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Coexistence curve of sulfur hexafluoride in the critical region. Ohrn, Kenneth Edward 1972

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( i ) THE COEXISTENCE CURVE OF SULFUR HEXAFLUORIDE IN THE CRITICAL REGION by KENNETH EDWARD OHRN B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFIMLENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n ENGINEERING PHYSICS i n the Department of PHYSICS We accept t h i s t h e s i s as conforming to the required standard from candidates f o r the degree of MASTER OF APPLIED SCIENCE THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1972 In presenting th i s thes i s in p a r t i a l f u l f i lmen t o f the requirements fo r an advanced degree at the Un iver s i t y of B r i t i s h Columbia, I agree that the L ib ra ry sha l l make i t f r e e l y ava i l ab le for reference and study. I fu r ther agree that permission for extens ive copying of th i s thes i s fo r s cho la r l y purposes may be granted by the Head of my Department or by h i s representat ives . It is understood that copying or pub l i c a t i on of th i s thes i s fo r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of Pf)-f5>ICS The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada Date Ift ^AR 12 ( i i ) ABSTRACT This t h e s i s studies the shape of the coexistence curve of s u l f u r hexafluoride i n the c r i t i c a l region. The di f f e r e n c e i n index of r e f r a c t i o n between the l i q u i d and vapour phases i s shown to be proportional to the dif f e r e n c e i n density. Thus the c r i t i c a l exponent " |3 " i s measured. These values were found from l i n e a r f i t s to log-log data: r 2 y3 * 0.339 +_ 0.003 - £ < 3 x 10 /3 = 0.347 +_ 0.002 - £ > 10* -3 T—Tc Here, "Tc" i s the c r i t i c a l temperature and C = T c The temperature range covered i s 3 x 10" 64 - 6 < 6 x 10" 2 . The c r i t i c a l index of r e f r a c t i o n (n c) i s measured, with the r e s u l t n c - 1.093 + 0.002 ( i i i ) TABLE OF CONTENTS I. Introduction. I I . History. A. General Introduction. B. C l a s s i c a l Theory. C. Modern Theory. D. Recent Experiments. I I I . Description of Optics. A. Q u a l i t a t i v e Explanation. B. Information from the Fraunhafer Pattern. IV. The Lorentz-Lorenz Relationship. V. Analysis of Optics. A. Coexistence Curve. B. Isotherms. VI. Experimental D e t a i l s . A. The Sample C e l l and F l u i d . B. The F i l l i n g Process. C. Temperature Control. D. Technique and Other Equipment. VII. Index of Refraction Measurement. A. General Description. B. Experimental Technique. C. Prism Index of Refraction. D. Analysis of Optics. E. External Angle Measurement. (iv) VII. Index of Refraction Measurement, (cont'd.) F. Data An a l y s i s . G. F i n a l Results. H. Si g n i f i c a n c e of Results. V I I I . Data A n a l y s i s . IX. Error Discussion. (v) LIST OF TABLES Page I. D e f i n i t i o n of C r i t i c a l Exponents. 14 I I . C l a s s i c a l and Ising Model Values of C r i t i c a l 15 Exponents. I I I . The C r i t i c a l Exponent "j3 n. 18 IV. Prism Index of Refraction f o r A = 6328 A. 49 V. Least Square F i t of SF g Refractive Index Data. 55 VI. V a r i a t i o n of Chosen Tc with Assumed Q Value. 63 1 I . INTRODUCTION. This t h e s i s i s the examination of the coexistence curve of su l f u r hexafluoride i n the c r i t i c a l region. An o p t i c a l interference technique i s used to measure the d i f f e r e n c e i n index of r e f r a c t i o n between the l i q u i d and vapour phases. The c r i t i c a l index of r e f r a c t i o n and the sum of the l i q u i d and vapour indices are also measured with a wedge-shaped f l u i d sample. The o p t i c a l method used allows considerable p r e c i s i o n to be reached i n the determination of the shape of the coexistence curve. A more thorough introduction to c r i t i c a l phen-(29) omena can be obtained from the a r t i c l e s of Fisher (1967), S e t t e ( 3 1 ) ( 1 9 6 9 ) , S m i t h ( 3 2 } ( 1 9 6 9 ) , Kadanoff et a l ( 5 ) ( 1 9 6 7 ) and H e l l e r ^ 3 0 ^ ( 1 9 6 7 ) . These lengthy a r t i c l e s o f f e r a d e t a i l e d p i c t u r e of theory and experiment i n c r i t i c a l phenomena. 2 I I . HISTORY A. General Introduction. C r i t i c a l phenomena research began c i r c a 1869 when Thomas A n d r e w s ^ r e p o r t e d that, above 304°K. he could not l i q u i f y carbon dioxide. Similar observ-ations on other materials showed t h i s behavior to be a basic property common to most materials. Research has continued on these higher-order phase t r a n s i t i o n s . The general behavioral s i m i l a r i t i e s known as c r i t i c a l phenomena are shown by widely d i f f e r e n t kinds of phase t r a n s i t i o n s : l i q u i d - g a s ; binary mixtures; order-disorder transformations i n binary a l l o y s ; f e r r o -magnetic-paramagnetic systems; anti-ferromagnetic-paramagnetic systems; polarized-unpolarized f e r r o -e l e c t r i c s ; super conductors-normal conductors; and superfluid-normal f l u i d . This t h e s i s i s concerned with the liquid-gas t r a n s i t i o n of a simple f l u i d . I f we examine the i s o -therms of a t y p i c a l f l u i d such as SFg, we f i n d a region where two phases c o - e x i s t . Such isotherms are shown i n F i g . l . The dotted curve's'top i s the c r i t i c a l region. The curve i t s e l f i s the locus of points g i v i n g the composition of the l i q u i d and vapour phases. Section B of t h i s chapter derives the shape of the coexistence curve using van der Waals' equation and the Maxwell equal area r u l e . 3 F i g . 1 The general behavior of the isothermal curves of a r e a l gas, and the general shape of the coexistence curve. 4 This t h e s i s seeks to measure the shape of the coexistence curve of SF^ to a higher degree of accuracy than previous experiments. Similar coexistence curves e x i s t i n other systems. Certain p a i r s of f l u i d s , c a l l e d binary f l u i d s , mix i n any proportion above a c r i t i c a l temperature (Tc). Below t h i s temperature, they separate i n t o two .phases whose composition i s given by a coexistence curve. The d i f f e r e n c e i n t h e i r concentrations decreases with i n -creasing temperature and disappears at the c r i t i c a l point. Fig.2a shows such a coexistence curve f o r a binary f l u i d system. A ferromagnetic material (Fig.2b) at low temp-erature maintains a magnetic moment per unit volume when the magnetic f i e l d i s removed. Such magnetization decreases r e g u l a r i l y as temperature i s r a i s e d and goes to zero at the Curie temperature. Similar observations : can be made f o r other types of c r i t i c a l systems and the s i m i l a r i t y i n the behaviour of q u a n t i t i e s describing t h e i r equilibrium properties can be e s t a b l i s h e d . The r e v i v a l of i n t e r e s t i n c r i t i c a l phenomena has both t h e o r e t i c a l and experimental b a s i s . Improve-ments on the van der Waals equation have proved d i f f i c u l t since a dense gas or a f l u i d i s i n t r a c t a b l e mathematic-a l l y . The major obstacle, evaluation of c o n f i g u r a t i o n a l i n t e g r a l s , has seen some progress r e c e n t l y . Also, 5 O X£ » 1 Coneentration of F l u i d A (X A) Fig.