OPTICAL STUDIES OF CRITICAL PHENOMENA IN … STUDIES OF CRITICAL PHENOMENA IN FLUIDS b~ Nicola Fameli Laurea in Fisica. Cniversità degli Studi di Padova, Italy, 1991 -1 TUESIS SCBlIITTED

OPTICAL STUDIES OF CRITICAL PHENOMENA IN FLUIDS

b~

Nicola Fameli

Laurea in Fisica. Cniversità degli Studi di Padova, Italy, 1991

-1 TUESIS SCBlIITTED IY P.IRTI:\L FCLFILL51EYT OF

THE REQCIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

THE FACC'LTY OF GRADCATE STC'DIES

Depart nient of P hysics and Ast ronomy

M e accept this thesis as conforming

to the required standard

THE L'SIYERSITY OF BRITISH COLclIBIA

SI- 2000

@ Nicola Fameli. 2000

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Abstract

.-\cciirate optical techniques were employed to study the behaviour of the biriary liquid mist ure n-heptane+nitrobenzene (C7H16 + C6 H5-V02) and of the pure Aiiid 1.1-difluoroethylenc (C2 H2 F2) near t heir respective critical rcgions.

It is founcl t h . in the region of redoced terriperature. t = (Tc - T)/Tc < 3 x IO-". thc orcler parameter lof [ for the bina. mixture follo~vs a power law Lori z t'. with a leading eïponent 3 = 0.367 rt 0.006. which is higher ttiaii the predicted theoretical value of .3 = 0.336 k 0.002. .-\ careful study d t h rffect of refractive inclex gradients caused by the Earth's gravitational firld. potrritial variation of the optical thickness of the saiiiple cclls chie to wttit ig of the ceil ivalls by one the phases. and the long thermal equilibration t im!s of hiriary niistures has lailed to resolve the disagreenlent between .j iiii~iisiirccl on IL-licp tarie+ni t ro benzeiie and its t heoret ical vdue. A significarit foatiiro of tliis itivestigatiori is the novel application OF an optical techniqiie ( t lir i~iiagc plarir technique) for stiidying critical phenornena in transparent I~iriary licpids.

T h Lorentz-Lorenz funrtion. L. of 1.1-difluoroethyleiie was measiirccl in ordrr to dctcrniine the fluid's clensity from its refractive incles. The quantity C. is foiitid to var- by about 1.4% with density of the Ruid. p. with a gentle riiirsiriiiirn at a clensity slightly tiigher than the critical density. pc. The drlrisity rneasiircrnerits were then usecl to constriict the coesistence ciirvc for t tiis triaterial.

T h cocsistcnce curw of C2 Hz& was measured with the oLorentz-Loren~' rsprririirnt ancl ivith a new apparatils conibining tmo cornplenientary optical riietliods. the prism ce11 technique and the focal plane technique. into one. Tho rnciisiirerrients carried out on Ci H2F2 in this thesis seme as a test of this apparat us. The orcier parameter data of C2 H2F2 are clescribed acctirately II!. a scaliiig power law in terms of the reduced temperature t. ivith the cri t ical rsponents at t heir t heoretical ~a iues . The coexistence c u n e data tiikcri sini~iltaneously with the two combined techniclues agree d l with each over t lie ahole range of temperature investigated.

Contents

Abstract i i

List of Tables

List of Figura viii

Acknowledgments xiv

1 Introduction 1 1.1 Featiires of critical plieriornena . . . . . . . . . . . . . . . . . . 1 - 1.2 Siimniary of work . . . . . . . . . . . . . . . . . . . . . . . . . i -

1.2.1 Binary licpici niistiire . . . . . . . . . . . . . . . . . . . t

1 . 2 2 Purc fliiiti . . . . . . . . . . . . . . . . . . . . . . . . . 10 1 .:3 Oiitli~ie of t h t hesis . . . . . . . . . . . . . . . . . . . . . . . 1 1

2 Theoretical background 13 2 . Iritrocliictiuri . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 2.2 Scalirig . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . 14 2.3 lIo<icls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Rcriorriializat ion group t heory . . . . . . . . . . . . . . . . . . 19

3 General experimental features 25 :3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Temperature control and measurernent . . . . . . . . . . . . . 23

3 2 . 1 Thermal control . . . . . . . . . . . . . . . . . . . . . . 26 3 2 . 2 Temperature measurement . . . . . . . . . . . . . . . . 30 3.2.3 Thermometer calibration . . . . . . . . . . . . . . . . . 3 1

3.3 Op t ical investigation techniques . . . . . . . . . . . . . . . . . 3 1 3.3.1 Prism ce11 technique . . . . . . . . . . . . . . . . . . . 32

B Physical properties of the substances studied

C Relation between 14 and An

D Bending of light by a refiactive index gradient

E Technical drawings of apparatus

F Photographs of apparatus

List of Tables

Theoretical values of the critical exponents. The resiilts Froni the classical. mean fielci theories (MFT) are reported for coni- parisort witti the predictions of the modern theory. . . . . . . . 24

Paranleter wlues for a nonlinear least square best fit of Ao = B& to the volunie fractiori data OF Fig. 4.11. Quantities in

. . . . . . . . . . . . . \)rackets twre helti fired during the fit. 59 Parmieter values for a nonlincar Ieast square best fit of 10 = ~ ; t " . [vit h t' = (T - Tc)/T (fit C) and 10 = BotJ( 1 + BitA + & t 2 l ) (fits D . E . F) to the volume fraction chta of Fig. -1.1 1.

. . . . . . Qiiantities in brackets were held fised duriiig the fit. 61 Paranictrr values for a nonlinear least square hest fit of 10 = ButJ (1 + BltA + B#A) (fit G). AO = BotJ (1 + B i t A + &Pn) (fit H). and 10 = BotJ (1 + &tA + B;P') (fit 1) to the vol- iinic fraction data of Fig. 4.11. Qiiantities in brackets w r c lielcl fisecl during ttic fit. . . . . . . . . . . . . . . . . . . . . . 64

Rcsults of a qiiüdratic fit to the Lorentz-Lorenz data of 1.1- clifluoroet hylene (C2& F?). . . . . . . . . . . . . . . . . . . . . 84 Parameter values of the fit of equation (5.3) to thc prisin ce11 data on C2 H - 6 . Quantities in parent heses were held fisecl tluring the fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Pnrarrieter values of the fit of equation (5.3) to the focal plane data on C2H2F2 taken with the combined cell. Qiiantities in

. . . . . . . . . . . parentheses were held fisecl during the fit. 90 Critical temperature. melting point and boiling point tempcrat ures of the substances studied. . . . . . . . . . . . . . . . . . 105

B.? Refractive index (at the D line of the spectrum of sodium and at 20°C)9 density (at 20°C, referred to the density of water at -4°C) and molecular weight of the substances studied. . . . . . 105

List of Figures

Qiiiilitiitive coexistence curve of a pure fluid. Following the dasliccl line froni point -1 to point B brings the fluid froni the wpoiir state to the liqiiid state continuoushj. witli the Huid

.? always reniaining iri one phase. . . . . . . . . . . . . . . . . . - Qualitative (and not to scalt.) pressure-teniperatiire pliase cliagrarii of a pure fluid. The point C. at temperatcire Tc and prrsstire P,. is the critical point. Siniihrly to the phase di- agrarii i r i Fig. 1.1. followirig the clasheci linc from point .-\ t.o point 5 hrings the Hiiici froni the vapour statr to the liq- i i i d stkite corrtirruously. witli the fliiid always remaining in one pliasr. The t hicker line rcpresents here t hc coesisteiice ciirve antl corresponds to the projection of the curw of Fig. 1.1 orito t h (P. T) plane. . . . . . . . . . . . . . . . . . . . . . . . . . 4 SIagrict ization-tenipernt ure curw for Fe nicasiired t ~ y Ciirie ( frorri refcrerire [3]). witli the magnetization on the orclinate i i d t lie trniperature on the übscissa ases. . . . . . . . . . . . 4 -4niagat's density -temperat cire curves for CO-. The horizontal ilsis is the terriperature antl the density is on the vertical asis. This figure is reprocluceci froni refcrence (31. . . . . . . . 5 .A typical cpalitatiw coesistence curw of a binary liquid mixture. 6

C'oarse graining of a square lattice into block spins. The black c.irc1f.s are the original. unrenormalized spins. while the white circles correspond to the *new' spins after a transformation R has been applied. . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 Graphic representation of the space of Hamiltonians and the Rows given by the renormalization group transformation K

9 9 (takcn from (1141)). . . . . . . . . . . . . . . . . . . . . . . . . --

. . . Top view cross section of a typical multistage thermostat. 27 Teniperature control and monitor stages of the apparatus used

. . . . for esperiments on both pure fluids and binary liquids. 29 Illustration of the prism cell. . . . . . . . . . . . . . . . . . . . 33 Schematics of the prism ceIl experimental principle and apparatus (not to scale). . . . . . . . . . . . . . . . . . . . . . . . . 33 Data used to obtain a calibration of the micrometer scale on

. . . . . . . the inovable mirror for the prism ce11 experiment. 3-1 Effect of medge angles of sapphire windows on the nieasure-

. . . . . . . . . . . riicrit of the refracted angle (not to scale). 35 Illustration of the behaviour of the refractive index of a binary mixture as a function of ce11 height at T < Tc (a). at Tc (b). aiicl T > Tc. Below Tc. ,-O marks the position of the meniscus I~ctncen the two phases. . . . . . . . . . . . . . . . . . . . . . 37 Formation of Fraunhofer diffraction pattern duc to a non- iiriiforrn rcfractiw index profile. . . . . . . . . . . . . . . . . . 39 Gcoiiiet rical illiist ration of the phase difference and the for- riiütion of t hc Franunhofer cliffraction pattern. . . . . . . . . . 39

. . . . . 3.10 Srliciiiatic cliagrarn of the esperimental optical setup. 41

Binary liqtiid rnistcire of two chernical species H ancl S. below t heir consolute critical temperature. . . . . . . . . . . . . . . . 44 Illiistrating a binary liquitl misture with two critical ternper- itt lires. Tc. ,pp,, and Tc. o is the ~olurne fractioii of ont! of the sprcics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . -45 Scherriat ic drawing of the t herniostat eniployecl for the binary liquicl csperinient . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Corifiguration of heating foi1 elements arounïl the heater blork. 30 Temperature stability test with thermistors. The black diamoncls represent the room temperat ure (left hand scale). while the white ones represent the temperature measured by one of the therniistors (ch). . . . . . . . . . . . . . . . . . . . . . . . 51 Temperature stability test with the quartz thermometer. The black cliarnonds represent the roorn temperat ure (left hand scale). the white ones the temperature measured by the quartz t hermonieter (TtT). . . . . . . . . . . . . . . . . . . . . . . . 52

-4 qualitative diagram of the coexistence curve of a bina- liquid mixture. 4 is the concentration (volume fraction) of one of tlie liquids. . . . . . . . . . . . . . . . . . . . . . . . . . 53 The height. in the cell, of the meniscus between the two liquids as a function of temperature. for various concentration devia- rions froni the critical concentration. Height "O" indicates the niidclle of the cell. . . . . . . . . . . . . . . . . . . . . . . . . . 5-4 Illustration of the g l a s nianifold used to prepare the bina- liqiiitl samples. . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Top: photograph of a typical focal plane film representing a clat iiin taken at a panicular temperature T l , indicated in the boitom cliagram. The number of fringes that can be counteti froni the top one d o ~ m is proportional to the difference be- taecri the refractiw indices of the two phases. Bottoni: il- Iiistration of the tinie line foIlo~vet1 during the tlatiini stiown - - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ I ~ O V C . 03,

Ttic cocsistence curve for the binary niistiire n-heptane+rii- trohrrizcne plottecl as the volurne fraction Aorr as a functioii of tlie recliicctl temperature (with the ases swpped as it is often donc to plot coexistence curves). Tlie iriset is an enlargenient of thc critical region of the curve. . . . . . . . . . . . . . . . . .58

-1.11 Log-log plot of the order parameter versus the reduceci tcmptmtiire t. The slope of the curie as t tends to zero cor- rcspoiids to the esponent 3. ;\ lirie with slope 0.3'16 is also tlrann for coinparison. . . . . . . . . . . . . . . . . . . . . . . 61

-1.13 Sriisitivc log-log plot. A Q ~ ~ / ~ ~ rersus t. of the cocsisterice chta oti rl-lieptane+nitrobcnzene. The value uscd for the critical rsponcnt is 3 = 0.3'16. A sensitirity scale is also draw-n in the grapli to indicate the slope the data would preferentially take. ar rc tliey plottecl ai th the ïaliie of the esponent corresponding to the iridicated slope. The size of the error bars is comparable to the data scatter. . . . . . . . . . . . . . . . . . . . . . . . . 63

4-14 The coexistence cume for the binary mist ure cyclohesane+aniline plottctl as the volume fraction An as a function of the absolute tempcrature.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.15 Sensitive log-log plot. h / t P versus t, of the coexistence data on cyclohexane+aniline. The data is plotted using a critical exponent ,d = 0.330. The sensitivity scale drawn in the graph inclicates the siope of the data wit h 3 = 0.350 would definitely sliow up in this type of graph. The size of the error bars is coniparable to the data scatter. . . . . . . . . . . . . . . . . . G5

4.16 Qualitative picture of the refractive indes profile of a bina- niisture as a function of ce11 height at T < Tc: (a) ideal case: (b ) distortecl by Earth's gravitational field. . . . . . . . . . . . 67

4.17 Sleasiirement of the refractive index profile üt a temperature T > Tc after the sample tvas shaken to hornogenize the phase. 68

4.18 Measlirement of the refractive index profile at a temperature T < Tc aftcr the süniple was taken from slightly above to sliglitly below critical. Black diamoncls: about 30 niinutes aftcr the terriperature change: white diamoncls: abolit 50 hotirs latcr.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G9

-1.19 S lcasiirerrierit of the refractivc iricles profile at a temperature T > T, after tlic sarriple was tieated frorii below to almve critical. Tlic sample \vas not shakeri to speed iip phase Iioirio- gtineity as in Fig. 4.18. . . . . . . . . . . . . . . . . . . . . . . 69

-1.20 Loiver phase of a bina- liquid niisture settirig the sarnple ce11 . . . . . . wlls iirici surrouncling the upper phase completely 71

Sensitive log-log plot. AorI/tJ versus 1. of the coesistence data on n -ticptarie+nitroberize~ie takcri \vit h a 3-nim ceIl (black dianioncis) and a 10-mm ce11 (white clianioncls) to stucly the rffect of a wetting filin on the data. (The value usecl for the critical clsponent is J = 0.326 and. as in analogous graplis. the clata scatter and the size of the error bars are comparable.) . . 63

Illustratiori of the combined c d . . . . . . . . . . . . . . . . . . 78 Tlic conibinecl ce11 esperiment setup. . . . . . . . . . . . . . . 79 Qualitative illustration of the coesistence curve of the pure Hiiid anci the procedure follo~ved to take LL data in the prisni cell csperinient. . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Temperature and density region where the data for the LL nieasurements are collected. Due to its large density gradients the crit ical region must be "circumna\-igated?' to take accurate d a t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5'2

S. 5 Measured density de pendence of the Lorentz-Lorenz function of lJ-difluoroethylene. The line s h o w is a quadratic fit to the data. The fit parameters are in table 5.1. The size of the error bars is comparable to the scatter in the data. . . . . . . 84

5.6 Variation of the weight of the prism ce11 with tirne, when the ce11 IW initially at a temperature about 5.8" below room temperature.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

" - 02. r Slcasurecl LL data wit h (white diamonds) and wit hout (black

diainoncIs) cor:ection on the ce11 mass. The size of the error bars is comparable to the scatter in the data. . . . . . . . . . . 86

3.8 Cocsisterice cuwe of 1.1-difluoroethylene as measured in the prisni ce11 esperimerit. . . . . . . . . . . . . . . . . . . . . . . 87

3.9 Correction sensitive log-log piot of l p * / t J vs t of the coeris- tewc data of 1.1 -difluoroct hylene meaçured in the prism ce11 espcrinirnt. The value of J wüs held fised at 0.326. The size of t h error bars is comparable to the scütter in the data. . . . 58

3.10 Sirriiiltaneoiis prisiii cell-type (circles) ancl focal plane (fi)

data oii 1.1 difliioroctiiylene nieasiireci wit h t hc (:ornbiriecl ceil . . . . . . . . . . . . . . . . . . . . . . . . . . . . tqxrinicnt. S9

1 Log-log plot of prisni (circles) and focal plane (a) clata on 1.1 -difhorocthylene nieasured with the combinecl ce11 esperi- rricnt.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Gel A collcrtion of esperiniental (black clianionds) ancl tlicoretical (white) valiles of the critical esponent .j in the past 35 years. Thc csperinierital data are frorn nieasuremerits on binary iiq- iiids only The rcsults obtaiiied by the CBC laborato- of critical ptieriomena are the dottccl circles. . . . . . . . . . . . . 93

.-\. 1 Gcoiiietry of configuration used to assess the attenuation of csternal teniperatiire Htictuation by a styrofoam laver. . . . . 103

D.1 .\Iode1 of a medium with a vertically varying indes of refractiori. n(z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

E. 1 The ce11 holder. . . . . . . . . . . . . . . . . . . . . . . . . . . 112 E.2 The inner cylindcr. . . . . . . . . . . . . . . . . . . . . . . . . 113 E.3 The outer cylincier. . . . . . . . . . . . . . . . . . . . . . . . . I l 4

F.l Some of the n-heptane+nitrobenzene samples . From left to right: 1-mm, %-mm, and lû-mm samples . . . . . . . . . . . . 116

F.2 The cell holder with a 1-mm sample . . . . . . . . . . . . . . . I l i F.3 The inner cylinder and cap rvith heating foils . . . . . . . . . . 118 F.4 From left to right: the ce11 holder . and the inner and outer

cylintlers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Acknowledgment s

1 woiild like to espress my appreciation to al1 the people 1 have had the fortune and pleasure to interact with during the course of this work and who have helped improve greatly the qurlity of this thesis.

In particular. 1 would like to thank Professor Douw Steyn for the numer- oiis helpful injections of energy and enthusiasm 1 have been lucky to receiw ( l i t ring clisciissions wi t h him.

1 apprcciate the time aiid availability for discussions and help dedicated t o rile by ni!. clear friend Dr. Bruce Todd.

Sincere thanks go to the teclmical staff of the departrnent. particularly to George Bahiriger for his help (and patience) cloring my instriiment-construction t i nios.

1 wotild like to acknowleclge the aclvice ancl support of Professor Thonias Ticdjc. Hcacl (II the Department of Physics and Astronorny.

Ttianks to riiy friencis Fetlerico Biagi. Giorgio Delfitto. h r c o Ekretti. E l l m Fowler. R.obcrt Lee. Renzo llufato. and Matteo Sarto for tlieir nioral i d spirit uiiI support.

Iiifinitc thanks to niy wife. Luisa. for lier iinfailing support and for being iirr inwluable critir of my life and work tlirough al1 these years. ancl to n- parrrits. Olindo anci Stefania. and sister and brother. Fedcrica and Gio~arini. for k i n g thcrc for me. always. To them 1 tledicate this work.

;\ii<l t lia11 k - p i t O niy su penisor Professor David Balzarini.

[.-\clclcd after the final clefense] I am intlebted to niy siipcrvisory commit- t w . Profcssors Iari .-\ffleck. Mark Halpern and Lorne Whitehead for their Iielp arid guitlarice through my doctoral work.

Chapter 1

Introduction

1.1 Feat ures of crit ical phenomena

Ttir nork rcportcd in this thesis is basecl on csperinients investigating tlic so d l c c l critical region of Auicls. The niain purpose of the csperiments is t u (+ital)lish t h correct sliape of the coexisterice ciirw of two kiricis of Auici sytcwis: ;i pure Hiiid and a binary liquicl niist ure. It seerris wort hwhile to giw a hrid (Ipscript iori of t lie features whidi rriake critical plie~ionienü interesting for physicists arid to introduce the definit ion of somc of the physical quanti ties iisrtl for t lie description of t hese phenomeiia.

As the tcmperatiire of ii liqiiid in equilibriuiii with its owri wpour is raisecl. t i w clcrisities of tlic two phases approach one another. At a teniperaturc cxllrd criticd temperuture. and usually indicated wit h Tc- the two clensit ies roincidc aricl ahore Tc there is no longer a distinction betwen the two states of iiggrcgation. but only one Auid in its gaseous state. .As shown cjualitatiwly in Fig. 1.1. t tie solicl curw is the locus of points corresponding to the density of tiic vapoor (part of the curw to the left of pc) and the liqiiid (to the right of p,) for p liases at t cniperatiires belon- Tc. Any t hernioclynaniic change taking the systcin across the coexistence curre at any place different from (p,. Tc) is a first ortler phase transition. At the critical point (p,. Tc) there ceases to t)c R distinction hetiwen the vapour ancl liquid phases and a continuous t rarisit ion betwen the phases oçcurs. This is usually referred to as a second d e r phase transition. -4 striking feature of th is phenonienon is that one cari bririg the system from point A in the vapour state to point B in the liquid stüte nithout erer observing a discontinuity between the two phases. This

Chapter 1. Introduction

Figiirr 1.1: Qiialitatiw coecsistencc curw of a piire fluid. Following the (liisti~~l lirie frorn poirit A to point B brings t tie fliiid from the vapour state to t h . liquitl state continuously. with the fluicl alwqs reniaining in one phase.

\vo~ild br: actiiewd by following the dashed line joining point -4 to point B in Fig. 1.1. .-\nother vicw of the critical point can bc haci through the (P. T) diiigriiiii in Fig. 1.2. whcrc P is t hc pressure in a closeci ressel containing the pliasr?;. iind T is the temperature. In this picture. the critical point C is thc loratrtl at t h encl of a line of dzscontirnrous. or first order. phase transitions hotwcrn t h vapoiir and liquid phases. The h i e frorri A to B illustrates iiiiot lier pat ti t hat noiilcl take the system continiiously froni vapour to liqiiid.

This feat tire has at tracted the interest of physicists For niore t han 130 ycw-s. siart iiig witli Ttiornas .4ndrens. the discoverer of the critical point. HP iiotccl in Iiis 1870 paper on the critical point of CO?:

-As a direct result of his esperiments. he [T. Andrews] con- cliides that the gaseous and liquid states are only widely sepa- rated forms of the same condition of matter. and may be made to p a s into one another by a series of gradations so gentle that the passage shall nowhere present any interruption or breach of continuity Frorn carbonic acid as a perfect gas. to carbonic acid


Figiirc 1.2: Qualitative (and not to scale) presstire-teniperiittirc pliasc dia- g r m i of ;i piirc Hiiid. The point C. kit temperntiirc Tc anci pressure P,. is t h . critical point. Siniilarly to the phase diagram in Fig. 1.1. following the (lasiid lirir from point A to point B brings the fluid from the vapour state to t lie liqiiitl statc conti ir~iiody. \vit h the Auid always rernaining in one phase. Tlic t hickcr line represents here the coesistence curve and corresponds to the piojcctiori of t hc ciirrc of Fig. 1.1 onto the (P. T) plane.

as a perfwt liquicl. the transition rnay be accomplished by a con- tiriiious proccss. ancl the gas aiid liquid are only distant stages of a long scri~s of coritinuous physical changes [l]."

