THE DEPOLARIZED DOUBLET SPECTRA OF MOLECULAR LIQUIDS : COMPARISON OF THEORY AND EXPERIMENT

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THE DEPOLARIZED DOUBLET SPECTRA OF MOLECULAR LIQUIDS : COMPARISON OF

THEORY AND EXPERIMENT

G. Enright, G. Stegeman, B. Stoicheff

To cite this version:

G. Enright, G. Stegeman, B. Stoicheff. THE DEPOLARIZED DOUBLET SPECTRA OF MOLECU-

LAR LIQUIDS : COMPARISON OF THEORY AND EXPERIMENT. Journal de Physique Colloques,

1972, 33 (C1), pp.C1-207-C1-213. �10.1051/jphyscol:1972136�. �jpa-00214926�

JOURNAL DE PHYSIQUE

Colloque C1, supplt!ment au no 2-3, Tome 33, Fkvrier-Mars 1972, page (21-207

THE DEPOLARIZED DOUBLET SPECTRA OF MOLECULAR LIQUIDS : COMPARISON OF THEORY AND EXPERIMENT (*)

G. D. ENRIGHT (**), G. I. A. STEGEMAN and B. P. STOICHEFF Department of Physics, University of Toronto, Toronto, Canada

Rbum6. - Le

spectre depolarise de plusieurs liquides molkulaires contient un doublet ktroit, d'espacement 1

GHz,

centre

a

la frkquence #excitation. Une theorie due

a

Leontovitch et Rytov predisait que de tels spectres Btaient dOs a des fluctuations dans l'orientation des molkules aniso- tropes causkes par des ondes de cisaillement fortement atthukes. Une analyse des spectres obser- ves expkrimentalement en termes de cette theorie est donnk

:

l'accord quantitatif avec l'experience est souvent considkrable nkanmoins il existe certains aspects que la thkorie n'explique pas de fa~on satisfaisante. Certaines thkries plus rkcentes sont brihvement expliqukes et comparees avec les observations experimentales, l'accord qualitatif entre ces thkories et l'expkrience est similaire

a

celui obtenu avec la theorie phenomenologique de Rytov.

Abstract.

- The depolarized spectra of many molecular liquids have been found to contain a narrow doublet, with spacing of -

^{1 GHz,}

centred at the exciting frequency. A theory due to Leontovich and Rytov predicted such spectra as arising from orientational fluctuations of aniso- tropic molecules caused by heavily damped shear waves. An analysis of the observed spectra in terms of this theory is given

:

it is found that there is considerable quantitative agreement, and several areas of disagreement. More recent theories are also briefly discussed and compared with the observations. They are found to give qualitatively similar results to those of the phenomenolo- gical theory of Rytov.

I. Introduction. - The interpretation of the depo- larized spectrum of molecular liquids has recently become a subject of great interest to both experimental and theoretical physicists. Experiments have shown that the portion of the depolarized spectrum which lies within - 100 cm-' of the exciting frequency origi- nates from a number of possible mechanisms, for example molecular reorientation, intermolecular col- lisions, etc. The narrowest component of this spectrum is attributed to scattering from molecular reorienta- tion or diffusion

:

here relaxation times lie in the range to 10-l2 s. Recent high resolution studies [I], [2]

of the depolarized spectrum in the frequency region near the exciting frequency have revealed a depolarized doublet for certain scattering geometries. The origin of this doublet which has been observed in many molecular liquids has been afocal point of a number of recent theoretical papers.

We have measured depolarized doublets in the spectrum of 20 molecular liquids. Starunov, Tiganov and Fabelinskii [l] were the f i s t to interpret similar spectra as due to scattering from shear deformation fluctuations, or in other words to scattering from transverse sound waves. Our observations of the

(*)

Research supported

in

part

^by the

National Research Council of Canada.

(**)

Holder of National Research Council of Canada

1967

Science Scholarship,

1968-1971.

polarization characteristics and line shape of the spectrum, as well as detailed investigation of the k-vector dependence of the doublet splitting have confirmed their interpretation. Our initial analysis of the spectra showed good agreement with many features predicted in the theories of Rytov [3] and Leontovich [4]. Further experiments, specifically those on the temperature dependence of the depola- rized spectrum, have uncovered some discrepancies between experiment and theory.