- 2a Coexistence curve of a binary f l u i d system 1 Temp i / Ti / M i \ n -Fig. 2b Spontaneous - magnetization 6 thermodynamical problems i n the c r i t i c a l region have occurred. For example, the Taylor s e r i e s expansion of the free energy about the c r i t i c a l point has proven to not be a wise t h e o r e t i c a l procedure. Useful c r o s s - f e r t i l i z a t i o n between branches of physics has occurred. For example, Onsager's exact s o l u t i o n of the two-dimensional Ising model of a f e r r o -magnet may be tra n s l a t e d simply and d i r e c t l y into the quivalent model i n f l u i d physics, the two-dimensional l a t t i c e gas. These important l i n k s , the new t h e o r e t i c a l under-standing and the refinement of experimental techniques have made c r i t i c a l phenomena a subject of revived i n t e r e s t . 7 B. C l a s s i c a l Theories. C l a s s i c a l theories, based on the existence of a t t r a c t i v e forces between p a r t i c l e s , consider cooperative e f f e c t s and are able to predict the existence of c r i t i c a l p o ints. Theories of t h i s type are: van der Waals' theory f o r f l u i d s integrated with the Maxwell equal area r u l e i n the two-phase region; the ferromagnetic theory based on the Weiss notion of molecular f i e l d and the Bragg and Williams theory f o r s u b s t i t u t i o n a l s o l i d s o l u t i o n s . To show how c l a s s i c a l theories can predict c r i t i c a l points, consider the following development of van der Waals' equation: ( ? + a p * X p - b ) = KT . ( i i - D We define Pc, Tc, pc as the c r i t i c a l pressure, temper-ature and density and we rewrite Equation ( I I - l ) : We equate c o e f f i c i e n t s of Equation (II-2) and the follow-ing "equation ( I I - 3 ) 8 These equations r e s u l t : ~3p c = - £ , ( I I - 4 ) 3pi - %. + 1£Tc (n-5) - p c = ~ - % . (II-6) r <xb Equation ( I I - 4 ) gives, on inspection: . b p c ^ 7} . (II-7) Combining Equations (II-5) and (II-6) to eliminate Pc, we obtain: Substituting Equation (II-7), we obtain: "RT C ~ 8 a . . ( I I - 9 ) Since Equation (II-7) gives: we can rewrite Equation (II-9) to give: a p e - RTc Substituting Equation (II-7) into Equation (II-6), we 9 obtain: a zi bx (11-10) Using Equations (II-7), ( I I - 9 ) , (11-10) i n the following forms: , ex.~ 3?c , 2 j k e 3 p t and multiplying Equation ( I I - l ) by 2 7b/a, we obtain: ( n - i i ) This gives: i - 8 _ | ^ ) _ 3 ( p / f t f . F ( ^ y r c (11-12) Equation (11-12) i s a universal equation of sta t e | implying a "law of corresponding st a t e s " . Any two f l u i d s with the same values of P/Pc, T/Tc, p/pc may be said to be i n corresponding states. A double power serie s expansion of Equation (11-12) about p/jP c = T/Tc = 1 gives: tn T c (11-13) Here, P'(T/Tc) has two i n t e r p r e t a t i o n s . If T>Tc, i t i s the pressure on the c r i t i c a l isochore. I f T<Tc, 10 i t i s the vapour pressure on the coexistence curve: When T < Tc, we set: t [Pjtf - 6 (T^ = = O (11-14) and we get: (11-15) (11-16) Equation (11-15) gives, upon rearrangement: p « p c [ l t 2 ( - 6 ^ ] T c Looking at P - p isotherms we can see that the two roots of t h i s equation describe the - p j points on the isotherms oc-cupied by the actual l i q u i d or. vapour density. Maxwell's equal area r u l e ensures that, f o r example, AB=BC and there-fore that the symmetry implied by Equation (11-16) i s i n d i c a t i v e of the shape of the coexistence curve. We may therefore write: V / i f J / i ! pt p p L * P c ( l -p v = p c ( l - 2 ( - £ V / a J (11-17) (11-18) This gives us the more usual formulation of the coexistence curve. or p L - p v ~ 4 ( - e V ^ . ( i i - i 9 ) The equation of van der Waals gives us another parameter - the c o e f f i c i e n t of isothermal c o m p r e s s i b i l i t y » i 3 it P o • on the c r i t i c a l isochore. Pc PjO i s dimensionless and: * 6" • (11-20) showing the divergence of the c r i t i c a l isochore compress-i b i l i t y as T —>• Tc. (2) Data compilation by E.A. Guggenheim i n 1945 suggested that a representation of the coexistence curve could be: pL~pv - 3.5 ( - € . ) ° . (11-21) This empirical equation obtains from a reduced-variable p l o t of a l l a v a i l a b l e data on various gases. Thus c l a s -12 s i c a l theory f a i l s f o r f l u i d s . In 1907, Pie r r e Weiss suggested large i n t e r n a l f i e l d s i n ferromagnets and derived: £ r / ( x ) « £ * ( l - f e * % . . . ) - ( H - 2 2 ) Here JX = average projected magnetic moment, jiL = molecular dipole moment, L(x) = Langevin's function, = coth(x) - 1/x and WT T I W (n-23) If we expand L(x) i n a power se r i e s about x=0, an analogy develops between Weiss' ferromagnetic equation and van der Waals' gas equation. We obtain: JTTC. O \ T C 1 5 ; ( 1 1 - 2 4 ) I f we' neglect higher order terms, the bracketed fa c t o r vanishes. Then spontaneous magnetization (H=0) can occur, with: "3" ( n - 2 5 ) This i s to be compared to Equation (II-7), the van der Waals equation. A further analogy can be drawn. If T>Tc, we may evaluate the i n i t i a l s u s c e p t i b i l i t y X c i n a ferromagnet by using Equation (11-12): This i s the Curie-Weiss law and i s to be compared to Equation (11-20). The two symmetric roots which occur correspond to magnetization "up" or "down". No extension or refinement of c l a s s i c a l theory was able to account f o r the coexistence or magnetization curves' exponents being other than Neither was there any accounting f o r the c o m p r e s s i b i l i t y or s u s c e p t i b i l i t y diverging f a s t e r than (T - T ^ " 1 . Thus c l a s s i c a l theory f a i l e d and more modern developments were needed. C. Modern Theory. came with Onsager's paper on the two-dimensional Ising model i n zero f i e l d . Each atom on a l a t t i c e s i t e can be i n a spin up or a spin down state. Interaction i s between neighbouring p a i r s only and i s +_ J depending on whether the spins are p a r a l l e l or a n t i p a r a l l e l . Spon-taneous magnetization i s predicted. At the c r i t i c a l point, the s p e c i f i c heat i s l o g a r i t h m i c a l l y i n f i n i t e . (11-26) The f i r s t step away from the c l a s s i c a l theories No exact s o l u t i o n has been found f o r the three 14 dimensional model. Computer approximations have been made whose r e s u l t s are u s u a l l y stated as l i m i t i n g ex-ponents. I f *-*o d IUK -x. y (11-2 7) we say, -to - a s x— > O (11-28) Following conventions established by M.E. Fisher a set of l i m i t i n g exponents are set out i n Table I. TABLE 1 D e f i n i t i o n of C r i t i c a l Exponents Exponent Magnet F l u i d j3 Mo £ (-6^ Ap T (€>0) Po ir' (6<0) A> A (€>o) C M ^ (- 6r* Cv cr *« C M » (-fey*' Cv S_ (fi-o) H * M 8 Here,-£ = T c . This t h e s i s i s the measure-ment of j3 f o r a f l u i d , using a p r e c i s i o n o p t i c a l method. (4) T.D. Lee and C H . Yang show that the mathemat-i c a l discussion of the l a t t i c e gas model i s isomorphic, to the magnetic problem i n which a spin points up or down. This strengthens the fluid-magnet analogy. In f a c t , analogies to superconductors, sup e r f l u i d s , f e r r o e l e c t r i c s , binary a l l o y s , and binary f l u i d s can be discussed i n the same c o n t e x t ^ \ Thus the modern era of c r i t i c a l phenomena begins. In Table I I are l i s t e d the c l a s s i c a l values (28) derived by Landau , by expanding the thermodynamic p o t e n t i a l i n a power serie s about the c r i t i c a l point. . Also r e s u l t s of computations using the l a t t i c e gas model are shown. TABLE II C l a s s i c a l and Ising Model Values of C r i t i c a l Exponents Exponent Landau Theory 3-D Ising h 0.313 + 0.004 r <t 1 1.250 +. 