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Optical study of the critical behaviour of pure fluids and binary mixtures Närger, Ulrike 1990

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O P T I C A L S T U D Y OF T H E C R I T I C A L B E H A V I O U R OF P U R E FLUIDS A N D B I N A R Y M I X T U R E S By Ulrike Narger Diplom, Technische Universitat Munchen, 1983 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES PHYSICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH CO LU MBIA March 1990 © Ulrike Narger, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Physics The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Abstract Optical techniques were used to study the critical behaviour of the pure fluids C H F 3 , CCIF3 and Xe, and binary mixtures He-Xe and nicotine + water. We find that for all these substances, the order parameter is described by a power law in the reduced temperature t = (Tc - T)/Te with a leading exponent 0 = 0.327 ± 0.002. Also, we determine the first correction to scaling exponent to be A = 0.43 ± 0.02 for the pure fluids and A = 0.50 ± 0.02 for the He-Xe system. The coexistence curve diameter in CHF3 and CCIF3 exhibits a deviation from recti-linear diameter, in agreement with a modern theory which interprets this behaviour as resulting from three-body effects. In contrast, no such deviation is observed in Xe where, according to that theory, it should be more pronounced than in other substances. In the polar fluid CHF3, the order parameter, isothermal compressibility and the chemical potential along the critical isotherm were simultaneously measured in the same experiment in an effort to ensure self-consistency of the results. From the data, two amplitude ratios which are predicted to be universal are determined: TQ /T~ = 4.8 ± 0.6 and DQT^BQ'1 = 1.66 ± 0.14. In the binary liquid system nicotine + water, the diffusivity was measured both by light scattering and by interferometry. The results agree qualitatively, but differ by a factor of w 2. From the light scattering data, the critical exponent of the viscosity is found to be *„ = 0.044 ± 0.008. The interferometric experiments on Xe and He-Xe furnish a direct way to maesure the effects of wetting: From the data, the exponent of the surface tension is found to be n = 1.24 ± 0.06. The similarity of the order parameter and compressibility in Xe ii and a He-Xe mixture containing 5% He indicate that the phase transition in this He-Xe mixture is of the liquid-gas type rather than the binary liquid type. in Table of Contents Abstract ii List of Tables viii List of Figures x Acknowledgments xiii 1 Introduction 1 1.1 Critical Points 1 1.2 Order Parameter, Susceptibility and Critical Isotherm 4 1.3 Universality of Amplitude Ratios 6 1.4 The Coexistence Curve Diameter and Dipolar Interactions 7 1.5 Diffusivities 9 1.6 "Gas-Gas Equilibrium" 11 1.7 Outline of the Thesis . . . 14 2 Theory 15 2.1 Critical Phenomena in Pure Fluids 15 2.1.1 Critical Exponents . . 15 2.1.2 Van der Waals Equation of State 16 2.1.3 Scaling 19 2.1.4 Lattice Gas, Ising Model and Series Expansions 22 2.1.5 Renormalization Group 23 iv 2.1.6 Implementation of the Renormalization Group 27 2.1.7 Monte Carlo "Experiments" 28 2.2 Subtleties and Applications 29 2.2.1 Corrections to Scaling 29 2.2.2 Universal Amplitude Ratios 30 2.2.3 Long Range Forces 31 2.2.4 Asymmetric Lattice Gas and Deviation from Rectilinear Diameter 31 2.3 Dynamic Critical Phenomena: Diffusivities 33 3 General Experimental Considerations: Temperature Control and Op-tics 37 3.1 Temperature Control 37 3.2 Optics 40 3.2.1 Prism Cell Experiments 41 3.2.2 Focal Plane Interference Technique . 43 3.2.3 Image Plane Interference Technique 48 3.2.4 Gravitational Rounding 55 3.2.5 Light Scattering Experiments 56 3.2.6 The Correlator 58 4 Experiments 61 4.1 Freon Experiments 61 4.2 Nicotine + Water Experiment 62 4.3 High-pressure experiment 63 4.3.1 Cell Design 64 4.3.2 Pressure Handling System 68 -4.3.3 Dimensioning of the Gas Handling System 72 v 4.3.4 Filling the Cell 73 4.3.5 Thermal Control of the Cell 76 5 Freon Experiments: Results and Discussion 78 5.1 Lorentz-Lorenz Data 78 5.2 Order Parameter Measurements 82 5.2.1 Order Parameter of CC1F3 82 5.2.2 Order Parameter of CHF 3 83 5.3 Coexistence Curve Diameter for CHF 3 and CC1F3 89 5.4 Compressibility of CHF 3 93 5.5 Critical Isotherm of CHF 3 95 5.6 Critical Temperature of CC1F3 and CHF 3 97 5.7 Discussion 99 6 Nicotine -f- Water Experiment: Results and Discussion 105 6.1 Order Parameter 105 6.2 Diffusivites . 109 6.2.1 Fringe Data . 