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Optical studies of pure fluids about their critical points Tiong, Pang Kian 1994

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OPTICAL STUDIES OF PURE FLUIDSABOUT THEIR CRITICAL POINTSbyPANG KlAN TIONGBSc, The University of Toronto, 1983MSc, The University of British Columbia, 1986A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT of PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1994° PANG Kian Tiong, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of_______________The University of British ColumbiaVancouver, CanadaDate____DE-6 (2/88)11ABSTRACTThree optical experiments were performed on pure fluids near their critical points.In the first two setups, CH3F andH2C:CF were each tested in a temperature-controlled,prism-shaped cell and a thin parallel-windows cell. In the prism cell, a laser beam wasadditionally deflected by the fluid present. From the deflection data, the refractive indexwas related to the density to find the Lorentz-Lorenz function. Critical temperature (7),density, refractive index and electronic polarizability were found. In the secondexperiment, a critically-filled, thin parallel-windows cell was placed in one arm of aMach-Zehnder interoferometer. Fluid density was monitored by changes in the fringepattern with changing cell temperature. The aim was to improve on the precision of 7,:T(CH3F)=(449O87±OOOO2)C; HC:CF)=(2974l9±OOOOl)C; and, to study thecoexistence curve and diameter as close to 7 as possible. The critical behaviour wascompared to the theoretical renormalization group calculations. The derived coefficientswere tested against a proposed three-body interaction to explain the field-mixing term inthe diameter near the critical point. It was found thatH2C:CF behaved as predicted bysuch an interaction; CH3F (and CHF3) did not. The third experiment was a feasibilitystudy to find out if (critical) isotherms could be measured optically in a setup whichcombined the prism and parallel-windows cells. The aim was to map isotherms in aswide a range of pressure and density as possible and to probe the critical region directly.Pressure was monitored by a precise digital pressure gauge. CH3Fand CHF3were tested111in this system. It was found that at low densities, the calculated second and third virialcoefficients agreed with reference values. However, the data around the critical pointwere not accurate enough for use to calculate the critical exponent, ô. The calculatedvalue was consistently smaller than the expected value. It was believed that the presentsetup had thermal isolation problems. Suggestions were made as to the improvements ofthis isotherm cell setup. Lastly, a joint project with the Department of Ophthalmology,UBC to assemble a vitreous fluorophotometer is discussed in Appendix F. The upgradingof the instrument took up the initial two years of this PhD programme.ivTABLE OF CONTENTSABSTRACT iiTABLE OF CONTENTS ivLIST OF TABLES viliLIST OF FIGURES xiACKNOWLEDGMENT xviiChapter One: INTRODUCTION 11.1 The Three Phases 11.2 The Critical Point 31.3 Phase Transitions 51.4 Objectives 6Chapter Two: THEORY 82.1 The Van Der Waals Equation 82.2 The Virial Equation of State 112.3 Mean Field Theories 112.4 The Order Parameter 132.5 Classical Universality 132.6 Power Laws 152.7 Scaling 192.8 The Renormalization Group 22V2.9 Universality Revisited 242.10 Corrections to Scaling 262.11 Other Methods 30Chapter Three: SETUPS AND PROCEDURES 323.1 Foundations 323.2 Temperature Control 333.3 Optics 373.4 Prism Cell Theory 393.5 Prism Cell Setup 413.6 Prism Cell Protocol 423.7 Image Plane Cell Theory 473.8 Image Plane Cell Setup 513.9 Image Plane Cell Protocol 523.10 Isotherm Cell Theory 553.11 Isotherm Cell Setup 583.12 Isotherm Cell Protocol 613.13 The Fluids 623.14 Computer Programmes 64Chapter Four: PRISM CELL RESULTS 654.1 Ronchi Calibrations 664.2 afldae 674.3 Coexistence Curve, p 72vi4.4 Diameter, Pd 76Chapter Five: IMAGE PLANE CELL RESULTS 805.1 Fringe Count Conversion 805.2 Coexistence Curve, p 825.3 Diameter, Pd 895.4 Discussion 97Chapter Six: ISOTHERM CELL RESULTS 1016.1 Calibrations 1016.2 Fringe Count Conversion 1036.3 Virial Fits 1066.4 Isotherms 1126.5 Discussion 117Chapter Seven : CONCLUSIONS 122REFERENCES 127Appendix A: DERIVATIONS OF p AND Pd 137Appendix B: DERIVATIONS OF n IN THE PRISM CELL 141B.1 Theory 141B.2 Calculations 146Appendix C: ISOTHERM CELL DRAWINGS 147Appendix D: INSTRUMENTATION 151D. 1 Digital Pressure Gauge 151D.2 Autocollimator/Telescope 152viiD.3 Quartz Thermometer 152D.4 Null Voltmeter 152D.5 Water Bath/Circulator 153D.6 Decade Box 153D.7 Sapphire Window 154Appendix E: ANALYSIS OF ISOTHERMS 155Appendix F: VITREOUS FLUOROPHOTOMETRY 163F. 1 Introduction 163F.2 The Fluorophotometer 163F.3 Uses 164F.4 Upgrading 165Appendix G: ABBREVIATIONS AND SYMBOLS 168viiiLIST OF TABLES2.1 Comparison of MFT and ROT critical exponents forliquid-gas, (d,n) = (3,1) systems. 293.1 Characteristics of the fluids used. 624.1 Polynomial fits to [4.11 using 2 as reference. 664.2 Electronic polarizability, ae (A3) and coefficients, L012 offits to forH2C:CF. (Figure 4.1) 694.3 Electronic polarizability, ae (A3) and coefficients, L012 offits to e for CH3F. 714.4 Fits to p* [4.21 to find T (C). =(O325}, A={05} and?A is retained. 734.5 Fits to Pd to find p (X iO moFcm’3). 774.6 Summary of the derived critical parameters. 795.1 Fringe count (N) conversion constants for [5.11 and [5.2]. 815.2 Estimates of 7 (in C) using polynomial curve-fitting fromFigures 5.1 and 5.2 for CH3F and H2C:CF respectively. 835.3 Some fits results using [5.3] for CH3F data.{T4j=4490867C (Table 5.2) 875.4 Some fits results using [5.3] forH2C:CF data.{T29}=2974186C (Table 5.2) 91ix5.5 Fits to Pd using [5.4] for image plane data for {7} andcr={089}. The fit in Figure 5.10 uses the resultsmarked. 925.6 Fits to Pd using [5.4] for image plane and prism data at{TJ and ={089}. The fit in Figure 5.11 usesthe results marked . 925.7 Fits to Pd using [5.4] for image plane and prism data at {Tj. 965.8 Fits to Pd using [5.4] for the image plane data of CH3F at u={0’89}. 965.9 Effective polarizability, a, [5.5], and criticalpolarizability product, 1006.1 Fit to [6.4] of p (moFcm3)vs N for CHF3. 1036.2 Fit to [6.4] of p (molcm3)vs N for CH3F. 1046.3 Three CHF3 runs with fit results, reference values rim and% differences for RT, B and C. a and b arevan der Waals constants. 1086.4 Fits to data for which p>p >268x iO molcm3 forCHF3 at Tr=259505C. Percentages arecomparisons to reference values shown in Table 6.3. 1096.5 Two CH3F runs with fit values, reference values [1Th and% differences for RT, B and C, and, a and b. [c3,Mlo] 1106.6 Averages of ôe and d8 calculated using [6.7). 120E. 1 Results of fits to the CII3F isotherm at Tr=449024C,xusing [E. 1] under variation <iv> forp={8765x1O3molcm3}and ô={482}. 157E.2 Results of fits to the CHF3 isotherm at T=2595O5C,using [E.1] under variation <ii> for ô={482}. 158E.3 Results of fits to the CHF3 isotherm at Tr=299505C,using [E. 1] under variation <i> forp{753x1O3molcm3}. 159E.4 Results of fits to the CH3F isotherm at Tr=449024C,using [E. 1] under variation <iii>. 160xiLIST OF FIGURES1.1 A general P-V-T surface for a simple substance whichexpands on melting. [W3] 21.2 (a) Projection of the P-V-T surface onto the P-T plane. 4(b) The projected P-T phase diagram. [W31 41.3 Two paths on a P-V-T surface. [W3] 61.4 Isotherm, Isochore, and Isobar. [w3J 72.1 P-V diagram (isotherms) of a van der Waals gas. T1> 7> T2. [W3j 92.2 Classical universality as indicated by the Law ofCorresponding States [2.6] of eight gases. [SI] 152.3 Sample of a flow diagram in RGT parameter space ofHamiltonians. Critical flow lines are darkened.They converge to the critical fixed point. ii 253.1 Sectional view of the thermostat. The cell and heater blockscan also be configured cylindrical (Not to scale.) 363.2 n or p profiles with height, H in a cell for T> T(supercritical), T 7 (critical) and T< T, (subcritical). 383.3 Optical setup for the prism cell. The thermostat itself isencased in styrofoam and an outer plywood box. 423.4 The Prism Cell. (Approximately full size) Sapphirexliwindow size=(1 dia X ¼ thick)in; cellvolume=(1225±001)cm3.[I’ll] 433.5 Flowchart for the prism cell protocol. 463.6 Density profiles with respect to mirror, M2 for(a) T> 7,; (b) T> 7; (c) T T; (d) T< T; (e) T 7. 483.7 The aluminium cylindrical image plane cell.(Approximately full size) Sapphire windowsize=(l dia x 14 thick)in. 523.8 The Mach-Zehnder interferometer for the image planecell setup. Unlike Figure 3.3, the styrofoam andplywood boards enclose some optical elements as well. 533.9 Top section of the outer water jacket containing theisotherm cell. (See Appendix C for details.) 573.10 The isotherm cell setup. The bleed-line, ht is a stainlesssteel hypodermic tubing of dimensions,((}46 CD X O27 ID)mm [P12]. 594.1 E (cm3mol’) vs p (mol.cm3)forH2C:CF.The verticalline represents the position of PC. A 1 % spreadabout the maximum is shown. 684.2 e (cm3mol1)vs p (molcm3)for CH3F. The verticalline represents the position of p. A 1 % spreadabout the maximum is shown. 70xlii4.3 Coexistence curve (v) and diameter (x) (molcm3vsprism cell temperature, T. (C) forH2C:CF. 744.4 Coexistence curve (v) and diameter (X) (mol cm3) vsprism cell temperature, 7’,. (C) for CH3F. 755.1 Sensitive plot of Tr vs (p)fl for CH3F to estimate T.13={0325} 825.2 Sensitive plot of 7. vs (o)” forH2C:CF to estimate Tj3={0325} 835.3 Log10 p vs log10 t for CH3F image plane data. The solidcurve is a best fit for 7={449O867} and13=:{O.325}. (Table 5.3) 8454 Log10 (r p) vs log10 t for CFT3F image plane data.(Table 5.3) 3={O325} 855.5 Log10 p* vs log10 t for CH3F image plane (+), prism(D) data and best fit (-) for b=b3={O},= {0325}. (Table 5.3) 865.6 Log10 p vs log10 t forH2C:CF image plane and best fitfor 7={2974186}, i3={0325}. (Table 5.4) 885.7 Log10 (t p*) vs log10 t forH2C:CF image plane andbest fit forb,2,3={O}. (Table 5.4) fl={0325} 895.8 Log10 p* vs log10 t forH2C:CF image plane (+), prism(LI) data and best fit (-) for {7,3}. (Table 5.4) 90xiv5.9 Pd vs t of image plane and prism cell data:CH3F(+);H2C:CF((>) 935.10 Diameter, Pd vs t <00035 for CH3F. The straight line isthe best prism cell data fit. The curve is a fitto image plane data. (Table 5.5) Prism data (LI);image plane data (x). 945.11 Diameter, Pd vs t< 0004 for CH3F. The straight line isthe best prism data fit. The curve fits to prismand image plane data. (Table 5.6) Prism data (C]);image plane data (x). 955.12 b0 vs a2. ()H2C:CF; (v) CH3F; (A) CHF3; (X) Xe;(LI) other fluids. 1-2J The solid line is anapproximate linear correlation. 985.13 ap vs a2. ()H2C:CF; (v) CH3F; (A) CHF3; (X) Xe;(El) other fluids. [iu-21 (-) approximate fit. 996.1 Fit to [6.4] of p (molcm3)vs Nfor CHF3.The solid line represents the quadratic fit.The horizontal line is p p. 1046.2 Fit to [6.4] of p (molcm3)vs N for CH3F. 1056.3 CHF3 isotherms at T,.=259400C (A), 259505C (G) and264090C (v). The vertical line is p=p. Theslope is the ideal gas law at T=264O9OC. 107xv6.4 Isotherms of CH3F at 448904C (+) and 449O24C (<D).The vertical line is p=p. 1136.5 Percentage pressure difference between the isothermsof CH3F at 449O24C and 4489O4C.The vertical line is p=p,. 1146.6 Compressibility, K (psi-1) vs p (mol.cm3)plots of fourCHF3 isotherms. X 266436C ; 259400C;+ 259505C ; El = 264O9C. 1166.7 Compressibility, K (psi) vs P (psi) plots of four CHF3isotherms. x 2&6436C; 259400C;+ 259505C; El 2&409C. 1176.8 Log10 K vs log10 jp I plot of the I=4489O4C isotherm ofCH3F. 1186.9 Effective exponent, ôe vs t using [6.7].(>=>zp <0. The solid horizontalline is 3e=482, the RGT value. 119B. 1 Z through the prism cell. A small causes an equallysmall I’2. (Angles are exaggerated; not to scale.) 141C. 1 Top section view of isotherm cell. 148C.2 Heater block for the isotherm cell. 149C.3 Top section view of the outer water jacket. 150E. 1 Effective exponent, ôe vs t I for fits to variousxviranges. El <a.i> ; + <b.i>; 0 <ci>;o= <a.iii> ; X = <b.iii> ; v <c.iii>. 161E.2 Effective exponent, 6 vs t I for various fits.El <1.i> ; + <2.1>; <3.1>;<1.iii> ; x = <2.iii> ; v= <3.ili>. 162F. 1 A block diagram of the (originally) assembled vitreous fluorophotometer. 167xviiACKNOWLEDGMENTTo my father, BOON THYE,my sisters, Lily, Maureen, Molly, Shirley,nephews, Mark, Galen, and nieces, Angela, Megan, Caitlyn,godchildren, Tiffany, Ryan, and good friendswho gave me this privilege of being here for so long,I love you all and thank you dearly for being patient.To my friends, teachers and colleagues,I thank you too for putting up with me. I can be a pain!To CANADA,I thank you for teaching me a few “invaluable lessons”. Some ofthem even overlap in their didacticism but I enter them below (in a more or lesschronological order of realization) as a reminder of what I shall miss on leavingLotusland. For surely, I am not to bring them with me!a. “Good Government” = Patronage.“My way or the doorway!”y. NIMBY = “Not In My BackYard”.ô. The desires of the few far outweigh the needs of the many. STRIKE!C. Cordiality is but skin-deep.How to be poor amid the rich.How to join the UI Tennis Club and the Welfare Ski Team. While they last!0. A sweet mouth and a smooth tongue will get you far.“There’s got to be a bandwagon I can climb on to!”K. How to live in “the best country in the world” and not pay the price.X. Problems? Throw money at it; strike up a committee; write a non-binding report.The “good life” is an entitlement, not due to diligence or chance privilege.p It is nice to have all the “rights” but not the responsibilities.Singapore is boring, sanitary. You don’t even have to “Take Back the Night”.o “Stand up and fight! We want the right to STRIKE!”0. No, you can’t always get what you want!1CHAPTER ONEINTRODUCTION1.1 THE THREE PHASESIn the study of matter, the first characteristics taught are that “matter can exist inthree phases”: solid (s), liquid (1) and gas (g). Usually, the gaseous phase of a substanceexists at high temperatures (7) and high volumes (y)t; whereas, solids occur when T andV are low. The liquid phase appears in the intermediate ranges of T and V. Theoccurrences of these phases can be summarily displayed on a phase diagram or athree-dimensional P-V-T surface like Figure 1.1.A P-V-T surface represents the states of a substance as a function of pressure, P,V and T in mechanical, chemical, and thermal equilibria. Different regions of the P-V-Tsurface are described by different equations of state, each of which relates reversiblechanges in energy of the system to the three state variables.There are ranges of F, V and T where the substance is easily recognized to be ineither the solid (s), the liquid (1), or the gaseous (g) phase. Between such ranges of thestate variables, any two of the three phases may coexist. Such (F, V, 7) points form aV is molar volume in units ofcm3mol’. In later chapters, density (p) in units of molcm3,will be usedinstead. Thus, p= V1. A list of abbreviations and symbols used is presented in Appendix G.The words, “substance” and “system” are sometimes used interchangeably although in some contexts,the latter may include the setup containing the sample (substance).2pcoexistence curve, (or, phase separation line, or, binodal line psi) when liquid and gasphases (l-g) coexist. Note that the line marked s-l-g in Figure 1.1 is the triple line iwialong which all three phases are present simultaneously.Figure 1.1 A general P-V-T surface for a simple substance which expands onmelting. [W3]31.2 THE CRiTICAL POINTThe l-g coexistence curve is seemingly a parabolic arc [(35] lying in a plane lyingthe P-T plane. If projected onto this plane (as shown in Figure 1.2), the two branches ofthe coexistence curve “merge” into a line (now called the vaporization curve or thevapour pressure curve) to maintain mechanical equilibrium. This line terminates at apoint which is the extremum or inflexion point of the coexistence curve. This point iscalled the critical point and is denoted by (P,V,7) or Note that the tripleline is reduced to a triple point in this perspective.Figure 1.2 shows the fusion curve as the projection of the P-V-T surface onto theP-T plane. As no similar termination point has yet been found between the solid (s) andthe liquid (1) phases, the fusion curve is shown extending upwards. In this case, phasetransition or separation must occur at the appropriate (P, V, 7) values.Another characteristic of the liquid-gas coexistence curve and the critical point isthat the dichotomy which establishes what is liquid and what is gas can be circumvented.In Figure 1.3, there are two reversible paths to change the system from point i (whichlies in the gaseous phase) to pointf (which lies in the liquid phase). Path 1 crosses thecoexistence curve twice, entailing continuous condensation of the gas until the state istotally liquid atf Whereas, by moving around the critical point along path 2, no phaseseparation occurs. Hence, the question of when the gas changes to liquid cannot beuniquely answered.The point to be noted here is that for T> T, i.e. above the critical point, a “gas”cannot be liquefied by increasing pressure, P. No phase separation can occur so that4(a)(b)Ts-i (fusion curve)critical point1-g (vaporization curve)triple point-Ts-g (sublimation curve)Figure 1.2 (a) Projection of the P-V-T surface onto the P-T plane.(b) The projected P-T phase diagram. [w3JppSgg5there is no clearcut distinction between liquid and gas. Such states can be designated (bythe more neutral term,) fluid. In terms of the density, p, approaching 7 from below (i.e.increasing T towards Tj, p (or V1) and p (or V) approach the same value, p (or Vj.Thus, at the critical point, T,Pz=Pg=Pc . [1.1]Hence, the liquid and the gaseous phases are no longer visually distinguishable at 7,.The physics of the P-V-T surface around the critical point is an example ofCritical Phenomena. It is the Critical Phenomena of liquid-gas coexistence in pure fluidsthat is the subject of this study.1.3 PhASE TRANSITIONSAs previously discussed (and shown in Figure 1.3), it is possible to take a systemalong a path around its critical point so that no phase change is observed. However, ifthe path crosses a phase boundary and a phase change is seen, a rwst-order phasetransition is said to have occurred because there are discontinuous changes in entropy(with non-zero latent heat), internal energy, and density at the phase boundary. [El]If instead, T is increased towards 7, the liquid and the gas branches of thecoexistence curve meet at the critical point, and a smooth or continuous transition is saidto have occurred. This is called a second-order phase transition where the free energyand its first derivatives are continuous functions of the state variables. wii As entropy isa continuous function, there is no latent heat in this transition. w1,K4]6p1.4 OBJECTIVESThe aims of this work are to:i. study the Lorentz-Lorenz relation, , and the polarizability, ae, of methylfluoride (CH3F) and 1,1 -difluoroethylene (H2C:CF2);Figure 1.3 Two paths on a P-V-T surface. iwi7Figure 1.4 Isotherm, Isochore, and Isobar. [W3]isochore ( V constant)isobar (p constant)isotherm (T constant)ii. investigate the coexistence curves and the diameters of CH3F andH2C:CF usingoptical interferometric methods;iii. map the (near-) critical isotherms of fluoroform (CHF3)and CH3F in as wide arange of density (p) and pressure (F) as possible;iv. compare the results to theory;v. determine if such optical methods are suitable for measuring the critical isothermsand the critical exponent, 8.8CHAPTER TWOTHEORY2.1 THE VAN DER WAALS EQUATIONThe simplest equation of state is that of an ideal gas made up of non-interacting,point particles. The three state variables are related thus:PV=RGT , [2.1]whereR0=831441 Jmo1’K’ is the universal gas constant, with V being the molarvolume (cm3mo11). This adequately describes a real gas at high temperature and lowpressure where no phase transitions can occur.The first successful description of (first-order) fluid phase transition was made byvan der Waals [Wi] in 1873. His empirical equation of state is:T [2.2]v2Rwhere a and b are (phenomenological) molecular characteristics of the substance. Thecorrection term, aV2, accounts for the long-range attractive portion in the inter-particleinteraction, whereas, the correction term, b, accounts for the repulsive portion arisingfrom the fact that molecules are not point-like and must occupy volume which reducesthe “free space” available. [W3]90pIII/IVFigure 2.1 P-v diagram (isotherms) of a van der Waals gas. T1> 7> T. w3]By expanding [2.2], the resulting cubic equation in V has three roots if T< 7.(Figure 2.1.) As temperature, T, is raised towards 7, these roots become degenerate atT, i.e. the two branches of the coexistence curve coincide. As T is increased further sothat T> 7, the isotherms become rectangular hyperbolae which can be described by theideal gas law of [2.1].critical point (p fri,T1TLII/i 3bI,F10A problem arises for the states in segment BD in Figure 2.1. The positivegradient implies that in an isothermal volume expansion, pressure (P) also increases. Thisimplies a negative compressibility (K) which is mechanically unstable. Hence, [2.2]cannot properly describe states in BD for T< T. Phase separation must occur to statesA and E, the points of equilibrium or coexistence.States in segments AB (supersaturated gas) and DE (superheated liquid), althoughmechanically stable (negative gradients), are however, thermodynamically unstablebecause their free energies are not minima. States in these two segments are metastableand have finite lifetimes. Phase separation may not occur immediately depending on themagnitudes of characteristic density fluctuations (that will drive any instability towardsequilibrium on the coexistence curve). pcjThe inadequacies of [2.2] in these segments are eliminated by using the Maxwellequal-area construction. [88,M2,R1,sl] This method is used to define the density values of Aand E as intercepts of the subcritical isotherm and the coexistence curve. It relies on thecondition that in equilibrium, coexisting phases must have equal Gibbs free energy whichentails equal temperature (T, i.e. isotherm), equal pressure (P, i.e. mechanical stability),and equal chemical potentials (I.L):PIIgHence, points A and E are joined by a straight horizontal line which divides out equalareas “under the P-V curve” with respect to pressure, P, (for the same Gibbs freeenergy). [K3,R1]112.2 THE VIRIAL EQUATION OF STATEA logical step towards an equation of state for a real gas is to make a virialexpansion of an isotherm of [2.1]. In terms of the density, this virial equation of stateis written as:P(p)=RT{p + B(T)p2 + C(T)p3 +... } . [2.3]B(T), C(T), etc, are respectively the second, the third, etc, virial coefficients. These arerelated to and can be calculated from intermolecular potentials. [IU] For values of density,p, that are not too large, only the first few terms of [2.3] need be kept. All virialcoefficients go to zero for an ideal gas. For the van der Waals potential, B(T) =b-aIRGT[R1,W3] where a (jsicm6.mo12)tand b (cm3mo1’) are the molecular parameters definedin [2.2].2.3 MEAN FIELD THEORIESThe van der Waals equation turns out to be an exact description for systems withthe following potential, pczlu,siiT<TU(r) — 2.4]I — rr0where U(r) acts on a molecule at a point, r. The limit, r0, is a characteristic size of theparticle. Analytical equations of state, like [2.2], are examples of mean field theoriesThe unit of pressure, P is pound per square inch (psi) for ease of computation and concordance withthe digital pressure gauge discussed in Chapter 6.12(MFT) in which the interaction potential is replaced by a constant - usually the averagevalue (Uj of the potential due to all other particles acting on the one. This may beinterpreted as one particle interacting equally with all other particles, resulting in aneffective self-consistent approximation of the many-particle interaction. LK3] In this way,long-range density (p) fluctuations and correlations are neglected. This substitution islogical insofar as classical theories (like van der Waals’) have the inherent assumptionthat inter-particle interactions are long-range.Mean field theories (MFT) provide the historical, the qualitative, as well as thephenomenological perspectives to understanding phase transitions. Other examples ofMFT are the Weiss theory on ferromagnetism [woj, and the Ginzburg-Landau theory ofsuperconductivity w7i. These “classical” theories are adequate as qualitative descriptionsbut fail quantitatively as they break down in the neighbourhood of the critical point.Thus, there exists a region, 9, around the critical point within which MFT do not work.This is due to three interrelated phenomena which arise when a system approaches itscritical point: [Si](a) There is an increase in density fluctuations, p.