0.001 1 1.31 + 0.05 S 3 5.2 ,+ 0.15 These values are from Kadanoff et a l . (1967) B. Widom i n 1965 introduced the " s c a l i n g law" to the t h e o r e t i c a l d i s c u s s i o n . Certain r e l a t i o n s h i p s are proposed between the c r i t i c a l exponents. d + r + a p - Z , (11-29) vSince 0,^,^ ' and & can i n p r i n c i p l e be measured 16 by the method of t h i s t h e s i s , a precise t e s t of the sc a l i n g laws i s p o s s i b l e . D. Recent Experiments. As i n t e r e s t i n c r i t i c a l phenomena grows, new methods develop and new analyses of o l d data are done. Conventional methods, measuring pressure and density, become very d i f f i c u l t near to the c r i t i c a l temperature. Good r e s u l t s have been obtained by Weinberger and ( 7) (8) Schneider and Habgood and Schneider f o r Xenon (9) and also f o r SF^ by Atack and Schneider . The p r e c i s i o n of these methods, however, l e f t much to be desired i n the region near c r i t i c a l . O p t i c a l methods are convenient f o r the study of f l u i d s , and considerable ingenuity has gone into other methods. 0. Maass^ 1 0^ studied C0 2 with bouyant b a l l s and Weinberger and Schneider^ 1 1^ measured the count rate as a function of height i n a sealed sample of Xenon mixed with a r a d i o a c t i v e t r a c e r . G.D'Abramo, F.P. R i c c i and (12) F. Menzinger studied the Ga-Hg binary f l u i d system by a method which i s e s s e n t i a l l y neutron radiography. (13) H.L. Lorentzen studied CO^ near c r i t i c a l using a prismatic vessel with v e r t i c a l prism a x i s . He measured a h o r i z o n t a l r e f r a c t i o n angle Q (Z) proportional to |0(z). A s i m i l a r method has been used by Schmidt, Traube and S t r a u b ( 1 4 - 1 6 ) . H . P a l m e r ( 1 7 } used a plane-parallel-windowed vessel i n a Z-Type Schlieren o p t i c a l system to study ethane, CC>2» and Xenon. He neglected the coexistence curve. This t h e s i s uses a prism method s i m i l a r to that of Lorentzen to measure the indices of r e f r a c t i o n of the l i q u i d and vapour phases of SFg as a function of temperature. Table I I I i s a summary of determinations of the c r i t i c a l exponent S f o r various systems and methods. 18 TABLE I I I THE CRITICAL EXPONENT " j} " VALUE 0.333 0.33 0.345 .+ 0.015 0.344 +. 0.01 0.346 +_ 0.008 0.362 0.370 0.354 + 0.010 0.368 + 0.005 0.349 + 0.006 0.348 +_ 0.002 0.354 +, 0.007 0.340 + 0.010 0.373 + 0.005 SYSTEM compilation Xenon Xenon-analysis of (7) c o 2 Xenon compilation f o r f l u i d s compilation f o r ferromagnets He 4 C r B r 3 N 20 ~~ c o 2 CC1F-3J 3-Methylpentane-Nit r o ethane Ga-Hg REFERENCE Guggenheim Weinberger and Schneider Fisher Lorentzen (2K1945) (7)(1952) (3)(1964) (13)(1953) D.A. B a l z a r i n i (18)(19) (1968) M. V i c e n t i n i -Missoni et a l (20)(1970) Roach and Douglas (21)(1966) Ho and L i t s t e r (22)(1969) Sengers, Straub and V i s c e n t i n i -Missoni Wims, Mclntyre and Hynne D'Abramo, R i c c i and Menzinger (23K1971) ( 2 4 X 1 9 6 9 ) (12X1972) 19 I I I . DESCRIPTION OF OPTICS A. Q u a l i t a t i v e Explanation The l i g h t source used was a helium-neon laser ( X = 6328 A) whose f l u x was attenuated by standard methods ; to below 0.1 mW. F i g . 3 shows a schematic diagram of the apparatus. The incident f i e l d was rend-ered uniphase over the c e l l aperture by means of an inverted telescope and pinhole f i l t e r . A glass c e l l with p a r a l l e l windows was f i l l e d with the f l u i d under study. The average density p was made as close as possible to the c r i t i c a l density p c -The c e l l was arranged i n s i d e apparatus that maintained temperature within 0.0002 degrees Centigrade f o r several hours. A d i s t i n c t i v e interference.pattern was observed on the back f o c a l plane of an objective lens. If p = pc, then earth's g r a v i t a t i o n a l f i e l d ensures that p> pc at the c e l l ' s bottom and p * j^c. at the top. Therefore, f*'^ o c c u r s a t some height i n the c e l l Z (T). Accurate f i l l i n g puts Z near to the o o centre of volume of the c e l l . I f T >T , the f l u i d i s c i n the one-phase region and the density d i s t r i b u t i o n resembles the sketch of Fig.4. The interference pattern formed was the subject of t h i s t h e s i s . The f i r s t reference to t h i s method of (25) studying f l u i d s i s Gouy (1880). Detailed analysis and d e s c r i p t i o n of experimental d e t a i l s i s found i n the (18) (19) th e s i s of D.A. B a l z a r i n i and i n Wilcox and B a l z a r i n i CROSSED TDCARIIERS 1 M ' C R O scope OBTecn«/e L£M5 1 1 s— LENS (-focal len 34k W-T'5 F i g . 3 Optical Apparatus Schematic pi ASKS "PLANE. WAVE h= 6328 -5 M O 21 F i g . 4 Density D i s t r i b u t i o n , T > T. g. 5 Formulation of the Fraunhofer Pattern 23 At Zo(T), the maximum density gradient occurs because the f l u i d ' s c o m p r e s s i b i l i t y i s larg e s t there. Fig.5 shows a plane wave incident on the c e l l . A ray bundle entering the c e l l at Zo i s r e f r a c t e d downward.. Above and below, bundles at Z + and Z are re f r a c t e d through a smaller angle, and are focused to a common point on the F-plane. There, they i n t e r f e r e with s h i f t e d phases. Assuming anti-symmetry of the density about (o) the f r i n g e minima should be n u l l s . However, small c e l l imperfections, such as s l i g h t departures from absolutely p a r a l l e l windows, can cause loss of contrast of f r i n g e s . B. Information from the Fraunhofer Pattern. If a f i l m i s slowly transported past a v e r t i c a l s l i t i n the F-plane while simultaneously the temperature i s r a i s e d through the c r i t i c a l region, a representation of I(k, T) versus T i s obtained. Fig.6 i s one such f i l m , taken as data f o r t h i s t h e s i s . Any v e r t i c a l section through the f i l m represents the Fraunhofer pattern at that temperature, save f o r e f f e c t s of l a g . If the f l u i d i n the c e l l had a constant density., gradient, the angle of r e f r a c t i o n would measure the constant c o m p r e s s i b i l i t y . However, the density gradient changes with temperature and, therefore, the c o e f f i c i e n t of c o m p r e s s i b i l i t y , which depends upon density, i s not constant. The most strongly r e f r a c t e d f r i n g e of the pattern, nevertheless, should be a measure of the Figure 6 Kymograph f o r SF & 25 maximum c o m p r e s s i b i l i t y . As the f l u i d approaches T