LIGHT SCATTERING IN DENSE MEDIA—ITSTHEORY AND PRACTICE

GJURO DEELI

Department of Biocolloidal Chemistry, Andrija Stampar School of PublicHealth, Faculty of Medicine, University of Zagreb, and Institute "Rug/er

Bogkovié", Zagreb, Croatia, Yugoslavia

ABSTRACT

Theories of molecular light scattering (elastic, inelastic and non-linear scat-tering) have been described, also the experimental procedures for light scatteringstudies. An attempt has been made to compare theoretical and experimentalwork in this field. It is concluded that the measurements of spectral distributionof scattered light are capable of giving much new and additional information tothe theory of liquids. Light scattering will, it is hoped, form an important tech-nique for investigating the structure of dense media as well as that of polymerand colloidal solutions.

INTRODUCTION

In the history of light scattering it is evident that in its early days thetheory developed much faster than the experimental information could begathered. This situation was mainly the result of experimental difficultiescaused by the lack of suitable instrumentation, such as intense light sourcesor highly sensitive photodetectors. By the rapid development of experimentaldevices one would expect that the gap between the theory and experimentwould gradually disappear. The situation, however, seems to be just opposite.Superspecialisation among scientists has led to a separation of the theoreticaland experimental work, as can be observed especially in the field dealingwith dense fluids. In studying these systems, most of which appear in theliquid phase, it is not possible to formulate simple models as in gases orcrystals. This difficulty has compelled theoreticians to use more and morerefined mathematical methods, most of which are inadequate for directnumerical evaluation. Even for the simplest liquids it is necessary to intro-duce approximations which may cause such deviations from reality thatcomparison with experimental data is often impossible. The theoreticiansare, therefore, devoting most of their interest to simple systems which are,as a rule, experimentally hardly accessible. The experimentalists are, on theother hand, working predominantly with complicated systems which arenot very suitable for testing theories. In doing so, they often work withouta sound theoretical basis, and the data collected are predominantly of anempirical nature.

327

GJURO DEELHAware of this situation, Frisch and Salsburg' have recently undertaken

an important task to collect facts on the statistical mechanics of liquids.Although their book is devoted primarily to simple liquids, it represents anup-to-date compendium of theoretical and experimental data which areuseful for discussing other, more complicated systems. One chapter dealswith light scattering, and its scantiness, resulting from lack of experimentaldata, is the best challenge for experimentalists to intensify their work onliquids.

The aim of this lecture goes beyond the scope of the book of Frisch andSalsburg. It is an effort of an experimentalist to apply the existing theorieson light scattering in liquids to systems which are much nearer to macro-molecular chemists—to substances which, in normal laboratory conditions,are liquids. In the growing field of light scattering investigations on macro-molecules more and more attention is being paid to the problems of inter-molecular interactions, as for instance in concentrated solutions or binaryand multicomponent mixtures. Satisfactory theories capable of interpretingexperimental phenomena encountered in these systems are still lacking.Here, less complicated systems of similar properties, such as pure liquids,may facilitate the theoretical approach and let us know how to handle morecomplicated systems both theoretically and experimentally. A unifiedpicture of the present state of investigations in the field of molecular lightscattering will be given, and an attempt will be made to show to what extentthe theoretical results can be applied in experimental work on pure liquids.There is another practical reason which makes the scattering in liquids quiteimportant. Pure liquids serve widely as standards in calibration of lightscattering photometers, and exact knowledge of light sca'ttering propertiesof liquids should be a part of everyday practice in a light scattering laboratory.

THEORY OF MOLECULAR LIGHT SCATTERING

Theory of elastic scatteringThe reason why optically homogeneous systems like liquids and gases

scatter light lies in the fact that molecules exhibit thermal motions. Thus,the dense medium appears to be homogeneous only when observed macro-scopically. A microscopic approach shows that this thermal motion producesfluctuations in thermodynamic functions leading to fluctuations in therefractive index. If we are considering a volume element V, small in com-parison to the wavelength, but big enough to obey the laws of statisticalthermodynamics, then the Smoluchowski—Einstein2'3 theory gives as aresult

R(90) = (ir2/2))V <(A)2> (1)

where R(90) is the Rayleigh ratio measured at a scattering angle of 90 degreesand defined as

R(90) = J(90)r2/J0V (2)

where J(90) is the intensity scattered by the volume V at 90 degrees, J0 isthe intensity of the incident unpolarized beam, and r is the distance betweenthe volume and the detector. Formula 2 can be replaced by a more convenient

328

LIGHT SCATTERING IN DENSE MEDIA

expression. If n is the number of scatterers (particles, molecules) in V, and1(90) is the intensity scattered by a single scatterer, then J(90) = nt(90), andwith the number density N = n/V one obtains

R(90) = NI(90)r2/J0. (3)

This expression is more customary in single particle scattering and in thework with macromolecules. The symbol in equation 1 stands for thewavelength of light in vacuo, and <(Ac)2> is the mean-square fluctuation ofthe dielectric constant c (measured at optical frequencies) about its meanvalue.

The whole problem of solving the scattering problem is now to evaluatethe fluctuation term <(Ac)2> of a given molecular system. In general thissystem exhibits optical anisotropy and the fluctuation Ac has to be regardedas a tensor denoted by Ack.

As known4, it is possible to separate Ack into two parts:

ik Ajk + Acs, (4)

the first term belonging to the so-called isotropic fluctuations and thesecond term being connected with anisotropic fluctuations. Here öik is theKroneker delta, and the components of Acs = 0 for i = k. The totalintensity of scattered light should therefore be proportional to

J <(Ac)> = <(Acis)2> + <(Acs)2> (5)

assuming that Acis and Ac are statistically independent.Kielich5 was the first to give a convenient expression for the Rayleigh

ratio, written in a slightly different form:

R(90) (7t2/10))(5F + l3Fanis), (6)

where F and Fans are the isotropic and anisotropic molecular scatteringfactors respectively. Factors F and Fanis are proportional to mean-squareisotropic and anisotropic fluctuations. This formula is valid if the incidentbeam is unpolarized and the scattered intensity is measured without placinga polarizer before the detector.

A more generalized expression valid for light of arbitrary polarizationand various scattering angles has been deduced5'6 and can be written inthe form:

R(O) = (2/5)L4) [5F cos2 + (3 + cos2 iQs)Fanjs] (7)

Here fl is the angle between the electrical field vectors L and A of theincident and scattered beams, respectively. The indices i and s in anddenote arbitrary orientation of both vectors.

Most of the light scattering experiments are performed by orientingboth F1 and ! perpendicularly and parallelly to the scattering plane definedby the propagation vectors of the incident and scattered beams. By workingwith linearly polarized incident beams and measuring linearly polarizedcomponents of scattered radiation it is possible to measure four combina-tions of Rayleigh ratios which we could name partial Rayleigh ratios":

VRh, hRV, hRh, where the indices v and h denote vertically and horizontally

329

GJURO DEEL1oriented electric vectors, respectively. Instead of using the notation 1R,we may take a simpler and more customary set of symbols introduced byKrishnan7: J/, H, Vh and Hh, where the indices denote the polarizationstatus of the incident beam, and the capital letter stand for the orientationof the polarizer before the detector. It is easy to see that subsequent formulaemay be derived from equation 7:

—V(O) = (it2'5A4) (5F1, + 4Fanis) = 0i'u3 tj IIi\ 24\ E' ITh 0VhU) — iiytU! = 7t /'LO aii is

Hh(O) (m15)) [5F cos2 0 + (3 + cos2 O)Fanjs] �2 = 8.If the incident beam is unpolarized (index u) or the polarizer is not placedin the scattered beam (capital letter R), four other partial Rayleigh ratioscan be measured:

V(6) R(O) I<(O) + Vh(O) = (7t2'5)L4) (5F1 + 7Fanis) (11)

H(O) = Rh(O) = Hh(O) + H(O)(7r25)L4) [5F cos2 0 + (6 + cos2 0)Fanjs]

Finally, the total Rayleigh ratio R(6) can be represented as a sum of fourpartial Rayleigh ratios:

R(O) = [Jç(8) + Vh(0) + H(O) + Hh(O)]/2= (2/10,4) [5(1 + cos2 O)F + (13 + cos2 O)Fanis]

From equations 8 to 13 it is obvious that some of the partial Rayleighratios are functions of the scattering angle (Hh, H, Rh, Re), whereas the othersare constants independent of 0.

