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NBS TECHNICAL NOTE 925

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

Overviews and Justifications

for Low Gravity Experiments on

Phase Transition and Critical

Phenomena in Fluids

QC

100

.15753

no. 925!

1976
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Uonal Bureau of Standards

IT ~ o 1876

Overviews and Justifications for Low Gravity

ao Experiments on Phase Transition and

5^3 Critical Phenomena in Fluids

^^^ ^

e , ^. Q

(o

M. R. Moldover and R. J. Hocken

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

and

R. W. Gammon and J. V. Sengers

Institute for Molecular Physics

University of Maryland

College Park, MD 20742

Sponsored by

NASA-Lewis Research Center 21000 Brookpark Road

Cleveland, Ohio 44135

>f " *

°*£AU Of

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson, Secretary

Edward O. Vetter, Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

VA .S. NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Acting Director

Issued October 1976
National Bureau of Standards Technical Note 925

Nat. Bur. Stand. (U.S.), Tech. Note 925, 99 pages (Oct. 1976) CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1976

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. C13.46:925). Stock No. 003-003- Price $1 .70 (Add 25 percent additional for other than U.S. mailing).
TABLE OF CONTENTS

Page No.

1. Phase Transition and Critical Phenomena in Fluids:

Scientific Questions and Interactions with Other Disciplines. 1

1.1 Introduction. i

1.2 The Relationship of "Phase Transition and Critical

Phenomena in Fluids" to Other Disciplines. 3

1.3 Important Questions in Phase Transition and Critical

Phenomena in Fluids. 7

2. Opportunities Provided by a low-g Environment. 12

2.1 Introduction. 12

2.2 Averaging Errors in "Bulk" Experiments. 15

2.3 Limitations on Optical Experiments due to Gravity Induced

Refractive Index Gradients. 27

2.4 Light Scattering Measurements and Some Gravity Related

Limitations. 39

3. Limitations in critical-regional experiments due to

modifications of fluid properties by a gravitational field. 52

4. Conclusions. $0

Appendix A. Parametric equations of state for fluids near the

critical point. 64

Appendix B. Calculation of density profile. 74

Appendix C. Calculation of correlation length. 77

Appendix D. Physical constants for various fluids. 86

Appendix E. Parameters used for xenon in this report. 87

Nomenclature 89

References 93

iii

Overviews and Justification for Low Gravity Experiments

on Phase Transition and Critical

Phenomena in Fluids

M.R. Moldover, R.J. Hocken, R.W. Gammon,

and J.V. Sengers

Important scientific questions concerning pure fluids and fluid

mixtures near critical points are identified and are related to the

progress of several disciplines. Consideration is given to questions

about thermodynamic properties, transport properties, and the complex

nonlinear phenomena which occur when fluids undergo phase transitions in

the critical region. We discuss, quantitatively, the limitations to

answering these questions by experiments in the earth's gravitational

field. The distinction is made between practical limits which may be

extended by advances in technology and intrinsic ones which arise from

the modification of fluid properties by the earth's gravitational field.

The kinds of experiments near critical points which could best exploit

the low gravity environment of an orbiting laboratory are identified.

These include studies of the index of refraction, constant volume

specific heat, and phase separation.

Keywords: Critical point; gravity effects; phase transitions.
1. Phase Transition and Critical Phenomena in Fluids ; Scientific Questions

and Interactions with Other Disciplines .

1.1 Introduction

The justification for space experiments in the area of "Phase Transition

and Critical Phenomena in Fluids" is essentially scientific. This particular

area of science has strong interactions with solid state physics, fluid physics

and other areas of physics. We may expect that scientific advances in these

areas will bring, indirectly, long term technological and economic benefits;

however, it cannot be said that the space experiments in themselves will produce

such benefits. Therefore the justification for space experiments in

phase transition and critical phenomena in fluids depends upon first establishing

that this is an important, active field of scientific research and secondly,

upon establishing that space experiments are likely to have a major impact on

its further development. A large part of the importance of this area of

research results from its initimate relationship to many areas of solid state

science. This will be discussed in the next section. In order to develop a

rationale for the argument that space experiments can have a major impact in

this area of research, we first discuss the important unanswered questions on

the subject; we then describe how gravity imposes severe limitations in

finding experimental answers to these questions; and, finally, we indicate to

what extent these limitations can be reduced by conducting experiments in

space.
1.2 The Relationship of "Phase Transition and Critical Phenomena in

Fluids" to other Disciplines .

Many of the research subjects described as included in research on

"phase transition and critical phenomena in fluids" are intimately related to

similar subjects in solid state physics. The experimental phenomena observed

and the theoretical ideas used to describe both fluids and solids have a high

degree of overlap. Indeed, this overlap is recognized by the "Physics and

Astronomy Classification Scheme, 1975" adopted by the Abstracting Board of the

International Council of Scientific Unions. This classification scheme,

which is used for grouping scientific articles for publication and information

retrieval, groups "phase equilibria, phase transitions and critical points"

together for both solid and fluid systems. We expect that important experiments o

fluid systems, such as those to be defined in this study, will have a substantial impact on corresponding areas of solid state science. The basis for our

expectation is the strong interrelationship in recent history between the

study of phase transition and critical phenomena in fluid and in solid systems.

The interrelationship arises from analogies between many phenomena which occur

near the critical point of the liquid-gas phase transition in pure fluids and

phenomena which occur at a corresponding "critical point" of phase transitions

in very different kinds of physical systems. A list of these different kinds

of physical systems would include ferromagnets near the Curie point, antiferromagnets near the N£el point, certain ferroelectrics near the ferroelectric-paraelectric phase transition, alloys exhibiting order-disorder

transitions (e.g. 8-brass) , crystals exhibiting order-disorder structural

transitions (e.g. ammonium chloride), binary liquid mixtures near consolute

points, multi component fluid mixtures near plait points, and helium near its
superfluid transition. (For general references on these subjects see:

Elcock, 1956; Stanley, 1971; Heller, 1967; Fisher, 1967).

In each of these physical systems, there is a "quality" (such as the

difference between the densities of the liquid and vapor phases of a fluid)

which distinguishes between two phases which coexist in thermodynamic equilibrium. As the critical point is approached this quality gradually disappears.

The gradual disappearance of this quality is the most obvious feature which

distinguishes phase transitions with critical points from other sorts of phase

transitions such as melting, sublimation, or crystallographic phase changes.

The diverse physical systems with critical points have a number of

experimental properties in common near their respective critical points. They

all exhibit a specific heat anomaly, long thermal relaxation times, a marked

sensitivity to external fields and impurities, and the quality which is

disappearing shows large long-lived fluctuations. The density fluctuations in

a pure fluid are responsible for the striking visual effects called critical

opalescence. Analogous fluctuations in solid systems manifest themselves in

ways such as enhanced neutron scattering, ultrasonic attenuation, and

electrical resistivity.

The existence of diverse physical systems displaying analogous critical

phenomena provides the experimenter with the opportunity to choose both the

best material and the best technique for studying one or another aspect of

critical phenomena. For example, the size of fluctuations is rather easily

studied in magnetic solids with neutron scattering, even very far from the

critical point. The same technique cannot be used extremely close to the

critical point because the fluctuations become much larger than the wavelength

of the neutrons available. On the other hand, fluctuations in fluids may be
studied easily quite close to the critical point using scattered light. We

wish to emphasize that extremely close (often near 99.9% of the

critical temperature) to critical points, all experiments on solids are limited

by lattice strains which result from impurities, vacancies, etc. Thus

experiments with fluids (which continuously "anneal" themselves) offer the best

opportunities for closest approach to the critical point.

Since critical phenomena occur in a wide range of systems, one might expect

that their theoretical explanation does not depend upon the detailed nature of

the interatomic forces in each system but rather, could be based upon an

accurate treatment of interactions which contain only those few general

properties of the true interatomic forces which are needed to make critical

points occur. Accordingly, similar theoretical treatments of microscopic

interactions have been used to interpret data on these diverse systems. The

van der Waals model of a fluid, molecular field model of a ferromagnet, and

the Bragg-Williams model of a binary alloy are closely related in approach and

yield similar predictions concerning thermodynamic properties near the critical

point of each of the corresponding systems. There is an exact correspondence

between the lattice gas model of a fluid and the Ising model of a ferromagnet.

This correspondence has been exploited as a plausibility argument for taking

a wide variety of theoretical results based on lattice models of critical

phenomena and applying them to describe real fluids as well as real solids

near critical points. Interestingly, several important recent advances in the

theory of lattice models are outgrowths of the "renormalization group"

technique, a technique developed for problems in theoretical high energy

physics (Ma, 1973).
Other aspects of phase transition phenomena in liquids have solid state

counterparts. Away from the critical point itself, phase transitions in both

fluids and solids may be initiated by nucleation processes or may occur

spontaneously through a spinodal decomposition mechanism. The theoretical

description of these processes' make use of the concepts of interfacial energy,

bulk free energy, and diffusion. The concepts are applicable to both fluid

and solid systems. The time scales for phase changes are very different for

fluid and solid systems, thus facilitating complementary experimental studies

which benefit the understanding of both fluids and solids. (Cahn, 1968;

Langer and Bar-on, 1973; Schwartz et al., 1975).

In summary, basic scientific studies of phase transitions and critical

phenomena in fluids are closely related to the study of similar phenomena in

solid systems. It is also quite likely that future developments in chemical

engineering will exploit the progress now being made in the understanding of

thermodynamic and transport properties of fluids near critical points. In the

study of pure fluids, equations of state for the critical region have been

developed recently which are much more accurate representations of data than

are standard engineering equations. (Levelt Sengers e_t al. 1976). These new

representations have very few parameters which must be adjusted for each fluid;

thus, they require fewer experimental measurements for reliable predictions.

The extension of these new equations to special mixtures has been demonstrated

and work on extending them to mixtures of engineering interest is in progress.

(Leung and Griffiths, 1973; D'Arrigo et al. , 1975). Similar advances have

occurred and are occurring in the correlation and prediction of transport

properties in the critical region. Thus we can expect that advances in the

scientific understanding of phase transition and critical phenomena in fluids

will influence chemical engineering practice.
1.3 Important Questions in Phase Transition and Critical Phenomena in

Fluids .

We will consider some of the important scientific questions in the

area of phase transition and critical phenomena in fluids which are amenable

at least to a partial answer by experiment. Naturally this cannot be done

without some reference to current theoretical ideas. We will first discuss

pure fluids and then fluid mixtures. Perhaps the single most important

question in this area is: exactly how much alike are the superficially

analogous phenomena occurring in the large variety of systems showing critical

points? Rigorous renormalization group calculations which apply to large

classes of model systems indicate that static correlation functions and all

the thermodynamic properties (each of which can of course be calculated from

the correlation functions) are "universal" (in the sense that the same

description applies to each) asymptotically close to the critical point. More

specifically, they predict that the correlation functions will depend upon the

dimensionality of the system under consideration (e.g. whether it is two

dimensional like a membrane or three dimensional like a crystal) and the

dimensionality of the "order parameter" (or the "quality" which vanishes as

phases become identical at the critical point). Thus the correlation

functions for a fluid (where the order parameter is the scalar difference in

density between coexisting phases)' will differ from correlation functions for

isotropic magnets (where the order parameter is a three-dimensional vector)

.

(Wilson and Kogut , 1974).

Variables such as the lattice structure (e.g. hexagonal close-packed or

cubic) and the presence or absence of second nearest neighbor interactions

etc. are expected to be irrelevant in determining the functional
form of the asymptotic expansions of correlation functions and thermodynamic

functions; however these other variables will determine the numerical value of

the critical temperature itself and numerical values of various amplitudes in

the expansions for correlation and thermodynamic functions. At the present

time, nearly all thermodynamic experiments on pure simple fluids indicate small,

but experimentally significant, differences between the measured properties and

those calculated for three dimensional lattice models with a scalar order

parameter. (Levelt Sengers and Sengers, 1977). It is possible that these

differences indicate that fluids do not belong to the same "universality" class

as the lattice models with a scalar order parameter. It is also possible these

differences indicate that the amplitudes of the corrections to asymptotic behavior

are quite different for fluids than for the lattice models studied. An experimental

distinction between these two possibilities would be of great value in understanding

the range of applicability of this important idea of universality. It is possible

that this distinction could be made by measurements of thermodynamic properties

closer to the critical point than is now possible in experiments carried out

in the earth's gravitational field. The very same situation exists to a lesser

degree when different pure fluids are intercompared. The apparent differences

between fluids are smaller (hence subject to greater experimental uncertainties)

than the differences between fluids as a class and lattice models; however, the

same question exists regarding their origin. Again, the answer could result

from experiments carried out closer to the critical point than presently

possible.