c from above, t h i s f r i n g e extends to la r g e r angles and diverges as £-*• O . For negative £, (T< Tc), there i s , at Zo, a meniscus i n the f l u i d which i s a d i s c o n t i n u i t y of density between the vapour and the l i q u i d phases. Thus, for £ < 0 , the pattern fades away, beginning with lowest order f r i n g e s . Each f r i n g e seems to diverge to i n f i n i t e angle at a p a r t i c u l a r temperature. The most re f r a c t e d band i n the Frauhofer pattern i s formed by rays which pass j u s t above and below the meniscus. For £ > 0 , ( s u p e r c r i t i c a l ) the highest angle f r i n g e was formed by rays which t r a v e l l e d the c e l l above and below the region of highest gradient. For £ < O ( s u b c r i t i c a l ) there i s a d d i t i o n a l o p t i c a l path d i f f e r e n c e caused by the d i s c o n t i n u i t y at the meniscus. The l i q u i d -vapour density d i f f e r e n c e may be measured by p l o t t i n g t h i s fading of f r i n g e s as a function of temperature. This t h e s i s measures the coexistence curve of s u l f u r hexafluoride by measuring the liquid-vapour r e f r a c t i v e index d i f f e r e n c e at temperatures below Tc. A d d i t i o n a l information on the actual indices of r e f r a c t i o n was obtained from the scheme described i n Chapter VII. 26 IV. THE LORENTZ-LORENZ RELATIONSHIP. Refractive index, "n"» and density "p" are re l a t e d by the Lorentz-Lorenz formula, with "L" a's defined by t h i s equation: p*-'. = p L ( i v - D The assumption that L i s a constant i s suspect close to Tc. The Lorentz-Lorenz function may be regarded as a constant only i n a homogeneous medium i n which the c o r r e l a t i o n length associated with density f l u c t u a t i o n s i s much smaller than the incident wavelength. These f l u c t u a t i o n s do e x i s t and give r i s e to phenomena such as c r i t i c a l opalescence. Assuming L constant, we can expand the Lorentz-Lorenz formula i n a Taylor s e r i e s about j3c • p ' - p-pc From the r e f r a c t i v e index measurements of Chapter VII, the following values of a.^ and a^ were c a l c u l a t e d : n c = 1.093 0.002, a1 = 1.0195 +. 0.0002, a 2 = 0.0215 .+ 0.0002, Where a i = ( n c + D ( n c 2 + 2) , (IV-3) 6n c and a 2 = ( n c 2 - l ) ( 3 n c 2 - 2) . (IV-4) 12n c 2 * (IV-2) 27 The v a l i d i t y of the assumption of constant L i s c r u c i a l . Some t h e o r e t i c a l analyses, notably Larsen, (27) Mountain, and Zwanzig conclude that the v a r i a t i o n i s small enough to have l i t t l e e f f e c t upon the meaning-f u l l n e s s of these experiments. Comparisons on Xenon of independent determinations of index of r e f r a c t i o n and (26) density by Chapman, Finnimore and Smith (1968) show a very s l i g h t r i s e i n L as Tc i s approached from above. They "...conclude that (1) i t i s u n l i k e l y that L f o r Xenon va r i e s by more than +. 1% throughout the e n t i r e f l u i d , range, (2) there i s no evidence that a large anomaly e x i s t s near the c r i t i c a l point They c a l c u l a t e t h e i r errors i n determining L to be 0.5%. c l e a r . However, the approximation i s not a bad one and index of r e f r a c t i o n data continue to be useful i n coexistence curve i n v e s t i g a t i o n . Thus the exact dependence of L on 2 8 V. ANALYSIS OF OPTICS. In t h i s chapter, an analysis i s made of the Fraunhofer pattern described i n Chapter I I I . The r e l -a tionship of the pattern to the density d i s t r i b u t i o n ^ ( z ) and to the shape of the coexistence curve i s shown. Geometrical o p t i c s are used i n t h i s a n a l y s i s . A. Coexistence Curve. Light from the He-Ne la s e r (6328 A) was f i r s t focussed i n t o a pinhole. The r e s u l t i n g expanded wave-fro n t was passed through a lens to obtain a plane wave-front t r a v e l l i n g down the o p t i c a l axis of the system. The i n t e n s i t y of t h i s wave front was attenuated with crossed p o l a r i z e r and i t ' s s i z e made much smaller than a c e l l width with various masks. The c e l l ' s v e r t i c a l axis was made perpendicular to the o p t i c a l a x i s . The density d i s t r i b u t i o n p ( z ) v a r i e s i n s i d e the c e l l (Fig.4). This gives r i s e to a varying index of r e f r a c t i o n dn/dz, as described by the Lorentz-Lorenz formula. A d i f f e r e n t i a t i o n of S n e l l ' s law gives: See Fig.7. Planes of constant r e f r a c t i v e index are hor-i z o n t a l and i t i s assumed that the path's curvature i s small i n the d i f f e r e n t i a t i o n which gives Equation V - l . A ray which enters h o r i z o n t a l l y (9 = 0) i s then 29 INC IDENT PLANE WAV/6 F i g . 7 Analysis of o p t i c s f o r coexistence curve determination 30 bent through the small angle 0 ; - (V-2) dn Here, n and - j - are evaluated at the p o s i t i o n Z and 9 ^ i s the i n t e r i o r angle. the outside angle 0 O . Using unity as a i r ' s r e f r a c t i v e index and assuming ©i £ s i n O i : The symmetry of the density d i s t r i b u t i o n i s such that the and Z above and below the o p t i c a l a x i s . Two rays that traverse such positions are r e f r a c t e d through the same angle and are focused at the same point i n the f o c a l plane of the lens. The o p t i c a l path d i f f e r e n c e w i l l c o n s i s t of two parts. There i s a di f f e r e n c e i n e f f e c t i v e path length due to the di f f e r e n c e i n r e f r a c t i v e index at Z + and Z~ . There i s also the standard d i f f e r e n c e due to a i r path i n e q u a l i t i e s . Since the ray at Z t r a v e l s through the le s s dense medium: At the c e l l boundary, S n e l l ' s law i s used to f i n d (V-3) r e f r a c t i v e index gradient i s equal at p o s i t i o n Z + (V-4) where n + and n"~ are the r e f r a c t i v e indices at Z + and Z . 31 Using n a i r = 1 and s i n 9 £ 9 and 9 ^  j£ i f (V-5) Destructive interference occurs when ( V - 6 ) Using the Lorentz-Lorenz r e l a t i o n _ n(j>)-nc - (nc-/)-a,-p* (V-7) where and P * _ 4>nc (V-8) (V-9) and n c i s the c r i t i c a l r e f r a c t i v e index, we obtain phase di f f e r e n c e by multiplying by o it o ( v - i o ) . ( V - l l ) Figures 8a and b i l l u s t r a t e the phase d i f f e r e n c e equation f o r sub and s u p e r c r i t i c a l density d i s t r i b u t i o n s . The phase di f f e r e n c e i n each f i g u r e i s represented by the crosshatched area. The f i r s t term i s the area F i g . 