-4s t tic critical temperature is approached from below. esperimcnts show t l iar t lie dcnsity clifference between the two phases oheys a potver larr- relation likr the f'ollo~virig. in the limit that T -t Tc:

The accurate determination of such power lan and in particular of the erpo- ricnt .J' is one of the main results of the present work.

An arguably even more interesting feature of these phenomena is that they are riot restricted to pure fluids. There is a common behaviour underl- ing fi-hat appear to be very different physical systems. The discovery of the

Clrapter 1. Introduction

Figiirr 1.3: Ifagnctizatioii-tcnipcratiire ciirve for Fc nieasurecl t- Ciirie (frorn rrfrrrricr [ 3 ] ) . witti the magnetization on the orclitiate aiicl tlie teriiperature oti t licl :ibscissa ases.

imritic.;il point of a piire fluicl was sooii followed by that of the critical point of ;i frrroniagnet (Fe). The magnetizatiori. JI. of a piece of Fe wnishes above ii rwtaiii critical ternperatiire Tc. that has corne to be known as Curie tan- p n t lire [?]. 'r Ioreover. it is esperinientally wrificd t hat as Tc is approached frorii t~clow the following powr law holds:

( \ Y ~ P I - P Tc is 11011. the Curie temperature) and the value of .i is the same foiiiid in the pure fiuid case. aithin esperimental error. The sirnilarit- of thc shapr of the esperimental curws. which can be not icecl in Figs. 1.3 and 1.4. iil tirit ii t cliffereiit temperat ure ranges. is quite st riking aiid suggests t hat t l i ~ r < > is more thnn just a coincidence to these events. Other esamples of physical systems with critical points are: antiferromagnets. ferroelectrics. hinary alloys. siiperconductors. superfluids.

Orle other esample should be described since it one of the siibjects of this thesis. Two difierent substances in their liquid state-a system known as a b i r iun~ liquid mixture-display a critical behaviour. analogous to a pure fluid. through r he phenornenon of mutual diffusion.

Chapler 1. Introduction

F r 1 4 : hriiagat's clensity-ternperatiirc curvcs for CO?. The horizontal i i~is is tlic tctnperatiirc and the clensity is on the vertical mis. This figure is rrprodiicccl froni refererice [3].

Giwi two liquicls. calletl for convenience H ancl Y. there can bc a temperat tir(.. Tc. callecl the consolute temperature. bclow wliicti t lie liqiiids separate iri two phases. the lighter one abose the hearier one. with a distinct rnenisctis h~trwcri ttieni. Above Tc. it becomes more energeticall~ fasourable for the systorii to be iii one phase. thus cliffusion occiirs. the meniscus between the liqiiicls bliirs anci cwntually clisappeürs. with the tivo species settling iri orle single licpid phase. The phase ciiagram in Fig. 1.3 represents clualitatiwly tlic co~sistcnre curvc of a bina- liquici misture. in terms of the concentration of oiir of tlic two species versus temperature.

0 t h physical qiiantities are foiind to behave similarly from one systern to ii~iotlier in tlic neighbourhood of the critical point and to follow r.vell- rc~prodiicible pomer laws. They are usually expressed in terms of the reduced trinperature t = (Tc - T) /Tc and they describe:

the constant volume specific k a t . cl-:

Chupter 1 . Introduction

One-phase region: liquids H and N

completely miscible

Figiircl 1.5: A typicül cpialitative coesistence ciirw of a binary liquid niistiire.

t t i ~ isotlirrrnal comprcssibility q- (or siisceptibility k~ iri ttic ferrornag- rirtic case):

KT = rgltl-? ( 1.4)

t lit. clieniical potent i d . p(p . Tc) dong the critical isot hem:

t tic correiatioti length. <: < = <;1t1-''

Ii(~sicir the alreacl- mentionecl density difference. Ap:

T11~ latter qiiantitj- is more generally referred to as the order pammeter. It is givrri by different cluantities for different physical systems. such as l p for piire Ruids and the magnetization. Jl. For ferromagnets. The esponents a. J. -#. &. and Y are called critical exponents (other critical esponents ni11 be ericoiintered in Chapter 2) and A& r;. Do, <& and Bo are know as critical

Chapter 1. Introduction 7

amplitudes. The f sign in some of the amplitudes indicates quantities to be considered below (-) or above (+) Tc.

The power laas given by equations (1.3), (1.4, and (1.6) show that the spccific heat . the compressibility and the correlation lengt h are singular at the crit ical temperature. These singularities have proven to be very important in the clevelopment of the modern t heory of critical phenornena.

-4s theoretical determinations of 8 have becotne more precise. the need for prcçisc ancl accurate measiirements of 3 have become iricreasingly more importarit. It is in this framework that the nieasurements reportecl in this tticsis fit.

Fluids are particularly siiitecl for the study of the critical region in that t l i i y do not present problerns related to lattice structure imperfections or dcfects ---a cornnion occurrerice in solicls, Moreover. transparent fluids can be prol)ccl by optical methods which are in general quite acciirate ancl "clean". Chic of tlic clrawl~acks of iising fltiicls on Eartli is the influence of the grav- it;itioiial field. n-hicli cari inask the phcnomcnon one is trying to observe. Ho\vcwr. as i t si11 be sliown. the opt ical niethods employed reniain satisfac- rody acSciiratr cfcspite gravitational cffects.

1.2 Summary of work

1.2.1 Binary liquid mixture

Critical temperatures

Tlic. first datuni ricecled to nieasure the coexistence ciirw of a binary liquicl rriistiirr (arid of a pure fluicl. for that niatter) is the critical ternpcrature of t lie syst mi. Phsical divergences prevent the esperinienter froni access- iiig tlic rritical point directly Thus. one cati only take measurements in a iic.iglii>oiirhoocl of the critical point. termed the critical region. and then try to in fer iii format ion about the physically inaccessible region by extrapolation froni the available data. The critical temperature. Tc. is obtained in this bsliiori. The alailable literature on the subject suggests that different anioiirits of impurities are responsible for a variation in the critiçal tempera- t lira of a niixture. In particular. the critical temperature varies linearly wit h the iimoiirit of impurities in the systern (see. for esample. references [4]: (51. aiid [ 6 ] ) . .As data were taken from different samples of the same bina. misture. slightly different TCs were found. These variations in Tc were used to


assess the level of influence impurities might have had on the reçt of the mea- surcments. The critical temperature data available from the experiments of tliis work were analyzed in terms of the predictions on the behaviour of Tc due to the presence of water and acetone as impurities in Our samples.

Coexistence curve

Iri t tic biriary liquid mixture experiment the coexistence c ime was nieasured For t h ~ systern n-heptane+iiitrobenzeiie and in particular the value of the ospoiient. 3. that characterizes it (n-heptane is a non-polar f'luicl. while riitrot)erizerie is strongly polar). The critical phenornena laboratop of CBC Ilils appliccl its opticiil rnethods to the study of the critical region of binary riiistiires hefore [il. At the tirne of those studies. the value of .3 foiind \vas in ilgrrcnierit a i th an older ..trend" predicting .7 = 1/3 or slightly larger. but ricwbr t lieorctical (levelopmerit in t lie field of phase transitions aiicl crit ical 1) timoriiena. ciilniinat ing in the early 1970s wit h \Vilson's rcnornializat ion groiip tlieory. woiilcl ~rlictate" a lower value of j. It secrried fit. thcreforc. t o cwry mit a stucty on a binary liquid misturc system to obtain a more ( l d i ~ i i t iw csperiinental mluc for 3.

As anticipated in section (1.1). optical tecliniques werc ernployd for these riiriisiircrritlnts. Thcse tcchniqiies have a history of siiccessful results for the past tliirty years [Ï. S. 91 and will be clescribed in cletail later in the text. The riirt tiod kriowri as fucal p h e technique is the principal tool of investigation of tlir c«rsistericr currc of a binary liquid mixture. The incles of refriiction cliffmmc bctwcen the two liquids nas mcasured as u function of temperattire aiitl t lir crit iciil esponent 3 nas obtained by fitting the t heoret ically predictecl p u w r law to thc data. Also. it mas possible to find the range of ralidity of qtiatiori (1.7) in ternis of the reduced temperature 1. and compare it the fiiidirigs of 0 t h esperinienters. Given the precisioti required to clistinguish het~vccri a 1 /3 rsponent and the slightly lower value of around 0.396 preclictcd t q - the rrioclcrn theories. an improvecl esperirnental setup aas made for these nirasii rcrrients. In part icular. the thermal cont rol was improved and sarnples with a longer optical path were employed. in order to reduce the error in t lie intles of refraction measurements. The performance of the newly built t licrriiostat rvas verified both in terms of its thermal gradients and its t hernial st ahility wit h noticeably improved results 114th respect to the older apparatus. h i n g a t hiçker sample, to increase the optical path, considerably reduces the scat ter in the coexistence curve data, thereby producing more reliable results.

;\ri analpis of the possible sources of systematic errors needed to be carriecl out as thoroughly as feasibie to ensure maximum accuracy of the col- let ted data and of the final results. The effects influencing crit ical phenornena studics in binary liquid mixtures are:

0 thernial gradients dong the sample: this is an aspect of the apparatus that \vas improved upon by the construction of a new thermostat. The performance tests on the entity of vertical thermal gradients in the saniple ce11 when it is placed in the thermostat suggest that thermal iiniforrnity and stability conclitions are within the tolerances for the r~quirctl acciiracy of tliese esperimeiits.

a t lie Eart h's gravitat ional field caiising a vertical gradient in t hc incles of rciraction: to attempt obsrrration of any gravitationally inducccl rrfrac t ive irides gradient. another optical intcrfcromet ric technique. kriowi as inrnge plane technique. was ernployecl possibly for the first titrir in tliis riianrier (no siniilar results w r e Foiincl in ttie literatiir~). Ttir rcsiilts ohtained are very helpful for the determination of' the scalr of r licsc graclierits and t heir influence or1 tlie final value of the critical ospurictrit J.

0 quilibratioii tinic at each temperature: it takes tinie for a systeni to rcacli tqiiilibriiini after tlie teniperature is ctiüiiged. .\leasurerrients stiorild only be macle after equilibration is achieved. Ttie same tech- niclil" tmployccl to measure the effect of grarity coiilcl he used to stutly r tic rqiiililmtion time issue and rule out the possibility of t his effect corriiptirig our data.

O wttirig of the ce11 walls by the sample: if one of ttie tu-O phases of a t~iriary mixture wets the other phase as w l l as the wall of the ce11 t hat contairis them. then the effective optical liglit pat h involvecl in liglit trarisniissiori through the ce11 is different from the nominal value giveri t)y the manufacturer. An estimate of the influence of this effect cari I>e made by performing measurements \vit h different iioniinal ce11 t tiickncsses.

O t lie relation betu-een refractive indes and coricentration: t his relation is neetlecl to obtain the order parameter of the particular system studied. -4 reasonably complete search through the literature has provided information on the question of the relation between refractive indes and


concentration of Our binary mixture. The findings helped estirnate the level of influence of this effect on the study of the coesistence cuwe reported.

The above influences on the observed critical phenoniena were checked with a firie-toothetl comb and analyzed mith great care.

1.2.2 Purefluid Lorentz-Lorenz coefficient

Ttir orclcr parameter of preference for pure Huids is the ciensity clifferencc ht\vecii t lit. licliiid and vapour phases. In the optical stuclies of this work. the orclcr parameter lias to br obtained from measurements of the difference in iritles of tefraction between the phases. through the Lorentz-Lorenz relatioii. nhicti relates the refractive index. n. of a fluitl to the fluicl clensity. p [IO]:

To 1~ i i l ~ l ~ to tisr it one niust first cletermine the Lorentz-Lorenz coe@cirnt C II>- tiiriisiiriiig the clensity. p. and the refrective indes. r i . of tlic piirc fluid i r ~ d e p e d e ~ t t l y of one m o t her and inserting t hem into equation ( i .8). Tticsc riicasiirrriicnts wre obtained witli anottier optical techniqiie callecl the prlsîn vell technique.

Tlir piire Huid stiidiecl in this thesis is 1.1-difliioroettiylerir ( G H 2 F 2 ) . wliose crit ical teniperat ure of aroiind 30°C is accessible wit h relative case. Tlir riieiisiircmcnts of Cc, rrifi as acciirate as obtained in this tvork ronstitute a riuvrl resiilt. as a research in the literature has faiiled to reveal aiiy other st iiclies ori t hc Lorentz-Lorenz funct ion of t his compound.

Coexistence curve

The prisni ce11 technique !vas also used to study the coesistence ciine of 1.1 -tlifliioroct hylerie arid analyze it in terrns of the scaling relation ( 1.7) men- tioried above in section 1.1.

The rriain espected difference between the coesistence ciin-e of binary licliiid mistures and pure fluids is the estent of the a s p p t o t i c region. that is. the estent to which the scaling relation (1.7) is valid in terms of the reduced teniperature. t. While there is no theoretical estirnate of the range of the


asymptotic region, experience seems to suggest that in pure fluids such a relation holds true for a smaller range of t than For binary liquids. Beyond that range, corrections to the simple power law relation must be addcd to achievc a proper description of the coexistence curve.

Accurate coexistence curve data on both a bina- liquicl and a pure fluid alloa a careful study of the importance of correction terms in the inter- pretütion of the data as well as in the estimate of the value of the critical rsponcnt 3.

Iri general. grüvity affects pure fluids at their critical point more than it clocs binary fluith. This effect is quite cvident when taking data with the prisni cc11 technique in a way that mil1 be described in due course. Because of tliis effect data cannot be reliably taken closer to the critical temperature thaii abolit 25 niK or 10-" in terms of t. To obtain a more coniplete mea- siircnient of the coesistence curve ttie focal plane technique nas used as was rlorir for the binary liquid coexistence cune.

Tticsc corriplenicntary prisni-ce11 ancl focal-plane meastirenierits are car- rii4 oiit in a siiiglr piece of apparatus. called combined cell. in nhich both kirids of rricasurerricrits cari be clone simultaneoiisly This nowi type of ce11 rirriinivcrits ttie probleni of having to use two clifferent samples witli possibly diff~rcrit iirrioiirits of impurities in them. niaking data arialysis ancl interpre- ta t ion easier ancl niore reliablc.

Cltimately. the two kincls of data can also be cornpareci aniorig each other to assess t lie problem of wetting of the ceIl windoas by the liqiiicl phase of tlic piire fliiid. This effect coulcl alter the effective length of the light path iii tlir cc11 and providc a source of systeniatic error in the data. Sirnulta- ritBous nieasiirerrierits wit h the combined ceIl can verify the presence of sornc irirasiiriibk wet t irig.

1.3 Outline of the thesis

Bryoncl t tic? present chapter. this thesis is organized as folloms. Chapter 2 is n summary of the status quo of the theoretical work on crit-

ira1 phenomena. starting from the scaling ideas to the latest niost accredited Renormalization Group Theory results. Some more definitions of the quantities used. besides those already given in this chapter. are also introduced in Chapter 2.

The main esperimental features of the experimental techniques employed


for this work and which are common for both the binary liquid and the pure fluid experiments are described in Chapter 3. This chapter also has a description of the t hree opt ical techniques used in the experiments: the prism ce11 technique. the focal plane technique and the image plane technique.

The subject of Chapter 4 is the experiment on the binary liquid mix- turc n-lieptane+nitrobenzerie, including those details of the apparatus which differ from tlie common features introduced in Chapter 3. and a thorough aiialysis of the sources of systematic errors.

Cliapter 5 presents the pure fluicl experiment with sections on the Lorentz- Lorenz nieasurements and tlie rneasurements of the coesistence curw of 1.1- difliioroetliylene.

Fiiially. Chapter 6 is a sunimary of the whole thesis with ari evaluation of t lir results ancl t heir possible consequences.

The appendices contain interesting. relevant niaterial which nevertheless woiild tiwe iinnecessarily encumbered t hc niain test of t his t hesis.

Chapter 2

Theoretical background

2.1 Introduction This rliaptcr is dcdicatcd to a description i r i broad strok~s of the modern tlirory of critical plienoniena and its principal resiilts that arc of interest for t licl prc.sciit csperimcntal norkl.

I have clcliherately electcd not to w i t e iinythirig about ahat is riormail- kriowii as t lie classical t heory of crit ical plieriomeriii'. Despi te t lie iniportarice o f t tic liistorical tlevelopnient of the classical theories. it is the resiilts of the riiodi!rii t lieory t hat the esperimental efforts are confronted witli. 1 have tliorrfor<l prcfcrrcd to omit aiiy mention of \'ail der \\kals'. Landau's or later. prca-sding. t tieories. wliose quantitative preclictions of the critical csponents. for wiriiplc. wcrc not iri agreement witli esperiinental cvidence. The classical rtwlts are only siirnmarized in a table (Table 2.1).

Iristrad. the chapter is centretl on the clevelopment of the idea of sculing ; i i i ( i ori iiii o~crview of renormalitation gmup theory.

' T h notes iri tliis chapter corne froni a rariety of sotirces. the most significant of which a r v [ I l . 2. 13. 14. 15. 16. 17. 181.

'Iri the theory of phase transitions ancl critical phenornena. the term "classical" is riot iisctl as against "quantal". but in opposition to the morlern' theory. which actually rriariages to proclucc more correct results.

Chapter 2. Theoretical background

2.2 Scaling -4 successfiil theory of critical phenornena should be able to explain the universal character of the power laws that were esposed in section 1.1 and also to calculate the values of the critical exponents a, 3. 7, 8. aiid u that govern t liose laws. The divergences in the specific heat , the isot hermal susceptibil- ity aricl the correlation length, in particular. should arise naturally from the tlicory. The scaling ideas that started to flourish in the mid '60s were the first theoretical effort capable of incorporating these singularities3.

The so-called scaling hypotliesis stems from the idea that the free en- wgy. f. of a systerii mith a second order phase transition can be tliought of as the suni of a factor that behaves regularly at the critical temperature iiiid m o t hcr factor t hat contains the singiilar behavioiir. The hypot hesis is t hat tlir sirigular part of the free energy is a honiogeneous fiiriction of the tc3riipc1ratiirc aiid of one of the important fields. For critical phenornena in Hiiicls the fielcl of clioice is usually the cheniical potetitial of the substance. p. Tlicrcbfore. oiic assunies that there esist two iiiimbers. u, and c l t . siich that foi aiiy positivc A. j ( p . t ) ' witli t = (Tc - T ) / T c obeys the rclatiori:

w l i c w the subscript s indiciites the singular part of the free rnergy. From t lit) properties of the homogeneous Functions and t heir clerivatiws and the rolat ion br t w r n t tic various t herniodyiamic funct ions ancl the derivat ives of t h frw rrierg' with respect to its variables. the critical esponents can be ~~sprcsscd in ternis of thc exponerits a, ancl al [16. 191:

"This -sinylarity' story resernbles the discovery of black holes. in reverse. There. the theory (Schwartzchild's solution to Einstein's equations) containeci a singularity that turrierl out to be real in the form of black holes. Here, the measured. i.e.. nul. singularities had to bc taken as a guide for a formulation of a complete theon

Chapt er 2. Theoretical background 15

The scaling equation (2.1) can be put in a form that makes it easier to O bt ain the power laws for the various t hermodynarnic quantit ies by choosing X = t - l / a t . In this case

.Aiiother formulation of the scaling hypothesis is that al1 the singularities at thc critical point stem from the divergence of tlie correlation lcngth. < as a fiiiictioii of JTc - TI and the conjecture that, at the critical point. ( is the only rrlcvarit Imgth of the problem. The correlation length is a measure of the rarigc of tlic density fluctuations (in the fluicl phenornena case). The scaling h!ptliesis then states that the details over srnall scales arc not important t u iiridrrstancl t hc physics of tlie phenonienon. Froni this pictiire. scaling rrlations cari bc obtained:

a s 1 t 1 = ITc - TI/Tc tends to zero. The correlation lengtti is linked to the h s i ty -(lerisi ty correlütion function G( r ) . which is usecl to tlescribe the density fliictiiatioris i n the systeni. On cluitt. general groiiricls it (:an be slioan t hikt G ( r ) clecéq-s esponentially for large clistances. that is to say

as 1. -+ x. Sirictl< becomes infinitely large at the critical point. while G ( r ) is still riit~;isured to clccéty to zero for large distances. i t is reasonable to espect t h the correlation function fa11 off as an inverse p o w r l a s in the vicinity of t lie crit ical point:

G ~ ( ~ ) r - d - 2 - ~ ( 2 3 )

for r + x. ahere d is the climensionality of the system and q is another criticai esponent.

In format ion on the correlation function can be obtained t hroiigh light scattering esperiments. The information cornes from the intensity I ( 0 ) of

Chapter 2. Theoretical background 16

the light scattered at an angle, 8, with respect to the forward direction. The scattering intensity I ( 0 ) is determined by the fluctuations in the index of refraction, ancl therefore in the density, of the fluid and it is proportional to t lie quanti ty

I ( B ) x ~ ( k ) = / d m i k - ' ~ ( r ) (2.9)

nhcre k is the shift in the wave vector of the radiation. Equation (2.9) is the Fourier transform of the real space density-density correlation function G(r) . Tlirough tlie fluctuation-compressibility relation [2]

the scatteririg intensity in the for\~itrd direction (0 = O) is proportional to t coi~ipressibility.

