A wavelength. Figures 10 and 11 support the contention that re-
fractive index and temperature are linearly related in the range from 20°C

to 80°C.

BIBLIOGRAPHY

O" U"! «h u N
o O O O O

10.
11.
12.
13.
14.
15.

16.
17.
18.
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20.

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:oxzuxzoxc
CODDUU'JU

Mountain, ., J. Research NBS, 22A, 95 (1968).
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Gornall, W. S., G. I. A. Stegeman, B. P. Stoicheff, R. H. Stolen and
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Landau, I. D. and Lifshitz, E. M., Statistical Physics, Pergamon Press,
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Landau, I. D. and Lifshitz, E. M., Electrodymanics of Continuum Media,
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dam -

 

 

 

APPENDIX A

CALCULATION or THE REFRACTIVE INDEX AT 5145 3.

The refractive index for A = 5145 3 was calculated using the
equations given in the Bausch and Lomb Abbe 3-L Refractometer instru-
ment manual. The refractive index at A = 5145 3 (115145 3) was calcu-
lated for all silicones studied (MM, MD3M, MDjM, D.C. 200 fluid 100 cts.,
D3, 05, 09, and 015) at five different temperatures. The equations used

were

 

"514° ° = A, + (51435 nm)2 (A'])
A’ = 0.52354 x 10° (nF - nC) (A—2)
B’ = 0.52364 x 10‘° B’ (A-3)
(nF - nc) = A + BC , (A-4)

where A, B and C are constants given in the DiSpersion Table of the Bausch
and Lomb Abbe 3-L Refractometer. A, B and C depend on the refractive index
of the sample and on the drum readings of the refractometer obtained during
the measurement of no.

115

 

116

The dispersion, (nF - "C) of the sample is the difference in the
refractive index at wave-lengths of 656 and 486 nm. Tables 2-4 presents
the values of n5145 X for the five temperatures at which the refractive-
indices were measured. Tables A-l to A-5 give the values of n5145 K for
the temperatures at which all the scattering measurements were taken.
Figures A-l to A-9 display curves of the refractive index versus tempera-

5

ture for MM, MD M, MD M, D.C. 200 fluids, lOO cts. and TO cts., 03, 05’ D9

3 7
and 015' For each silicone the slope and intercept with the n5145 X axis
was calculated,

° = A + BT (A-S)

"5145 A 0

Values of A and B are presented in the table for all silicones.

The temperature co-efficient of the refractive index (%$_9 decreases

' C
with increasing molecular weight for linear silicones. For cyclic silicones
there is also a decreasing trend of %%—- with increasing molecular weight

C

which can be seen quite clearly in Figure A-lO.

 

ll7

TABLE A-l
Temperature Dependence of the Refractive Index for A = 5145A.
. . o__ '4
Liquid. MD3M . "5145A - 4.60947 x 10 TC + 1.4055
TC(°C) n5145A
24.75 1.3941
29.32 1.3920
34.21 1.3898
38.88 1.3876
43.84 1.3853
48.60 1.3831
53.30 1.3810
58.25 1.3787
62.90 1.3765
67.80 1.3743
77.60 1.3698
TABLE A-2
‘. . o- ‘4
LlQUld. MM,n5]45A - -5080064 X 10 TC + 1.3930
24.75 1.3787
29.32 1.3760
34.20 1.3732
38.88 1.3705
43.84 1.3676
48.60 1.3649
53.30 1.3621
58.25 1.3593
62.90 1.3566
67.80 1.3537
77.60 1.3480

118

TABLE A-3
Liquid: MD7M ; n5145; = -4.1780 x 10’47C + 1.4100
Tc(°c) n5145A
24 75 1.3977
29.32 1.3978
34.21 1.3957
38.88 1.3938
43.84 1 3917
48.60 1.3897
53.30 1.3878
58.25 1.3857
52 90 1.3837
57.80 1.3817
77.50 1.3775
Liquid: 0.0.200 fluid, 100 cts.; n5145; = -3 79733 x 10'41C + 1.4155
24.75 1.4052
29.32 1.4045
34.21 1.4025
38.88 1.4009
43.84 1.3990
48.50 1.3972
53 30 1.3954
58.25 1.3935
52.90 1.3917
67.80 1.3899
77 50 1.3852

Liquid: D.C. 200 fluid, 10

n5145/1 ‘

16m)

24.
29.
34.
38.
43.
48.
53.
58.
62.
67.
77.

