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An optical study of the critical behaviour of CHF₃ Stein, Michael Paul 1986

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AN OPTICAL STUDY OF THE CRITICAL BEHAVIOUR OF CHF3 By MICHAEL PAUL STEIN B. S c , The Univ e r s i t y of Manitoba, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1986 © Michael Paul Stein, 1986 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f PHYSICS  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Dat e APRIL 29, 1986 -7Q \. To the memory of my grandmother, Clara Stein, and my grandfather, Ross McNaught i i ABSTRACT The coexistence curve for CHF3, measured in the reduced temperature range 4 x ICT6 < -t < 3.5 x l O - 2 by f ocal plane interference methods, has been analyzed in terms of the correction to scaling prediction ( p A - P v ) / P c = B 0 | t | ^ ( l + B i M A + B 2 | t | 2 A ) to yield values of the cr i t ical exponent 6 , the cr i t ical temperature T c > and the amplitudes B Q , B 1 and B 2 . An image plane interference technique has been employed in the reduced temperature range 1.7 x 10 - 5 < t < 2 x lO-1* to obtain pressure-density isotherms for CHF3. These isotherms are used to determine a value of the crit ical exponent y defined by the compressibility power law expression < T = r 0 | t p . i i i TABLE OF CONTENTS Chapter Page ABSTRACT i i i TABLE OF CONTENTS i v LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS v i i i 1 INTRODUCTION 1 2 THEORY 7 2.1 Introduction to Theories of Phase Transitions 7 2.2 C l a s s i c a l Theories 14 2.3 Modern Theories 22 3 OPTICS 28 3.1 Introduction 28 3.2 Fraunhofer D i f f r a c t i o n ' 30 3.3 Image Plane Interference 35 3.4 The Lorentz-Lorenz Relationship 36 4 THE EXPERIMENT 38 4.1 The C e l l 38 4.2 Temperature Control 41 4.3 Experimental Procedure 44 i v Chapter Page 5 RESULTS 48 5.1 Fraunhofer Diffraction Pattern 48 5.2 Image Plane Pattern 56 6 CONCLUSION 61 REFERENCES 63 v LIST OF TABLES Table Page 1 . Experimental and t h e o r e t i c a l values of the c r i t i c a l exponents a, B , y and 6 20 v i LIST OF FIGURES Fig- Page 1. The pure f l u i d phase diagram 8 2. Some possible paths on the phase diagram 10 3. Ideal gas isotherms 15 4. Van der Waals isotherms 17 5. Density p r o f i l e s of a pure f l u i d i n the earth's g r a v i t a t i o n a l f i e l d 29 6. The Fraunhofer d i f f r a c t i o n pattern 31 7. The o p t i c a l set-up 34 8. The c e l l 39 9. The temperature co n t r o l c i r c u i t 43 10. The pulse-width modulation c i r c u i t used to co n t r o l the bath temperature 45 11. Fringe pattern samples 49 12. Log-log plot of order parameter, ( p ^ - p v ) / p c , versus reduced temperature, t = (T - T )/T 53 c c 0 • 327 13. Sensitive log-log p l o t of ( p ^ - p v ) / p c | t | versus reduced temperature, t = (T - T^/T^ * 54 14. Plot of P Q - P versus reduced temperature, t = (T - T )/T 58 15. Plot of isothermal com p r e s s i b i l i t y , K T , versus reduced temperature, t = (T - T £)/T 59 v i i ACKNOWLEDGEMENTS It has been a pleasure to work i n the c r i t i c a l phenomena lab at UBC I am very g r a t e f u l to Dr. David B a l z a r i n i for th i s opportunity and for h i s supervision of my work on t h i s p r o ject. Friends have helped a great deal. Among them, I would l i k e to thank Barbara Frisken, my dance partner; Nick Mortimer, my draftsman; and Tony "Spud" Noble, my auto mechanic. I o f f e r a s p e c i a l thanks to my colleague John de Bruyn, who has allowed me to tap his knowledge and experience i n the lab. His assistance and encouragement were invaluable. I a p p r e c i a t i v e l y acknowledge f i n a n c i a l support received from the Natural Sciences and Engineering Research Council of Canada. Thanks also to Dr. Birger Bergersen for h e l p f u l comments and c r i t i c i s m s , to Mark Halpern for c u t t i n g and pasting, and to Antoinette Tse, who typed t h i s manuscript. F i n a l l y , I wish to thank my parents, Paul and Helen, and my s i s t e r , Shannon. Their u n f a i l i n g support of my work has been of great benefit to me. v i i i 1 Chapter 1 INTRODUCTION This thesis describes two optical experiments that were performed to study the liquid-vapour phase transition in the pure f l u i d CHF3. In the f i r s t experiment, the subcritical temperature dependence of the difference i n liquid and vapour densities, - P v > w a s determined from an analysis of a Fraunhofer diffraction pattern produced by a collimated laser beam that traversed a thin CHF3 sample c e l l . In the second experiment, the sample c e l l was placed in one arm of a Mach-Zehnder interferometer; the resultant interference pattern, recorded as the sample was heated through T , was used c to obtain pressure-density isotherms. These in turn yielded the temperature dependence of the isothermal compressibility, K ^ , , along the c r i t i c a l isochore. Near T^, both quantities, - P v an& K T> a r e expected to obey power laws: where t = (T - T^/T^ is the reduced temperature. The parameters 8 and y are c r i t i c a l exponents which are predicted to be universal for a l l pure fluids and, in fact, for other types of systems as well. A major objective of the two experiments was to determine values for 3 and y. ? x " P v ~ | t | P (1.1) (1.2) 2 The measurement of these and other c r i t i c a l exponents i s p a r t i c u l a r l y d i f f i c u l t i n pure f l u i d s , for which the c r i t i c a l region i s known to be very small; i . e . , the power law behaviour expressed i n Eqs. (1.1) and (1.2) appears to hold only for |t| < 10 - 5 . Furthermore, s i g n i f i c a n t density gradients near the c r i t i c a l point make conventional pressure-volume methods awkward and u n r e l i a b l e . O p t i c a l interference methods, on the other hand, when used i n conjunction with a stable and precise temperature co n t r o l system, are p a r t i c u l a r l y w e l l - s u i t e d to making high p r e c i s i o n measurements i n the c r i t i c a l region of pure f l u i d s . The d e f l e c t i o n of i n d i v i d u a l l i g h t rays traversing a t h i n sample i s s u f f i c i e n t l y small that each ray probes a region of e s s e n t i a l l y constant density^. The second major goal of t h i s work was to investigate the existence of corrections to Eqs. (1.1) and (1.2) that are expected when measurements are made outside the asymptotic c r i t i c a l region. Current theories predict that, as the distance from T^ i s increased, i t i s necessary to modify the power law dependences of thermodynamic quantities by the addition of so-called c o r r e c t i o n - t o - s c a l l n g terms. A d e t a i l e d analysis of the Fraunhofer d i f f r a c t i o n data was performed i n an attempt to confirm and measure the departure from power law behaviour exhibited by the density d i f f e r e n c e PA " p v " The remainder of t h i s introduction i s devoted to a h i s t o r i c a l survey which w i l l help to i l l u s t r a t e how the experiments of t h i s thesis f i t i n t o the o v e r a l l picture of c r i t i c a l phenomena. Research i n the f i e l d of c r i t i c a l phenomena was triggered i n the l a t e 2 1860's by the experimental work of Thomas Andrews , who found that 3 liquid-vapour coexistence i n C0 2 i s not observed above a c r i t i c a l temperature T c» 32°C. A t h e o r e t i c a l explanation of t h i s phenomenon was proposed by 3 van der Waals i n 1873. Van der Waals derived an equation of state that described various aspects of the liquid-vapour phase t r a n s i t i o n i n a pure f l u i d . The shapes of isotherms and the coexistence curve were predicted, as was the c r i t i c a l region behaviour of c e r t a i n thermodynamic properties. The van der Waals theory i s discussed i n Sec. 2.2. Similar developments took place i n the f i e l d of ferromagnetism^. The existence i n i r o n of a c r i t i c a l temperature above which spontaneous magnetization did not occur was f i r s t reported by Hopkinson i n 1890, and thoroughly investigated by Curie i n 1895. In 1907, Weiss formulated a theory to account for the e f f e c t . In many respects, the Weiss theory was analogous to that of van der Waals. In f a c t , they are both c h a r a c t e r i s t i c examples of large class of theories known as c l a s s i c a l or mean f i e l d theories of c r i t i c a l phenomena. Such theories have been devised for a v a r i e t y of systems ex h i b i t i n g phase t r a n s i t i o n s and, when f i r s t developed, they appeared to be i n approximate agreement with the l i m i t e d and somewhat crude experimental 4 r e s u l t s a v a i l a b l e at the time. Of p a r t i c u l a r importance was van der Waals 1 law of corresponding states — the f i r s t formal statement concerning the u n i v e r s a l features that pervade the study of c r i t i c a l phenomena. With the advent of more numerous and sophisticated experiments, however, i t gradually became apparent i n the early decades of the twentieth century that the predictions of c l a s s i c a l theories were not i n quantitative agreement with experimental r e s u l t s . The most convincing evidence of discrepancies between theory and experiment came i n 1945 from Guggenheim's^ 4 d e t a i l e d study of the coexistence curves of various pure f l u i d s . Guggenheim obtained a value of 8 = 1/3 which c l e a r l y disagreed with the c l a s s i c a l value 8 = 1/2. The c l a s s i c a l theories presented some important q u a l i t a t i v e concepts — p a r t i c u l a r l y that of u n i v e r s a l i t y — but lacked some fundamental Ingredients. Attempts to modify or extend these