The molecular scattering factors and Fan are of the greatest importancein describing a molecular system, since they can be simply evaluated fromthe absolute values of Rayleigh ratios. They can be theoretically evaluatedby the methods of statistical thermodynamics.

From equations 1, 5 and 6 one can deduce the relationships:

= V<(tc>and

— 5j7/( anis2anis — 13 \ 8ikFrom the two mean-square fluctuations the isotropic one can be readilycalculated. This was first done by Einstein3. The dielectric constant is takenas a function of density and temperature, and the final result is8:

F5 = kTKT(N/aN) + (RT2/NC) (E/T),where R is the gas constant, and C, the molar heat capacity at constantpressure. Coumou et a!.9 have shown that equation 16 can be written witha very good approximation as:

F5 =where k is the Boltzmann constant, T the absolute temperature, KT theisothermal compressibility, N the number density of molecules, and & =

330

LIGHT SCATTERING IN DENSE MEDIA

with n being the refractive index of the dense medium. Thus isotropicscattering arises from density fluctuations, the fluctuations in temperaturebeing negligible. The result of this phenomenological theory is very con-venient for numerical evaluation since equation 16 consists only of physicalconstants which are measurable. There have been attempts to calculateF in terms of mean molecular polarizability c. Kielich5 arrived at anexpression which in our definition of F has the form:

F1 = 16ir2N2 [(n2 + 2)'3] 2 1 + 4it [gfr) — N]r2 dr} (18)

with g(r) being the radial correlation function. As shown by Ornstein andZernike10, the expression within the brackets is equal to kTKTN and caneasily be determined experimentally. An important property of kTKTN is,as the system approaches the ideal gas state, kTKTN —+ 1. Expression (18) is,however, more difficult to handle than (17), since the molecular polarizabilityin dense media is strongly influenced by internal fields which are not easyto define at present. At low densities these effects may be neglected, but inliquids they have to be taken into account. It has frequently been overlookedthat instead of c, which is valid for separate, noninteracting molecules, onehas to use , the effective polarizability of a molecule placed in an internalfield1 1• A correction for changes in average polarizability owing to theincreasing density of the system has also to be taken into account1 1

The derivation of Fan in terms of useful physical constants is morecomplicated. Kielich5, and Pecora and Steele6 calculated <(Aes)2> andarrived at simpler expressions only in the case of axially symmetric molecules.This part of scattering can be explained as an effect of fluctuations in theorientation of anisotropic molecules. As shown by Benoit and Stockmayer' 2,in the case of dense systems with strong intermolecular interactions Fanisis proportional to the mean value5 <fiqZ(3 cos2 0pq — 1)> where 0pq is the

angle between the axes of symmetry of a pair of molecules. This factor isalso connected with an integral function A containing the orientationalcorrelation function1 3 The whole effect of the orientational correlations ofthe molecules may be condensed in a factor G which is related to A by:

G=1+JA (19)

For axially symmetric molecules A can be written in the form13:

=JJ(3 cos2 0pq 1)g2 (rn, Tq) dr dtq (20)

where g(2) (tn, tq) is the two-molecule correlation function, and TpTq are thevariables determining the position and orientation of molecules p and q.The important property of function A is that the value 1A = 0 indicates theabsence of angular correlation between molecules, whereas A <0 shouldshow a tendency to perpendicular orientation of the main molecular axesand 1A > 0 a tendency to parallel orientation.

Fanis depends also of what is called the optical anisotropy 52 of isolatedmolecules and is defined as:

331

GJURO DEELK( \2 i

(52 —— r k2 3! r —

—

2(c1 + c2 + L3)

where 1' o2 and stand for the polarizabilities in the directions of thethree principal molecular axes. Expressed in terms of the fluctuation theoryand defined as in equation 6 Fanis can be written as14:

Fanis = ((52 GIN) (Nê/ôN). (22)

In the practice of light scattering it is also customary to measure ratiosof scattered light intensities measured with certain combinations of polaroidsin the incident and the scattered beams. One can define:

D(6) = H(O)/V(6) = 3Fanis/(5Fis + 4Fanis) (23)

Dh(8) Vh(O)/Hh(O) 3Fanis/[5Fis cos2 6 + (3 + cos2 O)FanisI (24)

D(O) = H(O)/V(O) = [5F5 cos2 6 + (6 + cos2 O)Fanis]/(5Fis + 7Fanis) (25)

These quantities are frequently called depolarization ratios, although theterm 'polarization ratios' would appear to be more appropriate15. Fromall polarization ratios D is the most suitable for detection of optical aniso-tropy, since it is not dependent of scattering angle, and D—÷0 if (52

It is also useful to define an apparent optical anisotropy:

A2 Fanis/Fis = (52G/1TKTN = 5D(9O)/[6 — 7D(90)] (26)

which passes into (52 as the system approaches the ideal gas state, i.e.kT,cTN — 1 and G — 1.

From the foregoing formulae it is obvious that the isotropic light scatteringcannot be measured directly. By using one of the polarization ratios it ispossible, however, to eliminate Fanis and obtain the isotropic part. Equation13 may be written as:

R(O) = R(O) + Ranis(O) (27)

with

R15(6) = (2/2A4) (1 + cos2 O)F15 (28)

and

Ranis(O) = (m2/10)) (13 + C052 6)Fanis (29)

It is customary to eliminate Fanis by using D. For 0 = 90 degrees onearrives at the expression

R15(90) = R(90)

where [6 + 6D(9O)]/[6 — 7D(90)] is the well-known Cabannes factor.

332

LIGHT SCATTERING IN DENSE MEDIA

R values determined from (30) may serve now for comparison with thetheoretical values derived from:

R1(90) = (72/2))kTKT(NaS/aN) (31)

If the liquid is a multicomponent mixture of molecules exhibiting differentpolarizabilities, then additional fluctuations in the dielectric constant haveto be taken into account. It was shown by Einstein3 that the additionalscattering effect is due to the fluctuations in concentration. As a first approxi-mation, it is possible to suppose that the concentration fluctuations arestatistically independent of both the density and anisotropic fluctuations.In the case of a binary mixture the isotropic part in equation 27 can be separa-ted into two parts, as shown by Kirkwood and Goldberg16:

R(O) = Rd(8) ± R(O) (32)

where Rd stands for density fluctuations and R for concentration fluctua-tions. In order to evaluate R one has to determine Rd and subtract it fromR calculated by equation 3017. Rd can be calculated by18:

R — R KT(N?/0N)T 33d — d, 0KTo(N/äN)T,o

(

where the subscript zero denotes the pure solvent and other nonsubscriptedvalues stand for the solution. It has been proposed by Bullough19 that anextra term has to be included in the isotropic Rayleigh ratio of mixturesleading to an expression of the form

= Rd[1 + 4(nNcr/Nc;PT1

+ R (34)

Generally R depends on the interactions between molecules and deviatesfrom linear mixture rules, so we can write:

Kc/R = (1/M) + 2Bc (35)

with

— 2 2i 2 i4T— floUflIUCp,T/1o1Awhere M is the molecular weight of the solute, c is the concentration of thesolute in g/ml, and NA Avogadro's number.

A satisfactory theory explaining the anisotropic part of scattering inmulticomponent systems is not available at present. Although experimentaldata indicate2022 that in binary systems Fan factors of both componentsare additive, it is difficult to give an exact solution of the problem sinceuseful theoretical models for internal fields in multicomponent dense systemshave not been developed explicitly.