The decay of the range of density fluctuations (or the pair correlation

length) may be measured by measuring the angular distribution of light,

X-rays, or neutrons scattered by a fluid. The temperature and density

dependence of this quantity is predicted to be universal, but a significant

8
test of universality cannot be made until two conditions are met: 1) The

wavelength of the incident radiation must be short compared with the range of

correlation. 2) The wavelength of the incident radiation must be long compared

with the range of the interatomic forces. Neutron and X-ray scattering

experiments easily satisfy the first criterion; however, the second criterion

is not well satisfied in present experiments. Thus, a somewhat arbitrary

separation of the observed scattering into a part due to critical fluctuations

and a part due to short range order in the fluid is now required prior to a

test of the predictions of universality (as defined above). The best

experiments to date seem to indicate significant differences between the

predictions of lattice models and the behavior of real fluids (Warkulwiz ejt al . ,

1974; Lin and Schmidt, 1974). Experiments with scattered light are now unable

to satisfy the criterion mentioned above that the wavelength be much smaller

than the correlation length. The correlation length increases rapidly as the

critical point is reached. In a low gravity environment, the critical point

could be approached much closer than is now possible; thus a much tighter test

of this aspect of universality will be possible.

Other questions which may be answered by experiments defined within this

study pertain more specifically to aspects of fluid dynamics and structure.

Hence the answers are less likely to have as broad an impact on condensed

matter science than would a clear answer to the question of universality.

Nevertheless, the properties of fluids are sufficiently alike at critical

points that questions about large classes of fluids may be answered with an

experiment on one or two fluids.

Questions of considerable interest are: What is the nature of the small

anomalies which occur in the viscosity (Sengers, 1973), dielectric constant
(Hartley et al. , 1974), refractive index (Hocken and Stell, 1973; Stell and

Hoye, 1974; Bedeaux and Mazur, 1973), and diameter of the coexistence curve

(Weiner e_t al. , 1974) (i.e. the average of the liquid and vapor densities) as

the critical point is approached? There are theoretical and experimental

controversies on the nature of the anomaly for each of these properties. It

is reasonable to believe that the more definitive experimental results

available from low-g experiments would have a significant impact on the

theories for these properties in fluids. The experimental nature of the

stronger critical point anomalies in sound attenuation (Thoen and Garland,

1 1974) and thermal conductivity (Sengers, 1973) is somewhat better understood

from earth based experiments, hence, low-g environments are less likely to

have an impact on the theory of these properties.

The dynamics of the process of macroscopic phase change in pure fluids is

poorly understood in the vicinity of the critical point. A variety of

questions remain to be answered. We will briefly mention a few. There is at

least one observation which suggests standard nucleation theories are

fundamentally wrong near the critical point in pure fluids (Huang et_ al_. , 1975)

Is this observation correct? Other earthbound experiments are necessary in

this area, but only in a low gravity environment will it be possible to obtain

a homogeneous, macroscopic volume of a fluid under conditions sufficiently

close to critical to yield a definitive study of this problem. The spinodal

decomposition mechanism of phase separation occurs in alloys, glasses, and

binary liquid mixtures (Schwartz et^ al . , 1975). Does this process occur in

pure fluids? If it does, it is quite likely that the study of spinodal

decomposition in pure fluids (where many of the macroscopic parameters of the

theory are well understood) will clarify the understanding of spinodal

10
decomposition in these other kinds of systems. Are there critical point

anomalies in the process of bubble and droplet growth (or evaporation and

condensation)? Do they influence the time of equilibration of macroscopic

two-phase samples of pure fluids as has been suggested in two publications?

(Dahl and Moldover, 1972; Brown and Meyer, 1972). What will be the dominant

mechanisms for macroscopic phase separation upon cooling a pure fluid below

the critical point if bouyant forces on bubbles and droplets are greatly

reduced? Many other questions are possible in this poorly understood area

of phase separation.

In fluid mixtures, important scientific questions analogous to the ones

above exist. Specifically, the question: Do real fluid mixtures have the

same "universal" thermodynamic and correlation function behavior as lattice

models and/or pure fluids, is unanswered. This question is more difficult to

answer for mixtures because they have additional thermodynamic variables. For

example the "quality" which disappears at the critical point may be thought of

as being either a mass density difference or a composition difference between

coexisting phases. In principle, if the critical point is approached closely

enough, either of these variables (or certain others) could be used to answer

the question of universality. Questions analogous to those above may be asked

about weak anomalies and transport properties in fluid mixtures. The role of

gravity as an experimental limitation to answering these questions is much less

clear in the case of mixtures than in the case of pure fluids. This will be

discussed briefly below.

11
2. Opportunities Provided by a low-g Environment .

2.1 Introduction

In this section we will discuss in detail the manner in which gravity

affects phase transition and critical phenomena experiments in fluids. We will

see that gravity limits the closeness with which the critical point may be

approached in all earthbound experiments. Thus a low-g environment will

provide an opportunity for conducting experiments closer to the critical point

than is possible on earth.

Measurements of the equilibrium properties of pure fluids near the critical

point can in principle encounter two distinct kinds of limitations because of

the earth's gravitational field. One kind of limitation is essentially a

technical one. All experiments measure average properties over some finite

height. Since the variation of fluid properties with height becomes

increasingly large as the critical point is approached, this averaging causes

increasing errors as the critical point is approached. In practical cases an

important averaging error occurs even for optical measurements (which at first

thought might be expected to average over a height of a wavelength of light

~0.5 ym) . The size of this "averaging error" depends both upon the property

measured and the technique used to measure it. We will consider representative

cases below. We will find that for nearly all experiments the closeness of

approach to the critical point is limited by averaging errors. These errors

may prevent the answering of many of the questions we raised in the last

section.

A second kind of limitation to accuracy in the measurement of equilibrium

properties of pure fluids which is imposed by the presence of the earth's

gravitational field is an unavoidable or "intrinsic" limitation. Relatively

12
simple considerations show that close enough to the critical point the

correlation length (or the size of the density fluctuations) for the fluid is

limited in the earth's gravity field much as it would be limited if the fluid

were placed in a small container. This means that the properties of the fluid

itself are altered by the gravitational field. No improvements in experimental

technique will enable earthbound experiments to overcome this limitation. To

date there are no experiments which are limited by this phenomenon, but it

appears that this limit will be approached in the near future.

Gravity enters into the study of the dynamics of phase changes in pure

fluids and most fluid mixtures by causing relative motion of the two phases

which are almost invariably of different densities. Thus in a 1-g environment

a bubble or droplet will rapidly travel to the top or bottom of a macroscopic

sample after it has grown to a size of the order of 10 um. There are a

variety of "levitation" techniques for studying droplet and bubble growth

which circumvent sedimentation. Unfortunately, none have been demonstrated as

being appropriate for studies of pure fluids near critical points where precise

control of temperature and pressure are also required.

If the study of binary liquid mixtures were conducted at equilibrium near

their critical (consolute) points, "averaging errors" similar to those which

occur in pure fluids, would appear. In practice, most binary mixture studies

have been conducted at constant temperature but not in diffusive equilibrium

so that a different kind of error is present. The diffusion constant tends

towards zero at the critical point and diffusive equilibrium may take days or

weeks in samples a 1 cm high moderately close to the critical point. It has

recently been shown that the "pressure diffusion coefficient" diverges at the

consolute point of binary liquid mixtures (Greer et> al .

, 1975; Giglio and

13
Vendramini, 1975). This divergence indicates "averaging" errors of a different

kind may be encountered in liquid mixtures.

We will now discuss these experimental limitations more quantitatively

and in more detail.

14
2.2 Averaging Errors in "Bulk" Experiments .

As a first and most simple illustration of averaging errors we now

consider a hypothetical experiment to measure the density of a pure fluid as

a function of height exactly at the critical temperature. Such a measurement

together with the thermodynamic relation for the gravitation contribution to

the chemical potential,

du = - mg dz (2.2.1)

(where g is the acceleration due to gravity, m is the molecular mass, z

is the height coordinate increasing upward, and u is the chemical potential

per particle) would make it possible to test the predictions of many model

equations of state.

These model equations of state all predict (see appendix A) a critical

isotherm of the form

\i-M

6 P D

c

C P c

P-P

c

p c Sign (p-p ) (2.2.2)

(Here u, P and p are the values of the chemical potential, pressure, and number

density respectively, and the subscript c indicates the critical value; D is

a numerical factor of order unity which varies from fluid to fluid and 5 is a

"critical exponent" which will be the same for all fluids if universality holds for

fluids in the critical region.) Fig. 2.1 shows the outcome of an ideal experiment oi

this kind with xenon. This figure was constructed using the numerical values for

the parameters listed in appendix D. Note that the density can change by more

15
p- p r IN %

p-Pc IN %

Fig. 2.1. The variation of the reduced density of xenon with height at the

critical temperature. The calculated curve is displayed on two different scales.

Note that near the critical density, a pressure change of 10~6 p will cause the

density to change 5% and a pressure change of 10~9 p will cause the density to

change 1%.

16
than 5% as a result of a pressure change, AP, which is only one one-millionth

of the critical pressure. At equilibrium this pressure change occurs (at

normal earth gravity, g ) with a height change from 0.25 mm below the meniscus

to 0.25 mm above the meniscus. We have chosen xenon for this numerical

illustration (and others in this report) because it is one of the few fluids

for which relatively complete data exist in the critical region. It is inert,

available in high purity and has a critical temperature near room temperature.

(C0„ and SF, are other well studied fluids which are likely candidates for

low-g experiments.)

One practical instrument for the measurement of fluid densities in thermal

equilibrium at g is a float densimeter (Greer jst al. , 1974) . In this

instrument the bouyant force on a float of known density is balanced with a

spring or magnetic restoring force. The deflection of the spring or the

current in the electromagnet is then a measure of p, the difference between the

average float density and the average of the fluid density over the float's

height. Quantitatively,

/p. , ..(z) a(z) dz

_ t luia /o o i\

/a(z) dz

where a(z) is the cross sectional area of the float as a function of height.

The outcome of a hypothetical experiment with a 1 mm high cylindrical float is

shown in Fig. 2.2 (left). The average density p (points in Fig. 2) differs

substantially from the local density at the average height, z, of the float as

soon as any portion of the float overlaps the meniscus. An alternative

17
presentation of the "data" is of interest. We may think of this experiment as

one which measures an average "susceptibility", xT

> of the fluid. This

susceptibility is thermodynamically related to the isothermal compressibility

and plays a role in the atudy of fluids which is analogous to the role of the

magnetic susceptibility near a magnetic critical point. By definition,

X = P Z ^ = Q Z ("y) %> = (^7) (2. 2. A)

In Fig. 2.2. (right) we compare xT

(z) with the average value of x computed from

our hypothetical experimental data by

mg \^z/

T

X - exp

- - I *Vl . (2.2.5)

It is interesting to note that the experiment which measures density by

averaging over a finite height is analogous to a spectroscopy experiment with

a finite resolution. According to Fig. 2.2, the density experiment has only

a limited ability to "resolve" the diverging susceptibility. Details smaller

than the resolution are blurred. Yet, some information may be recovered on a

scale smaller than the resolution if theoretical guidance is available. For

example, if the shape and relative spacings of the spectral lines present in a

spectral peak are known accurately in advance, their absolute spacing may be

determined to much greater accuracy than they can be resolved by the instrument

Similarly, if Eq. (2.2.2) were known to describe the density profile and

if techniques were of sufficient quality, accurate measurements of 6, D, and

18
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D* CO to Hro <j rt m. co n ^

OQ 11 3* Hi CO Co rt 3 OQ CO • 3 (D O CO rt cr1 w p. n> Hi O O H- c

rt 3 OQ 3

3* 3* o O rt rt Hi Hi H"

M> /-> ° l> OQ 3" Oi 3 O 3 (D O

11 H- rt o

(B OQ CO Hi

CD 3* '—' rt • 3*

Co ro H CO Hfl> 3 M OQ a, H- 3*

Hi OQ rt O rt Ca- ll 3d rr /~N

ro ••

O

X 3

in an 3 ro 3" 3 o n ro ro 3 H- \s < a

h- co ro 3 rt H.CO "6 3 » -

H- H- ex a.

a o\

rt < < N to ff ro co rt

ro rt D*

ro

O

hi

H- ro 2 <

co ro

n * 1 rt ro -^

rr CT ex

a c i

3" ro hi — o i

CO to rt ro < 3* ex

oq ro ro H 1 ex

Co Co fl ro

<! oq ro 3

h- ro ex CO rt ac Hco o rt

rt ex ro ^

h- ro ex O 3 K 3 co 3-

Co H- ro M rt 3 ^

Hi

Hro

HEIGHT (mm) o 1 1 1 I > m <

S>5 /•

—

H 3* / • - 2 • n • - o- • - •

r~oo ><m • > a

r~ > om2

COH

-<

mOm2

to

-<

HEIGHT (mm) o

19
p could be obtained with a float of, say, 1 mm height. Unfortunately, the

right hand side of Eq. (2.2.2) is only the first term in an expansion about

the critical density. The higher order corrections to Eq. (2.2.2) are not

known with certainty; thus, the interpretation of data of finite height

resolution becomes subject to the uncertainties arising from different choices

of correction terms. It is at this point that gravitational averaging makes

experimental tests of universality difficult.

