THE LIQUID-VAPOR INTERFACE OF SULFUR-HEXAFLUORIDE NEAR THE CRITICAL POINT

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THE LIQUID-VAPOR INTERFACE OF

SULFUR-HEXAFLUORIDE NEAR THE CRITICAL POINT

E. Wu, W. Webb

To cite this version:

E. Wu, W. Webb. THE LIQUID-VAPOR INTERFACE OF SULFUR-HEXAFLUORIDE NEAR THE CRITICAL POINT. Journal de Physique Colloques, 1972, 33 (C1), pp.C1-149-C1-154.

�10.1051/jphyscol:1972126�. �jpa-00214916�

JOURNAL DE PHYSIQUE Colloque C l , supplément au no 2-3, Tome 33, Féurier-Mars 1972, page Cl-149

THE LIQUID-VAPOR INTERFACE OF SULFUR-HEXAFLUORIDE NEAR THE CRITICAL POINT

E. S. WU and W. W. WEBB

School of Applied and Engineering Physics, and Laboratory of Atomic and Solid State Physics Corne11 University, Ithaca, New York, U. S. A. 14850

Résumé. - Des mesures du coefficient de réflexion de l'interface liquide-vapeur dans SF6 au voisinage de sa température critique Tc ont fourni une valeur de l'épaisseur effective de l'inter- face L' = (1,60 f 0,07)

A où

= 1 - T / T c . Des analyses préliminaires de la fonction de corrélation du spectre hétérodyne de la lumière diffusée quasi élastiquement par les excitations

+ O 0 3

thermiques de l'interface ont donné pour la tension superficielle :

= 39,6 26;:

^erg/cm2

dans le domaine 4 x 10-4 <

< 4 x 10-2 et pour la viscosité moyenne

ij

287 + ^{13(Tc - T ) rt 20 X 10-6 poise} .

On a ainsi pu tester la théorie de Fisk et Widom ; par ailleurs l'expression de Kawasaki pour le coefficient de diffusion thermique A

k~ T/6 nijt a été calculée et comparée aux mesures de lar- geurs spectrales de la raie Rayleigh.

Abstract. - Measurements of the reflectivity of the liquid-gas interface in SF6 near its critical

temperature Tc have given values of the effective interface thickness L' ⁼ (1.60 f 0.07) E O . ~ ~ ~ * O J J ~ ~ A where

⁼ 1 - T/Tc. Preliminary analysis of measurements of the correlation function of the hete-

rodyne spectrum of the quasi-elastic light scattering by thermal excitations of the interface yielded

-0 06

the surface tension a = 39.6 2;:: e1'34+0:03 ergjcmz in the range 4 x ^{10-4 4 e} < 4 x 10-2 and

- ^{7 -}

the average viscosity q = 287 + ^{13 (Tc - T ) rt 20} x 10-6 poise. The theory of Fisk and Widom is tested and the Kawasaki form of the thermal diffusivity A = k~ T/6 ni75 is computed and compared with the measurements of the spectral width of Rayleigh scattering.

1. Introduction. - We are measuring the reflec- tivity and the spectrum of quasi-elastic light scattering of the gas-liquid interface of sulfurhexafluoride near its critical point using methods evolved from those used in our laboratory by Gilmer, Gilmore, Huang and Webb [l] and Huang and Webb [2], [3]. This paper reports the first results of a still incomplete analysis of some of Our data. From the reflectivjty measurements we have deduced an effective thickness L' of the surface which can be identified as the correlation length 5

using the interface theory of Fisk and Widom [4].

From the spectrum of inelastic scattering of laser light by thermally excited surface waves, we have deduced the surface tension o and an average kinema- tic viscosity ^5.

Applying our values of ^o and ( and available data on the difference between the coexisting densities of the liquid and gas phases p, - p, and the compressibility rc, of the liquid phase supplemented by measurements above the critical temperature Tc, we have tested the consistency of the theory of Fisk and Widom and of certain scaling laws. Furthermore Our values of the shear viscosity Fj' and the coherence length alone give

the measure of the critical part of the thermal diffu- sivity A = k , T/6 ~cq( suggested by Kawasaki and supposed to dominate the dynamics of the decay of density fluctuations in the hydrodynamic limit near the critical point. The width î of the Rayleigh scatter- ing in the critical region provides a direct measure of these decay processes, but the data above Tc in SF, [5]

have appeared anomalous and these results contri- bute to a test of the conjecture that the non-singular back ground part of the thermal conductivity is responsible for the difficulty as it seems to be in Xe and CO,. Unfortunately our results seem to enhance the opacity of the situation.