Theory of inelastic scatteringSoon after the basic principles of Rayleigh scattering in dense systems

333

GJURO DEZELK

were delineated by Einstein, it became apparent that scattering effects inliquids should produce some spectral broadening of incident radiationfrequency. Those effects, predicted first by Brillouin23 and Madelstam24,were experimentally confirmed by Gross25 and others4. Brillouin made hisprediction by assuming that light in dense media is scattered on inhomo-geneities resulting from the propagation of sound waves, and arrived at theresult that the scattered light should consist of a doublet with componentssymmetrically shifted from the incident frequency v0 by8:

= ±2v0(v/c) sin (0/2) (37)

where v and c are the velocities of sound and light in the medium, and 0 isthe scattering angle.

Experimental work has revealed, however, that this doublet is only apart of the whole scattering. It has become obvious that some of the scatteringoccurs also at the incident frequency v0, and also that there is always somebackground scattering, symmetric around v0, whose frequencies normallyextend over much of the range Brillouin's doublet, the so-called wings ofRayleigh scattering'.

It is not intended to give here a review of the theory of inelastic scatteringsince this has been done very thoroughly in the book by Fabelinskii4, butonly to discuss some important points necessary for understanding recentexperimental data.

The first successful approach to explain the fine structure of Rayleighscattering was done by Landau and Placzek26. They started with the iso-tropic fluctuations <(&ls)2 > but instead of taking as a function of densityand temperature, they chose entropy and pressure as the independentvariables. In this way, they were able to separate isotropic fluctuations intoisobaric entropy fluctuations which do not propagate in liquids and arethe source of the central, unshifted Rayleigh component of the scatteredlight, and into isentropic pressure fluctuations (sound waves) which are thesource of the Brillouin doublet. The isotropic molecular scattering factorcan be written as:

RT2=NC

(ac/aT) + kTk5(N36/0N) (38)

where ic5 is the adiabatic compressibility and C, the molar heat capacityat constant pressure. Here the first term describes the unshifted Rayleighcomponent and the second belongs to Brillouin components.

Attempts have also been made to develop theories from the standpointof the molecular theories of liquids. Pecora and Steele6 have developed atheory valid for fluids consisting of nonspherical molecules and have arrivedat expressions for the elastic and inelastic scattering expressed in terms ofangular moments of the generalized pair distribution function of the fluid.It was shown that simply, if an additional time dependence of the electricvectors is considered, this leads to formulae containing frequency dependentvariables. However, the time dependent g(2) (tn,tq) functions could not beexplicitly obtained so far in terms of some measurable quantities. Mountain27

334

LIGHT SCATTERING IN DENSE MEDIA

showed that essentially the same results as obtained by Landau and Placzekcan be derived if density fluctuations are considered to be time dependent.Other theories28'29 did not elucidate the situation further, so a properconnection between theory and experiment remains a problem for futureinvestigations.

Although the general theory of Pecora and Steele includes orientationalscattering, some more approximate theories seem to be useful in explainingexperimental results. Among these, one has to mention the first theoryof orientational scattering given by Leontovich3° who calculated the timedependence of <(As)2> by assuming that small disturbances from equi-librium are linear. The basic result from his theory is the prediction that theshape of Rayleigh wings should be Lorentzian, and that the ratio of intensitiesof vertical and horizontal components in the wings, 'H and should be:

I I — 39H! V — 4

Later theories were developed predominantly by the Russian school4,and together with the theories of Pecora and Steele6 and the recent work ofPinnow et al.29 all predict the Lorentzian form of spectra and 1H/'v =

Theory of nonlinear scatteringBuckingham31 was among the first to suppose that intense light beams

might induce optical birefringence in isotropic media. This idea was furtherdeveloped by Kielich5 in his theory of molecular light scattering and sub-sequent papers, but was experimentally confirmed only a few years ago byseveral authors32' who used an intense laser beam. The whole topic wasrecently reviewed by Kielich34, so only a few of the more important pointswill be mentioned here.

If a molecular system is illuminated with an intense light beam, the scatteredintensity J becomes a nonlinear function of the incident intensity Jo:

I = S110 + S2I + S3I + ... (40)

Here S1 defines the normal linear Rayleigh scattering, and S2, S3,.., arethe factors defining higher order nonlinear scattering due to reorientationof molecules and nonlinear polarizability. The basic process in nonlinearscattering35 consists of irradiating a molecule with two quanta of frequencyv and scattering a single quantum of frequency 2v. This process could beobserved in H20, CCI4 and CH3CN by Terhune et a!.33 by means of agiant-pulse focused ruby laser beam.

Kielich36 has shown that nonlinear scattering can be treated formallyin a similar way to linear scattering. He could estimate the molecular scatter-ing factors for nonlinear second harmonic light scattering and showed thatit is possible to separate them into isotropic and anisotropic parts. Secondorder polarization ratios could be deduced for various types of molecules.Thus, for example, for liquid CCI4 the theory gives DV =4 in the case whenthe action of the molecular field was neglected. The measured value33 wasabout . If the effect of the molecular field was taken into consideration,the theoretical value could be lowered and brought into a better accordancewith the experimental value.

335

GJURO DEZELH

It is important to note that in dense media there exist strong molecularfields which can produce not only linear, but also nonlinear polarizationof molecules, even if the external fields are weak37. Therefore, the nonlinearscattering appears to be very sensitive to molecular interactions and structure.

The investigations of nonlinear scattering are connected with manydifficulties, most of them arising from the experimental side. The effects inmolecular systems are mostly small (of the order of 10—13 of the incidentintensity3 3) and one has to work with extremely intense laser beams, whichin turn can provoke some complications (e.g. shock waves and dielectricbreakdown34). However, in colloid and macromolecular systems the effectsare much more pronounced and therefore the nonlinear light scatteringappears to be a promising field for future studies.

EXPERIMENTAL PROCEDURES AND RESULTS

The most important part of the efforts in experimental light scatteringwork with liquids belongs to the determination of absolute Rayleigh ratios.Seven years ago Kratohvil et al.38 gave a critical survey of Rayleigh andpolarization ratios found in the literature in connection with the calibrationof light scattering photometers. A part of this discussion was dealing withthe reliability of the so-called high' values of Rayleigh ratios for liquids.The suggested best values for benzene at 20° are R(90 465 x 10_6 and(155—160) x 10' cm1 for 436 and 546 mt, respectively. Kratohvil et al.arrived at these values after careful critical examination of calibrationprocedures applied in every particular case of the reported Rayleigh ratiofor benzene. Since then the great majority of reports on Rayleigh ratios ofliquids have confirmed the reliability of the above data.

Although the discussion on proper calibration procedures of light scatter-ing photometers can be regarded as practically concluded, papers occasionallystill appear which do not satisfy the requirements of a reliable calibration.It is therefore assumed that a short review of these problems might be ofvalue for future work.

There are two methods which are predominantly used for the determinationof Rayleigh ratios: (a) the Brice working-standard method39, and (b) thestandard scatterer method. Although it is frequently recognized that Brice'smethod leads to reliable absolute intensities41, the experience from author'slaboratory has shown that this method can lead to systematic errors, ifthe geometry of the apparatus has to be changed, and this is not madeproperly. In some cases, it is difficult to track the sources of these deviations,and calibration with a standard scatterer appears to be the best and moststraightforward method. One of the most suitable standard scatterers isliquid benzene, which can be easily prepared in a pure, water- and dust-freeform. Rayleigh ratios are obtained from

R1(O) = C C, f(O) (41)

where C = Rj90)/15(90) is the calibration constant. Here I denotes galvano-meter deflections, the subscript s stands for the standard (in this particularcase it is benzene), and C is a complex relative optical correction factorconsisting of corrections for differences in the refractive index, volume and

336

LIGHT SCATTERING IN DENSE MEDIA

reflections. It is always necessary to determine C when the refractive indexof the measured liquid differs from that of the standard.