(b) There is an increase in compressibility, K.(c) There is an increase in the range of p-p correlation.The definition of the limits of 9?, (which is difficult to do experimentally ij.2,M61), and thecreation of theories to describe the “crossover” into regions where classical theories workare of current interest in Critical Phenomena. [A3A5,B1,c2,L2,T6,w121132.4 THE ORDER PARAMETERThe different versions of mean field theories may be “combined” using theconcept of the order parameter. [U] An order parameter is defined as an n-vector(n= 1,2... co) that is non-zero for T< T in the two-phase region. It makes a discontinuousjump to zero when the system crosses its coexistence curve into a one-phase region atT< T, i.e. in a first-order phase transition. It goes smoothly to zero at T=T, whenapproaching T, from below (in the two-phase region), i.e. in a second-order phasetransition. It remains zero for all single-phase states, and/or all states for which T> 7.It is a measure of the amount of ordering in the system at temperature, T. pc,pjIn liquid-gas systems, the order parameter to be used is the average differencebetween the densities of the coexisting phases, and reduced by the critical density, p. Itdefines the coexistence curve, and is given by:pS= PzPg. [2.5]2p2.5 CLASSICAL UNIVERSALITYThe van der Waals equation, [2.2] works well in describing properties commonto all gases and liquids but not precisely in describing the specifics of a particularsubstance. Nevertheless, its success suggested a characteristic called (classical)universality where different substances can be described by a general equation of state14in which material-specific parameters (such as the molecular parameters, a and b in[2.21) have been removed or reduced out of the equation. For the van der Waalsequation, [2.2] is reduced and rewritten into what is called the law of correspondingstates:+ 3pW2) (-;; — = 8Tw [2.6]by using the substitutions,Pw Tw=!; [2.7]for whicha 1 8aP= ; p=—; RT=—; [2.8]271,2 C 3b C 27band,PC =.3 [2.9]RGTCpC 8In this form, different substances with the same values of Pa,, Pw, and T, are saidto be in corresponding states, even though their critical points are totally different. [2.6]is a “universal equation of state” for ny substance (van der Waals gas) whose criticalpoint is known. [L2J [2.9] is also universal. [sI,w2] Thus, physically different systems can15T/Tbe categorized into universality classes described by one equation of state, andcharacterized by a certain set of parameters. This universality is illustrated in Figure 2.2for eight different gases. In this way, mean field theories predict critical behaviour thatis universal, independent of the characteristics of the specific systems.2.6 POWER LAWSIn Figure 2.2, there are no data-points near p =p. Also, in substituting the critical[2 1.4Figure 2.2 Classical universality as indicated by the Law of CorrespondingStates [2.6] of eight gases. [Si]16parameters of substances into [2.9], the derived values are neither ¾ nor the sameconstant value. [sll,w31 These show that [2.6] provides a qualitative description of thephenomenon outside the critical region,•It neglects density fluctuations, and thecorrelation between them (as mentioned in §2.3). This is an example of the failure ofmean field theories inIn 9, the order parameter, p as defined in [2.5] is found to obey the relation:p*= B0 t , [2.10]where t=(T-T)IT for T T, is the reduced temperature. The exponent, j3, is a criticalexponent; B0 is a system-dependent coefficient or amplitude. [A4]Other thermodynamic quantities follow similar power law dependences on thereduced temperature, t in critical region, For example, the dimensionless isothermalcompressibility, K, is given by:ic = --=[2.11]P T*The ± indicate whether the system approaches the critical point, T from above 7 alongthe critical isochore, p =p, or, from below 7 along the two branches of the coexistencecurve, respectively. The quantity,,is another critical exponent, and 0± aresystem-dependent amplitudes. CM] The coefficients, F0 and F0 are not equal although theexponent is independent of the direction of temperature change. ni This inequalityt is sometimes used interchangeably with t=(T-TjIT. Usage will depend on context: whether t isabove or below T. t is generally used in formulae.17represents the difference in the shapes of the two branches of K above and below 7,, i.e.they are not symmetric about 7.The specific heat at constant volume, ci,, on the critical isochore is given by:= FItI , [2.12]where a is the associated critical exponent, and F and F are unequal, system-dependentamplitudes. [A4,07] As before, the ± indicates the directions of approach of T towards 7.The specific heat at constant pressure, ci,,, is proportional to t ““, like the power law forthe isothermal compressibility, K in [2.11]. ijAn important gauge of the size of density fluctuations is the correlation length,. This particular parameter is not accounted for in classical theories since fluctuationsdo not exist in those models. The correlation length measures the extent of the densityfluctuations in the system so that any thermodynamic quantity observed at a scale lengthgreater than , remains unchanged.For T> T, the relevant length is the smallest length in the system at that valueof temperature, T. (The average inter-particle distance is usually chosen as a goodrepresentative). If the sizes of density fluctuations are of this extent, these fluctuationsare independent of one another, and cannot be “correlated”. The fluid remains stable, andno phase separation occurs. However, near the critical point,=, [2.13]where v is the critical exponent and are the unequal, system-dependent amplitudes.18[F1,M1] That is, the behaviour of different fluids near their respective critical points isdominated by size, , which extends even farther as each T is approached. Interactionsat length scales smaller than the correlation length, , need no longer be considered (inany fluid). In this way, these different fluids can be grouped as a class of substances thatbehave in the same manner as they approach their respective critical points.A noticeable effect not predicted by any classical theories is optical criticalopalescence as T—T. This is due to the correlation length, , growing to the extent thatthe relevant length scale is of the order of optical wavelengths. Regions of such sizefluctuate coherently, and the fluid becomes opaque as light scattering increases. ioi,ijThe last power law (of relevance to this work) describes the critical isotherm in thecritical region, 9t,.* = PC ji(p)—= I P(ii)—IT T2.14]API =DoVi =D0IAp*IJT P3 in [2.14] is the critical exponent. D0 is the system-dependent amplitude. Ip1,F1,M1,sulAs the correlation length, , diverges. So do the isothermalcompressibility, K ([2.11]), and the specific heat at constant volume, c, ([2.12]), wherethe power laws are the reduced temperature, t, raised to negative critical exponents. Suchnon-analyticity at the critical point is then characterized by the respective criticalexponents. This is the principal difference - non-analyticity in the equations of state andsingularity in some thermodynamic functions - when compared to the mean field theories19(MFT) which are always analytic. MFT (as such [2.6]) also do not predict the correctcritical exponents. (See Table 2.1.)2.7 SCALINGThe theory of Critical Phenomena of second-order, continuous phase transitions,utilizes the occurrence of fluctuations on all length scales. This leads to the idea ofuniversality or universal behaviour- independence from all microscopic details of thesystems under consideration- and universality classes of substances. Such behaviour doesnot occur in first-order phase transitions where length scales are well-defined by thespatial sizes of the fluctuations in the coexisting phases. pciWidom’s Scaling Hypothesis wsi was among the earliest constructed to includethese fluctuations. It states that only long-range correlations (large correlation length, ),as T—’7, are the cause of all singular behaviour (in thermodynamic functions). iai Itrelies on the assumption that the Gibbs potential is a generalized homogeneous functiont[1381 so that the thermodynamic quantities (derived from it) are also homogeneous functionsof the “distance” from the critical point, (P,p,Tj. This means that the form of theequation of state remains the same albeit the state variables, (P,p,T) have gone throughchanges of scale as T-’7,. n2o,M1,s1,w9] An example is the reduced van der Waals equationA function, f (x,y) is homogeneous if and only iff(X,Xy) = Xmf(x,y)for all values of the scaling parameter, X. m, called the degree of homgeneity, is unspecified.f (x,y) = ytmf (x/y, 1) = ym .7(xly)or, f (x,y) = xmf (1 Iy) = xm tyIx)The function has been scaled (reduced) by a factor of X= l/y or X= lJx in each of the above equations.Significantly, a two-variable function, f has been reduced to a scaled one-variable function, .?or [Slj20[2.6] in which the state variables are scaled by their critical values. (Figure 2.2)The critical isotherm, expressed in terms of the reduced chemical potential, ,under the Scaling Hypothesis becomes11*___ItI6 ItIi3?(.) is the universal scaled function and tp is as defined in [2.14]. This equation isanalogous to the results for the law of corresponding states, [2.6], shown in Figure 2.2.[B8,F1,M1] The various gases, appropriately scaled by their respective critical points, andreduced temperatures, t, fall along the same universal curve, .9Besides the power laws, the Scaling Hypothesis also predicts the scaling laws:+ y + 2/3 = 2+fl(1+ô)=2 ; [2.15]= p + = /3(5Like [2.6], they are independent of the various physical characteristics of individualsystems. [SilThere are other scaling laws that relate the critical exponents to the physicaldimension, d. [A4] (These are discussed later.) The above equations lead to an importantcorollary: at this juncture, only three critical exponents must be known in order todetermine the other two. iij It should be noted that the Scaling Hypothesis itself doesnot predict the value of each critical exponent; it only establishes their interrelationsthrough the scaling laws.Kadanoff pcij gave a physical interpretation of the Scaling Hypothesis when he21used it to describe Ising-like systems. In that work, he invented the idea of interactingspin blocks which were groups of individually interacting spins. The scale was thuschanged from that of single, interacting, nearest-neighbour spins, to a scale of interactingblocks (or groupings) of individual spins. The spin of a spin block was the mean or neteffective spin of all the individual spins in that block. In this way, the spin block becamethe unit of interaction, and the details of each spin within a block were no longerconsidered. pcii Thus, an individual-spin-to-individual-spin interaction Hamiltonian wasrevised to a block-to-block interaction Hamiltonian at a larger scale.By rescaling and re-grouping the effective spins of spin blocks into larger blocksmade up of these “individual” previous effective block spins, the relevant scale lengthbecomes longer than the correlation length, , even as T—’7. This is mathematicallyequivalent to moving away from the critical point as block sizes increase and the numberof interacting units (blocks) decreases. In this manner, details of the microscopic scaleare gradually eliminated. This procedure is sometimes called smearing or coarsegraining. rin It reduces the number of degrees of freedom, i.e. the number of interactingSPiflS. Wi]The previous discussion on , and the spin blocks implied a grouping together ofpure fluids with the same critical behavior. Similarly, the Scaling Hypothesis predictsthat there exist broad classes of substances (universality classes) which have the same setThe Ising model is a crude model of ferromagnetic material or analogous systems used in the studyof phase transitions. Atoms in a d = 1,2,3 lattice interact via the Ising coupling between nearest neighbours. Thespins (of the atoms) are coupled to a uniform magnetic field. The Ising coupling assumes that along some axis -usually the direction of the magnetic field - each spin takes on a value of either + 1 (up) or -1 (down). Also,the energy of interaction is proportional to the negative of the product of spins along this axis. n= 1 in this case.[P2]22of critical exponents (for each class), although their individual (physical) characteristics(like T) may differ significantly. This is because the microscopic interactions ofseemingly disparate systems are overwhelmed by the correlations occurring at the muchlarger length scale, . For example, the Weiss ferromagnet belongs in the sameuniversality class as the liquid-gas system, as do binary mixtures of pure fluids. wiiThrough [2.15], and comparing to results from experiments, mean field theories(MFT), and other model-independent computations, it is clear that other dependences likespatial dimension, d, must also affect the values of the critical exponents: different setsof critical exponents are derived for different sets of d and n (the number of terms in theorder parameter). MFT and the scaling laws are either incomplete or incorrect since dand/or n do not appear explicitly in their equations. unj2.8 THE RENORMALIZATION GROUPThe Nobel Prize-winning theory of the Renormalization Group Transformations(RGT) is a mathematically completet theory of Critical Phenomena. wioj It consists ofa set of symmetry transformations which operates in two steps:a) coarse-graining (which enlarges the spin block sizes); followed by,b) a change of scale (which shrinks them back to their original sizes).Hence, ROT comprises the Kadanoff interpretation of reducing the number of interactingHowever, the set of transformations does not form a mathematical group because no inversetransformations are defined. ivfl]A symmetry group is a set of transformations that are equivalent to combinations of translations,rotations and reflections. [Ti]23spin (blocks) and the Scaling Hypothesis. 8,M11RGT derives all the scaling laws and predicts under what circumstances and howthe Scaling Hypothesis itself, fails. It operates on functions in a parameter space madeup by P, p, T, ... In the parameter space, a fixed point is defined as a particular pointwhich when operated on by any elements of the group, maps back onto itself, i.e. it isinvariant under the group transformations. The characteristics of fixed points thus providethe quantitative asymptotic behaviour of systems near their critical points. If functionssuch as the Hamiltonian, free energy density or the correlation length, , are invariantunder the group, universality is then indicated. [A4,MI] The critical point must itself be afixed point. jiij This explains why the equation of state has a homogeneous, universalform, determined by the properties of that critical fixed point. [A4] The critical exponentsare then related to the properties under the set of symmetry transformations of points inthe neighbourhood of the fixed points in the (P,p,7)-parameter space. Hence, theexistence of fixed points is vital to Critical Phenomena.The net effect of repeatedly applying these transformations is to move a criticalor near-critical system about its critical fixed point. In this sense, the system flows alongin parameter space under each application of the transformations, carrying within therenormalized equations, all the scaled information of the previous steps. This producesa set of recursion relations showing the changes in scale. With these recursion relations,the form of the original equations can be maintained by rescaling them in such a way thatthe post-transform problem looks just like the pre-transform problem.As mentioned previously, RGT also explains under what circumstances certain24special types of singularities occur; for example, the “confluent” logarithmic singularityin the specific heat at constant volume, ci,, at 7. wu It also derives the hyperscalingrelation, dv=2-a [A4,uS,F1] which incorporates the spatial dimension, d, in a rigorousmanner. When this is included in [2.151, the number of critical exponents needed touniquely determine all others is reduced to two! Note that the hypersealing relation holdsfor d <4 but fails for d> 4. ii No scaling law with the order parameter dimension, n,explicitly included has yet been found. That it is an important dependence is seen in thedifferent critical exponents for systems of different n, (as mentioned previously).2.9 UNIVERSALITY REVISITEDThe concept of universality is automatically incorporated by the prediction thatin the critical region, 9, the system behaves according to d, n, and the long- orshort-range intermolecular potential involved. [A4,K1,wll] Since there is no dependence onthe details of the Hamiltonian of (microscopic) interaction, the idea of universality classesof substances naturally follows. For the previously mentioned examples of universalityclasses, pure and binary fluid, and ferromagnetic systems are Ising-like with spatialdimension, d=3, and order parameter dimension, n= 1. An n=2 example is the critical(A) point of superfluid helium whose order parameter is a complex number (quantumamplitude of He atoms). [A21In the flow-in-parameter-space picture, universality and universality classes ofsubstances are explained thus. Different systems exist at different points in the sameparameter space (as they have different critical points). Upon applying the appropriate25\Figure 2.3 Sample of a flow diagram in RGT parameter space ofHamiltonians. Critical flow lines are darkened. They converge to the critical fixedpoint. [Fl]RGT, each critical point flows to a renormalized critical point further along in parameterspace. If, after several iterations, all the renormalized critical points converge to thesame critical fixed point, then all these systems fall into the same universality class withthe same set of critical exponents, scaling functions, etc. (Figure 2.3)Different systems (of the same universality class), starting from different criticalpoints usually require different numbers of transformation (RGT) applications (i.e.iterations) to converge (if at all) to the critical fixed point. This is because their physicalcritical points in parameter space are at various “distances” from the critical fixed point.The speed of convergence depends on the scaling parameter used, the stability of thephysicalcriticalpointphysical manifoldt, h)renormalizedmanifoldfixed point J•(*26relevant fixed point, and the proximity to the physical critical point (if the system is notinitially its critical point). Hence, to form a general theory, the idea is to map theflows in parameter space, seek out all fixed points and their ranges of influence (i.e.attraction), and study their properties in these domains. n8j2.10 CORRECTIONS TO SCALiNGSome pre-RGT experiments indicated that the critical exponent, f, under theScaling Hypothesis, seemed to have a temperature dependence (which would violateuniversality). a’i,L2] As the system moves away from the critical point, departure from thesimple power law relation appears. One consideration of why such deviations occur isthat as T—’7, and the correlation length, , grows very large, the fluctuation scaleexceeds the physically finite scale of any experiment. Also, to maintain an experimentat T (over long periods) is difficult. An experiment is almost always or can only be doneat some (finite) distance from T. The question is: How close must “close to 7” bebefore being asymptotically closet so that only pure power laws hold true? u,mjA general form for any thermodynamic quantity, f, in terms of the reducedtemperature, t, includes a power series to account for the “displacement” from T, i.e.when the system is not within 9?. As such, the power series are corrections to theasymptotic power law. Thus: 1,s1]A (classical) qualitative guide with which the size, t of is estimated (i.e. where fluctuations becomesignificant and mean field theories break down) is called the Ginzburg criterion. It depends on fluctuationsbeing much smaller than . [G1,K2,L2,M1] Note that the size of like T is non-universal; it depends on thecoupling constants (of the interaction). [B8,13]27f(t) =a01t12 { 1 + a1tJ4’ + a21t14 ÷ } [2.16]a0 is the system-dependent amplitude, and X is the critical exponent. In the criticalregion, 9, [2.16] reverts to a simple power law form when the non-universalcorrection-to-scaling amplitudes, a1, a2, ... go to zero., ,... sometimes called the“gap exponents”, are predicted to be universal as well. W1,R3,w4]Specifically, the coexistence curve or order parameter, p, is now written as:= B0t3 { 1 ÷ B1tI’ + B2ItIA + } [2.17]Comparing to the form in [2.16], the gap exponents are replaced by z2 2A1 and i =i.in the above equation. [L4]On the critical isotherm, recalling the definitions for the reduced pressure, I.P*,and the reduced density, in [2.14],IT =D0p*ô {±i +D1IAp*I } . [2.18]The diameter of the coexistence curve is defined as the reduced average of thedensities of the two coexisting phases. It is given by:- P1 + PgIn the classical regime, the law of rectilinear diameter of the coexistence curve is: [Cl]Pd = 1 + A2t . [2.19]28This linearity is due to the reduced temperature (t) range being outside the critical region,,and obeying the analytic equation of state. iusi The minimum value of t for therectilinear diameter to hold true is system-specific and can be as large as 1L2,M61The possibility of many-body interactions having an effect must be considered dueto the absence of the hole-particle symmetry (in the lattice-gas model1)in real fluids. Tothis end, a field-mixing term which reflects the weak singularity in the specific heat atconstant volume, ci,, is added. [021 Lastly, with correction to scaling also included, Pd thenbecomes: [M3-4,w13]Pd = 1 + + A2t + + [2.20]These additional terms introduce a curvature to the straight-line portion as T-’Y. Suchdeviations have been observed in polar and non-polar fluids. n5,Di,o2,u,N1,P1,w5] The field-mixing t1” term is difficult to detect experimentally because its exponent is close to 1,and therefore of nearly equal magnitude to the t term. [o2,P1,w51 (See Table 2.1.)The departure from the straight line diameter is a good first indication of the sizeof the critical region, 9i., in terms of the range of the reduced temperature, t. It alsoshows the effects of (adding more) correction terms on the t range. However, as they arenot universal, the amplitudes of all correction terms vary from substance to substance.Hence, the size of the asymptotic t range for pure power law dependence will vary fromsystem to system, in general. [UIThe lattice-gas model was developed to parallel the Ising model (for ferromagnets) by assuming gasparticles can only exist in a lattice structure. The ± (up/down) spins in the Ising model is equivalent to thepresence/absence of gas particles at each lattice site. [L12,Mi]29As the t range increases, more correction terms need to be included. The enlargedrange does not redefine nor mean that ‘Y is larger than it is (for a specificsubstance). It simply means that the inclusion of more and more correction terms allowsfor T to be further away from 7. This can mean the inclusion of more data points, andyet, may still allow for the extraction of the critical exponents, etc. Thus, there is nolonger a need to get asymptotically close to the critical point. This is especially usefulsince ¶R has been estimated to be quite small in liquid-gas systems. ip2,H1,M6,N2,u] Notethat these correction expansions are poorly convergent because of the small magnitudeof the A gap-exponent.Unfortunately, including more correction terms and extending the data range maynot necessarily yield more accurate results. Firstly, in expanding the data range used inCRITICAL MFT RGT REFEXPONENT VALUE VALUEa 0 0l09 A1,405 0325 ± 0006B1-2,81 l24l ± 00043 482 ± 006 Mlv 05 0630 ± 0002 Vi0 0031 ± 0011A- 0496Notes:(a) Mean field theory (MF) values are calculated using [2.6].