The purpose of this paper is to review briefly the experimental observations, to compare them with existing theories and to note the areas of disagreement.

A rCsum6 of the experimental observations is presented in Section 11. Our analysis of these observations based on the theories of Leorltovich and Rytov who in fact predicted such spectra before their observation, is given in Section 111. Section IV deals with a number of recent theories and in particular how they compare with experiment. The situation to the present will be summarized in the last Section, Section V.

11. RBsum6 of observations. - The depolarized spectra of 27 molecular liquids have been recorded in our investigations to date. Typical spectra observed for most of these liquids are shown in figure 1 for various polarizations of the incident and scattered light. For the experimental details, the reader is referred t o ref.

[5].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972136

G. D. ENRIGHT, G. I. A. STEGEMAN AND B. P. STOICHEFI;

FIG. 1.

-

Spectra of quinoline for various polarizations of the incident and scattered radiation fields.

The total scattered spectrum consists of the usual central Rayleigh line and Brillouin doublet due to longitudinal waves, all superimposed on an intense background. This background is depolarized and is comprised of a very broad component, up to 100 cm-' in breadth and commonly referred to as the

⁽⁽

Ray- leigh wing

)>,

and sharp components up to 10 GHz in breadth which are the main features in the I: and I;

spectra shown in figure 1. The I: (and I;) spectrum consists of a pronounced doublet symmetrically displaced about the exciting frequency with peak separation much smaller than the breadth of the doublet. The I: spectrum consists of a single Lorent- zian centred at the exciting. frequency with weak doublets in the wings at almost the same separation as the normal Brillouin doublet. The integrated intensities of the I; and I: spectra are equal.

Measurements of the peak separations and line widths of the I; and I: spectra for the various liquids are given in ref. 151. Here only the main results will be summarized. The doublet separations in the I; or I; spectra are 1 to 2 GHz depending on the liquid and scattering angle. (This separation is only a fraction of the Brillouin peak separation due to longitudinal waves.) As the scattering angle is increased from 45O to 1050 the peak separation increases slightly

;

it decreases rapidly at larger angles, becoming negligible

towards the backward direction and leaving a broad Lorentzian line at 1800. This behaviour is shown by the spectra of figure 2. The breadth of this Lorentzian is equal to that of the Lorentzian in the I: spectrum.

The breadth of the I; (and I:) component is very sensitive to temperature as shown in figure 3. On the other hand, the doublet splitting is approximately independent of temperature except when the splitting becomes comparable to the linewidth.

FIG. 2.-- I; spectra of quinoline at various scattering angles.

The solid line represents the Rytov theory.

1 I I

FREQUENCY GHI

I I I

FIG. 3. - I: spectra of quinoline at three temperatures for 90°

scattering. The solid line represents the Rytov theory.

15 10 5 0 5 10 I5

FREOUENCY CHI

Various liquids with different physical properties

were selected for investigation in order to determine

the factors necessary for observing the I: and I f :

spectra shown in figure 1. A representative list of

liquids investigated is given in Table 1 along with

their viscosities and molecular dipole moments. All

but the last 3 liquids exhibited spectra similar to those

of figure 1

;

the last 3 exhibited very broad depolarized

spectra but without these sharper components. From

this table it may be seen that doublet spectra were

observed for liquids with viscosities ranging from 0.8

COMPARISON OF THEORY AND EXPERIMENT

Physical Properties and Derived Quantities of Representative Liquids (20

OC)

0-nitroanisole Nitrobenzene Quinoline Aniline

Dibromoethane Hexafluoro benzene Benzene

Nitromethane Isobutanol

Dipole moment (Debye) - 4.8 4.0 2.3 1.6 1.5 0.0 0.0 3.5 1.6

Viscosity (centipoise)

- 8.6 2.0 4.0 4.4 1.7 0.9 0.7 0.6 4.6

Shear frequency

(GHz)

-

0.37 0.50 0.90 1.05 0.25

0.49 - - -

Shear modulus (10' dynes/cm2)

-

1.4 2.5 6.6 8.5 0.9 4.0

Relaxation time (10-11 s)