109 6.2.2 Light Scattering Data Ill 6.3 Potential Sources of Error 117 6.3.1 Gravitational Concentration Gradients 117 6.3.2 Equilibration Times 118 6.4 Discussion 123 7 He-Xe Experiments: Results and Discussion 128 7.1 Experiments on Pure Xe 128 7.1.1 Lorentz-Lorenz Function 129 vi 7.1.2 Coexistence Curve Diameter and Critical Density 131 7.1.3 Coexistence Curve 134 7.2 He-Xe Mixtures 139 7.2.1 Coexistence Curve 139 7.2.2 Compressibilities 141 7.2.3 Wetting 144 7.3 Discussion 147 8 Conclusions 150 Appendices 156 A Error Analysis for Prism Cell Measurements 156 A. l Calculation of refractive index from refraction angle 156 A. 1.1 Error Estimates 158 B Critical Refractive Index and Critical Density of Nicotine + Water 160 B. l Critical Refractive Index Measurement 160 B.2 Densities of Nicotine + Water Mixtures 161 C Thermal Gradients in the Sample Cell 166 Bibliography clxviii vii List of Tables 2.1 Values of some critical exponents, as obtained from the van der Waals equation of state, experiments on pure fluids and renormalization group theoretical calculations 18 4.1 Critical parameters of He-Xe mixtures 72 5.1 Results of a quadratic fit to the Lorentz-Lorenz data of CHF3 and CCIF3. 79 5.2 Results of coexistence curve fits for CC1F3 83 5.3 Results of coexistence curve fits for CHF 3 85 5.4 Results of fits of the coexistence curve diameter for CHF3 and CCIF3 . . 92 5.5 Compressibility fits of CHF3 data 95 5.6 Critical amplitudes of CHF 3 100 5.7 Critical amplitude ratios 100 6.1 Results of fits of the order parameter of nicotine + water 107 6.2 Equilibration times for quenches into the spinodal and the nucleation re-gions of the phase diagram, for a binary liquid 121 7.1 Results of fits to the Lorentz-Lorenz function in Xe 129 7.2 Results of fits to the coexistence curve diameter of Xe 131 7.3 Critical density of Xe 134 7.4 Results of fits to the coexistence curve of Xe 135 7.5 Results of coexistence curve fits of a He-Xe mixture 141 7.6 Compressibility fits of a He-Xe mixture 143 viii 8.1 Comparison of critical polarizability product, order parameter amplitude and diameter slope for substances with different strength of dipolar inter-actions . . . . 150 8.2 Comparison of the order parameter exponent /? and the correction to scal-ing exponent A for the substances studied in this thesis 154 ix List of Figures 1.1 Projection of the equation of state of a pure fluid onto the P — T plane . 2 1.2 Phase diagram of a pure fluid in the T— /> plane (a) and the P— /> plane (b) 2 1.3 Phase diagram of gas-gas equilibrium of the first kind 12 2.1 Schematic illustration of the motion of the Hamiltonian of a real physical system under the influence of the RG transformation, in parameter space. 25 3.1 Thermal control system for conducting optical experiments on fluids close to the critical point 38 3.2 Refractive index profile as a function of temperature in a cell of critical overall density 40 3.3 Optical setup of the prism cell experiment 42 3.4 Optics of the focal plane interference technique. 44 3.5 Formation of the Fraunhofer diffraction pattern in the focal plane 45 3.6 Schematic of the nicotine + water phase diagram 46 3.7 Schematic of the optical setup for image plane interference experiments. . 49 3.8 Effect of a tilted reference beam on the interference pattern 51 3.9 Optical setup for light scattering experiments 57 3.10 Schematic diagram of the autocorrelator 59 4.1 Technical drawing of the high-pressure cell 65 4.2 Photograph of the high-pressure cell. . 66 4.3 Components of the windows for the high-pressure cell 68 x 4.4 High pressure gas handling system 70 4.5 Schematic phase diagram of the He-Xe system at 14.5°C 75 5.1 Lorentz-Lorenz function of CHF 3 80 5.2 Lorentz-Lorenz function of CCIF3 81 5.3 Coexistence curve of CCIF3 84 5.4 Coexistence curve of CHF 3 87 5.5 Coexistence curve diameter of CHF3 89 5.6 Coexistence curve diameter of CCIF3 91 5.7 Compressibility of CHF 3 94 5.8 Graphical method used to extract the amplitude of the critical isotherm . 