(b) is the critical exponent in the correlation function.(c) Each set satisfies all scaling laws, [2.151.(d) MFT values satisfy the hypersealing relation for d =4 only.(e) References (in right column) do not match specific rows.Table 2.1 Comparison of MFT and RGT critical exponents for liquid-gas,(d,n)=(3,l) systems.30the curve-fit, consideration must be given to possible “crossover” limits, i.e. crossingover into classical (mean field theories) regime where different (analytic) equations ofstate are used. [A3,C2,L4] Secondly, an increase in the number of data-points may result inuncontrolled (implausible) changes in the free parameters of the fit, especially when thenumber of these free parameters are steadily increased.2.11 OTHER METHODSThere exist other means of calculating critical exponents. For example, high- andlow-temperature series expansions (of coupling constants) 8,P2], perturbation, and othermodel-dependent methods, the c =4-d expansion derived from Renormalization GroupTransformations (RGT) [A4,F11. All pre-RGT techniques sought to test the validity ofuniversality in various models, how the critical exponents depend on the spatialdimension, d, and the dimension of the order parameter, n, and, if there were any otherdependences. Series expansions are the most widely applicable, producing “quantitativelyexact” results si that confirm that the only relevant dependences are on d and n. [Ml]Specific models for pure fluids have been designed by parametrizing the orderparameter in terms of parameters that, for example, measure the “distance” from thecritical point. [s3-4] Such models describe the thermodynamic behaviour relative to thecritical point. However, they do not predict the latter. They work well in the criticalregion, correctly incorporating the singular behaviour, predicting the criticalexponents, and deducing explicit relations for the scaled thermodynamic potential andequation of state. ii Unfortunately, certain ranges of the parameters sometimes produce31“unphysical” results. ii Other workers have attempted to bridge these parametric modelsin to the analytic equations of state outside (e.g. van der Waals). These are thecrossover models. [A3,A5,c2,H2,T6,w12]Another observation is that pre-RGT calculations show a critical behaviour whichvaries according to specific ranges or values of d. As stated before, d=4 seems to havespecial status. RGT is required to explain rigorously (the occurrence of) such restrictions:when, how and why such calculations fail at certain values or ranges of d.Thus far, RGT-calculated critical exponents have concurred with experimentalresults for systems in their 9?.32CHAPTER THREESETUPS AN]) PROCEDURES3.1 FOUNDATIONSThe basis of the present experiments is that close to their critical points, pure fluidcompressibility, K, [2.11] becomes very large. Layers in a column of fluid cumulativelyweigh on every layer below them. The variation of pressure with height results in adensity gradient. This gradient can be deduced from the refractive index, n which canbe (optically) measured; for example: in scattering, or in a direct laser beam deflectionmeasurement, or in an interferometric setup. [B9,21,G8,H5,L15,M18-20,S9-1O,12-13,W14) The latter twomeans of probing critical regions of pure fluids are the basic techniques used in thisstudy. They have been previously applied to the study of Critical Phenomena in severalpure t,-7,1o-14,D1-3,,N1-2,P4,s2], binary iu,N1,s5] fluids and liquid crystal systems un2-23,M17,P5,131.Other methods in use w27], such as equation-of-state experiments m or usingstacks of capacitors [P1,6-7] - which measures fluid density, p, as a function of thedielectric constant, e, via the Clausius-Mossotti relation [A6,P7-SJ-are resolution-limitedby gravitational roundingt.The advantage of using optical wavelengths as a probe is that it reduces the effectst Gravitational rounding effects are most pronounced when T is approached. As ic increases, the density(p) gradients are very high. Hence, p, as measured by sensors of macroscopic height (like capacitor plates) isno longer constant over that height. The resulting measurement is an average over that height. pc5,L5,M6,P1,s6]33of gravitational rounding by about a decade in reduced temperature, t (compared to usingcapacitor plates). 6] Gravitational rounding becomes significant when the correlationlength, , grows to the order of the optical wavelength, X, as T approaches 7,.Therefore, the sub-micron X, (as opposed to the fixed, finite height between capacitorplates) probes closer to the critical points by allowing for steeper changes in density, p,before the gravitational averaging effect becomes significant. 1,M6] This limitation isfurther reduced if horizontally thin cells are used. Reducing the optical path through thesample lessens the beam deflection, keeping the beam within the ensuing lens aperture.(See below.)3.2 TEMPERATURE CONTROLIn the present set of static Critical Phenomena experiments, the systems understudy are in thermodynamic equilibrium. The (sometimes substantial) lengths of timeneeded to achieve such equilibrium require precise temperature control and maintenanceof T. This is especially true in the study of the (near-) critical isotherm where T must beheld constant (as close to 7 as possible) throughout a run. An added diffculty is that thesystem “slows down” as p—p) Hence, temperature-control requirements are that T mustbe maintained at better than fractions of a milli-Kelvin over periods of hours or days.The fluid cell volume is usually machined out of a cylindrical or rectangularaluminium block. The cell block fits snugly in a larger aluminium block. Foil heatingThe relaxation or equilibration time for the system to re-establish equilibrium after a change in Tincreases with proximity to T. [D4,s7134elements rii are attached to the outer surfaces of this larger (cylindrical or rectangular)block which is called the heater block.To sense the transient, instantaneous temperature, Tb of the heater block, severalthermistors LFZ] are embedded at strategic points in the block- as close to the cell volumeas possible. All, except one thermistor (designated Th), are calibrated to measure theabsolute local temperature, Tb of the block, to check the constancy of Tb in time, and/ortemperature gradients across the block. Note that in equilibrium, Tb is assumed to be thetemperature of the cell (Tr) as well. This assumes that there are no temperature gradientsacross the block and the cell.Each thermistor (with a current-limiting resistor in series) is connected to an armof a resistance bridge (powered by a 1 35V mercury cell for stability). The opposing armhas a decade box resistance mi, while the remaining arms are balanced with equalresistances, R. The output of the bridge goes to a null voltmeter/amplifier Eachthermistor resistance (which is a function of its temperature) is determined by adjustingthe decade box to cause the connected voltmeter to read zero. The temperature of the cellis measured by a quartz thermometer (QT) [H4J which is calibrated using the InternationalPractical Temperature Scale of 1968 for which the triple point of pure water is definedas 273 16K (or 0010C) in a properly prepared triple-point cell of pure water uij. Allthermistors are calibrated against QT. (See Appendix D.3.)The thermistor, Th, is used to control the current (1) applied to the heating foils.It is also connected to a resistance bridge. The bridge imbalance is sensed and amplifiedby the voltmeter which drives an operational amplifier-power supply (OPS) [1(6] which is35the source of I. The decade box resistance, Rd is then used to set Tr - the temperatureat which the cell is required to reach and maintain thereafter. The OPS maintains Tb=T,.by sensing the magnitude and polarity of the bridge imbalance and reacts by adjustingI as follows:(a) if T< Tr, i.e. Th > Rd, OPS heats the block by increasing I;or, (b) if T T, i.e. Th Rd, OPS decreases I to zero and holds.If I is initially non-zero, the resistance of Th decreases continuously towards Rdas the block heatd up. The bridge imbalance falls with I until the bridge is balanced. Ifoverheating occurs, I is held at zero until Th increases to a value close to Rd. In thisway, the block temperature, Tb is directly controlled. To attain and maintain a specificcell temperature, Tr, Rd is increased or decreased to lower or raise Tr respectively.tThe block is encased in 5cm-thick, styrofoam insulation. The entire setup issurrounded by a “water jacket” (WJ) through which temperature-controlled water froma bath [P3] is circulated. The temperature of the water is usually kept between 05 to 1Celsius below Tb, depending on the proximity of 7 to the critical temperature, T, thespeed at which the next 7 is to be attained which in turn, is determined by the thermalmass of the block and the current 7..The purpose of such an arrangement is to carefully balance the temperatures ofthe water jacket and the heater block. There is a (controllable) lag time in the responseof the power supply to the (rates of) changes in Th, Tb and 7’,.. A water temperature, Tthat is significantly less than Tb can cause temperature gradients across the cell volume.Note that Tb and Tr are sometimes used interchangeably.36LEGEND : E l.35V mercury cell; TN = Thermistor-N = (1,2,...); W Sapphire WindowFigure 3.1 Sectional view of the thermostat. The cell and heater blocks canalso be configured cylindrical. (Not to scale.)37If Tn., is close to Tb and the feedback-heating system reacts too quickly, 7 may go intoan overheat-slow-cool-down cycle, oscillating like an “under-damped” system.Conversely, an “over-damped” reaction can also occur.It is sometimes necessary to shield the water jacket from air currents in thedarkroom by enclosing it in another layer of styrofoam which is then boxed up withplywood boards. The whole system - heater block, Styrofoam layers, WJ, circulator-bath,thermistors, electronic control circuitries, (excluding the cell) - will be referred to as thethermostat in later discussions and diagrams.3.3 OPTICSIn the setups described below, there are two basic optical arrangements. Bothrequire a He-Ne laser for illumination (X=6328A). [1v113J The laser beam is expandedthrough a spatial filter oj to approximately 4cm in diameter, and is collimated to obtainplane wavefronts. The intensity can be reduced by using polarizers. The beam is thensplit into two branches. One branch goes through the cell as the probe beam; it isdesignated Z before passing through the fluid, and Zd after. The other branch, goingaround the outside of WJ, plays the part of a reference beam (Z0).Z is deflected by spatial variations in the refractive index, n, which is related tothe density profile of the fluid in the cell. In the T 7 extreme, the fluid is in itsone-phase regime and the density is homogeneous throughout the cell. (Figure 3.6a) ForT 7, the fluid has phase-separated. The gas is demarcated from the liquid by aninterface or meniscus. The density of each phase is also homogeneous. (Figure 3.6e)38C.)SUPERCP T I CAL CRITICAL SUBCR T I CAL(supercritical),As Tr approaches T, the “corners” of the meniscus begin to round and the densityprofiles become non-homogeneous but remain monotonic. The density profile becomes“sigmoid”. (Figures 3.2 and 3.6c) Zd is altered significantly by the density gradientsimmediately above and below the meniscus. Deflection occurs over the width of the beamas well. These effects become more and more pronounced as 7.In the following sections, the setups for separate experiments and protocols fordata collection are described. The means for converting the collected data to usefulvalues of n, p, etc are also discussed.I I IREFRACTiVE INDEX, n or DENsITY, pFIgure 3.2 n or p profiles with height, H in a cell for T> 7T 7, (critical) and T< T (subcritical).393.4 PRISM CELL ThEORYA prism deflects an incident ray through refraction. The deviation, 0, of the raydepends on the incident angle, the prism (apex) angIe, 4), and the index of refraction, nof the prism material. If 4) and the incident angle are constant, while n is varied, 0 alsochanges in a predictable manner. [H5]For a fluid contained in a prism-shaped cell [slo,141 at 7. 7:,, the refractive index,n, can be deduced from 0. Knowing the fluid mass in the cell and the cell volume, anaverage density can be calculated. This density may be reduced by bleeding smallamounts of the substance out of the cell.For a dilute gas, there is a formula between the density and the index ofrefraction called the Lorentz-Lorenz relation: Lr)5,L7-8,p9,s8,12-14,T41112 11 4p n2+2 [3.1]3= 4 Nlimp..0,_1 (n,p)ltANA is Avogadro’s number. For gases, E is nearly a constant over a wide range ofdensities, and even extending into the liquid phase. Deviations of from constantbehaviour are a measure of the effect of the fluctuations of dipole moments from theiraverage values arising from motions of the molecules in the electric field (of the laserradiation). Determination of near the critical point may therefore be important.[3.1] is a version of the Clausius-Mossotti function for non-ferromagnetic, dielectric polarizability atoptical wavelengths. 1A6,P5I40If a, known as the electronic or optical polarizabifity, is constantt, SJ(n,p) isapproximately independent of temperature (7) and pressure (P). This is true at lowdensities for non-polar gases in which the electrons alone can react fast enough to opticalfrequencies. [G8,p9,s12-13,v2] In polar or dipolar molecules, however, polarizabilities due toatomic polarization and/or orientational symmetry of the molecules may have to be addedto c, to form a total polarizability in [3.1]. [A7,F4,P1jFrom the geometry of the prism cell, assuming normal incidence of Z, therefractive index, n of a fluid in the cell obeys the relation (derived in Appendix B):n = 28528705sin(O+) - O0016832cos(O÷) [3.2]+ OOOO8491+ O.OO11697/5i172010— sin2(O÷b)where 0 is defined as the deviation from the direction of the incident beam which passesthrough the thermostat without the cell in place (i.e. Z1). Symbolically, 0=Zd-Zl and=(20525±00lO)° is the prism angle uii. All terms except the first on the right-handside are correction terms from non-parallelism of the prism windows . This may be dueto uneven tightening of the flanges and/or imperfect manufacture or polish of the surfacesof each window. The required correction for the non-parallelness is less than ±1 minuteof arc for each window. (These corrections are measured using an autocollimator. rroi)Other parameters included in these correction terms are the prism angle, 4, and therefractive indices of air is 1 0002622 [C3] (at normal atmospheric pressure, 101 325kPaor 14.696psi, and a room temperature of 25C). The refractive index of sapphireAny frequency dependence in a is not expected to appear until the ultraviolet wavelengths. [A6,B2814117660226 [G6]. Both refractive indices are values interpolated from references.3.5 PRISM CELL SETUPThe prism cell volume is machined out of one end of an aluminium block (— 15cm long). A small bore is hollowed along the cylindrical stem into the cell volume. Theopen end of the bore serves as the seat for the needle of a valve (NV) used in filling,bleeding and sealing the cell. (The needle valve body is sealed onto the end of the stemby an indium gasket.) A hole threaded in the valve body allows a fill-line (from agas-handling and vacuum pump system) to be attached. Contact with the thermostat isestablished at the base of the cell volume segment and the cylindrical stem. Twoprotruding pins at the base of the cell act as “keys” that fix (in a consistent manner) theposition of the cell with respect to Z and the side of the thermostat where the cell sits.Indium gaskets [L9] are used to seal the two sapphire (A1203)windows on to thecell body. Tight seals are accomplished by tightening flanges around the edges of theoutside surfaces of the windows using #4-40 stainless steel hex-head bolts. (Figure 3.4)This tightening is done over several days by using a precise torque wrench on the hex-head bolts every few hours. The window for the input laser beam (Zr) is fixed so that Z1enters at normal incidence at its outside surface. The inside surface is inclined at to theinside surface of the output window. (Figure B. 1) Prism angle, 4, is measured by anautocollimator/alignment telescope.The exiting deflected beam, Zd, is re-directed by a mirror (M3 in Figure 3.3) intothe alignment telescope. Mirror rotation (about a vertical axis) is controlled by amcC -. \M3II II42-0-— L —‘ I — P—. C —-.SF Bi 4 B2LEGEND: PRISM= Laser CELLe = TelescopeSF = Spatial Filter THERMOSTATL Lens StyrofoamP = Polariod PlywoodI=Jris tB1,2 = BeamsplittersM1,2,3 = Mirrors---- /mc = micrometer screw Ml Z0 M2X = 6328nmFigure 3.3 Optical setup for the prism cell. The thermostat itself is encased instyrofoam and an outer plywood box.micrometer screw. uoj Traversing a path around the outside of the thermostat, thereference beam, Z0, provides a means of periodically checking the alignment (stability)of the optics throughout a run.3.6 PRISM CELL PROTOCOLThe micrometer screw (mc) is calibrated (in terms of the deviation angles 0) byusing a 50-lines-per-inch Ronchi diffraction grating in which line thickness equals linespacing. The grating is placed in the thermostat, at approximately where the centre ofthe prism cell should be. The ruled surface faces the deflection mirror (M3 in Figure3.3) and is perpendicular to Z0; the lines are vertical. Angle, 0, is then defined as thebeam deviation from the incident beam direction, i.e. passage without the Ronchi.As the micrometer is turned, a set of evenly-spaced diffraction spots moves acrossthe cross-hairs of the telescope eyepiece. The position of each spot can be related to theWindowsFlangesCylindrical stemfor thermalcontact withheater blockNeedle Valve43Figure 3.4size=(1 diaThe Prism Cell. (Approximately full size) Sapphire windowx ‘A thick)in; cell volume=(1225±001)cm3rnj4O boltsPositioning pinsFill hole44position of 4, the undeviated beam spot. By recording and numbering consecutive spotsand by correlating each spot with the micrometer reading, the angle 0 is found using thesimple grating diffraction formulaml = d sine [33]where d=254±5O=O5O8mm is the grating period, m is the integer order of a spot fromthe undeviated beam, and X=6328A for the laser wavelength. 0 or sin 0 is then fittedagainst the matched micrometer reading. This calibration is done before and after a runto detect any alignment changes which may have occurred during a run (which may lastup to two months).After cleaning and repeated flushing with the substance to be studied, the prismcell is over-filled by keeping it at Tr< T. The needle valve (NV) is closed; the fill-linedisconnected; the cell is weighed [M8]. The cell is then inserted into the thermostat toequilibrate at a Trz T. (The result is a state that is on the liquid branch of thecoexistence curve or in the liquid segment of the phase diagram.)Deflection, 0, is read after the cell has equilibrated with the thermostat. The angleposition of the external beam, Z0 is also read as an alignment check. The blocktemperature, Tb, is now raised by decreasing the decade box resistance (Rd); the systemequilibrates, and another deflection reading is taken. This increase-Tb-read-Zd cyclecontinues until Zd no longer changes when Tb is increased. The fluid density profile overthe beam, Zd, is homogeneous because the system has been heated into a one-phase state(liquid at start of the run). (Figure 3.2c)45The cell is then removed from the thermostat and weighed. The needle valve(NV) is then opened slightly to release some fluid (usually less than 100mg). Weight isrecorded every 30s for up to 3 minutes after retrieval from the thermostat, and up to 5minutes after bleeding. The change of weight over time is used to estimate thecondensation on the cell outer surfaces if Tr is below room temperature. The cell isweighed before and after each bleed in this manner. This procedure allows a check ofthe integrity of the seals, i.e. existence of leaks in the gaskets or NV. Lastly, beforereplacing the cell in the thermostat, Z0 Z (the undeviated beam through thethermostat without the cell) are read.When bleeding has reduced the average density, p, to the extent that the meniscus(T,.< T coexistence) falls within beam illumination, two “spots”, Zd and Zdj are seen inthe telescope: for deviations in the gas and liquid phases respectively. Smaller amounts(usually less than 50mg) are bled as p-p Equilibration time is lengthened as the fluidnears its critical point, Zd and Zdl are smeared as 7—*T, because of the high densitygradients due to the highly increased compressibility.Once p has been bled to less than p, the system is on the gas branch of thecoexistence curve. And since those parts of the coexistence curve, when the meniscus is“within sight”, have already been recorded, there is no longer a need to maintain themeniscus. That is to say, once reduced past p, the reading protocol is changed: 7. maybe maintained at 7 until the cell is completely bled.When the cell has been bled empty and air has entered the cell volume, Z0 andZd are again read. The cell is weighed with NV opened, after which it is evacuated. It46YESFigure 3.5 Flowchart for the prism cell protocol.47is then removed, weighed and replaced in the thermostat. Z0 and Zd are again read. Thisprocedure is repeated several times to check for any variation in the cell mass with airand without.The Lorentz-Lorenz “constant”, E, is computed as follows: the Ronchicalibrations are used to convert the micrometer (mc) readings to deviation angles, 0,which are converted to refractive index, n, according to 3.1. n is calculated for thephase readings only. The density, p, is found by using the measured fluid mass and theknown cell volume. These pairs of n and p are then substituted into [3.1] to find Thisis the reason for heating the system above coexistence (into the one-phase region) at eachp. Polynomial fits can then be found for against p or n.Coexistence data are retrieved by using the Ronchi calibrations to find°g and 0,.Then n and n, are calculated via [3.2]. Lastly, knowing (the polynomial fits of) , [3.1]can be converted to find p and p,. The coexistence curve, [2.10,171 and the diameter,[2.19-20] can then be calculated and plotted against the cell temperature, Tr.3.7 IMAGE PLANE CELL THEORYThe Mach-Zehnder interferometer wsi is an optical system suited to measuringrelative variations of n and p in a sample placed in one of its arms. As explainedpreviously, n varies with height (H) across the expanded probe beam, 2. (Figures 3.2and 3.6) The presence of the sample alters the optical path length. The superposition ofZd and Z0 give rise to an interference pattern. A lens ii then focuses the pattern acrossa slit placed in front of the film plane of a modified camera. (The film plane coincides48CD CbD CcD CdD CeDC-)/— I I I I I I I I I IREFRACTiVE INDEX, n or DENSITY, pFigure 3.6 Density profiles with respect to mirror, M2, for (a) T 7;,;(b) T> 7,; (c) T T; (d) T< 7,; (e) T4 7,.with the image plane of the lens, L2 in Figure 3.8: hence, an “image plane cell”.) Thefilm records the evolution of a vertical segment of the fringe pattern. The change in thefringe pattern can be related to the change in n and p in the fluid as 7 is raised.A distinct meniscus is seen when 7 7;,. Since the n or p in each phase (althoughdifferent from one another) is homogeneous (Figures 3.2 and 3.6), the fringes in eachphase are evenly spaced. As Tr is raised towards 7,, the density gradients increase aroundthe meniscus. Fringes begin to “grow out” or appear from the meniscus into each phase.The distance between adjacent fringes varies from very small near the menicus to large49(approximately the same size if Tr4 T) away from the meniscus.When Tr= 7, the meniscus disappears altogether and gradients of n and p are attheir largest (oo). Fringes are very closely packed. As T,. rises above T, these verytightly-packed fringes begin to “unfold”, spreading out vertically across the film. Figure3.6 shows the density or refractive index profiles with respect to the inclination of themirror, M2 (from Figure 3.8). The reference “zeroth fringe” (N=O) which appears atTr= 7, becomes more visible, easier to keep track of. Fringe counting is made withrespect to this zeroth fringe. In this sense, the ii and p measured are relative to n, andc respectively. Hence, if Ng and N, are the numbers of fringes that “disappear” into themeniscus, and, n and n1 are the associated refractive indices of the two coexisting phasesat a subcritical 7< T, the count begins from N=O 7 where the refractive index is n.For a cell thickness of 10 (which is usually less than 2mm) and probed at a wavelengthof X=6328A, the optical path differences sj in each phase are given by:1N, = (n,—[3.4]Ng (“c — flg)where N, and N are positive numbers. Combining these equations,fli_flg=(N;+Ng) , [3.5]50fli+flg2flc=(NiNg) . [3.6]Thus, by counting the number of fringes that appear out of or disappear into themeniscus, the coexistence curve and its diameter are found using [3.11 and the equationsderived in Appendix A. The order parameter or coexistence curve is then given by:= (N,+N8){ai +a2(N,_Ng).1t+... }[3.7]while the diameter is given by:Pd = 1 +a(N—N2)+.a2(Nj2+Ng2)() + ... [3.8]The a1 and a2 coefficients are given by:a=____________— { n2+2 }1 (n2 — 1)(2 + 2) C 2 — 1a= 3(2—3n)— 6np’2 (n—1)(n+2)[3.10]—isf” { , + 2 } + { ‘ + 22 n2-1 n2-1C Cwhere511tD=__. =___[3.11]dn C dn2c[3.11] is retrieved from the best-fitting polynomial fits to SE in ii or in p:(n) = L0 + L1n + L2n + L3n +[3.12]L’(p) = L0 + L1p + L2p + L3p +Such expressions are called refractometric or refractivity vfrial expansions of SE.