theories did not succeed. A new and more complete type of theory was required for a r e a l i s t i c explanation of c r i t i c a l phenomena. The f i r s t steps towards a new approach were studies of the Ising model. Developed i n 1925, the Ising model was i n i t i a l l y intended to describe ferromagnetic systems. I t has since turned out to be an extremely important model, applicable to a v a r i e t y of systems, including antiferromagnets, l a t t i c e gases and DNA molecules. In i t s simplest form, the Ising model consists of a l a t t i c e of spins, each spin i n t e r a c t i n g with i t s nearest neighbour; the coupling energy i s of constant magnitude, i t s sign depending on whether adjacent spins are p a r a l l e l or a n t i p a r a l l e l . The one-dimensional Ising model shows no phase t r a n s i t i o n , but the two-dimensional version, solved a n a l y t i c a l l y by Onsager^ i n 1944, does predict a phase t r a n s i t i o n . While an a n a l y t i c s o l u t i o n to the three-dimensional case has not been found, there are continuing e f f o r t s to r e f i n e numerical solutions to the problem. In p a r t i c u l a r , there has been considerable success i n c a l c u l a t i n g 3-D I s i n g model c r i t i c a l exponents which, i t turns out, are i n better agreement with experimental r e s u l t s than c l a s s i c a l exponents^. Developments i n modern theories pertaining to pure f l u i d s started i n g 1952 when Lee and Yang proved that the l a t t i c e gas model i s mathematically 5 isomorphic to the Ising model. More recently, i t has been shown that pure f l u i d s , binary f l u i d s , l a t t i c e gases and I s i n g - l i k e spin systems a l l belong to the same u n i v e r s a l i t y c l a s s ; i . e . , they each exhibit c r i t i c a l behaviour that may be treated by the same mathematical techniques, with the same res u l t s obtained for each type of system. Further s i g n i f i c a n t t h e o r e t i c a l developments were introduced with 9 10 Widom's s c a l i n g theory (1965) and Kadanoff's a p p l i c a t i o n of s c a l i n g theory to spin systems (1966). In 1971, Wilson*^ formulated the renormalization group approach, a powerful mathematical technique that i s used to c a l c u l a t e properties of systems near the c r i t i c a l point. The basic concepts behind these contemporary theories are outlined i n Sec. 2.3. There have been equally s u b s t a n t i a l improvements i n experimental methods used to study phase t r a n s i t i o n s . As explained e a r l i e r , the common goal i s to make precise measurements as near to the c r i t i c a l point as possible. For studies of pure f l u i d s , i t has been necessary to f i n d a l t e r n a t i v e s to the conventional pressure-volume measurements which become exceedingly d i f f i c u l t i n the c r i t i c a l region. Buoyant spheres were used by 12 Maass i n 1938 to study the density d i s t r i b u t i o n i n C0 2. In 1952, Habgood 13 and Schneider obtained s i m i l a r information for Xe using a r a d i a c t i v e 14 trac e r . More recently Pittman et^ al_ used a sample c e l l containing two small p a r a l l e l plate capacitors to make d i e l e c t r i c constant measurements i n 3He and a 3He - ^He mixture. Pestak and Chan^ have used t h i s capacitance method, with a four-capacitor c e l l , to make high p r e c i s i o n measurements i n N 2 and Ne. They obtained values of 8, y and correction terms that are i n good agreement with current theories. Yet another approach i s the l i g h t 16 s c a t t e r i n g method used by Guttinger and Cannell i n 1981 to i n v e s t i g a t e corrections to s c a l i n g i n Xe. Among the most elegant and accurate means of i n v e s t i g a t i n g c r i t i c a l point behaviour are methods using o p t i c a l interference techniques. L o r e n t z e n ^ was the f i r s t to t r y t h i s approach when, i n 1953, he examined the c r i t i c a l region density d i s t r i b u t i o n of carbon dioxide by measuring the d i f f r a c t i o n of p a r a l l e l beams of l i g h t passing through a prismatic sample 18 v e s s e l . In 1954, Palmer used a Schlieren o p t i c a l system and a sample c e l l with plane p a r a l l e l windows to study C0 2, CjHg and Xe. High p r e c i s i o n interferometric measurements s i m i l a r to those described i n t h i s thesis were 19 20 f i r s t performed by Wilcox and B a l z a r i n i ' i n 1968 to study the c r i t i c a l region properties of xenon. These techniques have since been used by 21 B a l z a r i n i and Ohm to measure i n SF g a temperature dependence i n the e f f e c t i v e value of B , thus demonstrating the need for corrections to s c a l i n g , 22 and by Hocken and Moldover to investigate Xe, SFfi and C0 2 s u f f i c i e n t l y close to t h e i r c r i t i c a l points to obtain values of p consistent with 23 t h e o r e t i c a l r e s u l t s . In 1983, B a l z a r i n i and Mouritsen used the same interference methods to examine un i v e r s a l r a t i o s of c o r r e c t i o n amplitudes i n Xe. While the existence of corrections to s c a l i n g i s now well-established, more data must be c o l l e c t e d i n order to make an unambiguous evaluation of the predicted quantitative nature of these corrections. Studies such as the one presented here should help to make such an evaluation possible. 7 Chapter 2 THEORY 2.1. Introduction to Phase Tra n s i t i o n s A pure f l u i d system i s one co n s i s t i n g of only one type of molecule. Its state may be defined by the thermodynamic variables pressure P, density p, and temperature T. Depending on the values of these v a r i a b l e s , the f l u i d can e x i s t i n any one of three possible phases: vapour, l i q u i d or s o l i d . The r e l a t i o n s h i p between a p a r t i c u l a r f l u i d ' s phases and the variables P, p and T may be i l l u s t r a t e d by a phase diagram. The phase diagram of a t y p i c a l pure f l u i d i s shown i n F i g . 1(a). The three phases are separated by l i n e s c a l l e d coexistence curves. Along these curves, two phases coexist i n equilibrium. At the point A, the t r i p l e point, a l l three phases coe x i s t . The point C, at which the liquid-vapour coexistence curve terminates, i s c a l l e d the c r i t i c a l point. The values of pressure, density and temperature at t h i s point are termed c r i t i c a l and are denoted by P c > p £ and T c« The r o l e of the c r i t i c a l point w i l l be discussed l a t e r . The l i q u i d and vapour phases of a pure f l u i d may also be examined i n the pressure density plane, as i n F i g . 1(b). The s o l i d l i n e i s the liquid-vapour coexistence curve. The dashed l i n e s are isotherms — l i n e s of constant temperature. The c r i t i c a l isotherm, T = T , i s the one which i s tangent to the top of the coexistence curve. The region in s i d e the coexistence curve i s thermodynamically unstable. A f l u i d cannot e x i s t , i n equilibrium, i n a state l y i n g i n t h i s region, say at point G with density p and pressure P . Rather, i t w i l l separate out into vapour and l i q u i d phases 9 of densities and . In the region outside the coexistence curve, there is no distinction made between liquid and vapour. The labels "liquid" and "vapour" in Fig. 1(b) refer only to the phase densities corresponding to the two sides of the coexistence curve. While i t is possible, as w i l l be shown, to take a system from state D to state E by traversing the coexistence curve and, hence, undergoing a phase transition, the same result may be attained by a process which circumvents the coexistence curve so that no phase transition occurs. For this reason, the system i s considered to exist off the coexistence curve simply as a homogeneous f l u i d . As an example of a liquid-vapour phase transition, consider the process depicted in the phase diagram of Fig. 2(a). The system starts at point J with pressure P and density p . Increasing the temperature, with O J the pressure held constant, takes the system to the coexistence curve, where phase separation occurs; i.e., the fl u i d separates into phases of density and p ^ , corresponding to the points labelled K. At this stage, heat may be added to the system Isothermally un t i l a l l of the liquid boils away. This heat is referred to as the latent heat of the transition. Finally, i f the temperature is increased further, the system moves toward point L as a homogeneous f l u i d . Another possible path on the phase diagram is shown in Fig. 2(b). Starting at point M with density p M and pressure ? M, the temperature is decreased, at constant density, un t i l the coexistence curve is reached at N, and the system separates into vapour and liquid phases with densities ^v = a t U* ^ ^ S t* 1 6 t e m P e r a t u r e *-s further reduced, the system moves down the sides of the coexistence curve, with the average density, p , remaining 2. Some possible paths on the phase diag 11 constant. I t i s often u s e f u l to study phase t r a n s i t i o n s by looking at the behaviour of a p a r t i c u l a r quantity c a l l e d the order parameter. As the name suggests, the value of t h i s quantity i s a numerical measure of the amount and kind of ordering associated with a p a r t i c u l a r state of a system. For pure f l u i d systems, the order parameter i s defined to be the difference i n l i q u i d and vapour den s i t i e s divided by the c r i t i c a l density, (p^ - P v ) / p c * * n t n e two examples described above, the order parameter i s discontinuous across the coexistence curve: i t s value i s zero when the system f i r s t reaches the coexistence curve, but non-zero as soon as phase separation takes place. Such processes, for which the order parameter jumps discontinuously from zero, are known as f i r s t order phase t r a n s i t i o n s . They are also characterized by a latent heat of t r a n s i t i o n . Now consider the path shown i n F i g . 