LIGHT SCATTERING SPECTROSCOPIC STUDlES OF
SILICONE FLUIDS

Dissertation for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
KATHERINE EMILY REED
1977

 

 

This is to certify that the

thesis entitled
Light Scattering Spectroscopic Studies
Of Silicone Fluids

presented by
Katherine Emily Reed

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Chemistry

   

Major professorK/

Date W7

0—7639

 

ABSTRACT

LIGHT SCATTERING SPECTROSCOPIC STUDIES
OF SILICONE FLUIDS
BY

Katherine Emily Reed

A Brillouin light scattering spectroscopic study of
both linear and cyclic polydimethylsiloxane liquids was
undertaken to examine the effect of polymer chain length
on various spectral parameters. Eight fluids were investigated
at seven temperatures between 20°C and 85°C. The refractive
index, Brillouin shift frequency and full Brillouin linewidth
at half-height were obtained experimentally. These properties
were found to be linearly decreasing functions of temperature
for all fluids. The sonic velocity, adiabatic compressibility
and sound absorption coeffcient were calculated and also
found to vary linearly with temperature for each silicone.

At low molecular weights the molecular configuration,
linear or cyclic, did not affect the acoustical properties.
However, as the chain length or ring size increased the
cyclic liquids exhibited larger shift frequencies and sonic
velocities and lower compressibilities than the linear liquids.

At a single temperature the sonic velocity was found

Katherine Emily Reed
to increase rapidly to a molecular weight of approximately
1000 where it continued to rise very slowly with a further
increase of molecular weight. Although experimental data
supported Rao's Rule, a simpler empirical equation,
relating the velocity linearly to reciprocal molecular
weight, was found to adequately represent the data. In
addition, the Brillouin shift frequency, full Brillouin
linewidth and sound absorption coefficient varied with
molecular weight in the same manner.

Landau-Placzek ratios were found to be relatively
independent of temperature. Results were in good agreement
with heat capacity data for silicones and other liquid
polymers.

The acoustical parameters have been extended more
than an order of magnitude in frequency in this study and
are found to be in qualitative agreement with data obtained

by ultrasonics.

LIGHT SCATTERING SPECTROSCOPIC
STUDIES OF SILICONE FLUIDS

BY

Katherine Emily Reed

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Chemistry

1977

\J

TO MY HUSBAND

DR. PAUL G. GERTENBACH

ii

ACKNOWLEDGEMENTS

The author wishes to thank Professor Jack B.
Kinsinger who served as research preceptor for this work,
Professor Richard H. Schwendeman who served as second

reader, and the Department of Chemistry which provided

financial support.

The generous gifts of silicone fluids from the

Dow Corning Corporation are gratefully acknowledged.

iii

Chapter

I.

II.

III.

IV.

V.

TABLE

HISTORICAL

LIGHT SCATTERING

SILICONE FLUIDS

THEORY C O O O O O

APPARATUS

OF CONTENTS

EXPERIMENTAL INFORMATION

A. The Brillouin Spectrophotometer . . .

B. Control and Measurement of Temperature

SAMPLE PREPARATION . . . . . . . . . . . .

REFRACTIVE INDEX

REFRACTIVE INDEX

VARIATION OF THE
WITH TEMPERATURE

VARIATION OF THE
WITH TEMPERATURE

VARIATION OF THE
WITH TEMPERATURE

VARIATION OF THE
WITH TEMPERATURE

VARIATION OF THE
WITH TEMPERATURE

VARIATION OF THE
WITH TEMPERATURE

CONCLUSIONS . .

LITERATURE CITED .

RESULTS AND DISCUSSION

BRILLOUIN FREQUENCY SHIFT
AND MOLECULAR WEIGHT . .

VELOCITY OF SOUND
AND MOLECULAR WEIGHT . .

ADIABATIC COMPRESSIBILITY
AND MOLECULAR WEIGHT . .

BRILLOUIN LINEWIDTH
AND MOLECULAR WEIGHT . .

ABSORPTION COEFFICIENT
AND MOLECULAR WEIGHT . .

LANDAU-PLACZEK RATIO
AND MOLECULAR WEIGHT . .

iv

Page

26
35

40

40

44

52

71

86

89

104

112
115

118

LIST OF TABLES

Table Page

1 Measured Temperatures of the COpper Block
and Sample Liquid at Various Controller Settings . 38

2 Measured and Corrected Temperatures Obtained
During Refractive Index Measurements
of Silicone Fluids . . . . . . .‘. . . . . . . . . 42

3 Refractive Index As a Function of Temperature for
MDM Observed at 5890 A and Calculated at 5145 A . 45

4 Refractive Index As a Function of Temperature for
MDZM Observed at 5890 A and Calculated at 5145 A . 45

5 Refractive Index As a Function of Temperature for
MDSM Observed at 5890 A and Calculated at 5145 A . 46

6 Refractive Index As a Function of Temperature for
MD9M Observed at 5890 A and Calculated at 5145 A . 46

7 Refractive Index As a Function of Temperature for
MDllM Observed at 5890 A and Calculated at 5145 A 47
8 Refractive Index As a Function of Temperature for
MD13M Observed at 5890 A and Calculated at 5145 A 47
9 Refractive Index As a Function of Temperature for
D. C. 200 Fluid, Viscosity Grade 12, 500 cts,
Observed at 5890 A and Calculated at 5145 A . . . 48

10 Refractive Index As a Function of Temperature for

D4 Observed at 5890 A and Calculated at 5145 A . . 48
ll Refractive Index As a Function of Temperatureofor

D6 Observed at 5890 A and Calculated at 5145 A . . 49
12 Least—Squares SlOpes, Intercepts at 0°C, and

Correlation Coefficients for Refractive Index

versus Temperature Plots . . . . . . . . . . . . . 51

13 Brillouin Shift Frequencies of Some
Substances at 30°C . . . . . . . . . . . . . . . . 53

V

14

15

16

17

18

19

20

21

22

23

24

25

Observed Brillouin Frequency Shifts, Velocities

of Sound, Full Brillouin Linewidths at Half—Height,
Sound Absorption Coefficients, and Refractive
Indices of MDM as a Function of Temperature . . .

Observed Brillouin Frequency Shifts, Velocities

of Sound, Full Brillouin Linewidths at Half-Height,
Sound Absorption Coefficients, and Refractive
Indices of MDZM as a Function of Temperature . . .
Observed Brillouin Frequency Shifts, Velocities

of Sound, Full Brillouin Linewidths at Half-Height,
Sound Absorption Coefficients, and Refractive
Indices of MDSM as a Function of Temperature . . .

Observed Brillouin Frequency Shifts, Velocities

of Sound, Full Brillouin Linewidths at Half-Height,
Sound Absorption Coefficients, and Refractive
Indices of MD9M as a Function of Temperature . . .
Observed Brillouin Frequency Shifts, Velocities

of Sound, Full Brillouin Linewidths at Half-Height,
Sound Absorption Coefficients, and Refractive
Indices of MD11M as a Function of Temperature . .

Observed Brillouin Frequency Shifts, Velocities

of Sound, Full Brillouin Linewidths at Half-Height,
Sound Absorption Coefficients, and Refractive
Indices of MD13M as a Function of Temperature . .
Observed Brillouin Frequency Shifts, Velocities

of Sound, Full Brillouin Linewidths at Half-Height,
Sound Absorption Coefficients, and Refractive
Indices of D4 as a Function of Temperature . . . .

Observed Brillouin Frequency Shifts, Velocities

of Sound, Full Brillouin Linewidths at Half-Height,
Sound Absorption Coefficients, and Refractive
Indices of D6 as a Function of Temperature . . . .

Linear Least-Squares Slopes, Intercepts at 0°C,
and Correlation Coefficients of the Brillouin
Shift vs. Temperature Plots for All Silicones . .

Velocity of Sound as a Function of Frequency
For Several Silicones at 25°C. . . . . . . . . . .

Linear Least-Squares Slopes, Intercepts at 0°C,
and Correlation Coefficients of the Velocity of
Sound vs. Temperature Plots for All Silicones . .

Ultrasonic and Brillouin Velocities of Sound
for Linear Polydimethylsiloxanes at 30°C . . . . .

vi

56

57

58

59

6O

61

62

63

68

72

77

78

26

27

28

29

30

31

32

33

34

35

36

37

38

39

Brillouin Velocity of Sound Data for
Selected Polymers and Other Liquids at 20°C . . .

Rao Numbers, Molar Volumes and Related
Quantities for All Silicones at 25°C . . . . . . .

Velocities, Densities and Adiabatic
Compressibilities for Siloxanes at 25°C . . . . .

Adiabatic Compressibilities for Some
Organic Liquids at 25°C . . . . . . . . . . . . .

Adiabatic Compressibility as a Function of
Temperature for MDM . . . . . . . . . . . . . . .

Adiabatic Compressibility as a Function of
Temperature for MDZM . . . . . . . . . . . . . .
Adiabatic Compressibility as a Function of
Temperature for MDSM O O O O O O O O O O I O O O
Adiabatic Compressibility as a Function of
Temperature for MD9M . . . . . . . . . . . . . .
Linear Least-Squares Slopes, Intercepts at 0°C,
and Correlation Coefficients of the Full
Brillouin Linewidth vs. Temperature Plots for

All Silicones O O O O O O C O I O O C O O O O O 0

Relaxation Times of the Silicone Fluids as
a Function of Temperatire . . . . . . . . . . . .

Activation Energies for Relaxational
Processes in Silicone Fluids . . . . . . . . . . .

Linear Least-Squares Slopes, Intercepts at 0°C,
and Correlation Coefficients of the Sound
Absorption vs. Temperature Plots for All Silicones

Absorption Coefficient Ratios for
Silicones at 25°C . . . . . . . . . . . . . . . .

Landau-Placzek Ratio as a Function of
Temperature for the Silicones . . . . . . . . . .

vii

79

83

87

86

90

90

91

91

97

101

102

109

112

114

Figure

\JO‘WDWN

m

10

11

12

13

14

15

16

17

LIST OF FIGURES

Block Diagram of the Brillouin Spectrophotometer
Three Rayleigh-Brillouin Spectral Orders . . . .
Cross-Sectional View of the Thermostatted Jacket
Calibration Plot for the Thermostatted Jacket .
The Light Scattering Sample Cell . . . . . . . .
Calibration Plot for the Thermocouple . . . . .

Refractive Index at 5145Atas a Function of
Temperature for All Silicone Fluids . . . . . .

Brillouin Spectra of MDZM and MD13M at 85°C . .
Brillouin Spectra of MDM at 25°C and 85°C . . .

Brillouin Shift Frequency as a Function of
Temperature for MDM, MDZM and MDSM . . . . . .

Brillouin Shift Frequency as a Function of
Temperature for MD9M, MDIIM and MD13M . . . . .

Brillouin Shift Frequency as a Function of
Temperature for D4 and D6 0 I o o o o o o o o o

Brillouin Shift Frequency as a Function of
Molecular Weight for Linear and Cyclic
Silicones at 80°C . . . . . . . . . . . . . . .

Brillouin Shift Frequency as a Function of
Reciprocal Molecular Weight for Linear
Silicones at 80°C . . . . . . . . . . . . . . .

Velocity of Sound as a Function of
Temperature for MDM, MDZM and MD5M . . . . . . .

Velocity of Sound as a Function of

Temperature for MDQM, MD11M and MD M . . . . .

13

Velocity of Sound as a Function of
Temperature for D4 and D6 . . . . . . . . . . .

viii

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27
34
36
39
41

43

50
54
55

64

65

66

69

70

74

75

76

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

Velocity of Sound as a Function of
Molecular Weight for Linear and Cyclic

Silicones at 80°C

Rao Number as a Function of Molecular

Weight for Linear Silicones

Velocity of Sound as a Function of Reciprocal
Molecular Weight for Linear Silicones at 80°C

Adiabatic Compressibility as a Function
of Temperature for MDM, MD

Full Brillouin Linewidth as a Function

2

M, MD

5

M and

Temperature for MDM, MDZM and MDSM . .