Tliere is siibstantial eviclence. t herefore. tliat the iticrease in the density fiiict iiatioiis. iri the range of the clensity-derisity correlation fiinction and t ho coniprcssibility at and near the critical point arc al1 interrelated. Con- siywritly. it is plausible to cspect that the critical csporicnts rl and v bc rc+itcvl to the therniodyiiiirriic esponents a. .i. 7 . ancl 6. Oiic cari use t h flii(.tiiiit ion-siisceptibility t heorem to arrive at tlic scaling relation

Tlic~re is a furthcr relation that siioiilcl be mentioned. which is the only rrliition involvirig the climensionality of the systein. Data show that the uni- wrsiil charactcr of critical esponents holds orily wit hin so-çallccl universülity (hss ias . Thil esponents appear to depcnd on t be climerisionality of the systern tiiiti oii the sytrimetry of the order paranieter. A universality class is identifiecl I)y t l iv the diniensionality ci of the system. by the rank. n. of the order param- iwr tcrisor of the system. ancl by the short or long range of tlie interactions. I t is iisiially iritlicatetl by the symbol [d. n]. For esample. fliiicl systems be- lotig to the [3. l] uniïersality class. since the- are three-dimensional systems arid t heir orcler parameter is a scalar. wit h short -range. van der Waals-like irit~ractions. I t is quite important. then. that a "proper" theory be able to ;iw-ount for a dirnensionality dependence of the crit ical esponents. \Vit hout going irito the cletails. Erom a dimensional analysis of the free energv emerges ariot her esponent relation that goes by the name of "hyperscaling" relation:


The scaling theory arrives at the important observation that at the critical point eacli physically important quantity has a natural size. When this size is accouritecl for the universal character of the phenornena emerges naturally froni the theory. Furthermore? through the scaling theory and the scaling relations one can deterniine al1 the critical e'rponents from the knowledge of only two of thcm.

2.3 Models Of course. orle neecLs a way to calculate those two esporients. whichever they iiiay I w . to get the whole set anci compare it with experiniental results. One fiiritlairirritd quaritity from wtiich the esponeiits can be obtained is the total f r w cwrgy of the systeni (that is to say. the surn of the regular and the sirigiilar parts). which. as statist içal inechünics teaches lis. oric can obtain frorii t lic partition fiinctiori of the systeni. Z. once its Hamiltoiiiari is knoari.

Tlicw iirc scvcral t lieoret ical rnoclels t hat serve t his piirpose in wrious wys. h t ttirrr is orip particiilar niciclel that has inspirecl niost of the othcr oiicbs. wiiicli is the Ising rnodetl ['O]. I t is the sirriplest riioclel of a niüny body systriii ;ml. althougti it originateci frorn the stiidy of a fcrroriiagrict. it can hl iised to rcprescrit a fiuid.

Iii thr Ising moclel the space is clividecl into a lattice of S cells of ecliial voliinir aricl iclentified by a single lattice point. Each lattice site represents ii p;irticlc' of the systeni aiid in the ferromagnetic case an Ising 'spin'. s ,

( i = 1. .... .Y). is associated witli each lattice point. the spin variable taking i i tlirr o f two possible values. sa\. f 1. to represent its niicroscopic statc. To 1)iiild a iisefiil Harniltonian function. interactions between spins miist bc in- ihdri l Iwtwcen ncarest neiglibours only. in the simplest formulation. aiicl are rcyrwwtctl t)y an interaction constant J . The Ising Hamiltonian is

~ r l i~ r i l the second suni is meant to be over all. and on15 nearest neighbour pair of spiris and H is an esternal applied field (magnetic field in ferromagnets and pressure or chemical potential in Fiuids).

ilkm corrcctly it should be referred to as the Lenz-Ising model.


It can be s h o w that the king model is isomorphic to the lattice gas model [21], the latter being a very crude but useful model of a Buid. where the continuous space is replaced by a lattice of sites. The Auid atoms or molecules are only allowed to sit at the lattice sites. The relation between the t w niodels is realized by having, for example, the spin +l state correspond to ari empty lattice gas site and the -1 to an occupied one. The interaction potential betcveen two occupied sites. i.e.. between two fluid molecules is gi~eri by the qiiantity ut, , such that

x i f i = j - J if i # j. i ancl j nearest neighbours (2.14)

O othcrwise

Hariiig a Hariiiltonian fuiiction like in equation (2.13). the partition functiori Ss is

Zs = Tr,,=,, e- H . y / k s T ( 2 . G )

kll hirig Bol t znianii's coristarit aricl T the absolutc temperatiire. The frce c11ii1rgy is t lieri g iwn by:

F = -kBTlnZs (2.16)

iri r tic. tliclrniotlyrianiic liriiit that the nimber of moleciiles. .\;. ancl ttic voliinie. I '. of tlir systern both tend to infinity in such a ivay that the ratio Y/\' rr~rtiains finite.

I t is possible to solw esactly botli the one dimensional and the two di- irirrisiorial Isirig rriodel$. Howewr. for the t liree climensional case only ap- prosini;itc soliitions esist. One \va\. to obtain numerical values of t hc critical ~yo r i r i i t s froni a tliree dinicnsional Ising motlel is by the hiqh-tempemtu7.e sctricls rsparision [l-l].

The free rnergy is cspanded as a fiiriction of the coiipling parameter .J/klIT. which becomes small at high temperatures. Sirice the therniody- riaiiiic fiinctions correspond to derivatives of the free energu. the- will also br reprcsentecl by series espansions in terms of that parameter. The critical rsporirnts are then obtained by studying the ratios of the successive coeffi- rirrits of the power series as a function of the reciprocal of the order of the

" ~ h e exact solution of the 2-D king mode1 bu Lars Onsager in 1914. n-as one of tfic milestones of the development of the modern theories of critical phenomena (see, for esartiple. rcference [2]. chapter 15).


coefficient ([NI and references t herein). The high-temperature expansion met hoci arrives at the Following values of the critical exponents

2.4 Renormalization group theory

Rriiorriializat ion groiip tlieory of critical plienomeria is acclainied as the most iicciira t c in the ciescript ion and understanding of the crit ical point feat ures o f iiniwrsality ancl scaling and ici the correct esplanation of the origin of the rrit i c d rspoiients ancl the calciilation of their values [Z].

TIic tlicory is basecl on a transforniatiori performed iteratimly on the Hamiltonian of the systern in order to reduce. or integrate out. the niimbcr of drgrcc~s oF frecdom rcquired for the description of the critical point char- xtt~ristics. Tlic transforrnat ion in some way renormaLizes the Harniltonian iit wdi step. hence the naine of the theorf-!

Rclatirig back to the Ising mode1 as a starting point. the typiol Haniil- toriian of a systeni esliibiting some kind of criticality depends on ii niimber of piiratrieters that are rclated to the relevant thermoclynaniic fields of the prihlcni: tcniperature and csternal magnetic fielcl in the fcrromagnet ic case or tmiperatiire ancl pressure iri the case of a pure fluid. In the discussion rtiat follows the niagnctic case notation will be usetl. as it is often the case iii t l io literaturc on the theory of critical phenornena.

The Harriiltonian usually appears in the partition furiction as ail esporient of t h forni -31/kBT. it is therefore conwnient to work with the quantity

= - 3 / k B T . rcferred to as the reduced Hamiltonian:

where the quantities 1; and h are respectively related to the temperature and csteriial fielcl in this nay: h- = .I/tsT and h = H / k B T . J being the interaction aniplitude of the Ising Hamiltonian. The parameter C does not

"For the sake of exactness, since the inverse of such transformation does not esist. the algebraic structure of the set of transformations is a semigroup.

Chapter 2. Theoreticai background 20

appear in eqiiation (2.13), but it is used to simplify the mathematical treat- ment. It can be interpreted as the contribution to the free energy from the short wavelength degrees of freedom that are intergrated out at each step of the renornidization transformation.

The common idea underlying renormalizat ion group treatments is t hat of cspressing the parameters describing the Hamiltonian in terms of some ottier. possibly simpler set of parameters. while keeping the physics of the problerri iirichanged. In the study of critical phenomena this is acliieved by ii process of coarse graiiiing of the short-lengt h-scale degrees of freetloni in b w u r of the longer length--seales that appear to be the important ones as t tic rritical point is approachecl. as w s ciiscussed in the previous section. 111 the process of coarse graining. the retluced Hamiltonian u(K. h. C) is rtq~rcscntetl by a point in ii space called syace of Hamiltonians or parameter S ~ « C P . Ii~io\vledgc of these paranieters full- determines the free energ- of t lir prohlcni iii t tic t herrnodynarnic liniit. Clianging the temperat ure. T. and tii;igiictic field. H rrioves the point around in thc parameter space. In t hc r t h t iwly siriiplr clsaniple (at least conceptually ) of a ttvo-climensiorial sqiiare liittic<*. uiith c.m stle tiow the parmieters gct rerior-rndized by ii coarsc grainirig op(wtioii. Rrfcrring to Fig. 2.1. the coarse-graining procetlurr cari be carried oiit hy dividing the lattice in blocks cach containirig a set of spins. Each Iilock is rcprrsoited by one value of the spin variable. choseri. for esamplc. hj- t l io iiiajority riile: if niost of the spins in the block are +l. theri the ivliol~ Ihck spiri will have value +l. ancl vice uersu. Thc partition function is tlicri calciiliited by taking the sum over the block spins insteacl of over al1 t h original spins. From a lattice nit ti spaciiig u . after trarisforming oncc the ricw latticc has spacing 2a. but to reniain faithful to the original probleni t liv t riinsfortrieïi lengt hs are rescaled so t hat the nea lat tice spacing a' equals tlic original spacing. This entaiis that a generic distance r be rescaled to 1.' = 1.12. in the particular case of Fig. 2.1. In general the length-rescaling \vil1 he r' = r / h d . trhere b is a coarse-grainirig factor. and ci the diniensionality of t tic +-stem.

Iii the general case. after a renornialization transformation R is carriecl out. r i t w couplings enter the picture. and the Hamiltonian will be:

Hcrice. the space of Hamiltonians. in the most general case. is an infinite di- riiensional space. since each point in it represents a Hamiltonian that depends on an infinite number of parameters.


Figiirc 2 . 1 : Coarse graining of a square latticc into bloçk spins. The black cir- c l ~ s m i the original. iirircnornialized spins. while the white circles correspond to t l i ~ .rirw' spins after a transforniation R has been applied.

X fiiiicliiiiicntal rolc in t hc space of Haniil tonians is playcl by fLsed poirits. 3 ' . iiaiiicl y poirits t tiat are invariant under renormalizat ion:

Frorii the rcmiling of the lengths after eacti step of the trarisforniatioii it follow t liat t hc correlat ion lengt h also rescales as:

nt1ir.h at the fisecl point iniplies that [(li*. h*. C'. ...) = <(f ie . h*. Cm. ...)/ b. w i t ti b > 1. The last equality is valid for either F * = O or <' = x. the fornier is krioaii as a trkial fised point. while the latter is a nontririal and more iiitiwst irig fiscd point. The divergence of the correlation lengtli is the tiall- mark of t he critical point. therefore it is reasonable to say that a (nontrivial) fisrcl point of a renormalizat ion transformation represent a crit ical point.

The drnwing iri Fig. 2.2 is helpful in describing how the t raiisformation R opcrates in the parameter space. -1 typical transformation R applied to the ptiysical critical Harniltonian (@O) in Fig. 2.2)-it can be iniagined to be ttic critical Hamiltonian of Fe. for instance-yields a different renormalized Harniltonian %(? which is also a critical Hamiltonian since the renormalized correlation length that was infinite at R(O) is infinite at ??fl('). too. This


Figiirr. 2.2: Grapliic reprcscntatioii of the spiicc of Hamiltonians and the flows giwri by t hc renoriiializat iori group transformation R ( taken from ([l-l]) ).

siiggclsts that in grmeral the physical Haniiltoriian is rrot ii fisccl point of tlic parariicter spare. ratlier siiccessive applications of R gerierates a trajectory of critiral points in parameter space. It is plausible to espect that this .r<~riorrii;ilization trip' in parameter space will end at a fisecl point. 2' of the spicc. In ttiat case the critical Hamiltonian of Fe is saicl to lie on the stable criticid trianifolcl of the fis4 point 2'. that is to say the set of points in the sparr of Hariiiltonians. which will flow to U' iinder iterative renormalizatiori t r:msfornint ions. Xaturally. starting points tlifferent froni the critical point OF hl. liko t hosc of water or n-heptane+iiit robenzene for esaniple. coiild lie or1 the sanie manifold and therefore land at the sanie fiseci point after ri~norni;ilization. A11 the critical Hamiltonians that belong the stable critical riiiiriifolcl of tlic sanie fised point are said to belong to the same unirersality cliiss. Any perturbation causing the initial Hamiltonian to end up in the t~asiri of attraction of a different fisecl point is called relevant. otherwise is saicl to be irrele*uunt.

Crider the reasonable sssumption of smoothness of the generally nonlincar transformation R. this can be expandeci around a fised point. considering then only linear terms. Based on the semigroup properties of the transformation R. it can be shown that the free energy per unit volume as a function


of the linear scaling fields t, h, ..., gj, ... transforms as follows,:

J ( t . h. . . - Y gj7 --. ) z b-dl f (bl"'t, bLA?. ..., b1%JjT -.-) (1.21)

nliere b is a rescaling factor. d is the dimensionality of the system. 1 is the riiirnber of iterations of the linearized R, and the As are constants that clefirie the eigenvalucs of the linearized transformation. By choosing bA1 = lit. the gcrieral scaling rcsult as in equation (2.2) is recovered:

Iiiivirig sct '7 - n = d / X 1 . 1 = X 2 / h l , and 0, = h , / X l . Tlic tcriiis lih? g, / t *~ play a particiilarly important role in the clcscrip-

tioii of tlic critical region "not so close" to tlie critical point. If siich ternis correspoiicl to irrelevant variables. o < O and the whole term will vanish as t = O is approwhrcl. If it is possible to cspand the scaling function giveii by ~qiiii t ion (2.9'2) in trrrns of t hese variableso for esample. the ordrr paraniPtFr Fur a piirc fliiitl is ticscribecl by

Tlir iiiiivrrsal t q o n e n t -O is normally called correction-to -scaling esponent m c l it is part iciilarly significant since the rnagnit ude of t his terni can interfere \rit Ii il correct clctermination of t hc niain critical esponent .j ['23]. It tias bcen sliosii tliat eqiiation (2.23) can be written as

wtiicti is the form of the order parameter with rvhich the data of the es- pwinieiits in this thesis mil1 be confrontecl. In the work presented here. the cwrrction -tu--scaling ternis play an importaiit role and cannot be neglected if a correct analysis of the data has to be carriecl out. The esperimental resiilts will show lion- the need to include correction terms to the simple scaling lew arises at different temperature ranges in bina. liquid niixtures and in ~ I L S P fluicls.

To conclude. numerical values of the renormalization group t heory crit icai rlsponeiit for a systern can be calculated by using a rnicroscopic model of the system. -1s we have seen. the three-dimensional Ising model in the form


Table 2.1: Theoretical values of the critical exponents. The results From the classical. mean field theories (4IFT) are reported for cornparison with the predictions of the modern theory.

Esponent SIFT Scaling [ld] RG [24] c1 O 0.105 O. 109 * 0.004

o f the lattice gas nioclei hüs turned out to be a very goocl approxirnatiori o f a fliiid. The renorinalization riiethod values of the critical esponents for tliis ~iiotlcl are suniniarized in table 2.1. diere the scaling thcory values aricl thcl -cliissical' vaiiies of the niean field theories (IIFT) are aiso reportecl for ( m i parison.

Chapter 3

General experiment al feat ures

3.1 Introduction TIiorc arc cotiirriori issues which niust bc aclclressed in order to carry oiit the i~spi~riirients rcportccl in t his t hesis. For esample. the t hermostiits ernployecl tor tlw piirc fliiid rspcrimcnts and for the binary liqiiid misctirc cspcrinients siiarc. sonic hasic characteristics. althoiigh with slightly cliffercrit geometries. I iorcw-w. t hc opt ical techniques employcl in the measurenients are the sarne for the two kirids of csperiment. This chapter is therefore a general description of t h apparatus ancl techniclues that procluceci the chta for this thesis. Tlir issiic of the trmperaturc control aricl rneasurement is clealt witli in the first silctiori. This is follo1v4 by a section tlescribing the tliree optical tech- niqi1c.s iiscd in the esperiments: the pBsrri cell. the focal plane. ancl th^ irnuge plane trt:li~iiqiies.

Bot t i riiet ric and imperial iinits are iised for the apparatus dinieiisions. Ttic itsc of non -stanciard units is duc to the star~darcl dimensio~is of aietal stock rtornially xailable in Canada.

3.2 Temperature control and measurement

A n rscellent control of the temperature and its precise measurement are iriiportant in critical phenornena esperiments for several reasons. The tlier- niodynarnic anomalies at critical points of either pure fluids or bina- liquid mistiires occur in a very narros temperature region in a neighbourhood of the critical temperature. Tc. More precisely. in terms of the reduced

Chapter 3. General experirnental features 26

temperature, t = (Tc - T'')/Tc the asymptotic critical region of pure fluids is typically observed for t < IO-". whereas for binary liquid mixtures for t < 10-728, 29. 301. In other words, in critical phenomena occurring a t roughly room temperature, like those studied for this thesis, experimental observations have to be made from as close as possible to Tc to about 0.03 l i froni Tc for pure fluids and to about 3 K for binary liquids. Saturally, the closer Tc can be approached. the more accurate. and scientifically useful a description of the critical regioii (and the more difficult to study it).

-4s the cri tical temperature in appr~ached~ the thermal equilibration time iiicreases [31]: and it is around 30 hours at our closest approaches to Tc. that is to SV at about AT = Tc - T = 150 pK. If equilibrium is not reached. one is iiot nicasiiring the physical quant ities iri the appropriate conciit ions.

Teriiperature uniformity is also very important in these esperinients to awid therrnal gradients in the sample fluids. Thermal gradients may induce riit.asiirable index of refraction gradients. which. as it will be discusseci later (SCP .*Rcsi~l ts and discussion" section of chapter 4). constitutc a source of st-stcmiat ic crrors in the final mcasurements.

Tlicreforr. siicccssfiil csperiments in critical phenorneria clepend criicially. airiorig ottier tbings. on precise. acciirate ancl stable teniperaturc control. Ilorr. quant itat iwly speaking. it is necessaq to control the ternperatiire \vit tiin lms than 1 ml\: for up to several c1.s coritinuoiisly.

3.2.1 Thermal control

T~niperatiirt. control of the recpired stability and iiniformity is acliieved by riicaiis of a thcrtnostat with two active thermal control stages. The satriple r . i b l l is coritiiined in the innerinost stage of the thermostat (tiatchecl region in Fig. 3.1). This stage is a copper or alurninuni block fit t ing snugly iri anot her rriassiw bloçk--ah copper or aluminum. The latter block is wrappeci in ci t h heiit ing wire or tieating foils 1321. through which eleçtrical heating is tlrliwred to the sample.

The inrier stage is surrourided by a roughly 50-mm-thick 1-r of high tlensity styrofoani to provide insulation (Styofoam SN [33]). The thickness of the intcrlayer of styrofoam is arrived at through the heat conduction argu- ment presentecl in appendiv A. For the purposes described in this thesis the irisdation provided by Styrofoam SM is very satisfactory and it circumvents t tic heat convection problern t hat air insulation alone would present .

Inner stage and styrofoam layer are contained in the second active heating

Chapter 3. General experimental Jeatvres

TOP VlEW (CROSS SECTION)

therrnistors

OUTER STAGE: thermaegulated water circulation stage (kû.011

INNER STAGE: electricallv heated

Figiirc 3 . 1 : Top view cross section of a typical multistage thermostat.

Chapter 3. General eqerimentul features 28

stage of the thermostat, which is typically a copper cylinder with 318"-OD copper tubing soldered to the outside of it. Temperature regulated water is puniped through the copper tubing from tank of a water circulator [34] (we iisc distillecl water to avoid growth of molds inside the circulator tank, hoses aiid t ubing). The specified temperature control sensitivity of the circulator is k0.02 K. Tlie temperature of the water in the circulator tank was measured aiid foilnd to be stable within &0.01 K. however the water travels for a few riicters at room temperature from the circulator to the thermostat and it is espected tliat the actual stability be within I0 .1 K. This constitutes the (varsr stage of ttie thernial control and it is regulated at a temperature. T, -,,,,,, ,. surh t hat Tc - Tc. ,,,, 51 K.

Tlic firw stage is the electrical heating applied to the inrier cylintler. The principlc on which the fine-tiinecl thernial control is basecl is the detection of' tlic inibalance (or error) signal from a Wheatstone hricige. Ttie electrical lwatiiig to the inner block is controlled by n iiegatiw Letlback electronic systciii. A t herrriistor enibecldecl in the nietal senses its ternpcraturc enci rtywcwts orw arni of a resistive DC \\'heatstone bridge. as illustratecl iii

Fig. 3.7. Tlic otlicr t tirw arriis iire two itleiitical standard resistors (their t~piciil tolerarice is 2% and t heir temperat ure coefficient betweeti 50 ancl 200 pp~i / 'C) and a clecacie resistarice bos (Tinie Elect ronics Ltd .. mode1 1051. wi t li ;i tcriipcrat iirc coefficient of 100 prnl0C) . The required teniperat ure is srt II! a certain valut? of the decade bos resistance. If this resistance is (liffiwnt Erom tlic ttierniistor's value. i.e.. the iriner stage temperature. the \\lir;itstorir t~ricige is imbalancecl. The error signal frorn the bricige is fed into and aniplified by n Hewlett -Packarc1 niillmeter (mode1 HP 419). whose out put is iripiit irito an operationai power supply (IiEPCO moclel OPS 40- o..; B) tliat controls the current applied to the heating wirc or the hcating foils o F tlir inner block. wliich is. in turn. in thermal contact with the sample tioldt~r.

Ttir tvliok tki~rmostat is enclosecl in a plp-ood bos. linccl with a '25-mm Liycar of higli density styrofoam. of the sanie kirid used between the outer and iririer stages. Diiring esperimental measurements the only apertures of the hos to the esperiment room are tlirough two srna11 holes for the water hoses to ancl froni the Irater circulator. Lastly. care lias beeri taken not to have the csperiment room temperature fluctuate too much. by keeping the door shut aiid the ligbt off at al1 times.

The measured temperature of the inner stage of this two-stage thermostat is stable within 100 ph: for several hours. A precise monitor of the temper-

Chapter 3. General experimental jeatures

Figiirtl 3.2: Teinperatiire control ancl nionitor stages of the apparatus iiscd for osperinicnts on both pure Riiicls and binary liquids.

Chapter 3. General experimental features 30

ature was carried out for the binary iiquid esperiment thermostat and the performance checks are reported in Chapter 4.

\\'indows in the plywood box, metal cylinders and styrofoam allow an espanded He-Se laser beam to traverse the sample.

3.2.2 Temperature measurement

Higti precision and accuracy in the measurement of tlie ternperature are of paraniount importance in these experiments. Two types of therrnometers iwrc iisecl: a Hewlet t -Packard quartz t hermorneter and several semiconduc- tor t licriiiistors [33. 361.