2(c). I t i s e s s e n t i a l l y the same process as the one i n F i g . 2(b), except that the system s t a r t s with density equal to the c r i t i c a l density p . The temperature i s lowered, with density f i x e d , u n t i l the coexistence curve i s reached at the c r i t i c a l point C, where T = T^ and P = P^ and the order parameter i s zero. Here, unlike the previous two examples, there i s no jump i n the value of the order parameter. As the temperature i s lowered from T c > phase separation occurs and (p^ - p^)/p c grows continuously from zero. Processes of t h i s type are c a l l e d second order t r a n s i t i o n s and are characterized by an order parameter that i s continuous through the t r a n s i t i o n . As w e l l , they have no associated latent heat. The experiment described i n t h i s thesis deals with a second order phase 12 t r a n s i t i o n i n a pure f l u i d . The discussion from t h i s point on w i l l focus on t r a n s i t i o n s of t h i s type. A major objective i n the f i e l d of c r i t i c a l phenomena i s to study the behaviour of thermodynamic properties as close as possible to the c r i t i c a l point. I t i s i n t h i s region, the c r i t i c a l region, that c l a s s i c a l theories tend to break down. And, as w i l l be seen, the success of modern theories i s determined l a r g e l y by t h e i r a b i l i t y to q u a n t i t a t i v e l y predict the way i n which thermodynamic properties approach the c r i t i c a l point. Furthermore, since the ultimate test of any theory i s experiment, considerable e f f o r t s are being made by experimentalists to devise methods of probing systems very near t h e i r c r i t i c a l points. A conventional means of c h a r a c t e r i z i n g the c r i t i c a l region behaviour of a system i s v i a c r i t i c a l exponents. As T approaches T , most thermodynamic qua n t i t i e s are expected to exhibit a power law dependence on temperature or other state v a r i a b l e . For example, the order parameter for a pure f l u i d may be expressed near T as (2.1) where 6 i s the associated c r i t i c a l exponent and t = (T - T )/T i s the c c reduced temperature. Other pure f l u i d properties expected to show c r i t i c a l power law behaviour include the following: (2.2) 13 for p = p (t > 0) or on the coexistence curve (t < 0); K T p^dP^T ~ I * ' (2.3) for p = p (t > 0) or on the coexistence curve (t < 0); P - P 1/6 ( (2.4) c for t = 0. In the above r e l a t i o n s , a, y a n a < 6 are the c r i t i c a l exponents associated with the s p e c i f i c heat at constant volume, C^, the isothermal compressibility, K „ , and the reduced density, (p - p )/p . The variables S l c c and V r e f e r to entropy and volume. C r i t i c a l exponents may be defined analogously for systems other than the pure f l u i d . In a ferromagnetic system, for example, 6 i s defined by where the magnetization m(T) i s the order parameter for such a system. As w i l l be demonstrated i n the following sections, c r i t i c a l exponents play a v i t a l r o l e i n the f i e l d of c r i t i c a l phenomena. Much work i s being done to develop theories that predict t h e i r values and experiments that measure t h e i r values. Since they emerge from theories as u n i v e r s a l constants, i t i s of p a r t i c u l a r i n t e r e s t to examine the c r i t i c a l exponents of d i f f e r e n t thermodynamic systems (e.g., pure f l u i d s , ferromagnets, binary m(T) (-t) 1 14 f l u i d s , l i q u i d c r y s t a l s , e t c ) as w e l l as of a var i e t y of materials f o r each type of system. Such comparisons should i n d i c a t e the extent to which the c r i t i c a l exponents are, i n f a c t , u n i v e r s a l . 2.2 C l a s s i c a l Theories C l a s s i c a l or mean f i e l d theories of phase t r a n s i t i o n s generally f a i l In making quantitative predictions that are i n agreement with experimental r e s u l t s near a phase t r a n s i t i o n . Nevertheless, i t i s i n s t r u c t i v e to examine such theories since they do i l l u s t r a t e many fundamental aspects of c r i t i c a l phenomena, and often serve as a s t a r t i n g point for more complex theories. The following i s a b r i e f survey of some of the c l a s s i c a l theories that p e r t a i n to pure f l u i d systems. The simplest model used to describe a pure f l u i d i s the i d e a l gas. Defined to consist of non-interacting point p a r t i c l e s , the i d e a l gas obeys the empirical equation of state P = p k B T. (2.5) Here k_ i s the Boltzmann constant and p i s the number of p a r t i c l e s per u n i t volume. Ideal gas isotherms, as shown i n F i g . 3, are s t r a i g h t l i n e s which i n t e r s e c t the o r i g i n of the pressure-density coordinate system. The i d e a l gas law predicts no phase t r a n s i t i o n s and i s r e a l l y only u s e f u l for d i l u t e gases. As the density increases, the in t e r a c t i o n s and f i n i t e s i z e of p a r t i c l e s cannot be neglected. In 1873, van der Waals* formulated the f i r s t equation of state that i s Fig. 3. Ideal gas isotherms (T^ < T 2 < T 3 ) . 15 16 app l i c a b l e to both vapour and l i q u i d phases, and exhibits a second order phase t r a n s i t i o n . The van der Waals equation of state may be developed by making the following refinements to the i d e a l gas law. To begin with, the pressure of a r e a l gas i s s l i g h t l y less than that of an i d e a l gas, by an amount proportional to the square of the density of p a r t i c l e s . As w e l l , the p a r t i c l e s of a r e a l gas are of f i n i t e s i z e , unlike the point p a r t i c l e s of an i d e a l gas. Hence, the volume of a r e a l gas w i l l tend to a f i n i t e volume as the pressure i s increased, not to zero as predicted by the i d e a l gas law. With these modifications, van der Waals obtained the equation of state (P + ap 2)(^- - b) = k B T (2.6) where ap 2 i s the pressure c o r r e c t i o n due to p a r t i c l e i n t e r a c t i o n s and b - 1 i s the high pressure l i m i t of the density. The constants a and b are c h a r a c t e r i s t i c of a given material and may be determined experimentally. F i g . 4(a) shows some t y p i c a l van der Waals isotherms. For T > T^, the isotherms have the expected shape. For T < T , however, there i s a region (indicated by dashes) on each isotherm with negative slope. Thus, the compressibility i s negative i n these regions. I f a system i s to be i n a mechanically stable state, though, i t s compressibility must be p o s i t i v e : p o s i t i v e pressure increments must produce p o s i t i v e density increments. To make the van der Waals isotherms more p h y s i c a l , a modification known as the 24 Maxwell construction i s employed. This technique i s based on the p r i n c i p l e F i g . 4 . Van der Waals isotherms. the Maxwell construction 18 that, for liquid-vapour coexistence, the Gibbs free energy of the two phases must be the same. As i l l u s t r a t e d i n F i g . 4(b), a portion of each isotherm below T c i s replaced by a h o r i z o n t a l l i n e segment, r e s u l t i n g i n a more r e a l i s t i c phase diagram, with the same basic features as the phase diagram of a r e a l gas ( c f . F i g . 1(b)). Hence, with the Maxwell construction, the van der Waals equation of state predicts liquid-vapour coexistence terminating with a second order t r a n s i t i o n . The c r i t i c a l values of pressure, density and temperature may be obtained from the equation of state, and from the conditions that, at the c r i t i c a l point, the c r i t i c a l isotherm has both a vanishing slope and a point of i n f l e c t i o n : ^ c = 0 One obtains P = a c 27 b 2 8a (2.7) 27b 1 19 In terms of the dimensionless parameters P = P/P , p = p/p and T = T/T c, the van der Waals equation of state may be written as ~2 1 (P + 3p - 1) = 8T. (2.8) The s t r i k i n g feature of Eq. (2.8) i s that i t i s independent of the material-dependent quantities a and b. In other words, the van der Waals equation, cast In t h i s form, may be interpreted to be u n i v e r s a l . In p r i n c i p l e , one needs only to know p c > P c a n < i T c f ° r a n y material to determine i t s equation of state. Furthermore, Eqs. (2.7) may be combined to y i e l d a u n i v e r s a l r e l a t i o n f o r the c r i t i c a l parameters: V 3 (2.9) k T p " 8 * B c K c Such u n i v e r s a l predictions are manifestations of the law of corresponding 4 states, postulated by van der Waals i n 1880. I t states that a l l c l a s s i c a l f l u i d s , described i n terms of reduced parameters such as P, p and T, obey an i d e n t i c a l equation of state P = f ( p , T) where f i s a u n i v e r s a l function. While the law i s not e n t i r e l y corroborated experimentally, i t has played a key r o l e i n advancing the understanding of the c r i t i c a l behaviour of various systems. 