- 8.6 3.9 3.9 2.2 2.0 1.4

< 1.0

< 1.0

< 1.0 (*) A value for

z,

is not available.

to 8.6 cp and with dipole moments from 0 to 4.8 Debye. of these two theories. The equations derived by Moreover, doublet spectra were not observed for Rytov for the depolarized spectra are

other liquids with similar viscosities and dipole

moments. Thus we conclude that the existence of such 1;

=

I;

=

c i ^cos (3

^-

^{o2 + T2(02 o2}

^-

^{o;)2 +}

spectra is independent of these quantities. Further- more the existence of the doublet is dependent neither

upon the structure or planarity of the molecules, nor + sin2 (B) ¹

their relative alignment in the liquid state. In summary, 2

⁰²

z2 + 1 only molecules with anisotropic polarizability exhibit

doublet spectra and the detection of such spectra is

^I:

(O

=

900)

=

dependent on the relaxation time z (or inverse line- 3(02 -cot)'

width) being > lo-'' s, and on the instrumental line-

_{2 2}

+

width. ( a 2

-

+

^o2

^-

^0;

^- Q w,)

111. Analysis in terms of the Rytov theory. - Leontovich in 1941 141 was the first to predict a doublet structure in the depolarized spectrum of a liquid. In his thermodynamic theory, Leontovich assumed that shear stresses in a liquid could be described by a symmetric stress tensor and that this tensor was coupled directly to molecular reorientation. That is, he assumed that shear fluctuations in a fluid are relieved by local molecular reorientation. Fifteen years later, Rytov [3] considered the general case of scattering from thermal fluctuations in isotropic, viscoelastic media. The depolarized spectrum of light scattered from liquids is a special case of this theory and the results are similar to those obtained by Leontovich.

Rytov's formulation which is based on fluctuation theory assumes a direct coupling between symmetric stress and strain tensors via a frequency dependent shear modulus. The strain fluctuations are further coupled to fluctuations in the dielectric constant by a frequency dependent photoelastic constant. He then assumed the same relaxation times for the shear modu- lus and photoelastic constant.

At the time of our initial investigations, the theories of Leontovich and Rytov were the only ones available for comparison with experiment. As a result, our most complete analysis which is presented here is in terms

Also

-

w T = , / $ k = 2 n v T , (3)

k

=

k , sin and

Here

C

is a constant

;

0 is the scattering angle

;

o is the shift in frequency form the incident light

frequency

; z

is the shear and thus the photoelastic

relaxation time

;

u, is the Brillouin frequency for

scattering from longitudinal sound waves at an

angle of 900

;

and

y

is the shear viscosity. The variable

o, is related by the infinite frequency shear modulus

,u,

and the density

p

to the wave vector k of the

fluctuations (eq. (3)). Furthermore, k is related to the

scattering angle 0 by the Brillouin equation (eq. (4))

It is because of eq. (3) and (4) that w, is called the

shear wave frequency and the doublet at + ^{oT that}

C1-210 G. D. ENRIGHT, G. I. A. STEGEMAN AND B. P. STOICHEFF

occurs in the I: (and I! ) spectrum is labelled as

originating from scattering by thermal shear waves.

We have been able to fit our experimental spectra to the profiles generated by the computer from eq. (1) and (2). The solid line in figure 2 denotes the best fit of the theoretical profile, suitably corrected for the instrumental linewidth of

^N

800 MHz, to the experi- mental spectra of liquid quinoline at room temperature.

From such analysis, the shear wave frequency o, and the relaxation time z were evaluated and are given for a few liquids in table I. (Here it may be noted that it is possible to evaluate z from the I; spectrum, and thus independently of the doublet 1: spectra.) When o, was plotted as a function of sin (8/2), figure 4, a striking confirmation of eq. (3) and (4) was obtained. This diagram also shows that the experi- mentally observed doublet is not proportional to sin (812). From this we conclude that meaningful data can be deduced from such depolarized spectra only after proper numerical analysis and that the values of oT and z depend on the particular theory used in the analysis.

SIN I8121

FIG. 4. -The observed peak frequency v, (A) and shear fre quency VT (a) as a function of scattering angle for quinoline.