119

TABLE A-4

-3.54373 x 10’

4

Tc

5

 

CtS.

n5145A

1.4066
1.4050
1.4032
1.4015
1.3997
1.3979
1.3962
1.3944
1.3927
1.3909
1.3874

120

TABLE A-S
Intercept and Slope for the Index of Refraction-

Temperature Relationship n5145 A = A + BTC.

an °
Liquid A 8 = (M)

1 51C
15-1
C
MM 1.3921 -5.8006
MDBM 1.4055 -4.5095
MD7M 1.4100 -4.l780
100 cts. 1.4155 -3.7973
105 cts. 1.4155 -3.5437
3n °
. . _ 5145 A
L1qu1d A B - ( 3T 1
C
1
(““1
0C
03 1.4015 -5.4510
05 1.4110 -4.5325
09 1.4250 -5.7219
0 1.4181 —3.8623

15

121

 

 

1.38
1.37-
1.35-1
1.35-
x = 514571
1.34. o A = 5890A
1.33— i f T T 1 l 1 1 l T T
25 30 35 40 45 50 55 50 55 70 75 80

Tc(°C)

Figure A-l. Refractive Index Versus Temperature for 1114.

 

122

 

 

1.40-
1.39 q
n
1.38 . o x = 5145 A
o x = 5890 A
1.37 1 T I | l 1 1 1 T 1*
25 30 35 40 45 50 55 50 55 70 75 80

Tc(°C)

Figure A-2. Refractive Index Versus Temperature for MDBM.

 

“as--

 

 

123

 

1.40-
C
1.39-
n

o x = 5145 A
o A=5890A

1.38-

1.37

 

f T W I l l l T l 1
25 30 35 40 45 50 55 50 55 70 75
T,(°C)

Figure A-3. Refractive Index Versus Temperature for MD7M.

 

“

 

 

 

124

1.41d

1.40-

 

 

 

 

=N%A
= 5890 A
1.39-
1.38 T l l T T I T l r I 1
25 30 35 40 45 50 55 60 65 70 75 80

15°C)
Figure A-4, Refractive Index Versus Temperature for 0. 0.200 Fluid,
100 cts.

 

 

‘

H.

 

 

125

1041‘“

1.40*

1.39--1

 

1.38

 

 

 

25 31) 3‘5 40 45 5'0 55 5'0 5'5 7'0 75 8'0
[5°C)
Figure A-5. Refractive Index Versus Temperature for D.C. 200 Fluid

5

10 cts.

 

126

1.37-i

1.36-

 

1'35.. l T
70 , 70.5 80 80.2

 

 

7cm)

Figure A-6. Refractive Index Versus Temperature for D3.

 

 

127

 

 

1.40 a
1.39 -
-n
o A = 514571
c x = 5890A
1.38 -
1.37 _ ' [ l I l I l 7 l 1
25 30 35 40 45 50 55 50 55 70 75 80

14°C)

Figure A-7. Refractive Index Versus Temperature for D .

 

 

 

‘14

 

128

 

 

 

1.41 -
1.40 -
O
n o
o x -- 5145A
0 “589071
1.39 -
1038 '
25 311 3'5 3045 51151550557075
rem

Figure A-8. Refractive Index Versus Temperature for 09.

 

 

129

 

 

1.41-
1.40-
" o x = 5145 A
0 x = 5890 A
1.39-
1°38 r l r T I 1 I f V I
25 30 35 40 45 50 55 50 55 70 75

Tc(°C)

Figure A-g, Refractive Index Versus Temperature for 015.