20 A more stringent test of the van der Waals theory i s to see what values i t predicts for the various c r i t i c a l exponents. For example, consider the c r i t i c a l exponent y, defined i n Eq. (2.3). From the equation of state, Eq. (2.6), the isothermal com p r e s s i b i l i t y i s found to be < T = , P ( 1 - b P > . (2.10) Sabp^ - 2ap -Hc^ T + bP The temperature dependence of near the c r i t i c a l point may be found by se t t i n g P = P c and p = p c to y i e l d KT + x " x - T ~ t " 1 < (2-11) c c Thus, the van der Waals equation predicts y = 1. Experimental r e s u l t s confirm that the com p r e s s i b i l i t y does, i n f a c t , diverge at the c r i t i c a l point, but with 1.2 < y < 1.4. The other exponents may be calculated i n a si m i l a r fashion. The r e s u l t s are l i s t e d i n Table 1, together with values determined by experiment, and values predicted by a modern theory known as the renormalization group, which i s discussed i n Sec. 2.3. Table 1 25 c r i t i c a l experimental mean f i e l d RG exponent value theory  a 0 - 0 . 2 0 0.110 8 0.3 - 0.4 1/2 0.325 - 0.328 y 1.2 - 1.4 1 1.237 - 1.241 6 4 - 5 3 4.82 There are numerous other c l a s s i c a l theories, each with i t s own equation of state. T y p i c a l examples are the D i e t e r i c i equation a l P / k T Pe ( 1 - b l P ) = p k B T and the Berthelot equation (P + a 2 p 2 / k B T ) ( l - b 2p) = p k B T where a^, b^, a 2 and b 2 are experimental constants l i k e a and b i n van der Waals' equation. Another equation of state i s obtained by making the s o - c a l l e d v i r i a l expansion i n density: P = k B T [p + B(T)p 2 + C(T)p 3 + . . . ] . The temperature-dependent c o e f f i c i e n t s B and C are c a l l e d the second and t h i r d v i r i a l c o e f f i c i e n t s , and may be calculated from the intermolecular p o t e n t i a l . For example, B(T) = b - a/k T for the van der Waals equation of a state. Conversely, an experimental determination of the v i r i a l c o e f f i c e n t s can be used to obtain information about the intermolecular forces of a p a r t i c u l a r system. 22 A l l such c l a s s i c a l theories are the same as the van der Waals theory i n terms of t h e i r predictions f o r the values of c r i t i c a l exponents. The inadequacy of these theories became apparent i n the early 1900's when c a r e f u l measurements were f i r s t performed on pure f l u i d systems. In 1945, Guggenheim"* obtained the u n i v e r s a l coexistence curve equation from liquid-vapour coexistence data of a va r i e t y of f l u i d s . This r e s u l t provided some evidence for van der Waals' law of corresponding st a t e s . At the same time, however, Guggenheim's value of 8 = 1 / 3 indicated that c l a s s i c a l theories could not be r e l i e d upon to predict values of c r i t i c a l exponents. Since then, experimental r e s u l t s have f u l l y confirmed that the c l a s s i c a l c r i t i c a l exponents are i n c o r r e c t . A more r e a l i s t i c analysis of c r i t i c a l phenomena has required the development of modern theories of phase t r a n s i t i o n s . While no attempt w i l l be made to give a d e t a i l e d discussion of these theories, the following section w i l l o utline t h e i r basic approach. 2 .3 Modern Theories How does one accurately describe the behaviour of a macroscopic thermodynamic system? This, i n p r i n c i p l e , i s a formidable task. Such a system consists of on the order of IO 2 3 p a r t i c l e s . The state of each p a r t i c l e i s determined by the values of s i x degrees of freedom — three f o r p o s i t i o n and three for momentum — making the t o t a l number of degrees of •p _ f 1 / 3 = 3 . 5 ( 2 . 1 2 ) c 23 freedom for the system enormous. I t i s c l e a r that s i m p l i f i c a t i o n s of some sort are required i n order to make any progress. Furthermore, i t i s reasonable to expect that the nature of the s i m p l i f i c a t i o n s made w i l l u l t i m a t e l y determine the extent to which a p a r t i c u l a r theory succeeds i n making r e a l i s t i c p r e d i c t i o n s . The basic assumption made by c l a s s i c a l theories of c r i t i c a l phenomena i s that i n t e r a c t i o n s i n a system are l i m i t e d to those between each p a r t i c l e and an average f i e l d due to a l l other p a r t i c l e s ; long range f l u c t u a t i o n s and co r r e l a t i o n s among p a r t i c l e s are neglected. Despite t h i s coarse approximation, c l a s s i c a l or mean f i e l d theories are remarkably successful: they provide a reasonably accurate d e s c r i p t i o n of a v a r i e t y of systems, while avoiding e n t i r e l y the problem of 1 0 2 3 or so degrees of freedom. Nevertheless, i t i s evident that a s a t i s f a c t o r y understanding of c r i t i c a l phenomena cannot be gained from c l a s s i c a l theories alone. To be v a l i d , a theory must provide a quantitative d e s c r i p t i o n of what happens when a system i s taken a r b i t r a r i l y close to i t s c r i t i c a l point. In p a r t i c u l a r , such a theory should y i e l d values of c r i t i c a l exponents that agree with those found by experiment and, as demonstrated i n the previous section, c l a s s i c a l theories f a i l In t h i s respect. The reason for t h i s f a i l u r e l i e s i n the f a c t that the approximations made by c l a s s i c a l theories break down i n the c r i t i c a l region. To see why t h i s happens, consider the density f l u c t u a t i o n s i n a pure f l u i d . Far from T , the single phase present i s extremely stable and density f l u c t u a t i o n s are small. C l a s s i c a l theories work i n t h i s region. As T^ i s approached, however, the energy difference between l i q u i d and vapour phases diminishes and the c o m p r e s s i b i l i t y of the system diverges. I t becomes more 24 probable that regions of higher or lower than average density w i l l form. The closer the system gets to T c > the larger the extent of these coherent regions of independent density f l u c t u a t i o n s . Eventually they become large enough to scatter v i s i b l e l i g h t , causing an e f f e c t known as c r i t i c a l opalescence. This e f f e c t i s not predicted c l a s s i c a l l y since, as mentioned above, c l a s s i c a l theories ignore f l u c t u a t i o n s and expect the properties of a macroscopic system to correspond d i r e c t l y with the properties of small groups of p a r t i c l e s . This assumption must be replaced with a more r e a l i s t i c one i f the c r i t i c a l behaviour of systems i s to be f u l l y understood. To deal with these aspects of c r i t i c a l phenomena, modern theories introduce a thermodynamic quantity, the c o r r e l a t i o n length E, that i s a measure of the extent of f l u c t u a t i o n s i n a system. In other words, the thermodynamic properties of a system, i n a p a r t i c l a r state, w i l l not appear to change provided that they are not measured over a scale smaller than £. The c o r r e l a t i o n length i s a state v a r i a b l e . Far from T , £ i s t y p i c a l l y on the order of i n t e r - p a r t i c l e spacing. Close to T , £ i s expected c to diverge according to the power law I ~ | t p . (2.13) The exponent v i s another c r i t i c a l exponent, one that does not emerge from c l a s s i c a l theories since they do not account for f l u c t u a t i o n s . Modern theories predict v » 0.63, a value that i s i n good agreement with experimental r e s u l t s . The f i r s t theory to e f f e c t i v e l y incorporate the concepts of 25 9 fl u c t u a t i o n s and c o r r e l a t i o n length was developed by Widom i n 1965. Widom's so- c a l l e d s c a l i n g theory i s based on the p r i n c i p l e that thermodynamic quantities are homogeneous functions of t h e i r distance from the c r i t i c a l point; i . e . , as a system approaches i t s c r i t i c a l point, the associated thermodynamic properties change t h e i r scale but not t h e i r f u n c t i o n a l form. Among the predictions made by th i s theory are the un i v e r s a l c r i t i c a l behaviour of various classes of systems and c r i t i c a l region power law dependences of thermodynamic q u a n t i t i e s . Also predicted are a number of use f u l r e l a t i o n s h i p s between the c r i t i c a l exponents. For example: a + y + 28 = 2 a + B(6 + 1) = 2. In f a c t , acccording to the theory, a l l of the c r i t i c a l exponents may be expressed In terms of two fundamental parameters. The p r i n c i p l e of s c a l i n g underlies most of the contemporary theories of c r i t i c a l phenomena. In 1966, Kadanoff''"^ applied s c a l i n g theory to I s i n g - l i k e spin systems by describing the spin l a t t i c e i n terms of i n t e r a c t i n g spin blocks. A block of spins replaces the Individual spins as the basic unit of i n t e r a c t i o n , an average spin being associated with each block. As the c r i t i c a l temperature i s approached, and the c o r r e l a t i o n length increases, the block s i z e i s also increased, i n such a way that the relevant thermodynamic quantities rescale without changing t h e i r f u n c t i o n a l form. Increasing the block s i z e i n t h i s manner i s equivalent to reducing the 26 e f f e c t i v e c o r r e l a t i o n length, or moving away from the c r i t i c a l point. With t h i s procedure, the c r i t i c a l region may be examined mathematically by I t e r a t i v e l y reducing the large number of degrees of freedom per unit c o r r e l a t i o n length. A general and mathematically complete theory of c r i t i c a l phenomena known as the renormalization group (RG) was formulated by Wilson** i n 1971. In simple terms, t h i s theory incorporates the concept of s c a l i n g and, as i n the Kadanoff p i c t u r e , provides a p r a c t i c a l method for analyzing c r i t i c a l behaviour by systematically reducing the number of relevant