Full Brillouin Linewidth as a Function
M and MD M

Temperature for MD

Full Brillouin Linewidth as a Function
Molecular Weight for Linear and Cyclic

Silicones at 80°C

Full Brillouin Linewidth as a Function
Reciprocal Molecular weight for Linear

Silicones at 80°C

Temperature Dependence of Relaxation
Times for MDM and MD

Sound Absorption Coefficient as a Function
M and MD M

9

M, MD

13

M

Temperature for MDM, MD

Sound Absorption Coefficient as a Function
M and MD13M

Temperature for MD

9

M, MD

6

2

Temperature for D4 and D6

Sound Absorption Coefficient as a Function
Molecular Weight for Linear and Cyclic

Silicones at 80°C

Sound Absorption Coefficient as a Function
Reciprocal Molecular Weight for Linear

Silicones at 80°C

ix

11

Full Brillouin Linewidth as a Function
Temperature for D4 and D

11
Sound Absorption Coefficient as a Function

5

13

MD M

of

9

80

84

85

92

94

95

96

99

100

103

106

107

108

110

111

CHAPTER I

HISTORICAL

LIGHT SCATTERING

The phenomenon of light scattering has been of interest
to laymen and scientists for hundreds of years. Tyndall's
success in simulating "blue sky" atmospheric scattering in
the laboratory by passing white light through a gold sol
and observing blue light perpendicular to the incident beam
led to concentrated efforts to devise a workable theory of
light scattering (1). He erroneously attempted to explain
scattering using the laws of reflection, but it was Lord
Rayleigh (2,3,4) who correctly identified it as a diffraction
process and developed the foundation of elastic scattering of
light for transparent, optically isotropic, non-interacting
particles which were small compared to the wavelength of
light. Later Smoluchowski (5) suggested that due to thermal
motions of molecules in a liquid, the local density
fluctuated through a region on the order of the wavelength
of light. Using statistical thermodynamic fluctuation theory
Einstein (6) derived a mathematical relationship between the
total intensity of scattered light and density fluctuations.
Mie (7) in 1908 develOped a general electromagnetic theory of
light scattering for non-interacting, spherical particles
of arbitrary size which could be applied to systems in which
the scatterers were large compared to 10’ the wavelength of

incident light.

The frequency distribution of the scattered light was
first examined by Brillouin (8) in 1922. He treated the
density fluctuations (also called sound waves, acoustical
waves or phonons) which propagated throughout the medium
with no directional bias. He maintained that the light
scattered by these travelling fluctuations would undergo
frequency shifts of equal magnitude analogous to the Doppler
shift of classical physics and consequently derived the

following equation:
Aw = i 2w°(Vs/c)n sin 8/2 [1]

where wois the incident frequency, Aw is the Brillouin
frequency shift, c is the velocity of light, 8 is the angle
of observation, V8 is the velocity of sound in the medium,
and n is its refractive index. These Brillouin modes were
observed first by Gross (9) in 1930. In Gross' work and all
subsequent experiments an anomalous unshifted peak was also
observed and was attributed for several years to a technical
error such as dust. This central component was explained by
Landau and Placzek (10) in 1934. By a fortuitous change of
the two independent variables from.density and temperature
(used by Einstein) to entropy and pressure, they were able
to effect a separation of the total intensity into two
contributions: entropy fluctuations at constant pressure
and pressure fluctuations at constant entropy. The entropy
fluctuations were very slow, non-propagating thermal

fluctuations, dissipated by thermal diffusion and, therefore,

had lifetimes dependent on the thermal diffusivity of the
scattering medium. The unshifted intensity, commonly called
the Rayleigh peak, was due to these isobaric entropy
fluctuations. The pressure fluctuations, on the other hand,
were propagating and gave rise to the Brillouin lines, one
on either side of the Rayleigh component. The lifetime of
the propagating acoustic wave was determined by the rate at
which it was damped by losing energy to other modes of motion
and to overcoming the intermolecular frictional forces and
was in turn dependent upon the sound absorption coefficient.
Debye was responsible for extending the theory of light
scattering to dilute solutions of macromolecules (11,12).
After his initial suggestion many others made contributions
to the thermodynamic theory of Rayleigh light scattering by
small Optically isotropic molecules (13,14,15). In the 1960's
equivalent but more successful approaches were developed
including an extrapolation of Van Hove's (16) theory of
neutron scattering to light scattering (l7) and the introduction
of the phenomenological method of linearized hydrodynamics
(18-24) in order to implicitly consider molecular interactions.
The development of intense, monochromatic radiation
sources in the form of gas lasers in the early 1960's led to
a surge in experimentation in light scattering spectroscopy.
In addition, development of high resolution interferometers
contributed to the increased accuracy of the experimental
data and facilitated the observation of yet another intensity

component called the "relaxation mode" or "Mountain line".

When it exists, it is evidenced by a low intensity

background upon which the Rayleigh and Brillouin peaks are
superimposed. The source of this added intensity is theorized
to be coupling between the external translational modes of
the scatterer and its internal vibrational modes. Energy

in the form of acoustical waves in the medium is absorbed as
vibrational energy and is later released thermally as the

molecules relax back to their ground states.
8 ILICONE FLUIDS

Since the silicones became industrially important in
the 1940's, a wealth of information pertaining to their
chemical and physical properties has been obtained and
published. The silicones are organosilicon polymers in which
the silicon atoms are bound through oxygen atoms to form
chains or rings and have the following general form:

RRR

I I I

-Si-O-Si-O-Si-O- (32510)n

III.
The usage of the term silicone became common when it was
believed that there would be far reaching analogies between
the chemical behaviors of carbon and silicon. The repeating
unit R2810 appeared to correspond to a ketone RZCO. Later it
was found that the 81:0 bond was unstable compared to the
=0 bond and instead silicon formed single bonds to oxygen

resulting in polymeric compounds. However the term."si1icone"

remained firmly implanted. There are many different silicones

but our attention in this work is restricted to the
polydimethylsiloxane (PDMS) fluids and the following

notation is used to designate specific molecules:

Cyclic: D Linear: MD M
y x
CH 3 (‘2H 3 ('IH ('21-! 3
Si-O H C-Si- O-Si O-Si-CH
3 3
I Y I I I
CH3 CH3 CH3 CH3

A number of both cyclic and linear molecules have been
synthesized and fractionated to high purity from a polymer
mixture of low molecular weights. In addition liquids
containing a distribution of molecular weights are
designated by their viscosities in centistokes. Several
authors have compiled the information published in the
literature concerning the silicones (25,26,27).

The structure of PDMS in absence of solvents or diluents
is determined by the Si-O and Si-C bond distances and
Si-O-Si and C-Si-C bond angles. It has long been known that
the Si-O interatomic distance is considerably'smaller than
the sum of the atomic and ionic radii. The relatively large
'valence angle of the siloxane oxygen coupled with the large
size of the Si atoms compared to those of C is thought to
result in a long Si-CH3 bond distance whose effect is to
push the methyl groups so far from the Si-O-Si axis that
'virtually unrestricted rotation about the siloxane bond is

;possible. The small Si-O bond distance and large Si-O-Si

lyand angle have been explained using a resonance description

of the bond including covalent, polar and double bond
contributions (28). The presence of double bond character
would explain the short Si-O bond and the large Si-O-Si
angle. The partial donation of a lone pair of electrons
from oxygen to the unfilled 3d levels of silicon (dn+pn)

is thought to be responsible for the double bond character.
The Si-C bond distance, on the other hand, has been found
to be almost equal to the sum of the covalent radii and is
probably basically a covalent single bond.

It is believed that the pure liquid molecular conformation
is partially helical with siloxane bonds oriented toward
the screw axis and methyl group pointed outward (29,30). This
stable helix is believed to be due to the internal dipole
compensation caused by the polar component of the siloxane
bond giving rise to weak dipoles which align in the helix in
such a way as to have Opposing dipoles facing one another.
Further evidence consists of entropy of dilution of the
siloxanes in benzene and heat of mixing data (31). The high
values of the entropy and lack of chain dependence on the
heat of mixing both indicate that the liquid structure of a
linear PDMS is destroyed upon solvation, so is considerably
more ordered than many pure liquids.

PDMS rings with more than three -D- units are puckered.
D2 is an unstable ring due to strain. Isolation of rings
containing as many as thirty atoms has been accomplished.

The physical behavior of the cyclic molecules is very different

from that of their linear counterparts (25,26). Several

cyclic compounds are crystalline near room temperature.
The temperature coefficient of viscosity is not as low for
the cyclic as it is for the linear conformations (32). Also
the viscosity of the cyclic molecules increases more rapidly
with molecular weight than in the linear series (33).

The intermolecular forces in the polydimethylsiloxanes
are unusually weak and lead to high compressibilities as
well as other anomalous physical properties, including a low
temperature coefficient of viscosity, low freezing and pour
points, high vapor pressures and many others (34-38). The lack
of change of viscosity with temperature has been interpreted
in terms of the chemical structure Of the linear molecules
(29). Two Opposing effects, the expansion Of the helical coils
and the increase in volume, which occur upon heating, are
thought to somewhat compensate for one another resulting in
very little net change in the mean intermolecular distances.
However, this reasoning is speculative only and the physical
factors responsible for the unusual behavior of the siloxanes
remain unclear. Other investigators (39,40,41) have shown
that the methyl groups also demonstrate virtually unhindered
rotation and this is hypothesized to cause large space
requirements for the methyl groups which in turn cause
molecular volumes to be large (32) and intermolecular
forces to be small.

Ultrasonic and viscoelastic techniques have been
applied to the silicones to determine their acoustical

properties. Weissler (42,43) was the first to investigate

the relationship between the velocity of sound and chain
length and reported that at a molecular weight of 1000 the
velocity reached an asymptotic value. Other workers have
pursued this further (44,45,46,47) and have attempted to
correlate their results with relaxation theories. Also
Brillouin measurements were made recently on selected linear
and cyclic polydimethylsiloxanes in order to elucidate the
temperature and molecular weight dependence of their

thermodynamic and acoustical properties (48).

The intent of this Brillouin scattering study was to
examine spectroscopically several linear and cyclic
polydimethylsiloxane liquids in order to determine the
effects of chain length and molecular configuration
on various acoustical properties including sonic velocity
and sound absorption coefficient. It was hoped that this
information would provide insights into the factors

responsible for the weak intermolecular forces and unique

liquid characteristics of this series. Further, it was
hoped some information regarding the molecular motion
responsible for phonon prOpagation and attentuation could

be obtained.

CHAPTER II

THEORY

THEORY

Light is scattered by a medium because it contains
optical inhomogeneities. Inhomogeneities in pure materials
are known to be caused by fluctuations in the Optical
dielectric constant. Static nonuniformities would not
produce a frequency shift in the scattered light. However,
time dependent thermal motions of molecules in a dense
fluid create a frequency spectrum in the scattered light
which is characteristic of the resulting time-dependent
fluctuations. Light scattering theory has developed from
hydrodynamic, quantum mechanical and statistical mechanical
treatments.

When an electromagnetic wave strikes a scattering
particle, it interacts with the polarizability and induces
an oscillating dipole in the direction perpendicular to that
of the incident electric vector. The magnitude of the
induced dipole moment is directly proportional to the
particle polarizability and the magnitude Of the incident

electric vector is_given by

+

"x = axy By“ [2]

The elctromagnetic wave, propagating in the z direction
with its electric vector in the y direction, induces a

dipole in the x direction proportional to the deformability

10

of the particle electron distribution. In general, the
polarizability is a second order tensor but only the diagonal
elements contribute when scattered light is polarized.
The intensity of the scattered light, L), for an
isotropic sample is given by the Rayleigh equation (2,3)
16w“|5|2

Ie= IO

 

2 u (l + cosze) [3]
r A

where I0 is the incident intensity, r is the distance from
the scatterer to the point Of Observation, A is the
wavelength of light in the medium, 8 is the scattering
angle measured from the direction of propagation Of the
incident radiation, and U’is the average isotropic

polarizability given by

l
OI= §(°‘xx+°‘yy+ (122)- [4]

A Clausius-Mossotti type equation was used by Bhagavantam (49)
to express E'in terms Of the mean fluctuation Of the
dielectric constant about its average value
_- V<|Ae|>
a = ---- [5]
4n
where V is the small spherical scattering volume.
Substitution from Equation [5] into [3] gives
n2V2<(A€ 13>

I = I ’1 + cos2 6
8 o r21“ \ 0): [ ]

 

which is the initial equation used by Einstein and by

Landau and Placzek in their statistical thermodynamic

11

treatments Of the intensity Of scattered light. For
simplicity Equation [6] is usually written for 9 = 90°.
Clearly the mean square fluctuation of the dielectric
constant is the only unknown and cannot be directly
evaluated. Instead it is evaluated in terms of two other
linearly independent thermodynamic properties.

