THE LONG RANGE CORRELATION LENGTH AND ISOTHERMAL COMPRESSIBILITY OF CARBON DIOXIDE NEAR THE CRITICAL POINT

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THE LONG RANGE CORRELATION LENGTH AND ISOTHERMAL COMPRESSIBILITY OF CARBON

DIOXIDE NEAR THE CRITICAL POINT

D. Cannell, J. Lunacek

To cite this version:

D. Cannell, J. Lunacek. THE LONG RANGE CORRELATION LENGTH AND ISOTHERMAL

COMPRESSIBILITY OF CARBON DIOXIDE NEAR THE CRITICAL POINT. Journal de Physique

Colloques, 1972, 33 (C1), pp.C1-91-C1-95. �10.1051/jphyscol:1972117�. �jpa-00214907�

JOURNAL DE PHYSIQUE

Colloque CI, supplkment au no 2-3, Tome 33, Fkvrier-Mars 1972, page C1-91

THE LONG RANGE CORRELATION LENGTH AND ISOTHERMAL COMPRESSIBILITY OF CARBON DIOXIDE NEAR THE CRITICAL POINT

D. S. CANNELL and J. H. LUNACEK

Department of Physics, University of California, Santa Barbara, California, 93106

RBsumB. -

Nous avons mesure la dependance angulaire et lYintensit6 totale de la lumihre diffu- s6e par le gaz carbonique prhs du point critique, montrk que la theorie COrnstein et de Zernike est inadaptee et determine la dependance en tempkrature de la compressibilitb isotherme

ainsi que la longueur de corrklation <. Les exposants critiques relatifs

et

< sont, au-dessus de la temp&rature critique,

1,219

0,01 et

0,633

0,01 respectivement. L'exposant dkcrivant l'ecart

la theorie COrnstein et Zernike est

0,074

0,035.

Abstract.

- We have measured the angular dependence and total intensity of light scattered by carbon dioxide near its critical point, shown that the Ornstein-Zernike theory is inadequate, and determined the temperature dependence of the isothermal compressibility

and the long range correlation length <. The exponents describing the divergencesof

and t, above the critical temperature, are

1.219

0.01 and

0.633 k 0.01 respectively. The exponent describing the departure from the Ornstein-Zernike theory is

0.074

0.035.

It has been predicted by Ornstein and Zernike [I], and verified experimentally [2], that to a high degree of accuracy the angular distribution of the intensity of light scattered by a pure fluid near its critical point is given by the expression

where

is the isothermal compressibility, A is a constant which can be evaluated at any temperature,

is the angle between the electric field of the incident light and the scattered wavevector, and the scattering wavevector K

2 k, sin(0/2), where k , is the wave- vector of the incident light in the fluid and 0 is the scattering angle. The above expression serves to define the 0 - Z correlation length 5. As the critical point is approached both ~2 and

diverge. The 0 - Z theory predicts the4exponents

and y, describing the diver- gence of t and rc, to be related by 2

Since both

5 and

can be determined independently by measu-

ring the asymmetry and total intensity of light scattered by the fluid, such measurements offer an ideal method of checking the exact validity of the theory. In addition accurate measurements of the correlation length t

can be combined with measurements of the spectral width of the Rayleigh component [3] to check recent predictions by Kawasaki [4] concerning the dynamic behavior of critical fluids.

We have made accurate measurements of both the difference and magnitude of the intensity of light scattered through the angles 0,

17.50 and

0,

168.50 as a function of temperature in carbon dioxide near its critical point. The measurements were made on the critical isochore over the temperature range 0.023 OC < T

T, < 10.00 OC. In addition we measured the total scattered intensity over essentially the same range in temperature. Since the anisotropy in the scattered intensity becomes very small at higher temperatures [2], the measurements were made using a two channel differential technique with the appa- ratus shown in figure 1.

The scattering cell was constructed of beryllium copper and optical glass. Indium was used as the sealing material. Its temperature was controlled to

0.001 OC, and measured using a platinum resis- tance thermometer calibrated by the National Bureau of Standards. The cell was filled to within 0.1 % of the critical density with carbon dioxide containing less than 50 p. p. m. of impurities. The density was determined by measuring the meniscus height as a function of temperature, using the data of Michels, et al. [5], for the shape of the coexistence curve. This technique also serves to determine the critical tempe- rature, and yielded the value T,

31.000 + ^{0.001 OC.}

The critical temperature was also determined by observing the disappearance of the meniscus. This occurred between 31.000 OC and 31 .0010C. We thus took T, as 31.000 OC. The apparatus as shown could be used to make either differential scattered intensity or total scattered intensity measurements. The laser beam was spatially filtered and focused in the tem- perature controlled scattering cell. A differential scattered intensity measurement was made by removing

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972117

C1-92 D. S. CANNELL AND J. H. LUNACEK

FIG. 1. - Schematic diagram of the apparatus.

the beam splitter and blocking the transmitted beam.