degrees of freedom i n a system near i t s c r i t i c a l point. U n i v e r s a l i t y i s predicted by demonstrating that c r i t i c a l region behaviour depends only on the dimensionality of the system and the order parameter, and not on the d e t a i l s of the Interaction Hamiltonian. The q u a l i t a t i v e features of phase t r a n s i t i o n s are explained i n terms of system dimensionality and cooperative degrees of freedom. Perhaps of greatest importance, the RG approach has proved to be an extremely successful technique f o r c a l c u l a t i n g r e a l i s t i c values of c r i t i c a l exponents. Another s i g n i f i c a n t feature of RG theory i s that, as shown by 26 Wegner , i t predicts deviations from simple power law behaviour i n measurements made away from the c r i t i c a l point. Such corrections a r i s e because while the c r i t i c a l point value of the c o r r e l a t i o n length i s predicted by s c a l i n g theory to be i n f i n i t e , experiments are necessarily performed at f i n i t e distances from the c r i t i c a l point. The r e s u l t i n g departure from power law behaviour i s seen i n the RG predicted f u n c t i o n a l form of a thermodynamic var i a b l e f ( t ) , 27 f ( t ) = k\t\~\ 1 + aj_ | t | A l + a 2 |t| A2 + . . . ) , (2.14) where \ i s the c r i t i c a l exponent associated with f ( t ) . The exponents and A 2 are expected to be u n i v e r s a l , as are c e r t a i n r a t i o s of the amplitudes a^, ag, ... . For the case of the pure f l u i d coexistence curve, Eq. (2.14) becomes \ " P v • B p | t | P ( l + B 1 | t l A + B 2 |t| 2 A '+ . . . ) . (2.15) ^c S i m i l a r l y , the isothermal c o m p r e s s i b i l i t y < T takes the form < T = r 0 ItPU + r x | t | A + r 2 | t | 2 A + . . . ) . (2.16) I t i s a major objective of current c r i t i c a l phenomena experiments, i n c l u d i n g those presented i n t h i s t h e s i s , to obtain s u f f i c i e n t l y high p r e c i s i o n data that the status of c o r r e c t i o n s - t o - s c a l i n g predictions may be accurately assessed. 28 Chapter 3 OPTICS 3.1 Introduction This chapter describes the o p t i c a l methods used to study the c r i t i c a l behaviour of the pure f l u i d sample. There were two lnterferometric techniques used, each one providing information about d i f f e r e n t aspects of the t r a n s i t i o n . In the f i r s t method, discussed i n Sec. 3.2, a Fraunhofer d i f f r a c t i o n pattern created by the sample was used to determine the c r i t i c a l exponent 6 . Sec. 3.3 describes an o p t i c a l technique used to obtain the c r i t i c a l exponent y. In t h i s case, the sample was placed i n one arm of a Mach-Zehnder interferometer. The Lorentz-Lorenz r e l a t i o n s h i p i s discussed i n Sec. 3.4. Both' of the o p t i c a l methods used are based on the following properties of a pure f l u i d sample having an average density equal to the c r i t i c a l density. At a given temperature, the sample exhibits a height-dependent density p r o f i l e p(z) due to the e f f e c t of the earth's g r a v i t a t i o n a l f i e l d * . T y p i c a l examples of p(z) are i l l u s t r a t e d i n F i g . 5. In F i g s . 5(c) and 5(d) the sample i s i n the one phase region (t > 0). In F i g s . 5(a) and 5(b) the temperature i s below the c r i t i c a l temperature (t < 0); the d i s c o n t i n u i t y i n p(z) at z = Z q occurs at the i n t e r f a c e separating the l i q u i d and vapour phases. I f the average density of the sample i s i n fact equal to the c r i t i c a l density, the height Z q at which the maximum density gradient occurs w i l l be approximately temperature-independent. The shape of the density p r o f i l e changes as the distance from T i s varied, but at any given 29 Fig. 5. Density profiles of a pure fluid in the earth's gravitational f ield. (c) t > 0 (d) t » 0 30 temperature p(z) i s assumed to be antisymmetric about the height Z q. Hence, dp/dz i s symmetric about z^. As p(z) changes with temperature, so does the r e f r a c t i v e index n(z), i n accordance with the Lorentz-Lorenz r e l a t i o n s h i p (Eq. (3.5)). This temperature dependence of the r e f r a c t i v e index may be exploited to obtain information about the liquid-vapour phase t r a n s i t i o n . In p a r t i c u l a r , the o p t i c a l methods described below have proved to be e f f e c t i v e i n examining the t r a n s i t i o n very near T , at least f o r |t | ~ 1 0 - 6 . 3.2 Fraunhofer D i f f r a c t i o n To understand how a pure f l u i d sample can produce a Fraunhofer d i f f r a c t i o n pattern, consider the arrangement shown i n F i g . 6. A plane wave beam of laser l i g h t s t r i k e s the sample with normal incidence. Using Snell's law, i t i s found that a l i g h t ray traversing the sample i s deflected according to d9 L _ d £ dx n(z) dz ' where x i s the h o r i z o n t a l distance i n the c e l l . If the sample thickness L o i s s u f f i c i e n t l y small that the ray Is not bent appreciably before emerging from the sample, the d e f l e c t i o n angle 9^ may be written as ^o dn r — i i " L ~ n(z) dz J _ ' (3.2) z = z ± 3 2 w h e r e z ^ i s t h e i n c i d e n t h e i g h t . A t t h e c e l l b o u n d a r y , a f u r t h e r r e f r a c t i o n t a k e s p l a c e . S n e l l ' s l a w may b e u s e d t o f i n d t h e e x i t a n g l e ( 3 . 3 ) T h e r e f r a c t i v e i n d e x n i s r e l a t i v e t o t h a t o f a i r , a n d t h e s m a l l a n g l e a p p r o x i m a t i o n s i n 9^ « 9^ h a s b e e n m a d e . H e n c e , a h o r i z o n t a l l y i n c i d e n t r a y l e a v e s t h e c e l l a t a n g l e 9 q g i v e n b y E q . ( 3 . 3 ) . A r a y R q p a s s i n g t h r o u g h t h e r e g i o n o f t h e c e l l w h e r e t h e d e n s i t y g r a d i e n t i s a m a x i m u m i s b e n t t h r o u g h a m a x i m u m a n g l e Q^. S i n c e d n / d z i s s y m m e t r i c a b o u t Z q , f o r e a c h r a y R + t h a t t r a v e r s e s t h e c e l l a b o v e z , t h e r e i s a r a y R~ t h a t t r a v e r s e s t h e c e l l i n a r e g i o n b e l o w z w i t h t h e o o s a m e d e n s i t y g r a d i e n t . R a y s R + a n d R~ e m e r g e f r o m t h e c e l l p a r a l l e l t o e a c h o t h e r , a s s h o w n i n F i g . 6 . T h e e n t i r e b e a m may b e d e c o m p o s e d i n t o s u c h p a i r s o f r a y s . B e c a u s e o f t h e v a r y i n g d e n s i t y d i s t r i b u t i o n , r a y s R + a n d R~ w i l l h a v e t r a v e l l e d u n e q u a l o p t i c a l p a t h l e n g t h s . I f t h e y a r e s u p e r i m p o s e d i n t h e b a c k f o c a l p l a n e o f a n o b j e c t i v e l e n s , t h e y w i l l p r o d u c e a n i n t e r f e r e n c e s p o t . T h e i n t e n s i t y o f t h i s s p o t d e p e n d s u p o n t h e p h a s e d i f f e r e n c e o f t h e t w o r a y s , w h i c h i n t u r n i s a f u n c t i o n o f t h e i r d i s t a n c e f r o m Z q . I f a l l s u c h r a y p a i r s a r e c o n s i d e r e d , t h e r e s u l t i s a F r a u n h o f e r d i f f r a c t i o n p a t t e r n p r o d u c e d i n t h e f o c a l p l a n e F o f l e n s 1^ . T h e p a t t e r n w i l l c o n s i s t o f a v e r t i c a l s t r i p o f b r i g h t a n d d a r k i n t e r f e r e n c e s p o t s . F o r t h e c a s e t < 0 , t h e d i s c o n t i n u i t y i n p ( z ) a t t h e l i q u i d - v a p o u r i n t e r f a c e ( z = z ) p r o d u c e s a n a d d i t i o n a l p h a s e d i f f e r e n c e i n e a c h p a i r o f 33 rays R + and R~. I f N i s the number of interference fringes at a given temperature T < T , the dif f e r e n c e i n l i q u i d and vapour r e f r a c t i v e indices i s given by n = N l (3.4) where K i s the laser wavelength and L q i s the c e l l thickness. Thus, the order parameter for the system, (p^ - p v ) / p c > may be obtained by measuring N as a function of temperature and r e l a t i n g r e f r a c t i v e index to density with the Lorentz-Lorenz r e l a t i o n s h i p . I t was i n th i s manner that 8 was determined for the CHF3 sample. The o p t i c a l setup used i s shown i n F i g . 7. The l i g h t source was a helium-neon laser (X = 6.328 x 10 - 7 m). The beam was attenuated by a p o l a r i z e r . A microscope objective, I^ , focussed the beam onto a 10 micron pinhole P which, i n turn, was positioned at the f o c a l point of a converging lens, Ig . An e s s e n t i a l l y plane wave beam was thus generated. The sample c e l l , f i l l e d to the c r i t i c a l density and placed within the temperature c o n t r o l system, was positioned i n the path of the laser beam, with the c e l l windows perpendicular to the beam. The Fraunhofer d i f f r a c t i o n pattern produced i n the f o c a l plane F of lens was recorded continuously on photographic f i l m as the c e l l temperature was increased i n steps from several degrees below T^ to a f i n a l temperature above T^. The d e t a i l s of the experimental procedure are outlined i n S e c 4.3. H e - N e l a s e r P H R e f e r e n c e b e a m b l o c k e d a t B, f o r F r a u n h o f e r d i f f r a c t i o n e x p e r i m e n t . s a m p l e 35 3.3 Image Plane Interference As described e a r l i e r , the density of a pure f l u i d varies considerably with height near the c r i t i c a l temperature. Thus, at a given temperature, l i g h t rays that traverse the sample w i l l t r a v e l o p t i c a l paths of varying lengths. Rays crossing the lower region of the sample, where the density and r e f r a c t i v e index are greater, are retarded r e l a t i v e to rays that cross the upper region of the sample. I f these rays are then combined with the rays of an undisturbed reference beam, the r e s u l t i n g interference pattern, seen i n the image plane of a converging lens, d i r e c t l y maps the r e f r a c t i v e index of the sample as a function of p o s i t i o n . Since the r e f r a c t i v e index i s a maximum at the bottom of the c e l l and decreases monotonically with height, each fringe i n the v e r t i c a l d i r e c t i o n represents a change i n o p t i c a l thickness of one wavelength. From an analysis of the fri n g e spacing, i t i s possible to determine the temperature dependence of the compressibility along the c r i t i c a l isochore. This, i n turn, y i e l d s a value of the c r i t i c a l exponent y. The d e t a i l s of t h i s analysis are given i n Sec. 5.2. The e f f e c t s discussed above were achieved by placing the sample c e l l i n one arm of a Mach-Zehnder interferometer. F i g . 