The I; spectrum was also obtained and analysed at different temperatures. A temperature independent shear wave frequency is characteristic of all the liquids studied, with the exception of quinoline and analine in a limited temperature region as shown in figure 5.

The observed dispersion is not understood at the present time. Note also the comparison of the calculat- ed and observed spectra of quinoline a t several diffe- rent temperatures given in figure

3.

There is very good agreement at 72 OC and 22.3 OC, but obvious disagreement at

-

21 OC. (In the calculated spectrum at - 21 OC, z was chosen as the value obtained from the I: spectrum and oT was set equal to the temperature independent value shown in figure 5.) Similar discre- pancies between theory and experiment have been observed for other liquids when o, z - ^1.

FIG. 5.

-

The shear wave frequency VT of quinoline as a function of temperature at two scattering angles.

The behaviour of z from the I; spectrum (900 scattering) is shown as a function of temperature in figure 6. As would be expected, z increases with decreasing temperature. This behaviour is typical of all the liquids studied. We were also able to examine the validity of eq. (5) for a large number of fluids.

Typical results are summarized in table I. In general,

pa

z/y is always less than unity. This result is in contradiction to both the IZytov and Leontovich theories and possibly implies that the total viscosity does not relax with the time

^2.

Another related test is to examine the temperature dependence of zly.

If Rytov's theory were valid, z/y

= p ~ l

should be approximately a constant or increase with decreasing temperature. Our results illustrated in figure 7 show that z/q is indeed independent of temperature for

FIG.

6.

-

The relaxation time z versus temperature for quinoline.

COMPARISON OF THEORY AND EXPERIMENT C1-211

FIG. 7.

-

717 for quinoline (a) and zT/q for nitrobenzene (b) as a function of temperature.

quinoline but ~Tlu] is a better constant for nitrobenzene.

This result has been noted previously [6] and is believed to be related to the magnitude of the molecular dipole moment. When the value of

z

is compared to other relaxation times one finds

z

roughly equal to the dielectric relaxation time,

z,

(see table I). This infor- mation is useful in discussing relaxation mechanisms (see for example ref.

[7]).

IV.

-

Comparison with recent theories.

-

Since the time of the first observations of the depolarized doublet spectra there has been considerable interest in recasting the theory in new forms and several theoretical papers have appeared. Here, we shall briefly review the results of the various theories, without deriving them or justifying their basic assump- tions, and compare them with the experimental obser- vations.

Volterra [8] has formulated a thermodynamic theory taking into account not only the reorientation of molecules, but also their arrangement into new equilibrium positions, as mechanisms for relaxation of shear stresses and anisotropy. With this additional process he is able to account for the broad background of depolarized scattering on which are superimposed the sharper features of the

1;

and

1;

spectra. The equa- tions for the intensity contours of the I: and

1;

spectra are similar to those of Rytov but with the modification that the frequency shift of the doublet is given by w,/(l + ^D)% instead of or. The parameter D is the intensity ratio of the sharp component to the back- ground and is >, 1 and is slightly temperature depen- dent. The fit to observed spectra of quinoline is as good as with the Rytov theory and somewhat better at the lower temperatures.

A new approach to the problem has been attempted by Ben-Reuven and Gershon, Keyes and Kivelson and Andersen and Pecora. These authors have inde- pendently developed microscopic theories and have given molecular expressions, wherever possible, for

parameters used to explain the spectra. Ben-Reuven and Gershon

[9]

have used the Mori continued frac- tion method to represent the relaxations of the coupling of molecular orientation to hydrodynamic shear modes and longitudinal modes. They have derived expressions for the 15 and I: spectra which describe the general features of the observed spectra.

However, the fit of the theoretical and observed spectra is not satisfactory, as indicated by the example in figure 8. There are 3 adjustable parameters in this theory

:

the shear wave frequency a,, the line width due to the shear mode (i.e. in 1 : spectrum) y K and the line width due to molecular orientation (i.e.

in the I: spectrum) ri. The calculated spectrum appears to be insensitive to the value of o, in the range 0 to 2 GHz. A choice of y,

=

Ti

=

3.8 GHz gives the observed dip (after convolution with the instrumental width) but the doublet splitting is much larger than observed. Also, by changing

y,

to the value 1 GHz the observed splitting may be reproduced, but the dip is then less than that observed. Changes in the model used for calculating the strength of coupling to the sound waves may improve the agreement between this theory and experiment.

