Ill SCATTERING

Benjamin Chu*

Department of Chemistry·, University of Kansas, Lawrence, Kansas

and

Paul W. Schmidt

Department of Physics, University of Missouri, Columbia, Missouri

1. Introduction I l l 2. Theoretical Background 112

3. Xenon 114 4. Helium 116 5. Other Systems 116 6. Conclusion 117

References 117

L Introduction

The purpose of this chapter is to summarize the few available classical light scattering data on simple dense fluids.

During the past few years there has been increased interest in the light scattering from gases and liquids. Laser research has inspired the deve- lopment of very high-resolution interferometers [1], optical heterodyne systems [2], and homodyne self-beat techniques [3], which are capable of detecting exceedingly small frequency changes. The combination of ingenious modern detection methods with the high intensity and narrow line width (temporal and spectral coherence) of a laser source provides a greatly increased resolution in measurements of the angular and spectral distribution and the depolarization of scattered light. In studies of Brillouin spectra [4] in liquids, the Landau-Placzek ratio [5] for the intensities of the central and Brillouin components, the Brillouin shifts,

* Alfred P. Sloan Research Fellow.

I l l

and the linewidths [6-10] have been investigated. In addition, the theory of the spectral distribution of the scattered light has been developed for the critical region of both one-component systems [11] and binary fluid mixtures [12]. Experimental data have been published for sulfur hexafluoride [3], carbon dioxide [13], aniline-cyclohexane [14] and polystyrene-cyclohexane [15]. Nonlinear light scattering effects caused by very intense laser beams have also been studied [16-20]. However, none of the recent studies deal with a condensed monatomic rare gas system or any other simple dense fluid, except for the work of Woolf et al.

[21], who used the technique of Brillouin scattering to investigate the attenuation and velocity of high frequency acoustic phonons in liquid helium below the λ-point. It is nevertheless quite evident that with the techniques now available [1-3, 18, 21, 22], we are well within the realm of doing many worthwhile studies of the light scattering from simple dense fluids.

2* Theoretical Background

Classical theoretical formulations of light scattering have been deve- loped from two different standpoints. Rayleigh's theory [23] of dipole scattering assumes negligible interaction between molecules, with each molecular dipole scattering independently. The total scattered intensity then is the product of the number of scatterers and the intensity per scatterer. Rayleigh's theory is a single-particle scattering theory, which predicts a constant angular distribution of scattered radiation for perpendicularly polarized light. The theory has been extended to include anistropic particles [24, 25], which can be studied in terms of depolarization ratios [26-29]. Rayleigh's theory does not apply to liquids, in which the molecules do not scatter independently. For dense media, von Smoluchowski [30] and Einstein [31] used the thermodynamic theory of fluctuations to obtain a light scattering formula, which can be expressed [29]

where R is called the Rayleigh ratio; r is the distance from the sample to the detector; Ιη(θ) is the intensity of the unpolarized light scattered per unit solid angle per unit sample volume; Θ is the scattering angle (that is, the angle between the incident and the scattered beam); I0u is the intensity of the unpolarized incident light; λ0 is the wavelength of light in vacuum;

pQ is the number of particles per unit volume; €opt is the dielectric constant

at the frequency of the scattered light, h = 4πλ~^τ sin (0/2); λ = λ0/ι>

is the wavelength of light in the medium; v = ( e ^ )¹/² is the refractive index; and L(h) is a quantity which describes the effect of intermolecular interference and which for a given h value is the same for both x-ray and light scattering (see Sections 2.1 and 3.1 in the chapter by Schmidt and Tompson, this volume). The factor (1 + cos² Θ) accounts for the unpolarized incident light. Equation (2.1) neglects orientational correlations and molecular anisotropy [32-35], which are unimportant for simple dense fluids.

General discussions of light scattering theory are presented in standard texts [36-38]. Kratohvil [39] has written extensive reviews of recent developments in classical elastic linear light scattering. Frisch and McKenna have investigated the theory of multiple light scattering [40].

In light scattering, the shortest wavelength λ ordinarily possible in the sample is of the order of 2000 A. If the scattering apparatus permits measurements of scattering angles approaching 180°, the maximum value of h attainable with light scattering is about 6 X 10~³ A^{- 1}. On the other hand, in small angle x-ray scattering, the longest wavelength convenient for experiments is 1.54 A, and the smallest accessible scattering angle is about 1 X 10~³ radian, corresponding to a minimum value of h of the order of 4 X 10~³ A^{- 1}. Thus, light scattering measurements give L(h) for small values of hy while x-ray scattering gives L(h) for larger h values. The range of h possible with x-rays barely overlaps the values attainable with light scattering. In a few cases [41-43] x-ray and light scattering data have been combined to give L(h) over a greater range of h than would be possible with x-ray or light scattering alone.

When a system is far from its critical point, L(h)~kTPoß

where k is Boltzmann's constant, T is the absolute temperature, and ß is the isothermal compressibility (see Section 3.1, Schmidt and Tompson chapter, this volume). As a system approaches its critical point, there is a large increase in the scattered intensity. This effect is known as critical opalescence (see Section 3.1, Schmidt and Tompson chapter, this volume).