Siiiiilarly to the circuit used for the cotitrol of the rurrent to the electric tiwtrr i r i tlie thermostat (section 3.2.1). each therrnistor for the teniperature tiioiiitar is criibetldecl in the inner block rnetal and is one of the arms of a DC \\lic.iit.stonc bridge. the reniaining arms heirig again two identical stan- iliird rcsistors aricl a cleciide rcsistor box. The bridge signai is the input of iiiiot litlr Hcwlet t -Packarc1 nullnietcr ancl amplifier whose out put is rnonitored ~ i i tlic riieter's front panel and on a cliart recorder. Tlic value of' the (lecade rcbsistor tliat ritills the bridge output voltage corresponds to the valiie of the t ti<wiiistor rcsistance and. t brougli t tir t liermisror calibrat ion equatioii. to t h c l irilicr stagp arid saniple temperature. The "terriperature monitor" part o f tlir rirctiitry is s h o w in Fig. 3.2.

Ttic quartz tticrniomcter probe is corisi<lerably bulkier ttian any of the t ticrniist ors uscd ii1ic1 cloes not provide as acciirate a tenipcrat iire measurc- riicbiit. ospecially for the data wry close to the criticül temperature. Sonie cblitdcs oti t l i ~ qiiartz thermometer reliability were clone with the biiiary liqiiid clspcrinient t herrnostat anci are rcported in Chapter 4. Thcrefore. oncc t t i t y are cali t~ratcd (sec section 3.2.3). for wliicti proce<liire the quartz ther- riioiiictrr is indercl ver- useful. oniy the thermistors are employed for rcliable t criqwritt urc measurenierits.

To twsiire op tiniuni thermal coupling between the iriner block met al and t hc t hcrniistors. the latter are gluecl. \vit h high thermal conductivity eposy [37] irito copper holts. which are. in turn. screwed into the metal. -4 fiirthcr point in favour of the thermistors. is that the tirne they take to respond to the sr~iallest teniperature change occurring during the esperiment is niuch sliortcr than that taken by the quartz therniometer.

Chap ter 3. General experimentai features

3.2.3 Thermometer calibration For rna~irnum accuracy in our measurements both the quartz thermometer probes and the thermistors were calibrated at the triple point of pure wa- tw [38. 39: 401. This is one of the fundamental defining fixed points of the International Practical Temperature Scale (IPTS-68) and its temperature lias bceri assigned tlie value of f0 .01 OC on the IPTS and of 273.16 K on the Kelvin t liermodynamic scale.

'\Ioreover. since the thermistors neecl to be calibrated at two tempera- tiirrs. at Icast. iri order to have a calibration equation for them. the quartz t licr~riorrietcr and a t lierrnostat \\WC iisect to calibrate the t herniistors at ctif- fwriit teniperatiires. More details on tlie calibration procedure are given in t tic I~iriaq liquid mistiire experinient chapter (Chapter 4) . where the newly t~ i i i l t t lierrriostat is described in depth.

3.3 Opt ical investigation techniques

Thcl critical rcgiori of hoth pure fiuids and binary liquid inisturcs cari he st tidircl by oliserwtioii of the optiçal properties of the Riiici iincier investiga- t iori [9. -11. -42. 30. 43. 44. 45. 46. 471.

Tlir ordcr parametcr for pure fiuids is typically the riiass dcnsity difference t ~ w v c v w tlie Iicliiici arici wpoiir phases of the fliiicls'. l p = p~ - p l - . \\é can tliorrforr riirasiire the index of refraction. n. ancl then relate it to the density. p. ttiroiigli the Lorentz -Lorenz (later referred to as LL) relation (101 :

Siriiilarly. orle of t lie acceptecl order parameters in hiiiary l i c p i c l mist tire st iidies is t lie concentration difference. A S = .YL - Sr-. of one of the species Iwtwecri t lie two phases. lower (L) and upper (L). Tlius. by measiiring the in- (les of rcfraction difference. An. between the liquicls. ive tiare a nieasurement of A.\' by the direct proportionality relation:

10ccasionally the molar volume difference is used as order parameter. but ic produces a much l e s synimetric coesistence curve than the density difference. therefore the latter is preferred when the data is analysed for cornparison with the king (lattice gas) mode1 r~sults [48].


tlerived from the LL relation applied to the two liquids under the assumption that indes of refraction and density show no anomalies at the critical point iiritl uncler the hypothesis of additivity of volumes upon mising (appendix C lias a derivation of relation (3.3)).

It should be emphasized that the degree of approsimation of relation (3.2) coulcl lcad to systematic errors in the measurements of the coexistence cuve of biriary fluicls. This issue is discussed more in depth in the chaptcr describ- iiig the binary liquid esperirnent (Chapter 4).

This scction tlescribes the t hree optiçal techniques useci in the reported cspcriirierits. The prisni ceil technique is used for measuring both the LL corfficierit. L. aiid the coexistence ciirve of pure fluids. The focal plane tectiir ique also is rniployed to nieasure the coexistence ciirw of pure fluids ( h t i:loscr to the critical temperature than possible witti the prisni cell) as wcll iis of bi~iary rnistiires. The image plune technique cari be iised to stiidy t t i t b iritlcs of refraction. ancl hencc the clensity ancl concentration profiles. hot 11 iit~ovc and bclow the critical tcniperature.

3.3.1 Prism ce11 technique

0 1 1 ~ iicttcls to tlcttwriine the LL coefficient. L. appearirig in relation (3.1) for ;iiiy siit~sta~icc whose critical region is to be studied by rneans of its indes of r(hfr;ictioii bchavioiir in ttiat region. The clensity. p. and thr rcfractive incles. r i . o f thc l Hiiici have to be measured indepenclently of each othcr ancl usecl in iyiiat ion (3.1 ) to obtain ari esperimentel determination of Ç. This is realized II? t h pris111 rcll wperiment [Ag].

Tticb fliiicl is int rodiicccl into a hollow . aluminiini. prisrri-stiapctl cell. wliicti is slioivii in the drawing in Fig. 3.3 and used in the setiip shown schematically i i i Fig. 3.4. The cc11 walls (pcrpendicular to the plane of the page) are two sitpphire wintlon-s ttiat form an angle of about 20". The ce11 is fillecl nith the Hiiitl to Iw studiecl and a spatial-filtered. espanded üncl colliniatecl He-Sc 1asr.r heairi passing t hrough the ce11 emerges froni it. berit through a certain iiliglr. 8. \vit h respect to the strnight beam coming froni the laser. The beam wwrging from the ce11 is directed into an autocollimating telescope (David- sori niodrl D2i.j) by means of a movable mirror. The mirror is adjusted by a rriicrometer screw to bring the laser spot coming from each fluid phase to t tic crosshairs of the telescope eyepiece. There is then a one-to-one relation l)rtween the micrometer reading and the angle of refraction of the 632.8 nm He Se laser lirie through either the liquid or vapour phase. The microrne-

Chapter 3. General experimental features

PRISM

CELL BODY

VALVE

Figure 3.3: Illustration of the prism ceil.

The rmostatic Housing Movable

Mirror

He-Ne Laser

Telescope U

Figiirc 3.4: Schematics of the prism ce11 experimental principle and apparatus (riot to scale).

Chapter 3.

n V) Ci C tu

m œ

'CI L u

a

Figure : 3 5

Generul ezperirnental features

I

-

-

-

O=arcsin(mÀ/d) - m=O, rl, k2, ... : -.

-40 -20 O 20 40 60 micrometre scale

Data iised to ohtairi il ccalibration of the micronicter sciile on the riioval)lc riiirror for the prisrri cell esperiment.

t i ~ rtwli~igs arc iiieasiirecl witti respect to the reading of the beam coniing straiglit tlirougli the therniostat when thc ce11 is rcnioveci. This is calletl the T ~ ~ ~ I I C C berirri.

Tu d i b r a t c the adjustable mirror micronieter and hence bc able to obtairi iiii i\rigl~ frorii its reaclings. a 50-lines/inch diffraction gratirig is inserted i r i

t l w tliwriiostat iri place of the cell. Then. the position of the cliffractiori p i t t m i rriasima is reacl froni the micronieter scale and plot ted agairist the diffriict ion i i i i~~in ia angle calc~ilatecl from the relation:

alirrv rl = 0.508k0.002 mni (If30 inch). A = 632.9 nm. ancl .Y = H. k2. . . . . k\,.,. Ixing the highest visible maximum on either side of the central riiasirniini. -4 calibration eqiiation is then found that relates the refractiïe iirigle 0 to tlir micrometer readings. .As Fig. 3.5 shows. this relation is linear.

By measuring 6, and using Snell's lan applied to our particular optical systrrri cve are able to measure the refractive indes. nfiuid. as a function of t h angle. 8.

n f luid = nfluicl(@) (3-4)

Chapter 3. Geneml experimental Jeatures

Figiirc 3.6: Effect of wcdge anglcs of sapphire winclows on the nieasurement of thp rcfractccl angle (not to scale).

iit t 1i;it part iciilar clensity. Tlic sappliire ivinclow surfaces arc not perfectly pir;illcl. hu t sliglitly nctlgecl. which affects tlic cleterniination of r l f l , l l d throiigh t~piiit ion (3.4) . as illiistratcd iii Fig. 3.6.

I l i tlit ' idcal case of perfectly parallel surfaces. ccliiation (3.1) is esplicitly:

siri(cr + O ) f l u d = na1r

sin fi

~ v l i c w f i is the prisni angle nieasiirccl at (20.6853 f 0.0167)". If tlie wetlge mglos. o, and a,. of thc straight ancl tilted window. rcspectiwly. arc consicl- c w t l . t h rcliiation (3.5) becorncs. to first order in n , ancl nt:

12 a I ~ j i ~ ~ ~ d = - (sin(ci + 0 ) f sin CL

wlirrr n , = (0.00-42&0.0035)" and nt = ( O . O l X f O.OO3a)". na = 1.000262194 is tlie refractive index of air [50]. ancl n, = 1.7660026 the incles of refraction of sapphire [XI.

Ttieri the density is decreased by bleeding some fluid from the ce11 and the measurement is repeated until the ce11 is empty.

Chapt er 3. General experirnentai features

To measure the density of the Ruid, p, the volume, I-', and the mass, rn,,ll,

of the evacuated ce11 are measured first. The volume is measured by filling the ce11 with piire water? whose density can be precisely found out: and by tveigliing the water-filled cell. If the mass of the empty ce11 is known, the i-oliinie of tlie ce11 c m be determined. The distortion of the volunie of the ce11 \vit ti pressure is negligible for the pressures used in these experirnents. The tlierriial expaiision of the ce11 aluniinum body in the range of temperatures of tlicsc niecuiircnients would cause a volume variation of 0.2 %%. The variation i i i the Lorentz-Lorenz data causetl by sudi variation of the ce11 volume is less tliaii t h overall scatter in the data. The measured volume of the prisni ce11 is ( 12.066 k 0.003) cc.

Tlirri. at cadi filling (or rather. emptying) stage. the cell-plus-fluid sys- trrii is wcighed on a prccision chemical balancc (scnsitivity of 0.0003 g). t litwby yiclcliiig t hc qiiaiitity:

i i ~ i < l lirtirc ttir clcnsity of the fluid is giveri by:

Tho ialiies of r l f l u i d aricl p,lu,d froni Equatioris (3.6) ancl (3.5) are iisecl in rcilatiori (3.1) to yield values of the Lorentz-Lorenz function. L. This is iisiially plottcd agairist the clensity p. to study its behavioiir in the range of dwisitics espectccl to be investigateti. for esample. for the cocsistencc crirve o f t h fltiid.

To rricasiisr t lie coesistence ciirw \vit h t hc prism ceIl tcchniqiie. the sam- plrl in the ce11 is prcpared at the critical tiensity and tlie angles of refraction Fsoni t h liqiiid ancl vapour pliases are rneasured to yield the refractiw indes of tlit! t ao phases at different temperatures. Frorn the LL relation. pi- and pr iirr t hrn ol~tained. froni which. in turn. the order paranieter. Ap = p~ - p l - .

arid ttic diameter. pd = ( p L + p l - ) / 2 . are cletermined.

3.3.2 Focal plane technique

The focal plarie technique is one of the fundamental esperimental techniques iised by O u r laboratory to investigate the coexistence curw of either pure fliiids or bina- mixtures. It allocvs data to be taken closer to the critical teniperat ure t han the prism ce11 technique does.

Chapter 3. Generd experimental jeatures

F r 3.7: Illiistration of the behariour of the refractitp irides of a binary iiiisttirc as a function of ce11 hcight at T < Tc (a). at Tc (b). and T > Tc. Brlow Tc. rriarks the position of the rneniscus betwen the two phases.

Chapter 3. Generaf experimental features 38

This experimental method is based on the detection of variations that occur in the index of refraction profile of the liquid contained in a ce11 as its temperature varies. -4s indicated in Fig. 3.7a, when the temperature of the systeni is below Tc? there is a step discontinuity in the index of refractiori, as a fuiiction of height in the cell? a t the rneniscus between the two phases (eitticr the vapour and liquid phases of a pure fluid or the two liquids of a biritiry mixture). -4s the ternperature is raised above the critical temperature. a fraction of one phase transforms or diffuses into the other and vice versa. .As a result. the profile is no longer discontinuous. taking iiistead a sigrnoidal sliiip. as illustratcd in Fig. 3.7b. and 3.7c2.

Oricr again. the refractive incles profile of the saniple is probed using ii rolicreiit 1)eani of light froni a 631.8-nm He-Ne laser. space-filtered. es- piriclcd aiid colliinatecl to a diameter of about 25 rnrn. Fig. 3.10 shows the c-o~itigiiratiori of the optical bencti for bot11 the focal plane and the image plarir c~spcriiiicnts. Light passing t hrougli the ce11 is bent due to the gradient of t h rrfriict iw iiides n ( r ) in the sariiple by an angle 0 giwn by:

i i t i t lor oiir ty pical csperiniental condit ions equat ion (3.9) can bc i ritegratcd to yit4d t hc total cleflectiori angle inside the ce11

wlirro L is t h ceil t hickness (appenclix D esplains briefly how t his c m happen t tiroiigh the application of Snell's la+ hfter esiting the cell. the beam is iigiiiii cIrflt.cted according to Snell's law. so that. if tî is the the ariglc of the h i 1 1 1 leavitig the ce11 wit h respect to the origi~ial clireçtion

L cln 0 = --- na,, cl:

Eqiiat ion (3.11) gives the total bending angle of the light passing througii a cc11 of t liickness L at a height 2. where the refractive index profile has a gradient d n / c k . Referring to Fig. 3.8. two rays encountering the sigmoiclal

-The refractive index profile at temperatures T 2 Tc is describeci by the mror function" ~r f ( z ) = Ji erp(- ( ' )d( . under the assumption that the profile is describahle by an odd function of height with respect to the rniddle of the cell. that the profile is a step Function before diffusion starts to occur. and that the systern is unbounded.

Chapter 3. General experimental featwes

Sompie CeIl

Figiirtx 3.5: Forriiation of Fraiinhofer diffractioii piittcrn due to a rion-tiniforni

refraction angle

Figure 3.9: Geometrical illustration of the phase difference and the formation of thc Franiinhofer diffraction pattern.

Chapter 3. Generd experirnentai features 40

profile at points with equal absolute value of the curvature (e.g.? (+) and (-) in Fig. 3.8) are bent through equal angles. Those rays are then focused to the sanie point in the focal plane of a lens (7, 8, 91. The difference in optical path lcngth betwen ray (+) and ray (-) is given by,

'Ln L?.' = - [L(n- - 11, + (2- - 2+)nai,sin0]

X (3.12)

aith 11- aiid r i + rcpresenting the index of refraction of the Buid at heights 2-

iiiitl zI in the sample. respectively. Combining equation (3.12) witli (3.1 1). the optical path difference between the rays upon arr i~ing at the focal plane of t h lcris can bc re-written as

A Fraiiiihofer ditfraction pattern:' t hen appears at t hc focal plane. wit h min- ima occiirring iit angles wlicre L* is a i ~ odd multiple of ii.

A gcmitat rical tlcscriptioii of the sliowii iii Fig. 3.9. where L n ( - ) and qiiaiit ity L (r1- - r1.J corrcsponcls to h i c l sidr.. twtwen :, and 2-. w h i k

formation of the diffraction pattern is -L(clri/ck) are plottcd versus 2 . The t lie arca iinder the cime on the riglit the quant ity

is (yual IO the arw of the rectangle boiincled by :+ ancl :- ancl the lines Lclrilds = O and Lcfnld: = Ldnlckl,. A The difference between tlicse two arcas is eqiial to (,\/'l;;)c~ and it is crosshatched in Fig. 3.9. Becaiise the aiiglr of rrfract iori is proportional to - L (tlnld:). the angles çorresporitling t o tilt. niininia of the Fraunhofer diffraction pattern caii liaw the following ospliiiiatioti. Eaeti time the crosshatctied area in Fig. 3.9 increases by 2;r ariot her niininiiinl is formecl.

-4 canicra fitted with a clock motor to allom continiioiis film transport is placecl so that the plane of the film eoincides ~vith the focal plane of the lens.

Tlic diffraction pattern is recorded on photographic film. The number of fri~igcs. Y. detected from the diffraction pattern at each run. that is. at

' SIore accurately the diffraction iringes one observes in this setup are knon-n as Gouy fririges. after the first observer of the formation of Fraunhofer iringes caused by an index of refract ion gradient [52]

Chapter 3. General experimental features

Legend: 6 Thermostat 1 He-Ne Laser 7 Focussing Lens 2 Beam Expander and Spatial Filter 8 Carnera on Focal Plane 3 Collimating Lens 9 Camera on lmaae Plane 4 Polarizer 6% Beam spiitters 5 Expanded Beam M's Mirrors

Figiirc 3.10: Schcrnatic diagriirri of the csperimental optical setup.

o d i tctiiperaturr. is tlirectly proportiorial to the cliffercnc~ in rcfractiw incles t)rt\vwn t lir two phases. l n = ril - n-.

Thorefore. ii Art-vcrsiis-teniperatiire grapli represents the coesisterice ciirw of ~ i t l ic r the pure fliiid or the binary niistiire iincl it is suitable for ii 11 iritlirrct nirasurernetit of the rsponent .J.

3.3.3 Image plane technique I i i this technique the plane of the recording photographic film is placecl on t h image plane of the Fraunhofer lens also used in the focal plane tech- nicliic. aiid the whole refractive index profile of the sample is imagcd on the fi lrr i . This is achie~ed by placing the thermostat and sample in one of the beanis of a ILch-Zehnder interferorneter (see. for example. [12]). the "Sam- ple bcarn". wt-hile the other beam. the ~~reference beam". travels through air. The situation is depicted in Fig. 3.10.


A He-Ne laser beam, approximately %-mm in diameter, is prepared as in the focal plane technique and split by the first beani splitter, BI, into the sample and reference beams. -4s shown in Fig. 3.7, the refractive index varies considerably with the ce11 height, when the temperature of the sample is iii the neighbourhood of the critical temperature. The bottom part of the sample has a larger optical thickness, hence the light of the sample beam ttiilt traverses the bottom part of the sample is retarded more than the light passing tliroiigh the top part.

Ttierefore, wtien the waves at P and P' are reconibined at the beani spli t ter B2 ancl iniaged by means of the lens. horizotital intcrference fringes arc observeci at the image plane of the lens.

Tlic iricles of refraction as a Function of height is niapped this way. This tcv.liriiqur is also w r y ~iscful to stuciy tlic samplc ccpilibration timc at a part irtilar tempcrat urc.

Chapter 4

Binary ~iquic~ experiment

4.1 Introduction Tliis cliaptcr describes the first of the two main esperinients on which this tlitsis is tiased. Sonic of ttic data reportcd were taken with a tliermostat t l i a t \ w s \ ) i d t ex rwuo for ttie cspeririierit. The details of critical plienomena iri hiriary liqiiid rriistiircs are introtliiced in section 4.2. with ttie motivation for ttiis crperirncnt in section 4.3. Somc dctails oti the constructiori of the t lirwiiostat aiicl preparation of the saniples arc g i w n in section 4.4. Finally tlir loiig section 4.5 describes the data that w r e taken. how they were an- ;ilyzrd. t lie results t hcy prodiiccd aritl some considerat ions on t lie po tcnt ial s y s ~ ~ ~ i a t i r mor s m~ounterccl.

For the samc reasons anticipatcd in the previous ctiapter's introdiiction. I ~ o t li mrtric iiritl imperial systern iinits arc iised for the apparatus dimensions.

4.2 Criticality in binary liquids and order parameter

Ttic liquid-liquid. or binaq liquid. critical point is an esample of critical hcliaviour in Ruids that has been and still is studied both theoretically and ~sperinientally [27. 53. 34. Z3].

Refcrring to Fig. -4.1. a binary liquid is typically composed by two chemical conipounds in their liquid state and the critical phenornenon in question is their mutual miscibility as the temperature is varied through the criticdo

Chapter 4. Binary liquid experiment

I

N-rich phase

Figiiro 4.1: Biiiary licluid misture of two chemical species H and S. below t h i r consolutc critical temperature.

or r~onsolute. teniperature. Tc. I n t ho niost cornriion iristance. binary fluids esist as two iiiiriiiscible pliascs

t wlo~v T;.. wtiile ;ibove such temperat iire the two liquids beconic coriipletely riiiscit~lr arid only a single liquid phase csists. \\*lien the licluids are in the two p h w mgion. ii ireIl-clefincd menisciis Forms and is easily visible between the Iicwier liqiiicl at the hottom of the sarnple ce11 and the lighter one floating or1 top of it. -4s the temperature is raised and broiight abow Tc. the menis- riis I~rcoirics t~lurred until it erentually disappears leaving the two liq~iids cmiplrtely triiscti iri orle phase.

Tticrr do esist binary liquid systems nhich display an inverse behwiour. iiiitti(~l>. niisci bili ty below the crit ical temperüt iire and inirniscibility above. Tliry iiscially belong to the category of systems with two critical points. lorcer arid upper. a i t h the mixture having its two-phase region between the t ao critical tcmperatures. -4s depicted qualitatively in Fig. 4.2. these are ~.iill(ltl (*losecl-loop systerns. due t o the aspect of their phase diagram [56]. Ttic niisture stuciied in this thesis. however. is of the kind described in the tcst.

111 the binary licluid esperiment described in this thesis. the mixture n- tirpt ane+nitrobenzene ms studied [XI. the former being lighter floats on the latter nhen the system is below its critical temperature of about 18.9"C.

If the two phases are called L; and L as in Fig. 4.1. the order parameter of clioice is usually the difference between the concentration of one of the

Chapter 4. Binary liquid expen'ment

F i 4 . : Illustratirig a biniiry liqiiid mixture with two critical ternpera- turos. T,.. ,,pp,!r i m i Tc, lotvt!r. o is the volurrie fraction of one of the species.

spwios. cg.. rr tioptaric. offi iri phase L'. ancl its concentratiori iri ptiase L:

A t ttiis point t h qiicstion ariseç: what kind of conceritratiori is it best ro dioost) as tlir orcler parameter'! h s s fraction. volunie fraction. rriole frart ion'.'