II. Reflectivity measurements to determine the effec- tive thickness. - The gradua1 transition in density in the critical interface between the liquid and gas phases was predicted by van der Waals [6] during the last century and subsequently reconsidered by Cahn and Hilliard [7] and was first observed by Gilmer, Gilmore, Huang and Webb [l] in 1965. Intuitive concepts suggesting that the effective thickness was a measure of the correlation length were put on a firm

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972126

Cl-150 E. S. WU AND W. W. WEBB basis in modern theory of critical phenomena by

Fisk and Widom in 1969 141. They gave a relation between the surface tension

the effective thickness

the compressibility K,(T) or better

where y is the chemical potential and p the density and the coexisting liquid and the gas densities

that can be written

This result implies the relation between the critical exponents defined above

The theory of Fisk and Widom also provides a recipe for calculations of the profile of the density in the interface and gives a result based on a reasonable assumption for the equation of state. With such an assumption they obtain a profile described by an algebraic combination of transcendental functions of the parameter x/2 L' where x is distance perpendicular to the center of the interface. This profile is, however, remarkably similar to an error function except for its exponential tails eXlL' at large x. Knowing the profile, it is possible to compute the optical reflectivity as a function of the single parameter L' (in the notation of Huang and Webb) defining its thickness and the square of the difference between coexisting indices of refraction which is just proportional to Ap2. Fisk and Widom also showed their length L' to be identical with the correlation length < defined in terms of the correlation function of the bulk fluids.

Huang and Webb [2] measured the interface reflec- tivity of critical mixtures of cyclohexane and methanol and found that their measurements of L' at several light wavelengths were consistent with the assumed density profile. Warren and Webb [8] measured ^o and found their results combined with measured L' were consistent with eq. (1) within a factor of two.

Their fit of eq. (2) required y' = 1.22, a reasonable value in the absence of data and just consistent with lattice gas calculations. The effective thickness also appeared to be in reasonable agreement with the available data on the correlation length t, again to a factor of two. Zollweg et al., at this conference, report

measurements of a that combine with published data on t, Ap and (apldp)), in a vvay similarly consistent with eq. (1) [9].

Now assuming the Fisk and Widom profile, we have analyzed Our measurements of the reflectivity of the interface in SF, between coexisting gas and liquid in the range 0.032 < AT < 1.386 to determine the effective interface thickness. Our value of Tc is deter- mined independently to 5 Cl.001 K. Figure 1 shows these reflectivity measurements, with the straight line of slope 2 ,f? indicating the result expected of a perfectly sharp interface. Figure 2 is a log-log plot of the corresponding values 'of effective thickness as a function of E. The results may be written

The greatest difficulty in analyzing these measurements is the determination of the position of the asymptotic line representing the reflectivity of the hypothetical sharp interface. This difficulty is responsible for about

FIG. 1. - Logarithmic plot in arbitrary units of the data on the.

temperature dependent reflectivitjr R of the diffuse interface in SF6 near T,. The straight line of slope 2 P

0.666 is the best fit Fresnel reflectivity of the sharp interface proportional to the

refractive index difl:erence squares.

THE LIQUID-VAPOR INTERFACE O F SULFUR-HEXAFLUORIDE NEAR THE CRITICAL POINT Cl-151

I

IO-' 6 1 IO-'

Reduced Temperature r = 1-T/T,

RG. 2. - Logarithmic plot of effective thickness L' as a function of reduced temperature. Typical uncertainties are shown.

The line is the least squares fit given by eq. (3).

half of the stated error ; the rest is statistical error.

However, several alternative least squares fitting procedures with various choices of data weighting were, in the final analysis, al1 found to coincide within the statistical error. The results are subject to confirmation of the assumed shape of the scaled density profile by measurements at other light wavelengths ; these measurements are now underway. The value of the critical exponent is sensitive to the choice of profile which has only been checked once before, on a critical mixture 121.

The definition of L' is chosen so it is the characteris- tic length for the exponential tail of the density profile and thus should be equal to the correlation length { according to Fisk and Widom. Thus

The only available data on 5 in SF, is the determina- tion by Puglielli and Ford [IO] who obtained their values from measurements of the total scattered power above Tc along the critical isochore

These values are in exact agreement with ours. We would have expected Our value below Tc to be about a factor of two smaller from classical and lattice gas arguments. Thus we perhaps should regard these data as suggestive of L' - ^{2 5} rather than exact agreement ; although the critical exponents are clearly consistent.