Einstein (6) expressed the fluctuation Of the
dielectric constant as fluctuations in the density and

temperature:

Be 86
48 = (-) AD + (-) AT [7]
8p T 3T p '

He then assumed that the contribution made by (as/8T)p
was negligibly small compared to (as/8p)T. By eliminating
the AT term, Einstein developed the following equation for

the intensity Of light scattered at 90°.

nzv 38 2 2
I = I —- (—) 9 km . [8]
90 O r21“ 8p T T

In this equation k is the Boltzmann constant and ST is the
isothermal compressibility. The assumption that the change
in dielectric constant with temperature at constant density
is negligible has been experimentally determined to be
valid for some substances but nOt for others (50).

The experimental verification of the existence Of a
frequency spectrum of scattered light led Landau and Placzek
(10) to attempt a separation of the intensity into terms
which corresponded to the lines of the Rayleigh-Brillouin

triplet. By changing independent variables from density

12

and temperature to entropy and pressure,

as as
A6 = (--) m + (—) AP, [9]
as P 39 S

and proceding in the manner of Einstein, they succeeded in
making the separation. The fluctuations in entropy and
pressure have been shown to be statistically independent

of one another (51), so the mean square dielectric constant

fluctuation is given by

as as
<(Aef> = I-—fP<IAsf> + I-—?S<(Apf> [10]
as 8P

where the angular brackets indicate an ensemble average
over the initial states of the system.
Each term in [10] has been evaluated by statistical

thermodynamics (51) and the results are

as as 8T 88 T
(SE11, = (5;)P(;;)P = (3;)1, EVE; ’ [111
<(Asf> = kpvcP , [12]
as as 8p 38’
(3;)5 = (3;)SISEIS = (3;)5 p85 [13]
and ' <(Api> = kT/VBS [14]

where CP is the heat capacity at constant pressure and

BS is the adiabatic compressibility. By substituting for

the appropriate terms into [10] we obtain

as kTZ as p28 kT
2 S
<(A6) > = (-Y$-*- + (-¥g --- l15]

By using Equation [7] and the fact that (ET/3p)S =

13

(aP/aT) T/Cvpz, it is found that
p
as as 85 SP T
(-) = (--)T + (-) (-) -—- - [15]
3p 3 3p 3T 9 3T9 cvp2
Then assuming that (as/8T)p<< (ac/8p)T (53), the same
assumption made by Einstein, results in elimination of

the second term in [16], so that the final form Of <(Ae)2>

can be written

 

 

as kT’ ‘ae pzeskr
<(A€)2> = (-—I; + (-I; . [17]
3T pVCP 8p V

Now, substituting this expression into [6] we Obtain an

intensity equation containing two terms

 

I =I

nzv as M2
90 O IZAH

as
(3T);-C- + (;-); pszfig] . [18]
D P D

Rayleigh Brillouin

The first term involves fluctuations with temperature at
constant pressure and represents a nonpropagating mode.

This component of the scattered light was thought to
correspond to the unshifted Rayleigh line. The Rayleigh
linewidth is proportional to the rate of decay of the
fluctuations and broadened somewhat by the damping thermal
dissipative processes. The second term involves fluctuations
with density at constant temperature or entropy which are
propagated through the medium. They can be viewed as
successive compressions and rarefactions which propagate

or travel in a wavelike manner. This component of the

14

scattered intensity was thought to correspond to the mode
which is shifted in frequency by an amount proportional
to the velocity of the thermal waves. Since scattering
can occur from waves travelling in opposite directions,
this intensity component is actually divided between two
lines which together are called the Brillouin doublet and
are equally shifted in frequency from the Rayleigh line.
The lifetimes of these propagating thermal sound waves are
determined by the rate at which they are damped by losing
energy to other modes of motion and intermolecular
frictional forces.

The remarkable similarity between the Einstein total
intensity equation and the Brillouin intensity term in
their equation led Landau and Placzek to propose the intensity

ratio which bears their names:

Iggggfeln IR+ZIB I (n2V/r21“)[(ae/ap);] pszBT

O
Igangigfigiaczek 21B Io(w2V/r21“)[(86/8p);] pszBS

 

 

This intensity ratio is usually rearranged to the following

form and is called the Landau-Placzek ratio.

H
II
IA?
I
I-'
II
IS“
I
H
II
.<
I
I-‘

[20]

a:
H
'm
U)
0
<

Thus absolute scattering intensities are not necessary to

obtain heat capacity ratios of the medium.

15

The statistical thermodynamic method as delineated
above provides only a qualitative treatment Of light
scattering. It is based on equilibrium thermodynamics
and as such its conclusions are not completely accurate.
For example, this approach does not predict or take into
account any frequency dependence or dispersion of any of
the parameters in the equations. Cummins and Gammon (53)
have added correction terms for dispersion to [18] and [20]
but we must turn to the linearized hydrodynamic theory of
irreversible thermodynamics for a detailed development of
the time and frequency dependences Of the thermodynamic
and hydrodynamic variables.

The hydrodynamic theory of fluctuations differs from
the thermodynamic approach because it examines the behavior
in time of density fluctuations and takes into account the
effect of density fluctuations in adjacent volume elements.
The function which contains information about the fluctuations
is the generalized structure factor, S(£,w), and is the
space and time Fourier transform of the density-density
correlation function. Here A is the change in the wave
vector and m is the shift in the angular frequency of the
scattered light. The intensity of the scattered light is

directly proportional to S(E,w) as given by
+ +
I(R,w) = Io(a29“N/2nc“R2) sin2¢ S(k,w) [21]

‘where N spherically symmetric molecules of polarizability

a scatter light at an angle ¢ measured from.the electric

16

+
vector of the incident wave to R, the point of Observation.
The angular frequency of the scattered light is 9, the
shift in the angular frequency is w, and the change in the

wave vector is A which is defined by
+ + -
k = k12 sin (6/2) [22]

where 8 is the scattering angle measured from the direction
of propagation of the incident light and R1 is the wave
vector of the incident light. In order to Obtain the
correlation function from which the generalized structure
factor is deduced, it is necessary to solve the linearized
hydrodynamic equations for the time-dependent, kth Fourier
component Of the density. The following develOpment is
essentially that of Mountain (18-23) and details of the
calculations are omitted in order to more clearly
demonstrate the method.

The linearized hydrodynamic equations are the continuity
equation, the longitudinal part of the Stokes-Navier equation,
and the energy transport equation. The continuity equation
assumes that matter appearing in a volume element does so

by flowing through a boundary of that element and is given

by

8p1

-— + podiv V = 0 [23]

at
where p, is the equilibrium value of the density, pl is the
time-dependent density increment, and V is the flow velocity.

The Navier-Stokes equation is

17

 

av c3 C3800 4
Do(-) + (--)gradp1 + grad T1 - (~ns+n ) grad div V = 0
at 'Y Y 3 B
[24]
and the energy transport equation is
3T CV(Y‘1) 301 2
pocVI—I - —— (-) - Av T1= o [25]
at B at

where T and T1 are the equilibrium and time-dependent
o
values of the temperature, 8 is the thermal expansion

coefficient, A is the thermal conductivity,n and nB are

S

the shear and bulk viscosities, y is the heat capacity

ratio, and Co is the low frequency limit of the sound velocity.
The first step in Mountain's adaption of Van Hove's

neutron scattering method is to obtain the Fourier and

Laplace transforms of the linearized hydrodynamic equations

of motion and next to solve the resulting equations for

the Fourier-Laplace transform Of the density, designated as

n(k,s) and given by

32+ (a+b)kzs + abk“ + c§(1-1/y)k2
n(k,s) = n(k)

 

53+ (a+b)kzsz+ (Cfik2+abk“)s + aCfik“/y

[26]
where a = A/poCV, b = (408/3 + nB)/po and s is a dummy
variable used to evaluate the Fourier and Laplace transforms
of the density fluctuations. The inverse Laplace transform of
n(k,s) is n(k,t), the time-dependent kth Fourier component
of the density. In order to obtain the inverse Laplace

transform of [26], it is necessary to find the roots to the

18

cubic equation in the denominator, called the dispersion

equation:

s3 + (a+b)k2s2 + (C3k2+abk“)s' + aCfik“/y = 0 .

[27]

An approximate solution to [27] consisting of a power

series of coefficients is preferred to the exact algebraic
one because it is more easily evaluated and interpreted in
terms of the physical parameters characterizing the system.

The lowest order solutions to the dispersion equation are

 

4n /3+n 1 l A

s=riCok-!5 S B+—(--—---—)k2 [28]
p, ‘3va CP

3 = - lkz/pOCP . [29]

The real component of the first solution is the halfwidth

at half-height of a Brillouin line

4nS/3+nB l A A
F k2 = 8 ---- + -(- - -—) k2 [30]
DO 90 CV CP

and P is the temporal absorption coefficient or lifetime
of the fluctuation.
It is now possible to determine n(k,t), the inverse

Laplace transform of n(k,s) and the lowest order terms are

c -c c

n(k,t) = n(k)[-L¥exp(-Ak2t/pocp) + Jlexp(-rk“I-.)cos:c,,kt] .
c c
p p

[31]

19

The kth component of the density-density correlation

function can now be constructed from n(k,t) as indicated by

F(k,t) = n(-k)n(k,t) [32]

C -C C
= <n(-k)n(k)>[fE—-yexp(-lk2t/pOCP) + -yexp(-Fk2t)cosCokt] .
C C
P P

[33]
It is now possible to write the generalized structure
factor, which is directly proportional to the scattered

light intensity and related to F(k,t) as follows

S(k,w) = 2Re£mdt exp(iwt) F(k,t) [34]

Since
S(k,w) = S(k)O(k,w) [35]

where
S(k) = <n(-k)n(k)>. [36]

the function O(k,w), the ordinary structure factor,
contains the frequency distribution of the scattered
light and is

c -c 2Ak2/p c
0(k,w) = P V ° P

 

 

2 2 2
cP (1k /poCP) + w

CVI: rk2 I‘kz
__ + ] [37]
cP (Pk2)2+(w+Cok)2 (Pk2)2 +(w-Cok)2

 

 

All terms are easily recognized as Lorentzian in shape since
-1

Iterm°=(A+Bw2) . The first term corresponds to a density

fluctuation relieved by a thermal diffusion process where

A/poCV is the thermal diffusivity. This is a nonpropagating

20

mode; therefore, there is no shift from the incident
frequency, and information about the Rayleigh line is
contained in this term. The second term represents a
prOpagating decay in the fluctuation and results in two
Brillouin components which are equally shifted from the
incident frequency.

The frequency spectrum of light scattered by most
pure substances cannot be accurately predicted by this
analysis. With the exception of liquid metals, most
materials have been found to exhibit experimental sound
absorptions which are always larger than classically
predicted. Two general processes are thought to be
responsible for the excess absorption and are dispersive
relaxation phenomena. The first is a thermal relaxation in
which there is a weak coupling between the internal
vibrational and external translational degrees of freedom
of the molecule. The equilibrium distribution of energy
is disturbed by the periodic compressing and decompressing
of volume elements by the travelling sound wave. The exchange
of energy between the internal and external modes which
occurs while the system is returning to equilibrium may be

expressed by a simple rate equation
dE(Ti)dt = -[E(Ti) - E(tfl‘/T' [38]

where T' is the relaxation time and Ti and T are the
tenperatures of the internal and external degrees of freedom.

In light scattering theory the relaxation is described by

21

introducing a frequency dependent bulk viscosity
= 0 I ° I
”B nB + nB/(l + le ) [39]

or by adding an additional hydrodynamic equation for a
new state variable, the internal degree of freedom. The
second type of relaxation process assumes that the passage
of a sound wave actually induces a structural change in the
molecule and the relaxation time is determined by the
rate at which the molecule returns to its initial structure.
Mountain (19) has extended his hydrodynamic treatment
of the spectral distribution of scattered light to include
a thermal relaxation process which can be described by a
single relaxation time. He introduced a bulk viscosity
consisting of two terms, one dependent and the other
independent of frequency. The modification, therefore,

was incorporated into the Navier-Stokes equation which became

 

3V C2 CfiBpo
Do(-) + (-1-) grad 91+ ( ) grad T1 - (4nS/3+nB) grad div V
at Y Y

t
- J nWt-t') grad div V(t')dt = 0

[40]
where n3 is the frequency independent part and n' is the
Fourier transform of the frequency dependent part. The
procedure to obtain the generalized structure factor was
identical to that previously outlined so only the significant
results are presented here. The most striking difference

appears in the ordinary structure factor as an additional

22

intensity mode.