8 I l l u s t r a t i o n of phase d i f f e r e n c e Equation ( V - l l ) under the curve between Z_ and Z +; the second terra subtracts the area of the hatched rectangle. Since minima occur when , the minima are g r a p h i c a l l y constructed i n the manner shown. At s u b c r i t i c a l temperatures, there i s a d i s -c o n t i n u i t y i n the density d i s t r i b u t i o n at the meniscus. The angles of the lower order minima diverge to " i n f i n i t y " depending on the " i n f i n i t e n e s s " of the gradient there. It i s seen that the liquid-vapour density d i f f e r e n c e information needed to measure the coexistence curve i s represented by the number of fri n g e s which have "disappeared" under the " i n f i n i t e " gradient at the meniscus. A kymograph made of the Fraunhofer pattern as temperature i s slowly r a i s e d reveals the t o t a l number of f r i n g e s . Fig.6 i s a photographic p r i n t made from one such actual —5 kymograph with £ = -8.1 x 10 The number of fr i n g e s can give the coexistence curve only i f the Lorentz-Lorenz formula i s valid' with constant L and i f the r e c t i l i n e a r diameter of the c o e x i s t -ence curve were constant. That i s , there are errors i n r e l a t i n g n l i q - nvap t o j^1- ~ * These are discussed i n Chapte Tr r IV. It i s easy to see that i f N i s the number of fri n g e s present at a p a r t i c u l a r temperature, then the dif f e r e n c e i n index of r e f r a c t i o n between the l i q u i d and vapour i s given by: n L - n v - N-A I 1 ( V - l l ) where " A " i s the laser wavelength (6328 A) and i s the c e l l thickness. These determinations of the coexistence curve a l l measure N, the t o t a l path d i f f e r e n c e between rays tra v e r s i n g above and below the meniscus. Chapter IV discusses the Lorentz-Lorenz r e l a t i o n ship and evidence of i t s a p p l i c a b i l i t y . 35 VI. EXPERIMENTAL DETAILS. A. The Sample C e l l s and F l u i d Sulfur hexafluoride (SFg) was chosen as the ex-perimental f l u i d because of i t s convenient properties. It i s o p t i c a l l y transparent, i t s c r i t i c a l temperature i s approximately 45°C. and i t s c r i t i c a l pressure i s about 37 atmospheres. It i s f a i r l y easy to construct experiment-a l vessels to handle that kind of pressure and the r e -l a t i v e l y low temperatures are easy to reach and maintain. Two types of c e l l were used i n t h i s experiment. One was constructed from pyrex glass tubing and the other, more elaborate c e l l , has a metal body made of Kovar and plane p a r a l l e l sapphire windows brazed to the body with copper. The glass tubing was sealed at one end and joined to an 8-10 cm. piece of 2-mm. diameter c a p i l l a r y . Two opposing outer faces of the tube were polished and made o p t i c a l l y f l a t . No method was devised to p o l i s h the inner faces, however. Thus the i n t e r i o r of the c e l l i s quite rough. This introduces a va r i a b l e o p t i c a l path length that reduces the accuracy of the measurements near the c r i t i c a l temperature. The metal c e l l , however, has plane p a r a l l e l windows whose f l a t n e s s i s suspect only to the degree that small warps may have been introduced during the brazing operations of i t s construction and under the high pressures encountered near c r i t i c a l . 36 Fig.9 shows the two experimental vessels a f t e r f i l l i n g . B. The F i l l i n g Process. C e l l f i l l i n g was accomplished with the apparatus shown i n Fig.10. A Welch #1400 Vacuum pump was attached to a valve and connector assembly, as was a Matheson le c t u r e b o t t l e of SFg. During the f i l l i n g of the metal c e l l , a V i r t i s thermocouple vacuum guage was also attached. The glass c e l l s , a f t e r annealing, were thoroughly : cleaned with Sparkleen, tap water and research grade acetone and then baked overnight. A brass gas plug was then d r i l l e d to the diameter of the c a p i l l a r y stem and attached to the vacuum manifold with t e f l o n tape i n the threads. Torr Seal low vapour pressure epoxy r e s i n was then used to connect glass stem to brass, and 20 hours were allowed f o r s e t t i n g and curing. Afte r 8 hours, the vacuum pump was started and l e f t on f o r the remaining 12 hours. After curing the epoxy, the c e l l was pressure tested with SFg. Leaks at the glass-brass j o i n t could be detected at t h i s time. Liquid SFg was then allowed to completely f i l l the c e l l , to act as a f l u s h . The l i q u i d was then pumped away. The process was repeated. The c e l l was then pumped down, undisturbed, f o r 8 hours. Pressures as low as 0-3 microns were attained during t h i s stage with the metal c e l l . These precautions ensured VACUUM PUMP 00 F i g . 10 C e l l ; f i l l i n g apparatus 39 that a l e v e l of p u r i t y of experimental f l u i d was reached that was commensurate with the accuracy desired of the measurement of the coexistence curve. Li q u i d was next transferred into the c e l l and the c e l l ' s valve closed. The c e l l and i t s valve were then removed from the manifold. SFg was bled out of the c e l l , using the needle valve, u n t i l s u f f i c i e n t SFg remained to ensure c r i t i c a l density could be reached. L i q u i d nitrogen was used to freeze the c e l l ' s contents and the glass c a p i l l a r y was pinched o f f and sealed. Rough observations of meniscus movement were made on the sealed c e l l . The 6-8 cm. of remaining c a p i l l a r y were then s e l e c t i v e l y pinched o f f , according to c a l c u l a t i o n s made on the temperature at which the meniscus "disappeared". Eventually, c e l l SFg-3A was constructed to such accuracy that meniscus movement was absent at temperatures 0.09°C below c r i t i c a l . ( £ S 1 0 ~ 4 ) A s i m i l a r procedure was used f o r the metal c e l l . However, the needle valve assembly was not removed from t h i s c e l l . C. Temperature Control The c e l l s are placed insid e apparatus shown schematically i n Fig.11. The water c i r c u l a t e d i n the brass outer can tubing i s regulated to within 0.05°C. The r e l a x a t i o n time from the outer can to t h e r e a t sink i s about 3.5 hours. The 40 second hunting cycles of the CELL 40 W I N D O W S T H E R M I S T O R S H£AT SINK - COPPER OTUNDER WOUMO VRVTTVI I4EATIS.G COIL., SMSCDOED THERMISTORS. S T Y R O P O A K I I M E O L A T O R water bath are reduced to about 0.0005°C at the inner mass. During more s e n s i t i v e parts of the experiment, a plywood box was used to enclose the apparatus and prevent d r i f t i n g at the inner mass due to room temperature changes. Thus rough temperature co n t r o l i s achieved. Precise temperature co n t r o l i s achieved e l e c t r o n -i c a l l y . Tungsten wire i s inserted into t e f l o n spaghetti and i s wound i n a double s p i r a l i n grooves cut i n t o the,, heat sink. Thermistors are epoxied into s t e e l or brass screws which screw in t o tapped holes i n the heat sink. An error signal lis derived from one thermistor, which i s one arm of a d.c. bridge. The bridge output i s amplified by a d.c. a m p l i f i e r . The amplified detector output i s then conditioned by a lead-lag network, which makes the loop gain i n f i n i t e at zero frequency and provides the r o l l o f f needed to ensure s t a b i l i t y . This conditioned d.c. bridge output drives the c o n t r o l input of a stab-i l i z e d power supply, which provides power (SlOOmw) to the tungsten heating wire. A given temperature can be selected by se t t i n g a decade resistance box on one arm of the d.c. bridge. The network w i l l hold the therm-i s t o r ' s resistance to within 0.05J1 , which corresponds to 0.0015°C. (1.5 millidegree) almost i n d e f i n i t e l y . Fig.12 shows t h i s network i n schematic form. Near Tc» i n order to reach equilibrium, temperature c o n t r o l i s mandatory f o r periods of 12-24 hours. Correcting f o r the i n s t a b i l i t y of room temperature, REGULATED "POWER SUPW.Y. F i g . 