The correction factors have been thoroughly reviewed41—43, and it hasbeen noted that both the refractive index correction44 and the volumecorrection42 are important for the determination of Rayleigh ratios by thestandard scatterer method. For the special geometry when the detector 'sees'within the edges of the incident beam Hermans and Levinson45 have shownthat the total correction should be equal to the ratio of the squares of therefractive indices. Most authors simply use the n2-correction of Hermansand Levinson without further checking. It could be shown41 that the n2-correction really consists of a product of relative refractive index and volumecorrections. Data in Table 1 may be taken as an example. They were derived

Table 1. Relative optical correction factors for the Oster-Amincophotometer at 25 using a rectangular cell. n1 = refractive

index of water n2 = refractive index of benzene.

0(m1t) (n2/nl)2 C C e;c546 1268 1171 1083 1270436 1282 1181 1087 1283

C;, = C, 2:C, calculated from the formula of Carr and Zimm. J. Chess.Phys. 18, ll6 ('1950).

C:. = C2:C C calculated from the forula of Kerker et al.. J. Pa/veerSci. A2, 303, case 11(1964).

for the geometry of the Aminco photometer from equations given in previouspapers41'42. It has been proved14 that the geometry of this photometersatisfies the requirements for the n2-correction. If the detector sees' past theedges of the incident beam, the n2-correction appears to be too low. So, forexample, in the case of the Brice-Phoenix photometer with standard apertureswhen working with the standard scatterer method the real correction shouldbe 2 per cent higher than the n2-correction for the system benzene—waterat 436 mt. Since this method of determining Rayleigh ratios is a relativeone, the reflection correction43'46 is normally cancelled out41. It can beconcluded that in the case of a proper geometry (i.e. when the detector 'sees'within the edges of the incident beam) equation 41 can be simply written as:

R1(O) = (n/n)2 R(9O)/I(9O) 1(O) (42)

When polarized Rayleigh ratios have to be determined, one can proceedin various ways. With the Brice working-standard method one can use theBrice original formula, but because of definition 13 there should be anadditional factor in the formula40. Another method consists in measuringthe attenuation factors of the polarizing filters (or prisms) placed in boththe incident and the scattered beams14. One can measure it directly or fromthe scattered intensities of any liquid by adding galvanometer readings.

Another correction appears to be important when working with thestandard scatterer method. That is the correction for the different sensitivityof the detector to light beams polarized in different planes. CD is defined41

337

P.A.C.—23/2-3—H

GJURO DEZELR

as the ratio of galvanometer readings for light directly incidenting onto thedetector, i.e. CD = I(O)/I,(O). Therefore all readings made with a horizontallyoriented polarizer before the detector should be multiplied by CD. Thiscorrection factor varies for different detector tubes and wavelength, and inour experience it can take values between O9 and 11. This difference insensitivity might influence Rh values, a fact not taken into considerationin the past. It can be deduced that the polarization correction factor for Rhamounts to [1 + D(9O)]/[1 + D(9U)/CD]. In the case when Dh(90) = 04and CD = 11 this factor is 103 and may play a role in precise determinationof Rh. In most cases, however, the differences in polarization sensitivityare 2 per cent or less, so that correcting R becomes unimportant.

It is interesting to note that several years ago Rozhdestvenskaya andVuks47'48 described a different method for the determination of Rayleighratios of pure liquids. They combined lateral scattering measurements withtransmission measurements made both on pure liquids and polymer solu-tions with the same liquid as the solvent. From the ratio of laterally scatteredintensities and from the difference of the logarithms of the transmittedintensities Rayleigh ratios were determined. The authors obtained datanearer to the group of 'low' values. On the basis of these results Vuks49has recently made some theoretical considerations in trying to prove the

EU

x00.00)a)>,0

Figure 1. Angular dependence of the Rayleigh ratios R(O), H(fi and H1(O) of benzene at 25Cand 546 and 436 mp. Points are experimentaL curves theoretical.

338

2

——

h—-

30

60

90

120

151)

tJ—

30

60

90

12

0 15

0 0

Figu

re

3. A

ngul

ar d

epen

denc

e of

the

pol

ariz

atio

n ra

tios

D(O

),

D(O

) an

d D

(O)

of b

enze

ne a

t 25

°C a

nd 5

46 a

nd 4

36 m

tt. P

oint

s ar

e ex

peri

men

tal, c

urve

s th

eore

ticaL

Figu

re 2

. A

ngul

ar d

epen

denc

e of

the

Ray

leig

h ra

tios

V(O

), V

,,(O

),

Vh(

O)

and

H,(

O)

of b

enze

ne a

t 25°

C a

nd 5

46 a

nd 4

36 m

u. P

oint

s ar

e ex

peri

men

tal,

lines

theo

retic

al.

436

Ben

zene

, 25

0 A

0(m

)546

Q

(ii)

o

o(O

) .

S

Ben

zene

25

° A

0(m

j.)54

6 43

6 V

(O)

0 D

V(O

) S

S

/-1v

(O)

S

S

D

0 --

----

--0

- _I_

_ - —

s- -

- --

_. S

S

g6

L', 0 I:

10

V) 0 0 C 0 C

N

C oO

0

5

ri-I

z z cu

-

•0

----

----

-M--

-

GJURO DEELR

exactness of Slow' values. It appears, therefore, to be important to examinethe reliability of their experimental procedure. Their method has the advant-age of eliminating the n2-correction, but one can deduce easily that theprecision of the method is mainly influenced by the precision of transmissionmeasurements. In this particular case the main part of the error arises fromthe transmission measurement of pure benzene. For a cell of 100 cm pathlength which was actually used one can estimate an error of at least 10 percent. Since it may be assumed that the other three intensities have an errorof several per cent, the total error might be about 20 per cent which by farexceeds the precision of the methods discussed previously and puts somedoubt on the reliability of their results.

By studying the angular distribution of the scattered intensities it can beshown that for fluids formulae 8—13 and 23—25 hold without any exception.This behaviour was observed frequently on gases and liquids'8' Theonly deviation noted56 could later be attributed to remarkable experimentalerrors57. An illustration of the angular distribution of all measurable lightscattering quantities can be seen in Figures 1 to 3. The points are the valuesmeasured on benzene14 and the curves are theoretical. The calculation oftheoretical values will be discussed later.

By concluding this discussion it should be emphasized that Rayleighratios discussed here represent the integrated intensities of the wholespectrum of scattered light. Mostly, experiments are performed with nospecial precautions for elimination of Raman frequencies, since the amountof those is normally below the experimental error58. Therefore, all Rayleighratios are a measure for the integral electromagnetic energy scattered witha spectral distribution around the incident frequency v0.

COMPARISON OF THEORY AND EXPERIMENT

The crucial point in the evaluation of theoretical light scattering data isthe calculation of molecular scattering factors F and Fanjs.

For comparison with experiments F can be derived from equations 28and 30:

= (24/1t2)R(90) (43)

may be derived directly from the measured values of V,(8) or H,(O)by equation 9.