[ 2Ak2/pOCP ]
kAkZ/poCP)2+w2J

 

0(k.w) = (1-l/Y) +

 

[ICi-C§)k2-(V2/C§-l)(C3/V2T2+C§k2(l-l/Y)']

C3/V“T2+ Vzk2

 

 

[[1-c3/v2(1-1/Y)][v2k2+c§/vzri - (Ci-C§)k2]
+

C3/V“tz+ Vzk2

 

 

PB F
x + B [41]
§+(w- Vk)2 P;+(w+Vk)2

As in [37] all terms are Lorenzian in character. The first
term is identical to the Rayleigh mode obtained in the

absence of relaxation. The second term is also a nonpropagating
mode of decay and is related to the coupling between the
internal and external degrees Of freedom of the molecules.

The last term corresponds to the phonon modes or Brillouin
lines and contains terms in T, the relaxation time of the

new thermal process. Not only the intensity but also the
linewidth of a Brillouin line is altered by the presence of
relaxation. The linewidth is not a simple additive property

as can be seen by comparing the following expression

23

c3 ak2 blkz
2r = ak2 + bokz - (——)(-—-) + (-——-———-)(l-ak21) [42]
B v2 y 1+V2k2T2

with equation [30]. In light scattering experiments the
unshifted relaxational line is a very broad low intensity
line upon which the Rayleigh-Brillouin triplet is
superimposed and is recognized by a low intensity between
the Rayleigh line and its associated Brillouin lines.

The intensity ratio in the case of relaxation is a
much more complicated expression than.the classical
Landau-Placzek result because the unshifted intensity of
the new thermal mode is included when evaluating Ic' and

the Brillouin intensity is also affected:

 

I (l-l/Y)(Ci-C§)k2-(V2/C§-l)[(CZ/V212)+C§k2(l-l/Y)]

21 [1-cfi/VZII-1/Y)][v2k2+c§/vztz]- (cfo-cfim2

. [43]
Equation [43] reduces to the classical Landau-Placzek
value when phonon frequencies are small, that is, Vkr <<l.

And when the phonon frequencies are large, th >>l, the

result is

 

IC 7 C2-C3
-- = (Y'l) 1 + (‘-')( m )

v-1 c: [44]

In actual practice it is difficult to compare the theoretical
prediction with experiment because there is usually large
error involved in determining the intensity Of the second

nonpropagating mode because it is so broad.

24

Mountain (20) has also deduced the ordinary structure
factor for a thermal relaxation by introducing an equation-
of motion for the thermal property. A comparison of the
two approaches gave identical results for a structural
relaxation but slightly different ones in the higher order
corrections for the case of coupling between the internal
and external degrees of freedom.

Litovitz and Herzfeld (54) have develOped a classification
scheme for liquids based on the behavior of their sound
absorption coefficients, a, with temperature. All fluids
except liquid metals have been shown to exhibit sound
absorption coefficients significantly higher than predicted
by the Stokes equation of classical absorption
oz 3

- -1 Ac
v3p0n+4w )I/P)

 

OLclass [45]

2
3
where the first term represents absorption due to viscosity
and the second due to heat conduction. Liquid metals are
alone in that they are the only fluids for which heat
conduction is an important source of absorption. Liquids
are divided into the following classifications:
Group I : Normal Liquids - Monatomic and Diatomic
Liquids, Liquid Metals
da/dT = 0
Group II: Kneser Liquids - Most Organic and Some
Inorganic Liquids
da/dT > 0

Group III: Associated Liquids - Water, Alcohols

da/dT < 0

25

The normal liquids have absorptions only slightly greater
than the classical value. Kneser liquids have a positive

temperature coefficient and have a ratio a/aclass which

generally falls between 3 and 400. It is thought that the
excess absorption is due to a thermal type of relaxation.
Associated liquids, on the other hand, are characterized
by a negative temperature coefficient, have a ratio a/aclass

between 3 and unity, and probably owe their excess absorption

to a structural relaxation process.

CHAPTER III

EXPERIMENTAL INFORMATION

APPARATUS

A. THE BRILLOUIN SPECTROPHOTOMETER The Brillouin
spectrOphotometer used for all light scattering measurements
was designed and characterized by S. Gaumer (55). However,
in order to elucidate the instrumentation necessary to
obtain Brillouin spectra and because modifications have been
incorporated recently, a block diagram, Figure l, and brief
description of the instrument follow. Further details as well
as design considerations are available in the literature (55).
All Optical components Of the system were mounted on
a vibrationally insulated aluminum slab which served as an
Optical bench. The light source was a frequency stabilized
Spectra Physics argon ion laser. The most intense single
mode of the 5145 A line was chosen as the incident frequency.
The mode was selected with the aid of a Spectra Physics
Optical Spectrum Analyzer and an external etalon. The laser
cavity itself contained a solid etalon which, when tuned,
scanned the modes Of the laser lines under examination. The
analyzer output was displayed on a Tektronix Storage
Oscilliscope which allowed the selection of a single mode.
Typical laser output powers were in the 500 mw range. The
exact polarization of a polarization rotator attached to the
laser was calibrated by a piece Of glass with known

refractive index mounted at its Brewster angle.(55)

26

27

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28

The laser light beam impinged on an adjustable
mirror mounted on a triangular rail so that its position
could be changed to accomodate different scattering angles.
The mirror itself could be rotated and tilted on its mount
in order to direct the light beam precisely along a
carefully defined optical path. The purpose of this mirror
was to deflect the incident light across a fully rotatable
Bridgeport table to its center where the scattering sample
was placed. Two pinholes were mounted exactly Opposite
one another on a platter which extended the full diameter
Of the table. Pinhole positions were adjustable in the
vertical and horizontal directions with micrometer screws.
At the center of the platter was a holder for the scattering
cell holder.

Light scattered from the sample initially passed
through two variable apertures located at the ends of a
one inch diameter pipe. The aperture diameters were set at
one millimeter and 0.5 centimeters repectively. The pipe
_ was fitted through an opening in a light-tight aluminum
box. The light next impinged on a 100 millimeter focal
length acromatic antireflectively coated lens which
collimated the light and directed it to the interferometer.
A stray light collector located between the lens and the
interferometer served to block all but the light emanating
from the scattering volume of the sample. The aluminum
housing was connected to the casing of a Tropel Model 350

Fabry-Perot type interferometer by a bellows.

29

The interferometer was mounted on a base plate
designed to be adjustable both vertically and horizontally.
The interferometer consisted Of two optical flats which
were one inch in diameter and polished to a flatness of
1/100. Each was coated on one side with a broad band
dielectric to approximately 95% reflectivity and on the
other side with an antireflective coating. The mirrors
were glued into steel holders that were screwed to their
mounts in the interferometer housing. In actual operation
the rear mirror was stationary but its position could be
adjusted slightly with three micrometer screws. The front
mount was able to slide along invar rods so that the distance
separating the mirrors, and therefore the free spectral
range, could be adjusted. This mirror mount was attached to
three matched piezoelectric crystals and linearly moved
small distances while scanning. The crystals were subjected
to a linear voltage ramp supplied by a Tropel Model 351
Ramp Generator. Maximum voltage attained was 700 volts which
corresponded to a maximum displacement of the mirror on the
order of one micron. The rear of the interferometer was
attached by another bellows to a second light-tight aluminum
box.

After analysis by the interferometer the light impinged
on an antireflectively coated, 1000 millimeter focal length
lens and was focused at a one millimeter diameter pinhole
located at the rear of the box. Immediately behind the

pinhole was a piece of polarizing film rotated to allow only

30

vertically polarized light to pass. A shutter separated
the optical detection train from the EMI photomultiplier
tube. The tube was cooled to -15°C by a Products For
Research refrigeration unit. The signal from the
photomultiplier was amplified by a Keithly high speed
picoammeter whose signal was in turn fed to a Sargent
Welch SRG strip chart recorder.

A new precise alignment procedure was developed in
order to obtain Brillouin spectra of consistently high
finesse. In this work values as high as 70 were Obtained
with 60 being a typical value. The few instances of lower
finesse were directly attributable to the instability Of
the Optics to mechanical disturbances transmdtted from other
laboratories. In the absence of such mechanical disturbances
as well as significant changes in the temperature and
humidity, the optics and especially the interferometer were
stable over long periods of time. This represented a
significant improvement over previous capabilities because
it allowed an entire series of measurements to be made at a
consistently high finesse, thereby reducing the instrumental
linewidth and decreasing the intrinsic measurement error.

The instrumental alignment was performed as delineated below.

1. Check mode structure and polarization of laser.

2. Define optical axis by placing in each aluminum
box a pinhole which has been mounted and
constructed for this purpose. Adjust position of

He-Ne alignment laser so that its light beam

31

passes precisely through the center'of each
pinhole.

Rotate the Bridgeport table to 180° and

adjust its pinholes in the light beam which

now defines the Optical axis.

Secure the collimating lens to the rail in the
first aluminum box. Align lens so that the
transmitted beam passes through the pinholes
located in each box and so that the light
reflected from the front and center surfaces is
colinear and passes back through the center

of the pinhole mounted on the Bridgeport table.
Insert and secure the aluminum light pipe and
close apertures to their minima. Position
aperture centers on the Optical axis, then set
diameters at 1.0 and 5.0 millimeters for the
front and rear respectively.

Position the pinhole which precedes the
photomultiplier at the rear Of the second
aluminum box.

Screw in mounted interferometer mirrors. Locate
approximate mirror separation desired and lock
the front mirror. Measure true mirror separation
with Vernier caliper. 7
Adjust interferometer base both vertically and
horizontally until the reflection from the

front mirror passes back through the pinholes

10.

11.

12.

13.
14.

15.

32

and lens along the optical axis.

Insert focusing lens and adjust so that the
image is centered on the rear pinhole.

Roughly align interferometer mirrors by
adjusting micrometer screws on rear mount

until emerging light pattern is circularly
symmetric.

Check alignment of focusing lens and rear
pinhole and make small adjustments if necessary.
Remove the two alignment pinholes and close
both aluminum boxes and the interferometer
housing.

Rotate Bridgeport table to 90°.

Adjust angle of movable mirror and position so
that argon ion laser beam is reflected and
passes through the previously aligned pinholes
on the Bridgeport table.

Fine tune the interferometer by examining the
light scattered at 90° by a strong Rayleigh
scatterer. Set ramp generator at its fastest
scan rate and feed photomultiplier tube signal
to oscilloscope. Nudge micrometer screws gently
to ascertain their correct positions as indicated
by maximum intensity of displayed Rayleigh lines.
Finally, adjust each piezoelectric with its
control on the ramp generator again tuning to

maximum intensity of scattered light.

33

16. Positioning the sample cell completes the
alignment procedure. Secure the scattering cell
in the base of the temperature control block.
With the three compressed spring adjustments
allow the transmitted beam to pass through the
far pinhole and the reflected beam to pass

back into the laser beam.

All spectra were obtained by allowing the recorder to
run at maximum Speed as well as maximizing the time for one
complete scan of the interferometer, in this case 1000
seconds. As shown in Figure 2 a typical spectrum consists of
many spectral orders each of which is composed of a central
or Rayleigh component and two equally displaced Brillouin
lines. Since the voltage ramp to the interferometer
piezoelectric crystals is not perfectly linear, especially
at the beginning and end of the ramp, only the middle two
spectral orders of each scan were used to extract the
necessary data. Experimental parameters from four orders
were averaged for each fluid at each temperature and were in
good agreement with one another.

A preliminary spectrum was examined for each fluid in
order to determine the proper mirror separation and to
ascertain whether there was a depolarized component in the
scattered light. For this work a proper mirror separation was
judged to be that at which the distance between a Rayleigh
line and its adjacent Brillouin line was equal to the

distance separating Brillouin line of adjacent spectral orders.

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34

 

 

 

 

 

 

 

 

 

 

35

In order to ensure that the Brillouin lines were associated
with the Rayleigh lines nearest them, the initial mirror
separation was taken to be very small so that spectral
orders were separated by a large distance. Since a plot of
vB/frs versus d is linear and passes through the origin,

the SIOpe was determined from the results of the preliminary
spectrum. Then the mirror separation corresponding to one
third of the free spectral range was calculated. All mirror
separations used were approximately 1.40 centimeters. The
depolarized component Hv of the scattered light was found to
be negligible in all fluids. The pV depolarization ratios

of other fluids in the series have been reported (48) to be
on the order of 10.2 indicating very low HV intensities.