[B15,21,24,s12-13J Note that X110 (X110)2,i.e. since a1 a , the terms with a2 are muchsmaller in magnitude than those with a1 in [3.7] and [3.8]. Therefore, terms with higherpowers of the X/10 may be omitted.3.8 IMAGE PLANE CELL SETUPThe image plane cell is made from a cylindrical aluminium block. The smallamount of fluid (<0.6cm3)is contained between two sapphire windows L061, spacedapproximately 2mm apart. Indium gaskets seal the windows to the cell block. u.j Thewindows are held down by flanges which are tightened with #6-32 stainless steelhex-head bolts. A fill-hole is drilled from the top of the cell block into the cell volume.it is sealed by needle valve (NV) which is tightened onto the block by an indium gasket.A fill-line (soldered onto NV) connects the cell volume to a gas-handling and vacuum-pump system.The thermostat and the surrounding optical elements are all enclosed withinstyrofoam and plywood to prevent heat loss through convection over the water jacket52Needle valveCylindrical body6-32 boltsWindowsFlangesFigure 3.7 The aluminium cylindrical image plane cell. (Approximately fullsize) Sapphire window size=(1 dia X ‘4 thick)in.surfaces. (See Figure 3.8.) This also helps prevent “distortions” of the beams by reducingthe convection currents along the beam paths.3.9 IMAGE PLANE CELL PROTOCOLThe magnification, M, of the lens ci (L2 in Figure 3.8) is first determined. Thisis done by substituting a transparent object (a microscope slide with lines painted on) atthe cell position. M is the ratio of image size (at the film plane) to object size (at the cellsite). M is usually approximately -3 x, depending on the position of the lens with respectto the cell.LEGEND:_________= LaserL1,2 = LensesMl ,2 = MirrorsBl,2 = BeamsplittersP1,2 PolariodsSF Spatial FilterX = 6328nmM2 Z0 B2---. LiL2_____LED 0SlitMotorized CameraFigure 3.8 The Mach-Zehnder interferometer for the image plane cell setup.Unlike Figure 3.3, the styrofoam and plywood boards enclose some opticalelements as well.The cell on the fill-line is dipped into temperature-controlled water in a large glassjar. The cell volume is evacuated, then flushed with the fluid. This is repeated severaltimes. In order to fill to approximately pt,, the water temperature, T, is kept slightlybelow the fluid critical temperature, TC. Using the needle valve (NV) to control the flow,the fluid can be seen liquefying at the bottom of the cell volume. When the meniscus hasrisen to (approximately) the centre of the cell volume, NV is closed.T is then raised slowly to slightly above 7. If the cell is critically filled, the0!-,SF VL153P1PlywoodStyrofoamzi M2LiBi— P2HERM0STAT[ CELL454meniscus begins to fade without significantly changing its height in the cell. That is, themeniscus remains at the centre of the cell even as it slowly disappears! If it falls, the cellis “under-filled”. To increase the density, T is lowered below 7 again, and NV isopened to allow more fluid into the cell. T is raised through 7 again. If the meniscusnow rises as temperature is increased, the volume is “over-filled”. NV is opened to bleedthe cell slowly. (This can be done efficiently by lowering the pressure in the gas-handlingsystem by using a liquid nitrogen “cold trap”.) In this manner, the average fluid densityclose to p, is achieved. NV is then tightened and the fill-line is crimped near where theline enters the valve body. It is then snipped just above the crimp. The cell is theninstalled in the thermostat. The system comes to equilibrium at Tb=TTC.The polaroids (P1 and P2 in Figure 3.8) are temporarily removed. If the two armsof the interferometer are initially perfectly aligned iioi, uniform, destructive interferenceappears across the camera slit. By tilting the mirror, M2, slightly, the dark imagechanges into seemingly concentric fringes. The (dark) central spot is moved up, beyondthe top of the camera slit. The spot size shrinks as the tilt of M2 is increased. Whenapproximately 10 horizontal fringes appear across the entire slit, P1 and P2 arere-inserted into the beam paths: P2 is adjusted to maximize image contrast, followed byP1 to reduce image intensity to an acceptable level for the high-contrast, black-and-whitefilm pcij. The motorized [u161 slit-camera is then activated.Once the cell block temperature, Tr is stable (as noted by the thermistor-bridgevoltmeter [1-13] output on a chart recorder piij), a time-of-reading “marker” is flashed ontothe film by lighting an LED placed just in front of and just below the camera slit. The55temperature 7. is recorded against time. Several readings are made at the same stabilizedTr to ensure that fringes are not varying (due to T gradients, leaks, etc). If 7 T, 7. isincreased after (as little as) 2 hours after the last increment. When closer to T, thesystem may be left to equilibrate over a period of up to 2 days.Tr is never increased immediately after a reading has been done. A few minutesis needed for the LED marker on the film to advance beyond the slit. Fringe changes canthen be clearly recorded on film when Tr is increased. This is to ensure that there is noloss of fringe count, especially when close to 7.3.10 ISOTHERM CELL THEORYThe density (p) of a fluid in a prism cell can be deduced by beam deflectionmeasurements if the Lorentz-Lorenz relation, .E, [3.1] for the substance is known.However, because of the larger prism cell optical path, there is a limit as to how closelythe critical point can be approached (because of gravitational rounding). On the otherhand, although the image plane cell measures p relative to pr,, it can more closelyapproach the critical point before gravitational rounding affects the beam. Separateisotherms (for different temperature ranges) can be extracted from each set of data.However, two separate setups may produce different estimates of T for the samesubstance (due to different amounts of impurities in each setup, for example). Hence, thetwo data-sets will have to be “adjusted” before they can be collated and combined.A “better” system is proposed: juxtapose the two types of cells in the same setup(cell block) in order to avoid differences in conditions. Such an isotherm cell can be56hollowed out of an aluminium block with a parallel-windows (image plane) compartmentconnected to an angled-window (prism) compartment via a small bore. (See Figure 3.9and Appendix C.) The two cells act complementarily in the sense that p can be deducedfrom the image plane side when the system is very close to 7, while p can be measuredfrom the prism side further away from T before being affected by gravitational rounding.Fringe count (from the image plane side) can be directly correlated to the prism side dataif simultaneous reading are made. Hence, the span of density that can be measured in thissetup is increased.More importantly, both compartments contain the same sample at the same p and7’,.. Both the prism side and image plane side data are then collected from the samesample which is always under the same experimental conditions. There is no longer aneed to “adjust” data from different sources or setups in order to compare results.Data from the parallel-windows side are analysed in the same manner as that fromthe image plane cell setup. With Lorentz-Lorenz relation, [3.1) and critical data providedby the prism cell experiments, p is found from:NAI 1 NA [3.10 0where N, X, and 10 are defined previously in [3.4], and, a1 and a2 are defined in [3.9] and[3.10] respectively.The apex angle of the prism side is =(l9744±0024)° (as measured by thereticle reflection method). The formula to convert from deflection angle, U to refractiveindex, n is given in Appendix B:57oooJSShTE ooarD_________ ___DSlYROAM/4/ SS ftA/4/ DBL\IHEA /\/ D\ fT_ _ _ _______ _ _ _YALE %// 12____DP CL/_/\T/\STYflOFO//______D_ _DJoop”‘pool+ 2 2 2 2 + 2Figure 3.9 Top section of the outer water jacket containing the isotherm cell.(See Appendix C for details.)n = 296O9925sin(O+4)—O000lOl6cos(O÷4i) [3.14]+ 00013170— OOOO69O9/31172O1O- sin2(8+#)where the correction terms are present for the same reasons as in the prism cell.There are several difficulties to overcome in order to extract the (critical)58isotherm directly. Firstly, the maintenance of temperature over long periods of time isessential, especially when scanning the fringe changes and bleeding the cell when in thecritical density region 1R. [ti,s7] Secondly, fluid must be bled from the cell periodicallyin order to reduce the density. This means that the needle valve (NV) on the cell blockmust be accessible. Also, the pressure (P) must also be monitored continuously. Thelatter two stipulations mean that thermal isolation may be compromised, and temperaturegradients may result in the cell through these intrusions.Yet another temperature problem may arise. If the room temperature issignificantly lower than T, the small volume bled out may condense and flow back intothe cell creating temperature gradients and/or pressure transients. This problem may bealleviated by increasing room temperature to greater than 7, or, heating the bleed-line.Enclosing the thermostat and optical elements is not possible in this setup due toconstraints on the positioning and the accessibility to optical elements on the prism side.3.11 ISOTHERM CELL SETUPThermal leaks from the cell to the outside is reduced by using 304 stainless steel,Gauge #26 hypodermic needle tubing [1’12] as the bleed-line. The tubing can withstandpressures of up to 3000psi (or 20000kPa), and at the same time, has a low thermalconductivity [C3] so as not to generate temperature gradients in the cell. The tubing linksthe cell to a very precise digital pressure gauge (DPG) which uses a silicon pressuretransducer (SPT) [M9].The gauge is raised on a platform so that the sensor is approximately at the levelof the cell volume centre. This reduces systematic errors caused by “head pressure” if59M6 ,mc /-,—tMl B2 M2 tBS/ -. n -- \M5zdf I1iM3CellsDPGLiOLED______Slit0 Motorized CameraLEGEND : B#z=Beamsplitter; DG=Dial Gauge; DPG=Digital Pressure Gauge; ht=hypodermictubing; I=Iris; L#=Lens; mcmicrometer screw; M#=Mirror; MV=Metering Valve; NV=NeedleValve; P#=Polaroid; SF=Spatial Filter; tC/Pr=microComputer/Printer; 0 =Laser; e =Telescope;R=Bleed Reservoir.Figure 3.10 The isotherm cell setup. The bleed-line, ht is a stainless steelhypodermic tubing of dimensions, (046 OD x 027 ID)mm. [P12]the sensor is below or above the level of the cell volume. The gauge is factory-calibratedto read in kPa or psi. It is equipped with an IEEE-488 port from which (digitized)pressure data can be passed to a compatible microcomputer via its General PurposeInterface Bus (GPIB). [PS] The microcomputer reads and stores the DPG output, activates60the LED (to “time-mark” the film), and notes the time of the reading.In Figure 3.10, all components from the gauge to the gas-handling (bleed) systemare shown enclosed in a plywood box. A thermistor temperature controller (similar to theblock heater controller) regulates the current to a coil heater so that the ambienttemperature within the box is raised to well above 7,. In this setup, only a small lengthof the hypodermic tubing is exposed to a room temperature of approximately 27C. Allother parts of the tubing are inside the thermostat or the heated box.Two needle valves on either side of the cell block act as shut-off valves.(Appendix C.) A copper tube (with a bigger bore) is attached to the one valve. It servesas a faster pump- and fill-line. The bleed-line hypodermic tubing is soldered onto theother valve. A metering valve (MV), and another needle valve (NV) isolate the cell andgauge volumes from a reservoir (which is a lecture bottle to contain the released fluid).MV controls the amount and rate of each bleed precisely. NV acts as a shut-off valve.Due to restrictions on the sizes of the optical table, the cell block and thethermostat, and, the constraints on machining, the centres of the two compartments areset 2 inches apart. However, to ensure a “safe separation” during the simultaneouscollection of the two sets of data, the optical alignments are arranged such that eachmeasurement is carried out on opposite sides of the thermostat. (Figure 3.10) In thisway, one set of optical elements (image plane side) is kept away from unintented contactduring a reading of the prism side. Also, “stray light”, when the prism side is read willnot over-expose the fringe data on film.613.12 ISOTHERM CELL PROTOCOLCalibrations are carried out with a “calibration block” which simulates the actualcell block in shape and physical sizes. In place of sapphire windows, the prism side holdsa 50-lines-per-inch Ronchi grating at approximately the compartment centre. The gratingis used to calibrate the micrometer screw (mc) which controls the rotation of thedeflection mirror (M6 in Figure 3.10). To simulate the image plane compartment,microscope slides (with symbols inscribed on their surfaces) act as the cell volume.Focussing on these symbols, the image plane and the magnification of the lens (L2) iciare found. (The sizes of the symbols are measured using a travelling microscope.)The entire system - cell block, gauge, and reservoir- must be evacuated forseveral days because of the fineness of hypodermic tubing. After flushing several timeswith the fluid, the cell is over-filled (as in the prism cell case). The needle (NV) andmetering (MV) valves are closed; the copper-tube fill-line is crimped and snipped off.The cell block is inserted into the thermostat to equilibrate. The needle valve with thetubing attached is opened so that the fluid fills the sensor (SPT) volume as well.The microcomputer is programmed to light the LED, read the pressure (F) fromthe gauge and cell temperature (Ti) from the quartz thermometer hourly. It prints theresults immediately on completion of the readings. A (prism side) reading cyclecomprises the following ordered steps:(a) measure the prism-side beam deflection of the beams, 4 and Z0;(b) enter these results on the microcomputer;(c) microcomputer reads P and Tr, and lights the LED;62(d) microcomputer prints time of reading, 7;, P, Z0 and Zd; and,(e) bleed the cell by opening MV.Monitoring thermistors are used to check the stability of 7;.The fme metering valve (MV) can achieve very slow bleed rates in order toprevent the loss of fringe count which can occur if the density (p) changes too quickly.If this happens, a “fringe jump” or discontinuity is seen on the film. The rate of bleedis observed on the pressure gauge. When a sufficient amount has been released, MV isclosed and the system is left to equilibrate again. (Equilibration time is longer when thesystem is close to 7;.) This cycle continues until the fluid density equals the reservoirdensity. MV and NV are then left totally open.3.13 TIlE FLUIDSCRITICAL CHF3 CH3F H2C:CF UnitsPARAMETER257 44.9 297 C481 628 44583 xlO3kPa7•4985 88444 64964 XlOmolcni7OOl4 34033 64035 gmol’-82 -784 -857 C- 1552 - 141.8**- 144 C286 14397 2619 x103 gcm3165 18584 1382 Debyes[M16J [M15] [D8J7;PCPCMol WtBoiingFreezingDensity**Lo, DipoleMoment riiNotes: Measurements were made at 1O1325kPa () and at 25C (f) or at 20C (ff) u17,c3,MIo]Table 3.1 Characteristics of the fluids used.63The basic criteria on the fluids that can be tested in all the abovementioned setupsare that:i. they must be transparent,ii. cells can be constructed to withstand pressures of the order of orgreater than the critical pressure P, and,iii. temperature can be maintained for the given critical temperatureT over long periods.The selected fluids are: p17,c3,M1o]i. CHF3- called trifluoromethane or fluoroform or Freon23®;is colourless, nonflammable, nontoxic;of better than 98% purity;used as a refrigerant;ii. CH3F- called fluoromethane or methyl fluoride;is colourless, low-order-toxicity, flammable;of better than 99% purity;used as a propellant in admixtures; and,iii. H2C:CF - called 1, 1-difluoroethylene or vinylidene fluoride or Genetron-l 132A®;is colourless, flammable, nontoxic;of better than 99% purity;used in preparation of polymers and copolymers.CHF3 and CH3F are purchased from the Matheson Gas Company pnj. The sample ofH2C:CF is “on loan” from the Chemistry Department, UBC.643.14 COMPUTER PROGRAMMESCalculations are made using several commercially available computer programmesfor personal microcomputers and packages from the university’s UBCNet, MTS-GeneralFORTRAN library. These are:i. LOTUS® 123 Spreadsheet Lull:used for data entry on a microcomputer;does multiple regression or polynomial fits;provides errors of fitting coefficients for estimating error propagation;provides a goodness-of-fit multiple-correlation coefficient, R; and,provides simple plots for quick, interactive viewing;ii. MTS-G DOLSF nij:FORTRAN-callable, polynomial-fitting package;automatically finds the best-fit degree of the fitting polynomial;iii. MTS-G NL2SOL/NL2SNO ini:FORTRAN-callable, state-of-the-art, non-linear-curvefitting package; and,iv. MTS-G RPOLY1 ij:FORTRAN-callable package;finds the roots of polynomials.LOTUS® 123 is used to estimate possible linear or linearized fits. The results arethen used as initial values in the FORTRAN curve-fitting programmes. LOTUS® can alsobe used to find the standard errors of the FORTRAN fit results and to generate simpleplots.65CHAPTER FOURPRISM CELL RESULTSH2C:CF and CH3F are examined in the prism cell experiment. The aim is toestablish the Lorentz-Lorenz relation, in [3.1] for each sample in terms of densityranges that include their respective critical density, p, and to find the refractometricvirial coefficients of [3.12]. The density is reported in molcm3 to compare withreferences (where available). The values of the molecular weights used in calculations(of each sample) are those in Table 3.1 (as they are not measured as part of the prismcell experiments). The wavelength used is the He-Ne laser X=6328A. The conversionformula of the deflection angle, 0, to refractive index, n, is derived in Appendix B.The goodness-of-fit criteria used are the multiple-correlation coefficient, R, andF statistical tests at confidence level, p, (where applicable). The latter are also usedto test the suitability of adding more (correction-to-scaling or higher order) terms to fitsto N data-points (as R generally increases with the number of terms). The FORTRANpackage, DOLSF, determines the appropriate degree or order of the fitting polynomialautomatically. That is, an test should not be necessary when DOLSF is used.Values placed between braces indicate that that parameter is held constant at avalue of x. For example: a = {0 11 } means that the parameter, a, is set equal to a valueof 0 11 until otherwise changed.664.1 RONCifi CALIBRATIONSThe micrometer screw (mc) controlling the deflecting mirror (M3 in Figure 3.3)is calibrated using a 50-lines-per-inch Ronchi grating. The fitting equation is:sinO = 1245669291x103m [41]=+ f1u + f2u + f3u +where the multiplicative constant is found from [3.3]; integer m is the order of thediffraction spots; { J:i=0, 1,2,...} are the coefficients of the polynomial fit; 0 is thedeflection or deviation from the incident beam direction, Z; and,u = (micrometer reading of Z) - (micrometer reading of the mth diffraction spot)The results of the polynomial fits in the calibrations for the two runs are givenin Table 4.1. Fits of up to second-order polynomials are carried out. AsJJj, the fitsare nearly linear: R 1 and F 1. Other F<0.1tests indicate that additional terms are notneeded. This is also evident in the larger uncertainties in the coefficients of higher-orderH2C:CF CH3F000009 ± 000007 - 00011 ± 0000040l336 ± 00001 013154 ± 0’00006- 00019 ± 00002 000003 ± 000015R2 0999997 0999997m -14to59 -16to550 -1°to4°13’ -1°8’to4°Table 4.1 Polynomial fits to [4.11 using Z as reference.67terms. These results are consistent with previous calibrations Nij.Within run, (CH3For112C:CF), there is little difference between using Zand using 4 as reference because J(Z1)=J(Z0), and [f(Z1)-J(0I <3 X lO. Only theintercepts, J are different (although numerically small). These different values representthe different positions of the designated m=O diffraction spot with respect to Z,. Z ischosen so as to be consistent with the definition of 0. Also, Z0 is deflected by additionalexternal optical elements which are individually susceptible to (unintentional) changesduring the experiments; whereas Z goes directly through the cell as shown in Figure 3.4.The difference between the CH3F and H2C:CF runs (in the values of f0,12) is due tochanges when the optical elements are realigned at the beginning of each experiment.4.2 AM) aeFigures 4.1 and 4.2 show the best polynomial fits of the Lorentz-Lorenz relation,, to density, p, (according to [3.12]) forH2C:CF and CH3F respectively. A 1-percentspread about the maximum of the data-points is shown next to the vertical axis in eachplot. The results of the fits are listed in Table 4.2 forH2C:CF and Table 4.3 for CH3F.They will be used and/or compared to the results of image plane data (of Chapter Five).Note that the refractometric virial coefficients, L 0,1,2,3 (in [3.121), do not all have thesame units.In the calculation of density, p, during the H2C:CF run, the error in mass dueRecall that Z, is the incident beam, 4 is the beam that has been deflected by the cell (filled or unfilled).Thus, Zd=Z if the cell is removed (absent). 