Several theories have predicted that in the critical region L(h) has the approximate form [44-46]

L(h) =kTPoß(l +L²Ä²)-! (2.2)

where L is independent of h and has dimensions considerably greater than intermolecular distances.

In Debye's treatment of critical opalescence [45], the persistence length LD = \/6L is introduced. For a van der Waals gas at its critical

density and near its critical temperature Tc, in Debye's theory LD can be approximated by the relation

L 2 = ID' D (T/T.) - 1

where lD is a quantity which is determined from the second moment of the intermolecular pair potential energy and which has the magnitude of intermolecular distances.

In Fixman's theory of critical opalescence [46]

L2 = io-i/2(£7>0jS - 1)

where / is a parameter which is expected to be of the order of magnitude of atomic dimensions. In the critical region lD ~ (4/15)^1/2/. When a system is near its critical point, L and LD become very large, while / and lD are nearly constant. When L(h) has the form given by (2.2), a linear relation is obtained when the reciprocal of the scattered intensity is plotted as a function of h². This type of plot is often referred to as an Ornstein-Zernike plot or an Ornstein-Zernike-Debye plot and is frequently used in analyzing the scattering data in studies of critical opalescence.

Recent theories suggest that in the immediate neighborhood of the critical point, L(h) must deviate from (2.2) at small values of h [47].

Light scattering, which permits measurements at relatively small h values, is an advantageous method for looking from deviations from (2.2).

Although these deviations have been observed in a binary fluid mixture [48], the experimental light scattering evidence cannot yet be considered conclusive.

3* Xenon

Murray and Mason [49] studied the turbidity of xenon (and ethylene) near its critical point. They determined the turbidity τ from measure- ments of the transmitted light, using the relation

r = (lld)\n(I'

in

where d is the optical path length through the scattering medium and / ' and IQ are the transmitted and incident intensities, respectively. Pure samples were analyzed in a mass spectrometer, and transmission measu- rements were made immediately after vigorous stirring of the fluid inside the bomb. Furthermore, photoelectric techniques were used.

All indications show that their transmission measurements are acceptable except in the immediate neighborhood of the critical point when T — Tc is less than a few hundredths degree, when the effect of multiple scattering may be appreciable in a 10-mm i.d. bomb, even for one- component systems. Also, near the critical point, the time required to establish equilibrium may be very long, and density gradients because of gravitational fields cannot be avoided. Thus the system probably was not at equilibrium, and the mean density p has no meaning. At tempera- tures further from the critical temperature, the same effect exists, but it becomes less important. Murray and Mason used a polychromatic light source, complicating the analysis when the dependence of L(h) on h becomes appreciable, since A is a function of λ. Figure 1 shows a plot of transmittance (Γ/Ιό) versus the mean density p at various tempe- rature intervals Δ T = T — Tc above the critical temperature Tc of xenon [49].

1.00

0.75 .. o

0.50

0.25

1.0 I.I 1.2 (a)

16.63

*>

16.61

1.0 I.I 1.2 p gm /cm

(b)

FIG. 1. (a) The variation of transmittance Γ/Γ0 with density p along isotherms above the critical temperature Tc of xenon, (b) The liquid-vapor coexistence curve. (Murray and Mason [49, p. 1403].)

4+ Helium

In their light scattering studies of liquid helium, Lawson and Meyer [50] determined the ratio IJIg , where It is the scattered intensity from liquid helium, and Ig is the intensity scattered in the same apparatus by helium gas under one atmosphere pressure at its boiling point. They compared their observed values of (IJIg)oh8 with the values (IJIg)CSbiG

calculated by using experimental values of the refractive index and isothermal compressibility in a relation equivalent to Eq. (2.1). Their results are shown in Table I. In the values of (IJIg^s > ^{t n e} second

TABLE I

VALUES OF OBSERVED AND CALCULATED RATIOS Ii\Iga

T(°K) 4.2 2.1 1.5

(^lÄ)obs

4.0

l.o 0.76

(■*i/-'a)calc

3.8 1.0 0.7

° Data from Lawson and Myer [50].

significant figures are shown as subscripts because of the relatively high experimental uncertainty. Within this uncertainty, the calculated and observed intensity ratios were in agreement, and no anomalous effect was detected as liquid helium passed through the λ-point.

5* Other Systems

Drickamer and his co-workers have studied the light scattering from ethylene and ethane near the critical point [51, 52]. Critical opalescence has recently been investigated in carbon dioxide [53-55] and in sulfur hexafluoride [54]. Rayleigh scattering of 6943 A laser radiation in nitrogen gas at atmospheric pressure has also been measured [56]. A number of investigations have been made of the light scattering from organic liquids [27-29, 57-61]. These studies will not be discussed here, since we are restricting ourselves to the consideration of simple dense fluids.

Note added in proof: Thibeau et al. [62] have recently reported measurements of the depolarization and intensity of the Rayleigh light scattering in argon at 20°C and pressures up to 800 atm.

6. Conclusion

As there are almost no reliable light scattering measurements on simple dense fluids, light scattering data from simple dense fluids under almost any conditions of temperature and density would be of considerable interest. Since the techniques are available, some reliable

data can be expected to become available in the near future.

ACKNOWLEDGMENTS

The authors wish to thank the National Research Council of Canada and the publishers of The Physical Review for permission to use Fig. 1 and Table I, respectively.

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