A wide variety of realistic experiments have encountered the same density

resolution problems we have illustrated here with the hypothetical density

measuring experiment described above. For example Hohenberg and Barmatz [1972]

have analyzed in detail the effects of gravity averaging upon measurements

of the constant volume specific heat and of the low frequency velocity of sound.

It turns out that both experiments are strongly affected by gravitational

averaging at the critical temperature when the meniscus is within the

experimental cell (see Fig. 2.3). The shortest suitable calorimeters

constructed to date are about 1 mm high and the shortest low frequency

velocity of sound resonator is under 4 mm high, so that both of these

experiments will not resolve the density dependence of the quantity measured

within about 4% of the critical density at normal earth gravity.

The density resolution limit p scales with sample height, h, and gravity

according to the relation

p - p mg h p 1/6 1/6

1 p ' < d p

; { m g

; u./.b;

c c o

&

o

The scale factor for the density profile at the critical temperature, D P /(p m g ),

3 varies among 17 fluids (Appendix D) from 910 m in He to 10,400 m in H . For

20
°<j /><f*M*0 o

Q. 03 •H •—\

CJCO Q.1

t-l 3

Q. 60

^~/ •H CO >^ • ^— 4-1 03 /~* • •H •H CN o 03 x: r- c H on

a) i-H 13 s^

00 T3 <r oi o (1) • 00 u CM CM 6 3

-a ^ "i

a>

X vD

t-i 0) 03 4-1 • <a- > d > o• >• C 0) >H Pi o K O "0

03

O 03 00 03 P^ z. cfl 0) £. e X P-. Ll) 4-1 o Q •Hc rH 4-1 Q 3 CX Cfl UJ t-i •H £

4-1 S O a) rH CO 3 a 3 W

'fr

4-1 • O• UJ co fci X o or 0) X) 1 rH 3 o O CO •H e

M-l •^ bO 00 •H •-> M o

0J 3 X

dI a03 •H 3

0) >,3

aiu o

00

X

cO c u •H • OJ cu CO U *"~ > 4J • 6i

<:

X Ph O e o o

4-1

M • MH CO !-i • a) Td

CNJ X) 0) . >-( 4J • o ex

oc CO •H 3 X3

Ph

M CO

21
xenon, it is 1310 m, a fairly low value among those fluids which are likely

candidates for low g experiments. Because the exponent 6 is between 4 and 5

it is clear that a substantial reduction in gh is required to improve density

resolution at the critical temperature. Certain optical experiments

(whose limitations we will discuss below) have averaging heights of micrometers

rather than millimeters, thus optical experiments on earth become potentially

competitive with bulk experiments when the latter are carried out in an -3

environment of 10 g.

We have illustrated the density resolution limits at the critical temperatures arising from vertical averaging. The actual resolution is limited at

other temperatures as well. The shape of the resolution limited region is

shown in a qualitative fashion in Fig. 2.4. The true form of the region, of

course, depends upon the particular property being measured and upon the

techniques used. It is straightforward to estimate the extent of the gravity

affected region along the temperature axis as we have done for the density

axis. We will do so in a qualitative fashion here. Then we will indicate

how a more precise criterion for averaging errors could be used to define the

gravity affected region precisely. Such a precise criterion is formulated in

Appendix C.3.

A qualitative idea of the gravity excluded region can be obtained by

noting that the scaling equations of state (Appendix A) indicate that

asymptotically close to the critical point all thermodynamic quantities

(which usually are functions of two variables, say temperature and density)

may be expressed in the simple form rP f(6). Here r and are parametric variables

which are related to the temperature and density by nonlinear transformations.

Roughly speaking, r is a measure of the distance from the critical point and 9 is

a measure of the distance "around" the critical point from the coexistence curve. The

22
10"

5x10,-5

I-

-5x 10

,•5

r= 10"

30

LU

-20

-6 -2

(P p

c

)/p c in%

Fig. 2.4. Qualitative location of thermodynamic states in the temperature-density

plane which are inaccessible to bulk experiments because of gravitational

averaging. Averaging errors for a 0.3 cm high sample are substantial beneath the curve marked r = 10*"^. Averaging errors for a 0.03 cm high sample are substantial

beneath the curve r = 10~5.

23
exponent p depends upon the property under consideration, and f(6) is an

analytic function of 9 which also depends upon the property under consideration.

In those cases (including the important quantities C , K,^ and x ) where p <

and where f(8) is not a very strong function of 8, the r dependence of the

properties becomes dominant near T . It follows that divergent quantities will

assume approximately the same value along a curve of constant r and such

curves have been sketched in Fig. 2.4. It is also true that the gravity

excluded region for many experiments is also approximated by a curve of

constant r. We have already estimated the gravity limit for measurements of

X_ and C in a sample at the critical temperature, thus locating point B

on Fig. 2.4. To find the maximum temperature on the same contour of

constant r (point A on Fig. 2.4) we note that in one model equation of state

a contour of constant r is identical with a contour of constant xT (Appendix

A, Eq. A. 13). Thus in this model xT has identical values at B and A. (In

other realistic models the value of xT will be nearly the same at B as at A)

Now the power law expression for xT on the line Ap* = is

c c

We equate this expression for xT with the expression for xT along the

critical isotherm obtained by differentiating Eq. (2.2.2)

1-5

PcX

T _ _1_ i P L

P c 1 1-6 _ 1 / gh \ 6

p

2 D 6 p c c

1-6

= _1_ / gh \

D 6 WDH

o /

T - T = r( E.)~Y (2.2.8)

T

c 24
Using the values of the constants from the appendices we find that the temperature

limit for bulk susceptibility measurements is roughly 0.028 K for a 1 mm high

sample of xenon and that this limit scales with gravity and sample height as

(gh/g H ) • If a contour of constant r is used as a gravity limit criterion

o o other experiments will encounter gravity limits at some value of (T-T )/T which

also scales as (gh/g H ) .

From the work of Hohenberg and Barmatz [1972], the constant volume specific

heat of a 1 cm high cylindrical sample of xenon begins to deviate from the

specific heat of a zero height sample by several percent within 50 millikelvin

of the critical temperature; for a 1 mm high sample significant error will

occur within 12 mK of T . Naturally the exact temperature depends on the

criterion of accuracy, but with a fixed criterion of accuracy the temperature

n ft s of closest permissible approach to T will scale as (gh) *

. Thus a

substantial improvement in temperature resolution is_ possible by doing C

experiments in a low-g environment. In particular, the determination of T in

specific heat experiments would be greatly facilitated in a low g environment.

Instead of approximating the region of severe gravitational averaging by

a curve of constant r (as we have just done), a precise calculation could have

been done. The calculation would involve the following steps: 1) Identify

the quantity to be measured (say Q) and the precision with which the quantity

is to be known; frequently the desired precision may be expressed as a small

fraction, p, of the quantity Q. 2) Examine the measurement technique to

determine over what range of heights it averages (say ^ z ^ h) and with what

function (say w(z)) it weights measurements at each height. 3) Compute the

values of p and T for which the inequality

*/

h

w(z) Q (p(z), T) dz - Q (p(h/2), T) £ p Q(p(h/2), T) (2.2.9)
is satisfied. Here we have assuned that the weighted average of which the

experiment measures will be assigned to the thermodynamic state (p(h/2), T)

at the midheight of the experiment. Exactly this calculation is carried out

in Appendix C where Q is taken to be the correlation length and w(z) is

assumed to be unity. A somewhat more complex calculation is required if Q is

a quantity measured by experimentally taking a temperature derivative (such

as the constant volume specific heat). This problem is discussed by Hohenberg

Barmatz [1972].

We have just completed several illustrations of how gravitational

averaging influences the measurement of thermodynamic quantities. It is

important to note that most measurements of transport properties will also

be subject to limited density and temperature resolutions in the

earth's gravity. For example the density dependence of the viscosity or of

the thermal conductivity cannot be measured within 4-5% of p , if the viscosity

or conductivity apparatus is 1 mm high at earth-normal gravity. It is also true

that the density dependence of the turbidity (or total light scattering

intensity) cannot be measured within a few percent of p if the scattering

volume is 1 mm high (Leung and Miller, 1975) because of the vertical averaging

that occurs in the scattering volume. On the other hand quite different

limitations apply to the measurement of thermodynamic and transport properties

by optical techniques with very fine spatial resolution. They are

discussed in the subsequent section.
2.3 Limitations on Optical Experiments due to Gravity Induced Refractive

Index Gradients.

As a fluid is brought near its critical points, its coefficient of

isothermal compressibility diverges and, in a gravitational field, the fluid

becomes compressed under its own weight. The density gradients thus produced

place strong limitations on the validity of measurements, even with optical

probes, very near the critical point. The most serious limitation may be

termed a "thick cell effect" and applies to all optical experiments including

measurements of the phase, intensity and spectral characteristics of light

passed through the fluid. Light which is directed horizontally into an

optical cell filled with fluid near the critical point will be deflected downward by the index of refraction gradient (resulting from the density gradient).

The angle of deflection is proportional to the thickness of the fluid layer in

the cell. This effect is precisely the same, one which enables the sun to be

seen above the horizon several minutes after it has "set" according to

astronomical calculations. As the critical point is approached, the density

gradients and the deflections become so large that light "rays" pass through

layers of fluid of widely varying density. Then it is no longer possible to

relate the intensity, the spectrum, or the phase of the light emergent from the

cell to the density or any other local thermodynamic variable. The measurable

optical properties of the fluid become complex gravitational averages. In this

section we discuss and quantify these limitations for optical experiments

using a combination of analytical and numerical techniques.

Consider a sample of a dielectric fluid, in a cell with plane parallel

optical windows, with an average density, p , closely approximating the

critical density. When the sample temperature is held constant near T , the
fluid is compressed under its own weight and an equilibrium density distribution,

p(z,T), is formed. Since, in a gravitational field, the reduced chemical

potential y is directly proportional to z, the density profile is directly

given by the equation of state p(y,T). Furthermore, if current theory is

correct (Appendix A) such profiles are antisymmetric about some density

p (p is not necessarily the critical density and in general will be a cen cen °

function of the temperature. For this analysis we will assume that p = p ) cen c which occurs at the height where the compressibility has a maximum. The plane

at this height, which we call the centrus

,

c will serve as the origin for our

o vertical coordinate system. In Figure 2.5 we represent such a profile

o schematically for some supercritical temperature T > T . (All of our analysis

will deal only with the one phase region.) Because the refractive index of the

fluid is related to its density by the Lorentz-Lorenz (Lorenz, 1880; Lorentz,

1952; Hocken and Stell, 1973; Larsen et al. , 1965) formula, there is a

corresponding refractive index profile n(z,t) in the sample. Now we

c illuminate the optical cell with a monochromatic horizontal (normal to the z

axis) plane wave of light, and examine the trajectories of rays through the

profile at various heights.

In Fig. 2.6 we illustrate this situation with a simple ray picture. A

ray enters the cell horizontally at height z' above the centrus and is

refracted downward on a curving trajectory until it emerges at a height z.