-Theory of Ben-Reuven 8. Gershon

1 1 1 I I I 1

6 4 2 0 2 4 6

FREQUENCY GHz

FIG. 8. - The observed spectrum of quinoline at 0 = 9 0 0

and T = 22.3 O C (dots) is compared to the theory of Ben- Reuven and Gershon (solid line).

Keyes and Kivelson [lo] have investigated the coupling of molecular reorientation to longitudinal and shear hydrodynamic modes and the resulting effect of the reorientation on the polarizability auto- correlation function. Their derived spectra also reproduce the general features in the observed spectra.

Again, however, a detailed comparison (figure 9) including the instrumental breadth shows an unsatis- factory fit of the observed and calculated spectra. The expression for the 1: spectrum can be written as the difference of two Lorentzians, one with a half width r,,

describing the uncoupled molecular reorientation

(i.e. I: at 1800) and one with a (negative) intensity

G. D. ENRIGHT, G. I. A. STEGEMAN AND B. P. STOICHEFIF

...

^This experiment

-

^{Theory o f Kivelson}

8 Keyes

I I 1 1 I 1 1 I

8 6 4 2 0 2 4 6

FREQUENCY GHz

RG. 9. -The observed spectrum of quinoline at 0 = 900, T = 22.3 OC (dots) is compared to the theory of Kivelson and

Keyes.

with both intensity and half-width proportional to k2.

In calculating the spectrum shown in figure 9 we adjusted the coupling constant

^y,

so that the observed splitting, o,, agreed with the observed value, and found that

y, = y

the shear viscosity. However, the calculated dip was only - ^{50 %} of that observed.

Also, no value of

y,

would reproduce the observed dip. Finally, with

y, = y,

the angular dependence of

o,

does not show as good agreement with experi- ment as does the Rytov theory (figure 10).

Andersen and Pecora [ I l l use Mori's theory of fluctuations to develop a two variable theory to explain the depolarized I: spectrum, and a three variable theory to explain in addition the broad depolarized background. In the two variable theory, the velocity and the symmetric part of the stress tensor are the slowly relaxing variables. Also, it is assumed that the dielectric constant tensor is symme- tric and proportional to the symmetric part of the stress tensor. The resulting

1;

spectrum is the same

8

degrees

- 45 60 75 90 105 120 141 176

QUINOLINI:

22.3'C

( i Theory of Rytw

Oo ,I .2 .4 .6 .8 ^I I ¹ .O SIN ( 8 / 2 I

FIG. 10. - The angular dependence of the observed splitting for quinoline at 22.3 ^{O C} is compared to the theories of Rytov (I)

and Keyes and KLivelson (11).

as that predicted by Leontovich and by Rytov except for a small term dependent on the antisymmetric part of the stress tensor. This term slightly increases the intensity dip at o

=

0 and results in a value of

p, z/y

< 1. These authors compare the observed frequency shifts and linewidths for quinoline and aniline at various scattering angles (taking into account the instrumental linewidth'). The agreement with experiment is very good as shown in table 11 taken from their paper.

In a similar calculation, Ailawadi, Berne and Forster 1121 derive the I: spectrum from the stress tensor autocorrelation function. They show that the inclusion of angular momentum into the ordinary hydrodynamic equations leaids to an anti-symmetric part of the stress tensor. The

1:

spectrum is similar to that of Pecora and Andemen, but with additional terms proportional to

^{(0, T)'.}

Since this parameter is < 1 in all liquids for which doublets have been observed and because of the !similarity of this spectrum to that of Pecora and Anclersen's, good agreement with experiment can be expected.

Chung and Yip [13] have: calculated the spectrum based on a hydrodynamic approach. They have used

Test of Angular Dependence of Depolarized Scattering from Q~in~oline at 22.4 OC Predicted

by

Two Variable Theory

v s

(exp.1 GHz -78

_f

.04 -84 f .02 .89 + .02 .95 f .03 1.01

_f

.03 .93 + ^.03

v, (theor.) GHz

-

.775 f .025 ,850 + ^.025

.925 + ^.025

.950 + ^.025

.950 f .025 37.5 + .025 .650 + ^.025

.OOO + ^.025

rs(theor.)