In calculating F one can start from the approximated equation 17 withthe parameter (N8e/N)T which can be best determined from the measured(5/P)T values, as it follows from thermodynamic relationships. A veryconvincing proof of this was given by Coumou et al.9 and by others40' 59In the past, however, owing to the lack of experimental N3e/ÔN data, thisquantity was usually determined by one of the relations between the refractiveindex and density. Einstein3 used the Lorentz—Lorenz function;

(n2 — 1)/(n2 + 2) = eN (44)

340

LIGHT SCATTERING IN DENSE MEDIA

c being a constant containing the polarizability of molecules, and arrivedat the expression:

(N8/ôN)T = (,2 — 1) (n2 + 2)73 (45)

At that time the experimental values of Rayleigh ratios were systematicallytoo low because the refractive index correction was not applied. As the ratiobetween these incorrect experimental values and the theoretical onescalculated by equation 31, with NE1s/N calculated from (45), amountedin most cases to roughly the factor (n2 + 2)2/9, Ramanathan6° and Rocard61were led to an artificial assumption that the factor (n2 + 2)29 arises fromthe presence of the medium surrounding a molecule and that the propertiesof this medium do not undergo changes from fluctuations in small volumeelements. Therefore, this factor should be regarded as a constant duringthe differentiation process62. So they actually made use of the so-calledNewton—Laplace function:

n2lcN (46)and obtained

(NE/N)r = fl2 — 1 (47)

It is interesting to note that Vuks49 has recently tried to derive (Nle/N)Tusing a similar assumption, i.e. by assuming the reacting field fluctuationsto be zero, and arrived at the expression:

(N!NT = (n2 — 1) [3n2/(2n2 + 1)] (48)

Equation 47 gives too low values of Ncc1aN if compared with the experimentalvalues. The values from equation 48 are somewhat higher, but still signifi-cantly low. On the other hand, equation 45 tends to give higher values.From other relations between the refractive index and the density theGladstone—Dale function yields:

(Ne/0N)T = 2n(n — 1) (49)

and leads to lower values. The best results could be obtained from Eykman'sempirical formula44'63' which gives after differentiating:

(N8/8N)T = (n2 — l),[1 — (n2 — 1)/2n(n + 04)] (50)

The values of N1'N obtained by different formulae are compared withexperimental values in Table 2. It is obvious that none of these derivationsleads to fully satisfactory results.

The main shortcoming of all the preceeding formulae seems to be in thesupposition that the molecular polarizability is constant. More refinedtheories64'65 show that in dense media molecules are compressed and theresult is a decrease in c at high pressure. If the density dependence of polariz-ability is taken into consideration, functions of the form66:

(n2 — 1)j'(n2 + 2) = N[1 + S(N, T)] (51)

are obtained, where S expresses the density and temperature dependence

341

Tab

le 2

. Val

ues o

f(N

ic a

N)

calc

ulat

ed f

rom

dif

fere

nt r

elat

ions

betw

een

refr

activ

e in

dex

and

dens

ity f

or so

me

liqui

ds

no

i.0(m

q)

n E

xper

imen

tal

(Nc/

N)T

L

oren

tz—

Lor

enz

New

ton—

Lap

lace

G

lads

tone

—D

ale

(NPe

/N)T

A

(N

&iN

)T

A°

(Nec

N)T

A

° E

ykm

an

(NcN

) A

° V

uks

(NcN

)T

A/

benz

ene

carb

on

tetr

achl

orid

e et

hano

l w

ater

23

23

25

25

546

546

589

589

1503

1460

13

60

1-33

3

1655

1455

10

26

0859

1782

+

77

1559

+

72

1091

+

55

(197

8 +

14

1259

—

24

1132

—

22

085(

1 —

17

0777

—

96

1-51

2 —

86

1343

—

77

((97

9 —

46

0888

+

34

1614

—

25

1429

—

18

1034

+

08

((93

4 +

8-7

1546

—

66

1376

—

54

1003

—

22

091(

1 +

5-9

A -

di

ffer

ence

fro

m e

xper

imen

tal

valu

e.

Tab

le 3

. Val

ues O

f(N

c/PN

)T ca

lcul

ated

fro

m eq

uatio

ns 5

4 an

d 55

as c

ompa

red w

ith e

xper

imen

tal

data

for s

ome l

iqui

ds. (

t 25

C

= 58

8 m

j.t)

(N/N

) ex

peri

men

tal

(1 +

bN/a

)c (

NE

/N)

equa

tion

54

A%

B

d (N

/8N

) eq

uatio

n 55

A

°/

benz

ene

carb

on

tetr

achl

orid

e et

hano

l w

ater

1601

1440

" 10

26

0859

"

1097

1093

10

76

1160

1600

1410

10

14

0843

—01

—21

—

12

—19

0897

0913

—

0885

1577

1409

—

0865

—15

—22

—

-

+07

Rön

tgen

and

Zeh

nder

".

Cou

mou

et a

l.9. m

easu

red a

t 546

mja

. cor

rect

ed t

o 58

8 m

p.

De8

eliô

) C

alcu

late

d fro

m d

ata

of W

axie

r et

A

% d

ifft

ence

from

expe

rim

enta

l va

lue.

LIGHT SCATTERING IN DENSE MEDIA

of . The density dependence of the Lorentz—Lorenz function is confirmedby the experiment. One of the first empirical forms of equation 51 was givenby Rosen67:

(n2 + 2)/(n2 — 1) = (a/N) + b. (52)

Here a and b are constants for a given temperature and frequency. Theoreticalwork68'69 could show that b is associated with the radial correlation functiong(r) discussed before.

Recently Eisenberg70'71 has shown that the changes in the refractiveindex with pressure and temperature can be represented by a function ofthe form:

(n2 — 1)/(n2 + 2) = ANBe_CT (53)

where A, B and C are constants.By differentiating equation 52 one obtains1 1:

(N/aN)T = (n2 1) (ii2 + 2)3(1 + bN/a). (54)

and from equation 53 it follows:

(N0c/f3N)T = B(n2 — 1) (n2 + 2)3 (55)

Table 3 gives values of (N3/N)T calculated from both equations. Theagreement between the calculated and the measured values is generallybetter than for the functions without the density correction. In view of thefact that functions (52) and (53) can be associated with the theory of refractiveindices in dense media, they seem to be superior to other refractive indexfunctions. Thus it is apparent that the derivation (N?ic!ÔN)T is less sensitiveto internal field changes and models, but mostly to density effects on thepolarizability c. This is indicated by the results of calculations with polariz-ability functions having differently defined internal fields but no densitydependence. Thus, for example, by taking the Lorentz ellipsoidal field72the polarizability function can be written in the form:

— 1 = [1 + (n2 — 1)A1]1 (56)

where the summation has to be performed over three axes of the polarizabilityellipsoid, and A are the shape factors73. One gets the result:

(NEIâN)T = (n2 — 1) [1 + (n2 — 1)A]1(57)

From this formula, by taking the values of the parameters from literature74,the value (N5/aN)T = 174 is obtained for benzene, a value near the valuecalculated for the Lorentz spherical field (Table 3).

An additional proof of the validity of the fluctuation theory of isotropicscattering combined with the idea of density dependent polarizability canbe found in temperature dependence of P. in Figure data alcuhted

343

0

00)

Figure 4. Variation of experimental and calculated isotropic Rayleigh ratios of benzene at546 mt with temperature.

from equation 31 with (N8s/ôN) derived from equation 54 are comparedwith the experimental data derived from the paper of Ehi et al.7. They arein excellent agreement within the experimental error.

For the calculation of Fans the data for 52and G should be known. Thereare two methods which may be used for the determination of the opticalanisotropy ö2. The first method is based on the measurements of polarizationratios on vapours and the evaluation from the formula:

= 5D(90)/[6 — 7D(90)] (58)

which follows from (25) and (26). This method was frequently used in thepast58, but data found in the literature are hardly satisfactory. Most of thedata may be regarded as obsolete and new measurements with moderninstrumentation are urgently needed. In particular the wavelength depen-dence of D and ö2 should be measured. It was estimated'1 that ö2 mightdiffer as much as 10 per cent for wavelengths 546 and 436 mt and this mayproduce remarkable errors in determination of Fanis.

The second method consists of measuring H in diluted solutions with asolvent of low optical anisotropy. This method was developed and extensivelyused by Bothorel and collaborators76. Here the molecules of the solute areassumed to approach the gaseous state when the solution is sufficientlydiluted. It is supposed that the molecules of the solute do not interact withthe molecules of the solvent.

344

GJURO DEELI

Benzene

o exptt. R5• calc.