The output angle of the real ray will be proportional to the average refractive

index gradient 'seen' by the ray and its phase shift will be determined by an

average over its optical path. Also shown in the same figure are two ideal

rays which we call 'thin cell' rays. Both are equivalent to the real ray when

the cell is infinitely thin or the gradient so small that z' ?w z. They are
*^' 2

XT (z,T)orXT (M.T)

/>.Xt

Figure 2.5. Schematic of density and compressibility profiles,

29
THIN CELL RAY

EMERGING AT

SAMEZ

CENTRUS

REPRESENTATION OF DENSITY PROFILE

Fig. 2.6. Rays through a density profile. "Real ray" denotes the path

traversed by a light ray incident horizontally upon an optical cell filled

with fluid near the critical point. The other rays are defined as follows: the

angle equivalent ray (TER) is that thin cell ray which is refracted at an angle

equal to the real ray. It traverses the cell horizontally at a height (z 4- Az)

where the local refractive index gradient is equal to the average gradient seen

by the real ray; the height equivalent (ZER) is that thin cell ray which traverses

the cell horizontally at height z.

30
defined as follows: the "angle equivalent" ray (TER) is the thin cell ray which

is refracted at an angle equal to the real ray; it traverses the cell

horizontally at a height (z + Az) where the local refractive index gradient is

equal to the average gradient seen by the real ray . The "height equivalent" ray

(ZER) is that thin cell ray which traverses the cell horizontally at height z.

These thin cell rays are introduced for comparison purposes since for them

the relation of optical observables to thermodynamic quantities is particularly

simple (Estler et al. , 1975). For instance, if we expand the Lorentz-Lorenz

formula, we find to a good approximation that

p - p c n - n = n -, (

) + ... = n Ap * + .. c 1 p 1

9 9 (2.3.1)

(n - l)(n + 2)

/ - c c (n

1

- ).

6 n

c

Then the index gradient becomes

n / -12

H

c V

nn * dz 1 dz " IT (^*L ~ TT X T (2.3.2) kh

k

where y* is the reduced chemical potential, xT the reduced susceptibility

and H

q = Pc /(p

cmg), the scale height for the fluid being studied. In the thin cell

limit the ray ZER simply comes out at height z with an angle proportional to

XT

at that height, i.e.

31
XT

(z) = -f^

11' f°r the ZER, (2.3.3)

where d is the cell thickness.

Fig. 2.7 offers a comparison between angles of rays as a function of z

as determined from equation (2. 3.. 3) and angles determined numerically by

tracing rays (Born and Wolf, 1975) through the same density profiles. The

profiles were generated using the Stony Brook parametric equation of state (Wilcox

and Estler, 1971) with parameters obtained from actual profile measurements on xenon

(Estler et al. , 1975). The parameters used are given in Appendix E. The

T-T

C —D computation was performed with AT* = — = 5x10 . The solid line represents

c the thin cell results for the ZER (Eq. 2.3.3) and terminates in a rounded point

at an angle of about .17 radians, that is, off the graph. The dashed line is

the result of the ray tracing computation. The crosshatched region contains

rays that enter the cell close to the centrus above and below and emerge at

approximately the same height but at very different angles; the observer sees

a bright band at this height with a darker band above it. In this region the

gradient is so large that the rays are simply bent right out of the dark region

and into the bright band.

Fig. 2.8 shows the same results for the optical phase as a function of

height. Again the solid line is the thin cell result, i.e. neglecting any beam

bending effects, (cf. (2.3.3) and the dashed line is the result of ray tracing.

The optical phase relative to that at the centrus (A<j>) is simply related to

the density in the thin cell approximation:

A<}> = - k d n

x p*(z) (2.3.4)

32
.05

Z

(cm

.05

///////////////////'

OUTPUT ANGLE (RAD)

THIN CELL

THICK CELL

t = 5x 10" 6 d = 3mm

XENON PARAMETERS

Fig. 2.7. The angles of emergent rays as a function of height. The solid

line represents the thin cell results for the ZER (eq. 2.3.3) and terminates in

a rounded point at an angle of about .17 radians, that is, off the graph. The

dashed line is the result of the ray tracing computation. The crosshatched

region contains rays that enter the cell close to the centrus above and below

and emerge at approximately the same height but at very different angles.

33
-.10

XENON PARAMETERS

d = 3mm

t = 5x 10

ACTUAL SHAPE OF

DENSITY PROFILE

THIN CELL PHASE

PROFILE

REAL OBSERVED

PROFILE

10

OPTICAL PHASE

Fig. 2.8. Optical phase as a function of height. The phase profile calculated

in the thin cell approximation (solid curve) has the same shape as the density

vs height profile. The observed profile is represented by the dashed curve. The crosshatched area corresponds to the crosshatched area in Fig. 2.7.

34
Hence, the solid curve is, except for sign and units, simply the existing

density profile. The dashed line is then the apparent density profile deduced

from actual measurements and in fact such "kinked" profiles have been reported

in the literature (Lorentzen and Hansen, 1966).

Figures 2.7 and 2.8 offer a good visual picture of what happens to light

traversing a fluid near its critical point. They do not, however, offer

simple quantitative guidelines as to which regions of the thermodynamic

space are inaccessible to the earth bound experimenter. To provide these

guidelines we found it convenient to compare the real ray with the angle

equivalent ray (TER) rather than to the ZER as in Figs. 2.7 and 2.8.

As our computations progressed we discovered numerically a simple

relationship between Az (the height difference between the TER and the

real ray, see Fig. 2.6) and the output angle of the real ray (or TER since

they are defined to be the same) . We then derived this expression for Az by

solving the ray tracing equations analytically in a medium where dn/dz was a

slowly varying function of height. We found that if

2 dn(z) dni d n

di ^di^z +

T"2lz (z " Z o

} ' < 2 - 3 - 5 > o dz o

then

n

l

d * Az w-s". X 3Kn" T' (2.3.6)

e

As long as equation (2.3.6) holds, the real ray carries average fluid

35
information equal to the local information at a height Az above the height of

the real ray's emergence. This equation is valid both above and below the

centrus.

We then performed computer experiments to test the domain of validity of

(2.3.6) using numerical parameters appropriate for xenon. We found that (2.3.6)

breaks down when Az reaches a certain value nearly independent of the temperature

and the fluid studied. The results are plotted in Fig. 2.9 for xenon. In this

figure the solid dark line represents the Az that would be calculated from

(2.3.6) using the fluid's maximum compressibility. That is

2 2 n d

a n d -y

Az = —

v = —=— r AT* (2 3 7) max 3 Hn X T 3 Hn ! • ' * U.J./;

c max c

The region above and to the right of this curve is nonphysical: no real

thermodynamic states exist in this region. The area below this curve and to the

left is physical but mostly inaccessible to the- earth bound experimenter. On

this plot the critical point is at infinite Az. The dashed line at Az « .002 cm

represents the contour where (2.3.6) becomes incorrect by 1%. Above this line

the errors grow very rapidly. The region below this line is experimentally

accessible and here the properties of the interference pattern may be simply

related to local thermodynamic properties of the fluid. As a simple rule of thumb

we find that the observed interference pattern can no longer be simply related

to local thermodynamic fluid properties when the reduced compressibility

36
N

4 -

3 -

2 -

1 -

-7

THIS REGION IS PHYSICAL BUT ONLY

ACCESSIBLE IN SPACE

\ /

A.Z max /

THIS REGION IS -

NON PHYSICAL

—

1% ERROR LINE. 1g. '\

I -""

-

THIS REGION IS PHYSICAL AND

EXPERIMENTALLY ACCESSIBLE

AT1 g.

I I I *- I -5

log AT*

Fig. 2.9. Thermodynamic states accessible to optical experiments. The

states with sufficiently small values of the scaled compressibility Xt are accessible. Large values of %— do not occur to the upper right of the solid

curve (i.e. far from the critical point).

37
becomes larger than

Cell Thickness A.rnmax

e cm.

UlCt mil

1.0 2.6xl0 322

0.1 2.6xl05 220

* 3Hn

X

T

= (.002 cm) 1 . (2.3.8)

max n, d

It is convenient to tabulate some examples for comparison with gravitational

limits provided elsewhere in this report. We do so here for xenon

Ap . AT* min min

milliradians (At* = 0) (Ap* = 0)

0.0471 1.6xl0"

4

0.0119 3.3xl0~

6

Thus any optical experiment using a 1 cm thick cell is subject to limitations

comparable to a bulk experiment in a 0.1 cm high cell at g . An optical

o experiment in a 0.1 cm thick cell is subject to limitations comparable to a bulk experiment in a 3 um high cell at g or a 3 mm high cell at 10~ 3 e . o

° 6

o Optical cells with properties suitable for critical point experiments (Hocken

et al. , 1975) have been fabricated with a thickness of 3 mm. It seems likely

that 1 mm thick cells could be made. The feasibility of making much thinner

cells should be examined. One may expect that fluid samples in extremely thin

cells would be subject to contamination from the cell walls because of the large

surface to volume ratio.

38
2.4 Light Scattering Measurements and Some Gravity Related Limitations .

Scattering measurements provide a means to measure the space-time

dependence of fluctuations. Since much of the fundamental picture of critical

fluctuation phenomena deals with the distance scale over which fluctuations

are correlated (space) and the time scale over which fluctuations grow and

decay (time) scattering spectroscopy continues to be an important tool for

studying critical phenomena.

Near the gas-liquid critical point a fluid exhibits large fluctuations in

the density. The magnitude of these fluctuations is proportional to the

compressibility. The spatial extent of these fluctuations can be characterized

by a correlation length £. As shown in Appendix C this correlation length is

related to the compressibility by

5 - e (r x x*)

2 n (2.4.1)

o 1

where x T

is the dimensionless symmetrized compressibility defined in A. 2. The

constants £ and r are the amplitudes in the power laws (C.4) and (A. 4c) for the

correlation length and compressibility along the critical isochore. The constant

K is a microscopic distance of the order of the range of the intermolecular forces

The spectral intensity as a function of scattering angle and frequency is

proportional to the spatial and temporal Fourier transform of the time

dependent correlation function of the order parameter. In a one component

fluid the integrated intensity of the scattering is proportional to the

static structure factor

39
S(k) = f dR e lkR

G(R) (2.4.2)

where the correlation function G(R) is defined in Appendix C. The wave

number k is related to the scattering angle 6 by

k = 2 k

Q

sin j

(2.4.3)

where k is the wave number of the incident radiation,

o In principle S(k) is a function of temperature AT*, density Ap* and wave

number k. However, according to the scaling hypothesis the structure factor

near the critical point can be written in the form (Fisher, 1967)

S(k) = ? 2_n HW) (2.4.4)

where £ is the correlation length. This reduction of the description is valid

for any thermodynamic path in the AT* - Ap* plane of constant scaling variable,

1/B

i.e. AT* proportional to |Ap*| . Here we shall consider specifically the

properties of scattered light at the critical isochore Ap* = 0.

The scaled function i>0aO is known theoretically (Tracy and McCoy, 1975)

for the 2 dimensional Ising model for all values of k£. However, the

mathematical form of this function for fluids is not known except for the fact

that is must approach a constant in the limit k£ -> and that it must vary as

(kO~2+n for k5 > ».

It has turned out to be very difficult to determine a definitive value for

the exponent n. A precise knowledge of this exponent has a direct bearing

40
on the question of universality of the pair-correlation function.

In order to determine this experimental n with any accuracy, measurements

are needed for k£ sufficiently large so that the function ^(k£) can be

approximated by its asymptotic behavior (Tracy and McCoy, 1975). Due to

limitations caused by multiple scattering and the gravity induced density

gradients, it has been impossible to satisfy this condition for earth bound

light scattering experiments. It has been possible to reach sufficiently

large values of k with neutron scattering (Warkulwiz et_ al ., 1975); however,

here one has difficulties in satisfying the condition that k must be large

compared to the range of the intermolecular forces.

Of the several available scattering techniques (light, X-rays and neutron

scattering), optical light scattering using laser sources seems to be most

feasible one for space experiments. It is the only scattering technique with

sufficient spectral resolution to examine the narrow spectrum of the

fluctuations, can be done with easily portable sources (lasers), requires no

radiation shielding, and uses windows compatible with other optical records

desired of the sample behavior. The principle drawback to light scattering

measurements is multiple scattering: the extinction of the beam gives a severe

lower limit on the AT* which can be reached. Numerical estimates for this

limit are presented later in this section.

Using optical beating techniques, one can measure the spectrum of the

quasi-elastic Rayleigh line . (Cummins and Swinney, 1970). The width of this

line is related to the diffusive decay constant of the fluctuation of the order

parameter. A survey of the current status of the experimental work on

Rayleigh scattering near the critical point of fluids has been given by

Swinney and Henry [1973].

41
In interpretating such experiments one usually introduces the following

assumptions. First it is assumed that the fluctuations still decay

exponentially in time so that the dynamic structure factor has the form

2r

S(k,u>) = S(k) -5 ^y— (2.4.5)

r + w

s

where T

g is the decay rate of the entropy fluctuations. (Swinney and Henry, 1973).