96 5.9 Drift of the critical temperature of CHF3 as a function of time 98 6.1 Order parameter of the binary liquid system nicotine + water 107 6.2 Plot of the reduced order parameter of nicotine + water 108 6.3 Evaluation of the diffusivity from interferometric data in the nicotine + water system 110 6.4 Concentration dependence of the diffusivity 110 6.5 Diffusivities of nicotine + water as a function of reduced temperature, as measured in the interferometric experiment 112 6.6 Diffusivities of nicotine + water as a function of temperature, as measured in the light scattering experiment 114 6.7 Evaluation of the effective exponent zefj 116 6.8 Schematic phase diagram of nicotine + water 119 6.9 Concentration profile in a binary liquid cell after a quench into the nucle-ation region of the phase diagram 122 -xi 6.10 Comparison of the diffusivities obtained by two different methods in the nicotine + water system 124 7.1 Lorentz-Lorenz data of Xe 130 7.2 Coexistence curve diameter of Xe 132 7.3 Coexistence curve of Xe, prism cell experiment 136 7.4 Coexistence curve of Xe, interference experiment 138 7.5 Coexistence curve of a He-Xe mixture 140 7.6 Compressibility of a He-Xe mixture 142 7.7 Meniscus width of a He-Xe mixture and of pure Xe as a function of reduced temperature 145 8.1 Order parameter amplitude Bo vs. diameter slope Ai for a variety of fluids 151 8.2 Diameter slope A\ as a function of critical polarizability product for a variety of fluids 152 A. l Refraction geometry of the prism cell experiment 157 B. l Determination of the critical refractive index of nicotine + water: Plot of the number of interference fringes as a function of the cell's rotation angle 162 B.2 Temperature dependence of the density of nicotine + water mixtures for a variety of compositions 164 xii Acknowledgments I would like to thank Dr. David Balzarini for his advice and assistance during this work. I am very grateful to Dr. John de Bruyn for many discussions, instruction and assistance on the experiments, and for a critical reading of this thesis. Many thanks also to the technical staff of the Physics Department, for their excellent work and helpful advice in many instances. I am very grateful to UBC and the Killam Foundation for several years of scholarship support. Finally I would like to thank my friends in the Physics Department for many inter-esting discussions, help and encouragement. And of course, I am grateful to Dan for his unfailing support, encouragement and interest. xiii Chapter 1 Introduction 1.1 Critical Points The thermodynamic state of a pure fluid at pressure P and temperature T can be de-scribed by an equation of state [1,2] / ( P , 2 » = 0, (1.1) where p is the density. This equation defines a two-dimensional surface in the three-dimensional space (P, T, p). In some regions on this surface, the system is not thermody-namically stable. These regions are bordered by lines called the coexistence curves. The projection of these lines onto the P — T plane is shown in figure 1.1: There is a region of solid phase at low temperatures and high pressures, a region of gas phase at high temperatures and low pressures, and in between a region of liquid phase. At the triple point all three phases coexist. The liquid-vapour coexistence curve ends at a critical point (TC,PC). Therefore it is possible to take the system from any point in the liquid region to any point in the vapour region without crossing a phase boundary. Figures 1.2a and 1.2b show projections of the coexistence curves onto the T — p and P — p planes. In the T — p plane, the coexistence curve divides the plane into a single-phase region (above the curve) and a two-phase region (below the curve) in which liquid and vapour coexist. In general, the critical point of a system is characterized as a point in the parameter space at which a continuous transition between two phases occurs [3, 4]. In pure fluids the critical point is the end point of the liquid-vapour coexistence curve, at which point 1 Chapter 1. Introduction Figure 1.2: Phase diagram of a pure fluid in the T — p plane (a) and the P Chapter 1. Introduction 3 the liquid and vapour densities become the same, and in binary liquids it is the point at which the concentration difference between the two coexisting phases goes to zero. Close to the critical point, phenomena occur which are universal in two respects: • They are present in a wide range of very different physical systems: Pure and binary fluids, magnets, superconductors, etc. • Certain thermodynamic quantities diverge or go to zero close to the critical point as powers of the temperature difference or density difference from the critical values, with exponents which are quantitatively the same in very different systems; their values depend only on a few very general characteristics of the system. In the last two decades, great progress has been made both experimentally and the-oretically towards an understanding of critical behaviour and a determination of the critical exponents to very high accuracy. As theories make more and more precise pre-dictions and as large computer power allows for more and more sophisticated numerical calculations, there is a need for very accurate