-R (6-7D)I(66D)

1 4

1 2

10

20 40 60 80T, °C

LIGHT SCATTERING IN DENSE MEDIA

The orientation correlation factor G can be derived in principle fromany of the physical quantities which depend on optical anistropy of mole-cules. This can be done by measuring the depolarization of scattered light,electric, magnetic and streaming birefringence. Generally speaking, allmethods consist in comparison of the data measured with the same substancein liquid and vaporous states. In some special cases (benzene, carbondisulphide11) it could be shown that agreement with experiment can beobtained. As an illustration, data for benzene are given. In Table 4 physicalconstants and scattering factors necessary for the evaluation of all scatteringquantities from equations 8—13 and 23—26 are given14. The calculated data,compared with the experimental ones in Table 5 are in remarkable agreement.

Table 4. Physical constants and scattering factors for benzene at 25C

G calculated from Kerr constantsG calculated from polarization ratios.

The same method was applied for calculating curves in Figures 1—3. Thedifferences between the theoretical and the experimental values vary from2—12 per cent. A part of this can be ascribed to rather high experimental errorsresulting from the use of polaroid discs which, as known, allow results ofonly 5—10 per cent accuracy. The other part of the differences comes fromdifferent values of G obtained by different methods. It is probable thatimprovement of experimental techniques would lead to a far better agreement.

For most other liquids, however, neither 2 nor G are known with muchreliability. Light scattering measurements may provide, however, muchinformation about the apparent optical anisotropy L\2 and the product ö2G.Especially useful is the quantity C52G, since, if we suppose that ö2 is a molecularconstant, it is possible to get information on changes in G, and thus gainmore insight into the orientation correlation and structure in liquids. Acompilation of data from various sources9' 14,41,59 for a series of liquidsis presented in Table 6 together with 52G values evaluated from polarizationratios and anisotropic Rayleigh ratios by expressions:

= kTKTN6 -.-7D(90) (59)

and

— 102NRanis(90)(60)13(N/'N)

where Ranis is calculated either from equations 27 and 31, or directly from:

345

GJURO DEZELR

Ranis(90) R(90) 6 ±6D,(90)One can see that good agreement is obtained for highly anisotropic liquidswith D5 > 04 where Ô2G values vary less than 5 per cent. For liquid withlow anisotropy the variations are appreciable which is mainly caused byrather low precision of D5 data.

With known ö2 one could determine G. However, literature data on arerather incomplete, and if found, mostly without known wavelength depen-dence. So the determination of G can be performed at present only semi-quantitatively. An attempt is shown in Table 7. Here ö2 values are takenfrom various sources. The data of Stuart58 are based on measurementswith vapours with white sunlight, those of Massoülier17' 78 were measuredwith white light of a mercury lamp but corrected to 546 m.i, and the data ofClement and Bothorel79 were measured at 546 mi in cyclohexane solution.G was calculated from ö2G data obtained by equation 59. If we considerStuart's data to be a little too high for strongly anisotropic liquids becausethe wavelength correction was not applied and since it seems to be wellestablished now that ancient data are as a rule too high, mostly owing tostray light, we can expect G values to be in reality somewhat higher. Here

Table 5. Rayleigh ratios', polarization ratios and apparent optical anisotropy ofof benzene at 0 = 90° arid 25°C

calculated experimentalG = 053 G = 058 sentioctagonal cell cylindrical cell

0=546mt s%

R, 155 164 161 162 15T' 221 230 232 225 14Fl, 89 98 97 95 26V 176 181 183 184 27V 45 48 48 51 22H5 4•S 48 45 48 44H,, 45 48 48 47 19D, 040 042 042 040 20D5 025 027 025 026 38Dh 100 100 100 108 25A2 064 070 069 063

R 435= 436mj.t461 465 465 03

V 615 642 686 655 24H, 256 279 300 260 26T' 487 501 53.5 508 11V, 128 140 140 146 14H5 128 140 140 140 14Hh 128 140 138 136 13D, 042 044 044 042 19D5 026 028 026 029 21D,, 100 100 102 104 16A2 067 074 075 069

° All Rayleigh ratios are in cm and have to be multiplied by 10.6.4 standard deviation of angular measurements.

346

Tab

le 6

. Com

pari

son o

f Ô2G

eva

luat

ed fr

om e

xper

imen

tal

data

on

D(9

0) a

nd R

an s

(90)

for

2 =

546

mj.t

at r

oom

tem

pera

ture

.

Rj9

0) x

106

(cm

')

expe

rim

enta

l D

(90)

R

1(90

x 1

06 (c

m 1

)

equa

tion

31

Ran

js(9

0)

106

(cm

eq

uatio

n 27

eq

uatio

n 59

2G

eq

uatio

n 60

eq

uatio

n 61

benz

ene

162

040

600

102

168

175

172

tolu

ene

176

048

521

124

194

195

195

nitr

oben

zene

65

8 07

4 51

0 60

7 53

4 54

0 53

9 ca

rbon

di

suip

hide

84

6 06

56

122

724

915

892

896

carb

on

tetr

achi

orid

e 55

3 00

49

511

042

012

0087

01

2 cy

cloh

exan

e 45

6 00

49

421

035

011

0081

01

1 is

o-oc

tane

51

5 00

47

455

060

0093

01

15

0096

n-

hexa

ne

532

0073

44

7 08

5 02

1 02

3 02

1 n-

octa

ne

485

012

394

091

022

017

021

n-de

cane

49

5 01

5 37

9 11

6 02

0 01

6 01

9 n-

hexa

deca

ne

575

026

342

233

018

015

016

met

hyl—

ethy

l ke

tone

41

2 01

47

314

098

044

036

042

met

hano

l 24

2 00

51

207

035

034

050

036

wat

er

0865

00

76

0796

00

69

043

021

040

(1

tIl z z

00

Tab

le 7

. V

alue

s of

the

orie

ntau

onal

corr

elat

ion

fact

or G

for

var

ious

liq

uids

The

ori

gina

l da

ta gi

ven

as it

2.

Cal

cula

ted

from

the

axes

of t

he p

olar

izab

ility

elli

psoi

d.

Stua

rta

ö2x1

02

Mas

soul

ier

Clé

men

t-B

otho

rel

2G>

<10

2 St

uart

G

M

asso

ulie

r C

lém

ent-

Bot

hore

l

benz

ene

36

31

37

168

047

054

045

tolu

ene

40

—

194

048

- —

nitr

oben

zene

5.

1b

—

5-34

10

5 ..

carb

on

disu

iphi

de

134

129

—

915

068

071

—

carb

on

tetr

achl

orid

e 0

—

—

012

—

- —

cycl

ohex

ane

04

—

—

011

028

—

iso-

octa

ne

08

—

0093

01

2 -

n-he

xane

13

05

9 02

7 02

1 01

6 03

6 07

8 n-

octa

ne

12

084

024

022

018

026

092

n-de

cane

11

11

2 02

2 02

0 01

8 01

8 09

1 n-

hexa

deca

ne

—

—

018

018

- .

100

met

hyl—

ethy

l ke

tone

26

•—

04

4 01

7 —

LIGHT SCATTERING IN DENSE MEDIA

Massoulier's data for benzene and carbon disulphide seem to be the mostreliable at the moment. A puzzling effect can be observed with normalalkanes. Here Clement and Bothorel's data seem to be the best since theyare nearest to the G values (G > 1) which could be expected for those chain-like molecules76. Massoulier's data indicate augmented D values in n-alkanevapours which cannot be explained without assuming a strong interactionbetween molecules. It is interesting to note that D data. preceeding Mas-soulier's data80'81 are even higher.

An interesting method of determining G has recently been proposed byLalanne82. He has applied Kielich's method83 which corrects for differencesin the internal field of both the pure liquid and dilute liquid solution andhas arrived at some preliminary results which are in accordance with thedata for carbon disuiphide presented in Table 7, but differ much in casesof benzene and n-decane. It seems that Lalannes results are afflicted withthe same difficulties as the data in Table 7, i.e. they are based on inprecisephysical constants.