Secondly, one uses the assumption of dynamical scaling to write the decay rate

r in the scaled form

s

r = H 4v (kO (2.4.6)

s 1

It should be pointed out that these assumptions are not valid rigorously, and

that one expects to see deviations if the critical point is approached

sufficiently closely. Theories for the scaled function ^ (kC) have been

developed by Kawasaki and coworkers and by Ferrel and Perl; for a survey of the

literature the reader is referred to the article by Swinney and Henry [1973].

The various theories differ in their conclusions, but it has been impossible

to discriminate between the theories on the basis of earthbound experiments.

For light scattering measurements we need to consider the limitations on

the attainable experimental precision due to gradients in the density Ap*,

turbidity of the sample, and gradients in the correlation length £. Our

purpose here is to estimate how much the accessible range in AT* and Ap* can

be extended by conducting light scattering experiments at reduced g.

42
We first consider the variation in density. In a light scattering

experiment close to T one can consider using a weakly focused beam with a

diameter of ^ 100 um. Stronger focusing would require much lower power levels

to avoid local heating particularly at window surfaces. Thus there exists a

practical height or spatial scale limit for critical point light scattering

measurements of about 100 um.

The resolution of light scattering experiments has been exploited in earthbound experiments by moving the beam as a function of height, thus obtaining

local equilibrium isothermal scattering intensities (White and Maccabee, 1975)

and spectra (Swinney and Henry, 1973; and Kim, et al. , 1974). In order for

the density change over a height to be within a precision p of the average

density in that region for a sample in a gravitation acceleration g*(g* = g/g ),

we require

dApl (2.4.7)

dz

Using the restricted cubic model and Eqs. A. 13 and B.7 of the Appendices, we

find

k 1

ha*

K

2 vY (2.4.8)

o L

43
For xenon , using parameters of Appendix D, we find

r ;> 6.2xl0" 4 (h£^)- 84 (2.4.9)

with h in meters. Thus this density limit is a contour of constant r in the

(AT*, Ap*) plane as discussed in Section 2.2. The following

table gives examples of the lower bounds to r and corresponding AT* = r (with

Ap* = 0) and Ap* (with AT* = 0), for p = . 01 = 1%, h = 10~ m = lOOym corresponding to precision scattering experiments on earth (lg) , worst case Shuttle

accelerations (10 g) and best case Shuttle accelerations (10 g)

Table 2.4.1

g* = g/g

o

r = AT*

(Ap* = 0)

Ap*

(AT* = 0)

1 1.3xl0"5 1.9xl0"2

io3 3.9xl0"8 2.4xl0"3

lO"

6 1.2X10"

10 3„0xl0"4

Next we examine the effect of the strong (critical opalescense) scattering

of light in the sample. The density fluctuation scattering cross section per

unit volume per stearadian of a fluid is given by

dr u . dp d i

44
where X is the wavelength of the incident beam (in vacuum) , e is the optical

dielectric constant at A and K^, is the isothermal compressibility and <j> the

angle between the incident light polarization and the scattered wavevector.

Near the critical point K_ has a strong temperature dependence leading to

the result that the scattering intensity diverges like K . For k = 2tt/A << £ ,

this dipole cross section can be readily integrated over all angles to give the

scattering attenuation coefficient called the turbidity, t

t= T? ( 4f>

2 V *r (2 - 4 - u>

For the present estimates we have neglected the angle dependence of the

scattering due to the correlation length E, (see Puglielli and Ford, 1970 and

Cannell, 1975).

Like the cross section, the turbidity follows the temperature dependence

of BL,. Using the cubic model (Appendix A. 3) we can write for K

k

2

P c r"Y K = (~^) ~z- (2.4.12)

a

2 c

p

—

Y

Since close to the critical point p =. p , we find that x is proportional to r

In light scattering studies the intensity corrections due to turbidity

become severe for a path length £ such that x£ ^ 1. At approximately the same

x£ limit the corrections due to multiply scattered light reaching the detector

become severe. Thus we see that scattering experiments are limited to studying

the samples for AT* - Ap* such that x < £

We have evaluated the turbidity and the resulting limitations for the case of xenon using the parameters of Appendix E at \ = 6328A (He-Ne laser) , and

45
and approximating (p—) « ( e -1). We find

dp C

-3 -1.19 -1

t = 2.51x10

J r ' (m ). (2.4.13)

For an optical path I (m) we conclude that r must satisfy

r £ 6 v 53xl0' 3 £* 84

. (2.4.14)

_2 Typical optical cells at present have I = 10 m (1 cm). More specialized cells

_3

have been made with I = 10 m (1 mm). It may be feasible to work with

-4

£ = 10 (lOOum) . With such a thin cell one must carefully discriminate against

scattering from the inner cell walls. These distances have been used to

calculate the lower bounds of r and corresponding AT* and Ap*. For xenon and

A = 6328A we find:

Table 2.4.2

a Turbidity approach bounds at A = 6328A in Xenon

£(m) r = AT* Ap*

(Ap = 0) (AT* = 0)

10" 2 1.4xl0~4

4.4xl0~2

10" 3 2.0xl0~ 5

2.2xl0"2

10" 4 2.8xl0"6 l.lxl0"2)

46
Comparison of Tables 2.4.1 and 2.4.2 shows that at lg, a 1 mm optical

path cell of xenon has both the density profile and turbidity limit of about

r = l.xlO . This limit corresponds to AT « 3 mK. In order to take advantage

of decreased density gradients at 10 g one would need a cell with optical path

-4

less than lOOym = 10 m. The ability to fabricate such a cell has not been

demonstrated. Thus we see that turbidity and multiple scattering place a severe bound to scattering experiments closer to the critical point. On earth at lg this

has not been as severe because the sample density gradients have caused most of

the sample to be off the critical density giving reduced scattering and

turbidity. In a low g environment one expects to see the sample uniformly

opalescent and becoming quite opaque to visible wavelengths as the critical

point is approached. The fact that the sample is uniformly opalescent at low g

should make the analysis of the influence of multiple scattering much more

manageable. The new theoretical work on multiple scattering (Reith and Swinney,

1975; Bray and Chang, 1975) will be helpful in the analysis of low g scattering

data.

It is clear that for studies close to the critical point one needs to

reduce the observed turbidity. This can be done with the choice of A or choice

of fluid with a small defractive index. Since the turbidity varies as X~ , the

turbidity at a given AT* could be greatly decreased by using longer wavelengths.

Small continuous He - Ne lasers exist at 1.152ym and 3.391ym. Photomultipliers

still work at lym so this choice would be favored for spectral measurements and

would decrease the tubidity by 10. 9x and the r bound by 7.5x. At 3ym photoconductive

detectors would be required. The quantum efficiency may be so low that beating

spectra of the critical fluctuations may not be possible. Also the choise of window

materials becomes more restricted. The following table shows the turbidity

bounds of AT*, Ap* in xenon: for electromagnetic radiation with a wavelength of

3ym.

47
Table 2. A.

Turbidity approach bounds at X - 3.391y in xenon

Urn) r = AT* Ap*

(Ap* = ) (AT =0)

10~2

A. 8xl0~

7 6.0xl0~3

10~ 3 7.0xl0"8

3.0xl(T 3

10"4 U0xl0~8

1.5xl0"

3

_3

Table 2. A. 3 indicates that at 3.391um we could exploit the 10 g environment

—5 —8 —2

with lOOum path cells, over a new range of AT* = 10 to 10 and Ap* = 2x10

q _f. to 2x10" . With 6328A we could only go in to AT* - 10 .

Alternatively we can consider going to a fluid with lower dielectric

2 constant, e = n . The best choice (lowest e) would be the rare isotope of

3 helium,' He . Here n,~ no = « 1.0108 in the fluid near T . (We have taken n for

6328 c c

A 3

He from the data of Edwards and Woodbury [1963] and scaled it for He using the

Clausius- Mosotti relation with the critical densities from Appendix D) . Thus at

9 9 6328A we expect t„ /t„ = (e„ -1) / (e„ -1) = 1/153. One could further reduce t tt by

He xe He xe He

moving out to lym or 3ym. Thus, the use of helium would help overcome the

increase in turbidity encountered upon approaching the critical point at the

cost of doing experiments at cryogenic temperatures (3.3 K)

The angle dependence of scattering intensity and linewidth closer to the

critical point are of interest because of questions concerning the theory of

the fluctuations in the non-hydro dynamic region. The angle dependence of

intensity is written in scaled form in (2. A. A). At a fixed AT*, Ap* the angle

dependence arises from the function i^(x) with x = k£ and k = 2 k sin 8/2.

Similarly, the scattering angle dependence of the decay rate r is given

48
by (2.4.6) and comes from x in the function i|/ (x) . If the measurements are

to be used to determine £ or to test the form of ^(x) or ^_(x) it is essential

that we know x precisely. If £ varies over the sample or specifically over the

diameter of the probing beam in the sample due to gravity gradients then x is

correspondingly poorly defined. Assuming for the moment that k is well

defined through suitable collimation of the incident and scattered beams, the

measurement precision of x = k£ will be limited by the uniformity of £.

-4

We ask that £ be constant to a precision p over a height h = 10 m = lOOum

or

(§) T

h * P5 (2.4.15)

Using the results of Appendix C for (~r~) T one can use (2.4.15) to give

r(8) and then transfer to AT*, Ap*. We have done this for xenon with p = 0.1,

-4

h = 10 m giving the lower bound AT*, Ap* contours shown in Fig. 2,10 for

g* = 1, 10 and 10 . Comparing the figure to Table 2.4.3 shows that even

if the turbidity limit is lowered by changing to 3um wavelengths we will

-3

begin to be limited for g* =» 10 by the correlation length gradient.

There is a further effect of turbidity in determining x. This occurs

because k is well defined only for non-attenuated incident and scattered

waves. Attenuated waves give a spread to k. To have k defined to 1% we

need k > 10t. Since k - x/5, we require

t 1 ^ (—-) Z. (2.4.16)

49
10

r4

10

-5

10

-6

I10

-7

10

-8

rrrr

|Q LUU J J_i _L

9\^/CTG

1 1 I I I I ll 10 10

rZ

P-P,

Fie. 2.10. Contours of — -r* h = p for xenon with h = lOOym, p = .01.

Experiments with a lOOym diameter beam in the region below and to the right

of each contour are severely affected by averaging over correlation lengths

at the values of g* indicated.

50
This is least restrictive for large x. The condition (2.4.16) gives a lower

bound for r which is converted into AT* for Ap* = and Ap* for AT* = 0.

Table 2.4.4 gives the results for X = 6328A and X = 3.391ym for the case x = 1,

Note they are independent of g.

Table 2.4.4

Scattering vector approach bounds

X r = AT* Ap*

(Ap* = 0) (AT* = 0)

6328A 6.3xl0~

7 6.6xl0

-3

3ym 1.6xl0"81.8xl0

-3

The results in Table 2.4.4 indicate that the change in wavelength from

6328 to 3um would keep the smearing of k to an acceptable limit down to the

_-} -4

turbidity/multiple scattering limit for i of t:he order of 10 m or 10 m.

We can summarize the results reported here by stating that with the most favorable

geometry of lOOum path cells and 3ym radiation light scattering experiments could

3 3 approach closer to the critical point by 10 in r corresponding to 10 in AT*

1 -3

and 10 in Ap* in a 10 g environment. At present, experimental limitations

— f\ would prevent us from fully utilizing the minimum accelerations of 10 g. Even

at 3ym a requirements remains from ~\ to achieve the minimum r allowed by dz J density and turbidity we must fill the cell accurately to p to within .02%.

This is certainly possible but requires more care than does sample loading for

earth bound experiments.

51
3. Limitations in critical-region experiments due to modifications of

fluid properties by a gravitational field .

In the preceding sections we have commented on the technical complications

that are encountered in various experimental methods near the critical

point of a fluid in a gravitational field. However, in addition,

intrinsic limitations in earth-bound experiments exist due to the fact

that the gravitational field modifies the fluid properties in the immediate

vicinity of the critical point.

As mentioned in Section 2.4, upon approaching the critical point the

increase of the compressibility is accompanied by an increase in the

size of the fluctuations that extend over a correlation length K. If

the system were homogeneous and in true thermodynamic equilibrium the

compressibility and, hence, the correlation length would actually diverge

at the critical point. However, the presence of the gravitational field

prevents the fluctuations from growing indefinitely and compressibility

and correlation length will in fact- remain finite. Thus the presence of

the gravitational field causes round off effects which change the

nature of the thermodynamic behavior in the immediate vicinity of the

critical point. It is the purpose of this Section to estimate the range

in temperature and density where experiments will be affected by these

round off phenomena and to elucidate how this range depends on the

magnitude of the gravitational field.