experiments to test the theoretical pre-dictions. Thus, experiments are carried out to approach the critical point as closely as possible so as to measure the true asymptotic behaviour. The advantage of performing experiments on systems in their fluid state (i.e., on gas-liquid systems or on binary fluids) over experiments on solids is that the constituent particles are free to move within the system, and complications due to the underlying crystal structure, lattice defects etc. do not have to be taken into account. Also, trans-parent fluids can be probed using optical methods. However, in binary liquids and pure fluids the effects of gravity can hinder the observation of true critical behaviour as dis-cussed below. In order to approach the critical point as closely as possible, systems and experimental methods have to be chosen carefully so as to reduce gravitational effects. Chapter 1. Introduction 4 This can be attained by matching the densities of multi-component systems, or by choos-ing experimental techniques which are inherently less susceptible to gravitational effects. Optical interferometry is an example of such a technique which minimizes the effects of gravity. Whenever this method was not applicable, as for measuring diffusivities using light scattering, we chose a binary liquid system which is closely density matched. As we will demonstrate in this thesis, these methods are precise and furnish results that are easy to interpret. 1.2 Order Parameter, Susceptibility and Critical Isotherm In order to characterize the different thermodynamic phases, one defines an "order pa-rameter" \P which has the property that, upon approach to the critical point in the two-phase region, it goes to zero continuously, and it is identical to zero in the one-phase region. For pure fluids, the order parameter is proportional to the density difference between the coexisting liquid and gas phases oc pi — /?„), whereas for binary liquids (of molecular species A and B) it is proportional to the concentration difference of one species (say A) in the two coexisting phases I and 77: \& oc x\ — x^. Close to the critical temperature Tc, ^ obeys a power law with an exponent 0 in the reduced temperature t = (Te- T)/Tc: = B0tp. (1.2) Experimentally, the exponent /3 was found to be a have the same value for a variety of systems [1]. These systems are said to belong to the same "universality class". The universality class of a given system is determined by its dimensionality d, the number of components n of the order parameter, and whether the interparticle forces are short-or long-range. Both binary liquids and pure fluids have d = 3, n = 1 (scalar order parameter) and short-range, van der Waals-like interactions and thus belong to the same Chapter 1. Introduction universality class. This is also the universality class of the three-dimensional Ising model, in which each site in a three-dimensional square lattice is occupied by a spin which can point either up or down and interacts with the other spins in the lattice. As the Ising model is easier to handle theoretically than microscopic theories of gases, most theoretical results have been obtained for this model. There is a one-to-one correspondence between the variables of an Ising model with those of the so-called "lattice gas" model [5]. In the lattice gas model, each site of the lattice can either be occupied by a molecule or it can be empty, with interactions between occupied neighbouring sites. The lattice gas is thus a model of a real fluid. Besides the order parameter, there are also other quantities that exhibit power laws with universal exponents close to the critical point. One of them is the generalized isothermal susceptibility xt above (+) and below (—) the critical point: x f = r f C F * ) " 7 * (1.3) For pure fluids, xt OC (dp/dP)T is the compressibility, whereas for binary liquids, xt OC (dx/d(i)T is the "osmotic compressibility" (with p, the chemical potential and x the concentration of one of the species). Also, along the critical isotherm, the chemical potential p — pc is expected to obey a power law in the reduced density: (1.4) Here, pc is the critical density, pc = p(Tc, pc) the critical chemical potential, and 6 is another universal exponent. The exponents /?, 7 and 6 have been calculated for the three-dimensional Ising model both by high-temperature series expansions [6, 7] and e-expansions [8, 9]. Recent results obtained by these methods agree well and yield values of /? in the range from 0.325 to = D0 P~ Pc Chapter 1. Introduction 6 0.327 and 7 + = 7 " between 1.237 and 1.241. The quantities B0, TQ and D0 defined in the above equations are called the critical amplitudes. They are nonuniversal, i.e., their values depend on the particular system under consideration. 