12 Elec t r o n i c temperature control network the d r i f t i n temperature over such a period can e a s i l y be kept to less than 0.0002°C by enclosing the apparatus i n a V plywood box. Thus precise temperature co n t r o l i s achieved. A second thermistor i s used as a monitor. The c i r c u i t uses another d.c. bridge, d.c. detector, and a chart recorder. This thermistor i s c a l i b r a t e d with a platinum resistance thermometer, which i n turn i s c a l -ibrated i n a t r i p l e point c e l l . The bridge components are accurate enough to give absolute temperature to within 0.02°C. The thermistor's c h a r a c t e r i s t i c s are determined to such p r e c i s i o n that the error i n temperature i s S £ t (o.ot V € £ - . Much of the data f o r the metal c e l l was taken using a Dymec di g i t a l - r e a d o u t quartz thermometer. The accuracy claimed f o r t h i s machine i s 0.0001°C. for short-term work. However, there i s a d r i f t i n i t associated with changes i n room temperature. There i s 0.001°C. d r i f t f o r a 1.0°C. d r i f t i n room temperature. D. Other Equipment and Techniques. The c o n t r o l network was maintained at e q u i l i -brium below Tc fo** several hours while the sample c e l l - - 4 came to equilibrium. For fc. < 10 , longer periods were used, up to 8-10 hours. A chart recording was made of the monitor thermistor's resistence during t h i s time. 44 The Fraunhofer pattern was focussed onto a 35 mm. camera body used as a f i l m transport. A small motor and approp-r i a t e gearing allowed f i l m to be drawn past a narrow s l i t i n the F-plane at about h. rev. per hour. When equilibrium was established, the laser was turned on and the camera was sta r t e d . The temperature was then swept slowly upwards to above Te and the r e s u l t i n g kymograph made (Fig.6). The resistance of the monitor thermistor at equilibrium, or the quartz thermometer readout, and the number of fri n g e s counted on the f i l m form the raw data f o r the c a l c u l a t i o n of 6 f o r the coexistence curve. 45 VII. INDEX OF REFRACTION MEASUREMENT. A. General Description. In order to measure the c r i t i c a l index of r e -f r a c t i o n and to measure the temperature dependence of n L . + n v (the average index of r e f r a c t i o n ) , the following scheme was devised and c a r r i e d out. The scheme involved construction of a glass c e l l i d e n t i c a l to those involved i n the coexistence curve measurement. A 30°„ crown glass prism was cut and inserted i n the c e l l . Fig.13 i s a schematic diagram of the c e l l and prism, showing t h e i r o r i e n t a t i o n . The c e l l was then f i l l e d with. SFg to c r i t i c a l density and inserted i n the temperature co n t r o l apparatus. The apparatus was l a i d h o r i z o n t a l and mounted on a r i g i d platform. The He-Ne laser was used as a l i g h t source. Fig.14 shows the arrangement between c e l l and l i g h t beam used i n t h i s determination. Since the c e l l i s h o r i z o n t a l , i t could be e a s i l y arranged to impinge the beam on the mutual i n t e r s e c t i o n of the meniscus and prism edge so that three beams emerge from the c e l l . One beam passed through only the meniscus and thus emerged l a t e r a l l y undeviated, to serve as a zero-angle reference. The other two beams passed through the l i q u i d and vapour and prism and thus were r e f r a c t e d through angles which were c h a r a c t e r i s t i c of the density of t h e i r respective media. The emergence angle was measured with a f r o n t -s i l v e r e d mirror mounted on an arm attached to a spectro-46 C E L L . •A "POLISHeD F A C E S FACES F i g . 13 C e l l and prism o r i e n t a t i o n F i g . 14 Beam-prism o r i e n t a t i o n 47 scope turntable. The beams were r e f l e c t e d into a t e l e -scope which was focussed at i n f i n i t y . Both turntable and telescope were r i g i d l y clamped to o p t i c a l r a i l s , the same r a i l s to which the l a s e r , c o l l i m a t i n g devices and c e l l holder were attached. B. Experimental Technique. A seri e s of measurements of the emergence angles of the beams was made. Temperature c o n t r o l was obtained from the same e l e c t r o n i c apparatus as previously described i n Chapter VI-C. Readings were taken from as low a temperature as possible and proceeded upwards at regular small i n t e r v a l s . A serious drawback to t h i s scheme involved the glass c e l l s . As i n Fig.13,^when the prism was inserted into the c e l l , a c e l l end had to be cut o f f to accomodate the prism. Subsequent j o i n i n g of a 2mm. c a p i l l a r y tube stem onto the c e l l l e f t considerable r e s i d u a l stress i n the region of the j o i n between the pyrex c e l l body and the c a p i l l a r y tube. Glass c e l l s used i n the coexistence curve measurements were annealed overnight to remove these stresses before f i l l i n g was accomplished. However, since the prism's crown glass melts at a temperature below that of the annealing temperature of pyrex, these index of r e f r a c t i o n c e l l s could not be s i m i l a r i l y made s t r e s s - f r e e . Thus, as temperature was r a i s e d , the i n -creasing pressure i n s i d e the c e l l became s u f f i c i e n t 48 to cause cracks to form, r e s u l t i n g i n a minor ex-plo s i o n . Three c e l l s were destroyed i n t h i s manner. However, considerable data was obtained from each c e l l before i t s destruction. Annealed coexistence curve c e l l s have considerably greater strength. Future users of t h i s scheme are cautioned to obtain pyrex or quartz prisms and anneal t h e i r c e l l s i f pressures i n the neighborhood of SFg (310-550 psi) are encountered. The raw data was analyzed on UBC's IBM 360/67. Two extrapolations were made, one to y i e l d a c r i t i c a l temperature Tc, the other to y i e l d the c r i t i c a l r e f r a c t i v e index, n c . A l e a s t squares f i t routine was used on the l a t t e r to get a l i n e a r f i t to the data. C. Prism Index of Refraction. The index of r e f r a c t i o n of the prism was determ-ined experimentally. It was also checked against f i g u r e s given f o r crown glass by both the manufacturer and by a standard reference t e x t . Extrapolations of f i g u r e s from the Chemical Rubber Company Handbook of Chemistry and Physics gives a c o r r e c t i o n f a c t o r f o r the index of r e f r a c t i o n at the wavelength of our He-Ne laser (X = 6328 A). This c o r r e c t i o n f a c t o r was also applied to nn (n f o r the sodium "D" l i n e ) as given by the manufacturer. A further experimental determination was made by the standard elementary o p t i c a l technique of the angle 49 of minimum de v i a t i o n . The laser was used as a l i g h t source f o r t h i s measurement. Table IV shows the r e s u l t s . TABLE IV Prism Index of Refraction f o r A = 6328A SOURCE INDEX CRC Handbook 1.515 ± 0.0005 Manufacturer 1.517 _+ ? Experimental 1.518 +. 