As explained in Section 2.2 the gravitational field induces a

density gradient in a fluid near the critical point. When the density

does not change too rapidly as a function of height, one may assume that

52
the local thermodynamic properties of the fluid at a given level are the

same as that of a macroscopic homogeneous system with the same values of

density and temperature. Under these conditions the density profile is

determined by the equations given in Appendix B and measurements of the

local fluid properties in a gravitational field does provide information

on the thermodynamic behavior of a homogeneous fluid in the absence of

a gravitational field.

The assumption of local thermodynamic equilibrium is justified when

the fluid is locally homogeneous over distances of the order of the

correlation length, but will break down when the fluid properties begin

to vary over distances of the order of the correlation length. In

Fig. 3.1 we show calculated density profiles for xenon in the earth's

-4

gravitational field at three reduced temperatures, namely AT* = 10 , —fi —ft 10 and 10 . The existence of a density gradient implies that also the

correlation length E, varies as a function of the height. In Fig. 3.2 we

show the correlation length £ as a function of the height for xenon in

the earth's gravitational field. The curves are calculated in the

approximation that the assumption of local thermodynamic equilibrium

remains valid. However, in the dashed part of the curves the correlation

length varies so rapidly that the system can no longer assume local

equilibrium states that are homogeneous over the range of g. Under

these conditions the laws of thermodynamics no longer suffice to specify

the nature of the equilibrium states. The complications that arise when

the macroscopic thermodynamic relations can no longer be applied at the

local level, will be referred to as non local effects.

Such effects will enter when the correlation length C starts to

vary over its own height h = £. It thus follows from (C.19) that for

measurements conducted with precision p non local effects will be encountered

when c

„
a: AT* = I0~

4

b: AT* = I0"

6

c: AT* = I0"

8

Ap in %

Fig. 3.1 Calculated density profiles for xenon in the earth's gravitational

field and assuming local thermodynamic equilibrium. In the dashed

part of the curves the density varies so rapidly that the assumption

of local thermodynamic equilibrium no longer applies.

54
(0 \ X \ If) 1 1 1 I 1 1 l\ 1 o o o o o o o o oo o o o o o o ' o o o If) o If) o If) o If) * CO CO CM CM T— r»

If) o V ' f o o — +-> CU

*i

cu 4- CO

•r— ZJ T3

r— x:

r<3

M X> c CU o CU CO

•r— > co

+-> s- cu

«D 3 s4-> o D.

•i

Q. > cu 3

«3 -C CO

S- +-> C7> CU

<0 4- jQ

CO

O s 'i

SZ -M , +-> s- •i— E 1.

J- fO 3

03 a.

O) co * 1 -o c

a, ° a> cu o

x: sz •r— '"tL

4-> CO 4-> fO . r C -o =5 m •i— cu

4->U c x: 3 o 4-> r— c 4-

cu C. X t—i CU x:

i_ +-> o • 4- E3

4-O

+J r- x: S- CU a> XI ai

•i

•r— c

a) r— CO x: 3

S- 4- cx r— O cu «33 C o +-> o •i— CJ

•1— E <o +J fO u c CU c >> -C 3 "O 4-> 4- OE a

rc cu c

re

to X:

fO +-> 2 £ S

UJ1 <a o

-a x: o • E ll

+-> o co co CD r— -*: +J C (O o N O

cu CO cu cu

CU s- 4- < H

1- c: E3

X) 4-

CU ' o CO c

1- co o i

+J fO »r- rtJ <a 4-> CJ f— cu Q. o

ai c E 1

s_ o 3

s- CO e o 4- CO o o •r— fC c

CO

CO

a>

55
a

9?

(3.1)

3Z' ^ P

In terms of the parameters r and of the restricted cubic model

equation of state, it follows from (C.16) that non local effects can be

avoided only if

( J 91 \(2o\ V92

\p vV2/i L 2e6b

2

2e2 ^- e2 ) + 0-3e 2 )0-b9

2 e2)

(3.2)

Using the xenon parameters given in Appendix E, we conclude that at the

critical temperature, AT* = (6 ' = ±b

2 ~

), non local effects will be

encountered unless

/a *\ v+36

t

. 0.163

Ap*| ^ 0.007 JM = 0.007

\yi (3.3)

In Fig. 3.3 we indicate the region in the AT* - Ap* plane where the

behavior of the fluid is modified by non local effects and, hence, where

its properties will be fundamentally different from a fluid in the

absence of gravity.

Near the critical isochore, Ap* = 0, condition (3.1) becomes

unrealistic because of the rapid variation of 9^/8z and we should,

instead, consider the integrated form

4f<P (3.5)

The distance tz over which the change in £ is smaller than p is

P

derived in Appendix C. In particular we require

56
0.5rIDo

I

-0.4

0.01 0.02 0.03

( P-P c ) / Pc

Fig. 3.3 Region in the temperature-density plane where the fluid properties

are modified by non local effects for g* = 1 (earth's gravitational

field) and g* = 10~ 3

The curves refer to experiments for xenon

with a precision at the ^% level (p = 10 ).

57
5

- : !AT*r « 2Az

p (3.6)

where Az is defined in (C.13). For xenon we obtain

P

_J

'AT*| * 0.4xl0" 6 (£*)^+66 = Q.4X10" 6 (SlfAS

This equation determines the intercept of the contours in Fig. 3.3 with

the AT* axis.

It follows from (C.lla) and (3.2) that the system will only satisfy

the conditions for local thermodynamic equilibrium when

5 s 1 .5xl0-

6 (jL)^* * - l.SxlO"

6 &)°- 29 m (3.8)

When the critical point is approached more closely, the increase of the

range of the fluctuations will be suppressed by non local effects. It

seems plausible to assume that the correlation length E, in a fluid in

a gravitational field cannot grow any further when (9£/3z) ~ 1 in the

local equilibrium approximation. Hence, in a fluid in a gravitational

field, the correlation length will always remain finite and of the order

of

^

Snax

s l-5xl0" 6 g*" " 29

m (3.9)

58
It should be noted that the maximum correlation length attainable will

only increase inversely proportional to the cube root of the gravitational

field.

In this section we have considered the intrinsic limitations of

thermodynamic experiments near the critical point of a fluid in a

gravitational field. At present, more stringent limitations are imposed

by the complications associated with the state of the art of the various

available experimental methods. Nevertheless, current experimental

techniques have developed to the point that they are on the verge of

reaching the region of these intrinsic limitations in earth-bound experiments,

59
4. Conclusions

In this report we have shown that gravity causes experiments to

become inaccurate when the critical point is approached sufficiently

closely. We have considered the ranges in AT* and Ap* which are inaccessible

in a variety of earthbound experiments and how these ranges shrink when

gravity is reduced. Some of our conclusions are summarized numerically

in Table 4.1.

The critical point is a focal point of anomalous behavior of many

physical properties. In order to understand the nature of these anomalies

it is desirable to approach the critical point as closely as possible.

In fact, the most recent experiments (Hocken and Moldover, 1976; Balzarini

and Ohrn, 1972) indicate that at temperatures as close a 1 mK from the

critical temperature the anomalous behavior near the critical point of

fluids, although appearing to approach Ising model behavior still

differs from the behavior predicted by some theories for that model. A

disappearance of this difference at the critical point would have

considerable theoretical significance.

Experiments in a low-g environment will provide opportunities to

study the nature of the anomalies in a range of temperatures near the

critical temperature inaccessible in earthbound experiments. To

discuss which anomalies are best suited for study in space, it is

convenient to classify anomalies as "strong" and "weak" (Griffiths and

Wheeler, 1970). In this rough classification, quantities which behave

like (T-T ) p on the critical isochore are called strongly anomalous if

c

p is greater than, perhaps, 0.5. Thus the isothermal compressibility,

the constant pressure specific heat, the thermal diffusivity, and the

correlation length exhibit strong anomalies in pure fluids. In contrast,

60
•OI

o> o 1 Ot o o

X X

in m

S

<g o

l-i 00 to 01

01 o a r-( 9

rHO

KM

C

CO J=

3 T3"

•H U-l > 01

c

o u

>» >H B w u 4 H « -H •a .h -o H 01 4 4 U M

M h 10

,3 ° H O

j: 1

S B j: 60 01 60 H * <N H a ITI 4J • X 01 O

S <M CM JJ a o o a o u 60 01 iH u a) C u -H

J= o H «

i-l 60 <H O rH .-< H H <W <d a H at 01 0. 0)

.e

4J en a

Pu

oi o

o 01

JS 1 60 O

•u en -h t>

eg oi

ft. SB

4J -H H 0) ,

U CD 4J

0) 01 -H

O. h r-l u a B.-H no b j3

I o

•H <H

01 »•» u jE a u to u

01 -H -H 60 UH II S a »h u oi B«kH .

o o c

CJ o <

6: J5

u C

a) 01 O -8

01 oo

00 5

> en n H VD 01 w

s

** 4J u 4J

CO •o JJ to

hi a J= o

UH •H 60 o

01 •H

oi i-3

61
properties for which p is less than 0.5 (or even negative) exhibit weak

anomalies. Examples in pure fluids are: the velocity of sound, the constant

volume specific heat, the dielectric constant, and the shear viscosity.

In general, strongly anomalous quantities can be studied sufficiently

well on earth (particularly with refinements of optical techniques

mentioned in Chapter 2) so that low-g experiments do not seem desirable at

present. If the technology advances to the point where the non local

effects of Chapter 3 are encountered, this statement would have to be

revised. On the other hand, those properties which exhibit weak anomalies

and which must be studied in bulk samples are excellent candidates for

scientifically valuable low-g experiments. The study of these weak

anomalies in pure fluids is hampered by the strong variation of fluid

density with height on earth.

The weak anomaly in the index of refraction is a particularly

important one for low-g studies. Thorough understanding of the index of

refraction anomaly is of interest in itself; however, it is of even

greater interest in its impact upon the interpretation of equation of

state data obtained by optical techniques. Other good candidates for

study in a low-g environment are the C and viscosity anomalies which,

with present technology, must be studied in bulk samples at least several

tenths of a millimeter high.

In the areas of phase transition phenomena in fluids, low-g studies

of spinodal decomposition, nucleation, and macroscopic phase separation

all seem appropriate. At this time, it is hard to present a detailed

justification of low-g studies because the phenomenology is not as well

developed in these areas as it is for the thermophysical properties. We

62
believe exploratory experiments which exploit the long sedimentation

times available in spacelab are appropriate. We agree with the Overstudy

Committee (Dodge et al . , 1975) that very simple experiments will be

extremely important. The Overstudy Committee described one such experiment

(Dodge et al . , p. 9) which we heartily endorse:

Take a set of samples of a one-component fluid with different

densities. Included should be a sample at subcritical density, a

sample at the critical density and a sample at supercritical

density. Let the temperature vary either continuously or discontinuously through the transition temperature and take photographs at regular temperature intervals. Compare these photographs

with those obtained at earth under the same instrumental conditions.

Conduct similar experiments for a set of samples of a binary mixture

at concentrations smaller than, equal to and larger than the

critical concentration.

63
Appendix A. Parametric equations of state for fluids near the critical

point .

A.l Introduction

Let p be the density, V the volume, P the pressure, T the temperature,

u the chemical potential, BL, = p (3p/9P) the isothermal compressibility

and C the heat capacity at constant volume. It will also be convenient

to introduce a "symmetrized" compressibility

XT

= (3p/3y) T

= P 2 KT

(A.l)

The thermodynamic properties are made dimensionless by expressing them

in units of appropriate combinations of the critical temperature T , the

critical density p and the critical pressure P . Specifically, we

define

T* = T/T , p* = p/p . P* = P/P,

(A. 2)

v*= up

c /P

c

, xf = X,-P c /Pc' C

V

= C

VTc/VP

c

Note that the reduced specific heat C* is taken per unit volume. In

addition we define the differences

Ay* = {u(p,T) - y(p

c ,T))pc /P

c

(A. 3a)

AT* = (T-T

c )/T

c (A. 3b)

Ap* = (p-p

c )/p

c

(A. 3c)

64
The behavior of various thermodynamic properties is described by

power laws when the critical point is approached along specific paths in

the AT* - Ap* plane. One customarily defines four critical exponents

a, 3, Y» and <5 associated with the thermodynamic behavior of fluids near

the critical point. (Rowlinson, 1969). The exponent a describes the

divergence of C along the critical isochore.