 

 

 

.i.\

 

 

 

130

-5000 ‘

  
 
    
 

 

 

 

3"5145 A 0 Linear
B"""T""
a c 4 Cyclic
“4.00 " \ . \A
-3000

WW I l I I I I l l I F T V
100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400

Molecular Weight

Figure A-10. Variation of the Temperature Derivative of the Refractive Index

with Molecular Weight for Linear and Cyclic Compounds.

APPENDIX 8
'VARIATION or JV vITH TEMPERATURE Tc

The ratio JV. of the intensity of the central Rayleigh component
to the intensity of the Brillouin doublet, was calculated for all sili-
cones at various temperatures ranging from 20°C to 80°C. JV was found
to decrease monotonically with temperature for all silicones, as estab-
lished in Figures B-l to 8-9. Figures 8-5 and B-9 show some scattered
points. which may be the consequence of sample contamination. 03 was
sol id at room temperature. its melting point being close to 63%; hence
it was very hard to remove all dust partic1es.

131

 

 

132

 

 

 

 

 

 

 

 

 

0.3-
.n 4
o o o N + e
0.2-
0.14
20 30 40 50 60 70 80
Figure 84. Jv Versus Temperature for M. Tc(°C)
0.3-1
_n O
u 0- o o -
o 2.. 0 V o
T
0.14 '
000 I I r I ' fi
20 30 40 50 . 50 70 80

Figure 8-2. Jv Versus Temperature for MD3M.

 

 

 

 

 

O.ll

.

 

 

 

 

 

133
0.3-
0.2-
0.1..
0'0: I I I I t
20 30 40 50 50 70 80
Figure ‘B-3. Jv Versus Temperature for MD7M. Tc(°C)
0.3..
O
O
00 2 ‘-
0.1-
0~0 I I I I I r’
20 30 40 50 60 70 80

Figure 8-44 Jv Versus Temperature for D.C. 200 Fluid,100 cts. TC(°C)

 

III A. 11.1 A 11111

134

 

 

 

 

 

 

3.0-
200- 1...
Jv if
i
1.0-
0 I' I ‘1 I I I
20 30 40 50 60 70 80
Figure 8-5. Jv Versus Temperature for D.C. 200 fluid, 105 cts. TC(°C)
<3
0.4 -
JV to
(3
002 d
o I I I I 1 I
20 30 40 50 60 70 80

Tc(°C)

Figure 8-6. Jv Versus Temperature for D3.

 

an 11:1.III‘

 

135

 

 

 

 

0.3 -
0.2 -
Jv
0.1 -
0 I I I r I I
20 30 40 50 60 70 80
Tc( c)
Figure B-7. Jv Versus Temperature for 05.
0.3 " o
O
002 q
Jv
0.1 -
F I I I I
20 30 40 50 50 70 80
0
Tc( c1

F19ure 8-8. JV Versus Temperature for 09.

136

0.5-

 

0.2

 

 

1cm)

Figure 8-9. Jv Versus Temperature for 015.

APPENDIX C
LASER-BRILLOUIN-VELOCIMETER

Static fluid properties can be measured by Brillouin scattering.
Possibly certain dynamic fluid properties such as boundary layer thick-
ness and flow velocity[55] can also be measured by this technique.

The Doppler-shifted frequency 00, caused by the fluid velocity v

can be expressed by the equat1on[55.57. 8]

“D ' a" [17' (35C ' 31)] 2 (C‘])

where n is the index of refraction of the fluid, A0 is the wavelength
of the incident radiation in a direction specified by the unit vector 61
and fisc is the unit vector of the scattered radiation. Eq. C-l can be

rewritten in terms of angles as (see Figure C-l)

ID = £31— Sin(8/2)Cos [90° - (3 + 972)] , (02)
0

where e is the angle between the scattered and incident radiation and a is
the angle between the incident radiation and the velocity vector 7. In

our experimental arrangement a is equal to 180°;

137

138

hence,
_ va
“D - -;—-Sin(0/2)Sin(180 + 0/2) (C-3)
0
or
-3_".‘L 2
vD-Ao Sin e/z . (c-4)

I" 1.1111“ "Mt!