B. CONTROL AND MEASUREMENT OF TEMPERATURE A cross-sectional
view of the thermostatted jacket used to thermally insulate
and heat the scattering cell is shOwn in Figure 3. It was
designed primarily by Gaumer (55) and modified as follows.
An additional hole was drilled through the epoxy shell into
the copper block to accomodate a thermometer. Asbestos tape
served to insulate the copper block from a five foot strand
of 28 gauge nichrome resistance wire wound around it. Another
layer of asbestos tape was wrapped around the wire and
secured by silicone tape. A voltage was supplied to the
nichrome wire by a YSI Model 72 Temperature Controller. A
thermistor inserted into the copper block and coated with

Wakefield Thermal Compound provided temperature information

36

Mercury
Thermometer

Hole for

Thermometer \\\\\

\
gi‘éfiix I :ffl

Hole for

’////”Thermistor
Pflv

 

 

 

 

 

   
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

~\‘ rd ‘ ______ _.
Sample / Copper
Cell \\\\\\\ Jacket
\ I
N ' '——_‘_] Cooling
G1Ycerine L“‘F ”’T’COIls
‘\\\\\\ r,,..Heating
180° Wires
Viewing -~
Port \ \ U
1 _‘,,..Sample
___j L.— Cell
‘ ‘ ‘ Holder
Insulating I
Base ‘

Plate 7“~\. E]

/J |

Aluminum Bar on 1"::EJ\\\F‘Centering Pin

Rotating Table TOp

 

 

 

Figure 3. Cross-Sectional View of the Thermostatted Jacket

37

to the controller. It was found that the voltage supplied to
the wire was excessively high and melted the wire so a
Variac was used as a voltage divider. With the controller
bandwidth set at 0.1°C and a Variac setting of 20, the
temperature of the copper block was found to vary less

than 0.1°C.

Due to the viewing slot in the thermostatted jacket,
the sample was not completely insulated and it was therefore
suspected that at higher temperatures the sample temperature
would not equilibrate at the same temperature as the c0pper
block. For this reason the thermostatted jacket was
calibrated in terms of both the block and sample temperature.
Two calibrated, accurate etched-stem mercury thermometers
with divisions of 0.0l°C were used as temperature probes.

A Fisher NBS calibrated thermometer and an Owens-Illinois
thermometer were inserted to the proper depth into the
glycerine sample and the copper block respectively. The
calibration data are given in Table l. Plotting the copper
block temperature versus the glycerine temperature yielded
the calibration curve shown in Figure 4. The least-squares
slope and intercept of the line were used to calculate the
actual temperature from the observed block temperature in all

Brillouin studies.

38

Table 1. Measured Temperatures of the Copper Block and
Sample Liquid at Various Controller Settings

Controller Block Liquid

Setting Temperature Temperature
24.0°C 24.05°C 24.10°C
26.0 25.90 25.90
28.0 27.70 27.75
30.0 29.50 29.55
32.0 31.55 31.60
34.0 33.65 33.55
36.0 35.65 35.45
38.0 37.55 37.25
40.0 39.50 39.05
50.0 49.50 48.50
60.0 59.50 58.20
70.0 69.60 68.00
80.0 79.55 77.80
90.0 89.55 87.25

39

70

40

 

20

 

90 "
80 ‘
0
0
0
0
30 T
20

(3.) ilfllVUidWil X3018

Figure 4. Calibration Plot for the Thermostatted Jacket

LIQUID TEMPERATURE (°c)

40
SAMPLE PREPARATION

The following fluids were the generous gift of the
Dow Corning Corporation: MDSM, MD9M, MDllM' MD13M, D4 and
D6. The remaining fluids, MDM and MDZM were obtained from
Ohio Valley Speciality Chemical Company. All samples had
been analyzed by the respective companies to be greater than
99.5% pure and were used as obtained.

Since dust particles add intensity to the Rayleigh line
thereby invalidating the Landau-Placzek ratio, it was
necessary to clean the samples scrupulously. Each fluid was
forced under pressure through a 0.25 micron pore size
Millipore filter into the 1.0 inch 0.D. scattering cell
which had been attached to a 15 millimeter Fischer-Porter
joint as shown in Figure 5. The filtering process was
repeated until the liquid in the cell appeared totally free
of dust. The fluid was then degassed by the standard freeze-

pump-thaw method and then sealed under vacuum.
REFRACTIVE INDEX

Refractive index measurements of all silicone fluids
were made at five different temperatures with a Bausch and
Lomb Abbe 3-L Refractometer. The temperature of the prisms
was maintained by a Haake circulating bath temperature control
unit. A Doric Thermocouple Indicator Type T DS-350 with
digital temperature readout was used to determine the
temperature of the liquid between the prisms. The digital

voltmeter was calibrated by immersing the thermocouple in an

41

 

TR

1 v.”

 

 

272”

 

 

F1” I

(J Fisher-Porter Joint

 

/ Sample
Cell

 

A /Viewing
/ Window

1v.”

 

7/

 

 

 

Figure 5. The Light Scattering Sample Cell

42

ice bath and five additional constant temperature baths
whose temperatures were determined by the Fisher calibrated
NBS thermometer. The calibration data are tabulated in
Table 2 and plotted in Figure 6.

The refractive index data were obtained at 5890 A,
the sodium D line. The dispersion tables supplied by Bausch
and Lomb allowed calculation of the refractive index at
5145 A, the wavelength of the incident radiation.

Table 2. Measured and Corrected Temperatures Obtained During
Refractive Index Measurements of Silicone Fluids

Measured Corrected
-5.70°C 00.00°C
18.1 23.45
37.2 41.70
59.3 62.35
78.5 81.60

94.8 98.20

43

100—

THERMOCOUPLE READING (°C)
.5
O
1

 

 

° I I I F I
20 40 so no 100

mu: TEMPERATURE (’c)

Figure 6. Calibration Plot for the Thermocouple

CHAPTER IV

RESULTS AND DISCUSSION

REFRACT IVE INDEX

In order to calculate the velocity of the acoustic
wave at various temperatures, it was necessary to determine
the refractive index of the scattering fluid in the
temperature range of interest for the Brillouin study. All
measurements were made at 5890 A, the sodium D line of an
Abbe refractometer and since refractive index is a frequency
dependent property, dispersion corrections were made to
obtain the value at 5145 A, the Brillouin excitation
wavelength. Measured and calculated values of the refractive
index at 5890 A and 5145 A respectively are presented as
functions of temperature in Tables 3 - 11. Details of the
correction procedure are given in the instrument manual.

All of the silicone fluids exhibited a linear decrease
of refractive index with increasing temperature as shown in
Figure 7. The least-squares slopes, intercepts and
correlation coefficients are listed in Table 12 for the
refractive index-temperature data at both wavelengths. The
linearity indicates that there are no optical anomalies
occuring in this temperature region. It is interesting to
note that the molecules of cyclic conformation have
refractive indicies significantly higher than their linear
counterparts. Also, within the series of linear fluids, the

refractive index at a single temperature increased with

44

45

Table 3. Refractive Index As a Function of Temperature
for MDM Observed at 5890 A and Calculated at

5145 A
Temperature n5890 n5145
34.1°c 1.3790 1 0.0005 1.3824 1 0.0010
44.9 1.3732 1.3762
52.6 1.3689 1.3720
63.2 1.3630 1.3659
72.5 1.3582 1.3614

Table 4. Refractive Index As a Function of Temperature
for MD M Observed at 5890 A and Calculated at

5145 A2
Temperature n5890 n5145
34.1°C 1.3836 1 0.0005 1.3870 1 0.0010
44.9 1.3791 1.3823
52.6 1.3735 1.3769
63.2 1.3692 1.3723

72.5 1.3645 1.3677

46

Table 5. Refractive Index As a Function of Temperature
for MD M Observed at 5890 A and Calculated at
-5
5145 A
Temperature n5890 n5145
37.3°C 1.3885 1 0.0005 1.3921 1 0.0010
44.4 1.3841 1.3874
56.3 1.3795 1.3829
64.7 1.3762 1.3795
73.8 1.3712 1.3745
84.7 1.3636 1.3669
Table 6. Refractive Index As a Function of Temperature
for MD9M Observed at 5890 A and Calculated at
5145 31
Temperature n5890 n5145
34.0°C 1.3944 0.0005 1.3979 0.0010
43.3 1.3895 1.3925
51.1 1.3858 1.3892
63.2 1.3818 1.3848
72.5 1.3775 1.3808

47

Table 7. Refractive Index As a Function of Temperature
for MD M Observed at 5890 A and Calculated at

5145 A11
Temperature 115890 115145
34.0°C 1.3950 1 0.0005 1.3985 1 0.0010
44.9 1.3904 1.3939
51.1 1.3866 1.3900
63.2 1.3829 1.3859
72.5 1.3791 1.3821

Table 8. Refractive Index As a Function of Temperature
for MD13M Observed at 5890 A and Calculated at

5145
Temperature ”5890 n5145
34.1°C 1.3690 1 0.0005 1.3995 1 0.0010
44.9 1.3916 1.3952
52.0 1.3877 1.3913
63.2 1.3829 1.3862
72.5 1.3796 1.3827

48

Table 9. Refractive Index As a Function of Temperature
for D. C. 200 Fluid, Viscosity Grade 12,509 cts,
Observed at 5890 A and Calculated at 5145 A
Temperature n5890 n5145

34.0°C 1.4008 1 0.0005 1.4042 1 0.0010

45.7 1.3960 1.3996

52.5 1.3926 1.3962

62.5 1.3881 1.3917

72.5 1.3854 1.3885

Table 10. Refractive Index As a Function of Temperature
for D4°Observed at 5890 A and Calculated at

5145 A
Temperature n5890 n5145
34.0°C 1.3906 1 0.0005 1.3938 1 0.0010
44.6 1.3854 1.3889
52.0 1.3799 1.3834
63.7 1.3753 1.3784
72.5 1.3706 1.3739
78.1 1.3666 1.3695

49

Table 11. Refractive Index As a Function of Temperature
for D Observed at 5890 A and Calculated at

5145 51
Temperature n5890 n5145
34.0°C 1.3965 1 0.0005 1.3998 1 0.0010
44.0 1.3920 1.3951
52.1 1.3876 1.3910
63.2 1.3830 1.3863
72.5 1.3790 1.3822

78.1 1.3750 1.3779

 

 

 

50

3

El MD9M

O MDHM

D MD'I3M
. ll [as

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09.4

  

 

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if, u\\ §
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V MDM
- ll “V‘EAZIV‘
I MDSM
1.3600— ‘ D4
l W I l f I I x 1
40 50 60 7O 80
TEMPERATURE (°c)
Figure 7. Refractive Index at 5145 A as a Function of

Temperature for All Silicone Fluids

51

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52

increasing molecular weight to a chain length of about
thirteen silicon atoms where it then leveled off in an
asymptotic manner as indicated by the small spread in

refractive index over a large molecular weight range.

VARIATION OF THE BRILLOUIN FREQUENCY SHIFT

WITH TEMPERATURE AND MOLECULAR WEIGHT

When linearly polarized light is scattered by a
substance, the scattered light exhibits a frequency
distribution about the incident frequency which is character-
istic of the time-dependent fluctuations in the medium.
Entropy fluctuations and static inhomogeneties are responsible
for the scattered component centered around the incident
frequency. Light scattered by pressure fluctuations gives
rise to three pairs of symmetrically displaced Brillouin
lines corresponding to scattering from the longitudinal and
shear waves (56). Only the longitudinal doublet is regularly
observed in liquids. The magnitude of the Brillouin

displacement, VB, is directly proportional to V the velocity

SI
with which the presSure or sound wave travels through the

liquid, and is given by

v3 = 1 2vo(n/c)VS sin 0/2 [46]

where v0 is the incident frequency, n is the refractive index

of the medium, c is the speed of light in_vacuo, and 0 is the

angle of observation. Values of the frequency shift are

53

usually a few gigahertz (GHz).
The Brillouin shift is measured experimentally as
shown in Figures 8 and 9. For a_given liquid the frequency
shift changes with liquid molecular weight. The values
of the shifts as well as sound velocities and other parameters
are listed for various temperatures for each liquid in
Tables 14 through 21. Values of the Brillouin shifts at
30°C for the silicones and several other materials are

tabulated below. The low frequency shifts of the silicone

Table 13. Brillouin Shift Frequencies of Some Substances at 30°C

Material vB(GHz) Reference
Pyridine 6.3 57
Dimethylsulfoxide 6.0 57
PolyprOpylene Glycol 4.0 58
Polybutadiene 6.0 59
Polymethylmethacrylate 9.5 60
(amorphous)

n-Alkanes 5.0 56
Silicones 3.3-3.7 This Work

fluids compared to common solvents and polymers are paralleled
by low sound velocities as well and will be discussed in the
next section.