Altlioiigii tlicrc cloes not seeni to be a ciefinite ansiver. the establisfiecl t r c w l is t o corisicier the niost appropriate orcier parameter the quantity that wridrrs tlic corsistence ciirve more synirnetric. to be able to compare esper- iiiicwt ;il rrsults wit h the lattice gas rnodel preclictions [AS. 271.

Esp~rierictl siiggests t hat the ~olunie fraction usually yielcls niore syni- iiiiwir c-oc~sisterice ciirves. Sloreover. the cpantity that is usually nieasurecl iri oiir binary licliiicl (and in the pure Ruid for that matter) experiinents is tlir iritlcs of rcfraction difference between the two phases. which turns out to tw proportional to the volume concentration. as discussed below.

For tliese reasons. the volume fraction is the order paranieter chosen in t ht) hinary liquid esperiments reported in this thesis.

Chapter 4. Bina? liquid ezperirnent

4.3 Motivation and expectations Tlie decisioii to measure the coexistence curve of a binary liquid mixture stcnis from the need to clarify earlier results on the critical exponent J of biriary mixtures from the early 1970s [58. 271.

I t iippears quite clearly from earlier literature [58. 271 that until the late 1970s or even earIy 1980s the typical accepted experimental values for the critiral esponent J describing the order parameter dependence on temperatiirc iri the case of binary mixtures ivas somewhat larger than the ezpected t lieorrtical valiie for that exponent. In a compilation of experimental data on hiiiary liquid niistures. Stein and Allen [58] found that the data of the systenis t11c.y ;iiialyetl were al1 consistent with the critical esponent J = 0.34 rt 0.01.

Th! trend of 0 t h later esperinierital stuclies of coesistence curvcs of biiiary liquid niistures seemetf to bc confirniing the reriorrnalization group t Iioor(?t ical resiilt t hat the crit ical esporierit .i should range between 0.3'25 < . j < 0.327 [27. 14. 941.

Following the relatiw iinccrtainty of the cspcririiental rcsults obtairicri im I~iriiiry liqiiicls. it i m s tlioiigtit iiecessary to itivestigate t kiis tliscrepaiicy flirt tics.

Tlir optical riicthocls employed by this laboratory are particiilarly suitable For t h stiidy of the critical region of both pure Huids [59. 601 and binary rriistiir(1s [7. 961. but they did not scem to haw beeri iised to their potential y t irl prt~vioiis studitu: (and tkiere will certainly be room for improvcment iift (1s t tir prrscnt work).

I t scenictl thercfore fit to perform a stucly of the coesistcnce ciirw of the H+S biriary licpid niisture with an iniproveci thernial coiitrol systeni and r(riiic:td rrrors in the rneasurements so as to obtain a more precisc value of t tic) {.rit ical esponent J.

Onc woiiltl espect that. once re-mensureci. the value of .j wotilcl have a h t tcr - t han -aïerage probability of conforniing to arialogous "nioclern" niea- stiroriicrits arid thereforc with the most accredited t heoretical results. Sur- prisirigly. as described below. this aas not the case.

SlOE VlEW (CROSS SECTION)

OUTER CYLINDER: thennoregulated water circulation stage (3.04 K)

Figiircl -1.3: Schematic cIrawirig of the tlierrriostat cniployecl for the hinary l iqti it l csperirrierit .

4.4 Experimental details

4.4.1 Thermostat

I t is \wrtti\vhilc distinguishing the thermostat eniployecl for the more accti- rair data of this thesis from the typical thermostats usecl in the past for st iidirs of critical phenomena on bina- mistures. Althougli it \vas built ori t lie sanic principles of other t hermostats-as describecl in the General Esperinientai Observations' chapter-its design has some improved features o w r older clesigns. which gives us confidence that a higher degree of accuracy is acliievccl in the measurements taken wit h it .

The main thermostat is made of three nested cylinders (Fig. 4.3): the ce11 Ii«l<ler. t lie electrical heating stage (the inner cylinder). and the water jacket ( t hc out er cylinder) .

K e have chosen a cylindrical shape for the nrhole temperature control

Chapter 4. Binary lzquid experiment 48

apparatus in order to achieve a more uniform heating of the sample al1 around. This is also an improvement on earlier thermostats, which had a parallelepipedal outer stage wit h cylindrical inner stage. In most cases, our sample cells have an elongated vertical shape? as can be observed in the picture of the sample cells in Appendis F.

The ce11 holder

The holclcr is a 160-mm ta11 copper plinder with a 65 nim base dianieter aiid it is split irito t ao halves dong its a i s . Insiclc each half are rnilled out pockcts to accorrimodate the sample cells of various shapes and sizes (see picturrs in Appendir F) and also windons through wtiich the laser beani tisecl i r i oiir iriterferorneter passes.

Spcbcial carc \vas put into milling the faces of the tno hnlves. since it is wry iriiportarit that the thernial contact betneen tliem be as good as possible. oricrl agiiiri iri an effort to achieve the niost iiniform lleat distribution arounci t I i ~ sariiplc. The two halves are tield togcttier by sis 1/4-20 stainless steel s(.rws and t t i ~ t w facc surfaces are coated witli a tliiri 1-er of Thermal Coriipo~irid (Tticrrrial Cornpoiirid. part no. 120-9. \\'akcfielcl Engineering Ill(..).

Inner cylinder (electrical heating stage)

h s t iii t t i ~ d e r of cyliriclers is the electrical heating stage. also rcferrccl t o as irrtirr c!llinder: The purpose of this stage is to provitle the fine-tiineci rriiipcrat iire rcgiiiatiori of the t herniostat to the samplc.

Ttic riiairi body of ttie inner cylinder is a 'monolith' macle froni an ap- prosiriiatdy 230-rnrn long section of -1"-dianieter copper rocl. The ce11 holder fits sriiigly insitle a hole niachinetl dong the asis of the inner cylinder. which i i h lias siric1o~i.s niilled out of its na11 for the laser beam to pass.

Tlie clcrtriciil heatirig of this stage is üchievecl by means of heating foils wappccl aroiintl the main body of ttie cylinder [XI. Current is supplied to the heating foils from the electronics in a IV-. that is described in the -T tiernial control' section of the previous chapter.

Another conipotient of the inner cylinder is its cap. also made of copper. Its piirpose is to allow to completely enclose the ce11 holder inside the inner rylinder. Iii working conditions. the ce11 holder is contained in the inner

Clrapter 4. B i n a q iiquid experiment 49

cylinder and it is surrounded by the same thickness of material both a t the cylincler wall and a t its ends.

The temperature is measured by both a quartz thermometer and a set of thermistors. Of the latter there are two in the inner cylinder cap and t hrce in the bottom of the inner cylinder (one for the control circuit and two as trniperature monitors). To ensure the temperature reading is accurate. tlic thermistors are glued into copper bolts, with high thermal conductivity eposy [XI, and the bolts are screwed into either the inner cylinder cap or the hottom of the inner cylinder.

Outer cylinder (water jacket cylinder)

Tlic. [vater-jacket cylinder is the first heating and thernial regulation stage id it will be rcferrcci to as the outer qlinder. It has a diameter of 190 i t i n i ;ml a tieight of 285 rnni. The latter two diniensions are ciictated by tlw iircd to mininiize the effect of the teniperatiire oscillations of the outer q-liritlrr ori the inrier cyliiider. In Appendis A. I have describeci how such tliiiiorisinris of t tir oiitcr c'ylinclcr are arriveci at bascd on the characteristics of t l i ~ w t c r circiilators and on the physiciil properties of the materials iisecl. Thch oiitcr cylincler is ii rollocl and welcierl 118" -thick picçc of copper platc wi t li t\vo L/Y-t hick coppcr Hanges solderecl oii ends. .A lici. also 118"-thick. is scrowed to the flanges. The watcr jacket is a 3/5" OD copper tiibirig solclorrtl onto t~otti the cylinder csternal surface and the lids. The thircl figiircb in Appcridis E illustrates a side riew of the esternnl cylinder and the plwtograptis in Appendis F show the final product. \ \ h i o w s are ciit out of thcl wall of this cylinder in correspondence of the window of the otlier c*~li~ictcrs.

As alrcarly rricritioned. the entire unit. consist ing of ce11 holder. inncr c.yliriclrr. and outcr cylinder is cnclosed iri a plywood box. which is thermally insiilatd frorn the rooni temperattire by a styrofoani lining. The cletaileci tcchriical clrau-ings of the thermostat are reporteci in Appendis E.

4.4.2 Temperature uniformity

I test aüs perforrned to ensure that the possible vertical teniperature gra- tlitmt dong the heater block nas within tolerable margins. The heater block has two sets of heating foils. as illustrated in Fig. 4.4: one set is ivrapped aroilnd the cap of the heater block onl. FIcap in Fig. 4.4. while the other.

Chapter 4 . Binary liquid experiment

. . Thermostat

Figiircl 4.4: Configuration of heat ing foi1 elemcnts around tlic heater block.

H h,,ly. rriveloprs the body of the block. proviciing a inore symmctric ciistri- I)iitioii of' hcat to the cylinder than thc cap heaters. The two sets of lieaters m i tw c*oiinrrtr(i iri srrirs or i i s d incleprndmtly. riich onc as the only source o f litliit t~ the block.

For tlic gradient test. heat was supplieil to the block by provichg rur- rcmt only to t tic cap Iieaters. t hercby recreat irig a heating contiit ion niost proric t u prodiicing a large tcmpcrature gradient. Then. t hc teniperat tire \viis iii~iwiirccl witti 4 diffcrent thermistors both a t the top anci a t the bottom o f thcl hlock. One of the therrtiistors measiirecl consistently a niiich larger tcwipcratiire diffcrence betneen top aricl bottoni tlian the otlier three anci was riot consicIerecl very reliable. Tt was later noticeïl tliat t tie cfecade rcsis- t iirirc \)os associatecl with it did not give reprociiicible rcadings. possibly dile t o tlirty contacts. Frorii the reniaining threc tliermistors. the worst tempera- t iirr gradient rneasured was 2.4 x 10-' K/m. In the final arrangements used to takc mir rneasurements. heat was delivereci to the block only via the body 1ipiitcm. Hburly. so that the heating p o w r \vas triore uniformly distributeci on t tic cylincler i d 1 and. in turn. on the sample cell. It seems reasonable to rsprct that thermal gradients in these conclitions should be much smailer. al1 t h othcr parameters being the same.

Chapter 4. Binary liguid experirnent

Time (days)

Figiirr 4.5: Tenipcrature stability test with therniistors. The black dianionds rrpribsciit thc room temperature (left hancl scalc). while thc white ones rep- rtlseiit t lie tcmpcraturc mcasurcd by one of ttic thermistors (Tth).

4.4.3 Temperature stability

Tlic tciiiperatiirc stability of the new thermostat \vas checkrd simply by mon- i toriiig tlic teniperatiire of the heater block. as nieasured by t hc ttiermistors owr a tirrie span of iip to two weeks. The resiilt is reporteci in Fig. 4.5.

\\'hile the rooni tcmperatiire increased by about 1.3"C. the temperature iiicwstirccl by the tlierrnistor reniained stable aithin 0.3 mli. During the sanie tw-wek t irnr span. the quartz thermonieter nieasiirements show an iiicrcase hy atioiit 4 riil<. This is mainly due to the following factors. On the one liiind. thc ttierniotiieter probe itself is positioned in a w l l clrilled through the cwitrc of the iieater block cap ancl partially into the top of the ce11 holcier. i t is tlierefore closer to the ambient than the thermistors are. G i ~ e n the pliysical size of the probe. a stainless steel cylinder 25-mm long and 11 mm iii chr ieter . it nould have required substantial design niodifications to place th0 probe elsewhere on the heater block. On the other hand. both due to its position and to its construction. the quartz thermometer probe has a worse thermal coupling to the block tiian the thermistors and for the same rpason it represents a heat leah from the inner to the outer stage of the t herniostat . Thirdly. the quartz thermorneter electronics box. located in the


25.4 t - -+-

n

0 25 O- -

E 8 L 24.6 1

+ t $7 1 - /

24.2 - * I . t . -:18.875

O 2 4 6 8 1 0 1 2 1 4 Time (days)

Figiirc 4.6: Ternperature stability test wit h the quartz tticrniorneter. The I)l;tck t1i;inionds rcprcsent t lie rooni tcniperat lire (left hand scale) . t lie wlii tc orlcls t tic tcrnperat lire meosiireci by the quartz t heriiionieter (TQT).

(~sp~r i~ i i c t i t roorri. is subjcct to rooni tcmperat urc variations. which coulcl 1 ~ ; i t l t o trnipcraturr niisreadings. The link betwcen thc rooni temperattire ciriat ioti aricl the quartz t herrnometer readings appears clearly from the plot i i i Fia. -1.6. Sotne photograplis of tlie tlierrnostat ancl its wrioiis parts are rcyortcxl iii Appenclis F.

4.4.4 Samples

-4s iwcli part of the apparütiis lias to be assembleci with great care to ensure iiiiisiiiiiitii acciiracy of the results. the preparation of the saniple cleserws pirticiilar attention. The two liquids must be present in the sample ce11 in tlir riglit proportions so that the mixture is a t its critical concentration. if t h t ~ critical region is to be studied accuratel?: As illustrated in Fig. 4.7. if t h esterri is at a concentration or. different from the critical concentration. O,. then. as the teniperature is raised. the amoiint of one liquid. say the l o w r one. will decrease while the other increases. until a t the coexistence cbiirrc. only one liquid will be present. In practical terms this implies that tlic nieniscus betwcen the two liquids will. in this case. travel to the bottom (to the top if Ive had started from d l r in the diagram in Fig. 4.7) of the ce11 aiid ive will Iose track of it before the system reaches the consolute point.

Chapter 4. Binary lzquid ezperiment

Figiiril -1.7: .4 qualitative cliagrani of the coexistence ciirw of a bitiary liqiiid itiisttirc~. o is tlic concentration (volume fraction) of one of tlic Iicpids.

For rsiiriiplc. if the composition of the tnistiire is in error by about 2%. at hrst i t ail1 riot be possible to get. any closer to the critical temperature than i i f ~ \ v riil<. as showri in the graph of Fig. 4.8.

Ttic liqiiids w r e piircliased froni Fisher Scientific with statecl purity of 09% for iiit robenzene iincl 95% for ri-heptane. They nere clistilled neat (the fornicr iiiitlcr reclucecl pressure) ancl brought to a piirity ofbetter tlian 99.9 %. as est iriiatecl by gas cliroinatography as well as SIIR. .\ mixture \vas preparecl i n tlic proportioris of 40.6 at% 11-heptane and 50.4 nt% nitrobenzerie. as siiggc.stc.tl by earlier studies [61] and the sample \.as prepared following a iri~tliocl (le\-isecl and used by this laboratory in the past. with rery good rcsults [T l .

Thc liqiiids in the above proportions are introduced in a pyres glass manifold to which the sample quartz cell and a pyrex bulb are attachecl (Fig. 4.9). At first. the fluids are contained in the bulb and the manifold is connected to a vacuum piimp. By repeatedly freezing, pumping, and thawing the fluids. t h air is rernoved from the manifold and, eventuall- the latter is flamed-

Chapter 4. Binary Iiquid experiment

T-T C

Figiircl 4.8: Tlic hcight. iri the ceIl. of the mcniscus betwcen the tno liqiiids ai; i i fiiiictioii o f teriiperature. for various conccritration cle~iatioris froni tlic i-rit ird roncerit r;itiori. Heiglit "0" iridicates t hc triicicile of t hc cell.

TOP VlEW

SlDE VlEW

Quartz

Figure 4.9: Illustration of the glass manifold used to prepare the binary liquid sari1 p les.

Chapter 4. Binary lzquzd experiment 55

sealed off from the vacuum pump system. A drawback of the air-evacuation process is that it alters, however slightly. the composition of the mixture. Therefore. after this stage, the manifold is brought to a temperature T > T'y at which the sample only has one phase, and part of the sample is trans- ferrecl to the quartz cell. The whole manifold is the11 immersed in a bath at a teriiperature below. but very close to. Tc, and the formation of the meniscus scparating the two phases is monitored by visual observation. Knowing the critical composition of the mixture in ternis of weight fraction. the critical volume fraction is Foiincl (iincler the assumption t hat the mising volume is iicgligi ble-sec section 4 - 5 2 ) and therefore the expected position of the menis- ~ i s at the critical volume fraction is marked on the sample ceil to aid visual ot)scrt.atiori. If it is observed t hat the menisciis does not fortn at the .*critical iiiark". part of the top. i.e.. lighter. component is added or removecl to bring t h riienisciis clown or up. respectively. When le are sntisfied that we have a critically fiIlcd sarnple. we Hame-seal it off from the manifold. Ttie saniples iiro t ticrrkve preparecl üt the vapoiir pressure of thc mixture.

Srwnd ilifFcrriit saniple cells were prepared from the sanic rrianifold. In orilcr to ot~scrw 110th the rcgion far from the critical point aiid the ncar - (.ritical rcgiori witti relative ceris. the sanipk cells were chosen to have light patlis of 1. 2 . 5 . and 10 riirn. (The tolerances of the light paths are 0.01 mm.) Alore prrcisely. froni the ticscription of the focal ancl image plane techniques. t I i i b tiiirriber of fringes recorded at each teniperatiire decreases as t hc crit ical t clnipcrat lire is approacliecl. wtiereas the crror made in counting the fringcs is prxtically coustant. Hence. in an effort to reduce the relative error on the fririgc coiint on the data near the critical region. ahich are iisually more sigiiifi(mt. saniplru with longer light paths ( 5 and 10 mm) are iisecl in that rcgioti. -1s diita arc taken farther and farther from critical. the nuniber of fringrs at each clatiim increases greatly for the sanie ligtit path (2 300 at T, - T 2 0.1 1\: aith the 10 mm cell). In the temperature regioii farther froni T, t lir. 2 and 1 i i i r r i liglit path sample cells are employed.

4.5 Results

The data for the study of the coesistence cun-e of the mixture n-heptane+ni- t robenzene were taken over the course of several years wit h both the Focal plane and the image plane techniques. While the former was used in the past to stucly other bina- mixtures. the latter technique has been employed for


this sort of nieasurements for the 6rst time. The image plane data were very iiseful in deterrniriirig the importance of some potential sources of systematic errors.

4.5.1 Focal plane data for the order parameter

The collected coexistence curve da ta span 5 decades in terms of the reduced tenipcrature t . froni t 2 4 x IO-' to t = 4 x IO-'. From the consulted literatiirc it would appear as though most esperimenters do not estend tlieir nieasiirenierits as close to the critical temperature as in t his work [30. 62: 53. C3. 64. 7. 41. 6.5. 29. 661.

Data taking in the binary mixture esperiment is relatively slow. with tlir rate avcraging at about one experimental point eïery two t lqs . This is iiiostly duc to the ccpilibration tirne requirecl by the systeni to arrive at a s t cwly coricerit rat ion profile after a tempcrat lire change. especially for t lie data \.cary c:losc to the critical temperature. This point is trcatecl in cletail I)c~lowv i r i ;i separatc subsectiori. The tirne lirie follo\~etl when taking data is sIiowii iri Fig. 4.10. Thc photograph is a typical set of fringcs frorii a Fociil planc rspcrinicrital ruri and it represents one csperirnental claturn. The ([lot to S C B ~ ) diagram iii Fig. 4.10 inclicates thüt the systeni is prepared at it tcriiperaturc. Tl. belorv. but as close as possible to. Tc aiid left there for several Iioiirs or clqs to alIo\v it to reach equilit~rium. The temperature is tlien iricreasccf to t a k e t lie systeni abovc the critical temperature and let diffusion I~ct~vecw thc two species take place. As tliis happens. aricl as explainecl in Cli;ipt~r 3 . t lit> Fraunhofer diffraction pattern gerieratecl by t hc He-Se laser light iiit cract irig wi t li t tie changing index of refract iori profile is recortled on filrii. iiii osample of ivhich is iri Fig. 4.10 [il. The status of the system can I I C risiially tiionitorecl to ensure that when the first. ttiicker fringe iias been rc~rorclcd on the filni the datum has been taken and the temperature for the rirst orir can be prepared. The system is then brought to a temperature. T2 < Tl. itrid the procedure to take another point is repeated for as rnany data as are ncccssary to have a satisfactory coexistence cune . In the case of t lie rriist lire n-lieptanefnit robenzene. the data were taken so as to span the ;icccssit>le range of temperatures of the misture. The temperature range for this system is around 12.3"C and it is given by the difference betneen the cri tical consolute temperat ure and the freezing temperature of the species n-hcptanc.

time-

Figiirr 4.10: Top: photograph of a typical focal plane film rcpresenting a da- tiirii takeri ut a particular teniperature T l . indicated in the bottom diagrani. T l i ~ niinihcr o f fririges that can be counted froni the top one clown is pro- port ional to tlir clifference betrveen the refractive indices of the two phases. Bot torii: illtistration of the time line followed during the datum shown above.


Figiiro -4.1 1: The coesisteiice curw for the binnry mixture ri-

h c y t aric+riit rot)ciizcnc plottcd as the wliirne fraction l o i r as a function of ttir rctiiicctl teniperatiire (witli the XYCS swappal as it is often dorie to plot cwesistcnçc curvcs). The insct is an eiilargernent of the critical region of t lict c:urw.

T h c:ocsistence curw as i t is revealecl by the data froni sis data sets is stiosri in Fig. 4.1 1. Saturally. it is orily hiilf of the whole coexistence curve. siiiro it is ;t direct nieasure of the ordcr parameter. 10~~. namely the difler- r r l c : in thc volilme fraction of one of the species (n-heptane) between the iippikr aiid lowr pliases. It is plotteci as a functiori of the rcduced ternpera- turc t. iristeatl of the absolute temperattire T. in order to be able to report (la t i i froni several sets wi t h slight 1- different cri t ical ternperat tires. It will be disciissed belou* hiov the different critical temperature "problem" arises.

It skioiild bc recalled frorn Chapter 2 that. in the vicinity of the critical tetriperat ure. t lie coesistence curve is supposed to be described by the simple scalirig law:

M = &tJ (-4.2)

witli t = (Tc - T)/T,. The -vicinity7 to Tc: or t = O-the so-called asymp totic region-has generally been found to be the region with t < IO-' for t~inary liquid mixtures [27]. Translating this to the particular case of n-

Chapter 4. Binary lzquid experiment

Table 4.1: Parameter values for a nonlinear least square best fit of Ag = Botd to the volume fraction data of Fig. 4.11. Quantities in brackets were held fisecl cliiring the fit.