 

DFEW

THE EXPERIMENTAL SET-UP

The experimental set-up for velocity measurements is shown in
Figure C-l. A laser beam is passed through the tube in a direction opposite
to the velocity of the flowing fluid. After the fluid (ZOO-proof ethyl
alcohol) comes to Reservoir 1, a variable-speed pump circulates it to
Reservoir 2 then returns it through the system. The reservoirs are speci-
ally designed to smooth the pulsating flow from the pump and eliminate all
bubbles generated by the pump's rotary motion.

The tube is scanned at a distance of 35 inches (g-Z 1/40) from the end
where the fluid enters. Such a clearance is required to assure a developed
flow in the test section. Light scattered from the test section is then
collimated and passed through the optical and detection system described
in Chapter III.

DISCUSSION

The frequency spectrum obtained exhibits one Rayleigh peak at the

center and two Brillouin doublets, as in the static case. Figure

C-3

shows the Rayleigh peaks for the static and dynamic cases superimposed.

 

 

 

 

Spectrum for
Static Case

A.

 

 

 

 

Fig. C-3. Static and Dynamic Brillouin Spectra.

 

Spectrum
for
Dynamic
Case

Here, ”BS is the Brillouin frequency shift in the static case, ”Bd is the

Brillouin frequency shift in the dynamic case, and “D is the frequency

shift due to Doppler effect.
Then,

(C-5)

(C-6)

‘ '. :~

 

111.1.“ _
‘em.v.;_._ .

140

50,

v=1/Zv --v C-7
0( 1(de Bd1)- (1

Measurements were made with green light of wavelengths A0 = 5145 A,

5 1

or frequency 00 . 5.83 x 1014 Hz. Then, for e = 45°, 00:3.3 x 10 vsec' -
1 F.“

and for e = 135°: “D” 3.3 x 105 Vsec' . e = 45° and 135° are the minimum 1.

V.,“;

and maximum scattering angles available on the instrument. Therefore, in
the present case it was only possible to vary vD in the range from 103 to

105 v Sec", where V is expressed in cm/sec. Hence to measure velocities f1

 

in the range of 1 cm/sec to 100 cm/sec, the Doppler frequency shift would

have had to be in the range from 103 Hz to 107 Hz, and a detection technique
with very high resolving power would have been required. Therefore

Spectral range of the instrument utilized in the measurements described
above is in GHz range and it would have been nearly impossible to measure
shifts falling in the MHz range.

There are other heterodyning techniques which have very high resolving
power and are being used for Doppler-shifted frequency measurements [6,
7,58, 59.60.61). However, the Laser-Doppler-Velocimeter requires that
some foreign particles be present to effect light scattering. No such

particles are needed for the Laser-Brillouin Velocimeter.

5“? -' ' .

 

 

 

141

.y},

”In“
yw

 

 

 

Tr.

Detector

 

 

 

 

 

= ‘2—0" [v°(fisc "' I'1>1°):1

’1‘: [ii-(2 Sin %)T]

_ 0V 8'
or, vD - XS-Z Sin E-Cos (a + g) .

Fig. C-l. Vectorial Representation of Incident Light, Scattered Light

and ‘11-‘11. \In‘nn‘J-u

142

Test Section .1

20" / 3 5 ”
‘47

< J

 

 

  

 

 

. f

_ ._
1..“

1 Collimator and
Detection System

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(A) ("N ("1 ("N
Variable
Speed
Pump f;=—— I 1
Reservoir Reservoir
No. 1 No. 2

Laser beam perpendicular to the
plane of the paper.

 

_:] Detection System

 

Test Section

Fig. C-2. Experimental Set-up for Velocity Measurements.