The temperature dependences of the Brillouin shifts for
the liquid silicones are illustrated in Figures 10 - 12.
Over the temperature range examined each fluid exhibited a
linear decrease in shift frequency with increasing
temperature. This type of behavior is well established for
pure liquids (61-64). The least-squares slopes, intercepts

and correlation coefficients are listed for each silicone

54

MD13M

 

 

 

MDZM

 

 

 

 

 

 

 

 

 

Figure 8. Brillouin Spectra of MDZM and MD13M at 85°C

55

25°C

 

 

85°C

 

 

 

 

 

Figure 9. Brillouin Spectra of MDM at 25°C and 85°C

56

Table 14. Observed Brillouin Frequency Shifts (v ),

T(‘C)

24.6
34.4
44.0
53.6
63.2
72.8
82.4

Velocities of Sound (VS), Full Brillou n
Linewidths at Half-Height (F ), Sound Absorption
Coefficients (a), and Refracgive Indicies (n)

of MDM as a Function of Temperature

n5145 vB(GHz) VS(m/s) Ih(GHz) 6(cm-1)
1.3875 3.61 946 0.254 1340
1.3821 3.48 916 0.238 1300
1.3768 3.27 864 0.212 1230
1.3715 3.11 825 0.199 1210
1.3662 2.99 796 0.191 1200
1.3610 2.84 759 0.180 1190
1.3557 2.86 719 0.170 1180

Uncertainties in Measured and Calculated Quantities

10.1

10.001 10.03 110 10.005 120

57

Table 15. Observed Brillouin Frequency Shifts (v ),

T(°C)

23.6
34.4
44.0
53.6
63.2
72.8
82.4

Velocities of Sound (V ), Full Brillouin

Linewidths at Half-Height (PB ), Sound Absorption

Coefficients (a), and Refractive Indicies (n)

of MDZM as a Function of Temperature

n5145

1.3924
1.3870
1.3821
1.3772
1.3723
1.3674
1.3625

vB(GHz)

3.48
3.33
3.19
3.07
2.96
2.84
2.78

VS (Tn/S)

909
873
840
811
785
756
742

FB(GHz)

0.297
0.286
0.270
0.244
0.233
0.228
0.201

a(cm

1630
1640
1610
1500
1480
1500
1350

Uncertainties in Measured and Calculated Quantities

10.1

10.001

10.03

110

10.005

120

_1)

58

Table 16. Observed Brillouin Frequency Shifts (v ),
Velocities of Sound (V ), Full Brillou n
Linewidths at Half-Height (PB), Sound Absorption
Coefficients (a), and Refractive Indicies (n)
of MDSM as a Function of Temperature

T(°C) n5145 vB(GHz) VS(m/s) PB(GHz) a(cm'1)
24.4 1.3986 3.62 942 0.376 2000
34.4 1.3935 3.46 903 0.345 1910
44.0 1.3887 3.33 872 0.323 1850
53.6 1.3839 3.22 846 0.281 1660
63.2 1.3790 3.15 831 0.265 1590
72.8 1.3742 3.06 810 0.254 1570
82.4 1.3694 2.89 768 0.249 1620

Uncertainties in Measured and Calculated Quantities

10.1 10.001 10.03 110 10.005 120

59

Table 17. Observed Brillouin Frequency Shifts (VB),
Velocities of Sound (V ), Full Brillouin
Linewidths at Half-Height (P ), Sound Absorption
Coefficients (a), and Refracgive Indicies (n)
of MD9M as a Function of Temperature

T(°C) n5145 ' VB(GHz) VS(m/s) FB(GHz) 6(cm-1)
23.4 1.4017 3.67 952 0.438 2300
34.4 1.3970 3.59 935 0.416 2220
44.0 1.3928 3.46 904 0.406 2240
53.6 1.3887 3.38 885 0.378 2130
63.2 1.3846 3.26 857 0.362 2110
72.8 1.3804 3.21 846 0.334 1970
82.4 1.3763 3.10 819 0.312 1900

Uncertainties in Measured and Calculated Quantities

10.1 10.001 10.03 110 10.005 120

60

Table 18. Observed Brillouin Frequency Shifts (VB),
Velocities of Sound (VS), Full Brillouin
Linewidths at Half-Height (P ), Sound Absorption
Coefficients (a), and Refracgive Indicies (n)
of MDllM as a Function of Temperature

T(°C) n5145 vB(GHz) VS(m/s) FB(GHz) a<cm‘1)
23.6 1.4026 3.83 993 0.421 2260
34.4 1.3981 3.72 968 0.400 2070
44.0 1.3940 3.61 942 0.384 2040
53.6 1.3899 3.44 900 0.362 2010
63.2 1.3858 3.33 874 0.351 2000
72.8 1.3817 3.24 853 0.329 1930
82.4 1.3776 3.14 829 0.307 1850

Uncertainties in Measured and Calculated Quantities

10.1 10.001 10.03 110 10.005 120

61

Table 19. Observed Brillouin Frequency Shifts (vB),
Velocities of Sound (V ), Full Brillouin
Linewidths at Half-Height (r3), Sound Absorption
Coefficients (a), and Refractive Indicies (n)
of MD13M as a Function of Temperature

T(°C) n5145 vB(GHz) VS(m/s) FB(GHz) d(cm‘1)
24.0 1.4041 3.72 964 0.420 2180
34.4 1.3994 3.61 938 0.389 2070
44.0 1.3951 3.49 910 0.352 1930
53.6 1.3908 3.39 887 0.346 1950
63.2 1.3865 3.29 863 0.336 1940
72.8 1.3822 3.16 832 0.330 1980
82.4 1.3780 3.06 810 0.320 1980

Uncertainties in Measured and Calculated Quantities

10.1 10.001 10.03 110 10.005 120

62

Table 20. Observed Brillouin Frequency Shifts (v ),
Velocities of Sound (VS), Full Brillou n
Linewidths at Half-Height (P ), Sound Absorption
Coefficients (a), and Refracgive Indicies (n)
of D4 as a Function of Temperature

T(°C) n5145 vB(GHz) VS(m/s) FB(GHz) a(cm'1)
23.4 1.3997 3.31 860 0.462 2690
29.6 1.3963 3.30 860 0.460 2670
39.2 1.3911 3.24 847 0.444 2620
48.8 1.3860 3.16 829 0.378 2280
58.4 1.3808 3.10 817 0.360 2200
68.0 1.3756 2.97 786 0.318 2020
82.4 1.3678 2.74 729 0.274 1880

Uncertainties in Measured and Calculated Quantities

10.1 10.001 10.03 110 10.005 120

63

Table 21. Observed Brillouin Frequency Shifts (vB),

T(°C)

29.6
39.2
48.8
58.4
68.0
77.6

Velocities of Sound (VS), Full Brillouin
Linewidths at Half-Height (F
Coefficients (a), and Refrac

of D6 as a Function of Temperature

n5145

1.4020
1.3974
1.3928
1.3882
1.3836
1.3789

vB(GHz)

3.74
3.56
3.35
3.32
3.14
3.01

Vs(m/s)

971
927
880
870
826
794

0.531
0.515
0.463
0.417
0.375
0.359

), Sound Absorption
ive Indicies (n)

a(cm’1)

2730
2780
2630
2400
2270
2260

Uncertainties in Measured and Calculated Quantities

10.1

10.001

10.03

10.005

120

64

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in Table 22. Although both the slope and the intercept
vary irregularly from liquid to liquid, the lepe tends to
decrease as the chain length increases.

The Brillouin shift frequency at a given temperature is
also dependent on the molecular weight of the sample as
well as its chain configuration, linear or cyclic. The
Brillouin shift at 30°C is plotted versus molecular weight
in Figure 13. The data of Kumar (48) at the molecular weight
extremes are included to illustrate the behavior of the
Brillouin shift over a broader molecular weight range.
These data will also be included in subsequent molecular
weight plots. Several interesting features are apparent
in Figure 13. First, the curve is smooth and continuous
indicating that no gross structural changes occur as the
chain length increases. Also the lepe is much greater at
molecular weight values less than 1000. The cyclic molecules

exhibit an analogous trend. The small rings, D and D6' have

4
shift frequencies approximately equal to those of their
linear counterparts but the frequencies diverge as the number
of silicon atoms in the ring or chain increases. It was

found that the Brillouin shift at a single temperature was
inversely prOportional to the molecular weight of the liquid.
Figure 14 shows the linear relationship between the shift
and reciprocal molecular weight. Since MM and the 100 cts
fluid (MW 6400) both fall on the least-squares line, this

relationship apparently describes the behavior of the shift

equally well at high and low molecular weights.

Table 22. Linear Least-Squares Slopes,
and Correlation Coefficients of the Brillouin
Shift vs. Temperature Plots for All Silicones

Fluid

Slope

(6112/ “0).

-0.01615
-0.01217
-0.01175
-0.00978
-0.01216
-0.01135
-0.00940
-0.01470

68

Intercept
(GHz)
4.005
3.743
3.876
3.904
4.122
3.996
3.587

4.141

Intercepts at 0°C,

Correlation
Coefficient
0.9981
0.9956
0.9935
0.9967
0.9966
0.9999
0.9710
0.9886

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71

VARIATION OF THE VELOCITY OF SOUND

WITH TEMPERATURE AND MOLECULAR WEIGHT

The velocity of sound obtained from the Brillouin
shift frequency according to Equation [46] is related to
the liquid structure through the adiabatic compressibility,
BS:

8
VS = (1/088) [47]

where p is the density. In a light scattering experiment,
the fixed wave vector is scattered by a travelling acoustic
wave of the same wavelength as the incident light. The
liquid is undisturbed because no gradient has been introduced
and the measured frequencies and velocities are those of

the thermally driven fluctuations inherent in the medium.
Since the incident wave vector determines the frequencies
which will scatter light, it is possible to examine other
frequencies by changing the wave vector. This is done by
changing the wavelength of the incident light or more
commonly by varying the scattering angle. Dispersion in the
sonic velocity is said to occur when the waves of different
frequencies do not travel with the same velocity. Previous
ultasonic and Brillouin studies have not reported velocity
dispersion in the siloxanes (44,48,66). The results of Kumar
and Friedlander (48,66) are reproduced in Table 23 to
demonstrate that the velocities of the silicones are

independent 0f frequency over the measured range.

72

Table 23. velocity of Sound as a Function of Frequency
For Several Silicones at 25°C
VB (GHz) VS (m/sec) vB (GHz) VS (m/sec)
MM MD7M
50° 1.87 1 0.05 827 1 10 60° 2.73 1 0.05 1010 1 10
60° 2.21 824 75° 3.30 1000
90° 3.40 897 90° 3.91 1030
120° 4.12 887 105° 4.32 1010
135° 5.07 1010
D5 D15
75° 3.08 1 0.05 932 1 10 60° 2.86 1 0.05 1040 1 10
90° 3.58 932 75° 3.22 967
105° 4.19 971 90° 3.75 969
120° 4.58 973 120° 5.21 1090
135° 4.88 973

73

In an ultrasonic experiment a sonic wave of variable
frequency is introduced into the sample and its velocity and
spatial attenuation are measured. In this case the medium
is disturbed by the sonic wave. In spite of this difference,
velocities measured by the two techniques have been found
to be in close agreement for many liquid systems (50,63,65).

For every liquid studied the velocity of sound was
a linearly decreasing function of temperature as illustrated
in Figures 15 - 17. The data are tabulated in Tables 14 - 21.
The sloPes, intercepts and correlation coefficients
determined by the method of least-squares are listed in
Table 24. With the use of thermodynamic relations, it can

be shown that Equation [47] can be rewritten as

_ _ ’5
VS - [CP(Y 1)/a2T] [48]

where CP is the constant pressure heat capacity, y is the
heat capacity ratio , a is the coefficient of thermal
expansion and T is the absolute temperature (75). Since

CP' y and a vary only slightly with temperature, Equation
[48] correctly predicts that the velocity will decrease as
the temperature increases. The ultrasonic data of Weissler
(42) are presented in Table 25 in addition to the Brillouin
velocities. As can be seen from.the data, the Brillouin and

ultrasonic velocity values agree within a few percent.