Fit Region BO 3 Y-

ticptarie+riit rot>cnzcne. the region where the simple scalirig law sboulcl be valid rstcnds for aboii t 3 clegrees below Tc. It should be emphasized. how- i w r . ttiat the asyniptotic region is not known a priori. nor do we h t m a thorctical cstiniate of it. 1Ioreover. in certain circumstances experimcnts oii t~iiikiry niistiircs have indicated that the actual iisyrnptotic region extends orily to t 2 10-y' fronl the critical temperatlire [X. 67. 291. In vien of siich fiiicliiigs. iii t he arialysis of the present clata a narrower asyrnptotic region O < t < 10-" \vils chosen to fiiicl the critical temperature and the order pa- rairictt r r rritical csponent 3. Beond the asyniptotic region. corrections to sciilirig ronic! iiito play:

B w u s c it is physically inaccessible. the critical ternperatiire Tc miist be i i i f c w w i Frotii the measiired coexistence curw clata. A first estimate of Tc is gatlicrd t>y plotting the raw data as (10)''~ versus T in the (espccted) ;isyriiptotic terriperature range. where such a plot is liiiear. -4 liriear fit to t l i ~ data intercepts the Iiorizontal axis at (a first estimate of) Tc. This value o f t l i ~ critical trniperatiire is taken as an initial value for a nonlinear least sqiiartl fit of relation (4.2) to the coexistence curve data. In the fit. 3. Bo. ancl T, are iisecl as free parameten. Tc is only allowed to Vary wit hin the rriisonable range siiggested by a careful esamination of the coesistence curve ritwr the critical point. The best values of the parameters found by the rioniincar least square fit are given in table 4.1. -4s is apparent from the table. the fit obtained with 5 set at 9 = 0.326 (the theoretical value) does not seem to represent the data as well as the one where ,3 is unronstrained. lloreover. the best value of the exponent is found to be higher than espected at J = 0.367 f 0.002.

Chapter 4. Bznary liquid expetintent 60

Within each set of order parameter data the temperature was measured using the same thermistor, but different thermistors or other parts of the thermal control electronics were used in different sets. ,CIoreover7 as mentionecl below in section 4.4.4' different samples were used to collect the body of data for t tiis csperiment.

Systematic experimental studies [4, 51 on the effect of water and acetone inipuri t ies in clifferent binary liquid samples (met hanol+cyclohesane) show t h a percentage volume of water of about 0.1 in the mixture rvould alter the critical temperature by almost +4 K. rvhile 0.5% of acetone gives a in- crcasc of about 2 K. S o regular variation of the critical temperature with the diffvrrnt sample \vas fotincl ancl the ciifferent absolute values measured are al1 witliiri a fraction of a percent of one another. \+'hile it is reasonable to think tliat both watcr and acetone impurities are preseiit in the samplcs (the glus riiiiriifolds arc clcaned with both cListillec1 water and acetone hefore niaking tlic sariiplcs as clescrihed in section 4.4.4, it is presumcd that the anioiiiits o f tkiosc iiripuritics do not cliffer miich frorn one saniple to the 0 t h .

For t l i ~ s ~ rcasoiis. the actual critical corisoliite teniperat ure of f i litl~>tarit~+riitrol)~rizeiie varicd somewhat. \VMe For cach chta set it c m be d~wrrriinrrl to nithiri less than 0.5 r d < . its absoliite vaIiie c m only bc giveri as T,. = 291.7s K rt 0.03%(= 18.63"C).

Oricc a rrasoriably good estiniate of Tc has becn obtaineci. a log-log plot of ttir data in the fiil1 teniperature range is usefiil to see if arîy correction -

r o scalirig terins woiilcl bc necclecl. and shether they stiould be positive or iiogatiw. to interpret the data. Also. the dope of the data in the asyniptotic ri>giori in t l i ~ log-log plot corresponcls to the criticül esponent. A log-log plot for the data of Fig. 4.1 1 is reported iri Fig. 4.19. The sliglitly clecreüsing slopr at higticr d u e s of t indicates tliat some correction to the simple scaling 1;in is necdcd ancl that it will have CO have a negative coefficient. This (+ail I)e srcii by prrforniing a nonlinear least square fit of the scalirig laa iri

rclli\ti«ri (-1.3) to the cocsistence ciirve data in the whole range of temperature stiiclird. by iising the best values foiincl in the asymptotic range for the critical tmprrature. Tc. arid the cri tical exponent. 3. and leaving the coefficient BI frcr. wliile the correction-to-scaling esponent 1 is held fised at its theoretical value of 0.54 [6S. 691. The line througti the data in Fig. 4.1 1 corresponcls to t tiis fit ancl the parameters determinecl by the fit are reported in table 4.2. fit D.

-4 fit like C is useful in order to estimate how important correction terms would be in fitting the data. If the coefficient Bo and the exponent 3 of this

Chapter 4. Binary liquid experirnent

0.326 slope (t heoiy )

J

Figiirc 4.12: Log -log plot of the orcler paranieter hl[ versils t h reclucetl tritiprriitiirc t . Ttic siope of the ciirve ils t tends to zero corresporids to tlic wpoiitbrit .i. -4 liiie \vit Ii dope 0.326 is also (irawn for cornparisori.

1 4.2: Parameter values for a noniinear least square best fit of 10 = BottJ. w i t h t' = (T - Tc) /T (fit C) and Ao = BotJ(l + BltA + &tu) (fits D. E. F ) to the voliinie fractioii data of Fig. 4.1 1. Quantities in brackcts werc ticld fiscd during the fit.

Fit R~giori A BA. Bo BI J

i- C t l 0.367 I 0.002 BA = 1.9 k 0.04 O 1.3 D tC0.04 (0.367) 1.905 k 0.007 -0.96 f 0.03 4.8 E t50.04 (O. 326) 1.39 0.01 0 1 k 0 . 0 18.4 F 150.04 0.361 I0.002 1.82t0.03 -0.81 k 0.06 4.3

Chapter 4. Binary liquid experirnent 62

fit were to differ appreciably from Ba and found through fit B, it would be plausible to suspect that correction terms would have to be considered. Ho\vevero the d u e s of the parameter found for fits B and C are the sanie aitliin esperimental error. It is evident from the best fit parameters of fits D, E. and F that the scaling law with correctiori terms works better on the data wlien 3 is the value found by the simple scaling law used in the asymptotic regiori (D) t han when the theoretical value of ,d is imposed during the fit (E) or when the esponent is again treated as a free paranieter in the fit (F).

A differeiit way to perform the analysis of the data and arguably the ulti- iiliittl wri ficat ion of the importance of correct ion tcrms is achieved by plot t ing tlir data iri ii mariner that separates the contribution of the corrections ternis frorii t hc rest of t lie ternis in t tie relation (4.3). The correct ion-tcrm-sensiti~e forrii uf equation (4.3) iised is:

if rio c*orrcc*tiori ternis are riecded to fit the data and the value of the critical c>spoiiriit is .correct'. a plot of l o g ( ~ o l i / t J ) vcrsiis log t. which I will d e r to ils s c ~ ~ s i t i o e plut. \voiiid distribute the data dong a horizontal line. Departurrs froiii ;i zcro-slope line would then iridicate either an 'incorrect' valiie of .j o r t h . iiercl of correction terrils to fit the data (or botli!). Fig. 4.13 is a scwit il-e plot of t lie coesistencc tlata collected during the esperitnent with I L ticptiiiic+riit robcrizene. wherc t tie t heoretical valiie of .j = 0.326 \vas iised. It is cvicht once again that a value of .i largcr tlian the theoretical vdiie is ricccssap to -flattenW the dope of the data. This type of graph can b~ iisc~tl to pwforni a cross check on tlie values of the criticul tenipcratiire Tc iiri(1 . j prodiicrd hy a nonlinear least sqtiare fit to the raw coexistence data of Fig. -4. i 1. Starting froni some 'gooc1 guesses' for Tc aricl .i. one can then vary racli of tlieni iridividually step-by-step until the data of the sensitim plot lies on a tiorizontal line. The critical temperature will only affect the (lata ver? close to t = O. while changes in 3 will change the overall dope of tlir tlata. The values of the critical temperature and the critical esponent protlurecl bu the esperiment will then be the values that make a zero-slope plot. S loreover. if after t his stepwise analysis the data appcar distributed dong tliffererit dopes at different ranges of t. this would be an indication t bat correction ternis to the simple scaling law should be included in the fit.

The nioclern theory of critical phenornena being as widely accepted and siiccessful as it is. it seemed worthwhile to try some fits to the data by using

Chap ter 4. Binary liquid ezperimen t

Figiiro 4.13: Sensitive log-log plot. hlr / t " versus t . of the coexistence data on 11-licptarir+nitrobenzene. The talue iised for the critical espoilerit is .i = 0.326. A scnsitivit- scale is also tlrawn in the graph to indicate the slope t lie data \votiItl preferentially take. were t hey plot tetl \vit h the valuc of the esponent corresponcling to the indicatetl slope. The size of the error bars is coniparablc to the (lata scatter.

Chapter 4. B i n a q liquid ezperirnent

Table 4.3: Parameter values for a nonlinear least square best fit of Ad = Bot3 (1 + ~ , t ' + B?P) (fit G), A@ = Bot3 (1 + Bl tA + ~ ? t ' - ~ ) (fit H)? and Ad = B& (1 + BltA + ~;t"" (fit 1) to the volume fraction data of Fig. 4.11. Quantities in brackets were held fixed tluring the fit.

Fit Region $3 Bo B1 B ? B ; , B ; x2 C t50.04 (0.326) 1.33 k 0.01 1.5 f 0.3 -7 f. 1 12.0 H t50.04 (0.396) 1.32 k 0.01 2.3 rt 0.4 -6 5 1 11.1 1 t50.04 (0.396) 1.295 rt 0.015 i I l -9 k 1 9.6

t hr t lirorrtical value of 3 ancl aclding t v o correction ternis in the scaling law ( rrlat ioti (4.3)). I t Iiiu hceri suggested t hat clifferent esponents For the seconcl iwrrcction terni cari be tried when analyzing coesistence clata [;O. 27. 661. Fiirtticr fits to oilr coesistence data werc then tricci with the sccorid correctiori tc~iiis Iwiiig: . . . + B2tZA. with 1 = 0.54 as alreaciy rnentionecl. + B(,tl-".

Il w- w liorr! (i = O. 1 1 is the specific heat critical esponent. ancl . - + B2 t - a i t li (siipposrdl~~) .J' = 0.396. The ratioriale behincl keeping the choice of thc soi.oiid trrrti open is basctl on the esperiniental clifficulty of distinguishing tirtwwti a correction esponent thüt is slightly lürger ( 2 1 ) or snialler (1 - n or 2. j ) ttiari onr. The results of these fits arc reportcd in table -4.3.

A different binary liquid

Altlioiigti t h ability of this niethocl to nieasure the coesistencc curw of a I)iri;try liquicl rriist tire lias been acceptecl [Tl. it rvas necessary to re-wrify t hat tliis iiricspected result for the value of 3 \vas not causecl by an inherent. but so fiir iindiscovered Haw of the rnethod. To this end the coesistence curve of iiti~tlicr rriisturr was nieasured to get an estimate of 3 from another systeni. r'siiig esact ly the same apparatus (thermostat and t hermometers). the same rspcririiental rnet hod (focal plane rnethocl-Chapter 3) and tlie same data aiialysis approach as clescribed above. the coesistence cun-e of the binary mis t tire çyclohesane+aniline (later C+A) \vas nieasured. The results are stionn in Fig. -1.14 and Fig. 4.15. the (half) coesistence curve and the cor- rrction sensitive plot. respectively. The measured d u e of' orcler parameter critical esponent is 3 = 0.330 f 0.010. The available data for C+A are much


F i 4.14: Ttir cocsisteticr curve for the biiiary niisturc cyclohes- ;iric~+;liiiliiic! plotted as the volutiic fraction An as a fiincrion of the absolute t v r t i p m i t WP.

Figure 4.15: Sensitive log-log plot. h / t J versus t. of the coexistence data on cyrlotiesane+aniline. The data is plotted using a critical esponent 5 = 0.330. Tlic sensitivity scale drawn in the graph indicates the slope of the data with .i = 0.3.50 would definitely show up in this type of graph. The size of the crror bars is comparable to the data scatter.

Chapter 4. Bina? liquid ezperàment 66

less than those for H+X, hence the larger error estimate on In spite of this drawback, the data in the sensitive plot (Fig. 4.15) show quite clearly that the niethod can clistinguish between 3 = 0.330 and ,O = 0.330.

To summarize this section of the data analysis. it is observed that the rtieasiired critical esponent for the order parameter of the binary liquid misturc r~-heptane+nitrobenzene is 3 = 0.367 i 0.006. This value is higher than the t hcoretically predicted value of 3 = 0.326 rt 0.002. The coexistence curve is wcll ciescribecl by the scaling law of equation (-4.3) with one correction tcrrns. tvhile aclding estra correction terni cloes not iniprove on the fits nor tloc.s it seem statistically justifiable.

Dwausc of the iinespecteci result. clespite the trust iri the esperiiriental riictthod rhat lias prociiiced notable resiilts iti the p u t (sec. for esample. ttic. papcrs in [7. 9. 71. 60. 72. S. 731 to namc a ka). it is of paraniount itiiportaiicc to corisitier carcfully al! ttic potetitial sources of systctriatic errors tliiit liirk I~ohiiid csperinients at the critical point of Riiicls. This is the siibject of thcl iicst section.

4.5.2 Sources of systematic errors

Ttic potiwtially tiiiirriftil effccts of tempcratiirc graclients and hacl thcrrrial staldity in clsperirricnt on crit ical phenoniena harc heeri describcd ici earlier s t ~ t i«iis. Howwr. t here are ot her known soiirces of syster~iatic errors t hat c*orist itiitc <lmgeroiis pitfalls for t hesc esperinients. Serious at tentiori niust I )ib paid t« t lie effwt of gravitationally incliicerl concentration graclierits. t tic rqiiilil~rat iori t inic of a binary rnistiire. the possihle prescnce OF wctting of tlio satiiplc ce11 d l s by one of the phases. ancl the definition of the orcler pa ra r~ i~ t ~ r .

The effect of gravity

Lrirler t lie iiiflrience of Eart h's gravitat ional field the refractive indcs vertical profila uf a niistiire below its consolute critical teniperature is in principle distorted froni a simple to a *crookecl' step-function as stiown in Fig. 4.16 (31. -42. 741. It is conceivable that a profile as in Fig. 4.16b woulcl cause the appearance of diffraction fririges even when the system is at a teniperature T < Tc. if the distortion from a step-like profile as in Fig. 4.16a is appreciable. Howerer. no uiivanted' fringes were ever detected in any of the esperimental data sets. This can also be observed from Fig. 4.10 (top), where it can be

Chapter 4. Binary l i p i d experiment

Figiirr -4.16: Qiialitatiw picture of t hc

b

refractiw incles profile of a binary rriistiirr as a furiction of ceIl height at T < Tc: (a) icleal case: (1)) distorted hy Eart 11's gravitational field.

Chapter 4. Binary liquid ezperiment

55 O 2 4 6 8 10

fringe number

F i 4 . 7 : Sleasiirernent of the refractive indes profile at a temperatiirc T > Tc lifter the sarnple was shaken to homogenizc the phase.

swri t h thc filrri lias not recorclctl ariy fringes before the tcrnpcrature u s raisod a imv~ Tc.

To stiidy tliis point iri inore clepth. the image plarie rriettiod (Cliaptcr 3 . sc3rtiori 3.3.3) was ernployed bccaiise of its ability to map *tlirectly' the prof le of thp irides of refrnctiori. By coiinting the fringes as a fiinction of tlieir positiori oti the filni (the latter being relatccl to the height in the saniplr c - r b l l ) . the profile vari be plotted. as Figs. -4.17 ancl 4.18 illustrate. Profile rr~oilstIrcnlCnts w r c taken at a temperatiire abow Tc. after the sample was stiaketi antl tlierefor~ i n a situation where a Iiomogeneous indes of refraction is c*sprctrd (Fig. 4.17) antl below Tc. both shortly after ancl man- tiours after t h terriperatiire \vas lowered. shoivn in Fig. 4.18a antl 4.ISb. respectiwly. To rrisiirc tliat ttiis method is sensitive enough to reproduce a signioidal profile if oiic is present. a measurerncnt s a s macle of the profile after the ~ t i i p r r a t Lire \vas raised frorn bclow to above Tc. biit wit hout shaking the siiriiplr to Iiorriogenize the phase. In this situation a profile like Fig. 3.ïb is i~sprctrd. aiid is qiiite clearly measured by t his niethod as is apparent frotn Fig. 4.19.

Froni t tie rneasured profiles above ancl below the crit ical point. t here seems to be no cvidence of a dramatic deviation of the index of refraction from a lioniogeneous beliaviour. both in the one-phase and the two-phase regions.

This optical met hod has not been ernployed before to probe the refractive

O 5 10 15 20 fringe number

Figiirïl -1.18: IIeasurenient of the refractive incles profile at a temperature T < Tc aftcr the sarnple was tiiken from slightly abow to slightly belon r i t 1 . Black ciianiontis: about 30 niiiiiitcs aftcr t hc tcniperat lire change: wtii tv diairio ri&: about 50 haiirs iater.

i i i ( i [ ~ s profilr aricl. althoiigli its perfoririaricc riccds refinenient. it lias provcri w r y suitahlc for t licse measurenients.

Chvi t at ional cffects arc siipposecl to he more evideiit t lie larger t lie derisity tliffest~rice hctween the two species of the niisture [Z]. The densities of r l Iictptaiic aiitl nitrot>enzerie (sec Appendis B). wit h a ratio of ps/prr = 1.76.

i i i ~ t as closely niatchcd as those of ottier compounds [29]. however a stiiciy ciri iiri cveii triore clensity-mismatched niisture hiis revealed no influence of gr;ivitat iurial effect on the nieasurernents [jg].

Giwri the resiilts of oiir observatioris anci of tliosc rcported in the lit- witiirc. tlic infliiericc of the gravitational field on the irieasurenients of the ro(1sistrricr cu rw of H+S can be neglected.

Equilibrat ion

.-\ vcry important issue to address. as sereral iluthors have pointed out in t lie past [XI. is that of the tirne a binary misture takes to achiela equilib- riiirii. As a consequence of the physicnl properties of the critical point. the quilibration time gets longer the closer to Tc the teniperature is set. This is iiri<ierstandable. for esample. by recalling that the specific heat of the system diverges at the critical point. Howevei. more important in the determination

Chapter 4. Binary liquzd experiment

O 20 40 60 80 1 O0 fringe number

Figure 4.19: Measiiremcnt of the refractive index profile at a temperature T > Tc aRcr the saniple \vas heatecl froni beloa to abom critical. The s;iitiplr l ias not shakcn to speecl up phase honiogeneity as in Fig. 4.18.

o f t l i c b rqliilihratiori tiitie is the slowing clown of the niiltual cliffusioti lietwec~ri t t i ~ spccics as t lie cri t ical tcniperat ure is approached. The eqiiili hrat ion t inie issiic! is one of t h rcasoris one needs a very stable thcrnial control systcni. a s IWS rrientionetl earlier in this chnpter. It is ver' iniportant. thcreforc. to w i t a long enoiigh time for equilibriiirn to have occurrecl after a particular tcmipcrittiire has hccn set in the thermostat. An estimate of the diffusion co- dfiricrit for the ri- hept arie-nit robenme mist ure from the csperimentd data l i i i ~ !-irlclcd ;t value of about 7 x IO-l' ni's-l at l e s than LO-" I< from thc cari tical tmipcrat lire. This estimate is comparable to the coefficient foiincl ir i diffusion st udies on the nicotine+water system [56j. where the cells wcrc itlrrit ical to tlir cclls iiscd in this esperirnent.

Tlir iriiagc plane rnethod useci to measure the incles of refrection profiles i m iilso be iiseci to cstimate the ecpilibration time of the system. after a tcriiperature diange. This is clone by monitoring the interference fringes at t liv iniage plane of the focusing lens (Fig. 3.10 of Chapter 3) with time as the f ih i movrs in the camera. For our purposes. equilibriiim has been reacliecl whrn the fringes appear horizontal on the film. The film is usually marked at regiilar intervals to keep track of time on it.

Iri the niost significative measurement performed. the temperature tvas set at a value T, slightly above Tc and held there for about two hours. it aas

Chapter 4. Binary lzquid experiment

Laser bearn

/ meniscus

U .+UPP~~ phase

Lower phase

Figrirc 4.20: Lowcr pliase of a binary liquicl rnisture wetting tlie saniplc cc11 \~iills and siirroiiriclitig the iipper phase completely

tlirii cIccrcasctl to a tempmmire Tf below Tc. siicti tiiat 7', - Tf 55 x IO-" K. Ttitt systcrii iras kept at tliis tcrnperatiire and thc fringcs rccorclcd on film For sri-cral cl--S. \\'hile the chart recorcler track of the tlierniistor oiitpiits stio~v~id tliat Tf \vas reactiecl about 20 miiiiites after it a a s set. the fringes on ttir filrii do riot appcar to flatteri to a horizontal slope until about 30 Iioiirs latcr.

T h cmsistetice riirw data close to Tc nere al1 taken aftcr aii eqiiilibratioii tirtic. of iit~oiit 50 hours. ahile for the rest of tlic chta the equilibriuni tinie ttllowrirc \vas bctweeii 10 iirid 20 hoiirs. N'itli t liis rrieasurement ancl t hc iliff\isiori constant est iniatc nicntionecl abovc. it is fairlu safe to say t hat the issiir of cqiiilibratiori pas aell looked nfter. To ni' knocvleclge the image pliiricl nieasurenicnts have not been iisetl iri ttiis fasliion before in binary iiiist mas aricl constitute a n interesting iiivestigation tool wherever optical iritwiircrrients can be applied.

Wetting

Tticbrcl is evitlcnce that in binary mixtures one of the phases wets the walls of t lie ceil containing the sample. sometimes surrounding the ot her phase coniplctely approsimately as Fig. -1.20 shows [76. 771- The effect of this ptienonienon in the esperimental conditions of this n-ork is that the effec- tire t hickness of the light path the laser traverses inside the ce11 is different

Chapter 4. Binary liquid experiment 72

from the nominal thickness given by the manufacturer. Since the index of refraction. and hence the order parameter, is measured by the number of fririgcs detected at each temperature divided by the light path length. if the latter is ülterecl by the presence of a wetting film, the consequent refractive intles measurements will be affectecl and rendered wrong, unless the precise thickiiess of the wt t ing film is knoan.