(T*TC

, p = p c

) C

y/T*= £ {(AT*r a-lh (A. 4a)

the exponent 6 characterizes the shape of the coexistence curve

(T * T . p = p cxc

) Ap*

xc

= ± B |AT*| 6

, (A.4b)

the exponent y describes the divergence of the compressibility

(T * T

c

, p = p c) Xf = r (AT*)" 7 (A. 4c)

and the exponent 6 specifies the behavior of Ay along the critical

isotherm

(T = T

c

) Ay* = D (Ap*)|Ap*|

6-1

(A.4d)

The description of the thermodynamic behavior is based on a scaling

hypothesis (Widom, 1965; Fisher, 1967; Levelt Sengers, 1974, 1975). This

scaling hypothesis suggests that the equation of state upon approaching

the critical point will be asymptotically of the form (Griffiths, 1967)

65
Ay* = Ap* |Ap*|

6 ' 1 h(x) (A. 5)

x

= AT*/|Ap*| 1/B (A. 6)

The scaling hypothesis implies that the thermodynamic critical exponents

introduced in (A. 4) satisfy the relations

2 - a = 3(6+1)

(A. 7) Y

= 6(6-1)

so that only two exponents can be chosen independently.

The function h(x) in (A. 5) must satisfy a number of stability and

analyticity conditions (Griffiths, 1967). However, it turns out to be

very difficult to formulate an explicit mathematical expression for the

function of h(x) that would satisfy all required analyticity conditions

and which would be analytically integrable to calculate the specific

heat (Vicentini-Missoni et al . , 1969; Schmidt, 1971; Levelt Sengers

et al ., 1976).

These problems are solved by using parametric equations of state

(Josephson, 1969; Schofield, 1969). In the parametric equations of

state the relationship between the physical variables Ay*, AT* and Ap*

is given implicitly via two parametric variables r and 6. The constraint

that the scaling law (A. 5) is satisfied is met by the following choice

66
am* * r

65 H(e)

AT* = r T(e) (A. 8)

A P * = r

3 M(e)

The variable r is meant, in some sense, to describe a distance from the

critical point and the azimuthal variable 6 a location on a contour of

constant r. The critical isochore, the critical isotherm and the

coexistence curve are all curves of constant angle 9. In Fig. Al we

show the location of these curves in the Ay* - AT* plane and indicate

the meaning of the parametric variables r and G.

Parametric equations of state with various choices for the functions

H(9), T(9) and M( 9) have been used successfully to represent experimental

equation of state data in the critical region of fluids. The parametric

equations of state used in this report are the restricted linear model,

the restricted cubic model and the Wilcox-Estler equation of state.

67
THERMODYNAMIC

STATES

COEXISTENCE CURVE

cc

ULI X

h-o

<

cc

V CRIT CAL ISOCHORE

CRITICAL POINT

Fig. Al . Coexistence curve, critical isochore and critical isotherm in

the plane with coordinate axes tiv and t = AT*.

68
A. 2 Restricted linear model .

The restricted linear model equation of state corresponds to the

choice (Schofield, 1969; Schofield, Litster and Ho, 1969)

Ay* = a, r

66 e(l-e 2

) (A. 9a)

AT* = rO-b^e2

) (A. 9b)

Ap* = k

1 r

3 e (A. 9c)

with

b

2 =

6 - 3 (A.9d)

1 (6-i)(i-2e)

where a. and k1 are adjustable constants. In this formulation 8 = on

the critical isochore, 6 - ± l/b

1 on the critical isotherm and 6 = * 1 on

the coexistence curve. The sign of 9 corresponds to the sign of Ap*.

The restricted linear model yields for the compressibility

=r'- ©4 ;(??)•*] < a - io »

and for the singular contribution to the specific heat

*Cv

» sin 9 = a lkl (1-2g)(y-1)(^H --a (AJ1)

T* '

' 2a(«-3)

so that a contour of constant r may be interpreted as a contour of

constant anomalous specific heat (Schofield, Litster and Ho, 1969).

The restricted linear model has been used to fit experimental data

for magnets (Schofield, Litster and Ho, 1969) and fluids (Hohenberg and

69
Barmatz, 1972; Huang and Ho, 1973; Thoen and Garland, 1974; White and

Maccabee, 1975; Levelt Sengers and Sengers, 1975; Levelt Sengers et al.,

1976) . It can be integrated and also be fitted to the pressure (Murphy

et al ., 1973, 1975).

Restricted linear model parameters for a variety of fluids are

presented in Appendix D.

A. 3 Restricted cubic model .

The restricted cubic model equation of state corresponds to the

choice (Ho and Litster, 1970; Huang and Ho, 1973)

Ay* = a

2 r B6 e(l-e 2

) (A. 12a)

AT* = r(l-b

2

2

e 2

) (A. 12b)

a p * = k

2

r

6 e(l+c e 2

) (A. 12c)

with

b

2 = _JL_ , c

= £££dL (A.12d)

6 3-26 3-26

where a~ and k„ are adjustable constants. Just as in the linear model,

6 = on the critical isochore, 8 = ± l/b~ on the critical isotherm and

9 = * 1 on the coexistence curve.

The restricted cubic model yields for the compressibility the

simple form

X T

* = - r~ Y

(A. 13)

so that a contour of constant r corresponds to a contour of constant

compressibility.

70
When the restricted cubic model is fitted to experimental data the

quality of the representation is comparable to that of the restricted

linear model. Restricted cubic model parameters for a variety of fluids

are presented in Appendix D.

A. 4 Wilcox-Estler model .

The parametric equation introduced by Wilcox and Estler [1971] has

been used to analyze density gradient profiles in the extreme vicinity

of the critical point (Estler et al . , 1975; Hocken and Moldover, 1976).

It is defined through the equations

AW *= -

B r^ YV,e)W(e) (AJ4a)

Y

AT* = r e (A. 14b)

a p * = Y

6 (r,e) [1 + fi W(e)] (A. 14c)

with

W (e) = I

.-" 9/6x (A.14d)

1 - e/e

Y(r.e) = Y Q

r (l-e/e

Q

) A (A. He)

a = 1 - e o /e

x

This equation of state is designed such that

* -Y

X = r '

(A.14f)

(A. 15)

so that a contour of constant r = (l/xT

) corresponds to a contour of

71
constant compressibility as in the restricted cubic model. It has the

additional feature that AT* is a linear function of both r and @.

In this formulation lines of constant are straight lines as indicated

in Fig. A2. In this model = o corresponds to the critical isotherm , while

6=0 on the critical isochore and 0=0 on the coexistence boundary,

o x

J

For the special choice 3A = 3/2 the Wilcox-Estler model reduces to the

restricted cubic model (Estler etal. , 1975)

.

72
CRITICAL POINT

Fig. A2. Coexistence curve, critical isochore and critical isotherms in the

* 1 / plane with coordinate axis r = (1/xj ) and t = AT*.

73
Appendix B. Calculation of density profile .

We assume that at each level z the local chemical potential

y(p(z),T) equals the chemical potential of a system with uniform density

p = p (z) at temperature T in the absence of gravity. The conditions

under which this assumption is valid are discussed in Section 3 of this

report. Since in the presence of gravity the total chemical potential

is the sum of the local chemical potential Ti(p(z),T) and the gravitational

potential mgz, we have

u(p(z),T) - u( P (z ),T) = - mgAz (B.l)

where m is the molecular mass, if the chemical potential is taken per

particle and where Az = z - z . For convenience we take the reference

o | level z as the level where p = p . In terms of dimensionless quantities

we write

Az - - H Ap* < B - 2)

with

H = P

C /(P C mg) (B.3)

The quantity Au* is defined in (A. 3a) and the product p m is the mass

density at the critical point. The quantity H represents a scale height

for the chemical potential of a fluid in a field of gravity g. H

represents the scale height for the chemical potential in the earth's

gravitational field where g = g . In Appendix D we present the scale

factors for the critical isotherm for a number of fluids in the earth's

gravitational field, where

74
D H = DP /p mg ,,-',* o c

K c so (B.4)

(These scale factors for the critical isotherm differ from H by a

numerical factor, D, which varies between 1.2 and 3.2 among various

fluids). Introducing a gravity ratio

g* = g/gQ

(B.5)

we may rewrite (B.2) as

9

AZ = -(-£) Ay* (B.6)

Substitution of the scaled equation of state (A. 5) into (B.6)

yields the relationship between height and density at a given temperature.

In practice we either use the restricted linear model (A. 9), the restricted

cubic model (A. 12), or the Wilcox-Estler model (A. 14).

It follows from (A.l) and (B.6) that the density gradient

(9p*/8z)

T

is given by

In practice the density can be determined as a function of height

using a float densimeter (Greer et al . , 1974) or by measuring the capacitances

between a stack of horizontal conduction plates (Weber, 1970). The

density resolution in such experiments is limited by the variation of

the density over the height h of the float or the distance h between the

two capacitor plates. If the local density is to be obtained with a

precision p we must require that

75
|p*(.Az+h/2) - P *(Az-h/2)h h.|||*| ^p (B.8) 9Z

It thus follows from (B. 7) that errors due to gravity in determining

densities are avoided if

* H

Q

p

XT

= g*h (B.9)

or in terms of the cubic model parameters, using (A. 13),

h

^ Vy

r ;> /a^^V (b.io)

For xenon this condition reads

r^6.2xl0"4 (^) ' 84 (B.ll)

where the height h is to be expressed in m. In particular at the critical

isochore p = p

AT* a 6.2xl0- 4 (2^L)°- 84

(B.12)

and at the critical isotherm T = T

c

Ap*-.,£ 0.076 (i^l) - 3

(B.13)

76
Appendix C. Calculation of correlation length .

C.l. Definition of correlation length .

When a fluid approaches the gas-liquid critical point, its thermodynamic

state is accompanied by large fluctuations in the density. The magnitude

and spatial character of these fluctuations are described in terms of a

correlation function defined as (Stanley, 1971)

p

2 G(|R-R'|) = < p(R)p(R') > - p

2 (C.l)

where p (R) is the local (number) density at position R and

p the average equilibrium density which is independent of the position

R (not considering the presence of external forces such as gravity)

.

The zeroth order moment of the correlation function is related to the

isothermal compressibility by the fluctuation theorem

k

B

T x T

= P

2 /dR G(R) (C2)

The correlation function G(R) = G(Ap*,AT*;R) is a function

Ap* and AT* as well as of R (where R = |R - R'| ).

The spatial extent of the fluctuations is characterized by a correlation

length %. It is defined as (Fisher, 1964, 1967)

F

2 _ 1 /dR R

2 G(R)

JdR G(R)

The correlation length g. diverges at the critical point. In particular

along the critical isochore it follows the power law

(T ^ T

c

, p = p c

) £ = £ o (AT*)~

V

(C.4)

77
Another exponent n is introduced to specify the nature of the

dependence of the correlation function on the distance R. It is defined

such that at the critical point Ap* = 0, AT* =

G(o,o;R) «

; R

1

^

The exponent is zero in the classical theory of Ornstein and Zernike

(Fisher, 1964). In current theories n is small, but finite.

The correlation function exponents v and n are related to the

thermodynamic critical exponents a, 3, y,& introduced in Appendix A by

the relations (Fisher, 1967; Widom, 1974)

(C6)

(C.7)

(C.8)

The relations (C.7) and (C.8) are sometimes referred to as hyperscaling

relations (Levelt Sengers and Sengers, 1975).

Y ~ v(2-n)

3v == 2-ct

2-n

3

6-1

6+1

78
C. 2 Correlation length as a function of density and temperature .

The correlation length £ = £(Ap*,AT*) is a function of density and

temperature. The hypothesis of scaling for the thermodynamic behavior

can be extended to the correlation function G(Ap*, AT*;R) as a function

of Ap*, AT* and R (Kadanoff, 1966). The scaling hypothesis implies that

the correlation length £ can be written in the form (Sengers and Levelt

Sengers, 1976)

1

5 = C Q

R(x/x

Q

) (r

_1 xT

*)

U (C9)

i / ft where x = AT*/|Ap*| is the thermodynamic scaling variable introduced

in (A. 6) and x = - AT*/ Up* | 1 ^= B , where B is the amplitude of

o cxc

the power law (A. 4b) for the coexistence curve. The function R(x/x ) is

a universal function such that R(°°) = 1 at the critical isochore x = °°.