Although the absolute G-values cannot at present be determined withenough accuracy, much valuable information about molecular interactionsand structuring in liquids can be obtained by measuring the temperaturedependence of ö2G. Recently, Schmidt84 has reported on measurementsperformed for a series of liquids, and could show, in accordance with previousdata1 1, that ö2G, and consequently G, rises with temperature in case ofbenzene and other aromatics. This may indicate a gradual disordering withincreasing temperature for liquid aromatics, since it can be supposed withmuch certainty that for these liquids G < 1, indicating a prevalence of per-pendicular orientations between benzene molecules. For n-alkanes, however,G is decreasing with increasing temperature, and that could again be explainedwith gradual disordering of prevalently parallel intermolecular orientationsif we suppose that in the beginning G> 1.

o G from polarization ratios• G from kerr constants

13 Benzene

60T, °C

Figure 5. Dependence of G on temperature for benzene. G derived from polarization ratios andfrom Kerr constants.

349

GJURO DEZELK

o ExptL.

• Coic. R35-(NaCa[c. Rn1-(N IaN) NG constBenz ene

11

T (°C)

Figure 6. Variation of experimental and ca'culated anisotropic Ray'eigh ratios of benzeneat 546 mj.t with temperature.

This situation can be observed for benzene in Figures 5 and 6. Here againthe experimental values are based on the data of Ehi et a!.75. It is interestingto note the divergence in G values calculated from polarization ratios,equation 59, and from Kerr constants. The molecular theory of liquidsleads to the following expression for the Kerr constant5 corrected fordensity dependence of polarizability1 1:

K — (n2 — 1)(n2 + 2)(c — 1)( + 2)52G(62)—

12Ozn2kTN(1 + hN/a) (1 + bN/a)where index s stands for parameters measured in static fields. The curvederived from the Kerr constants measured by LeFêvre and LeFèvre85 looksunrealistic in view of the above discussion on the temperature dependenceof molecular orientations. This may be proved by calculating the temperaturedependence of anisotropic scattering. Figure 6 shows that good agreementof theoretical data calculated by equation 60 with experimental values canbe obtained only if a temperature variation of G is assumed.

Finally, a few words on the scattering from multicomponent systems.A series of papers was devoted to this problem (e.g. t7, 18.2263.8687.97)and the most important result of these investigations is the possibility ofdetermining activity coefficients and the excess free energy of mixtures.Experimental data based on high' values of Rayleigh ratios are in very goodagreement with theoretical relations and the data from other experimentaltechniques (like vapour pressure'' 22 and electrolyte activity coefficients87).The only point which is not fully elucidated is the validity of the Bulloughextra term C in equation 34. While Sicotte'8'88 considers that this termexplains correctly his experimental data, Pethica and Smart87 claim thatit is incorrect to include C in addition to equation 32. The final word has tobe left to future experimental work.

350

LIGHT SCATTERING IN DENSE MEDIA

The experimental achievements in the important field of spectral distribu-tion of scattered light will also be only touched here. These light scatteringtechniques started to expand rapidly after the invention of lasers. Afterthe first report of several groups89—91 some authors tried to perform morequantitative checking of existing theories.

One of the important points of the theory of nonelastic scattering is theratio of intensities of the central component, I. to the sum of intensities ofBrillouin lines, 'B' the so-called Landau—Placzek ratio. Landau and Placzek26predicted the expression:

'C/21B = (icT — 1C5)/K5 (63)

which followed from the approximated equation 38. It was found8' 92 thatequation (63) should be corrected for dispersion of the quantities s and(NoiaN)5, as derived by Fabelinskii4. It was also found that both theRayleigh and Brillouin lines are fully polarized93 which is proof that theybelong to the isotropic part of light scattering.

The measurements of the spectral distribution of scattered light renderinformation on the sound velocity (so-called hypersonic velocity) in liquidsand on relaxation times of vibrational degrees of freedom in liquids94'95.Laser measurements of the anisotropic part of spectrally displaced scatteredlight are rather scarce. Shapiro and Broida96 measured the wings of Rayleighscattering in carbon disulphide and arrived at results in agreement withtheoretical predictions. For 'H/'v they found the predicted value . Themeasurement of the orientational relaxation in liquids can be interpretedas a diffusion process depending both on the temperature and the structureof molecules29.

It may be concluded that the measurements of spectral distribution ofscattered light are capable of giving much new and additional informationto the theory of liquids. They will certainly be an important technique infuture for studying the structure of dense media as well as of polymer andcolloidal solutions.

In view of all the facts presented here we can see that very good agreementexists between theory and experiments for the isotropic part of light scatter-ing. However, in treating the anisotropic part some difficulties still remain.It is difficult at present to give much credit to the values of G calculated fromthe data available in the literature. One reason is certainly the incompletenessof physical constants which makes it almost impossible to calculate theoreticallight scattering quantities, except for a few liquids. Another reason is theinadequacy of present theories of anisotropic scattering. So, for example,it is not possible to give an explicit expression like equation 22 for calculatingthe anisotropic scattering factor of spherically symmetric molecules (suchas carbon tetrachioride) with 52 = 0. It is not advisable to make far reachingconclusions about the liquid structure and the intermolecular correlationsfrom present G data since it might be possible that the experimentallydetermined G data do not only consist of the functions of the form of equation19, but might include some terms arising from the nonlinear effects due toremarkable internal fields. It would certainly be possible to improve thissituation by collecting more data on more liquids. Complete sets of physicalconstants measured at the same temperatures and wavelengths, and carried

351

GJURO DEZELK

out with increased precision, would allow additional improvements inthe theories of light scattering in dense media.

References

H. L. Frisch and Z. W. Salsburg, editors. Simple Dense Fluids Academic Press, New York,N.Y. (1968).

2 M. von Smoluchowski. Ann. Physik 25, 205 (1908).A. Einstein. Ann. Physik 33, 1275 (1910).1. L. Fabelinskii. Molecular Scattering of Light Izdatel'stvo Nauka. Moscow (1965).S. Kielich. Acta Phys. Polon. 19, 149 (1960).

6 R. Pecora and W. A. Steele. J. Chem. Phys. 42, 1872 (1965).R. S. Krishnan. Proc. md. Acad. Sci. 1k 782 (1935).

8 H. Z. Cummins and R. W. Gammon. J. Chem. Phys. 44, 2785 (1966).D. 1. Coumou, E. L. Mackor and 1 Hijmans. Trans. Faraday Soc. 60, 1539 (1964).

10 L. S. Ornstein and F. Zernike. Physik. Z. 27, 761 (1926).Gj. Deeliá. J. Chem. Phys. 45, 185 (1966).

12 H. Benoit and W. H. Stockmayer. J. Phys. Radium 17, 21(1956).13 5 Kielich. J Chess:. Phys. 46, 4090 (1967).14 Gj. Deeliá and J. Vavra. Croat. Chem. Acta 38, 35 (1966).

W. Heller, W. J. Pangonis and N. A. Economou. J. Chem. Phys. 34, 971 (1961).16 J. G. Kirkwood and R. J. Goldberg. J. Chem. Phys. 18, 54 (1950).17 D. J. Coumou and E. L. Mackor. Trans. Faraday Soc. 60, 1726 (1964).18 Y. Sicotte and M. Rinfret Trans. Faraday Soc. 58, 1090 (1962).19 R. K. Bullough. J. Polymer Sci. 46, 517 (1960).20 c Clement and P. Bothorel. Compt. Rend. 251, 2323 (1960).21 D. J. Coumou, J. Hijmans. and E. L. Mackor. Trans. Faraday Soc. 60, 2244 (1964).22 c Ghita. Rev. Roum. Chin:. 13, 1019 (1968).23 L. Brillouin. Ann. Phys. (Paris) 17, 88 (1922).24 L. Mandelstam. J. Russ. Phys. Chem. Soc. 58, 831 (1926).25 E. Gross. Nature 126, 201, 400, 603 (1930), 129, 722 (1932).26 L. D. Landau and U. Placzek. Z. Phys. Sowjetunion 5. 172 (1934).27 R. D. Mountain. Rev. Mod. Phys. 38, 205 (1966).28 M. Bertolotti, B. Crosignani, P. Di Porto and D. Sette. Phys. Rev. 157, 146 (1967).29 D. A. Pinnow, S. J. Candau and T. A. Litovitz. J. Chem. Phys. 49, 347 (1968).30 M. Leontovich. J. Phys. USSR 4, 449 (1941).31 A. 0. Buckingham. Proc. Phys. Soc. B69, 344 (1956).32 R.W. Terhune, P. D. Maker and C. M. Savage. Phys. Rev. Letters 14, 681 (1965).