74

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Table 24. Linear Least-Squares Slapes, Intercepts at 0°C
and Correlation Coefficients of the Velocity
of Sound vs. Temperature Plots for All Silicones

Fluid Slope Intercept Correlation
(m/sec°C) (m/sec) Coefficient

MDM -3.948 1044 0.9980

MDZM -2.909 972 0.9952

MDSM -2.776 1002 0.9922

MD9M -2.286 1008 0.9959

MDllM -2.902 1064 0.9961

MD13M -2.684 1030 0.9994

D4 -2.157 926 0.9606

D -3.565 1069 0.9909

78

Table 25. Ultrasonic and Brillouin Velocities of Sound
for Linear Polydimethylsiloxanes at 30°C

Liquid MW Ultrasonic V Brillouin V
(m/sec) S (m/sec) S
MM 162.2 873 1 10 873 1 10

MDM 236.3 901 925
MDZM 310.4 919 884
MD3M 384.5 931 929
MDSM 533.1 942 918
MD7M 720 953 983
MD9M 829 939
MDllM 978 976
MD 3M 1126 966 949
100 cts 6400 985 975

In order to place in perspective the magnitudes of
the velocities of sound for the silicones, the Brillouin
velocity results for other polymers and a few organic
liquids are listed in Table 26. The siloxane sound velocities,
ranging from 875 to 975 meters per second, are significantly
lower than those tabulated. High sound velocities have been
empirically correlated with strong intermolecular forces
and low compressibilities which are thought to facilitate
passage of the sound wave. For this reason associated
liquids such as dimethylsulfoxide have higher velocities
than the nonpolar normal alkanes. In this respect the velocity
of sound is a qualitative probe of the extent of inter-
molecular interaction. This interpretation leads to the
supposition that the intermolecular forces in the siloxanes
are weak in comparison to many liquids and are weaker than
in the alkanes of corresponding chain length. Also, the
increase of the sound velocity with molecular weight as

shown in Figure 18 may be an effect of the increasing

Table 26. Brillouin Velocity of Sound Data for

7

9

Selected Polymers and Other Liquids at 20°C

Compound

Polymethylmethacrylate
(amorphous)

Benzene

Water

Polybutadiene

C14H30

Toluene

C12H26

Carbon disulfide

Polypropylene glycol 425

Acetone

Acetic acid

Ethyl alcohol

n-Hexane

Methyl alcohol

Carbon tetrachloride

Ethyl ether

Silicones

V

S

(In/see)

2870

1500

1480
1400
1350
1350
1310
1250
1200
1190
1160
1160
1110
1110
1000
1000

900 - 1000

Reference

60
53
53
56
56
53
56
53
58
53
53
53
53
53
53
53

This Work

80

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81

interaction between siloxane functional groups. In long
chain molecules there are a greater number of covalent
bonds between the siloxane units than in short chain
molecules. The decrease of the velocity of sound with
increasing temperature can also be attributed to the
increasing randomness of the liquid structure as the liquid
is heated.

A Rayleigh-Brillouin light scattering study of
polyprOpylene glycol 425, a polymer somewhat simdlar in
structure to the siloxanes, has been reported by Wang and
Huang over a broad range of temperatures (58). They
examined the behavior of the velocity of sound with
temperature as the polymer underwent transitions from the
glass to rubber and finally to the liquid state. The
behavior of the siloxanes very closely paralleled that of
polypropylene glycol in the liquid region. From 40°C to
90°C the velocity of sound in polyproPylene glycol varied
from 1000 to 750 meters per second and the siloxane
velocities fell in this same range. Also the temperature
coefficients of velocity are virtually identical for the
two types of polymer liquids.

The velocity of sound was found to depend not only
on molecular weight but also on molecular structure.
Figure 18 shows the smooth curve describing the relation-
ship between the velocity and molecular weight for both
linear and cyclic conformations. Since the velocity is

calculated from the Brillouin shift frequency, the same

82

functional form is expected for both, as is indeed the case.
The sloPe is higher at low molecular weights and the velocities
of the cyclic fluids exhibit similar behavior but are
somewhat higher than those of the corresponding linear
fluids. A recent Brillouin study of normal alkanes, C12 to
C36' reported by Champion and Jackson (56) revealed that
the sound velocity increased with increasing chain length,
although no attempt was made to delineate the mathematical
relationship. On the other hand, Wang and Huang (74)
report no molecular weight dependence on the velocity for
polypropylene glycol ranging from weight average molecular
weight 425 to 1000.

Although Brillouin and ultrasonic theories do not
contain any information pertaining to molecular weight,
a semi-empirical approach was developed by Rao. He observed

that for many unassociated organic liquids (67)

1 8V 1 3V

--<-—S) /-——<—)P [49]
vS 3T P v 3T

where V is the molar volume. In integral form this equation
is

R = v v [50]

where R is the Rao number. The validity of equation
[49] was checked for several silicone fluids and ratios
ranging from 2.7 to 2.9 were obtained. Rao predicted that

R would be a linear function of molecular weight (67).

83

Table 27 contains the molar volumes and Rao numbers for

all the silicones studied.

Table 27. Rao Numbers, Molar VOlumes and Related Quantities
for All Silicones at 25°C

Fluid VS(m/sec) p(g/cc) V(cc/mole) R
MM 894 0.7619 212.9 2052
MDM 945 0.8200 288.4 2830
MDZM 899 0.8536 364.0 3513
MD M 945 0.8755 439.3 4312
MD3M 932 0.9110 585.2 5717
MDSM 995 0.9134 745.6 7445
MD7M 950 0.9223 898.7 8837
MDilM 991 0.9305 1051 10,480
MD13M 962 0.9335 1206 11,900

The linear relationship between the Rao number and molecular
weight, MW, was verified for the silicones as shown in
Figure 19. Therefore, according to Rao, the velocity of

sound is given by the following third order equation

V

S 03[b(1/MW) + a]3 [51]

pa[b3(l/MW)3+3ab2(l/MW)2+3a2b(l/MW)+a3]

where p is the density and a and b are the sloPe and
intercept of the Rao number versus molecular weight plot.
However, for the silicone fluids, a completely empirical
first order equation

-4

vS = (-3.10 x 10 )(1/MW) + 852 [52]

also fits the data as demonstrated in Figure 20. The
contribution of the second and third order terms in Rao's
equation give numbers on the order of 10 and 0.1 m/sec

which are within the limits of experimental error for this

84

12.001

10.00—

8.00-

6.00“

Rx10'3

4.00“

2.004

 

 

I I I I I j
200 400 600 800 1000 1200

MOLECULAR WEIGHT

Figure 19. Rao Number as a Function of Molecular Weight
for Linear Silicones

85

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86

study. In order to determine which equation better
describes the behavior of these liquids, a much larger

data set would be necessary.

VARIATION OF THE ADIABATIC COMPRESSIBILITY

WITH TEMPERATURE AND MOLECULAR WEIGHT

The compressibility is inversely related to the
strength of the intermolecular forces or degree of
association so it is a probe of liquid structure. It is
possible to calculate the adiabatic compressibility of

a liquid using the velocity of sound and density from

2
1 v
5 /ps .

B [53]

Table 28 lists the compressibilities of all siloxanes
for which velocities of sound have been determined. The
compressibilities of the silicones are unusually high.
Values for some other liquids are given in Table 29

for comparison.

Table 29. Adiabatic Compressibilities for Some Organic
Liquids at 25°C

Liquid BS x 1010 (cmz/dyne) Reference
Pyridine 0.438 57
Dimethylsulfoxide 0.406 57
Decane 0.935 56
Dodecane 0.847 56

The associated liquids, pyridine and dimethylsulfoxide,

are much less compressible than the silicones and even

87

Table 28. Velocities, Densities and Adiabatic
Compressibilities for Siloxanes at 25°C

Liquid p(g/cm3) VS(m/sec) BS X 1010 (cmZ/dyne)
MM 0.7619 895 1.64
MDM 0.8200 945 1.37
MDZM 0.8536 399 1.45
MD3M 0.8755 946 1.28
MDSM 0.9110 933 1.26
MD7M 0.9134 996 1.10
MDgM 0.9233 951 1.20
MDllM 0.9305 992 1.09
MD13M 0.9335 963 1.16
100 cts 0.9579 990 1.06
D4 0.9500 872 1.38
D5 0.9528 933 1.20
D6 0.9621 980 1.08
D9 0.9756 1010 1.00
D 0.9736 1050 0.94

15

88

the normal alkanes are for molecules of corresponding
chain length. A possible explanation for these observations
lies in the large space requirement of the methyl groups.
Neutron scattering and nuclear magnetic resonance studies
have demonstrated the high mobility of the methyl group
hydrogen atoms which is attributed to rotation about the
Si-C and Si-O bonds (39,71,72,73). In other words the
molar volumes are large and the intermolecular forces
are weak. Also, nearly free rotation about the silicon
oxygen bond contributes to the overall mobility of the
chain. This freedom of motion is considerably more
restricted for cyclic molecules than linear ones as evidenced
by the smaller molar volumes (32) and compressibilities
of the cyclic liquids. The compressibilities of the cyclic
molecules fall more rapidly with increasing molecular
weight than do those of the linear molecules. The smallest
ring containing three silicon atoms is planar so there is
little rotation about the Si-O bond. As the ring increases
in size, it puckers and there is increased floppiness or
freedom of motion. Rings and corresponding chains containing
at least nine siloxane units have equal compressibilities.
The compressibility of the linear series initially
decreases with increasing molecular weight then reaches a
limiting value. The statistical model for polymers considers
the chain to be composed of molecular segments whose motions
are virtually independent of one another (76). The

asymptotic behavior of the compressibility may be related

89

to these segmental motions.

The compressibilities of four silicones were
calculated for several temperatures and the results are
tabulated in Tables 30 - 33. The listed densities have
been calculated from the reported temperature coefficients
of density for these liquids (26,68). Each fluid exhibited
a linear increase in compressibility with increasing
temperature as is shown in Figure 21. In general, the
compressibility and its temperature coefficient decreased

with increasing chain length.

VARAIATION OF THE BRILLOUIN LINEWIDTH

WITH TEMPERATURE AND MOLECULAR WEIGHT

The measured Brillouin linewidths were determined
directly from the experimental Brillouin lines. An observed
Brillouin line is actually a convolution of the true
Brillouin line and the instrumental response function, the
central of Rayleigh peak. Immediately prior to each series
of Brillouin measurements, the linewidth at half-height of
the instrumental response function was determined with the
use of a strong Rayleigh scatterer. The central line of a
Brillouin spectrum.was considered the instrument response
function because both had the same linewidth. Since the

instrument function and the observed Brillouin line have been

90

Table 30. Adiabatic Compressibility as a Function of
Temperature for MDM

Table

T(°C) p(g/cm3> BS x101°(cm2/dyne)
24.6 10.1 0.8140 10.001 1.37 $0.05

34.4 0.8039 1.48

44.0 0.7939 1.69

53.6 0.7839 1.87

63.2 0.7739 2.04

72.8 0.7639 2.27

82.4 0.7540 2.56

31. Adiabatic Compressibility as a Function of
Temperature for MD M

2

T(°C) p(g/cmg) BS x1010(cm2/dyne)
23.6 10.1 0.8521 10.001 1.42 10.05

34.4 0.8405 1.56

44.0 0.8301 1.71

53.6 0.8197 1.85

63.2 0.8093 2.01

72.8 0.7989 2.19

82.4 0.7885 2.30

91

Table 32. Adiabatic Compressibility as a Function
of Temperature for MDSM
3

T(°C) p(g/cm ) BS x1010(cm2/dyne)
24.4 10.1 0.8975 10.001 1.26 $0.05
34.4 0.8873 1.38

44.0 0.8774 1.50

53.6 0.8676 1.61

63.2 0.8578 1.69

72.8 0.8480 1.80

82.4 0.8381 2.02

Table 33. Adiabatic Compressibility as a Function
of Temperature for MD9M
. 3

M C) p(g/cm ) as x101°(cm2/dyne>
23.4 $0.1 0.9233 30.001 1.19 10.05
34.4 0.9128 1.25

44.0 0.9037 1.35

53.6 0.8945 1.43

63.2 0.8854 1.54

72.8 0.8763 1.59

82.4 0.8671 1.72

92

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93

successfully approximated by Lorentzian line shapes
(48,57), the observed Brillouin linewidth is the sum of
the linewidth of the true Brillouin line and the central
line:

exp = true + 54
PB PB PC [ ]

All linewidths reported in Tables 14 through 21 have been
corrected for the contribution of the instrumental
response function.