0.002 The experimentally determined value 1.518 _+ 0.2% was used i n the data a n a l y s i s . D. Analysis of Optics. The symbols used i n t h i s a n a lysis are shown i n Fig.15. n-^  - index of r e f r a c t i o n of a i r = 1.0002 , n 2 = index of r e f r a c t i o n of crown glass = 1.518 +_ 0.2% , n3 = index of r e f r a c t i o n of SFg , a = 30° . From Snell ' s law we get sin «*n^, » n 3 s in <j>i . ( v i i - i ) Since the beam i s aligned to impinge at 90° to the c e l l body, 0j i s then a constant and i s equal to <X . 50 rtt (air) F i g . 15 Schematic diagram of optics of index of r e f r a c t i o n determination scheme 51 Thus «h <k - f t ..net . m i _ 2 ) S i m i l a r i l y , since T:=(f>Z~~& >• II3 - n t • Sin <~f>3 sin (<pa~oL) (VII-3) Expanding s i n d) and using cos<^ = (1 -s in^f 2 - (\~ (vn-4) we obtain the f i n a l r e s u l t (VII-5) We now have, i n Equation (VII-5), the SFg index of r e f r a c t i o n as a function of the e x t e r i o r angle of emergence of the l a s e r beam. E. External Angle Measurement. Figures 16a and 16b show the angles © and 0 and also the physical layout of the apparatus used i n t h i s index of r e f r a c t i o n measurement scheme. For a plane surfaced mirror, i t can be shown that i f beams one and two are p a r a l l e l a f t e r r e f l e c t i o n , then © = 0/2. (Fig.16a). I f the telescope i s focussed at i n f i n i t y and kept MIRROR SURFACE F i g . 16a Angle d e f i n i t i o n s F i g . 16b Apparatus layout r i g i d l y mounted, then a l l beams f a l l i n g on the crosshairs are p a r a l l e l , thus a measurement of ©i, (for the l i q u i d ) and ©2 (for the vapor), can e a s i l y be accomplished. The spectroscope turntable allowed accuracy of about 1 minute. Therefore the accuracy of measured angles i n the v i c i n i t y of 7° i s about 0.5%. F. Data A n a l y s i s . The raw angles ©^ and ©2 were used to c a l c u l a t e nL and n v f o r A = 6328A. The temperature was also c a l -culated. Since the Lorentz-Lorenz r e l a t i o n when expanded gives . n ( p ) - n c + (hc-O - c L , -fp/jpc) , ( vn-6) ( n L - n v ^ cc (pc-pv) , ( vn-7) we can obtain a rough determination of a r e l a t i v e c r i t -i c a l temperature (T c) by p l o t t i n g ( n L - r w f ™ . T C O An extrapolation of t h i s information to (n L-n v) = 0, gives T c = 45.125°C. This graph i s Fig.17. A second graph i s then pl o t t e d , using t h i s i n -(2) formation. Guggenheim found, i n h i s analysis of SFg coexistence curve data the following evidence of a s l i g h t l y skewed d i s t r i b u t i o n . 55 (VII-8) Combining t h i s with Equation (VII-6) we obtain the f o l -lowing . Where (VII-10) From these determinations, n~.= 1.093 and thus r w = (l.iz MO""4) ( T C - T ) +- n c (VII-11) n L + n y Thus a pl o t of 7;—- v.s. (T c-T) w i l l extrapolate to n c at (T c-T) = 0. Such plo t s were made and the UBC LQF lea s t squares f i t routine was used to f i t the data to a l i n e a r equation. The following r e s u l t s were, obtained. Fig.17 i s the c r i t i c a l temperature extrapolation and Fig.18 i s the n c extrapolation. CELL rv RMS ERROR TABLE V SLOPE >-4 RMS ERROR -4 SF 6-Rx 1.09213 0.00004 2.68x10 0.05x10 S F -R2 1.09338 0.00002 3 . l 6 x l 0 ~ 4 0.03xl0~ 4 SF 6-R 5 1.09238 0.00006 2.87xl0""4 0.04xl0" 4 NO. OF POINTS 11 36 20 67 I 0 0 ft 0 " O 0 a <3 = u cc-23= UJ -oc-i— id-ux: 0 • 00" j O ^ " — z UJ a "tot-—r — o ^T--r-<>-W-!—-o-io--t-<Ho-o-0 ~ ::*-g:.jS-S4r^: —S-s-s—f-— . : £ m i _, - In lo j- !. -1 •ot-b*-i*-0 0 0 - 0 0 0 O - 0 00=) + 1 -i 1 1 1 1 r 0 00 D OL 0 03 O'OS a'Ut> 0'0£ 0 02 oooorxreoT-tz/fAN+iNn o'ocr i a'oor o'qi The small RMS errors i n d i c a t e that the data i s very well f i t t e d by a s t r a i g h t l i n e and that scatter i s quite small. G. F i n a l Results. A' weighted average of the c r i t i c a l index of r e f r a c t i o n was taken, using the number of data points f o r each c e l l as weight f a c t o r . The f i n a l r e s u l t then f o r the c r i t i c a l index of r e f r a c t i o n of SFg i s n c = 1.0929. Since the accuracy of the prism index of r e f r a c t i o n i s l i m i t e d to 0.2%, a more r e a l i s t i c f i g u r e would then be n = 1.093 + 0.002 . c — The errors i n temperature measurement are neg-l i g i b l e i n t h i s index of r e f r a c t i o n determination scheme. H. S i g n i f i c a n c e of Results. 25 The C.R.C. handbook gives n D =1.167 for SFg. This determination gives N L I Q at 25°C as 1.1655. Thus, aside from the necessary wavelength c o r r e c t i o n between the sodium "D" l i n e and the He-Ne la s e r ' s 6328A, there i s excellent agreement to standard data. 58 VIII. DATA ANALYSIS AND RESULTS. The main data analysis was done by least-squares f i t s to a log-log p l o t . As discussed i n Chapter IV and Chapter VI, the number of f r i n g e s present under the d i s c o n t i n u i t y at the meniscus and the Lorentz-Lorenz r e l a t i o n s h i p form the basis of t h i s a n a l y s i s . Given the d e f i n i t i o n of j8 , PL-PV = A / T C - T 7 * v T c ~ T j ^ ; ( v i i i - D given the f i r s t order expanded Lorentz-Lorenz r e l a t i o n s h i p , n L = n c + ( n c - i ) - a , £± ; ( v m - 2 ) and given the r e l a t i o n s h i p of Equation ( V - l l ) between the number of f r i n g e s present (N), the l a s e r wavelength( and the width of SFg i n the c e l l (Jl), n L - n v = N | (VIII—3) we can say pL-p* = X " N . (VIII-4) / ° c Jc-a,-(nc-t) Thus we can write (VIII-1) as: N * (-O^ . (VIII-5) 59 It i s well known that /3= ]/Q (see Table I I I ) . We can say roughly that, since (-£ ) = T c , N 3 * (TC-T) . (VIII-6) 3 An extrapolation of the plot N v.s. T should then y i e l d T c-With t h i s T c, we can proceed to more s e n s i t i v e analyses.,, Given Equation (VIII-5), we can see that £g 1 . (VIII-7) Thus a log-log p l o t gives a slope of 3 |3 -1 - ( ^ - ' v io 3 t - £ ) . ( V I I I . 8 ) And i f t h i s slope value i s "m": ' (VIII-9) Fig.19 shows one such graph produced by the IBM/360 - Calcomp f a c i l i t i e s at UBC. A cursory examination of t h i s data, compiled from several i n d i v i d u a l data determination runs on the metal c e l l and one on the glass c e l l SFg-3A, shows two regions of d i f f e r e n t slope. The slope of these two regions was found by using .