7 i l l u s t r a t e s the arrangement of the o p t i c a l apparatus that was used. A plane wave las e r beam was generated as described i n the previous section. The beam was s p l i t at B^, with one portion traversing the CHF3 sample before recombining with the reference portion at B 2. The lens placed behind B 2 formed a combined image of the e x i t plane of the sample at p o s i t i o n 0 and the reference beam at some p o s i t i o n 0'. A serie s of h o r i z o n t a l fringes was observed i n the image plane I of the lens L, , with maxima occurring at points where the phase 36 difference of the two beams was an integral multiple of 2%. The fringe pattern was recorded continuously as a function of temperature. A description of the data collection procedure is given in Sec. 4.3. The data for this experiment, and the Fraunhofer diffraction experiment, are presented and analysed in Chapter 5. 3.4 The Lorentz-Lorenz Relationship What was actually measured by these experiments was the refractive index or the difference of refractive indices. To convert refractive indices to densities, the Lorentz-Lorenz relationship was used. Derived independently by Lorentz and Lorenz i n 1880, i t may be expressed as n 2 pL , (3.5) n 2 + 2 where n is the refractive index, p is the density and L is the Lorentz-Lorenz function. For many applications, i t is sufficient to assume that L is a constant. This assumption i s known to be valid for a homogeneous medium in which the correlation length £ that characterizes the extent of density fluctuations is negligible compared to the wavelength X of the incident light. As explained earlier, though, a pure f l u i d in the vici n i t y of i t s c r i t i c a l point does not satisfy the above condition: the characteristic size of density fluctuations becomes large when the system i s near T^. Nevertheless, there is some evidence to suggest that the approximation of constant L may s t i l l be reasonable in the c r i t i c a l region. 37 27 In p a r t i c u l a r , Larsen, Mountain and Zwanzig use a t h e o r e t i c a l argument to conclude that, when X and E are of the same order of magnitude, L deviates from i t s o f f - c r i t i c a l value by approximately 0.02%. The experimental r e s u l t s 28 of Burton and B a l z a r i n i i n d i c a t e that, for CjHg, L varies by less than 0.1% i n the c r i t i c a l region. Under the assumption of constant L, the Lorentz-Lorenz equation may be expanded about the c r i t i c a l density p i n a Taylor series p - p p - p 2 n(p) = n c + a x (n, - 1) [— + a ^ — J + .. .J where ( n c + l ) ( , n c 2 + 2) c ( n c 2 - l ) ( 3 n c 2 - 2) *2 12n 2 c For the purposes of t h i s experiment, i t was s u f f i c i e n t to neglect second and higher order terms, and the expression n(p) - n c = a 1 ( n c - l ) ( ^ — - ± ) (3.6) H c was used to r e l a t e r e f r a c t i v e index to density. 38 Chapter 4 THE EXPERIMENT 4.1 The Sample C e l l The material studied i n both experiments was a 98% pure sample of trifluoromethane, CHF 3, commonly known as fluoroform. I t was chosen because i t i s a pure f l u i d with a second order phase t r a n s i t i o n that had not been previously investigated by an experiment of t h i s type. The fa c t that i t i s polar makes CHF3 even more i n t e r e s t i n g , as most c r i t i c a l behaviour studies of pure f l u i d s have been l i m i t e d to non-polar materials. Furthermore, i t has properties that made i t a p a r t i c u l a r l y convenient sample to use. I t i s o p t i c a l l y transparent and thus su i t a b l e for an experiment that o p t i c a l l y probed changes i n r e f r a c t i v e index. I t has a r e l a t i v e l y low c r i t i c a l temperature (T » 26°C). Temperatures i n t h i s neighbourhood were not d i f f i c u l t to reach and maintain with the temperature co n t r o l system used i n the experiment. F i n a l l y , i t was r e l a t i v e l y easy to design and construct a vessel capable of holding CHF3 under pressures i n the range of i t s c r i t i c a l pressure (P » 5 x l O 4 Pa). The vessel used to contain the CHF3 sample was the aluminum c e l l shown i n F i g . 8. Two sapphire windows, 2.54 cm i n diameter and 0.635 cm thick, were situated p a r a l l e l to each other on ei t h e r side of the c e n t r a l hole, separated by a gap of 1.84 mm. The gap siz e was determined by subtracting the t o t a l window thickness from the distance between the outer window faces. Indium wire, 0.76 mm i n diameter, was used to form a s e a l between the c e l l and the windows. Aluminum flanges clamped the windows to the c e l l , with 40 gaskets serving as a cushion between the windows and the flanges. The ten cap-head screws on each flange were tightened progressively to 0.8 Nm with a torque wrench. The c e l l was f i l l e d as follows. The sample entered the c e l l v i a a small hole that connected the sample space between the sapphire windows to a larger, tapped hole at the top of the c e l l . A s t a i n l e s s s t e e l needle valve, inserted i n the tapped hole, was used to seal and unseal the small hole during the f i l l i n g procedure. Once the c e l l was cleaned, assembled and free of leaks, i t was attached, v i a the needle valve, to a gas handling system. The c e l l and the gas delivery l i n e s were evacuated by 48 hours of pumping with a d i f f u s i o n pump and then flushed out with CHF 3, contained under pressure i n a c y l i n d e r . A meniscus appeared as the CHF3 entered the c e l l , which was at room temperature. A f t e r several flushes, the c e l l was f i l l e d slowly u n t i l the meniscus was just above the midpoint of the c e l l windows. The c r i t i c a l density was recognized as the density for which the meniscus remained stationary (before disappearing) as the sample was heated through Its c r i t i c a l temperature. With the c e l l s l i g h t l y o v e r f i l l e d , small amounts of CHF3 were bled out through the needle valve. A f t e r each bleeding, the motion of the meniscus was observed while the c e l l was immersed i n a temperature-controlled water bath that was slowly heated through T^. This procedure was repeated u n t i l there was no dlscernable motion of the meniscus, at which point the c e l l was removed from the gas handling system. 41 4.2 Temperature Control An important requirement of most phase t r a n s i t i o n experiments i s stable and precise temperature c o n t r o l of the system being studied, p a r t i c u l a r l y i n the v i c i n i t y of the c r i t i c a l region. This section outlines the procedure and equipment with which temperature c o n t r o l was achieved i n these experiments. The sample c e l l was situated in s i d e three containers, each insulated from the other by a 5 cm thick layer of styrofoam. The innermost container was a s o l i d aluminum "block" through which had been bored a hole of the same diameter as that of the c e l l . With the c e l l placed in s i d e the block, good thermal contact was maintained between the two pieces. Insulated heating wire was wrapped and glued around the outside of the block. The heating wire had a resistance of about 6 ohms. Two Fenwal thermistors were epoxied i n s i d e hollow copper screws which, i n turn, were inserted into holes tapped i n the block. A Hewlett-Packard model 2804A quartz thermometer probe was placed i n a hole i n the top of the block. Apiezon N grease was applied to the quartz thermometer probe and thermistor screws to improve t h e i r thermal contact with the block. Next, the aluminum block was encased i n a layer of styrofoam and inserted Into the second container, an aluminum "box". Machined i n the i n t e r i o r of the 1.5 cm-thick walls of the box was a channel system through which a temperature-controlled water-antifreeze mixture could c i r c u l a t e . This box was covered with styrofoam and placed inside the t h i r d and outermost container, a plywood box. Appropriate holes were cut i n the containers and 42 styrofoam to allow f o r the unobstructed passage of a laser beam through the sample c e l l during the experiment. The primary means of temperature co n t r o l consisted of an e l e c t r o n i c feedback system that regulated the temperature of the inner aluminum block. The e l e c t r o n i c c i r c u i t i s shown i n F i g . 