GHz

COMPARISON OF THEORY AND EXPERIMENT

C1-213

analogous electric circuits to take into account the effects of shear, orientation, and thermal relaxations.

The form of the spectrum obtained contains the broad background as well as the narrow line just as in Volterra's theory and Andersen and Pecora's 3 variable theory. The shape of the sharp component is similar t o the Rytov form, but the physical interpretation of the spectral components is different. They assume that shear relaxation occurs much more quickly than orientational relaxation and therefore associate the background line with the stress tensor and the sharper component with orientational relaxation. Their calcu- lations of the

1;

and I: spectra (including instrumen- tal effects) are in good agreement with the observed spectra for quinoline and nitrobenzene.

As already mentioned all of the above theories can produce the general features of the observed spectra.

When a detailed comparison is made to the observed 1

: spectrum of quinoline the only theories that produce the observed lineshape are those that are similar in form to the Rytov theory. Since these theories, at this stage of their development are not a significant improvement on the Rytov theory we

did not carry out a detailed analysis of the data as in section 111.

V. Summary. - So far an explicit temperature dependence has not been included in any of these theories. However, by allowing some of the parameters to vary in a suitable manner with temperature, agree- ment between theory and experiment may be improved.

Certainly, additional experiments at higher resolution are necessary, especially in the region

^o, z

- ^1.

From the present analysis of theory and experiment the following qualitative picture emerges. A disorder- ed quasi-lattice exists in a liquid for an average time

2.

Every

z

seconds, a fluctuation in the local free volume creates vacancies in the lattice with the resul- tant reorientation of individual molecules. Thus the lattice

⁽⁽

dissolves

⁾⁾

and transverse sound waves are damped out.

Note

added in

^proof. -

Kivelson and Keyes have recently corrected an error that appears in refe- rence [lo] (as in their accompanying paper). With this change the agreement with the observed spectra is greatly improved.

References

[I]

STARUNOV (V. S.), TIGANOV (E. V.) and FABELINSKI

(I. L.),

Zh. Eksperim. i Teor. Fiz. Pis ma Redakt., 1966, 4, 262.

[English transl

: JETP Letters, 1966, 5 , 260.1

[2]

STEGEMAN (G.

I.

A.) and STOICHEFF (B.

P.), Phys.

Rev. Letters, 1968, 21, 202.

[3]

RYTOV

(S. M.), Zh. Eksperim. i Teor. Fiz., 1957, 33, 514

and

669.

[English transl

: Soviet Physics JETP 1958,6,401

and

513.1

[4]

LEONTOVICH

(M.

A.),

ZZV. Akad. Nauk. SSSR Ser.

Fiz., 1941, 5, 148.

[English transl

: J. Physique (USSR), 1941,4,499.]

[5]

STEGEMAN (G. I. A.), and STOICHEFF

(B. P.), Phys.

Rev.,

to be published.

[6]

FRENKEL (J.), The Kinetic Theory of Liquids,

1946,

Oxford University Press, p.

296.

[7]

IVANOV

(E. N.), Zh. Eksperim. i. Teor. Fiz., 1963, 45, 1509.

[English transl

: Soviet Physics JETP, 1964, 18, 10411.

[8]

VOLTERRA (V.),

Phys. Rev., 1969, 180, 156.

[9]

BEN-REUVEN (A.) and GERSHON

(N.

D.),

J. Chem.

Phys., 1971, 54, 1049.

[lo]

KEYES

(T.)

and

ISNELSON

(D.),

J. Chem. Phys., 1971, 54, 1786.

[ l l ]

ANDERSEN

(H.

C.) and PECORA (R.),

J. Chem. Phys., 1971, 54, 2584.

[12]

AILAWADI

(N. K.),

BERNE (B. J.) and FORSTER

(D.), Phys. Rev.,

A,

1971,3, 1472.

[13]

CHUNG(C.

H.)

and YIP

(S.), Phys. Rev.

A,

1971,4,928.