D. 1. Mash, V. V. Morozov. V. S. Starunov and I. L. Fabelinskii. JETP Letters (USSR) 2,25 (1965).S. Kielich. IEEE J. Quant. Electronics QE-4, 744 (1968)." R.Berson, Yoh-Han Pao and H. L. Frisch. J. Chem. Phys. 45, 3184 (1966).

36 5 Kielich. Acta Phys. Polon. 33, 89 (1968).S. Kielich. Acta Phys. Polon. 33, 63(1968).

38 j P. Kratohvil, Gj. DeeliC. M. Kerker and E. MatijeviC. J. Polymer Sci. 57, 59 (1962).B. A. Brice, M. Haiwer and R. Speiser. J. Opt. Soc. Am. 40. 768 (1950).

40 j, P. Kratohvil, M. Kerker and L. E. Oppenheimer. J. Chem. Phys. 43, 914 (1965).41 Gj. Dee1iC. Groat. Chem. Acta 33, 99 (1961).42 M. Kerker, I P. Kratohvil and E. MatijeviC. J. Polymer Sci. A2, 303 (1964).' J. P. Kratohvil. J. Colloid Interface Sci. 21, 498 (1966).

CI. Carr and B. H. Zimm. J. Chem. Phys. 18, 1616 (1950).J. J. Hermans and S. Levinson. J. Opt. Soc. An:. 41, 460 (1951).

46 B. SedláCek. Chem. Listy 47, 1113 (1953).' N. Rozhdestvenskaya and M. F. Vuks. Ukr. Fiz. Zh. 9, 548 (1964).48 N. Rozhdestvenskaya. Vestn. Leningr. Univ. Ser. Fiz. i Khim. 19, No. 16, 42 (1964).

M. F. Vuks. Opt. Spektroskop. 25, 857 (1968).° W. H. Martin. Trans. Roy. Soc. Can. 17, 151 (1923).D. J. Coumou. J. Colloid Sci. 15, 408 (1960).

352

LIGHT SCATTERING IN DENSE MEDIA

G. Weill. Ann. Phys. (Paris) 6, 1063 (1961).S. Claesson and J. Ohman. Arkiv kemi 23, 69 (1964).R. C. C. Leite, R. S. Moore, S. P. S. Porto and J. E. Ripper. Phys. Rev. Letters 14, 7 (1965).H. Utiyama. J. Phys. Chem. 69, 4138 (1965).

56 T. V. George, L. Goldstein, L. Slama and M. Yokoyama Phys. Rev. A137, 369 (1965).R. D. Watson and M. K. Clark. Phys. Rev. Letters 14, 1057 (1965).

58 H. A. Stuart. Die Struktur desfreien Moleküls Springer-Verlag, Berlin (1952).G. Cohen and H. Eisenberg. J. Chem. Phys. 43, 3881 (1965).

60 K. Ramanathan. md. J. Phys. 1, 413 (1927)." I. Rocard. Ann. Phys. (Paris) 10, 159 (1928).62 s Bhagavantam. Scattering of Light and the Raman Effect Chemical Publishing Co., Inc.,

Brooklyn, N.Y. (1942).63 M. I. Shakhparonov. Methods of Investigating the Thermal Motion of Molecules and the

Structure of Liquids (in Russian). Izdatel'stvo Moskovskogo Univ., Moscow (1963).64 C. A. Ten Seldam and S. R. de Groot. Physica 18, 905 (1952).65 R. L. Amey. J. Phys. Chem. 69, 702 (1965).66 R. L. Amey. The Electromagnetic Equation of State Data in Simple Dense Fluids ee Ref. 1,

p. 187.67 J S. Rosen. J. Chem. Phys. 17, 1192 (1949).68 W. F. Brown, Jr. J. Chern. Phys. 18, 1193 (1950).69 P. Mazur. Advan. Chem. Phys. 1, 309 (1958).70 E. Reisler and H. Eisenberg. J. Chem. Phys. 43, 3875 (1965).71 H. Eisenberg. J. Chem. Phys. 43, 3887 (1965).72

S. Kielich. J. Phys. (Paris) 29, 619 (1968).M. Yasumi, H. Okabayashi and H. Komooka. Bull. Chem. Soc. Japan 31, 402, 669 (1958).' M. S. Malmberg and E. R. Lippincott. J. Colloid Interface Sci. 27, 591 (1968).J. Ehi, C. Loucheux, C. Reiss, and H. Benoit. P4akromol. Chem. 75, 35 (1964).

76 P. Bothorel. J. Colloid Interface Sd. 27, 529 (1968).' A. Massoulier. J. Phys. (Paris) 24, 342 (1963).78 A. Massoulier. Compt. Rend. B267, 132 (1968).

C. Clement and P. Bothorel. J. Chim. Phys. 61, 878 (1964).80 J Powers, D. A. Keedy and R. S. Stein. J. Chem. Phys. 35, 376 (1961).81 F. R. Dintzis and R. S. Stein. J. Chem. Phys. 40. 1459 (1964).82 J. R. Lalanne. Rapport de Ia dffusion Rayleigh depolarisée et des techniques de mesure de

l'anisotropie optique ii Iadetermination du parametre de correlation angulaire intermoleculairedes liquides (to be published).

83 5, Kielich. Poz. Tow. Przyj. Nauk 11, No. 2, 3 (1962).84 R. L. Schmidt. J. Colloid Interface Sci. 27, 516 (1968).85 C. G. LeFèvre and R. J. W. LeFèvre. J. Chen Soc. 2670 (1959).86 M. Giurgea, L. Ghita and C. Ghita. Rev. Rounl. Phys. Acad. RPR 7, 5 (1962).87 B. A. Pethica and C. Smart Trans. Faraday Soc. 62 1890 (1966).88 Y. Sicotte. J. Chim. Phys. 61, 1086 (1964).89 G. B. Benedek, J. B. Lastovka, K. Fritsch and T. Graytan. J. Opt. Soc. Am. 54, 1284 (1964).90 R. Y. Chiao and B. P. Stoicheff. J. Opt. Soc. Am. 54, 1286 (1964).91 D. 1. Mash, V. S. Starunov and I. L. Fabelinskii. J. Exptl. Theor. Phys. (USSR) 47, 783

(1964).92 D. H. Rank, E. M. Kiess, U. Fink and T. A. Wiggins. J. Opt. Soc. Am. 55, 925 (1965).

H. Z. Cummins and R. W. Gammon. Appl. Phys. Letters 6, 171 (1965).C. L. O'Connor and J. P. Schlupf. J. Acoust. Soc. Am. 40, 663 (1966).C. L. O'Connor and J. P. Schlupf. J. Chem. Phys. 47, 31(1967).

96 S. L. Shapiro and H. P. Broida. Phys. Rev. 154, 129 (1967).G. D. Parfitt and J. A. Wood. Trans. Faraday Soc. 64, 805, 2081 (1968).

98 W. C. Röntgen and L. Zehnder. Ann. Physik 44, 29 (1891).R. M. Waxier. C. E. Weir and H. W. Schramm. Jr. J. Res. Nat!. Bur. Std. Ser. A68, 489 (1964).

353