[ • I ! —— I I-— T i.o tn-O.M _ l 0.08 _J . L0GIN3/EPS) -10 0.12 - 0.16 0.2 0.24 0.28 _J 0.32 [ 0.36 01.4 _J B • - ! | • 'rx dtz • 3 § » 61 a l e a s t squares method. An a r b i t r a r y d i v i s i o n point was selected and the data above or below t h i s point was compiled. The "best s t r a i g h t l i n e s " through these points were found and the r e s u l t i n g slopes taken as a good estimate of the slopes of the two regions. Equation (VIII-9) was then used to f i n d f$ f o r each region. Further examination of Fig.19 shows that each data run i s i d e n t i f i e d by a d i f f e r e n t s p e c i a l symbol. An estimated error bar i s pl o t t e d above each point with the square s p e c i a l symbol. The error bar corresponds to the combined error of h~ extra f r i n g e and a temp-erature error of 0.0005®C. Only error bars greater than 5% are prin t e d . The c a l c u l a t e d values of p are accompanied by an estimate of RMS e r r o r . i f ( x i , y i ) are the data points, and i f the c a l c u l a t e d s t r a i g h t l i n e i s = m -xx +- b then (VIII-10) These r e s u l t s are found Crx = 0.000001 c r 1 = 0.000029' The larger p goes with the lower temperature. Thus, /S decreases close to T c. Chapter IX discusses e r r o r s . However, the low 62 values of Q~ i n d i c a t e that the data i s quite smooth and that s t r a i g h t l i n e s are good choices to f i t t h i s data. 63 IX. ERROR DISCUSSION. A. Choosing the C r i t i c a l Temperature. o Graphs of log (N /( - £ )) v.s. log (-£ ) show much v a r i a b i l i t y below log ( - £ ) ~ -4.0. This i s due i n part to small inaccuracies i n the choice of T c. As i n Chapter VIII (Eq. VIII-6), T c can be found by assuming ^3= 1/3, p l o t t i n g N ^ v.s. T, and ex-tra p o l a t i n g to N ^  =0. This procedure works well and i t s l i m i t a t i o n s were explored. Since the >^ found i n t h i s study i s not 1/3, and since t h i s i s compatible with other researchers' f i n d i n g s , a computer study was made of the v a r i a t i o n of extrapolated T c as a function of choice of /S . A simple least-squares l i n e a r f i t was done to varying numbers of data points. The r e s u l t s are presented i n Table VI. TABLE VI Va r i a t i o n of chosen T c with assumed ji value Assumed |3 Value 0.333 0.336 0 .339 0.342 0.345 10 points Tc 45.433 29 45.433 30 45 .433 33 45.433 36 45.433 log (-€) A Tc - 0.000 01 0 .000 03 0.000 03 0.000 < -4.7 30 points Tc 45.432 91 45.433 09 45 .433 29 45.433 49 45.433 log (-O A Tc - 0.000. ,18 0 .000 20 0.000 20 0.000 < -3.4 49 points Tc 45.432 60 45.433 03 45 .433 46 45.433 88 45 434 log (-O A Tc - 0.000 43 0 .000 43 0.000 42 0.000 < -3.0 69 points Tc 45.430 28 45.431 75 4 5 .433 21 45.434 69 45.436 log (-€ ) ATc - 0.001 47 0 .001 46 0.001 48 0.001 < -2.5 64 The extrapolated values of Tc show extremely small v a r i a t i o n . The values converge at /S » 0.339 with differences i n Tc being of the order of 0.0002°C. Accordingly, the l e a s t squares f i t was re-computed and P was found f o r Tc = 45.4333 ± 0.0002°C, using a l l data points below log (-£) = -.2.5. Thus a l l data below the slope change region i s used. These r e s u l t s are found, f o r log (-£)< -.2.5; Tc 45.43 31 45.43 33 45.43 35 Therefore the best value of j3 i s 0.339 ± 0.003. 2 The low value of 0- ' f o r Tc = 45.4333 indicates that t h i s Tc gives the st r a i g h t e s t l i n e f or the data involved i n the computation. U n t i l less experimental scatter i s possible we s h a l l have to be content with a s t r a i g h t l i n e f i t to the data below log (-6 ) =-2.5. B. Errors i n Relating Density to Refractive Index. As discussed i n Chapter IV, the r e l a t i o n s h i p of density to r e f r a c t i v e index i s approximated by the Lorentz-Lorenz r e l a t i o n s h i p . The d e t a i l s of t h i s r e l a t i o n ship are unknown. 0.336 16.8 x 10 0.339 2.9 x 10 0.342 4.3 x 10 65 However, i n the analysis of the o p t i c s , only the f i r s t term of an expanded Lorentz-Lorenz function i s used. Taking the second term, Eq.(IV-2), we obtain: f f c L I + (IX-l) Our data analysis attempts to f i t the following equation (IX-2) We obtain from Eq. ( I X - l ) , to second order: n L - n v = (IX-3) and from Eq. (IX-2) and Eq. ( V - l l ) : (IX-4) (2 ) Taking Guggenheim's well known r e s u l t s as approximations fit. = % H ) > I p. (IX-5) 66 • O r , ' ~,l ! (IX-6) Substitution of equation (IX-5, IX-6) into (IX-4) gives 4 (IX-7) Following the log-log data analysis scheme de-scribed i n Chapter V I I I and talcing |3 * V3 > A= and aj .= O.OZ , we obtain, upon cubing Eq.(IX-7): _ h £ * A3(-<0P" - 3.Q(-e) * o.ic>(-£f-(o.ooiJ(-6) 3 . T Y / • - / M ( I X - 8 J Since t h i s t h e s i s deals with a data range of 0.000001 t~ < 0.06,. the l a s t two terms are c l e a r l y i n s i g n i f i c a n t , thus we can say: r 3 (IX-9) Talcing logs of both sides and graphing, the slope of the graph i s taken as 3 |3 -1. Thiei largest (-£.) i s approximately 0.01. Thus the c o r r e c t i o n term i s completely n e g l i g i b l e , as the value 3 of N / ( - € ) i s comparatively huge and constant with (-€ ). Therefore, the data range of t h i s experiment allows us to ignore the second term i n the expansion of the Lorentz-Lorenz r e l a t i o n s h i p . 67 References (1) C. Domb; Physics Today, ,21, 23 (1968). (2) E.A. Guggenheim; J . Chem. Phys., 13, 253 (1945). (3) M.E. Fisher; J . Math. Phys., 5., 944 (1964). (4) T.D. Lee and C.N. Yang; Phys. Rev., 87, 410 (1952). (5) L.P. Kadanoff, W. Gotze, D. Hamblen, R. Hecht, E.A.S. Lewis, V.V. Palciauskas, M. Rayl, J . Swift, D. Aspens, and J . Kane; Rev. Mod. Phys., 39, 395 (1967). (6) B. Widom; J . Chem. Phys., 43, 3898 (1965). (7) M.A. Weinberger and W.G. Schneider; Can. J . Chem., 30, 422 (1952). (8) H.W. Habgood and W.G. Schneider; Can. J . Chem., 32., 98 (1954). (9) D. Atack and W.G. Schneider; J . Phys. and C o l l . Chem., 55, 532, (1951). (10) 0. Maas; Chem. Rev., ,23., 17 (1938). (11) W.A. Weinberger and W.G. Schneider; Can. J . Chem., 30, 847, (1952). (12) G. D'Abramo, F.P. R i c c i and F. Menzinger; Phys. Rev. L e t t . , .28, 1, 22 (1972). (13) H.L. Lorentzsen; Acta Chem. Scand., 2» 1 3 3 5 (1953). (14) E.H.W. Schmidt; "Optical Measurements of the Density Gradients Produced by Gravity i n C02> N 20 and CC1F 3 Near the C r i t i c a l Point", C r i t i c a l Phenomena, eds., M.S. Green and J.V. Sengers, N a t l . Bur. Std. Misc. Publ. 273, (U.S. Dept. of Commerce, Nat. Bur. Std., Washington, D.C, 1966), pp. 13-20. (15) E.H.W. Schmidt and K. Traube; Progress i n Internat-i o n a l Research on Thermodynamics, ASME (Ac-v ;ademic Press, New York, 1962). (16) J . Straub; Chem. Ingen. Technik, _39, 291 (1967). (17) H.B. Palmer; J . Chem. Phys. 22, 625 (1954). (18) D.A. B a l z a r i n i ; Thesis, Columbia U n i v e r s i t y , (1968). (19) L.R. Wilcox and D.A. B a l z a r i n i ; J . Chem. Phys. 48, 753 (1968). 68 (20) M. V i c e n t i n i - M i s s o n i , R.I. Joseph, M.S. Green, J.M.H. Levelt Sengers; Phys. Rev. B, 1_, 5, 2312 (1970). (21) P.R. Roach and H.D. Douglass, J r . ; Phys. Rev. Lett;*, 1 2 , 1083 (1966). (22) J.T. Ho and J.D. L i t s t e r ; Phys. Rev. L e t t . , 22,, 603 (1969). (23) J.M.H. Levelt Sengers, J . Straub, M. V i c e n t i n i -Missoni; J . Chem. Phys., J54, 12, 5034 (1971). (24) A.M. Wims, D. Mclntyre and Finn Hynne; J . Chem. Phys., 50, 2, 616 (1969). (25) Guoy; Compt. rend., 90, 307 (1880). (26) J.A. Chapman, P.C. Finnimore and B.L. Smith; Phys. Rev. L e t t . , .21, 18, 1306 (1968). (27) S.Y. Larsen, R.D. Mountain, and R. Zwanzig; J . Chem. Phys., 42, 2187 (1965). (28) L.D. Landau Phys. Z. Sowjun, ,11, 26, (1937). Zh.^eksp. Teor. F i z . , _7» 1 9 (1937). Phys. Z. Sowjun, .11, 545 (1937). Zh. eksp. Teor. F i z . , l_y 627 (1937). Collec t e d papers of L.D. Landau (ed. D. t e r Haar), Gordon and Breach,. N.Y., 1965. (29) M.E. Fisher; Rep. Prog. Phys., 30_, 615 (1967). (30) H e l l e r ; Rep. Prog. Phys., .3°., 731 (1967). (31) D. Sette; R i v i s t a Del Nuovo Cimento, 1_, Numero Speciale (1969). (32) B.L. Smith; Contemp. Phys., 10* 4» 3 ° 5 (1969). 

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