4 is the beam the goes around the outside of the thermostat.Symbolically, the deflection angle, 0=4 - Z1 when the (empty or filled) cell is present.6810.90010.65C)10.6010,7510.700N 10.6510.6010.55S10.500. 000Figure 4.1 (cm3mo11)vs p (molcm3) for H2C:CF. The vertical linerepresents the position of pt,. A 1 % spread about the maximum is shown.to the condensation of water vapour onto or evaporation from the prism cell outersurfaces is taken into account by observing the average rate of increase or fall of massreading over the time the prism cell is weighed. Such corrections are especially neededduring the beginning period of each run when the cell temperature, T,. (at each removalfrom the thermostat), can be more than 20C below room temperature. The maximumerror in p is approximately 8 x 106 moFcm3.Comparing this to the minimum calculatedvalue of p, the percentage error is less than 1 %.For H2C:CF, in the low density region, p <OOO5molcm3 (032gcm) or0.002 0.004 0.006 0.008 0.010 0.012DENSiTY, p in molcm’69Range p < 0005 p < 001 n < 11684•233 ± 0006 4230 ± 0005 423 ± 050L0 1068 ± 001 1067 ± 001 - 079 ± 0.01L 28±2 40±2 20±1L2- - 2540 ± 150 - 91 ± 05R2 086 084 084N 45 204 204Table 4.2 Electronic polarizability, ae (A3) and coefficients, L012 of fits toforHC:CF (Figure 4.1)n < 1 08, the best fit to 45 data-points is a straight line. Extrapolating this line to p =0,the electronic polarizability is ae (4233 ±0006)A. In extending the fitting range top <001molcm3(064gcm3)or n< 1.168, the best fit to 204 data-points is a quadraticwhich estimates ae to be (4230±0005)A. In this range, the data-points are scatteredby less than 03 % about the quadratic fit. That F(quadratic) > F(cubic) F<0.1 confirmsthat the quadratic is the most suitable fit to the data in this range.It should be noted that the value of ae is highly dependent on the degree of theresulting (DOLSF-chosen) best-fit polynomial which is determined by the range and thescatter of the data-points. The 1-percent limit (Figure 4.1) shows that the scatter in eitherfitting ranges is less than 2% (of the maximum). The position of the peak of the fittingquadratic is at a larger density than p. If the peak of a quadratic fit is hi at [B14,D1,N1],TheH2C:CF molecule is not spherical. o, which is a symmetric, second-rank tensor [P8], will havedifferent values along the diagonal. The calculated value of a is presumably a linear combination of the threediagonal components, ¼(2a1 + a,). [B24,2816 716 706 696 686.676.666.656.646.636.626.61Figure 4.2 E (cm3mol’) vs p (moFcm3) for CH3F. The vertical linerepresents the position of p. A 1 % spread about the maximum is shown.ae (in the same density range) is found to be (4230±0003)A,which is not significantlydifferent from the value found above (in the other fit).In applying [3.2] when the prism cell is filled with air at atmospheric pressure (atthe end of theH2C:CF run), the value of the refractive index, n, is estimated to be notmore than 01 % from the refractive index of air (used in Appendix B). For otherevaluations of n using [3.2], the most significant digit in the estimated error affects thefourth decimal place of n up to a maximum error of approximately 00O08.The fits to CH3F data are for p <OOl7moFcm3(058gcm3)or n< 1175. The70CEE8SiC-)0,000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016DENSITY, p in mo1cm’71vs p.QUADRATIC2639 ± 00036657 ± 0008100 ± 08- 690 ± 40082vs n.CUBIC2639 ± 00036657 ± 000810 ± 2- 660 ± 270- 1200 ± 9500082n < 1 175 QUADRATIC CUBICae 26±04 3±17L0- 0687 ± 0008 -2142 ± 0008L1 137 ± 08 18 ± 32L2-64±04 -10±29L3- 1±9R2 081 081Table 4.3 Electronic polarizability, ae (A3) and coefficients, L0123 of fits tofor CH3F.best polynomial fit is a quadratic (using DOLSF). If a cubic fit is forced, the derivedvalues of are not more than ±0004% from those derived from the quadratic fit to thesame 140 data-points. That is, the cubic fit very nearly coincides with the quadratic fitin this density range (as shown in Figure 4.2). However, the uncertainties inL0,123arelarger for the cubic fit (as shown in Table 4.3). An F<0.1 test of the cubic term showsthat it is indeed not required.p < 0017L0L1L2L3R272Extrapolating this CH3F quadratic fit to p = 0, e =L0= (6 657 ± 0 008)cm3mol1.Using this in [3.1], ae=(2639±O 03)A. These values are not significantly differentfrom previous low density experiments in whichL0=(6640±0007)cm3mol’ andae=2.63A3.n19,25] This last L0 point is shown (in Figure 4.2) on the vertical axis by Xbetween its error limits denoted by the arrows.It should again be stressed that and ae depend on the extrapolation of the fitand therefore, the degree of the polynomial selected by DOLSF as the best fit to thedata. This is dependent on the scatter of the data in the density range used in the curve-fit. Comparing against the 1-percent spread about the peak value (in Figure 4.2), thescatter of the data-points is not more than 2% in this range. If the peak of a quadraticfit is forced to be at p, [rn5,D1,N1], the average values of t() and ae are (6661±0003)cm3mol’ and (2641 ±0001)A3,respectively. These are not significantly different fromthose values in TableWith e for the two samples known, the coexistence curves, p and theirdiameters, Pd are calculated using the algorithm described in §3.6.4.3 COEXISTENCE CURVE, pFor curve-fitting purposes, the critical temperature, T, is estimated by rewriting[2.17] in terms of the reduced temperature, := 1-7/T (which is positive as Tr< Tj:Other more precise methods designed specifically to measure and a exist. [A1O,B25-26] Althoughit is not the aim of the prism cell experiment to measure these values, the results for CH,F do not varysignificantly from those found by using more precise methods.73ab0b1b2b3R2N7,’.H2C:CF- 000084 ± 0000030020 ± 0001- 014 ± 0’0313 ± 03-4+1099899699297419 ± 00020CH3F- 000203 ± 00000200274 ± O’0007- 0041 ± 0005011 ± 0’02-041 ± 002099994236449092 ± 00090Table 4.4retained.Fits to p [4.2] to find T’. (C). j={0325}, iX={05} and t3 isPz — Pg= a + b0t + b1t4 + b2t4 + b t3432[4.2]Note that p’. (in the definition of p in [2.5,2.10]) has not been determined at thisjuncture. Hence, the coefficients, b0123 are not the same as B0,123in [2.17].By allowing 7,’. to be a “free parameter”, the accepted value of 7 (in t) is thatwhich maximizes multiple-correlation coefficient, R, in the fit to [4.2]. The results of thefits are shown in Table 4.4 for critical exponents, fi={0325}, i={05}, and with thet3’ term included.ForH2C:CF, [4.2] is fitted to 99 data-points in the range, 247< 7.<297 C. If13 is increased, while ={0’5}, and the number of correction terms is kept constant, 7decreases. For example, if the t3 is retained, 7,’, falls from 29745C to 29739C as 13 isincreased from 03 to 035. However, if 3 is kept constant while the number of7429-c)V X V28-V X VV x VV X V1 26- V x vx vV x V25- Vw x VV— I0003 0.005 0.007 0.009COEXISTENCE CURVE and DIAMETER in molcmFigure 4.3 Coexistence curve (v) and diameter (X) (molcm3vs prism celltemperature, Tr (C) forH2C:CF.correction terms is increased, 7 generally increases as well. Considering all variationsof i3 (in the abovementioned range) and the number of terms (up to t3) simultaneously,R varies in the range, 099886<R2<0 9899. F tests show that all fit variations areacceptable. Adding more correction terms do not enhance the fit. The coefficients, a andb 0,1,2,3 do not vary significantly. Selecting the case in which R is maximum and with3 ={0325}, i={O5}, and retaining the t term, 7=(297419±OOO2O)C with t>For CH3F, 236 data-points are used in the fit in the range, 95<7<446 C. If13 is changed from 03 to 035, 7 decreases from 44’94 to 44’88 C when the t term754540c)35302520L)15ID0.001 0.003 0.005 0.007 0.009 0.011 0.013 0.015 0.017 0.019COEXISTENCE CURVE and DIAMETER in molcm ‘Figure 4.4 Coexistence curve (v) and diameter (x) (moFcm3vs prism celltemperature, T (C) for CH3F.is included in the fit, and A={05}. On the other hand, if 13={0325}, adding morecorrection terms (up to the t3 term) in [4.2] increases 7 from 4455 to 4491 C. Overthese variations, R lies in 09991<R2099994. The F value for 13={0325} with thet term is much greater than F<0.1 which implies a good fit. Lastly, testing thenecessity of the t term, FQ 3) > FQ 2Z) > F<0.5 which implies that the t3 term shouldbe retained. In this case, T. is found to be (449O92±O•OO9O)C with t> lO whenj3={0325}, z={O•5}, and, R is maximum.The lower limits on the reduced temperature, t, are well in line with the limitation76imposed by gravitational rounding which usually restricts t to be greater than 1O. i6JDue to the longer optical path (— 5mm) through the prism cell, when 7 is approached,the laser beam no longer measures the local refractive index across its width because theincreased isothermal compressibility, K of the fluid under gravity, creates large verticaldensity gradients. The beam spots are seen as smears or long-tailed “comets” whosecentres are difficult to pinpoint.Comparing [4.2] to [2.17], coefficient, a, should be zero in Table 4.4. There aretwo likely contribution to these “zero errors”. Firstly, the angle calibrations using theRonchi may not be exact, thereby, introducing a shift, (for example, f, in [4.11).Secondly, systematic errors can be compounded and propagated with each step ofcalibration and curvefitting. Dividing a by p, for each substance (using the results of p,derived below), this leading constant term becomes -O13 and -O23 for H2C:CF andCH3F respectively.4.4 DIAMETER, PdTo fit to the coexistence curve diameter, Pa, [2.19] is rewritten as:Pd= += a + a1t -I- a2t , [4.3]2pwhere the correction-to-scaling terms are omitted. This is adequate because the t rangeof the data lies outside the critical region, [M6] a is the critical exponent associatedwith the specific heat at constant volume, c, in [2.12]. a is held at {011} in the77H2C:CF CH3F297419 ± 00020 449092 ± 00090a0 1O00O ± OOOO7 10000 ± 00008a1 - 04 ± 03 055 ± 003a2 15 ± 04 O51 ± 004R2 O987 09997N 63 2196445 ± 0004 8765 ± OOO7Table 4.5 Fits to Pd to find PC (x 1O molcm3).following fits (in Table 2.1). p, then becomes the free parameter which is adjusted untila0= 1 ±s. This brings [4.3] in line with [2.19] when t=O. The error limits of P isapproximated as the maximum of pC(aO=1-s)- p(a0=1) and p(ao=1+s) - p(a0=1). Resultsare displayed in Table 4.5.ForH2C:CF,63 data-points are used in the fit to [4.3]. The result is found to bep=(645±O.OO4)X 1O molcm3or (04127±0003)gcm3.Considering the rectilineardiameter only, (i.e. omitting the t ‘ term), P is (6444±0004) x iO molcm3.For CH3F, 219 data-points are used, finding pC=(8765±OOO)X 1O molcm3or (O2983±OOOO2)gcm.For a rectilinear diameter fit to [2.19], the result isPC (8 79±0 O1)X iO mo1cm3.The rectilinear diameter results are less than 03% fromthe results of the fit with the field-mixing term present. The uncertainty limits coincide.The closeness of the results shows that in each case, deviations from therectilinear diameter are not apparent, i.e. the field-mixing t 1-a term has little or no78obvious effect on the values of p in each reduced temperature range. Conversely, findinga by using it as a free parameter (in the fit) is not justified as the nearly-equalmagnitudes of t’ and t terms make discerning between them difficult.These calculations are also n very sensitive to the variation of 7. This is testedby observing the changes in the fit coefficients, O1,2, and R for different values of T.The span and scatter of the data do not permit adequate resolution to allow a to be a freeparameter as well. Omitting the fielding-mixing term altogether for the rectilineardiameter does not change p, significantly.Comparing Figures 4.3 and 4.4 and the values in Table 4.5, the rectilinear slopes,a2 (in [4.3], seem to) differ significantly for the two substances. This is discussed in §5.4when the image plane data are included. For 112C:CF, the two beam spots, Zd and Zd,jdiffer in the amount of deflection as 7.—T. Zd increases more rapidly than Zdl decreasestowards p. Hence, there is a “hook” at the top end of theH2C:CF diameter in Figure4.3. This is likely due to the uncertainty caused by the smearing of the beam spot asTr’Tc, or, may be specific to the system. The field-mixing term should not produce sucha pronounced effect. (See Figures 5.1 and 5.2.)Using these values of PC and the respective values ofL012 (from Tables 4.2 and4.3) in [3.12], 4 can be calculated for the two substances. Lastly, using p, and 4 in[3.1] itself, establishes the critical refractive indices, C• The results are summarized inTable 4.6. These compare quite favourably with the reference values in Table 3.1 butwith improved precision. However, recall that the prism cell data lie further away fromT, because of the “thick cell” effect 6,s15] which limits how close to 7 measurements can79be made. The next chapter shows that the above results are in fact consistent.H2C:CF CH3F UnitsT 29•7419 ± 00020 449092 ± 00090 C11068 ± 00002 10895 ± 000021082 ± 002 &69 ± 001 cm3mol’6445 ± 0004 &765 ± 0007 x iO molcm304127 ± 00003 02983 ± 00002 gcm3Table 4.6 Summary of the derived critical parameters.80CHAPTER FIVEIMAGE PLANE CELL RESULTSH2C:CF and CH3Fare examined in separate parallel-window, image plane cells.The aim is to have the reduced temperature, t, approach 0 as closely as possible. Thewavelength used is the He-Ne laser, X=6328A. The magnification at the image (film)plane of the lens, L2 [1(8] (Figure 3.8) is M= —(305±005)x for theH2C:CF setup, andM= —(2•78±0•05)x for the CH3F setup. In the following analysis, parameters ornumbers enclosed by braces imply that they are held constant. Examples: {13} andj={0325} mean that the parameter, [3, is kept constant, or, kept at a constant value of0325 in the referred calculations.5.1 FRINGE COUNT CONVERSIONThe fringe counts, N8 are converted to the coexistence curves, p, and diameters,Pd, using [3.7] and [3.8] respectively. (Ng are positive numbers in the followingformulae.) Rewriting them as:= hi-f(Nj+Ng) ÷h2()(Ni+Ng)(Nj_Ng)o 0 [5.1]= G1(N,+N) + G2(N?—N)and81Pd = 1 + h1 - (N1 — Ng) + - h2 (.. )2 (N,2 + N2)10 2 [5.2]= 1 + G (N, — N) + G2 (N,2 + Ng2)where h1 and h2 depend on the critical parameters and the Lorentz-Lorenz relation,Tables 4.2 and 4.3 (for ) with Table 4.6 and the formulae in Appendix A are used tocalculate the dimensionless constants, G1, G2 and their error limits. The results are shownin Table 5.1.Only the first two terms of the Taylor expansions in p and Pd are retained asI G1 I I G I because 1 Xl ‘ (X10’)2even though I h1 I — I h2 I. For the image plane, thefringe count is not directly correlated to density, p, (whereas, p can be measured throughbeam deflection in the isotherm cell setup in §6.2). The values of p calculated using G1and 2 are smaller than those found using G1 only because h2 is negative. ForH2C:CF,using the critical parameters in Table 4.6, the maximum differences are given by:(p(G1) - p(G1,G2)) < 0002 and (pd(GI) - Pd(Gi,G2)) < OO15}12C:CF CH3F1 1536 ± 0002 177 ± 001 mmXli’ 4.119 ± 0005 358 ± 002 xiO4h1 4558 ± 0008 55O ± 001- 166 ± 004 - 118 ± 0051877 ± 0004 197 ± 0.01 x103G2 - 141 ± 004 - O76 ± 003 x107Table 5.1 Fringe count (N) conversion constants for [5.1] and [5.2].8244.90944.908- C44 907-044.908 -044.905 -tjQ 44.904 -44.903 -0C44.902— I I 1 I I I I I0 0.4 0.8 1.2 1.6 2 2.4(p) x 1O-Figure 51 Sensitive plot of 7vs (p*)lP for CH3F to estimate 7. j3={O325}.Similarly, for CH3F, the maximum differences are:(p*(G)- p(G1,G2)) < 000003 and (pdGI - Pd(GI,G2)) < 0006These differences become larger with increasing values of p or Pd’ i.e. with increasingt or the “distance from” the respective 7. These limits and an F<0.1 test show that G2can significantly affect the diameter, Pd. For p, an F>0.35 test shows that it is lessaffected by the inclusion of G2. In the analysis below, data are deduced using G1 and G2.5.2 COEXISTENCE CURVE, pAs mentioned at the end of the previous chapter, the thinner image plane cell8329.742 -29. 741929.7416D29.741729. 7416L) 29.741529.741429.741329.741229.7411 D29.74129.740929.7409- C29.7407-29.7406- C29.7405-.) 29.7404- C29.7403-29.7402-29.7401-29.74- I0 2 4 6x 10’Figure 5.2 Sensitive plot of T vs (j )“ for H2C:CF to estimate T.13={0325}.should be able to improve the estimate of the critical temperature, 7, as it “defers”gravitational rounding effects to smaller values of reduced temperature (t) than for theFIT H2C:CF CH3FLINEAR 2974171 ± 000022 4490856 ± OOOO17QUADRATIC 2974184 ± 000011 4490877 ± 000016CUBIC 2974186 ± 000012 4490867 ± OOOO16PRISM 297419 ± 00020 449092 ± 00090Table 5.2 Estimates of 7 (in C) using polynomial curve-fitting from Figures5.1 and 5.2 for CH3F and H2C:CF respectively.84-0.6-0 7—0.8-0. o—1—1.10-1.2-1.3-1.4-1.5-1.6-1.7-5.2-2.4log10 tFigure 5.3 Log10 p vs log10 t for CH3F image plane data. The solid curve isa best fit for 7={449O867} and j3={O325}. (Table 5.3)thicker prism cell. As a first estimate of T, two sensitive plots of the image plane p’ datacan be done. These are 1og0(t p) vs log10 t and T vs (p* where T is the celltemperature (=Tr) and I3={O325}. In dividing out the t term in the first type of plot,data closer to T (i.e. for small t) should scatter about a horizontal line if the value of 7used (in t) is “accurate. Such plots will be shown later.Figures 5.1 and 5.2 are examples of the second graphing method. Only the imageplane data nearest to the expected value of 7 are shown. The plotted data-points are notaffected by any chosen 7. They can be fitted to a polynomial function which extrapolatesto the temperature axis at T. Linear, quadratic and cubic fits are tried. The fitted values—4.8 -4.4 -4—3.6 -3.2 -2.8850.190.180.170.160150,140.130.120.110.1‘—‘ 0.090.080.070.060. 050.040.030.020.010-0.01Figure 5.4 Log10 (tp*) vs log10 t for CH3F image plane data. (Table 5.3)I3={O325}.of 7 for CH3F andH2C:CF are shown in Table 5.2. They differ by less than 1% fromthose found from the prism cell. The precision is also improved.The curve-fitting equation for the coexistence curve (p) data is similar to [4.4].However, only the critical exponent, j3, is permitted to vary in some fits. The gap orcorrection exponent, ={O5} (from Table 2.1). The re-arranged equation is:= + b0t { 1 + + b2t ÷ b3t } , [5.31where b acts as an “error-trap” estimating any errors in the fringe count. Its magnitudeis expected to be small.-5.8 -5.4 -5 -4.6 -4.2 -3.8 -3.4 -3 —2.6log10 tU-0.1-0,2-0.3-0.4-0.5-0.6• -07-0.8-0.9—1—1.1-1.2-1.3-1.4-1.5-1.6-1.7-1.8-1.9Figure 5.5 Log10 p vs log10 t for CH3F image plane (+), prism (C]) data andbest fit (—) for b=b3={O}, i3={O325}. (Table 5.3)86Figures 5.3 and 5.6 show the results of fits to the image plane data of CH3F andH2C:CF respectively. The results for image plane with prism cell data are shown inFigures 5.5 and 5.8. Figures 5.4 and 5.7 are sensitivity plots of the first type mentioned.They illustrate the necessity of correction-to-scaling terms as the data veer away from theexpected horizontal line that would have indicated a pure power law is applicable to theentire data range.Tables 5.3 and 5.4 list some of the results of several fits under variousrestrictions. It is observed that-6-5 -4-3 -2 —1log10 t- O2 < b < 0487CH3F: Image Plane DataN 197 199 197 199 197t, 4x106 6x10 9x106 4x106 2x10t 3 x 10 3 x l0 3 x i0 3 x i0 3 x 10T 449078 449098 {T} {T} 449071f {0325} {0325} 0405 {0325} 0324b {0} - 00104 {0} - 00056 000053b0 1053 1288 2436 1’149 1053b1 1359 608- 0025 12•37 12•31b2 - 9910 - 3175 {0}- 1598- 7661b3 {0} {0} {0} 9823- 5&80CH3F: Image Plane Data + Prism DataN 428 428 424 428 428t 4x106 2x106 3xl0 4x10 4x10tm 1 X 10W’ 1 x 10_i 1 X 10_i 1 X 10_i 1 X 107 {T} 449086 449054 {T} {T}j3 0•359 {0325} 0347 {0325} {0•325}b {0} - 00044 {0} {0} - 00057b0 1539 1l92 1535 1114 1208b1 474 671 260 801 &48b2 -1836 -23•69 -372 -2913 -22•82b3 -2324 2913 -269 3&70 2797Table 5.3 Some fit results using [5.3] for CH3F data. {T} =4490867C(Table 5.2)in most of the fits in which b is used as a free parameter with correction term(s) whether7 and/or i3 are also free parameters. Hence, b remains close to 0 as expected.It is also found that pure power-law fits (i.e. all correction terms, b123 = {0}) do88-0.9—1—1.1-1.2-1.3-1.4-1.5I-1.8-1.9-2-2.1-2.2—5 6log10 tFigure 5.6 Log10 p vs log10 t for H2C:CF image plane and best fit forT={2974186}, 13={0325}. (Table 5.4)not work well for whatever value b takes. This implies that a significant portion of thedata is far enough removed from T that correction terms are required, especially whenthe prism cell data are included in the analysis as listed in Tables 5.3 and 5.4 and asillustrated in Figures 5.4 and 5.5 As discussed before, if the pure power law suffices,the graph will have been a nearly horizontal fit approximating b0.b0 does not vary significantly even when the fitting range (in t) is changed. Thisis expected if b0 is a system-dependent amplitude. The averages of the results in Tables5.3 and 5.4 are shown in Table 5.9. H2C:CF fits have more widely dispersed resultsbecause of the inclusion of its prism cell data which are more “scattered”, as seen by-5.2 -4.8 -4.4-4-3.6—3,2890.10-0.1L-0.200— -0.3-0,4-0.5-0.6log10 tFigure 5.7 Log10 (tp) vs log10 t forH2C:CF image plane and best fit for= {O}. (Table 5.4) I3 = {0 325).comparing Figures 5.6 and 5.8. Higher correction term coefficients, however, tend tovary widely with fit variations. This is likely due to the scatter of the data-points and theincreasing flexibility of the fitting function, as previously discussed.5.3 DIAMETER, PdLike [4.3], the equation used to fit Pd is of the form [Pu:Pd = a0 ÷ a1t° ÷ a2t + a3t”° [5.4]-8-6-4where t is the subcritical reduced temperature. Firstly, the law of rectilinear diameter is90-0.3-—-0.4 --0.5 --0.6 --0.7 --0.6 --0.9 -—1 -—1.1 --1.2 -Ilog10 tFigure 5.8 Log10 p vs log10 t forH2C:CF image plane (+), prism (0) dataand best fit (—) for {7,$}. (Table 5.4)tested where a1 =a3=0. Secondly, [5.4j is used with and without the correction term ina3. The exponent, o-=1-a (=089 for {a} fixed at the number in Table 2.1), is allowedto vary as a free parameter in some of the fits; while the gap exponent, ={05} is heldconstant.Table 5.5 lists the results of fits to theH2C:CF and the CH3F image plane dataonly, with i={089} and 7,={2974186} forH2C:CF and T={4490867} for CH3F.Table 5.6 lists the results when the prism cell data are also included. The data are plottedin Figure 5.9. In these ranges, a 1 (as expected) although the magnitudes and signs ofthea1,23 terms change with the addition of more fitting (correction) terms and more data-++I I I-5 -4 -3 -291H2C:CF : Image Plane DataNtnntmTb0b1b398lxlO-69x104297408{0325}{0}06992494{0}{0}1003 x 109x104{T29}{0325}{O}06642832{0}{0}1005x109x104297425{0•325}- 00082107813430- 6453674391003x109x104{T29}{0 325}- 00051907263468- 3668{0}1005 x 109x1042974250304- 00093506343942- 699271474H2C:CF Image Plane Data + Prism DataN 131 131 131 131 131ç, 3 x 10-6 3 x 10-6 3 x 10-6 3 x 1o6 3 x 10-6tmax 1 X 102 1 X 10-2 1 X 10 1 X 102 1 X 10-2T {T29} {T29} {T29} {T29} {T29}fi {0325} {0325} 0329 {0325} {0325}b - 00098 - 00022 {0} {0} {0}b0 0904 0629 0592 0710 0•570b1 1700 4286 4947 2581 5042b2 - 7546- 36389 - 435’49- 11556- 43856b3 {0} 106453 1317•42 {0} 1315•78Table 5.4 Some fit results using [5.3] forH2C:CF data. {T29}=29.74186C(Table 5.2)points (from the prism cell). This is due to the added “flexibility” of the fitting line asmore terms are included, i.e. more free parameters to vary. This effect is illustrated inFigure 5.10 (for the image plane fit to the CH3F data near 7). The straight line is the92H2C:CF N=149 7={2974186} : 78x103t512a0 09993 09966 09962 09976a1 {0} 065 074 - 024a2 085 {0} - 012 15603 {0} {0} {0} - 110a0a {0} 021 -816 .5.06*a2 046 {0} 1609 8.17*03 {0} {0} {0} 20.16*Table 5.5 Fits to Pd using [5.4] for image plane data for {Tj and o={089}.The fit in Figure 5.10 uses the results marked .H2C:CF:_N23309995{0}084{0}7={2974186} : 53x iO t51 x 1009984 09996 09991061 -006 100{0} 092- 100{0} {0} 150Table 5.6 Fits to Pd using [5.4] for image plane and prism data at {7} andi={089}. The fit in Figure 5.11 uses results marked .CH3F: N=199100127={4490867} : 4x10t34x1010012 10022 1.0021*a0a1a203CH3F: N=258 : 1={4490867} : 97x104t11x00 10017 09983 09994 0.9979*a1 {0} 096 065 2.23*a2 121 {0} 039 2.29*03 {0} {0} {0} 1.63*93I 021 01IFigure 5.9 Pd vs t of image plane and prism cell data: CH3F(+);H2C:CF(0)rectilinear fit to the prism data for large t.The data for small values of t fall below the rectilinear fit to the prism data (forlarger values of t), as illustrated in Figure 5.11. This shows the effect of the t’ termiiui which is explained as a repulsive three-body interaction [G2,N1-2,P11 between themolecules. The “scatter” (in the plot) is (likely) due to fringe count and other randomerrors. These problems reduce the reliability or accuracy of the correction terms whichaccount for the variations in magnitudes and signs when they are added to a fit.One variation to the fit is to let o= 1—cr be a free parameter, while fixing {7}.In this case, o tends towards 1 showing the difficulty of determining the critical1 06I 05I 04I 030 990 0.02 0.04REDUCED TEMPERATURE, r941 . 