The amplitude £ can be deduced from light scattering data (Chu, 1972)

.

In this report we have calculated the correlation length in the

approximation that R(x/x ) = 1 independent of x

1

5 = 5 (r

_1 xT

*)

^

(CIO)

This approximation, though not strictly valid, is adequate for the

purpose of this report (Sengers and Levelt Sengers, 1976). In this

approximation a contour of constant compressibility xS coincides with a

contour of constant correlation length £. For this reason we found it

convenient to calculate £ using the restricted cubic model equation of

state (A. 12) for which

79
e = 5 r"

v (C.lla)

AT* = r(l-b| e

2

) (C.llb)

ap* = k

2 r

6 e(l+ c e

2

) (C.llc)

with b

? and c again given by (A.12d).

80
C. 3 Correlation length as a function of height .

Once the density profile and density gradient profile has been

calculated from (B.6) and (B.7), the correlation length as a function of

height follows from (CIO). In terms of the parameters of the restricted

cubic model the relationship between the correlation length £ and the

height z at a given temperature AT* is determined by

5 = S |AT*r |1 - b| e 2| v (C12a)

az= -% a

2 |AT*| 36

|1 - b| e2

I -66

6(1-6^) (c . 12b)

9

The correlation length assumes its maximum value at the level

Az = (8 = 0) corresponding to the critical density. The height az at

which the correlation length will be reduced by a factor 1-p is determined

by the conditions

V ± b

2

_1

C

1 - 0-P)

lA ]" 2 -i b

2 "W (C13)

so that

t

2 H a_ i V d - p

Az 1 -1 fyS ( 2

} (ca4)

P

g b

2 V

v b/- b/ B6 P

- - 4 / (t-)

s

i AT*i

g

D

2

v

The rate at which the correlation length £ at arbitrary levels will

vary as a function of height is determined by

81
C4-(^feV-?«(&) T

(CJ5)

In terms of the parameters of the restricted cubic model this equation

becomes

m-(f)(P-) ,.-<*«> r V! i

(c . 16)

V

dZ

/T W\a

2 / [266 b

2

2

e

2 (l-e 2 )+(l-3e 2 )(l-b

2

2

e 2 )J

In particular at the critical temperature AT* = 0, (0 = ± b„ ) this

(|i) = + (Sl\rJl) r-(v+Bi) b2"

,. „ .

gradient will vary as

3

(b

2"-l)

with r(A p *) determined by

2 _wu 3

A p* = k

2

r

6 (b^ + c)/b

2

J (C.17b)

The correlation length is the fundamental length scale which then

determines the anomalous behavior of thermodynamic and transport properties

near the critical point. In actual experiments these properties are

measured as averages over a finite height h. Such experiments will be

reliable when the correlation length over this height does not vary

within the precision p desired. Thus in order to avoid errors due to

gravity we require that

82
|?(az + h/2) - (az - h/2)|s p 5(42) (C.I8)

which we approximate as

1 /9£\

5 V>zjT

h

Upon substituting (C.lla) and (C.16) into (C.19) we conclude that the

error? due to gravity will become appreciable unless

.Bi a ( gih

)( v

j H P "2

2

2t> 2

9

236 b

2

2 6 (!-n2)+(l-36 2 )(l-b

2

2

6

2

)

(C.20)

When combined with (C.llb) and (C.llc) this equation defines a range in

the AT* - Ap* plane where the measurements become inaccurate due to

gravity effects. In particular at the critical temperature AT* = 0,

this condition becomes

* ? j

f l (C.21)

where D is the amplitude of the critical isotherm defined in (A.4d). In

particular for xenon

83
|Ap*| *-0.16 C^1-)

- 23

(C.22)

where h is to be expressed in meters.

At the critical isochore Ap* = at T ^ T , (H/Sz) = and (C.19)

is not a good approximation to (C.18). The distance Az over which the

correlation length is reduced by a factor 1-p is given by (C.14). In

order to avoid gravity effects we must require that this distance be

larger than the height (strictly h/2) over which the properties are

measured

A2J * r

P 2 (C.23)

It thus follows from (C.14) and (C.23) that at the critical isochore

AT* | ^ f£^L 12

( v

}

b

2 « V2

[f

H

o

d 2

P

-, 1/36

(C.24'

In particular for xenon

-3 /q*h\ ' 65 AT*|, & 1.85xl0" 3 fi*ll\

VP J

(C25)

where h is to be expressed in meters.

84
Table C.l. Critical Region Parameters for a Number of

Fluids Assuming Universal Effective Exponents

Critica

P c MPa

1 Point Parameters

p c% T

c kg/nr K

Restri

Linear

k

l

cted

Model

a

l

Restricted

Cubic Model

k2

a?

OH,*

m

3

He 0.11678 41.45 •3.3099 0.924 4.58 0.818 4.05 906

4

He 0.22742 69.6 5.1895 1.021 6.40 0.904 5.66 946

Ar 4.865 535 150.725 1.309 16.1 1.160 14.2 2260

Kr 5.4931 908 209.286 1.309 16.1 1.160 14.2 1500

Xe 5.8400 1110 289.734 1.309 16.1 1.160 14.2 1310

P-H2 1.285 31.39 32.935 1.156 9.6 1.024 8.5 10410

N

2 3.398 313.9 126.24 1.361 18.2 1.206 15.1 2560

°2 5.043 436.2 154.580 1.309 15.6 1.160 13.9 2790

H

2 22.06 322.2 647.13 1 .622 21.6 1.438 19.1 8980

D

2 21.66 357 643.89 1.622 21.6 1.43819.1 7950

co

2 7.3753 467.8 304.127 1.436 21.3 1.273 18.9 3470

NH

3 11.303 235 405.4 1.573 21.4 1.394 19.1 7134

SF

6

3.7605 730 318.687 1.337 23.9 1.185 21.2 1730

CH

4

4.595 162.7 190.555 1.361 17.0 1.206 15.1 6260

C

2

H 4 5.0390 215 282.344 1.350 17.5 1.197 15.5 5540

C

2 H

g 4.8718 206.5 305.33 1.416 20.2 1.255 17*9 5520

C

3

H8 4.247 221 369.82 1.451 20.2 1.286 17.9 3830

Notes : a = 0. 100 6 = 0.355 b

2

2

= 1.3100

Y - 1. 190 6 = 4.352 >,

2 = 1.3909 c = 0.0393

V = 0. 633 n

= 0.12

The last quantity tabulated, DH = D P /(p mg ) is the scale factor for the density

O Q " ** vs height profile at the critical temperature. D is found from the relation:

D = a

lkl b

l

^ b

l ~ 1 ^' which ls taken from Sengers et al. [1976].

85
Appendix D. Physical constants for various fluids .

In this Appendix we present critical parameters and critical region

parameters for a number of fluids, taken from a survey prepared by

Sengers and Levelt Sengers [1977]. The restricted linear model and

cubic model parameters in this table represent informed estimates assuming

universal effective critical experiments corresponding to the range

5xl0~ 4 s|£T*! <; 3xl0~ 2

, |ap *! £ 0.25 (D.l)

86
Appendix E. Parameters used for xenon in this report .

Critical point parameters (Levelt Sengers et al., 1976)

P = 5.8400 MPa

c

3

mp = 1110 kg/m

c T = 289.734 K

c H = P /p g = 536 m

o , c c°o

Restricted linear model (Levelt Sengers et al., 1976)

a = 0.100 A = 1.71 In = 1.309

3 = 0.355 B = 1.827 a = 16.1

Y = 1.19 r = 0.0813 b

±

2 = 1.3909

6 = 4.352 D = 2.44

Restricted cubic model (Sengers and Levelt Sengers, 1976)

a = 0.100 A = 1.68 k

2 = 1.160

3 = 0.355 B = 1.827 a

£ = 14.2

Y = 1.19 r = 0.0817 b

2

2= 1.3100

6 = 4.352 D = 2.44 c = 0.0393

Wilcox-Estler model (Hocken and Moldover, 1976)

n = 1.1379 3 = 0.3293 6 = 0.1076

c o n '= 0.14 Y = 1-229 Y

3 = 0.323

c

' o

A = 4.4

Correlation length (Swinney and Henry, 1973)

v = 0.633 £ = 2.0xl0~10

m.

o

87
Acknowled gment s

The authors have benefitted from many stimulating discussions with

Dr. J.M.H. Levelt Sengers. J.V. Sengers is also indebted to Professor

J.M.L. Van Leeuwen for some valuable observations concerning gravity

effects in fluids near the critical point.

88
Nomenclature

A = amplitude of power law for specific heat C

a.- - parameter of linear model

a« = parameter of cubic model

B = amplitude of power law for coexistence curve

b. = parameter of linear model

b„ = parameter of cubic model

C = constant-volume heat capacity

C * = CT/VP V V c c c

c = parameter of cubic model

D = amplitude of power law for chemical potential along critical

isotherm

d = sample thickness

g(R) = correlation function as a function of distance

g = gravitational acceleration

2 g = 9.80 m/s = earth's gravitational acceleration (units)

g* = g/g = gravitation acceleration relative to its value at the

earth's surface

h = sample height

H = P /p mg = scale height for chemical potential

H = P /p mg = scale height for chemical potential on earth

k =. wave vector

IC = p (3p/9P) = isothermal compressibility

k = Boltzmann's constant

k, = parameter of linear model

k„ = parameter of cubic model

m = molecular mass

n = refractive index

89
n = refractive index at critical point

n

± = (3n/9p*)

T n = n = value of nn at critical point

c c 1

K

P = Pressure

P - critical pressure

c

r

P* = P/P

c AP* = (P-P )/P

c c

p = desired precision

r = parametric variable

R = spatial coordinate

R = magnitude of spatial coordinate

R = scaling function for correlation length (Appendix C.2)

S = structure factor

T = temperature in Kelvin

T = critical temperature

T* = t/t

c AT* = (T-T )/T

c c

t = AT*

V = volume

x = scaled thermodynamic variable AT*/J.Ap*| or scaled wave vector k £

x - B"1/B

o Y = parameter of Wilcox-Estler model

z = height

z = height at which p = p

Az = z-z

90
Greek letters

a = exponent of power law for specific heat C

6 = exponent of power law for coexistence curve

Y = exponent of power law for compressibility xT

T = amplitude of power law for compressibility x T

T

g = decay rate of entropy fluctuations

6 = exponent of power law for chemical potential along critical

isotherm

£ = dielectric constant

A= 1 - 9 /9 = parameter of Wilcox-Estler model ox

n = exponent for spatial dependence of correlation function

at the critical point

9 = parametric variable

o value of 9 on critical isochore in Wilcox-Estler model

value of 9 on coexistence curve in Wilcox-Estler model

x

£ = correlation length

£ = amplitude of power law for correlation length

A = wave length

y = chemical potential per particle

u* = yp

c /P

c Ay* = {y(p,T) - y (P c> T) >P

C

/P

C

v = exponent of power law for correlation length

p = number density

P c = critical density (particles/volume)

p = average density

P* = p/p

c Ap* = (p - p )/p„

c c

91
D = density of vapor or liquid at coexistence

cxc

*

/ p = P /P cxc cxc c

Ap* = (p - p )/p cxc cxc c c

a = cross section area

t = turbidity

XT

= (3p/3y)

T

= p 2 Kr[

X* = (3p*/9y*) = xT

P

c /Pc

co = frequency

92
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96
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U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR RICPORT NO. NBS TN-925

2. Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE Overviews and Justifications for Low Gravity Experiments

on Phase Transition and Critical Phenomena in Fluids

5. Publication Date October 1976

6- Performing Organization Code

7. author(S) M.R. Moldover, R.J. Hocken, R.W. Gammon,

and J. V. Sengers

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

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11. Contract/Grant No. NASA-No. C-62861-C

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45. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less {actual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) Important scientific questions concerning pure fluids and fluid mixtures near

critical points are identified and are related to the progress of several disciplines.

Consideration is given to questions about thermodynamic properties, transport

properties, and the complex nonlinear phenomena which occur when fluids undergo

phase transitions in the critical region. We discuss, quantitatively, the limitations

to answering these questions by experiments in the earth's gravitational field. The

distinction is made between practical limits which may be extended by advances in

technology and intrinsic ones which arise from the modification of fluid properties

by the earth's gravitational field. The kinds of experiments near critical points

which could best exploit the low gravity environment of an orbiting laboratory are

identified. These include studies of the index of refraction, constant volume

specific heat, and phase separation.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Critical point; gravity effects; phase transitions.

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