Light scattered through the angle OF was collected using the optics labeled L,, L,, and L,. The solid angle of collection was determined by the variable aperture A. The lenses L, and L, served to image the scattering cell on the slit S, thus determining that sec- tion of the beam from which scattered light was accepted. The collection optics for light scattered through the angle $ functioned in an identical manner.

The lenses L, and L; served to focus the scattered light from each channel onto the same spot on the photo- cathode of an RCA 7265 photomultiplier. The rotating chopper, CH, alternately accepted light from each channel. The photocurrent was passed to a lock-in amplifier whose reference was generated by the chopping wheel using a light bulb and photodiode.

The output of the lock-in amplifier was thus propor- tional to the difference in the intensity of the light collected by each channel. The intensity of light collected by either channel alone could be measured

by blocking the other channel.

In order to measure the total scattered intensity the beam splitter was replaced and the total attenua- tion of the incident beam due to the cell and the fluid was determined by comparing the reference and transmitted beam intensities 161. This was accomplish- ed by passing the beams through the chopper and allowing them to fall on a solar cell, whose output was then passed to the lock-in amplifier. The Glan- Thompson prism in the reference beam served as a

variable attenuator. The intensity of the reference beam was measured by blocking the transmitted beam.

In making either differential scattered intensity measurements or total scattered intensity measure- ments, the output of the lock-in amplifier can be written as

V = GF IF

G B I B (2) where GF and GB are the gains of the forward and backward channels. In the case of a differential scattered intensity measurement, I, and I, are the scattered power per unit solicl angle for the angles 8,.

and 8, respectively, and the gains are determined by the collection solid angles, lens, window and mirror losses, photomultiplier gain and the gain of the electronics. In the case of a total scattered intensity measurement I, and IF are the intensities of the inci- dent and transmitted beams i:n the cell, and the gains are determined by the reflectivities of the beam splitter, Glan-Thompson setting, solar cell sensitivity, lens, window and mirror loss,es, and electronic gains.

We now consider the measurements of the diffe- rential scattered intensity. For each temperature T i the ratio of the lock-in output with both channels open, to that with only the forward channel open is

where

and KF and KB are the scattering wavevectors corres- ponding to the angles 8, and 0,. For each pair of tem- peratures Ti and T j the ratio

was computed. Thus the measurements serve as an experimental determination of the set of ratios ft(Ti)/fr(Tj). If the correlation range is assumed to have the form to((T - T,)/T,)-", then the parameters to

and

can be determined from the measured ratios.

In order to obtain numerical values for the correlation range from each experimental point it is necessary to know the gain ratio GB/G', appearing in eq. (3).

This was obtained by using the best fit for

to calcu- late GB/GE using eq. (3), from data points at

The correlation range for all other data points was then calculated using this value for GB/GF. These experimental values together with the best fit to are shown in figure 2. Many of the experimental points which appear to be single points on the graph are the results of as many as five independent measurements.

A very similar procedure was utilized in interpreting

the measurements of the total scattered intensity.

THE LONG RANGE CORRELATION LENGTH AND ISOTHERMAL COMPRESSIBILITY C1-93

FIG. 2. -The long range correlation length of carbon dioxide along the critical isochore as a function of the reduced tempera- ture t = (T- To)/Tc. The triangles are from the X-ray data [ll].

In this case the &(Ti) used in the analysis of the data is given by the expression

where z(Ti) is the beam attenuation per unit length in the fluid, L is the pathlength in the cell, and I, and rF are the intensities of the incident and trans- mitted beam, in the cell. The effect of unequal trans- mission and reflection by the beam splitter, as well as the transmission losses due to the cell windows are included in the gains, GF and G,. The experimentally determined ratios thus take the form,

thereby providing a measurement of the difference in the attenuation per unit length, for any two tempera- tures. The attenuation per unit length

may be related to the isothermal compressibility by integrating eq. (1) over angles. The result is [6]

where a is 2 k i g2, and

is the wavevector of the incident light in the fluid. The quantity A is given by the expression A

[(n2 kB T/1:)

ac/ap)+], where

is the wavelength of the light in vacuum, kB is Boltz- man's constant, T is the absolute temperature, p is the density of the fluid and

the dielectric constant. The quantity [ ( p a ~ / d ~ ) + ]

[(n2 - 1) (n2

2)12/9 was cal- culated using the Lorentz-Lorenz relation and measur- ed values of the index of refraction [7], n. If the isothermal compressibility is assumed to have the form

then the parameters

and

can be determined from the measured ratios, however in order to do this it is necessary to know the correlation range. Figure

shows the fit and the experimental values for the

FIG. 3. -The susceptibility p,2 ^KT of carbon dioxide along the critical isochore as a function of the reduced temperature t = ( T - Tc)/Tc. The squares are from the static PVT data of [5].