Zollweg et al. have argued on a classical basis that corresponding correlation lengths should be weighted by the fluid density so pl 5; ⁼ p, 5, at corresponding temperatures [9]. If this conjecture were to prove to be true, the value of L' should be regarded as some weigh- ted average for the two phases. A preliminary study indicates that this effect would be negligible in the range of Our measurements.

III. Inelastic scattering from thermal excitations to determine the surface tension and viscosity. - Ther-

mal excitations of the interface have capillary-gravita- tional wave characteristics. In the absence of viscous damping they would be described by a dispersion relation of the simple form

where g is the gravitational constant and q = 2 ^TC/&

is the wave number of surface waves of wavelength As.

The inelastic scattering spectrum P(o, q) would yield information on Ap and a that could be sorted out from the q dependence. This sort of analysis was carried out by Katyl and Ingard on the free surface of ordinary fluids, in spite of difficulties due to low scattered intensities [Il]. However near the critical point both Ap and a vanish, so the amplitudes of thermally excited waves increase and so does the scattered intensity as one can see by applying the equipartition theorem. Inclusion of viscous damping in the disper- sion relation [12] leads to a 6 x 6 matrix equation that can fortunately be simplified by certain assumptions about the viscosities. The inelastic scattering spectrum then leads to information on Ap, o and an averaged kinematic viscosity G.

Huang and Webb [3] in 1969 analyzed the heavily overdamped waves (relaxation times up to 20 s) in critical mixtures of cyclohexane and methanol using known values of a and Ap and demonstrated that the viscosity did not diverge strongly in conformity with then new scaling arguments of Kadanoff and Swift [13].

At about the same time Bouchiat and Meunier reported measurements on CO, that have given 7, o and Ap assuming V = q,/p, = qg/pg with no strong temperature variation of q [14]. Notice that the kinematic viscosity ?

so defined is the appropriate average viscosity for critical Auids. In this conference Bouchiat and Meu- nier discuss a correction to the usual dispersion relations [15]. This correction is most important near critical damping where previous data had looked peculiar.

We have now carried out measurements of the spectrum of laser light inelastically scattered from SF, near its critical point to determine a and %. In this paper we report results of only a small part of Our data obtained from direct measurements of the autocorrelation function and the frequency spectrum of the heterodyne signal obtained by beating the light scattered by interfacial waves with a strong local oscillator at the laser frequency that was generated by a silicone grease coating added to the pressure ce11 window. A typical wave spectrum and corresponding correlation function is shown in figure 3. In the corre- lation spectrum, one sees the damped sinusoidal function corresponding to the damped thermally excited waves. The data analyzed cover the temperature range 0.112 < AT < 12.71 and the wave vector range 100 < q < 1 500 cm-'.

Analysis of additional data beyond these ranges

is still in progress. Most of Our data contains a mixture

Cl-152 E. S. WU AND W. W. WEBB

The previously mentioned correlation correction [153 has not been incorporated in Our present preliminary analysis since it is nearly negligible in the range of the data presented, i. e., ï/o 4 1. It is however necessary to retain the gravitational wzve term that appears as the first term in the numerator of the definition of y.

A least squares fit of the experimental data yields the two quantities ï and o which are used for numerical solution of the dispersion relations, eq. (9), for Sv and Sr to obtain numbers for

and for y as defined in eq. (7a) and (7b) and thus for o and ?j = (q, + qg)/2.

In principal Ap could also be determined but is actually only checked since it is already known to greater accuracy than is attained here

The temperature dependent surface tension o(&) is plotted logarithmically in figure 4. The data as least FIG. 3. - Example of data on recorded intensity autocorrela- squares fitted are represented by

tion function and corresponding heterodyne spectrum of the

intensity. This 200 Hz full scale spectrum was computed with a o = go

Saicor SA1 51, 200 channel analyzer and the 100 ms full scale

correIation function was computed with a Saicor SA1 42,

(10)

100 channel correlator. q

155.2 cm-1 and A T

0.780

= (39.6 ' ^i::)

erg/crn2 of heterodyne and homodyne signals requiring an

analysis that was not completed in time for this report.

Thus the results reported here are subject to some future refinement.

