THE SPECTRUM AND INTENSITY OF LIGHT SCATTERED FROM THE BULK PHASES AND FROM THE LIQUID-VAPOR INTERFACE OF XENON NEAR ITS CRITICAL POINT

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THE SPECTRUM AND INTENSITY OF LIGHT SCATTERED FROM THE BULK PHASES AND FROM THE LIQUID-VAPOR INTERFACE OF

XENON NEAR ITS CRITICAL POINT

J. Zollweg, G. Hawkins, I. Smith, M. Giglio, G. Benedek

To cite this version:

J. Zollweg, G. Hawkins, I. Smith, M. Giglio, G. Benedek. THE SPECTRUM AND INTENSITY OF LIGHT SCATTERED FROM THE BULK PHASES AND FROM THE LIQUID-VAPOR INTER- FACE OF XENON NEAR ITS CRITICAL POINT. Journal de Physique Colloques, 1972, 33 (C1), pp.C1-135-C1-139. �10.1051/jphyscol:1972124�. �jpa-00214914�

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JOURNAL DE PHYSIQUE Colloque Cl, supplément au no 2-3, Tome 33, Février-Mars 1972, page Cl-1 35

THE SPECTRUM AND INTENSITY OF LIGHT SCATTERED FROM THE BULK PHASES AND FROM THE LIQUID-VAPOR INTERFACE OF XENON

NEAR ITS CRITICAL POINT

(*)

J. ZOLLWEG (**), G. HAWKINS, 1. W. SMITH, M. GIGLIO (***) and G. B. BENEDEK

Department of Physics and Center for Materials Science and Engineering Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Résumé.

-

Nous présentons les résultats de la mesure de la longueur de corrélation

(T),

_{de la}

compressibilité ^(KT), de la viscosité (4 et de la tension superficielle ^{( O )}du xénon près de son point critique. Ces données ont été utilisées pour vérifier de façon détaillée la validité de plusieurs relations fondamentales reliant la divergence de la longueur de corrélation et la divergence de la conlpressi- bilité, pour vérifier aussi la disparition de la tension superficielle et la divergence de la conductivité thermique près du point critique.

Abstract. - We report measurements of the long range correlation length (0, the compressibility ^(KT), the viscosity (r), and the surface tension ^(CT)in xenon near its critical point. Using this data we have been able to examine in detail the validity of several fundamental theoretical relations which connect the divergence of the correlation range with the divergence of the compressibility, the vanishing of the surface tension, and the divergence of the thermal conductivity near thecritical point.

The central feature in the description of a system near its critical point is the long range correlation length

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which describes the characteristic distance over which fluctuations in the order parameter are spatially correlated. In the vicinity of the critical point the correlation length becomes very large.

Therefore the spatial integral of the correlation function for the order parameter increases extremely rapidly as the critical temperature is approached.

The divergence of this integral manifests itself expe- rimentally as the divergence of the equilibrium sus- ceptibilities of the system. Thus the precise nature of the divergence of the equilibrium properties is intima- tely connected to the magnitude and the temperature dependence of the correlation range. This connection appears explicitly in the static scaling theories of critical phenomena.

On the other hand, non-equilibrium properties of the system are also connected to the divergence of the correlation range. The development of dynamical scaling theories for transport properties near the critical point (expressed in detail in the mode-mode coupling theories of Kadanoff, Swift and Kawasaki)

(*) This research was supported by the Advanced Research Projects Agency under contract DAHC 15 67C 0222 with the Massachusetts Institute of Technology and by the National Science Foundation.

(**) Now at the Department of Chemistry, University of Maine, Orono, Maine.

(***) Now at the University of Milan, Milan, Italy.

shows that the long range correlation length plays a central role in understanding the transport coefficients.

The purpose of this note is to present Our measurements of the correlation length

<

in xenon along the critical isochore and along the coexistence curve and to relate these measurements to the observed behavior of the compressibility, the surface tension, and the thermal diffusivity. In this way we shall be able to examine the validity of theoretical predictions connecting the correlation range to both the static and the dynamic properties of the system.