The circles are our own determination of the absolute compressibility.

isothermal compressibility, which for convenience is presented as

The experimental points were obtained in the same manner as those for the correla- tion range. We also made an absolute determination of the compressibility by measuring the ratio of the forward scattered intensity at two different tempera- tures. As seen from eq. (1) this serves to determine the

of the isothermal compressibilities at those two temperatures. A different relationship involving the compressibility can be obtained by noting that the measured ratio R:j gives an absolute determination of the dzference in the attenuation per unit length for those two temperatures. In computing the ratio we were able to eliminate the effect of attenuation by the fluid, using our measured value for the diffe- rence in the attenuation at those two temperatures.

Using eq. (7) and the measured values for the corre-

lation range, this difference may be combined with

the ratio of the isothermal compressibilities obtained

from the scattered intensity to yield value for the

compressibility at these two temperatures. These

values are shown as open circles in figure 3. The ana-

lysis yields the following results for the long range

correlation length t, and the isothermal compressi-

bility

C1-94 D. S. CANNELL AND J. H. LUNACEK

The error bars given for t, and v, and for

and y,

were obtained by varying one parameter while adjust- ing the other so as to maintain a best fit, until the mean squared deviation of the fit from the experimental points was four times that of the best fit.

As may be seen from figure 2, the 0 - Z expression for the angular dependence of the scattered intensity accurately represents our data with the above power law dependences for t and

However, the theory also predicts that the exponents v and y describing the divergences of 5 and

are related by 2

Within the errors of our measurements this relationship is not satisfied. Consequently we conclude that the 0 - Z theory is inadequate to describe completely the long range correlation in a pure fluid near its critical point. This failure of the 0 - Z theory has been anticipated theoretically and a modified theory has been worked out [8]. The principal feature of the modified theory is the introduction of a new exponent

describing the departure of the angular dependence of the scattered intensity from that given by eq. (1).

In addition the relationship among the exponents becomes (2 - q)

y. This relation together with our values for

and

yields the result y

.074 + ^.035.

For the sake of consistency we reanalyzed our data using the modified theory and this value of

The resulting values for

and y differed from those given above by less than their experimental uncertainty, and the fits obtained in this manner did not differ significantly from those shown in figure 2 and 3.

Recent work by Kawasaki [4] has resulted in a closed form expression for the critical part rc of the Rayleigh linewidth r,

^{Ak2/(p, C,)} of a pure fluid near its critical point,

where

is the shear viscosity and T is the absolute temperature. The above equation was derived using the 0 - Z form of the correlation function, but for our range of Kt is very insensitive to a nonzero y < 0.1.

Using measured values for the thermal conducti- vity

A, it is possible to obtain values for the cri- tical part of the Rayleigh linewidth r c , from the measured values [3] r,. ^{Since y,.} has been measur- ed [lo] for CO,, our measurements of < provide the means of checking eq. (8). We find the predicted values of rc to be in excellent agreement with the measured values, the difference being less than 15 % in the range 0.023 OC < T - Tc < 4.42 OC with the predicted values being the smaller. We feel that the agreement is well within the experimental error for the linewidth measurements and corrections.

The primary factors affecting the accuracy of the differential scattered inten-sity measurements are, sta- bility of the optical system and detector, stray light, multiply-scattered light, heating of the sample by

the beam, attenuation of both the beam and the scat- tered light by the fluid, and noise due to the intrinsic fluctuations in the scattered intensity.

The stability of the optical system was enhanced by mounting the entire system

a massive cast iron table. The scattered light collected by each channel was imaged to the same spot on the photocathode of an RCA 7265 photomultiplier tube. In addition the optical system was designed so as to render the image positions insensitive to angular deviations of the mirrors. It was also necessary to temperature control the photomultiplier tube. The resulting system was sufficiently stable that the value of the difference in the scattered intensities measured at any particular temperature was reproducible to within f 0.05 % of the scattered intensity, throughout the course of the experiment.