Our procedure for fitting the data starts with the time dependent autocorrelation function rather than the frequency spectrum more commonly employed so is described briefly here. The autocorrelation function assumes the form

(6) with û = tan-' T/o and A = surface area where

and

We have defined the damping constant

s r

r = -

20

and frequency

as normalized values of respectively the real and imaginary parts of the complex function S = Sr + ^iSi

that are specified by the dispersion relation

-

p 2 [(1 + s)? - m Z T ] .

(PI + P ~ ) ~

Reduced

FIG. 4. - Logarithmic plot of the temperature dependence of the surface tension a obtained in preliminary analysis of cor-

relation spectra.

in the range 4 x < e < 4 x The asyrnrne- trical error estimate is reported in this preliminary analysis because it is possible that neglect of the homo- dyne term in the preliminary analysis may reduce both prefactor and exponent by the amount indicated.

This can be seen by fitting the data in the range of larger AT, i. e., 4 x IOd3 < E < 4 x IO-' which gives o = 3.42 e1.304 erg/cni2 with small statistical error.

The average kinematic viscosity is given by the equation

- v = 392 + 17(T, - T) -11 27 x 1 0 - ~ stoke (11)

THE LIQUID-VAPOR INTERFACE OF SULFUR-HEXAFLUORIDE NEAR THE CRITICAL POINT Cl-153 or for the shear viscosity Fj = V(p, + pg)/2

5 = 287 + ^{13(Tc - T) f 20} x IO-, poise . (12) The exponent p falls in the range of observations for other critical fluids. (See the forthcoming review by B. Widom [16], and [17] for an examyle.) Using this result for p and published data, we can test eq. (2) as follows :

Discussion.

Al1 of the factors in eq. (1) are known so to test for consistency we shall calculate a using eq. (1) and compare the result with the experimental value.

First the inconsistency of exponents can be, represen- ted by the difference

Values of a and p are given by Our work in eq. (10) and L' and v' by eq. (3) ; Ap and P were determined by Lastovka and Feke and previously by Saxman [5] as Ap = Ap, with

The values of (aplûu), = (aplap), Ë Y have been determined below Tc only for the liquid by Lastovka and Feke as [5]

and above Tc by Ford and Puglielli [6] from total light scattering as

Combining the exponents below Tc we find

Although error estimates are not given for y' and P,

it is not likely that A = O within the range of the estimated error with these data. The value of y' seems most suspect since it is uncommonly low ; setting it at 1.24, the upper limit of the experimental value [IO]

above Tc ; lowering p to its lower limit 1.28 ; lowering v' to 0.65, and raising 2/3 to .69 just makes A = O by stretching. We notice also that Fisher's equation for the exponent y describing the deviation from Ornstein-Zernicke theory gives an unusually high value for y assuming v = v'

The magnitude of the physical quantities can also be included to obtain by eq. (13) the computed value of ^c

o (calculated) = 63.5 ^{8 l . l} erg/cm2 (20).

which is to be compared with

a (experimental) ⁼ 39.6

erg/cm2 . _{(2 1)}

The calculated values of o are more than three times the experimental values in the measurement range. We might again suspect the compressibility data. However we might suppose that the discrepency is associated with the identification of L' (measured) as the correct length to insert in eq. (l), then apparently the L' of theory is too large compared to experiment. This is the same viewpoint that was reached above in comparing L' with the available data on r. A similar comparison can be made on Xe using the surface tension data of Zollweg et al., where experimental data on 5 can be compared with the value of L' calculated from o using eq. (1). In that case 5 (measured) = 1.4 L' (calc) in the range of measurement.

It is possible to say that Our results are consistent with Fisk and Widom only by exercising sufficient skepticism about the data of others. A clue that we can- not interpret is the enormous value of the exponent vy which could be indicating either important interna1 degrees of freedom (note that critical organic mixtures often have large values of y) or that y' is in error.

However Our measurements of o and L' reported here seem to be consistent with expectations from other fhids. It would be useful to have complete measure- ments of 5 for T < Tc and a repeat of the measure- ments of (dplap), as well as the final refined results of our measurements of a and the wave length dependence of the reflectivity to complete this analysis.

Finally, Our experimental values of the average viscosity and L' adds another facit to the problem created by the longstanding anomaly in the measured spectral width of the Rayleigh scattering above Tc in SF, [5]. The Rayleigh width TR is supposed to be given in the hydrodynamic regime by

where A is the thermal diffusivity, q the thermal conduc- tivity and C, the specific heat at constant pressure.