By observing the angular anisotropy of the intensity of light scattered from xenon, we have measured the magnitude and the temperature dependence of the correlation range

t

along the critical isochore in the temperature range

Our results along the critical isochore are shown in figure 1 and are summarized by the equation

Along the vapor side of the coexistence curve Our previous measurements [l] of show that the correlation range there obeys the equation

A .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972124

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Cl-136 J. ZOLLWEG, G. HAWKINS, 1. W. SMITH, M. GIGLIO AND G. B. BENEDEK

FIG. 1. - The temperature dependence of the long range correlation length along the critical isochore in xenon.

The relationship between the correlation range

5

and the isothermal compressibility (K,) is predicted by the Ornstein-Zernike theory to be

Here ~c, = l/(p, k, T) is the compressibility of an ideal gas, p, is the number density of the fluid, k, is Boltzman's constant, and R is the direct correlation range. In the Ornstein-Zernike theory the pair correlation function has the form (e-'lS/r) from which it follows that R is a constant near the critical point.

However, if the pair correlation function has the form (e-rlr/ri-s) as proposed by M. Fisher, then R will diverge weakly near the critical point.

I n order to investigate the validity of eq. (1) and to deduce the behavior of R near Tc, we have measured the compressibility rc, (or equivalently ( d p / d ~ ) ~ = p2 rc,) along both the critical isochore and the vapor side of the coexistence curve. This measure- ment is made by determining the absolute intensity of light scattered at very small angles in the ffuid near its critical point. The results for (apldp), are shown in figure 2. The upper curve corresponds to measurements taken along the critical isochore.

Here

The lower line corresponds to measurements of (aplap), along the vapor side of the coexistence curve.

This data obeys the equation

These results may be used to examine the validity of eq. (1). We first consider the limiting behavior as T approaches Tc. If v and y are the critical exponents describing respectively the temperature dependence

FIG. 2. - ( ô p / ô p ) ~ ) ~ as a function of temperature along the critical isochore (upper line) and along the vapor side of the coexistence

curve (iower line) for xenon.

of

5

and of (aplap),, then in the limit T -, Tc we expect from eq. (1) that y = 2 v. From Our measurements on the critical isochore y ⁼1.21

+

^{0.03 and}

2 v = 1.14

+

0.1. Despite the experimental uncertain- ties it is possible to estimate a value of Fisher's exponent y. See reference [16].

Secondly, we can use eq. (1) to calculate numerical values of the direct correlation range R at various temperatures in order to determine wheter R is in fact constant. From Our data we find that R = 4.9

+

0.1

A

both above and below Tc. (This value is slightly different from that previously reported [l] because the magnitude of (~?p/ap)~ has now been obtained more accurately). It is interesting to note that when this value of R is used in analyzing D.

Cannell's data [2] on the Brillouin components of xenon near Tc, the deduced magnitude and temperature dependence of

5

along the critical isochore are in excellent agreement with the results reported here.

From Our data, we have also computed the ratio of (aplap), on the critical isochore to (aplap), on the coexistence curve. We find asymptotically that this ratio is (4.1 & 0.2). This implies that for equal temperatures above and below Tc, the compressibility on the critical isochore is lai-ger than that in either phase along the coexistence curve by a factor of 4 . l ( P l ~ 3 ~ .

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THE SPECTRUM AND INTENSITY OF LTGHT SCATTERED FROM THE BULK PHASES Cl-137

We next examine the implications of Our measurements of

5

insofar as the transport properties of xenon are concerned. By measuring the spectral width of monochromatic light scattered quasi-elastically from a fluid, it is possible to deduce the total thermal diffusivity D = (Alpc,). Here A is the thermal conductivity, c, the specific heat per unit mass, and p the mass tiensity. Near the critical point cp diverges quite strongly, and as a result D approaches zero. This implies that entropy fluctuations in the fluid relax back to equilibrium ever more slowly as the critical point is approached. This increasing relaxation time manifests itself in an extraordinary narrowing of the spectral line width of the quasi-elastically scattered light. Despite this narrowing, the line width (which may be less than 100 Hz) can be measured quite accurately using the techniques of optical mixing spectroscopy [3].

The mode-mode coupling theories of Kadanoff, Swift [4], and Kawasaki [5] predict that in the hydro- dynamic limit the so-called « critical contribution » to the thermal diffusivity should Vary with temperature in accordance with the formula

where y is the shear viscosity of the fluid. This result lends itself to a simple physical interpretation. It States that heat is transported by a random walk process in which the elemental units carrying heat have dimension

r.