0061 0051 004I . 0031 0021 .00110 . 990 0.001 0.002 0.003REDUCED TEMPERATURE, tFigure 5.10 Diameter, Pd vs t < OOO35 for CH3F. The straight line is the bestprism cell data fit. The curve is a fit to image plane data. (Table 5.5) Prism data(D); image plane data (x).exponent, a. a0 still remains close to 1, while a1,23 have increased in magnitude.Conversely, allowing T to be a free fit parameter while u={O89} produces values ofTe that are vary significantly about the values in Table 5.2. Some results for image planeand/or prism cell data for these fit variations are shown in Tables 5.7 and 5.8. In almostall cases, a1 and a2 are comparable in magnitude when only one of them is present (i.e.where a1 = {O} a = {O} in any fit), or when both terms are present in any fit,a1—a,i.e. they tend to offset one another. This is because the field-mixing t° termand the rectilinear term in t are approximately equal in magnitude as o is close to unity.951 0021 001I0 . 9990 0.031 0.002 0.003 0.004REDUCED TEMPERATURE, tFigure 5.11 Diameter, Pd vs t< 0004 for CH3F. The straight line is the bestprism data fit. The curve fits to prism and image plane data. (Table 5.6) Prismdata ([]); image plane data (X).In fact, the exponents in the three lowest order terms in [5.4]- theoretically, 089, 1,l39, respectively - are not at all significantly far from unity.The last possible variation in [5.4] is to free both T and o in any fit. Such avariation is found to produce “unreasonable” results. It is also difficult to ensureconvergence in such fits. 7 and o stray meaninglessly and significantly from theirexpected values as such fits have up to six free parameters to adjust to suit the scatter ofthe data.I. 006I. 005I 004I 00396H2C:CF N=233 T={2974186} : 53x10t51 0-2J 0890 1037 1007 0860a0 09995 09998 09996 09989a1 {O} 094 100 100a2 084 {O}- 014- 136a3 {O} {0} {0} 208CH3F: N=418 : 7={4490867} : 4x10t11x10’a. 0890 0948 0999 0999a0 10014 10006 10007 10007a1 {0} 108 8772 3041a2 122 {0} -8665 -2918a3 {0} {0} {0}- 028Table 5.7 Fits to Pd using [5.4] for image plane and prism data at {T}.N 199 180 85t, 24x10 70x106 0t 36x10 34x10 30x107 449838 448981 44•7957a0 10011 10026 10014a1 {0} - 940- 170a2 046 1832 050a3 {0} {0} 34.44Table 5.8 Fits to Pd using [5.4] for the image plane data of CH3F atu={089}.975.4 DISCUSSIONFigure 5.9 shows that the rectilinear component of the diameter (away from 7)is dominant. Its coefficient, a2 increases with 7 for the different samples by the largerslope for the CH3F data. This is in agreement with studies of other fluid diameters. Ni2,P1] Figures 5.10 and 5.11 demonstrate the downward trend of the data-points, away fromthe rectilinear diameter extrapolation. This tendency is also in accord with the effect ofthe field-mixing, t term as t—’O. [P1]As previous stated, this t ‘ anomaly in the diameter alters the classical law ofrectilinear diameter near the critical point because real fluids lack the particle-holesymmetry presumed in the lattice gas model. Its derivation has been achieved byassuming the existence of a three-body interaction among the fluid molecules (02,P1]. Thelack of dominance of the a1 term over the a2 term is proposed as being due to thecompetition between the weaker three-body forces and the two-body forces. The relativestrength (ratio) of these two interactions is quantified by c p, which is called the criticalpolarizabifity product. If the three-body forces are of the Axilrod-Teller, dipole-induced-dipole, repulsive potential [A9,G9], it is predicted (up to the leading order ofmagnitude) that such three-body forces will result in I, and a2 being linearly related toeach other, and, b0 and a2 being each approximately linearly correlated with a p. Thelatter is a dimensionless number if i (moFcm3)is interpreted as the critical numberdensity by multiplying by Avogadro’s number, NA. The polarizability, (A3), definedas an effective polarizability is an enhancement of the electronic polarizability, a, forpolar molecules at their critical point, T. If the dissociation/ionization energy,‘d’ and982-1.9 -1.8 -1.71.6 -1.51.4 -1.3- I I I I I I I0.5 0.7 0.9 1.1 1.3RECTILINEAR DIAMETER AMPUTUDE, a0Figure 5.12 b0 vs a2. ()H2C:CF;(v) CH3F; () CHF; (X) Xe; (D) otherfluids. [N1-2] The solid line is an approximate linear correlation.the permanent dipole moment,,are known, the enhanced polarizability is calculatedto be jpij:2 = 2 + e,2+ [5.51e9Idl4BTCkB is the Boltzmann constant.Recall that a2 is the rectilinear term in [5.4], for T further away from T. Valuesof a2 can be found in Table 5.6. The quantity, b0, is the pure power-law amplitude in[5.3] for a range of values of t as close to 0 as possible. The values of b0 in Table 5.9D0CCVC0x990.3320.03 -0.028 -0.026-0.024-00.022 -0.02-0.018-0.016 -0.014-0,312 -0.01-0,008-C.)0.006-0.004 -0.5RECTILINEAR DIAMETER AMPLITUDE, a0Figure 5.13 ap vs a2. (<C>)H2C:CF;(v) CH3F; (A) CHF3; (X) Xe; ([]) otherfluids i1-2]; (-) approximate fit.are derived from pure power-law fits for t< 34 x 1O for CH3F and t< 88 x 1O forH2C:CF. Other data have been taken from Tables 4.2, 4.3, 4.6 and 5.2 in order todefine the polarizability products.It is found that CH3F does not conform to the hypothesized linearity between a2and b0 ipii when compared to other substances (as shown in Figure 5.12). Whereas, theresult for H2C:CF follows the expected behaviour more closely. Similarly, for thecorrelation of a to a in Figure 5.13. Other fluids in these plots (D), from low tohigh a2 values, are: Ne, N2,C2H6, CC1F3,C2H4, SF6 and Xe. p2,w1-2,L16,P1]The positions of the CH3F and CHF3 data-points in Figures 5.12 and 5.13 seemxDD 0VC0 7 0.9 1.1 1.3100to indicate a “common difference” for the two substances compared to the other samples.One such difference is that they both have large permanent dipole moments,,compared to the other substances. (See Table 3.1.) For example, it equals =165Debyes for CH3F. [B17] It is suggested that other many-body forces (besides the AxilrodTeller-type interaction) may play more significant roles on the behaviour of moleculeswith large permanent dipole moments, resulting in such “deviations”. 2-z5,2s,N1-2] Theexact nature and strengths of these interactions are however, presently unknown.Lastly, it should also be noted that the selected b0 and a2 values are dependent onthe curve-fitting ranges of t and the number of correction-to-scaling terms used. Data-points which are more scattered and/or the choice of the ranges of t which are closer orfurther away from T, will affect the values of these amplitudes. These factors make thecalculations and comparison of a1, p, (from different investigators) difficult to interpret.However, many other polar substances (with larger or smaller dipole moments) will stillhave to be examined to further this many-body interaction theory.Fit Variation CH3F H2C:CFJh0 [M14] 18584 1382 Debye[L13] 1247 1029 eVa 3•77 464 A30020 0018a2 12l 08410 163 l59Table 5.9 Effective polarizability, a1,, [5.5], and critical polarizabilityproduct, a1, p.101CHAPTER SIXISOTHERM CELL RESULTSThis isotherm cell was constructed to test the feasibility of studying as large aportion of P-V-T space as possible with the same sample by optical means. Technicaldrawings of the cell are shown in Appendix C and a schematic of the thermostat inFigure 3.9. In this arrangement, the prism method (Chapter 4) and the fringe method(Chapter 5) are combined. The use of one sample for a large range of densities avoidsthe problems of combining data from different samples in different setups and/ordifferent investigators. Some preliminary results are presented for CHF3 and CH3Ftestedin this two-compartment isotherm cell. The aim is to measure isotherms of pressure (F)and density (p) directly. The pressure unit selected is pounds per square inch (pSi).t Thelaser wavelength is X=6328A.6.1 CALIBRATIONSThe calibration of the micrometer screw (mc) directing the deflection mirror onthe prism side (M6 in Figure 3.10) of the isotherm cell is carried out in the same manneras for the prism cell, i.e. using a 50-lines-per-inch Ronchi grating. Using the definitionsgiven in Chapter 4 and [4.11, the calibration equation is:The conversion is 1 psi = 6894757kPa. Conversely, 1 kPa = 0 1450377psi. [M91102sinO = {_(1.9±o.4) ÷ (13154±03)u — (5±l)u2}xlo4 [6.1]where u is the difference in the micrometer readings between the incident beam, withno cell present and the deflected beam, Zd with the cell installed. The correlationcoefficient of this fit is R=0999999.The prism angle, 4 is found using a reticle reflection technique from an auto-collimator [D6J. The average reading is =(19744±0024)°. The 11parallelism” of thesurfaces of each window is also tested by the same method. For the normal-incidence,(Z1) entry window, the surfaces are wedged at -09 minutes of arc (-0015°); while the(Zd) exit, angled window surfaces are off-parallel by +08 minutes of arc (+0.013°).(Appendix B shows the diagram, definitions and derivations.)The same test is applied to the surfaces of the sapphire windows of the imageplane side. The wedge angle between the inner surfaces (which are in contact with thefluid) is less than 25 minutes of arc (0042°). The angle between the outer and innersurfaces of the (Z) entry window form a wedge of -2 minutes of arc (-0033°); for the(Zd) exit window, it is + 17 minutes (+0028°). The cell thickness, 10, is an average ofreadings measured with a micrometer screw gauge. It is (2 16±001)mm. Magnification(by L2 in Figure 3.10) at the film plane is found to be M=—(278±005)x.The Digital Pressure Gauge (DPG) is factory-calibrated to give a “0 psi” readingat 1 standard atmosphere (i.e. 146962psi) ii91 against a standard gauge. (For this reason,the stated unit of psisg.) However, since the gauge actually measures against the ambientpressure, all pressure data are “re-zeroed” by adding the average of the gauge reading103after it has been evacuated before and after a run. This uncertainty is measured to be lessthan ±035 psi (during and after evacuation of the gauge over several days). Assumingthis variation to be a normal daily average value, such a magnitude in the uncertainty willonly be significant near the end of a run when the pressure in the cell is low.6.2 FRINGE COUNT CONVERSIONThe data consist of fringe counts, N, interspersed among deflection angle data.(For example, deflection is measured every three hours on the half hour, while themicrocomputer takes gauge and temperature readings and marks the film by flashing theLED every hour on the hour.) The deflection data are converted into density, p, usingthe Lorentz-Lorenz relation, [3.11, the refractive index, n, [B.26] and the Ronchicalibrations, [6.1]. For CH3F, the values of the refractometric virial coefficients are thosegiven in Table 4.3. For CHF3, the coefficients (in molar units) are: 1-2](p) = 6905 + 376 p - 2800 p2 [6.2]=- 21 ÷ 52n - 24nLINEAR QUADRATICfo 756 ± 002 756 ± 002 X103Jj 2705 ± 0002 2711 ± 0003 x1051 - 3±2 xlO’°R2 099996 099996Table 6.1 Fits to [6.4] of p (mol cm3) vs N for CHF3.1040.0110.010.00929 0.0DB020.007• 0.0060.0050.004O 0030.0020.0010REFERENCE FITfo 8765 8756 ± 0011 X103Jj 2827 2834 ± 0012 xlO-5j - 8921 - 8 ± 4 x10•’°R2 099996-200 -100 0—300 100FRINGE NUMBER, NFigure 6.1 Fit to [6.4] of p (moFcm3) vs N for CHF3. The solid linerepresents the quadratic fit. The horizontal line is p =p.Using the paired data of deflection and fringe count (PQ), a fit of p vs N can becompared to the “theoretical” result calculated using [3.13] and the critical parametersTable 6.2 Fit to [6.4] of p (molcm3)vs N for CH3F.1050.0160.0150.014 -080.013 -0.012 -0.011 -0.01— I I I50 70 90 110 130FRINGE NUMBER, NFigure 6.2 Fit to [6.4] of p (molcm3)vs N for CH3F.found previously; (Table 4.6 for CH3F.) For CHF3, using T=(2&008±00O7)C,n=10806 and p=753X103molcm3 (0527gmol1)[N1-2], [3.13] is:=÷ 27O7x105N+ 1004x10’°N2 . [6.3]The measured data consists of 112 pairs of data-points. It is fitted to:p =f0 +f1N +f2N . [6.4]The results are shown in Table 6.1. fo,i do not vary by more than 04% from those in[6.3]. However, J is significantly different (-70%) from the “reference” value. ItsF I I I— I150 170 190 210 230106uncertainty is also large. As seen in Chapter 5, the linear term is much larger than thequadratic term, i.e.JjJ because of X/10. Although both fits have correlation coefficient,R 1 and F> 106 (i.e. they are good fits), an F<0.1 test shows that the quadratic termis not necessary to improve the fit. The quadratic fit is shown in Figure 6.1 (althoughthe linear fit is used to convert data later). The horizontal line, p=p, intersects thevertical N=0 line.The results of similar computations carried out for the paired data from CH3Fruns are shown in Table 6.2 and Figure 6.2. There are 199 pairs of data-points for thequadratic fit. The fit results do not differ significantly from the reference calculationseven though p, is not included in the range of densities tested.6.3 VIRIAL FITSThe virial equation, [2.3], is compared to a polynomial fit to the data.P= RTp+RTBp+RTCp÷P A0 + P ÷ 2 p2 ÷ 43 3 +0 RT 689x103A1 [6.51B 829x104 c 829x1cY4T 2 Twhere B and C are the second and third virial coefficients, respectively, the temperature,T is in Kelvins, and the units of the coefficients, {A : j=0,1,2,...} are psicm3mol.A non-zero A0 fit result indicates that there may be a zero shift error (most likely) causedby the atmospheric pressure adjustment of the gauge (as discussed before) and/or a10790080070080050040030020010000 0.002 0.004 0,006 0.008DENSiTY, p in molcm3Figure 6.3 CHFisotherms at Tr=259400C (A), 259505C(<>)afld264090C(v). The vertical line is p =p. The slope is the ideal gas law at T=264090C.density conversion systematic error, or, the density range of the fit is not appropriate forextrapolating to p =0. The numerical factors (in [6.51) are derived from converting metricunits to psi. [M9JAnother choice is to restrict the fit to the cubic van der Waals equation. Rearranging [2.2],P=-ap2+ RTp [6.6]1-bpwhere a (psicm6mol2)and b (cm3mol’) are the van der Waals molecular parameters108mentioned in §2.1 and §2.2. They may have small temperature dependences. [c3,R11Curve-fitting to [6.5] is divided into three segments for each run. The first is afit to all data-points in a run, i.e. spanning p. The second is a fit to data for which> The third is for P <pt, data. Three isotherms for which P <pt, are presented inFigure 6.3. The results of fits are listed in Table 6.3. “Theoretical” or reference valuesof the virial coefficients (if available [1)7]) and the percentage differences of the fits fromRUN #1259400±00001505 x iO-120 x 10.22341056 5514P<0.062+32498 ± 62486 84043- 190 ± 1- 1860219712220 ± 16012155056233 x 104767RUN #2 RUN #3259505±00003 264090±0•0002268 x iO 147 x 10103 x 10.2 958 x 1087 3567350 8753554 173p < 00048 p < 0008008±282 -5±62498 ± 8 2622 ± 12248693 249074044 5.3-187±2 -184±2- 18601 - 18541034- 05612560 ± 250 10986 ± 14012155 121319.46703 x 10 6841 x 105169 5446CmoFcm3mol cm3psipsidata-pointsmolcm3psiJmo11J. mol4cm3 moldcm3mold%cm6mol2cm6 mol2%____psi cm6 mo12cm3 mol’Table 6.3 Three CHF3 runs with fit results, reference values 7] and%differences for RT, B and C. a and b are van der Waals constants.TpinmPinax‘minmaxNFitA0RTBCab109the reference values are also included. Density, p, is deduced from fringe count, N, usingthe linear fit equations in Table 6.1.A fit to all data from the subcritical isotherm measured at T.=259505C requiresan eighth-order polynomial, i.e. of up to the p8 term. However, the values of RT, B andC are appreciably different from their respective reference values (as shown in Table6.3). Hence, a virial fit (to a subcritical isotherm) to a range of densities that includeswill likely give erroneous results, especially for the (higher-order) virial coefficients.Such deviations are caused by the inclusion of the data around the coexistencecurve, i.e. data at or near the coexisting densities, Pi and pg. The compressibilities orslopes around these densities increase rapidly. This affects the number of fitting termsrequired and the values of coefficients, A 0,1,2,...• This region should be excluded in theFIT #1Prnax 7x103A0 26 ± 14RT 2483±5- 02B-191±127C 158 ± 0330D-46±2N 382FIT #24 x 10-s3.5 ± 192441 ± 1518- 170 ± 6- 877.5 ± 25- 3860 ± 30180FIT #32 x 10-s31 ± 252470 ± 3007- 187 ± 130515 ± 42391molcm3psiJ mol’%cm3 mol’%X 10 cm6mol2%x104cm9mo13Table 6.4 Fits to data for which p>p>268x105moFcm3 for CHF3 atT=259505C. Percentage are comparisons to reference values shown in Table6.3.110virial expansion or the van der Waals equation.A possible means of avoiding such curve-fitting problems may be to use super-critical isotherms only or data-sets that span a wider range of densities and pressures (soas to minimize the effect of the data near the coexistence curve). However, the currentsetup is designed to hold up to approximately l500psi or— 2P which means that p <2pfor the substances tested. The same problem applies if the data are restricted to p >p.A0 is (usually) non-zero because of the large extrapolation required in such fits. A 0,1,2,...Table 6.5 Two CFT3F runs with fit values,differences for RT, B and C, and, a and b. [c3,Mtoreference values 7] and %Tr 448904 449024 Cp 007 003 X 10 molcm3Pmax 10•0 105 x103 moFcmP 276 82 psi870 871 psiN 207 252 data-pointsA0 -12 -7•7 psiR T 26004 26165 Jmol126444 26445 J’moit17 11 %16087- 16232 cm3mol117790- 17812 cmmol’9’6 8911700 11900 cm6mo1215380 15640 cmmo1..E____- 315700- 326700 cm9mo13688 688 x107psicm6mol’5138 51’38 cm3mo1B -CDab111are generally larger.For the low density region, P <pa, no more than the first few terms, 01234 (forexample), should be needed in fits to [6.5]. [jul The results for various ranges of P <pfor Tr=259505C are shown in Table 6.4. The percentage differences of the fit from thereference values (given in Table 6.3) are placed below the fit values. The uncertainty inA0 is comparatively large. This demonstrates the influence of the extrapolation to p =0and the dependence on the degree of the polynomial selected. However, A0 is effectivelyzero in each fit when compared to P.The RT term approximates how closely the ideal gas law, [2.1], is approached.As shown in Figure 6.3, this is true as p-0. The percentage deviation is small. Thesecond virial coefficient, B(7), does not deviate by more than ± 10% from the referencevalues in the density ranges shown. However, shortening the range of density values(shown in Table 6.4) significantly affects the derived values of the third virial coefficient,C(7), (listed in Table 6.3). These effects demonstrate the strong influences of the densityrange and the number of terms have on the results of the fit.Two isotherms of CH3F at T=449024C and 7.=448904C are measured. Theresults of the fits are presented in Table 6.5 and plotted in Figure 6.4. The virialcoefficients and van der Waa.ls parameters, a and b, do not vary significantly betweenthe two temperatures. a and b are not sensitive to the small differences in temperature.[c3,R1] (These parameters do not vary by more than 5% for a and 7% for b for the threeCHF3 runs reported in Table 6.3.) The reference values for CH3F are a=&806x 10psicm6’mol2 and b =52 64 cm3’mol1. [c3,M10l These differ from the fit results by 11 %112and 24% respectively.6.4 ISOTHERMSThe calculations above show that in the “classical” regime, away from the criticalregion, and phase transitions, this optical system can adequately make directmeasurements of isotherms. The isotherms shown and the calculations in the classicalregime attest to the consistency and reproducibility of the equipment. Three CHF3isotherms are plotted in Figure 6.3 and the two previously mentioned CH3F isothermsare plotted in Figure 6.4. Figure 6.5 shows that the percentage pressure differencebetween the two CH3F isotherms. The densities are (interpolated and) averaged becausethe two data-sets do not match exactly along the density axis. The percentage pressuredifference is small.A conspicuous characteristic of these isotherms is the nearly flat portion aroundThe gradients are small but not negligible. (See Figures 6.3 and 6.4.) It representsthe slow decrease in pressure as the fluid is being bled slowly because the celltemperature, 7 7,, or, that p is close to p, as the system passes through the criticalregion, 9,. The effect may also be due to the “slowing down” [S71 effect close to 7.Alternatively, the small slope may also be caused by thermal and mechanicalinequilibria in the system. Recall that the gauge and much of the capillary tubing (ht inFigure 3.10) are kept at T> T; while the middle portion of the capillary tubing is atroom temperature. The rest of the tubing and the cell are at 7 T. Hence, over the(longer) periods required to equilibrate when the system is near its critical density, pt,,1131.31.21.1•a I0.sX 0.80.70.60.30.20.100.006 0.0DB 0.01 0.012 0.014 0.016DENSiTY, p in molcm3Figure 6.4 Isotherms of CII3F at 448904C (+) and 449O24C (). Thevertical line is p =p.the effect of these temperature differences is more significant as the fluid state issensitive to thermal leakage into the cell through the tubing. It is also possible that thetime allowed for equilibration between bleeds is not long enough so that the system isconstantly re-equilibrating throughout this portion. (A small, slow leak in the shut-offV valves to the gauge could have the same effect although no leak was detected.) Thisuncertainty is due to instability of the fringes close to 7.Another characteristic of the plot is the smoothness of the curves near where thecoexisting densities, p, and p, are expected to be. That is, at phase separation, first-orderdiscontinuities are not apparent. This is likely due to the finiteness of the system, or, the0 0.002 0.0041140.90.8070,5050.40.3L)Z 0.20.10-0.1, -02-0.3-D 4-0.5-0.5-0.7-0.8-Q 90 016Figure 6.5 Percentage pressure difference between the isotherms of CH3F at449024C and 4489O4C. The vertical line is p=p.thermal problems as mentioned before. Such rounding or smoothing of the data meansthat determining the appropriate ranges when 7< T for curve-fitting will be difficult.The pressure-density curve-fitting methods and results are detailed in AppendixE. The results in general are discussed below. In nearly all possible variations of thefitting equation, the value of the critical exponent, , is lower than the RenormalizationGroup value of 4 82. This is partly because the measured isotherms are not the criticalisotherms. Also, the difficulty in selecting the appropriate range of densities to use (incurve-fitting) is not easily resolved.Another approach is to consider the isothermal compressibility, K, in the following0 0.002 0 004 0.006 0 008 0 01 0.012 fl,D1AVERAGE DENSITY, p in molcm’115form: 1A81KIT—d€I1PI [6.7]where d is assumed constant. By finding ôe for several supercritical and subcriticalisotherms, ô at t=O can be interpolated. The advantage of using x (as defined by [2.11])is that since it can be derived from the isotherm by differentiation, fits to [6.7] aresimpler because several free parameters are eliminated. However, this procedure dependson the method used in estimating gradients from data-points. The scatter of data-pointscan produce erratic results- for example, sign changes- even between consecutive data-points. The slope is estimated or smoothed by assuming that the gradient at a data-point,p, is the slope of a quadratic fitted to three consecutive data-points at Pk-1, Pk and Pk+1(after the data-set has been ordered according to density, p). Obviously-erroneous values(wrong signs, etc) are discarded.Figure 6.6 shows the plots of the compressibility against the density calculatedfrom four CHF3 isotherms. As the cell temperature, T,, approaches 7 (t— 0), the curvesare steeper near the phase transitions at Pi and p, than for those curves for which 7. thatare further from 7.Figure 6.7 plots compressibility vs pressure for the same four CHF3 isotherms.The K peak for the supercritical isotherm at 2&409C (0) is broadened, continuous andsmaller compared to the larger, sharper, narrower “spikes” for subcritical isotherms.This indicates that this (supercritical) isotherm varies continuously, without (first-order)phase transitions, as expected. A “displacement” error in the 256436C (X) peak is1160.150,14 -0.13 -0.120.11-0.10.09-> 0.090.0?