By comparing refractive index measurements taken ai th cells of different thickncsses. the effect of a possible wetting layer can be monitored. -\ssuming t Iic wtting 1-er that lorms has the same thickness independent of the nominal ligtit path of the cell, if the data taken with different cells overlap, within cqxrinieiital error. the influence of the wetting film is negligible. The cocs- i s t ~ ~ i w ciirve nieastirements aere performed with different ce11 thicknesses to hr a t h to tiiap the whole range of t a rdab le for n-heptanefnitrobenzene with rqiial ease aricl acciiracy. Those nieasurements are reported in Figs. 4.11 ;ml -4.1:3 aricl i t c m be observcd that the ciata indeed al1 st!ern to follow the siinir pattern. To cniphitsizc this. 1 have selectecl two sets of data taken with ii .? rw11 tirid a 10 nini cc11 itnd reportecl in Fig. 4.21. \\'ithin the acccinicy of t h iticasureiiierits the data overlap. iriclicating that if il wtting film is prcwnt. it is. liuwwr. urimeasiirable in t hese csperiments.

Iii any case. tliere arc indications from other obscrvatioris [33. 4. 31 and t h m t i r a l preclic:t ions [781 t hat wet ting behaviocir. iilthough present at al1 ttliiiptbratiires. iisiially beconies measurable at temperatures several degrecs 1)rlow the critical terriperature. a region of the coesistence c i i m that one is I~ss <:orirernrt& with wlien determining the critical espoiient J.

Correct order parameter

A s clcscribcci abovc. ttic indes of refractiori is used to cstract the voliinie fraction inforniatiori to determine the coexistence curie of a niistiire. In clo- iiig this ttirec assumptions are niade. The first is tliat the indes OF refractiori (lors not present ariy anomal. at the critical point. There are theoretical pr<dict ions [;O] and esperimental obsemat ions [SOI deüling wit h t his issue. tmth showirig thiit any anomaly in the refractive index is below 100 ppm. lcss tliati the resoliition of these esperiments.

The second assumption made is that of zero niixing mlurne. In other \vorcls. ahen the two individual species. -4 and B. are joined together to forrn the rnisture it is assumed that their volumes. \:A and lb : simply add. while in general i t is to be espected t hat l hiXture = + I I 114 ahere \ E is called

Chapter 4. Bina y liquid experiment

Figiiri. 4.2 1 : Sensitive log-log plot. Amrr/tJ versiis t. of t lie coexistence data on n lieptanc+tiitrobenzene taken with a %-mm ce11 (black cliamoncis) aiid a 10 riirii ce11 (white diarnonda) to stiidy the effect of a wt t ing film on the data. (Tlir value iisecl for the critical exponent is J' = 0.326 and. as in analogous griiptis. tlic data scatter and the size of the error bars art. comparable.)

Chapteî 4. Binary liquid expen'menl

the esccss volume and can be positive or negative. Experimental studies on two systems 162,811 where C i # O have shown t hat this had no consequences on the measured critical exponent 0. The first study has found that the critical aniplitiide Bo remained the same within 3 to 4%, while the second rccorcled an increase of up to 45% in Bo when the nonzero mixing volume is takeri irito account.

Thirclly, it is assumeci that the Lorentz-Lorenz relation remains valid for hinary riiistures to the sanie extent as it valicl for pure fl uidso for which it Iiolds nitliin about 1% Accorcling to an experiment aimed a t determining the ralidity of this assumption (621. the Lorentz-Lorenz relation is verified nitliin 0.5%. when the volume loss upon miring is considered.

Al t hoiigli no riieasurements were carried out on n-heptane-tnit robenzeiie to wrify the abow assurnptions. 1 have assumed tliat they are wlid for the rrsiilts obtained with this system.

Chapter 5

Pure fluid experiment

5.1 Introduction Th iywrirnental stiiciy of the critical region of the polar piire fliiid 1.1- rlitiiiorocthylerie (C2H2F2) is prcsented in this chapter. The riieasurenients w r o cï~rriccl mit iisi rig two tliffcrrnt apparat us: the prisiri ccll. alreacly dr- sc*rilwcl in Chaptcr 3 and the conibiriecl cell. clescribed in t his cliapter.

Tlic data ori the Lorentz-Lorenz function of H2F2 arc reported iri the first scct ion. Section 5.3 giws ari overview of where tliese trieasurernents fit in. Scct ion 5.4 lias csperimental detüils or1 the combineci ce11 euperimeiit and t h procccliircs follo~ved in collecting the data. Tlic resiilts are reportcd in WC t ion 5 . .5.

5.2 Order parameter and Lorentz-Lorenz re- lat ion

It tras c~splaitied in Chnpter 3 lion the coesistence curve of pure fluids çan h~ st iiclird nit h optical techniques. wit h very similar procedures to t hose ticscribeci in the prerious chapter for the case of binary liquid mixtures. It is actiially with the study of pure fluids (in particular senon) that the focal plane teclinique uras introduced in critical phenornena esperinientation (8. 91.

Recalling a few ideas from Chapter 3. the choice of order parameter to stiitly pure fluids is usually more restricted than it is for binary fluids. Both t lie clifference in niolar volume and in either mass or number density between

Chapter 5. Pure fluid ezperirnent

the liquid and vapour phases could be used as pure-fluid order parameter. Once again. however, the quantity that yields a more syrnmetric coexistence ciirw is to be preferred, which biases the choice toward the density difference. iisiially normalized to the critical density of the systenio p, 1-18, £321:

The relation becween the incles of refraction and the order parameter in t h pure Ruid case is givcii by the Lorentz-Lorenz (LL) relation:

w ~ i c w p is tlic iiiiiss clensitu of t tic fiiiicl and the quantity C is the LL coefficieiit or L L fiitiction. weakly (Icpenclent on p [IO]. There is no cviclence of any (liy~~riclciice oF L on T. The Lorentz-Lorenz relation is eniployd to obtaiii the ~iiliio o f the tlcnsity of the fir i icl frorn rriciisurcnients of the iricles of refraction pclrforiiicrl \vit ti the focal plane or the iniagc plaric tcchnicltics.

Tlir corfficiciit C nec& to bc cletcrminetl acciiratrly beforc it rail br iisecl i i i c~lii i i t ioii (5.1 ). This is <lotie by trieasriring the density. p. and t tic indes of rc4'rwtioii. ri. iiidcpericlcntly witli t h prisni rell trchiiicpe.

5.3 Motivation Foi.;il plaritl and prisni ceIl coesistcnw ciirw nicasiircnieiits on 1.1 -cliAuo- rocltliylciie acrc carriecl out to test a new type of csperiniental apparatiis iIwc~lopcd iii oiir lahoratory. This substance was chosen both for its easily m*cwit~ic crit ical teniperat ure ancl becaiise no acçurate data on its critical rvgion aiicl LL coefficient seem to bc awilable.

A rirw piecc of apparattis nas conccived and constructecl a f e n p a r s ago iv i t l i t lie purpose of niapping the whole P\'T space of a fluid [83]. One of the ownt iial goals of t his apparatus is the measiirement of the critical esponent 6. gowrning the crit ical isot herni. and t lie relative critical amplitude Do (see C hapt cr 1). Conibined wit h coesistence curve and compressibility measure- riients. which yielcl values of the critical esponent 7 and the amplitude ro. these data can then be used to test universal critical amplitude ratios. and scaling laws (36. 84.83. 861. The idea is to have an esperiment that combines

Chupter 5. Pure flvid experiment

the three techniques used (the prism tell? the focal plane and image plane) to- gethcr. so that different types of measurement can be performed on the same saniple. Due to the effect of gravitational rounding of the indes of refraction profile-this effect is more rnarked in pure fluids than in binary mixtures-. the prism ce11 technique alloas the coexistence curve to be measured only to about 3 mK of Tc ( t = 10-'): mhile with the focal plane technique one cilri estencl the range by almost anotlier decade in terms of t . to t = Ori ttie other hand, prism ce11 measurements can be used for measurements pcrforiried far froni Tc. However. when the measurements are performed on s;iniplm contained in different cells for different esperinients. the critical teni- pclr;itiirrs of tlie data sets n i q cliffer enough to niakc it impossible to compare t liciii riieaiiingfiilly withoiit adjustmerit of the data. .Ut hough tliis apparat us lias alrcacly proclucetl some data 183). however. a direct coniparisori of prisni rt.11 arid focal plane data had yet to be performecl.

\\*itli th[) combiiietl ce11 the influence of potcntial rvet ting effects on tlic tiic.;il plane (aiid itnage plane) cell \i.alls can be stiidird. as tlic tlata is taken ..in pariillcl" with prisrri ceil tlata. whirh should not suffcr froni wetting cffects siiiw it is I>asc!tl or1 the detectiori of tlic liorizorital cleviatioii of the light by t ho fliiid.

5.4 Experimental apparat us

5.4.1 Combined ce11

--\ (Irawirig of the combincd cell. iised for sonie of the pure Hiiicl nieasurenients. is sliown in Fig. 5.1. Lt is built from an aluniinurn. square cross section piirelleltyiped of climcn~ioris (101.1 x 44.45 x 44.45) m r d . Two charribers cvere iwat cd t>\- rtiilling clifferent recesses for the ce11 winclows and flariges. On one c.1iainbcr--which 1 will refcr to as the flat chamber-the windows are mountecl piirdlr4 to cadi ot her. the gap between t iie windoas irieasures 2.16 nim and it is hiiilt witii a tolerance of better than 0.10 mm. The windows of the ut lier clianiber-t lie prism chamber-form an angle B = (20.775 f 0.017)" witti one ariother. with one window parallel to the Rat chamber ones. The t ~ o cliarribers are in communication via a small bore between them. Seedle valves are fitted at each end of the ce11 for Auid filling and bleeding purposes tluririg the esperinients (831.

Chapt er 5. Pure Buid eqeriment

Figiiri? 3.1: Illtistratioii of the corribiriecl cell.

Oritl of t t i ~ pirccs of apparatus uscci for the pure Huicl measiircnients is the prisrii cc11 cIt.scril~ecl in some cletail in Chaptcr 3 . section 3.3.1. Thc wliolc iil~pariitiis is dcscribed in an earlier paper by Balzarini et (il. [-KI].

Esseritially. to coniplete the description of the pure fliiici esperinient ap- I>iiïi\tiis. the cc11 (prisrn or combinecl) is placecl in the centre of a t herrnostat opcwting «ri t hc same geiieral principles describecl in Chaptrr 3 . section 3.2.1. Urilikr tlic t~iriary liqiiicl esperiment. the pure Ruici esperiment thermostats

laid or1 tlicir side to respontl to clifferent esperiniental operrting nceds. siich as liavirig to take the cells out for bleeding ffuid and weigtiing very often diiririg the espcrirnerits. The temperature is rneasiired ai th both a quartz tticriiioriietcr ancl two thermistors, one on either side of the thermostat. Thc c.iiliI>ratioii of the thermometers is carried out as described earlier.

The opt ical setup for esperirnents with the combined ce11 must enable us to use the three techniques described in Chapter 3 on the same esperinient. This eritails quite a bit OF juggling of equipment on the optical table to find the

Chapter 5. P vre Puid experiment

1 & ~ e mer 8 Camera on F l Plan 2 Beam Expander and Spatial Filter cames on ,mam 3 Collimating Lens 4 Polarizer BS Beam splitters 5 Expanded Bearn C Combined cell 6 Thermostat MS Mirrors 7 F ocussing Lens f Telescope

Figure 5.2: The coinbined ceIl esperimcrit setiip.

twst c~speririieiital geonietry for al1 types of measiirernents that are iiiten~leci t o I)o wrrierl out. Tlie setiip is illiistrated in Fig. 5.2. The two siniiiltaneous rtiwsurerrlents are carriecl out on opposite sides of the thermostat to avoiti possit)le intcrfcrence of the prism ce11 data collection operations with the foral (or image) plane measiirements.

Xsicle froni a few logistic problems which Iiad to be omrconie (e.g. align- iiig a largcr nuniber of optical cornponents on a cro\vdeci optical bencti). the t~l<>i~st~r~Itlelit proceclure tesembles those made a i th a single prism ce11 or a single intcrfcrorneter. Nore difficulties are encounterecl when this apparatus is iised for the rneasurement of the critical isotherm [83]. nhich however is heyml t h e purpose of t his thesis.

Chapter 5. Pure flvzd experirnent

5.4.4 Samples The sample of 1.1-difluoroethylene used was purchased from Scott Specialty Gases in a lecture bottle of purity 99.4% (see appendix C for Further prop erties of this fluid). Preparing the sample for the pure fiuid experiments is a relatively easier process than for the binary liquid experiments. The çell-eithcr the prism or the combined cell-is first leak-tested, since it will tiavc to sustain pressures of up to about 150 atm (2 15 MPa). Then. it is eracuatcd and pumped on for days. Then, it is repeatedly flushed ai th the sarriplc gas ancl erxuated, in an effort to recluce the amount of impurities as i~iiicli as possible.

Piirticiilar carc must be put into the operation of filling the prism ceIl For the iiicasiirernents of the LL coefficient at higli clensities. Iri that situation. thtl ceIl is usually kept ininiersed in an ice. or colder. bath. while the fiuid is tr:irlsforrecl from the lecture bottle to the cell.

Cider tliesc conciitioris. the cell is filletl until its volurne is occiipied. ;ilitiost cmtircly. tq. the liqiiid phasc of the ffuicl. Referriiig to Fig. 5.3. the s!.stoiii is iit a derisity pld. on the coesistcrice curvc. near the liquicl pliase rryjoii of tlic pliase cliagrani. Clearly tlieri. if the ce11 is allonecl to hcat the Hii id wil l el1 transform to liquid and the pressure in the ce11 will increase clritrrmtically if the teniperat lire is raiseci.

5.4.5 Experimental procedure

.-1s iiiiticipiitd i r i the previous section. to takc LL data thc sarnple ceil is iriitially fill~d CO a derisity p' in Fig. 5.3. such that at the teniperatiire of t t i ~ iw hath tlic sample is nearly al1 liquid with only a srtiall aniount of txpour prcsent. The sample is thcri introdiiccd into the thermostat set at ;i trmiprratiire Tl < T'. In these conditions. the Iicpicl phase refractive in- clos. r l L ( p l . Ti). can be rneasiired. The temperature is ttien increased to TL anrl I I ~ (p'. &) is measured and foiind to be different from nL(& Ti) as long as TL < Tt . The procedure is repeated until. for temperatures abow Tl. the nieasurenient of nL(p' . T) is found to be practically independent of tririperature. The systern is then in the one phase region outside the co- csistericc ciirve and rneasurements of the refractive index in this situation yidd nL(pt. T t ) needed for the determination of L at density p'. The density of the fliiid is deterrnined using the known (previously rneasured) volume of the evacuated ce11 and its m a s , as described in Chapter 3. The ce11 is

Chapter 5. Pure Juid expen'ment

Figiircb .3.3: Qiiali t lit ive illiistrat ion of t hc cocsistcrice ciimc of t hc piire Huit1 ;in(! t tic) prorr~l i i r~ follo~vccl to take LL data in the prisiri ce11 esperinient.

t licii takeri out of tht. thermostat. the sample brought to ariother derisity p" I)y l~lc~cdirig oiit soirie Aiiid and the procedure repeatecl to obtain nt (p". T") and so ori. On the vapour sicle of the ciirve nieasiirerrients arc carriecl out sirtiilarly. tv tlecretising the temperature after cach clatum iint il rooiii tem- porat i i r r is rcadicvi. at which point the tcniperature is left iinchariged whi l~ Huit! is blcd at cacli point. The data used to calciilatc L are collecteci in t h stiaclrd area aroiincl the coesistetice curw as inclicated iri Fig. 5.4. The tiigh coriipscsssit~ility of the fliiid aroiind the critical poitit (f i . Tc) makes it (lifficiilt to obtain accurate nieasurements of n ( p . T). The region is thereforc s*;ivoi(lrd". as stio\vn near the top of the curve in Fig. .54. by hcating the saiiiplc to about one clegree above Tc.

Iri the coriibined ce11 esperiment. the sample is prepaseti at the critical cirrisity. p,. of the fluid. and inserted in the thermostatic housing for that iqwrin ie i i t . The prisrn chamber data are read via a micrometer that is calibratecl in the sanie rnanner described in section 3.3.1 of Chapter 3. Data are taken starting from the farthest temperature from Tc that one ivould like to he co~ered by the experimental run. At each set teniperature. the quartz

Chapter 5. Pure fluid experirnent

Figiirr 5.4: Teniprrattire aiid density region wherr. the data for the LL mea- siiri.riitbrits ;ire collectecl. Diie to its large dcnsity gradients the critical region iiiiist Iw ~~rircuriinai~igatecl!~ to take accuratc data.

Chapter 5. Pvre Puid experiment 83

thermometer as well as the thermistor values are recorded and micrometer readings are taken as the carnera and film in the focal plane of the lens riin continuously After each reading the temperature is raised slightly and after eqiiilibriiim has been reached again another measurement is taken. The procediire is repeated until the system has beeii brought above the critical tcnipcrature. Data collection on the prism chamber usually must stop at niore thün a niillidegree from Tc, due to the distortion in the beam procluced by t lie g r a ~ i t at ionally-induced density gradient in the fluid.

5.5 Results

Esp~rinicnt;il data on pure fluids using the techniques clescribed can rior- riially he takeii relatiwly more qiiickly than on binary liqiiicl niistures. The &it ii prrscntctl hcre are the product of sereral prisrn-ce11 and conibiriecl-ce11 riiiis. Somc of the data constitute the first available direct comparison of cwsistcricr data prodiicecî by the focal plane tectiniqiie and the prism ce11 t wtiiiiqw or1 t hts samr sampie of fliiitl.

T h riicasitrcwients of the LL fiinction. C. arc ciisctissccl first sincc it is ;i q~imt i ty ncccled for i l I I the subsequent rsperiments. In the prisrii ce11 cqwririiciit. the iiiiportancc of a careful analysis of t he ce11 weighing operation is rriipliasizcd as it is siispect~cl it can Iead to a systemiitic error in tiic r)wilsurcmmt of L.

5.5.1 Prism ce11 experiment: measurements of the Lo- rent z-Lorenz funct ion

T h collectccl Lorentz-Lorenz data for 1.1-clifluorocthylene are presentecf in tlir grapli iii Fig. 5.5. The LL function clefinecl by relation (3.1) is often foiincl to rsliibit a weak dependence on density around the coesisterice cume 1.56. 53. 871 witli a niasimum locatecl approsimately arouncf the criticai tiensity. Tlic present csperiment confirms this trend and shows that. altliough Ç

follou-s a roughly parabolic shape. its value is *-constant" within about 1.2%. .-\round the critical density p,. where an accurate deterniination of C is more criicial. the measured value does not vary by more than 1%. To obtain t lie ternperat ure dependence of the order parameter of 1.1-difluoroet hylene on t eiriperat ure from refract ive index measurements. the critical value L, = 0.1668 5 0.0017 cc/g aas used.

Chapter 5. Pure fluid experiment

Figtirr 5 .5: .\Ieasiirecl clensity clependerice of the Lorentz-Lorenz funct ion of 1. I ilifliioioethylcrie. The line s h o w is a quadratic fit to t h e ciata. Tlic fit pmiriicltcrs ;ire iri tiiblc 5.1. The size of the crror bars is conipiirable to the srattor iri t h data.

Grricrally. t he data siiggest t h t hc LL fiirictioii cari t x espariclcd in a pwrr serics as a furiction of p:

Tliv c h t a i r i Fig. 5.3 are sliown with a cpdrat ic fit through thcrn. with the fit paraiiirters giwn in table 5.1.

Tlic data at both ericis of the clensity range investigütccl are affectecl by Iiigrr rrrors tliiiti the data arouncl the critical clensity region. At the Iow di1risity end. the acciiracy in the cleterminat ion of L is mainly limited by hon

Table 3.1: Results of a quadratic fit to the Lorentz-Lorenz data of 1.1- difl~toroethylene (C2& F2) .

Substance Lo ( c c / g ) Ll (cc/g)' C2 ( c ~ / g ) ~ G H 2 & 0.1639 0.0142 -0.0166

~0.0002 *0.0007 ~ 0 . 0 0 0 T

L, ( c c / g ) O. 1668

*0.0017

Chapter 5. Pure fluid experiment 85

accurately the mass of the empty prism ce11 can be measured at the end of the riin. -4s the quantity one needs is the difference m - m,ri (see equation 3.8), whcre m is the mass of the cell+fluid system and rn,,~l is the m a s of the ernpty cell. the same tlegree of uncertainty on the empty ce11 mass yields a largcr inaccuracy in the low density than in the high density measurements. Lorentz-Lorenz data were collected at lower densit ies t han t hose reported in Fig. 5 .5 . but were not included in that graph. Due to the large errors in t lierii. tliey were not very nieaningful in the reported measurernent of L.

At the higli density sicle of the data. one must watch for iinother erperi- riirrital pitfall. At those densities the coexistence curve of 1. l-difluoroethylene is at ttwperatures inuch lower than the typical room temperature. The reportcd data w r e taken starting at the high density end. which meant main- tiiitiirig tlic cc11 at temperatures down to about +YC. Clearly. then. when tlir cd1 is mkcn oiit of the therniostiitic housing to be weighetl-an operation ttiat rriust only last a few minutes for the rcasoris mentionecl above-. con- (lcrisatiori. ilrici ttierl evaporation. of air moistiire on the ce11 bocly occitrs cpiitc riipidly t lirrrl~y tiindering one frorrl taking a precise reading of the ce11 weight . I pdori i i id a roiigli. but still significarit. stiitly of this effect at tlic end of thc i~spc.rii~ieiit. \i.lirii. with the crnpty ccll. 1 coulcl bring it to an- giwn terriper- iitilrP i i r l c l tlicn rrionitor its weight ori the precisioii balance as a fiinctiori of t i r w . This \ v i s clone at scveral temperat ures. ni t h cacli measurerrient being (w-rictl out for about 24 hoiin. Sot surprisingly. it was foiind that the dif- fmwtv Arn = rn, - nc,. between the rneasiired ce11 wcight as soon as the ce11 w;is takm out of the tlierniostat. n i , . and abolit one (1- latcr. m l clepends o r i thp ditfercrice between the cell's initial temperature in the thermostat ancl t l iv rooiii tvniperntiire AT = Tset - T'.,,n. The latter i w s measiire<l quite ofttlii duririg the esperimcnt and it avernged at about 23.3". Fig. 5.6 stio\u t l io cffwt at the highest AT measiired of about -3.8'. At AT = -5.8'. it appcars t h the ce11 actual weight is overestimatetl by about 0.05 g. This nuiilcl lead to an iinclerestimation of the value of C of about 0.5%. ,At lower ATs. thp cffict \vas less niarketl. From the measurements taken. a rough qiiaiititat iw correction factor. decreasing with decreasing AT sas estrapo- latrcl arid ilsec1 to "correct" the high density LL data. I report iri Fig. 5.7 a (miriparison betneen the corrected ancl uncorrected data.