Eq. (2) is simply the diffusion coefficient for the motion of « particles » of size g moving in a medium of viscosity y.

In the case of xenon, the total thermal diffusivity (Alpc,) has been measured as a function of temperature along the critical isochore by Henry, Swinney and Cummins [6]. In order to compare these measurements with the critical contribution (k, T/6 nyt) predicted by the mode-mode coupling theories, it is necessary to know the shear viscosity q of xenon. We have measured y (as well as the surface tension a) along the coexistence curve of xenon by observing the spectrum of light scattered inelastically from thermally excited surface waves on the liquid-vapor interface 171, [8]. Our results show that

= 1448

+

12(T, - T)] i I l microstokes [9]

P

from which we may evaluate the shear viscosity at T ⁼Tc and p = p,. Since the viscosity changes only slightly as the temperature increases along the critical isochore we can, with good accuracy, calculate k, T/6ny5. In figure 3 we plot the value of kg T/6nyr along with the experimental data of Cummins and Swinney for (Alpc,). In figure 4 we show the same measurements over a narrower range of temperature so as to focus attention on the data clo- sest to the critical point. It is clear from figures 3 and 4 that it is not correct to assume [6] that eq. (2) alone

FIG. 3. -The total thermal diffusivity (A/pcP) (upper curve) compared with the critical part of the thermal diffusivity as predicted by the mode-mode coupling theories (lower curve).

The temperature range includes the entire region where experimental data is available : (T- Tc) 6 6.0 OK.

0 . 5 k Dota of Henry,Cummins,& Swinney

/

Temperature ( O C )

FIG. 4. - (A/pcp) as compared with k~ T/6 nq< in the temperature range (T- Tc) < 0.600 OK.

accurately describes the measured thermal diffusivity.

In fact there is an additional contribution to the thermal diffusivity that is not included in the mode-mode coupling analysis. This additional term arises from the non-divergent or background part of the thermal conductivity (Abackground) and contributes an amount (Abackground/p~,) to the total thermal diffusivity.

In order to examine the accuracy of the mode-mode coupling theories one must determine whether (Abackground/pcp) is of such a size as to account for the difference between

The background thermal conductivity data of R. Tufeu, B. Le Neindre, and B. Vodar for A ,,,kg,,

,,,

in xenon has been recently analyzed by J. V. Sengers [IO]. The specific heat c, in the critical region may be obtained from the thermodynamic relation

by using our measurements of ( d p l d ~ ) ~ , the measure-

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Cl-138 J. ZOLLWEG, G. HAWKINS, 1. W. SMITH, M. GIGLIO AND G. B. BENEDEK

ments of Buckingham et al. [1 Il for c,, and the results of Habgood and Schneider [12] for ( d P / d T ) , on the critical isochore. The contribution of c, to c, is small and the primary temperature dependence of c, is determined by the divergence of (dp/dp)u),. From these measurements the magnitude and temperature dependence of the background contribution to the thermal diffusivity may be calculated. The results of this calculation are plotted in figure 5 along with the quantity k, T/6?cqt. Also showil js the sum of these two quantities, which is seen to agree remarkably well with the data points of Cummins and Swinney for the measured total thermal diffusivity. We therefore conclude that on the critical isochore of xenon the contribution to the total thermal difTusivity predicted by the mode-mode coupling theories is consistent with the observed values of (Alpc,) when account is taken of the background thermal conductivity.

Dafa of Curnmins 8 Sw~nney

FIG. 5. - Broken line : The Kawasaki critical contribution (kg T/6 nqt) to the total thermal diffusivity (Alpc,) on the

Reduced temperature ( I - ^T/Tc⁾

FIG. 6. - Surface tension a for xenon as a function of the reduced temperature (1 - T/Tc). 'The open circles represent Our measurements. The clo'sed circles correspond to the

corrected data of Smith, Gardner and Parker.