Kawasaki has shown that the critical contribution to A can be written in the hydrodynamic regime as

However it has been proposed that eq. (23) represents

only that part of the critical behavior of A due to the

singularity of 1 but that there is also a background

contribution to A/pCp identified as l,/pCp. Thus there

should be two contributions to ï,, one from the

background nondivergent part of 1, the rest from

eq. (23). Assuming L' = 5 below Tc and using 5

Cl-154 E. S. WU AND W. W. WEBB

measured by Ford and Puglielli above Tc with Our value of i j we have calculated A, obtaining

A, = 1.1 x 1 0 - 5 ( ~ c - ; T < Tc (24)

o Broun, Hamrner, Tschamuter and Weinzierl Saxman and Benedek A Lostovko and Benedek

Liquid ride of

__--

--.

5 4 3 2 1 0 1 2 3 4 5

FIG. 5. - Reproduction of data on the thermal diffusivity A deduced from the spectral width of Rayleigh scattering from SF.5 after Benedek et al. [5] with their calculation of the « back- ground» contribution to the thermal diffusivity A shown by a solid line and Our estimate of the singular part Ah

k~ T/6 zvt

shown by a dashed line.

and

These values are plotted on the now familiar figure of Benedek et' al. in figure 5. The data on T , are plotted as points. Benedek's calculations of the background are the solid lines and our calcillations of the singular part are the dashed lines. C1r:arly either contribution is nearly enough by itself above Tc. Below Tc it appears that the sum of the background and singular parts might be about right to account for the Rayleigh width fairly near Tc. Since this kind of approach has been reported to work very well for Xe and CO,, we must conclude that there is something wrong with Our knowledge of SF,. However this analysis probably demands more precision of the available data than is yet justified.

IV. Acknowledgements. - We are pleased to ack- nowledge helpful conversations with B. Widom and P. Hohenberg, with G. Benedek who suggested figure 5, and with G. Hawkins about the current state of the data on SF,. We are grarteful for support of this research by the National Science Foundation and the Advanced Research Projects Agency through the Materials Science Center at <:orne11 University.

References [l] GILMER (G. H.), GILMORE (W. C.), HUANG (J. S.)

and WEBB (W. W.), Phys. Rev. Letters, 1965, 14, 491.

[2] HUANG (J. S.) and WEBB (W. W.), J. Chem. Phys., 1969, 50, 3694.

131 HUANG (J. S.) and WEBB (W. W.), Phys. Rev. Letters, 1969,23,160.

[4] FISK (S.) and WIDOM (B.), J. Chem. Phys., 1969, 50, 3219.

[5] BENEDEK (G. B.), LASTOVKA (J. B.), GIGLIO (M.) and CANNELL (D.), Proc. Battelle Conf. on Critical Phenomena, Geneva, 1970.

161 VAN DER WAALS (J. D.), 2. Physik Chem., 1894, 13, 657.

171 CAHN (J. W.) and HILLIARD (J. E.), J. Chem. Phys., 1958,28,258.

181 WARREN (C.) and WEBB (W. W.), J. Chem. Phys., 1969, 50, 3694.

191 ZOLLWEG (J.), HAWKINS (G.) and BENEDEK (G. B.),

private communicatiom ; see also these confe- rence proceedings.

[IO] PUGLIELLI (V. G.) and FORD (N. C.), Phys. Rev.

Letters, 1970,25, 143.

[Il] KATYL (R. H.) and INGARID (V.), Phys. Rev. Letters, 1968,20,248.

[12] PAPOULAR (M.), J. Physique Radium, 1968, 29, 81.

[13] KADANOFF (L. P.) and SWIFT (J.), Phys. Rev., 1968, 166,89 ;

SWIFT (J.), Phys. Rev., 1969, 173, 257.

[14] BOUCHIAT (M. A.) and E EU NIER (J.), P h y ~ . Rev.

Letters, 1969, 23, 752.

[15] BOUCHIAT (M. A.) and MEIJNIER (J.), to be published, J. Physique, 1971.

[16] WIDOM (B.), to be published in

Phase Transitions and Critical Phenomena », IV. Domb and M. S. Green, Ed.

1171 SMITH (D. L.), GARDNER (P. R.) and PARKER

(E. H. G.), J. Chem. P,hys., 1967, 47, 1148.