Chapter 5. Pure Puid experirnent

Time (minutes)

Figiiro 3.6: \'iiriation of t h wcight of the prism ce11 wi th tiirie. wlieri the ce11

Figiirr 3.1: Sleasured LL data aith (white diamonds) and without (black di- arriorids) correction on the ce11 rnass. The size of the error bars is comparable to the scatter in the data.

Chapter 5. Pure jluid experiment

Figiirc 5.8: Coesisterice curve of 1 .l-difluoroet hylenc as nieasurecl in the prisrii ceil tbsperiniciit .

T1iv corrrctid clilta do yieicl a Iiiglier value of Lclrr.,fii. at higli derisities. siiggtlst iiig t liat i t lias ;i weakcr ciensity ~Iepencierice t han previously ohserwtl - - i n Fig. .,.a. Hoivewr. froiri thib increasccl scatter in the rneasuretiients. th<' th ta also sliow that a better stiidy of ttiis effect would l x recpired before (lrarving sound quarit itüt ive conclusions.

5.5.2 Prism ce11 experiment: coexistence curve

Ttic. cocsistciice rurve of &H2& was rneasurecl ovcr the rcciiicect temperature iritervd -1 x IO-' < t < 10-'. The pronouncecl tlensity gradients in the fliiici iicwiiig its critical region rencier it arcluous to take acciiratc chta at values o f t< LU-! Tlic coesistence curw is presented in Fig. 5.8.

Csirig the value of L mentioned in the preïious section. the data were fit t et1 to t lie espression:

\diete the correction esponent 1 iras held fiscd at its theoretical values of 0.34 and 0.50 and the critical amplitudes were treated as free parameters in the nonlinear least square fit to the data. The critical density p, sas nieasiired at 0.418 k 0.3% g/cc and it was found by extrapolation of the

Chapter 5. Pure jh id experiment

Table 5.2: Parameter values of the fit of equation (5.3) to the prism ce11 data on Crr H2 F2. Qiiantities in parentheses were held fixed during the fit.

c.ot~sist~ncc ciirw diameter. pd = ( p L + p \ - ) / 2 . as a ftiriction of temperattire. ri) tlic critical tcniperature. The fit valiies arc given in table 5.2. It cari tx observccl froni t tic correctioii-sensit ive plot in Fig. 5.9 t hat correct ion- to smling ternis arc ccrtainly rieedecl to interprct orcler parameter clata on C F . Tlir fit values show that the theoretical values of the esponents fit tlic (lata well itrid that a two-terni fit aoulci seeni more appropriate to ii i t orprct t hcsc data. -4 t hree-tcrm fit slioivecl rio iitiproverrient oii t lie tivo- trbrrii fit in the interpolation of the clata. The best estiniatc OF the rritical t iwpcwt iirc of 1.1--difliiorocthylerie from the prisni ceil csperiiiient is Tc = 302.9S f O.OOGC/c fi.

5.5.3 Combined ce11 experiment

Tlir chta of t tir combinecl cc11 esperirrient cowr a reducecl temperatiirr rangc. o f a littlc ovrr three clecatles 'sroiiii -4.3 x 10-"to 1.6 x IO-" It is possible to carry out riieasiirenierits over a much larger temperatiire span with this cywrinicnt . too. Once agaiii. however. the iiirn of the combineci cell mea- siireriirnts reportecl here is to compare focal plane ancl prisni cell-type chta ririir t hc cri t ical region of l .l-difiuoroet hylene taken ivit h the same sample in ordrr to st iidy more closely the differences hetween the results obtainecl tliroiigh the two techniques. The coexistence ciirve data of bath kinds is pre- sriited iri Fig. 5.10. The curves through the data correspond to a nonlinear h s t square fit. whose parameter best values are in table 5.3. The density diffcrcnce A p is obtained dzrectly from l n for the focal plane data. t hrough t lie Lorentz-Lorenz relation. yielding:

Chapter 5. Pure Puid ezperiment

Figiircb 5.9: Corrclct ion sensitive log-log plot of Ap'/td us t of t hc coexistence k i t ii of 1 . 1 -difi iioroet hylene mcasurecl in the prisni cc11 esperiment. The \ - i i l i iv of .i \vas h l c l fisecl at 0.326. The size of the error bars is comparable t O t l i c stzittcr iri the data.

- pnsm : ocal plane

Figure 5.10: Sirnultaneous prism cell-type (circles) and focal plane (JS) data on 1 . 1 -diHuomet hylene measured wit h the combined ce11 experiment.

Table 5.3: Parameter values of the fit of equation (5.3) to the focal plane data on C2H2F2 taken with the combined cell. Quantities in parentheses acre lielcl fised during the fit.

range B A Bo BI Y'

~r t i r r r k = 3 ( u L + 1 1 1 . ) / [ ( r 2 i + 2) (nt- + 9 ) ] and L = 0.1668 cc/g is the Lorentz- Lorrwz c«cfFicieiit fourid cnrlier with the prisni ccll espcrinicnt. The qiiantity k <*iiii tw nimsured a i t h the prism ceIl data from either the prisni ce11 es- poririiriit or the present one. It is Folind to he eclual to k = 0.6396 and (.otist;iiit wit hin 0.02%. The clata obey t hc t heorctical orcicr parameter law (cqriiitioii ( .7 . :3 ) ) with the critical esporient .j = 0.326 ancl orle corrcctiori terni with tlir mrrrction esponent fisecl 1 = 0.54. The critical aniplitiicles wcrc lrft frw. T h Ixst parameter ~ t l i i e s of the rionlincar least square fit to t h prisrii atid t h focal plarie data arc reportecl in table 3.3.

;\riiong t lir acluntages of t his esperinient is t hat t hc prisni-type data raii I)r ;iiialyzt.d more precisely bccause the critical temperature can be clc tcv-riiiticd via thc focal plane data. which can be tiiken closer to critical. \\'tirri t tiis ir; done ttie order parametrr data can be plottecl on it log-log plot iis iisiial.

Tlir goocl agreement betweeen the prisni-type data ancl the focal plant. (lata is rviclcrit iii thc log- log plot of tlie data wrsiis tlie recliiced teiiiperatur~. iis s l io~r i iri Fig. 3.1 1.

Chapter 5. Pure fluid experiment

II 5c X

*, 0.1 - B -- d =. prisrn 4 0.08: . c focal plane .

0.06 - 1 o.5 1 ow4 1

Figiirc 5.1 1: Log-log plot of prisni (circles) and focal plane ( r s ) data ori 1 . 1 difltioroet tiylcne nieasured wit h the combinecl ce11 esperiment .

Chapter 6

Conclusion

6.1 Binary liquid Tliv cari t i c x l rrgioti arid t hr corsistcricr rtirvr of t tir biriary licliiid rnistiirv rr liq)taticb+rii t robci~zcric ivi.;is stiidir!d to ii Iiigli dcgrrc of acciiracy by optical tiicbthods. Oiic of the goiils \vas to rririsiircb tlir critical osporicrit J. govcwiirig tlw p o w r Iiiw folionrd by ttic ordcr pararrietcir as tlic tcrripcwtiircb ;ippr«iiclic~ r l i v cmritii~iil tciripcratiirc Tc.

Tliv riioiisiircd valiie of .j is 0.367 k 0.006. wihich is consistcritly bigtivr r li;m t tic tlir«rrtic;il valiic of .ah = 0.326 rt 0.002. Thc critical consoliitc t t m - p w t tirch for t liis system is riieasiirecl kit Tc = 29 1 . Î8 I< * 0.03% (= 18.6JcC).

Iii iiri rffort to discover if iiri csperiniental Haw coiild iicrourit for the diffcrcwc~ t~cbtivccw t tit. tlicorct ical iind esperirncrital wliies of . j. t tic knowiii potriitiiil solirrcls of systcmatic error wcrc carcfiilly analyccl. Iii this prowss o ~ i ( ~ oî tlic optical techniques--the irnage plane tectiriicliie-ivas cniployxl to iiioiisiirc* ttic profile of the incles of rcfractiori of the hinary liqtiicl s;inipl~ iii ii u.+* tliat liacl not h e m trieci hcforc. The resiilts arc intercsting iri tliar t l i i b s liapc. o f t lie r~fractive incles vertical graclient can hc tnappeïl ciirwt ly tiy tliis tectiriicliie. Thc rclatively ricw itsc of this optical tool lias helprd riilr out tlie possibility that surfacc wctting by one of the phases coiild hr rc~sponsiblc for t hc discrepancy between the rncasurecl and the t hcoretically pr~dictccl csponerit A. The influence of other effects such as irnpurities in the sani ples. the gravitational field of Earth and the exact defiriition of t hc orïlrr pararrieter also appear to be too srnall to account for the high measured valiic of J.

Chapter 6. Conclusion

Figiirv 6.1: A c-ollcction of esperimental (black clianioncls) and theoreticai ( wliitr ) valurs of the criticül esponcnt j in the past 35 yars. The experimen- t il1 t l i i t il iirfb froni nieasiirenieiits «ri binary liqiiicls only The results obtaitied II!. t lit) I'BC 1at)oratory of critical phenornena are the dottccl circles.

A Ixief scarch of the past results hüs procluce the data in Fig. 6.1. Indecd. i t sccrris tliat cspcriirientiilists and theorists have slowly arrived at a good iigrrriiicrit iri the last coiiple of ciecacies. While it r n q look somealiat awk- \rml tliat tlic prescrit resiilts stand out of the pack. the resiilt is at l e s t as iic~.lliiitC as mitny othcr rneasiirement issueil by this laboratory. and whicti Iiiivc ticlpccl biiilci the crcdibility of the modern iinderstanding of crit ical ptictionicnn.

I t is conclucled ttiat the rrieasiirernent of 3 reporteci is accurüte ancl precise iirid pro~idrs food for thought for both esperinientalists. to look for other possible causes for the cleviation from nom. and theorists to Iielps us in that qilrst .

6.2 Pure fluid Tlic crit ical region of the polar pure fluicl 1.1-clifluoroethylene (C2H2F2) has h e m st iidied via ttvo opt ical investigation methods, called the prism ce11 and the focal plane techniques.

Chapter 6. Conclusion 94

In the prism ce11 experiment the Lorentz-Lorenz functiou of the fluid, relating the fluid density and its index of refraction, ww determined as a Iiiriction of densite in the density range below about 0.82 g/cc. It would not appear that tiiis quantity has been measured before. a t least. not to the acciiracy t hat t hese experiments produce.

The coexistence curve of C2H2F2 wvas also measured over a temperature range estendirig to about 28 degrees from the critical temperature. The critical temperature nieasured wit h t his experiment is Tc = 302.95 K=29.80°C. The data obeys t lie order parameter scaling relation wit h twvo correct ion tcrrris. iiritl the theoretical values of the exponents d and -5.

A riove1 esperirnent combitiing the three main optical techniques. focal m t l iniagc plane arid prism cell, in orcier to perforni measurenients of clifferent kititls or1 t hc sanie saniplc has been testcd. With the LL functioti nieasiirccl iii tlic prisni ce11 esperiment. the coexistence curw of -& F2 \vas measorecl sirrr rr l taneo~~~s l tvi t li t lie prisni ce11 and the focal plane met liods.

Tlit. fi rst &ita froni both niettiocls overliip quite well owr the wholr tern- pwt t u r r range irivcstigatecl. nanicly between t = 4.3 x 10-"a~icl t = 1 .G x 10-3.

Tlicl c-ot~sistrncr wrve data oii Ct2H2& taken with tlic conibincd esperi- iiirrit iiïo iiitc~rpolatcd well by the theoretical scaling relation with the espo- ricbiit ~riliics set at their t heorctical valiies.

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Appendix A

Heat flow problem on the binary liquid thermostat

hi t h tticrrriostat. a layrr of Styrofoani [33] of tliickness L attenuates ester- rial tmiperaturc fiiictuatioiis so that the interilal ternpcrütiirc can bc kept cgoristarit witliiri 10-"K (Fig. Al) . .A tirrie--periodic variation in t h csternal tiwiprraturc propügates towiircls thc interior of the thermostat in the Foriri of ii \vaw wit ti an arnplitiicie t hat decrcases esponentially wit h propagation t1ist;iiicr. 2. At the iriner slirface of the layer. the tieat flus flow irito ttie c q p o r layclr of tliickness. 1.

.\ri cstiiiiatc of ttie arnplitiide of the tetriperature fluctuations at the inner d g c ~ of the styrofoar~i layer can be obtained 1- solving the one-dimensional Iiclat coiitliict ion cqiiat ion stibjcct to the following hoiiridary contlit ions:

1. iit s = O and tinie t. the tenipcratiirc. T(0. t ) . is giwn by the espressioti.

whcre ; is the freqiiency of the temperature fluctuation of amplitude TI anil To (constant) is the ambient room temperature:

2 . iit the interior copper surface. the heat flus is v e v rapidly distributed tliroughout the copper layer so t hat its temperature can be calculatecl frorn t lie espression (Fourier's l m ) :

Appendix A. Heat pow probiern. . .

copper layer of thickness A

thermostat interior

Figiiro A.1: Geonictry of configuration iisecl to assess t h atteniiation of twcwi; i l tcrriperat iirc fiiictiiation t- ii styrofoarii Iqw.

n-tierc K is the thermal ciontliictivity of the styofoani. 1 is t h thickriess of ttir copper l q e r ancl (pS)c,,) is the specific heat of the copper per iiriit i-oliinic. The above expression assiirrics that the copper laycr is ttiiri rrioirgh for the tieat fioiving into it froni the styofoani to raise its tcbiiiprratiirr to the sarne valiie throiigtioiit its entire volume.

Th tiei\t f l i i s iri the styrofoani is governecl by t lic heat coridiiction q u a -

\i-hcw ( P S ) . ~ is the spccific hear per iiiiit voliimc of the styofoani. Tlic soliit ion ni+- bc mit ten in the forni.

t h = is(pS)s/r; 2 i; x a rcsult obtained by noting that. for S tyrofoarri SM. (pS)& 2 1 0 - ~ m-'S.

It follows froni the boundan; conditions in equations (-4.1) and (-4.2) and t lie above cqiiat ions t hat .

T(L. t ) = To + 2.4eiUt/(1 + Li) (A. 5)

Appendix A. Heat flow problem. . .

n-here

arid

To eriliance the attenuation of the temperature fluctuation at = L? it t raiispires that L is chosen such that e - a L < l ~ - 4 . In this case. equations (h.5) micI (-4.7) take the form

Tikitig ;iccoiint OF the specific heats per unit volurrie of copper and styro- fuarii. arid t h TI lias an obscrved fluctuation frcquency. J. of approsimately 0.2 s-l. it follows froni equation ( A G ) that ICI > 1. if 1 > IO-:' m. To es- r itiiiitc! a ..safc" tiiickncss for the styrofoam. such that the fluctuations iti

T ( L . f ) arc3 lws t hari IO-.' is suffices to choosc L stich t hilt c - " ~ < IO--'. A styrofoani t tiiçkrirss of ttic orcler of 33 nim satisfies the rcquirernent of keeping rlie iiitrriial trnipcratiire fluctiiatioris bcloa the requimcl levcl. if TI < 0.2 K. Tl10 42 nini thick styrofoam laver used in the csperinient is thcrcfore rnore tiiiiri ii(1ecltiatr to sçrecn the iriterior kotn the influence of cstcrrial fluctuations fit the Icvcl of 0.2 K or so.

Appendix B

Physical properties of the substances studied

Thcl followirig data Iiaw t~een taken from ref~rcnces 1501 aiid (371.

Table B. 1 : Crit ical teniperat ore. rrieltirig poirit aiid boilirig point tcrliperatiircs of t hc siihstariccs stiidiccl.

Substance Tc (w ?u (cl TB (cl n-heptanc (C; Hio) 340.2 -90.6 98.5

nitro t~enzene (Cs H5.V02) - - 3.1 2 10.8

iiniline (Cs &.Y) 699 -6 184. I cyçloliesane (Cs Hi-) 353.5 6.6 80.7

1.1 -tiiHuoroet hylenc (C'? H2 Fi) 309.75 - l -85.7

Table B.?: Refracti~e indes (kit the D linc of the spec- truni of sodium and at 20°C). density (at 20°C. referred t o the tlcnsity of nater at 4°C) and rnoleculür weight of t lie siibstances stucliecl.

Substance J rifi mol. a t .

nit robenzene 1 .titi62 1 ,2037 123.11 aniline 1.3363 1.0'217 93.13

cyclohesane 1.4266 0.7785 84.16 1.1 -difluoroet hvlerre 64.03

Appendix C

Relation between A$ and An

Iii Cliaptw 4 t tir rclation of direct proportionality bctwccri the iriclrs o f refraction (lifferencc betwecn the two liquicl phases of a hitiary niistrire aiid the difference iri volunie fraction of one of t h sp~cios \vas statecl witliout justification. It is deri\-cd ici t h iippriidis t)y nicans of ttic Lorentz-Lorenz rclation appliecl to the case o f ii biriary rriistiirc.

Cridrr thta ~issiimption of the additivity of t hc volilnies of t tie t w Iiqiiicls iipon rtiising. iri the one-phase rcgiori one lias

trlierc LH, .y is the Loreritz-Lorenz coefficient of the two separate licliiicis H and S. (equatiori (3.1) of Cliapter 3 ) . and the O[[ , .y = I i f . .J(l jl + I )) are t heir respective volume fractions. In the t1.o--pliase rcgion. relation (C.l) can be appliecl to the upper (C) ii~id lower ( L ) phases. to obtain

and

Appendix C. Relation between A@ and An 107

Subtracting equation (C.2) h m (C.3) and after some simple algebraic steps one finds the following relation:

K . 4 ) Tliroiigh the introduction of the quantities AdfI = &. - dH, C. aiid l n = ( n L - nu), equation (C.4) can be rewritten as = k h . nith k corresporidiiig to the quantity in squarc brackets in ecliiation (CA). In other studies [89, 901: the quantity k was foiiiitl to be constant aithin about 0.1% in the teniperature range of t lie two-phase region of the rr-hept ane+nit robenzene mixture. a l i ik the iiricertainty in Ln is between 1% ancl 10% in the same riingc. In the coriditions the experiment was carried out. it ivas tlirrcforr safc to consicler the volume fraction Ao siniplu proportiorial to An. Ir1 this case. the .constant' A. can he calciilatcd in t h particiilar case that n~ = nc- = n,. n, t~cirig the critical indes of rcft-art ion. w1iir.h can bc calculatecl froni the Lorentz-Lorenz rdation oricr t hr tritical cornposition of the niistiirc is knoivn.

Froiii t lit! ptiysical properties of the two species usecl. listcrl in the prcvioiis apperidis. the proportioriality coristaiit A. is foiincl to be: k = 6.045.

Appendix D

Bending of light by a refract ive index gradient

Iri t lie focal plane technique clescribed in Chapter 3. orle esploits t h c l fact ttiat a gradient of index of refractioii caii berid a liglit ray passing through thc rnccliuni cshibitirig such gradient. It w a s s ta t~c l that light encouritering a medium with an irides of r[,fract ion gradient (in t tic vertical direction, 2 . in our particular case) tlr~/d: gcts bent following the relation:

Equatiori (D.1) clerives froni ari application of Snell's law to a i i i c d i t i l i i a i t h varying indes of refraction. if the medium is t hoiight uf as st ratificcl in layers each with different refractive indices. eacti diffcring lrom its neighbours by ari amount dn. such as dn << 1 1 .

as skiown iri Fig. D.1. Applying Snell's laiv to the l-er sliown yields

nsinû = ( n + dn)sin(8 + dB) (D.2)

Sirice dn ancl dB are supposed infinitesimal. equation (D.2) c m be approsimated retaining only terms of order O(dn. dB). like this

Appendix D. Bending of fight by a refractive index gradient

Figiirc. D.1: .\Lo<lcl of a nicdium with a rertically varyitig index of refrüc- t ion. r l ( : ) .

\\Mi siriiple rcarratigiiig of the terms in the last qiiatiori. onc olitairis

1

Froiri the cliagrarii i t i Fig. D.1. the term cote caii be witten as: c:otO = cli/cly. Csing this in equation (D.3). equation (D. 1 ) is foiirid [9 11.

Appendix E

Technical drawings of apparat us

A ppendzx E. Technical drawings O/ appamtus

1 Dt rnen~~ons- rnillimetcrr (inchez1

2 Mi~teriai copper

3 Ouantiiy. 2

fhe University o f British Columbia. Vrtncouvcr. Cunodo Oepartment of Phyrlcs and Astronotny

Lobaratory of Cr~t~coi Phenomcnq

S.AMPLE CECL HOLDER

5hzet n o . [ille. 1, front and sccltons a l & - A ana 6-8

Jrarn by. Nick Fomeli Date 5 1 AUCUST 98

îhecked by- Scole 0 ?5

Figure E.1: The ce11 holder.

Appert dix E. Technical drawzngs of apparatus

1 /4-20 (countersiink) thrended hales e

Ihe University of Oritish Columbia. Vancouver. Canada Oepdrtmcnt a l Phywcs and Astronomy

Laborotary o f Crit~cal Phenornena

I IHERMOSTN INNER CrLiPIDER

Drown by: Nick Fameii Oatc S t AUGUST 98

Figure E.2: The inner cylinder.

A ppendzx E. Technical drawzngs of apparatus

- -

The University of Brrtish Columbia. Vancouver. Canada Oeportrncnt a l Physics and htronomy

Lobaratory of Crilical Phenornerio

rHERMOSTA1 OUTER CYLINOER

Shcel na . Lille: 1 , f r o n t clevation

Drarn by. Nick Famcii 1 Ode. 29 AUCUST 98

Figure E.3: The outer culinder.

Appendix F

Photographs of apparatus

Figure F. 1: Some of the n-heptanc+nitrobenzene samples. From left t o right: 1-mm. ?-mm' and IO-mm samples.

Appendix F. Photogmphs of apparatus

Figure F.2: The ce11 holder with a 1-mm saniple.

A ppendàx F. Photographs of apparatus

Figure F.3: The inner cylinder and cap with heating foils.

A p pendix F. Photogmphs of apparatus

Figure F.4: From left to right: the ce11 holder, and the inner and outer cylinders.

Documents

OPTICAL STUDIES OF CRITICAL PHENOMENA IN … STUDIES OF CRITICAL PHENOMENA IN FLUIDS b~ Nicola Fameli Laurea in Fisica. Cniversità degli Studi di Padova, Italy, 1991 -1 TUESIS SCBlIITTED