9. The control thermistor i n the block was balanced against a decade resistance box i n a Wheatstone bridge. The error s i g n a l was amplified by a Hewlett-Packard 419A DC nullmeter and fed into a Kepco OPS 7-2 power supply. The feedback produced a combined proportional and i n t e g r a l c o n t r o l which powered the heater on the inner block. During the course of the experiment, the temperature was changed i n a series of sweeps and waiting periods that systematically took the sample c e l l from an i n i t i a l temperature several degrees below T c, through T , to a f i n a l temperature above T . A ten-turn, 1 ohm variable potentiometer divided r e s i s t o r s and R2 i n the Wheatstone bridge. An e l e c t r i c motor drove a gear system that turned the potentiometer; when the motor was on, an imbalance i n the bridge was created i n a continuous and steady fashion. The rate at which the temperature changed was determined by the motor speed, which could be varied, and by the resistances of the bridge arms R^  and R 2, which could be set at i d e n t i c a l values of 30 ohms, 100 ohms, 300 ohms or 1000 ohms. A Commodore Pet computer was in t e r f a c e d with the motor and programmed to turn the motor on and off for desired lengths of time. The waiting periods, when the motor was o f f , allowed the c e l l to reach thermal equilibrium a f t e r a temperature change. F i g . 9. The temperature control c i r c u i t . 44 A secondary l e v e l of temperature control was also incorporated. A temperature-controlled water-antifreeze mixture was c i r c u l a t e d v i a foam-insulated hoses through the channels i n the walls of the aluminum box. A modified Forma model 2095 c i r c u l a t o r , able to heat and r e f r i g e r a t e , pumped the bath mixture through the system. The temperature of the bath mixture was regulated i n the following manner. C e l l and bath temperatures were monitored by separate thermistors, one i n contact with the inner aluminum block, the other with the aluminum box. These thermistors served as balancing resistance arms i n a Wheatstone bridge. The error s i g n a l from the bridge was amplified and fed i n t o the pulse-width modulation c i r c u i t shown i n F i g . 10. This c i r c u i t c o n t r o l l e d the bath temperature by a l t e r n a t e l y switching on, for appropriate time periods, the Forma c i r c u l a t o r ' s heater and r e f r i g e r a t o r . The c i r c u i t was adjusted to maintain the bath temperature at approximately 0.5°C below that of the c e l l . The bath temperature was stable to ± 0.005°C. The c e l l temperature was monitored with the quartz thermometer, which was previously c a l i b r a t e d against the t r i p l e point of water using a J a r r e t t water t r i p l e - p o i n t c e l l . The quartz thermometer had a r e s o l u t i o n of 0.0001°C. I t s output was continuously recorded by a chart recorder. With the co n t r o l system described above, i t was possible to hold the sample c e l l temperature stable to ± 0.0001°C for a period of days. 4.3 Experimental Procedure The data c o l l e c t i o n for the Fraunhofer d i f f r a c t i o n pattern experiment took place as follows. With the sample c e l l f i l l e d and i n place In the temperature co n t r o l system, the c e l l and o p t i c a l apparatus were aligned on an »100kn L E D R 2 • V C C R E S E T D ISCH T H R E S H 555 TRIG O U T C O N T R O L G N D 5 .6uF 0 .01pF : R 3 3 . 3 k f t > > 3 3 0 k f l 0 . 01uF • V C C R E S E T D I S C H T H R E S H 555 T R I G O U T C O N T R O L G N D I N . . o. | 4 1 4 8 i i : 6 . 6uF S E N S O R r-OvWCK-R E F E R E N C E — O v V N A O - t 1N751 I—yvVA-3 . 3 k Q R1 100kQ - W V , L R E L A Y 1 L E D [ 3 . 3 k Q •+15v - v W 1 3 . 3 k Q 1 0 k Q 1 0 0 k Q 1 N 4 1 4 8 : 5 . 6 u F 46 o p t i c a l bench as described i n Sec. 3.2. A motor and gear assembly was used to advance the f i l m i n a 35 mm camera body. The camera was positioned so that the f i l m , exposed to the d i f f r a c t i o n pattern through a narrow v e r t i c a l s l i t , was i n the f o c a l plane of lens ( F i g . 7). With the camera motor running, i t was possible to obtain a continuous record of the f o c a l plane d i f f r a c t i o n pattern as the sample was taken through i t s phase t r a n s i t i o n . An i n i t i a l wait of several hours allowed the sample to reach thermal equilibrium at a temperature about 10°C below T £ (t « -3 x 10~ 2). At t h i s point, the experimental room was darkened and the camera motor was turned on. Film was transported across the camera s l i t at a rate of 3 cm/hour. The Commodore Pet computer changed the c e l l temperature by switching on and o f f , for desired lengths of time, the motor that drove the variable potentiometer i n the control bridge. Temperature was increased towards T c i n steps; a waiting period a f t e r each step allowed the sample to e q u i l i b r a t e . The waiting times varied from several minutes to two hours, depending on the distance from T c and on the s i z e of the previous temperature increment. The s i z e of the temperature steps was systematically decreased so that there was roughly an equal number of data points for each decade of reduced temperature that was covered. For example, for t « -10 - 2, 0.1°C steps were taken; for t » -10" 5, 0.001°C steps were taken. To minimize thermal and o p t i c a l disturbances, the experimental area was vacated during most of the data c o l l e c t i o n period. The camera was reloaded with new f i l m approximately every 24 hours. The Fraunhofer d i f f r a c t i o n pattern data used for analysis was c o l l e c t e d over a period of eight days. Several preliminary runs were conducted to obtain an estimate for and to determine an appropriate beam i n t e n s i t y and su i t a b l e sweeping rat e s . The procedure for the image plane experiment was i d e n t i c a l to that of the f o c a l plane experiment, except that the camera was positioned with the f i l m i n the image plane of lens 1^  . Data was c o l l e c t e d over a four-day period f or temperatures i n the range 1.7 x 10 - 5 < t < 2 x 10~^. 48 Chapter 5 RESULTS 5.1 Fraunhofer D i f f r a c t i o n Pattern The Fraunhofer d i f f r a c t i o n pattern data, c o l l e c t e d i n the two-phase region over a temperature range of 4 x 10 - 6 < | t | < 3.5 x 10 - 2, were evaluated by examining the fri n g e pattern that was continuously recorded on f i l m . Samples of the f i l m are shown i n F i g s . 11(a) and 11(b). A v e r t i c a l s l i c e at any point on the f i l m represents the Fraunhofer d i f f r a c t i o n pattern at a p a r t i c u l a r temperature. Regions where the fri n g e pattern changes, with new fringes appearing at the top edge of the f i l m , i n d i c a t e temperature sweeps. Once the sample reached equilibrium a f t e r a temperature change, the frin g e pattern remained e s s e n t i a l l y constant u n t i l the next temperature step. Intermittent problems with e l e c t r o n i c s hindered the s t a b i l i t y of temperature con t r o l at times, causing f l u c t u a t i o n s such as those seen In F i g . 11(b). The chart recorder output of the quartz thermometer readings was used to assign values of temperature to each region of equilibrium. The v e r t i c a l bars on the f i l m are due to a l i g h t emitting diode that was momentarily turned on between temperature steps; these bars aided temperature i d e n t i f i c a t i o n on the fringe pattern. At any given s u b c r i t i c a l temperature, there was a c e r t a i n number of fringes N that were "missing" due to the i n f i n i t e r e f r a c t i v e index gradient at the liquid-vapour i n t e r f a c e . The number N decreased as the c r i t i c a l point was approached and the d i f f e r e n c e i n l i q u i d and vapour ind i c e s , n - n , Figure 11. Fringe p a t t e r n samples. (d) Image plane p a t t e r n (t » 3 x IO" 5) temperature-* 50 diminished. By the time the sample was taken through i t s phase transition into the one phase region, the last missing fringe appeared. This fringe, the zeroth fringe, represents the light that was most refracted at any given temperature. A value of N was measured for each of 91 subcritical temperatures. Fringe number was converted to refractive index difference using Eq. (3.4), where X is the laser wavelength and L q is the c e l l thickness. In turn, the order parameter, (p^ - p v ) / p c > was related to n^ - n^ via the approximate form of the Lorentz-Lorenz relationship, Eq. (3.6), to obtain n, - n v L (3.4) o (5.1) Eqs. (3.4) and (5.1) were then combined to yield a, (n - 1)L c o - 1.67 x 10 - 3 N, (5.2) where the following values of constants were used: X = 6.328 x 10-/ m 51 n =1.196 c (n + l ) ( n 2 +.2) c c a, = — r — - = 1.05 * 6n L = 1.84 x 10 - 3 m. o Since no tabulated value of n for CHFq was a v a i l a b l e , the value c * l i s t e d above i s an estimate that was obtained from the following c a l c u l a t i o n . The density of CHF3 at T x = 173.15 K i s known to be pl = 1.520 g/cm3, and the r e f r a c t i v e index at T 2 = 199.85 K i s known to be n 2 = 1.215. As w e l l , P c = 0.516 g/cm3 and T c = 298.95 K. A corresponding states expression of the form p - p T - T 1 / 3 " iH^^^T—) (5-3) was assumed f or the reduced density ( c f . Eq. (2.12)). The values of p^, T^ , p and T were substituted into Eq. (5.3) to determine the constant K , with c c o the r e s u l t K = 2.6, so that o p - p T _ T 1/3 — 2 . 6 ( - ^ ) . (5.4) p c c Eq. (5.4) was then used to f i n d p 2 = 0.564 g/cm3 at T 2. Substit u t i o n of and p 2 into the Lorentz-Lorenz r e l a t i o n s h i p , Eq. (3.5), yielded a value of the Lorentz-Lorenz c o e f f i c i e n t L = 0.243 cm3/g. F i n a l l y , t h i s value of L, 52 assumed to be constant (see Sec. 3.4) and the value of p £ were substituted into Eq. (3.5) to give the r e s u l t n c » 1.196. The coexistence curve data are shown i n F i g . 