The connection between oulr measurements of the surface tension and the behavior of the correlation range is expressed in the Fisk-Widom formula [15]

which relates the interface thickness (1,) to the surface tension ^oby :

critical isochore of xenon. Dotted line : The non-critical or

background contribution (Abacitprouna/pcp) to the total thermal Here pl and pv are the of the liquid and

diffusivity. The solid line represents the sum of the critical and VaPor phases res~ectivel~, b is the critical exponent

the non-critical contributioiis to (Alpc,) and the data points describing the temperature dependence of ( p , - p,),

correspond to measurements of (Alpcp) taken by Henry, and the parameter c is a consitant predicted to be of

Cummins, and Swinney. the order of unity. Furtherniore, Fisk and Widom Finally we shall examine the connection between

the divergence of the correlation range

5

and the vanishing of the surface tension a on the coexistence curve of xenon. We have measured ^oby analyzing the spectrum of light scattered inelastically from thermally excited surface waves on the liquid vapor interface [9]. Our data for the temperature dependence of o is shown in figure 6 as open circles. The closed circles correspond to the data of Smith, Gardner, and Parker [13], corrected by using the more accurate density formulae of Garside, Molgaard, and Smith [14].

These results may be summarized by the equation dynes/cm for 0.070 OK < ( T c - T ) < 5 OK.

demonstrate that in the limit T -+ Tc, L should be equal to the correlation range

5.

Using literature values and Our own measuremi:nts of each of the terms in eq. (3), we find that

a temperature at which the value of

5

is well known.

If we keep. c constant at this value and deduce the temperature dependence of Il from the behavior of ( d p / a ~ ) ~ , a, and ( p l - p,) then we find that

This value of the interface thickness is in good numerical agreement wjth Our direct measurements of the correlation range

t

along the vapor side of the coexis-

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THE SPECTRUM AND INTENSITY OF LIGHT SCATTERED FROM THE BULK PHASES Cl-139 tence curve for 0.025 OK < (Tc - T ) < 0.125 OK. along the liquid side of the coexistence curve and that We must conclude that the Fisk-Widom relation is in along the vapor side. Consideration of this effect in good agreement with the experimental results in the fact improves the agreement between the values of temperature range (Tc - T ) < 0.125 0K. Farther deduced from the surface tension data and the values from the critical point, the interfacial thickness L measured directly by observing the angular dependence is likely to be a combination of the correlation range of the intensity of scattered light.

References

[l] GIGLIO (M.) and BENEDEK (G. B.), Phys. Rev. Letters, [7] KATYL (R. H.) and INGARD (U.), Phys. Rev. Letters,

1969, 23, 1145. _. _, 1968,20,248.

CANNELL (D.) and BENEDEK (G. B.), Phys. Rev.

Letters, 1970,25,1157.

BENEDEK (G. B.), ⁽⁽Optical Mixing Spectroscopy with Applications to Problems in Physics, Chemistry, Biology, and Engineering », pp. 49-84, in Pola- rization, Matter and Radiation, Jubilee Volume published by Presses Universitaires, Paris, 1969.

[8] BOUCHIAT (M. A.) and MEUNIER (J.), Phys. Rev.

Letters, 1969, 23, 752.

[9] ZOLLWEG (J.), HAWKINS (G.) and BENEDEK (G. B.), Phys. Rev. Letters, 1971, 27, 1182.

[IO] SENGERS (J. V.), private communication.

[ I l ] EDWARDS (C.) LIPA (J. A.) and BUCKINGHAM (M. J.), Phys. Rev. Letters, 1968, 20, 496.

[121 H A B G O ~ D (H. W.) and SCHNEIDER (W. G.), Can.

. . . .

[4] KADANOFF (L. P.) and SWIFT (J.), Phys. Rev., 1968, J. ~ h e i . , 1954, 32, 164.

166. 89. [13] SMITH (D. L.), GARDNER (P. R.) and PARKER (E. H. C.),

-. 7 - -

J. ~ h e m . Phys., 1967, 47, 1148.

151 KAWASAKI (K.), Phys. Letfers, 1969, 3% 325, Phys. 1141 G~~~~~~ (D. H.), (H. v.) and ^SMITH Rev., 1970, A 1 , 1750. (G. L.), J. Phys. B (Proc. Phys. Soc.), 1968, 1.

[6] HENRY (D. H.), SWINNEY (H. L.) and CUMMINS [15] FISK (S.) and WIDOM (B.), J. of Chem. Phys., 1969, (H. Z.), Phys. Rev. Letters, 1970, 25, 1170 and 50,3219.

Henry (D. H.), Thesis (Ph. D.), Johns Hopkins ^[16]SMITH (1. W.), GIGLIO (M.) and BENEDEK (G. B.),

University. Ph-vs. Rev. Letters, 1971, 27, 1556.