-0.06-0.05-0.04-0.03-0.020.01 -0- I I I I’Ti .0 0.002 0.004 0.006 0.008 0.01 0.012DENSiTY, p in mol-cm’Figure 6.6 Compressibility, K (psi-’) vs p (mol-cm3) plots of four CHF3isotherms. x = 256436C; K 259400C ; + 259505C; El 2&409C.clearly indicated as it does not occur at a lower P (to the left of the other peaks).Figure 6.8 shows how K is “folded” about the critical density, p, by plottinglog10 I.ip* for the 7=44- 8904C isotherm of CH3F. The two solid lines illustrate thedifference in slopes for P <PC and > The expected sharp changes in slope at p andPi are not discernible from the data, i.e. pg and Pi are not easily defined.Figure 6.9 shows the scatter of &e for fits to various density ranges of CHF3 andCH3F isotherms using [6.7]. Almost all are below the Renormalization Group value of4-82 as Tr T. The fit ranges are selected by observing the spread of the data-points inplots of 1og0(K) vs 1og0(I I) (like Figure 6.8). Without other criteria, it is difficult0+0+00++0 +0o +0 -0++ 0000X0bcrx1170.150.14- 00.13 -00.12 -o !0.09- 000.06-0.07-0.06-00.05- 00.02-0 0.2 0.40.6 0 8PRESSURE, P in x iO psiFigure 6.7 Compressibility, x (psi-1) vs P (psi) plots of four CHF3 isotherms.x = 256436C ; K = 259400C; + = 259505C ; El = 2&409C.to determine any alignment of the data that will permit useful extrapolation to t-Oexcept to note that 6e will be less than 482 for these fits. There are too few supercriticalisotherms to note any trend for the t—0 case. The “best” fit comes from the subcriticalisotherm of CH3F at Tr=449024C (t— -2 x 10-s), in the ranges, 001O <p <0014molcm3and 870< P< 1037 psi. Table 6.6 shows the values of 8e and de averaged overall the isotherms reported. They are categorized according to the fluid used and whetheris positive or negative. ôe is generally less than 482.6.5 DISCUSSION118-3-3.2-3.4-3.6-3.8-4-4.2-4.4-4.5-4.8-5-5.2-5.4-5.6-5 8-2.4-2 -1.6-1.2-0,8-0.4log10 I IFigure 6.8 Log10 K vs log10 I I plot of the 7.=448904C isotherm of CH3F.= Lkp>O X =Figures 6.6, 6.7 and 6.8 are instructive in restricting the ranges to use for curve-fitting. They can be used to estimate the values of p or P where gradients begin tochange rapidly. These values may also serve as (poor) approximations of the values ofPt and p. They estimate the density limits, p and p. (See Appendix E for defmitionsof these limits.) The slopes in Figure 6.8 are observed to be nearly equal in the twoportions: the solid lines further away from T and the “flattened” portion nearer to 7.These slopes are the estimates of 6.As previously mentioned, insofar as a means for directly measuring isotherms ofpure fluids, the optical setup and technique described above produce quite consistent01195.5 -5-4.5- 0 x8 x3.5-xx3- 002.5 -0x2-x1.5-x1- I I I I I I I I I-0.0016 -0.0012 -0.0008 -0.0004 0 0.0004 0.0008 0.0012 0.0015REDUCED TEMPERATURE,Figure 6.9 Effective exponent, ô vs t using [6.7j. X ip>O; tp<OThe solid horizontal line is e=482, the RGT value.results in the low density regimes. However, near or within the critical region, 9t,, thecurrent setup may be experiencing thermal isolation problems. Despite the constructionof a large thermostat and the use of fine, stainless steel, hyperdermic tubing (ht) toreduce conduction into the cell block, certain physical restrictions were unresolved andunresolvable (with this setup and at this juncture). The pressure sensing device (DPG)and part of the tubing are outside the thermostat, and, are at different temperatures fromthe cell. This breach of thermal isolation is especially significant when the system isscanning the critical region,A possible improvement to the setup may be to embed the silicon pressure120transducer (SPT) itself within the thermostat in thermal contact with the cell block. Dataacquisition is then controlled in the “remote mode”. This option, however, is notavailable with this particular model of the gauge. Even if the option is chosen, thermalisolation problems may still be unresolved because it will still be necessary to have ableed-line exiting the system and regular accessibility to flow (MV) and shut-off (NV)control valves will breach the isolation. These effects may be lessened by allowing longerwaiting periods for thermal re-equilibration.Ideally, the pressure sensor (SPT) should be at a higher temperature than TTr in order to avoid phase separation within its volume. This is the rationale for thecurrent configuration. Hence, embedding the SPT in the thermostat or cell block may notbe the needed solution, assuming physical size constraints can be overcome.Another alternative is for the entire setup to be enclosed in a box within whichthe temperature is kept slightly above 7 and 7. However, this does not remove thepossibility of thermal gradients when the system is bled; especially when approaching orwithin the critical region, The beam deflection (on the prism side) must be readregularly and film cartridges must be periodically extracted from the camera on thede NCH3F 418 ± 040 (2 ± 5) X 10 5CHF3 3.43 ± 095 (4 ± 5) x 10 25386 ± 060 (10 ± 5)X104 11391 ± 060 (10 ± 6)x105 14Table 6.6 Averages of & and de calculated using [6.7].121image plane side. (See Figure 3.10.) These steps mean that accessibility which may causethermal isolation breach, must be taken into consideration again.The heater block has to be an amply large thermal reservoir to stabilize the cellblock temperature and prevent the occurrences of temperature gradients across the cellvolumes. For the current configuration, T. (as monitored by thermistors embeddedaround the cell block - Figure C.3), is highly stable. It is most affected by fluctuationsin the water temperature (T of the water jacket) which is affected by sudden ambienttemperature changes (like opening a window on a cold day).The thickness of the styrofoam insulation between the water jacket and the heaterblock (Figure C.3) may however, be changed to accommodate physical size and thermalresponse constraints of the setup. For example, if the heater block size is kept constantand the thickness of the styrofoam layer is reduced, an outer styrofoam casing envelopingthe water jacket can be devised if size restrictions can be adhered to. The addedinsulation should improve the block thermal stability although it is not clear that this willreduce the thermal isolation problems of the capillary tubing leading into the cell block;especially when the system is near critical.Lastly, pressure and density ranges must be extended to higher values. Theincreased span will better allow for expanded tests of classical theory.122CHAPTER SEVENCONCLUSIONSThree experiments were performed to investigate the behaviour of pure fluids neartheir critical points. Each experiment used the deflection of a laser beam and/or aninterferometric technique to follow density changes in a fluid contained in a cell. In thefirst experiment, a temperature- and mass-controlled, prism-shaped cell was used to relatethe density, p, to the refractive index, n, of a pure fluid through the Lorentz-Lorenzrelation, . In the second setup, density changes of a fixed amount of a pure fluid sealedin a thin, parallel-windows cell, were measured against temperature (Tj. In the thirdexperiment, the feasibility of measuring isotherms of pure fluids directly by combiningthe prism and the parallel-windows cells in one thermostatic block was studied. Threesubstances were individually tested in (some of) these setups: fluoroform (CHF3),fluoromethane (CH3F) and vinylidene fluoride (H2C:CF).H2C:CF and CH3F were separately tested in the prism cell apparatus. Thedeflection of a laser beam passing through the fluid sample measured the change inrefractive index as the fluid temperature and mass were changed. The Lorentz-Lorenzrelation, , the coexistence curve, p, the coexistence curve diameter, Pd, and the criticalparameters of each substance were deduced from the deflection data. ForH2C:CF, thecritical temperature, T, was found to be (297419±00020)C; the critical density, p,123was (O4127±OOOO3)gcm3;the critical Lorentz-Lorenz coefficient, S, was(1O82±OO2)cmmolthe critical refractive index, ne,, was (1 1068±00002), and theelectronic polarizability, ce, was (4233±OOO6)A.For CH3F, 7=(449O92±OOO9O)C;p=(O2983±OOOO2)gcnr3;=(669±OO1)cmmol’; n=(1O895±OOOO2), andae=(2639±00O3)A.These values were in accord with reference values elsewhere. Theprecision (for some parameters) was improved.In the parallel-windows, thin cell experiment, the cell containing the fluid samplewas placed in one arm of a Mach-Zehnder interferometer. Density changes relative tothe critical density of the fluid, were recorded on film as interference fringe (pattern)changes as the cell temperature was changed. H2C:CF and CH3Fwere separately testedto obtain the coexistence curves and their diameters.It was found that in the range of temperatures studied, correction-to-scaling termswere required in the analysis. The number of correction-to-scaling terms, the reducedtemperature range to consider and the role of the critical exponents became importantwhen the scatter of the data-points was taken into account. The leading power-lawamplitude, b0 was found to be stable whichever (other) components were varied in eachfit. However, correction-to-scaling amplitudes changed unpredictably and appreciably.The critical temperatures deduced from these data were in agreement with thosededuced from the prism cell data. The image plane cell values, however, had a muchgreater precision as the setup was designed to retrieve data at smaller values of reducedtemperature, i.e. to get closer to the critical temperature, T,. 7 was found to be(4490867±000016)C for CH3F and (2974186±OOOOl2)C forH2C:CF.124For the coexistence curve diameters, by combining the image plane data with theprism cell data, it was found that the diameters followed the classical rectilinearbehaviour away from T (of each fluid). The slopes of the rectilinear diameter were foundto increase with the critical temperature, T for different fluids.The slopes of the data appeared to increase when 7 was approached (frombelow), i.e. a tendency towards smaller diameter values. This demonstrated theincreasing importance of the field-mixing anomaly about the critical region, L02,Pt] Thedata were tested against a theory of three-body interaction proposed to explain theexistence and strength of this anomaly. Specifically, the derived values of the power lawamplitudes (b0) obtained from curve-fits to the coexistence curves, were compared to therectilinear diameter coefficients (a2), and, to the critical polarizability product, a1,p. Theresults from H2C:CF agreed with the proposal that the non-rectilinear diameter, fieldmixing, t term was due to a three-body-interaction force in competition against thetwo-body forces between molecules. However, the results from the CH3F fits did notfollow the trends proposed. Instead, they were more like those from CHF3 1-2j whichalso tended to be different from the expected linearities. Compared to the other molecules(used to test the validity of the proposal), CHF3 and CH3F have large permanent dipolemoments which are thought to imply that the influence of forces other than the threebody type suggested was significant.The third experiment was to test the feasibility of measuring isotherms directlyin an optical arrangement that combined the prism cell and the thin cell in a single entitycalled an isotherm cell. A precise pressure gauge was used to measure P directly (when125the sample was far from the critical region, The measurements were carried out inthe density range, 0< p <2p, and pressure range, 0< P< 2P for CHF3 and CH3Fseparately.At low densities, the “classical” parameters like the van der Waals molecularconstants, a and b, and the second and third virial coefficients at low densities, B and Crespectively, were calculated. For CHF3, the values of B and C were less than 2% and10% respectively, from reference values. For CH3F, however, the differences werelarger at 10% for B and 25% for C. Similar analysis was difficult for data in the highdensity ranges and for those in the vicinity of the critical density, p. This was thoughtto be caused by thermal isolation problems and (possible) “leaks” in the gas-handlingsystem at higher pressures. It was believed that in the current configuration, thermalisolation could not be readily improved and without compromising other characteristics.Alternative configurations such as thermally equilibrating the pressure gauge and the cellin one enclosure, are thought to face the same problems with the added ambivalence ofmeasuring pressure at subcritical temperatures (not necessarily close to T) when the twophases coexisted in the pressure-sensing volume of the gauge. Also, the requirement toaccess the optical elements to make periodic measurements made this form of thermalisolation difficult to accomplish. However, it was also believed that improvement mightbe achieved through a different measurement protocol when the system was in the criticalregion.Two algorithms were tested to find the critical exponent, , from the isotherm celldata despite the abovementioned problems when the system was in or near the critical126region. The first technique was to apply the equation of the critical isotherm withcorrection-to-scaling terms to a chosen range of data around the critical density. Thesecond algorithm was to fit to compressibility data (deduced from the pressure data)about the near-critical data. Both algorithms would then interpolate or extrapolate thevalues of an “effective” &e (from the fits) to t=O for the asymptotic critical exponent.The choice of data range was important in order to avoid “crossing over” into theclassical regime or requiring too many correction-to-scaling parameters. It was found thatthe main difficulty was in defining a “proper” range of data. These algorithms were notable to find ô because the quality of the data around p, was inadequate and the numberof supercritical and subcritical isotherms collected was not enough (for interpolation orextrapolation). The values of 5 found in curve-fitting were generally smaller than thevalue of 482 (expected from Renormalization Group Theory).Hence, the isotherm cell setup could directly measure fluid isotherms in the limitsof density and pressure physically possible for the apparatus, but away from the criticalregion. The technique was consistent and reproducible in the low density portions of theisotherms where classical parameters were calculated. However, the quality and analysisof the data near the critical point needed further improvement as the calculation of thecritical exponent, , was largely unsuccessful. 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M Dekker, NY (1969). p 321.W8 B Widom: 3 Chem Phys 43, 3898 (1965).W9 B Widom: Physica 73, 107 (1974).Wi0 KG Wilson: Phys Rev B 4, 3174 and 3184 (1971).Wil KG Wilson and J Kogut: Phys Rep 12, 75 (1974).W12 HW Wooley: Tnt J Thermophys 4, 51(1983).W13 B Widom and IS Rowlinson: 3 Chem Phys 52, 1670 (1970).W14 LR Wilcox and DA Balzarini: 3 Chem Phys 48, 753 (1968).W15 AS Williamson, PR LeBreton and IL Beauchamp:I Am Chem Soc 98, 2705 (1976).137APPENDIX ADERIVATIONS OF p AND PdA Taylor series of density, p, in terms of refractive index, n, about the criticalpoint (denoted by subscript, c) is:_=i1 (n_n)+!L.E (n—n)2+dnc 2dnSubstituting intoPi — Pg = (P1— p) — (Pg—P1— Pg (fl_flg) {L + + ;and, intoPiPg(PiPc)(PgPc)2 ‘P1 + Pg = 2p ++ { + (flg_flj2 } +The derivatives-coefficients at the critical point are found through a re-arranged LorentzLorenz relation, from [3.1]:n2-1 1P=+ 2 ‘(n)where 9(n) is now written as a power series (in n) whose coefficients are derived fromcurve-fits to the prism cell data when the fluid is in its one-phase region:= L0 + L1n ÷ L2n ÷ ...Then138=-I +eIdnlc afl’c C aeIcwhere, in terms of n and p,ap=______________an c (n2— 1)(n2 ÷ 2)=—2{fsn2+22== L + 2Ln + 3Ln +cI 1And,I=+ 2t’’ aP I + &O!I + a2f Idn2lc an2L C aeanIc C C ae2Ica2pI 3(2—=2p2an2lc-1)(n2 ÷ 2)2ap I=-2 CPCPCae an Ic (2 — 1)22L 2P{n2÷2 2=C }=2L2+63n÷...dn2lc139Hence, the DENSITY relative to p is given by:- =2p {hi(n-ii) + h2(n - n)2 + }while the COEXISTENCE CURVE is given by:PiPg(i’g) {h1 + + flg—+... }and the DIAMETER is given by:- PiPgPd = 1 + h1(n + flg—+ (ri1—+ (flg — n)2 } +whereh = ——=-p, n +21 2,p dn (,l.2— 1) (it2 + 2) 2— 1h = ld2p 3(2-3n) - 6npI’2 dn2 c (it2 —1) (it2 +2)2 (it2 — 1)2— { + 2 + { PC + 22 n2-1 n2-1C C140The propagated errors are:2_ 1 In2+21 [ + eI2{o}2](8h1) ——12 if I.+ {35n }2 [1 npe’ 12 + f 3n4+2÷2 121L l(flcl)J l(c2_1)2 (fl2+2)2f jand(8h2) = {38n }2 F 16n(2n —3n2 ÷2)12 4n(p5fI)2(n2 +2)Li (2 - 1)2 (n2 ÷2) + (n2 - 1)I npP” 12 + 1 ÷3,2 }2 ]+ l(n2_1)2f (n2-1)3I p n2t21+{o}2l__fI 6np 12 {2I{ fl2+21}]+ {o/}2{lfnC -1J2‘22 1F I £t” + 2 2 6 ‘ 12+ 12 p ‘tn: +2 I I I+{8PC}2[fl21I + {flC2i2c 1 1 -ii Ii141APPENDIX BDERIVATIONS OF n IN THE PRISM CELLB.1 THEORYAs illustrated in the ray tracing in Figure B. 1 below, the resultant deviation, 0,of a beam at normal incidence on the outside surface, 1, of the input sapphire windowof a prism-shaped cell can be calculated using simple geometry, trigonometry and Snell’sLaw of Refraction. iisLet each numbered interface and set of connected parameters be subscripted byj=1,2,3,4 as shown. Then= incident angle at interface, j;= refracted angle at interface, j;= deviation angle from original input, Z, at interface, j;= input window’s surfaces, 1-2, non-parallelness;= output window’s surfaces, 3-4, non-parallelness;= prism apex angle.Figure B.1 Z through(Angles are exaggerated;the prism cell. A small ço causes an equally small \t’2.not to scale.)ofciç00withLASERzFL1 2 3 4142Note that are positive for the inclinations shown in the figure below i.e. positive iftapered towards the prism angle.If n, n and na are the refractive indices of the fluid, sapphire and airrespectively, then proceeding from left to right:At interface 1: for normal incidence,[B.1]and in between:1112 = [B.2]At interface 2:flw2=f Sifl6)[B.3]= - P2and in between:[B.4]At interface 3:nsin V’3 =nws1n [B.5];and in between:[B.6]At interface 4:=Sfla)4[B.7]O4=—(ifrO3)Eliminating as many and w1 as possible:143[B.8]Snell’s Law at interfaces 2, 3 and 4 are respectively:= n sin(02 +n sin(+ ‘2) = flw sin(fr4 - [B.9]SflCc)4= a S1fl)4Eliminating as many and c as possible:n sin(02 + q.) = n sinq1 [BlO]sin(fr—= n sin(e+ + ; [B.11]71a sin&4 = sin fr4 [B.12]Expanding [B. 11]:Sin cit4 COS-COS 1J!4 Sin [B.13]= n sinb cos(02 + + n cos sin(02 ÷ q)Since4) < 1’ {orO4} x 1O, cos4) 1, sin4) 4)where 4) = {co,ço0} measured in radians, eliminating by replacing with 04, and rearranging [B.13]:Sm‘Po—‘Po—Sm (04 [B 14]= siu4 /n2- (c)2 + q cos#By using [B.8] and the above approximations,sin4 = (1 — qi0) sin(04 + 4) + (q, ÷ cos(04 + ) [B.15]which to first order is:144sin4 sin(04 + q5) + (q ÷ q?0) cos(04 + [B.16]Although it is usually the case that n < n, it is also true that n2 (nc’1)2. Hence,rewriting 04 as 0 for the total deviation, and reducing to first order only,ns1nI [B.17]i2—qJ!cos—q,0 }n InaRewriting [B. 17] as:A [B.18]sin#where A is the expression within the braces in [B. 17]. The error propagated in thecalculation of n is:2 = (6fl2÷ ( 8 2 + 2 [B.19]\tanwhere(8A)2 = + (4 s280 / I+(A )2+o)2[B.20]iaA \2+ I— ôn I + 1— 6na! \anw W145The partial derivatives are:—= cos(O+tb)— (p+p)sin(O+4) + sin2(O-’-) [B.21]ao 2BaA [B.22]=—+q Sm3OaA—= cos(O+) — .!!cos4 [E.23]8A= cos( [B.24]0(00EM -+ (0.-!!cos(f) ; [B.25]On0 Ba3EM P1--—coscf’ [B.26]B 2whereB =- sin2(8+4) [27]no146B.2 CALCULATIONSThe following values are used in the calculations:= 1000262194 ± 0000001740 [c3];= 17660226 ± 00010000 [G61.For the prism cell: ii=- (37 ± 4)” = - (000018 ± 000002) radians,=- (1’25” ± 6”) = - (000041 ± 000003) radians,(20525 ± 0010)° = 03582288 ± 00001745 radians,and,n = 285287O537sinT — 1’683193617x10cosq’ [B.28]+ 8490899495x10 ÷ 116967692x1OB’For the isotherm cell:=- (55.21 ± 1758)” = - (00002677 ± 00000852) radians,= (4813 ± 1749)” 00002333 ± 00000848 radians,= (19744 ± 0024)° = 03445925 ± 00004255 radians,and,n = 296O992552sinF - 101635502x104cosT [B.29]+ 1317044167x10 — 69O92O444x1OB’And, where[B.30]B’ = 31l72OO986-sin2’147APPENDIX CISOTHERM CELL DRAWINGSThe specifications of the isotherm cell and its thermostat are shown in thefollowing figures. The units used are inches (“) for convenience in machining and tocater to other restrictions. The diagrams are not to scale. The following are noted.In Figure C. 1, the cross-hatched areas are the 1 “dia x ‘A “thick sapphirewindows. cj Each window is held in place by a flange which is tightened to thealuminium cell block by ten equally (angularly) spaced, 6-32 hex-head, stainless steelbolts. Note that the shape of the flange for the angled window on the prism side isdifferent from the other three flanges. Teflon gaskets are used to “cushion” the window-flange contact; while indium 0-rings are used to seal each cell volume- between eachwindow and the cell block. [L9] The flanges (i.e. windows) are tightened slowly (over aperiod of several days) using torque wrenches. The two (7116)”-20 threaded holes at bothends of the cell block are for the needle valve bodies.In Figure C.2, the cross-hatched regions represent the heating foils irii glued ontothe heater block. Only one (of two) end-piece “plugs” is shown. The through hole is toallow for the protruding needle valve stem and body (from the cell block) and to accessthe valve during (filling or) bleeding. The block and end-pieces are made out ofaluminium. The thermistors wi are embedded in the middle of the block in a planeperpendicular to the cell axis. The quartz probe [144] (not shown) is placed through oneend-piece, in (thermal) contact with one end of the cell block.In Figure C.3, the cylindrical water jacket is made from (1/16)” thick copperplates turned to form a cylinder. The circles (not to scale) are the ¾” copper tubingsoldered onto the outside of the plate/body. The tubing circulates the temperature-controlled water from the Forma bath wi. The tubing is wound as tightly as possible tocover as much surface as permitted by the positions of the laser in/out windows andneedle valve access ports (through the end-pieces). The coverage is also limited by themechanical/thermal constraints during soldering. Each end plate is bolted onto the maincylindrical section using eight ¼ “-20 hex-head, stainless steel bolts. “T” represents someof the positions of thermistors (embedded in the same horizontal plane as the cellvolumes). Glass plates (large microscope slides) cover the laser in/out windows at theheater block and the outer water jacket to reduce convection problems.00C.)CCCC.)c)149Figure C.2 Heater block for the isotherm cell.DDDDDDDDDD150122 jm 2 2 2 2 H 2Figure C.3 Top section view of the outer water jacket.151APPENDIX DINSTRUMENTATIONDescribed below are some of the essential pieces of equipment used in this study.D.1 DIGITAL PRESSURE GAUGEMENSOR PRECISION INSTRUMENTS AND SYSTEMS, CORP.Model 15000 DPG II Digital Pressure Gauge with Silicon Pressure Transducer. [M9jSpan 0 to 1500 psi(sg)Total Instrument Uncertainty 001 % fsOperating Temperature Range 0 to 50 CCompensated Temperature Range 15 to 45 CTemperature Effectson Zero Shift < ± 00003 % fsC1on Span Shift < ± 00002 % fsC1Over Pressure Ratings 150 % fsGravity/Orientation Effectson Zero Shift Negligibleon Span Shift Negligibleon Linearity NegligibleSensitivityto Tilt Negligibleto Vibration NegligibleMaximum Zero Drift 0005 % fshr’001 % fs’wk1Maximum Span Drift 0003 % fswk1001 % fs(9odays)Maximum Mechanical Shock 5 X g for 9sCommunications IEEE-STD-488- 1978Calibration N.I.S.T.