12, which i s a p l o t of log [ ( p ^ - P y ) / P c l versus log | t | . If the order parameter obeyed a simple power law, the data would f a l l on a s t r a i g h t l i n e of slope 8. The curve i s nearly l i n e a r but close inspection reveals a s l i g h t curvature. The departure from power law behaviour may be examined i n greater d e t a i l i n the s e n s i t i v e p l o t shown i n F i g . 13. This graph shows the same data with 0 327 log [ ( p ^ - p v ) / p c | t | ' ] versus log | t | . The value of T £ used was obtained from a f i t t i n g procedure described below. In this graph, the leading power law dependence on temperature has been divided out, so that a s t r a i g h t h o r i z o n t a l l i n e would i n d i c a t e power law dependence of the order parameter. This may be the case for data point i n the region 5 x 10 - 6 < | t | < 10 - l t. Those points i n the region 10~^ < | t | < 5 x 10 - 2, however, c l e a r l y deviate from power law behaviour and i t i s evident that corrections to s c a l i n g must be considered to i n t e r p r e t the data properly. No error bars have been pl o t t e d with the data. Points at the r i g h t of the graph are i n s e n s i t i v e to errors i n measurement and the choice of T^. These factors become important as T^ i s approached. The scatter of the points i s a good representation of the s i z e of error bars on the points. A multi-parameter non-linear least-squares f i t t i n g routine was 29 executed by computer to f i t the f u n c t i o n a l form of the order parameter predicted by corrections to s c a l i n g theory, Fig. 12. Log-log plot of order parameter, (p^ - p^)/pc, versus reduced temperature, t = (T - T c)/T c. I I I I—I 1 1 1 I I I I I I I I I \ \ 1 1 1 1 — 1 1 1 1 1 1 I 1 I—I 1 1 1 p ~ Q o o V cs o U i 55 = B Q | t | P ( l + B j t | A + B 2 | t | 2 A ) , (5.5) to the experimental data. I n i t i a l l y , a three-parameter f i t was made, with both correction terms neglected (B, = B 2 = 0 ) . This yielded the r e s u l t s The f i t exhibited a systematic deviation from the data and gave a value of 8 that i s not i n good agreement with the accepted RG value, 8 = 0.327, thereby confirming that the data i s not well-described by a simple power law. An improvement was seen i n a four-parameter f i t i n which only the second correction term was neglected (B 2 = 0) and A was f i x e d to i t s RG value, A = 0.5. This f i t gave the r e s u l t s : 8 = 0.342 ± 0.001. T = 298.9506 ± 0.0002 K c B n = 1.644 ± 0.004. 8 = 0.330 ± 0.001 T = 298.9499 ± 0.0001 K c B, '0 = 1.50 ± 0.01 Bx = 0.31 ± 0.02. The best value of 8 was obtained from a five-parameter f i t with A = 0.5 and both c o r r e c t i o n terms included: 56 8 = 0.327 ± 0.002 T = 298.9497 ± 0.0001 K c B Q = 1.47 ± 0.03 Bx = 0.5 ± 0.1 B 2 = -0.4 ± 0.3. This f i t i s indicated by the s o l i d l i n e of F i g . 13. As w e l l , a f i t made with T c and 6 fi x e d at t h e i r accepted values, yielded a value of A = 0.48 ± 0.13, consistent with the RG p r e d i c t i o n of A - 0.5. F i n a l l y , an attempt to f i t the data with BQ , B^, B 2, 8, T c and A a l l free parameters gave erroneous r e s u l t s , i n d i c a t i n g that the p r e c i s i o n of the data was not s u f f i c i e n t to perform a meaningful six-parameter f i t . 5.2 Image Plane Pattern The films obtained from the Mach-Zehnder image plane experiment were used to plo t s u p e r c r i t i c a l isotherms of CHF 3, which, i n turn, were used to estimate the value of y, the c r i t i c a l exponent associated with the compressibility, K ^ . Data were c o l l e c t e d i n the one-phase region 1.7 x 10"5 < t < 2 x 10"^ with steps of At » IO"5 . Samples of the f i l m obtained from the experiment are shown i n F i g s . 11(c) and 11(d). The f r i n g e pattern maps the o p t i c a l thickness of the sample as a function of p o s i t i o n , with each successive fr i n g e i n d i c a t i n g a, change i n o p t i c a l thickness of one wavelength at a p a r t i c u l a r height i n the 57 sample. For every p o s i t i v e increment i n temperature, the sample became more homogeneous and the r e f r a c t i v e index changed more slowly across the v e r t i c a l length of the c e l l , causing the fringes to branch out from the c e n t r a l region of the f i l m . Isotherms were extracted i n the following manner. For each of 15 s u p e r c r i t i c a l temperatures, the distance z' of each interference f r i n g e minimum from the undeflected zeroth fringe (interference maximum) was measured using a t r a v e l l i n g microscope. These distances were converted to pressure v i a the r e l a t i o n p g.z' p 0 " p = - h i <5-6> where p c i s the c r i t i c a l density, g Is the acceleration due to gravity, M i s a magnification factor due to lens 1^  (Fig.7 ) and P q i s the undetermined reference pressure at the centre of the c e l l . Eq. (5.6) was obtained by c a l c u l a t i n g , at a given distance z = z'/M above the centre of the c e l l , the presure P due to the weight of the sample above z. Fringe number was r e l a t e d to density using Eq. (5.2). Three t y p i c a l CHF3 isotherms are plotted i n F i g . 14. The maximum value of I I oP was found for each isotherm i n order to calc u l a t e values of K^, along the c r i t i c a l isochore. The one-phase c r i t i c a l region temperature dependence of the compressibility i s i l l u s t r a t e d i n the log Kt versus log | t | p l o t of F i g . 15. A least-squares f i t of the power law K = r 0 | t | ^ to th i s data y i e l d e d a value of y = 1.26 ± 0.05, which i s i n <0 a. PL, I e CL, 20 15-10-5 0 -5 -10 -15 -20 H -25 -30 *. • • 20 * A * T - T C C O w • 0 . 0 0 5 1 A * * 0 . 0 1 7 9 A * A 0 . 0 2 6 7 A * 1 1 -10 0 i 10 I 20 P - Pc ) / P c ( X 10 " 3 ) 0 9 • t l M O o l-h I ro H on c 0) n n a. c o (0 a. rt ro •8 ro i p> rt C n ro H I H 30 "• Cn CO 59 F i g . 15. Plot of isothermal compressibility, K^, versus reduced temperature, t = (T - T )/T . c c agreement with the RG value y = 1.24. The r e s u l t i s also consistent with other experimentally measured values of y. There was not a s u f f i c i e n t number of data points to investigate corrections to s c a l i n g associated with K T . 61 Chapter 6 CONCLUSION Two o p t i c a l interference methods were employed to study the second order phase t r a n s i t i o n i n the pure f l u i d CHF3 . In the f i r s t experiment, a Fraunhofer d i f f r a c t i o n technique was u t i l i z e d to determine the difference i n l i q u i d and vapour r e f r a c t i v e indices along the coexistence curve for the reduced temperature range 4 x IO - 6 < - t < 3.5 x 1 0 - 2 . Refractive index difference was related to the pure f l u i d order parameter, (p^ - p v ) / p c , by means of the Lorentz-Lorenz r e l a t i o n s h i p . A multi-parameter f i t to the data of the corrections to s c a l i n g expression ^ ^ ^ B j t l ^ l + B j t f ^ B j t l ^ ) c gave a value of B = 0.327 ± 0.002, i n agreement with the predicted renormalization group value for t h i s c r i t i c a l exponent. The existence of corrections to s c a l i n g was confirmed and values of the c o r r e c t i o n amplitudes were obtained from the f i t . In the second experiment, a Mach-Zehnder interference pattern, produced over the reduced temperature range 1.7 x 10 - 5 < t < 2 x 10 _ 1 +, mapped pressure as a function of height i n the sample. From an analysis of t h i s pattern, pressure-density isotherms were obtained and used to determine the c r i t i c a l behaviour of the isothermal compressibility, < T > along the c r i t i c a l isochore. The expected power law, K = TQ | t | ^ , was f i t to the data with the r e s u l t y = 1.26 ± 0.05, which i s consistent with the accepted renormalization group p r e d i c t i o n for the c r i t i c a l exponent y. 63 REFERENCES 1. M.R. Moldover, J.V. Sengers, R.W. Gammon and R.J. Hocken, Rev. Mod. Phys. _51, 79 (1979). 2. C. Domb, Physics Today 21_, 23 (1968). 3. J.D. van der Waals, Doctoral d i s s e r t a t i o n , Leiden (1873). 4. J.D. van der Waals, Verh. Kon. Akad. Wetensch. Amsterdam (1880). 5. E.A. Guggenheim, J . Chem. Phys. 13_, 253 (1945). 6. L. Onsager, Phys. Rev. 65_, 117 (1944). 7. J.W. Essam and D.L. Hunter, J . Phys. C 1_, 392 (1968). 8. T.D. Lee and C.N. Yang, Phys. Rev. 87, 410 (1952). 9. B. Widom, J . Chem. Phys. 43, 3898 (1965). 10. L. Kadanoff, Physics 2_, 263 (1966). 11. K. Wilson and J . Kogut, Phys. Rep. 12C, 75 (1974). 12. 0. Maass, Chem. Rev. 23_, 17 (1938). 13. H.W. Habgood and W.G. Schneider, Can. J . Chem. 30, 847 (1952). 14. C. Pittman, T. Doiron and H. Meyer, Phys. Rev. B 20, 3678 (1979). 15. M.W. Pestak and M.H.W. Chan, Phys. Rev. B 30, 274 (1984). 16. H. GUttinger and D.S. Cannell, Phys. Rev. A 24, 3188 (1981). 17. H.L. Lorentzen, Acta Chem. Scand. 7_, 1335 (1953). 18. H.B. Palmer, J . Chem. Phys. 22_, 625 (1954). 19. L.R. Wilcox and D. B a l z a r i n i , J . Chem. Phys. 48, 753 (1968). 20. D. B a l z a r i n i , Ph.D. th e s i s , Columbia University (1968). 21. D. B a l z a r i n i and K. Ohrn, Phys. Rev. L e t t . 29, 840 (1972). 22. R. Hocken and M.R. Moldover, Phys. Rev. L e t t . 37_, 29 (1976). 23. D. B a l z a r i n i and O.G. Mouritsen, Phys. Rev. A 28, 3515 (1983). 64 24. J.C. Maxwell, Nature 1_1, 357 (1875). 25. M.J. George and J . J . Rehr, Phys. Rev. L e t t . 53_, 2063 (1984). 26. F.J. Wegner, Phys. Rev. B 5_, 4529 (1972). 27. S.Y. Larsen, R.D. Mountain and R. Swanzig, J . Chem. Phys. 42_, 2187 (1965). 28. M. Burton and D. B a l z a r i n i , Can. J . Phys. 52_, 2011 (1974). 29. The computer program was written by John de Bruyn and w i l l appear i n h i s Ph.D. t h e s i s . The program u t i l i z e s the f i t t i n g routine NL2SN0 supplied by the UBC computing centre. 

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