The systematic error caused by the collection of stray light increases as T

Tc is increased, due to the reduction in the scattered intensity. We estimated its effect by extending our measurements to

where the scattered intensity was one-half of that at T - Tc

10.0 OC. We then attributed all of the difference between the measured correlation range and the value obtained by extrapolating our fit, to the effect of stray light. The one-sided error bars on the three highest temperature points in figure 2 represent the possible effect of stray light at these temperatures, as calculated in this manner. At T - Tc

4.42OC the stray light contributed less than .03 % to the observed asymmetry of 0.23 %. The extremely low level of stray light was achieved by orienting the cell in such a manner that the background viewed by either set of collection optics was the opposite cell window, viewed in a plane containing no specular reflections.

For temperatures close to the critical the primary experimental difficulty arises from the collection of multiply-scattered light, which does not have the same angular distribution of intensity as does the singly scattered light. At T - Tc

0.023 OC we estimated the contribution of this effect to our measurement, by observing the asymmetry and intensity of the multiple scattering occurring immediately above the beam. If we correct for this effect the point at T - Tc

0.023 OC in figure 2 is moved to the point indicated by the X.

In order to eliminate any effects due to the heating of the fluid by the laser beam we attenuated the inci- dent beam until the quantities being measured were independent of beam power.

A systematic error present at all temperatures may

arise due to the attenuation of the beam, and the

scattered light, by the fluid. This error arises if the

path length in the cell is not exactly the same for both

channels. Fortunately, for our geometry this effect

changes all the measured values of t by essentially

THE LONG RANGE CORRELATION LENGTH AND ISOTHERMAL COMPRESSIBILITY C1-95

the same percentage, the maximum difference in the changes being less than 0.5 % and the changes being less than 4 %.

In making measurements of the scattered intensity or differential scattered intensity the principal contri- bution to the noise on the measured signal arises from fluctuations in the scattered intensity, and from shot noise generated by the photomultiplier. The intensity fluctuations are intrinsic, and are associated with the spectrum of the scattered light. As the critical point is approached the intensity fluctuations occur ever more slowly. We have calculated the signal to noise ratio

in terms of the observed photocurrents for the two channels,

and i,, where 6(iF - i,) is the RMS fluctuation in the difference. We obtain the result

S -

-

- ^(i, - i,) 4%

(9) [&(is + ^is) + ^-

where G in the photomultiplier gain,

the electronic charge, NF and NB the number of coherence solid angles subtended by the forward and backward chan- nels, rF ^and rB the spectral widths of the light scat- tered through the angles 6, and 8,, and To the averag- ing time. Due to the temperature dependence of

iB, rF and r B , the signal to noise increases rather strongly as the critical is point approached, until

Kt

1 for scattering at the angle 8,. Our observed signal to noise agreed with this expression, and by collecting many coherence solid angles we were able to eliminate signal to noise problems even at tempera- tures far above T,.

In conclusion, our measurements of the long range correlation length and isothermal compressibility of C 0 2 , indicate that the Omstein-Zernike theory is inadequate to describe completely the form of the long range correlation function in a pure fluid near its critical point. Our measurements together with the exponent relation (2 -

lead to the result

0.074

0.035. In addition, our measurements,.

together with measurements of the thermal conduc- tivity [9], Rayleigh linewidth [3] and shear viscosity [lo]

show that the critical part of the Rayleigh linewidth is adequately described by the dynamical theory due, to Kawasaki

Acknswlegements.

The authors wish to thank Professor Richard A. Ferrell for many valuable sug- gestions concerning the interpretation of our results.

We also acknowledge with thanks stimulating conver- sations with Dr. Marzio Giglio, and wish to extend special thanks to Professor Stuart B. Dubin who aided us immeasurably in the original design and testing.

of the apparatus. We also wish to thank Professor George B. Benedek for providing us with the tempe- rature controller used in the experiment. We were aided immeasurably in our efforts by the skill a n d meticulous craftsmanship of Mr. Hans R. Stuber.

References [I]

ORNSTEIN

and ZERNIKE (F.),

Amsterdam, 1914, 17, 793 ; Physik Z., 1918, 19, 134.

[2]

GIGLIO (M.) and BENEDEK

B.),

1969,23,1145.

[3]

SWINNEY (H.

and CUMMINS (H.

[4]

K A W ~ ~ A K I

[5]

MICHELS

BLAISSE

and MICHELS (C.),

Roy. Soc., 1937, A 160,358.

[6]

PUGLIELLI

and FORD

C.),

Letters, 1970, 25, 143.

[7]

STRAUB

Ph.

Thesis, Technische Hochschule,.

Miinchen,

r81

FISHER (M.

i9j SENGERS

(J. v.) and KEYES

[lo] KESTIN (J.), WHITELAW (J. H.) and ZIEN

[ l l ]

CHU

LIN (5. S.) and DUISMAN