-traceable sgResolution (span-dependent) ppmResponse Time 02 s* denotes ambient temperature changes of < 3C.fs denotes full scale.sg denotes “Standard Gauge”.152D.2 AUTOCOLLIMATOR/TELESCOPEDAVIDSON OPTRONICS, INC.Model D-275 Alignment Telescope and Autocollimator. 6]Magnification 24 xFocusing Range 16 to oo inField @ lOOft 20 inResolution 5 sOptics fully coatedReticle 15 1-mm concentric ringsTotal Field 30 mmD.3 QUARTZ ThERMOMETERHEWLETT’-PACKARD CO.Model 2804A Quartz Thermometer with HP 181 1OA Quartz Probes. 1)14]Measurement Temperature Range- 50 to 150 CAbsolute Accuracy ± 004 CResolution 00001 CAmbient Temperature Range 0 to 55 CHysteresis of Quartz Probe ± 0’02 CStability of Quartz Probe ± 0004 Cmth’Ambient Temperature Stability ± 0005 CStability ± 0003 Cmth’Calibration Temperature Error* ± 00002 C* denotes within 30days from last calibration*, and, measuredat same ambient temperature as during calibration.+ denotes only in the measurement temperature range.* denotes calibration by the Triple-Point Cell of Water. En]D.4 NULL VOLTMETERHEWLErr-PACKARD CO.Model 419A DC Null Voltmeter. [H3]Voltmeter OperationsVoltage Range ± 3 to ± 10Accuracy ± 2 % I fs I153Response Time (within is) 95Drift <05 Vday’Operating Temperature Range 0 to 50 CTemperature Coefficient < 005Noise < 03Amplifier OperationsMaximum Gain (range-dependent) 110 dBOutput Oto±1 VNoise 001 to 5 Hz<10 mVD.5 WATER BATH/CIRCULATORFORMA SCIENTIFIC, INC.Model 2095 Refrigerated and Heated Bath. [P3]Tank Capacity 284 1Heat Removal Capacity @ 20C 05 kW@OC 03 kWTemperature Range- 20 to 70 CTemperature Sensitivity ± 002 CPump Capacity @ 0 head pressure 75 lmin1Maximum head pressure capability 10 ftD.6 DECADE BOXTIME ELECTRONICS, LTD.Model 1051 Decade Box Resistance. rjResistance Range 001 to 106Zero Residual Resistance < 90Residual Resistance Stability < 3Maximum Voltage 250 VPower Consumption (per resistor) 1 WTemperature Coefficient 100 ppm C’Accuracy Ranges@001 10@01 5 %@1 1@10 (2 05 %@>100 (2 01 %154D.7 SAPPHIRE WINDOWGENERAL RUBY AN]) SAPPHIRE, CORP.Sapphire Windows for Cells. [061Chemical Formula A1203Density 398 gcm3Hardness 9 mohsPoisson Ratio 025Tensile Strength* 4 X 10 PaBulk Modulus 3.5 X 10’Young’s Modulus 3 X lO PaRigidity Modulus l45 X 1011 PaMinimum Rupture Modulus 44 X 108 PaThermal Conductivity 2•7 x 10 WmKThermal Expansion Coefficient*+ 84 X 10 K1Heat Capacity 418 X 102 Jkg’Specific Heat* 777 X 101 JmolK’Refractive Index** 17660226Reflectance** 00766958Transmittance** 084Absorption Coefficient** 32 X 10-2 cmt* denotes value at 24C or 25C.+ denotes value at 60° to c-axis.* denotes value at wavelength, X=6328A.For a typical (1 dia X ‘A thick)in sapphire window used in the image plane cell filled toapproximately l000psi 7 x 106 Pa, the centre of the window is displaced outward byapproximately ‘AX. rrj155APPENDIX EANALYSIS OF ISOTHERMSThis appendix is created to record the many calculations carried out on theisotherms in Chapter 6. The general discussion is left to the text in that chapter.The equation used for fitting to the isotherms is:— OIT = d0IAp*lo { i + djAp ÷d2IAp*IAIP } [Li]where Jp* is defined in [2.14], and the multipliers, d0,12 are different from in[2.18]. The magnitude of d0 should be approximately equal to PD0.P0 is a temperature-dependent pressure constant. In the following fits, the only exponent allowed to vary isô; the values of the exponents, 3 and , are those found in Table 2.1. Note that as themeasured isotherms are (likely) not exactly at the fluid critical temperature, there shouldbe a temperature dependence in [E. 1]. However, it is not explicitly shown as temperatureis assumed constant for each isotherm! Just as P0 is assumed constant in each fit.Each isotherm is divided into three ranges to be analysed separately:<1> p<p<pC; <2> p<p<p; <3>where, pOOO5 molcm3; p,,,OOl2moFcm3.That is, range <3> uses the data in ranges <1> and <2> simultaneously.Within each range, three fits are carried out:<a> fixd1=d2{O} ; <b> fixd2={O} only; <c> freed1 and d2.Within each fit, three variations are considered:<i> fix p, only; <ii> fix ô only; <iii> free p and &.A fourth variation, <iv> is to fix and PC to the values in Tables 2.1 and 4.6. Thisassumes the data are approximated by [E. 1]. It allows estimates of cL312 which can beused as initial values in other fits.156NOTE: The <METHOD> heading in the following tables definesthe method of fit by the triangular brackets.e.g.: <1.b.iil> is a fit that uses the range, p<p<p,fixes d2 = {O}, and uses p, and 6 as free parameters.Under fit <a.i> for which d1=d2{O} and p, is fixed, an “effective exponent”, ôe Cbe defined for each temperature. The pure power law 6 can then be estimated byinterpolating (or extrapolating) to t=O, i.e. T-’T. Figures E. 1 and E.2 show the ôevalues calculated using <i> and <iii> in various ranges with varying a number ofcorrection terms. Most ô values are below the Renormalization Group value of 482.Several overlapping criteria are applied to reject a fit:(A) 8<0 or 8>6 in <iii> and/or <i>(B) I - PCthe0If I > OO015 molcm3 in <iii> and/or <ii>(C) I P0 - > 100 psi in all fits(D) Id01 > 10 in all fits;(B) number of data-points, N< 10 in all fits.157< METHOD> P0 d1N p d0 d2<1.a.iv> 000600 864’52-63 000831 3923’1-<2.a.iv> 000887 868.71-10 000916 38117-<3.a.iv> 000600 86698-131 001097 5173’1-<1.b.iv> 000500 86479- 154100 000831 70435-<2.b.iv> 000887 70083- 1709 000898 55362-<3.b.iv> 000600 86659 -245131 001097 16294-<1.c.iv> 000600 86569 -72963 000831 32540 837<2.c.iv> 000891 86823- fr6353 001200 19553 7•24<3.c.iv> 000600 86648- 880131 001097 65637 1088Table E.1 Results of fits to the CH3F isotherm at T=449024C, using [E. 1]under variation <iv> for ={8765 x 10 moFcm3}and 5={482}.Table E. 1 shows the effect of adding more correction terms under variation<iv> for the CH3F isotherm at T=449024C. d0 increases rapidly with the number ofcorrection terms added; while d1 becomes more negative. P0 remains “stable”, slightlybelow P as 7< Ti,. Within each fit, the fit parameters, P0 and d0,12 do not vary widely(except for d0 with the addition of more correction terms).158<METHOD> p P0 d1 p,N p d0 d2<1.a.ii> 000550 69971- 000811103 000688 11355-<2.a.ii> 000780 700.56- 000734143 001000 21053-<3.a.ii> 000500 69992- 000741279 001000 26996-<1.b.ii> 000550 69975 -250 000759103 000688 72828-<2.b.ii> 000780 70025 - 164 00067185 000899 88254-<3.b.ii> 000500 69989 - 1•59 000739279 001000 53177-<1.c.ii> 000550 69981 - 1041 0’00726103 000688 63295 1410<2.c.ii> 0’00853 70065- &23 00075082 001000 19255 1041<3.c.ii> 000500 69996 -831 000740279 001000 30058 1006Table E.2 Results of fits to the CHF3 isotherm at 7.=259505C, using [E. 1]under variation <ii> for 6 = {4 82}.Table E.2 shows the fit results when 8 = {4 82} (variation <ii>) for theTr=259505C CHF3 isotherm. Adding more correction terms lowers the calculated palthough it remains approximately ± 10% from the reference value (quoted in §6.2). P0does not fluctuate widely, remaining less than 1 % from the value of P, (given in Table3.1). The changes in d012 are consistent with those in the previous table.159<METHOD> p P0 d1 ôN p d0 d2<1.a.i> 000600 69979 - 40473 000688 1294•6-<2.a.i> 000780 700.61 - 452143 001000 2450•1-<3.a.i> 000500 700’77- 326279 001000 45396-<1.b.i> O00550 69979 -237 437103 000688 39141 -<2.b.i> 000780 70060 402 398143 001000 58046-<2.c.i> 000780 70025- 958 338143 001000 25339 1352Table E.3 Results of fits to the CHF3 isotherm at Tr=259505C, using [El]under variation <i> for p={753x103molcm3}.Table E.3 shows the results of fits under variation <i> to the CFTF3 isothermat Tr=259505C. No <3.b.i>, <1.c.i> and <3.c.i> fits are found to be acceptable.Again, P0 remains close to the P. However, d0 values are smaller than those in TablesE.1 and E.2. The sign for d1 in <2.b.i> has changed.160< METHOD> p P0 d1 8N Pmax d0 d2 p<1.a.iii> 000751 86687 - 14219. 000874 3183- 000881<2.a.iii> 000880 867.51- 38763 001100 9238- 000740<3.a.iii> 000600 86748- 251131 001100 25050 - O00896<1.b.iii> 000751 86687 1•68 12519 000874 1889- 000879<2.b.iii> 000880 86751 150 32763 001100 4591- 000758<3.b.iii> 000600 86749 499 195131 001100 5138- 000896<3.c.iii> 000751 86659 -2682 29826 000937 25919 5977 000848<3.c.iii> 0OO650 86733 - 1046 12782 001050 4791 21•29 000891<3.c.iii> 000600 86740 - 1000 139131 001097 6706 1904 000894Table E.4 Results of fits to the CH3F isotherm at Tr=449024C, using [E. 1]under variation <iii>.Table E.4 displays the results of the fits to the CH3F isotherm at Tr=449024C,under variation <iii> d0 are significantly smaller than the values in all previous tables.The signs of d1 under fit <b> are positive. The values of 6e are significantly lower thanthe expected 482. The fits with the largest 8 have the smallest fit values of p, which aresignificantly less than the reference (Table 3.1) and prism cell values. The effect of therange used in the <3.c.iii> fit is indicated in the last three rows. Only P0 and p, remainrelatively “stable”, increasing slowly with the expanding range.1615-4.5- 0+x4- +35, V*+Q 2.5-s+001.5 -1— I I I I I I I I I0 0.0002 0.0004 0.0006 0.0009 0.001 0.0012 0.0014ittFigure E.1 Effective exponent,‘e vs Iti for fits to various ranges. D =><a.i> ; + <b.i> ; <c.i> ; =‘ <a.iii> ; x <b.iii> ; v<c.iji>1625-4.5- ±CV V4- ÷V +35 r V3-x +I I I I I I I I0 0.0002 0.0004 0.0006 0.0008 0,001 0.0012 0.0014JtjFigure E.2 Effective exponent,‘e vs for various fits. D <1.i> ; +<2.i> ; <3.i> ; <1.iii> ; X <2.iii> ; v <3.jjj>163APPENDIX FVITREOUS FLUOROPHOTOMETRYF.1 INTRODUCTIONA joint-project was proposed by Prof David Baizarini, Critical Phenomena Lab,Department of Physics, UBC and Prof lain Begg, Department of Ophthalmology, UBCto assemble and test a scanning system called a vitreous fluorophotometer (VFM). Thisproject was carried out as an MSc thesis experiment by me in 1986.a Over the followingtwo years, the fluorophotometer was upgraded. A portion of the initial part of my PhDprogramme was to assist a group of senior undergraduate students in this upgrading.However, that portion of the programme eventually became a “full-time” project insoftware writing. The following is a summary of the project and the work done by me.F.2 THE FLUOROPHOTOMETERVitreous Fluorophotometry (VF) is a non-invasive, clinical means of scanningthe blood-retinal barrier (BRB) of the human eye.b This method studies the state(permeability) of the BRB and the vitreous body by scanning for indications ofbreakdown in the integrity of the blood vessels in the choroid immediately behind theretina. This is done by detecting the presence of a dye immediately in front of the retinaand (its diffusion) in the vitreous body.The vitreous fluorophotometer, as was first assembled, is comprised of thefollowing principal components:(1) Nikon Zoom-Photo Slit Lamp Microscope;(2) logarithmic amplifier designed and built by the Physics Electronic Shop, UBC;(3) data-acquisition system (designed and built by me);(4) Gamma Scientific Digital Radiometer Model DR-2;(5) Gamma Scientific Photomultiplier Detector Model D-47A;(6) Gamma Scientific Scanning Photometric Microscope Eyepiece Model 700-1O-34A;(7) Spectrotech filters, SE4 and SB5;(8) Osborne portable microcomputer and printer;a PANG Klan Tiong: A Laboratory and Clinical Study on Vitreous Fluorophotometry, MSc Thesis,UBC (April 1986). 166 pages.b DM Maurice: A New Objective Fluorophotometer’, Exp Eye Res 2, 33 (1963).164(9) Cooper Vision Piano-Concave hard plastic lens;(10) Funduscein Fluorescein Sodium 25% dye.The basic principles are: a narrow beam of filtered (SE4) light from a slit lampis focussed into the back of the eye. The beam excites the spontaneous fluorescence ofliving cells and (if present) a dye called fluorescein. (The dye is injected into the patientless than 3 minutes from the first scan. The presence of the dye in front of the retinaindicates that the BRB is no longer intact.) The fluorescence in a small volume along theincoming beam path is picked up by the objective (of the slit lamp microscope) whichhas a 450-jim fibre-optic probe/conduit at its focus. The signal is conducted through aband-pass barrier filter (SB5) which screens out all other signals except that from thefluorescein. The filtered light is passed into a photomultiplier tube; then, to a radiometer,followed by a logarithmic amplifier. The slit lamp scans the axis of the eye, profiling thepresence of the dye in the vitreous body. The state of the vitreous (e.g. proportion ofliquefaction) can be gauged by studying the diffusion of the dye with time. This is doneby comparing profiles taken in successive scans (over several hours).The position of the slit lamp (relative to its initial position at beginning of a scan)is measured by a potentiometer (a geared turn-pot) attached to the lamp: the turn-pot gearruns on a rack which is attached to the base of the slit lamp. The voltage imbalance thenrepresents a (scaled) position of the excitation volume (at which the probe is focussed)with respect to the position of the slit lamp. This voltage imbalance and theaccompanying signal from the radiometer are paired and digitized by the data acquisitionsystem and stored in a microcomputer. The data acquisition system contains an analogue-to-digital converter chip preceded by an analogue multiplexer into which two amplifier-sample-and-hold inputs (for the position and radiometer signals) are connected. Thissetup is able to collect about 80 pairs of data-points per second. The lower limit ofdetection in vivo is 4.4 x 10 gml1 (of fluorescent signal).A “window” into the fundusc is created by using a plano-concave contact lens forthe entry of the excitation light and the exit of the signal. This lens is needed to ensurethat the angles of incidence and exit are constant during a scan. Otherwise, the curvatureof the corneal surface will redirect the input light in a manner which negates the positioncalibrations of the slit lamp. Without the piano-concave lens, such errors are non-linearwithin one scan; they also vary from scan to scan and patient to patient!F.3 USESThe aim of this technique is to provide a sensitive, non-invasive means toreproducibly monitor and quantify the state of the blood-retinal barrier. First applied toC The fundus is that part of the back of the eye farthest from the pupil.165the study (juvenile) diabetic retinopathy, it was found that in some diabetics, a smallmolecule like fluorescein can “leak” into the vitreous humour before bleeding becomesobvious (through other means of detection.) With this setup, it is possible to:(a) judge the intactness of the blood-retinal barrier;(b) detect early, non-apparent changes in the barrier;(c) study the state of the vitreous by observing the diffusion of the dye with time.Besides the blood-retinal barrier (BRB) and the vitreous, this means of detectingvascular-barrier changes is also being tested on the anterior chamber of the eye and (theaging of) the eye-lens. However, the current interest continues to be: to expand into thestudy of other pathologies where changes in the BRB are suspected to occur at differentstages of disease, i.e. to chart disease progression. Conversely, it can also monitor theeffectiveness of (new) drug therapies.’ Examples of pathologies studied are diabeticretinopathy, hypertension, pars planitis, and in our experiment, multiple sclerosise.The MSc project had the following objectives:(i) improve the slit lamp filtering and position-sensing systems;(ii) computerize data aquisition and storage (where before, the profiles were plotted);(iii) write the appropriate software to control and analyse the fluorescein profiles;(iv) study the applicability of this technique to multiple sclerosis.Our results were submitted to various conferences (e.g. CAP 1986), “open house” and“research days” displays, competitions and symposia.”F.4 UPGRADINGData collection continued even after the end of the MSc project. The aim was toenlarge the statistical sample so as to better study the correlation of the various stages ofmultiple sclerosis - relapses with and without progression, remission- to blood-retinalbarrier changes (and to blood-brain barrier changes). Upgrading of the fluorophotometerwas required, especially in the speed of data acquisition, the interactiveness or userd JG Cunha-Vaz, CC Mota, EC Leite, JR Abreu and MA Runs: “Effect of Sulindac on the Permeabilityof the Blood-Retinal Barrier in Early Diabetic Retinopathy”, Arch Ophth 103, 1307 (1985).Multiple sclerosis is a disease of the central nervous system. Increased retinal venous permeability hasobserved in cases with and without retinal periphlebitis. Hence, this study.Proceedings Paper: “Multiple Sclerosis Activity Studied by Vitreous Fluorophotometry”. Presentedat the Fourth Symposium of the International Society of Ocular Fluorophotometry in Luso (Coimbra), Portugal(May 1988). Printed in Ocular Fluorophotornetrv and the Future Edited by JG Cunha-Vaz and EB Leite. Kugler& Ghedini Pub. Amsterdam (1989). pp 125-129. ISBN 90-629-9054-1.166friendliness of the software (for the convenience of the eventual medical end-users) andthe speed of analysis by the dedicated computer system.A team of final-year undergraduate students was assigned to this task. My rolewas initially to assist (advise) in the rebuilding of the data acquisition system and thesoftware for the new (faster) computer, and to ensure conformity to clinical and (other)systems requirements. (The logarithmic amplifier became redundant in this process.) Theelectronics (data acquisition system) upgrade was successful. However, the studentsgraduated without completing the systems-controlling Due to scheduling (ofvolunteers and patients), rewriting the software became a priority. As I was aware of thetheories used in the analysis, my task was to readapt the systems-control and analysissoftware to the new data acquisition system and microcomputer. Also, as I was alreadyaware of the clincal measurement protocol and the data analysis, the software wasrequired to have:(a) increased interactiveness and user-friendliness (menu-driven displays),(b) simplified output of calculations (for readability), and,(c) simpler end-user instruction and documentation.The last requirement considered likely future modifications by technical personnels andthe degree of difficulty of use by non-technical, medical end-users (so as not to requireany inordinately detailed instructions of use or prolonged “training period”).The actual software design required assembly language programming (tomaximize the speed of data acquisition) and several modules linked by menus and sub-menus to run the different protocols in scanning, data file management, analysis, andoutput.g Along with assisting in the upgrading, I was also attempting to construct a “rotating disc viscometer”during the same period.h BM Ishimoto, RC Zeimer, JG Cunha-Vaz and MM Rusin: “Data Analysis and Presentation in VitreousFluorophotometry”, Jut Soc of Ocular Fluorophotometry Quarterly 2, 1 (Sept 1985).H Lund-Andersen, B Krogsaa, M Ia Cour and J Larsen: “Quantative Vitreous FluorophotometryApplying a Mathematical Model of the Eye”, hives Ophth Vis Sci 26, 698 (1985).167Figure F.1 A block diagram of the (originally) assembled vitreous fluorophotometer.Regulated LampPower SupplyTungsten BulbExcitation FilterSLIT LAMP BASEEyepiece &Fibre-Optic OREVitreous TChamber IAContactLensBarrier Filter168APPENDIX GABBREVIATIONS AND SYMBOLSThe following is a list of the frequently-used abbreviations and symbols in thetext. Only those that are mentioned in different sections and/or chapters are noted below.a van der Waals’ (substance-specific) molecular parameter inpsi. cm6 mo12;also, used as (subscripted) curve-fit parameter for p and Pdb van der Waals’ (substance-specific) molecular parameter incm3mol’;also, used as (subscripted) curve-fit parameter for p and PdB second virial coefficient for isothermB0 amplitude of the power law pc used as subscript to imply critical variable/parameterspecific heatC third virial coefficient for isothermCHF3 chemical formula for fluoroformCH3F chemical formula for methyl fluorided used as subscript to imply deflected beam variable/parameter;also, Ronchi grating spacing/constant = 0508 mmd spatial dimensionD0 amplitude of power law critical isothermDOLSF UBC best polynomial curve-fit programmeDPG digital pressure gaugef polynomial fit parameters/coefficients in calibration equationsF-distribution testing statistic at probability, palso, amplitude of power law c,g gaseous phase; also used as subscript in relevant variable/parameterh coefficients of Ntop* or Pd conversion equationsht stainless steel, hypodermic tubingH vertical height in cellH2C:CF2 chemical formula for 1,1 -difluoroethylene169I heating foil current from OPSionization or dissociation energy in eV1c8 Boltzmann constant = 0861736x 1O eVK11 liquid phase; also used as subscript in relevant variable/parameter10 cell thicknessLED light-emitting diodeLorentz-Lorenz relation or constantm Ronchi grating spot indexmc micrometer screw controlling deflection mirrors of prism cellsM magnification of lens (preceding camera)M# mirror index in diagramsMFT mean field theoriesMV metering valven refractive indexnofairn number of components (dimension) of the order parameterN fringe countNA Avogadro’s number = 6.022045 x 1023 mo11N number of data-pointsNV needle valveOPS Operational Amplifier Power SupplyP pressure in psicritical pressure in psiQT Quartz Thermometerr vectorial displacement; interatomic distance; a lengthR multiple correlation coefficient in linear curvefittingRG universal gas constant = 8.31441 JmolKR Resistor or resistance in bridge circuitRd decade box resistance that sets the 7 to be attained/maintainedRGT Renormalization Group Transformationscritical region about the critical points solid phase; also used as subscript in relevant variableSPT silicon pressure transducert reduced temperature = 1— 7170T temperature in KelvinTb heater block temperature; usually = Trcritical temperature in Kelvin7. cell block temperature; usually = TbT temperature of WJ or bath/circulatorTh thermistor used to control Iu difference between mc readingsU interaction potentialaverage (constant) interaction potentialV molar volume in cm3mol’V, critical molar volume in cm3moltWJ water jacket; temperature-controlled water circulatorZd deflected beam after traversing sample in cellZ1 incident beam on cellZ0 outside “reference” beama critical exponent for c,electronic polarizability in A3/3 critical exponent for p7 critical exponent for Kgap exponentisothermreduced density used in isothermscritical exponent for zXP0 angle between Z1 and ZdK isothermal compressibility in psi1X He-Ne laser wavelength = 6328Achemical potentialv critical exponent forcorrelation lengthp density in molcm3p critical density in moFcm3Pd diameter ofp*order parameter; coexistence curveu exponent = 1 —prism cell apex angle± superscript to indicate direction approach towards 7:+ for approaching 7 from above— for approaching T from below{x} implies parameter is held fixed at a value of x171

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