The behavior of the linewidth with temperature is
shown for each silicone fluid in Figure 22 - 24. The
linewidth was found to be a linearly decreasing function
of temperature. The least-squares slopes, intercepts and
correlation coefficients are_given in Table 34. Although
there is little information available in the literature
for comparison, the same linear decrease of Brillouin
linewidth with increasing temperature has been reported for
pyridine and dimethylsulfoxide (57). Hydrodynamic theory
predicts that the linewidth should be proportional to the

viscosity of a simple non-relaxing fluid:

4n /3+n 1(Y-1)
r = [-—§-———§ + ] E2 [55]

s
p pCP

 

where "S and "B are the shear and bulk viscosities, A

is the thermal conductivity and :2 is the square of the
wave vector. For most liquids the second term in Equation
[55] is very small compared to the first. For the silicones

the maximum contribution of the thermal conductivity term

94

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97

Table 34. Linear Least-Squares SlOpes, Intercepts at 0°C,
and Correlation Coefficients of the Full Brillouin
Linewidth vs. Temperature Plots for All Silicones

Fluid Slope Intercept Correlation
(GHz/°C) (GHz) Coefficient

MDM -0.001444 0.2837 0.9919

MDZM 20.001619 0.3378 0.9883

MDSM -0.002302 0.4222 0.9711

MD9M -0.002144 0.4925 0.9946

MDllM -0.002214 0.4872 0.9840

MD13M ~0.001605 0.4420 0.9396

D4 -0.003432 0.5567 0.9864

D -0.003946 0.6549 0.9889

98

was calculated to be 0.001 GHz, too small to be detected
in this study. Since the viscosity is known to decrease
slowly with temperature for the silicones (32), the
experimental results are in qualitative agreement with
theory.

The Brillouin linewidth was also found to vary with
molecular weight as shown in Figure 25. Since the viscosity
increases as the chain length is increased, this is not
an unexpected result. Figure 26 demonstrates that the
linewidth is a linear function of reciprocal molecular
weight.

The reciprocal of linewidth has the dimension of
time and may be described as a relaxation time or distribution
of relaxation times. Although the molecular relaxation
process occurring has not been interpreted for polarized
Brillouin scattering, it is possible to estimate the
activation energy for the process by assuming its temperature

dependence is described by the Arrhenius equation
l/1rI‘B = A exp(E/kT) [56]

where l/flI‘B is the relaxation time, A is a constant,

B is the activation energy, k is the Boltzmann constant and
T is absolute temperature. Table 35 contains the values of
the relaxation times at various temperatures for all the
silicones. The activation energy can be calculated from the
slope of the line obtained by plotting ln(l/nFB) versus

l/T. The linearity of such a plot is established in Figure

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27 for two silicones. The activation energies are listed

in Table 36.

Table 36. Activation Energies for Relaxational Processes
in Silicone Fluids

Fluid E(kcal/mole) Fluid E(kcal/mole)
MM 0.563 0 3.18

MDM 1.48 D4 1.91

MD M 1.34 03 0.774
MDEM 1.61 015 0.886
MD9M 1.20

MDllM 1.11

MD 3M 0.936

100 cts 0.742

Energies for several different processes have been determined
for the siloxanes and summarized in a paper by Flory (69).
The barriers to rotation about the Si-C and Si-O bonds are
approximately 1.5 to 2.0 and 0.4 kcal/mole respectively.

In addition, Flory's rotational isomeric state model

for the siloxanes defines three possible "states” for

each Si-O bond and energy barriers between these states are
about 1.0 to 2.0 kcal/mole. It is interesting to note

that the energies calculated from.the Brillouin linewidth
data fall in this region. The small cyclic molecules,
however, are characterized by higher energies which is

not surprising in view of their more restricted molecular

motions.

103

 

 

20.-

 

I

U l
2.8 3.0 3.2 3.4
1/1’ x 103 (’K)

2071

20.84

 

21.01 0

 

 

1

2.3 3.0 3.2 3.4
1/1‘ x 103 (’K)

Figure 27. Temperature Dependence of Relaxation Times for
MDM and MD 1 3M

104

VARIATION OF THE ABSORPTION COEFFICIENT

WITH TEMPERATURE AND MOLECULAR WEIGHT

The sonic absorption coefficient is calculated from
a = I‘B/2VS [57]

where PB is the full Brillouin linewidth and VS is the
velocity of sound. It is related to the efficiency of
energy transfer from internal vibrational to external
translational degrees of freedom. If this efficiency is
low, then a large absorption will occur. As discussed in
Chapter II, most liquids are classified into two groups

on the basis of the sign of their temPeraturecoefficient
of the absorption of sound. Kneser liquids exhibit positive
temperature coefficients while associated liquids are
characterized by negative ones. Also, ultrasonic theory
suggests that all liquids have absorption in excess of the
classical Stokes value. The Stokes expression considers
absorption due to viscosity alone and for light scattering

experiments it is approximated by

80202

 

n

a =
class 3vgp s [58]

where "S in the shear viscosity.
The absorption coefficients of the silicones are given
at various temperatures in Tables 14 - 22. The least-squares

slopes, intercepts and correlation coefficients are given

105

in Table T7- The absorption coefficient was determined

to be a linear function of temperature with a negative
slope as shown in Figures 28 - 30. The negative lepe
places the silicones in the associated category. However,
there is no evidence to support this implication. On the
contrary, the high compressibilities and vapor pressures
as well as other data indicate weak intermolecular forces.
The unassociated normal alkanes (56) also have.negative
absorption temperature coefficients. According to Herzfeld
and Litovitz (54) this apparent anomalous behavior is
characteristic of viscous liquids.

In a series of papers, Hunter and coworkers investigated
the behavior of the ultrasonic absorption coefficient of
various silicone polymers (44,45,46). They found that the
absorption increased with molecular weight, reached a
maximum value, and then leveled off as the molecular weight
further increased. The Brillouin absorption confirmed this
trend as indicated in Figure 31. They also noted that the
position of the maximum depended on the incident ultrasonic
frequency and shifted to lower molecular weights as the
frequency increased. The value reported by Hunter for the
molecular weight of maximum absorption was 6400 at a frequency
of 90 megacycles. The Brillouin frequency, however, is on
the order of 2000 megacycles and the corresponding molecular
weight is about 1000. As in the case of previous parameters,
the absorption coefficient was found to be linear in

reciprocal molecular weight as shown in Figure 32.

106

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Table 37. Linear Least-Squares SlOpes, Intercepts at 0°C,
and Correlation Coefficients of the Sound Absorption

Coefficient vs. Temperature Plots for All Silicones

Fluid Sloge _1 Intefcept Correlation
(cm C) (cm ) Coeff1c1ent

MDM -2.738 1380 0.9179

MDZM —4.460 1770 0.9197

MDSM -7.668 2150 0.9256

MDQM -6.652 2480 0.9642

MDllM -5.654 2320 0.9344

MD13M -2.830 2160 0.6689

D4 —15.16 3090 0.9789

D -12.27 3170 0.9500

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112

Hunter also pointed out that absorptions for the
silicones were less than the classical Stokes values and
that the deviation was more pronounced at higher frequencies.
The Brillouin absorptions are also lower than the Stokes

values as indicated by the ratio a/aclass listed in Table 38.

Table 38. Absorption Coefficient Ratios for Silicones

at 25°C
Liquid aIcm'l) a/a
. class

MDM 1300 0.317
MDZM 1650 0.251
MD M 1950 0.145
MngM 2330 0.0954
MDllM 2250 0.0758
MD13M 2200 0.0586
n 2700 0.263
02 2850 0.108

Hunter conjectures that the Stokes prediction is too large
because the viscosity undergoes relaxation. Evidence for

his position was provided by the d/a

class ratio varying from

2 at very low frequencies to 0.25 at high frequencies,
thereby indicating a frequency dependent viscosity. The even
smaller value of a/a at the hypersonic Brillouin

class
frequencies provides further substantiation for his hypothesis.

VARIATION OF THE LANDAU-PLACZEK RATIO

WITH TEMPERATURE AND MOLECULAR WEIGHT

The Landau-Placzek ratio is the ratio of the integrated
intensity of the central peak to the sum of the Brillouin

intensities and is denoted JV. It was determined for each

113

liquid as a function of temperature and the values are
listed in Table 40. This ratio was found to be relatively
insensitive to both temperature and molecular weight.
The large experimental error associated with the measurement,
in conjunction with the very small spread of the data,
prohibits more than a qualitative interpretation.

For a simple liquid the Landau-Placzek ratio is related

to the heat capacity ratio by

Jv = cP/cV - 1 [59]

The heat capacity ratios for several silicones were determined
by Weissler (42) and ranged from 1.31 for MDM to 1.22 for
MD11M' These values are in good agreement with the Brillouin
results. In addition, Wang and Huang found similar Jv

values for liquid polyprOpylene glycol and noted the very
slight decrease of J with temperature.

v
The high Landau-Placzek ratios of MD M and D are not

9 6
consistent with the values for the other liquids in the
series. Sample contamination by a small amount of high
molecular weight polymer would account for this observation.-
The other spectral parameters would be influened only

slightly since the principle effect would be to increase

the intensity of the Rayleigh line and therefore, Jv

114

Table 39. Landau-Placzek Ratio as a Function of Temperature
for the Silicones

T(°C) MDM MDZM MDSM MDgM
25 0.30 0.26 0.31 1.11
35 0.28 0.27 0.29 0.93
45 0.26 0.26 0.27 0.93
55 0.25 0.25 0.24 0.86
65 0.24 0.23 0.23 0.75
75 0.23 0.23 0.22 0.57
85 0.22 0.21 0.22 0.49
T(°C) MDll MD13M D4 D6
25 0.27 0.27 0.35 2.2
35 0.27 0.23 0.35 2.2
45 0.25 0.23 0.33 2.1
55 0.24 0.21 0.31 2.0
65 0.23 0.18 0.30 1.9
75 0.24 0.18 0.28 1.9
85 0.22 0.17 0.26 1.8

The relative uncertainty in the Landau-Placzek Ratios is 15%.

115

CONCLUSIONS

Brillouin light scattering has been used to study
the effect of molecular weight on spectral parameters in
a homologous series of polydimethylsiloxane fluids. Each
liquid behaved in the expected manner exhibiting linear
relationships of refractive index, Brillouin shift, velocity
of sound, adiabatic compressibility, Brillouin linewidth,
and absorption coefficient with temperature. The high
compressibilities and low sound velocities were evidence
of the weak intermolecular forces in the silicones. Sound
velocities agreed with those determined by ultrasonic
methods.

The Brillouin shift, velocity of sound, Brillouin
linewidth, and absorption coefficient were all found to
vary in the same way with molecular weight. Each parameter
increased rapidly to a molecular weight of about 1000 but"
increased only slightly as the molecular weight increased
further. An empirical equation was found which described
the given parameter as a linear function of reciprocal
molecular weight. It has been pointed out that a chain
length of eleven silicon atoms, corresponding to a molecular
weight of approximately 1000, is just long enough to form a
planar, unstrained ring (68). Since long polymer chains are
known to be coiled, these steric considerations suggest that

this does not occur below a molecular weight of 1000.

116

In summary:

1.

The first comprehensive Brillouin study of a homologous
series of silicone fluids has been completed and the
results are in agreement with reported ultrasonic data.
The acoustic properties of the cyclic fluids were
distinctly different from those of the linear fluids.
Sonic velocities and absorption coefficients in

cyclic liquids were higher than in linear ones.

This behavior was interpreted in terms of the more
restricted motions of cyclic compared to linear
molecules.

Adiabatic compressibilities for all the silicones

were found to be unusually high, indicative of their
weak intermolecular forces. The small ring liquids
were less compressible than the corresponding linear
ones but this difference disappeared for the higher
molecular weight liquids.

Estimated activation energies for relaxation

processes were found to be numerically the same as
calculated barriers to rotation about the siloxane
bonds.

The temperature coefficient of the sonic absorption
coefficient was negative for all fluids, placing

them in the category of associated and viscous liquids.
The classical prediction for absorption due to
viscosity alone was found to be too large for all the

silicones in contrast to most other liquids which

117

exhibit excess absorptions. This behavior has been
attributed to a frequency-dependent viscosity.

The lack of velocity dispersion, absence of a
Mountain line and low sonic absorptions indicate
that the silicones behave as simple fluids in

which there are no thermal or structural relaxations.

CHAPTER V

LITERATURE CITED

(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)

(ll)
(12)
(13)
(14)
(15)
(16)
(17)

(18)
(19)
(20)
(21)
(22)

J.

J.

unt‘ :9 b w: a» :z Q

‘

w "0"!)

L.
12.

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178 3552

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