1.3 Universality of Amplitude Ratios Scaling theory [4, 10] states that the three exponents firf* and 6 are connected by the two scaling relations, 7 + = 7 " and 7 * = — 1). Given /? and 7 ± , 6 thus can be calculated. One obtains 8 between 4.79 and 4.82 [11]. Scaling also predicts that, even though the individual critical amplitudes are system dependent, certain combinations of them are universal. For example, the ratios T~\/T~~ and DOTQBQ"1 are expected to be the same for all systems in the same universality class [12]. Measurements of these amplitude ratios have been reported earlier [13, 14]. In these publications, results obtained from different experiments were combined to determine the amplitude ratios. This method, however, is subject to errors due to the different data evaluation methods and different samples used in the different experiments and, in particular, susceptible to effects caused by different determinations of the critical temperatures, which affect the critical amplitude ratios considerably. For a consistent determination of the amplitude ratios, all the amplitudes should therefore be extracted from a single experiment. In this way, the critical temperature can be determined independently in the evaluation of the various quantities, and agreement is an important check on the consistency of the results. This approach was followed by Weber [15] and later improved by Pestak and Chan [16, 17] who used a stack of capacitors to measure the density as a function of chemical po-tential in Ne, N2 and HD. Their results are very self-consistent and show good agreement with theory. However, close to the critical point, their data are affected by gravitational rounding, which they correct for by analyzing it using the restricted cubic model [18,19]. Chapter 1. Introduction 7 We have measured these amplitude ratios in the pure fluid system CHF3, a strongly polar fluid. By using an optical interference technique [20, 21] which is less susceptible to gravitational rounding, we minimized gravitational effects [22]. Like the capacitor method, our method allows measurements of the coexistence curve, compressibility and critical isotherm in a single experiment. By confining the sample to a thin cell, gravity effects are negligible even close to the critical point, so that no corrections due to grav-itational rounding have to be made. We used a cell only 1.86 mm thick, which enabled us to approach the critical point as closely as \t\ « 10~6 without encountering appre-ciable errors due to gravitational rounding. For measurements on the critical isotherm, only reduced densities Ap* = \p — pc\/pc > 4 x 10 -4 were used for the evaluation, for which beam bending errors are less than 0.1% [23]. We have thus obtained values of the amplitude ratios Tq /T~ and D0Tq Bq'1 which are probably the most accurate to date. Our data are in excellent agreement with theoretical predictions. They also agree well with measurements performed on nonpolar fluids [17], and are thus in accordance with the universality principle. 1.4 The Coexistence Curve Diameter and Dipolar Interactions Another quantity of interest is the coexistence curve diameter p*^ defined as the average of liquid and vapour densities: . _ Pl + Pv , , Pd ~ — ( L 5 ) In early experiments, the diameter was found to vary linearly with t, a property known as the "law of rectilinear diameter" [24]. In the'framework of scaling theory, however, due to the lack of particle-hole symmetry in real fluids, there is a deviation from linear behaviour close to the critical point, and the diameter takes the form [25] pmd = A0 + A^t1-" + Art. (1.6) Chapter 1. Introduction 8 Recently, this expression has been interpreted by Goldstein et al. in terms of the mi-croscopic interactions of the system [26]. Within this model, the term A\-atl~a results from three-body interactions, and the amplitudes A\-a and A\ are related to the rel-ative strengths of three-body and two-body interactions. If the dominant three-body interactions are of the Axilrod-Teller type [27], then the order parameter amplitude Bo, the diameter slope A\ and the strength of the deviation from rectilinear diameter are expected to be proportional to the dimensionless quantity ctppc, the so-called "critical polarizability product" [28]. (Here ap is the molecular polarizability and pc is the critical density). Recent precise measurements on a number of nonpolar fluids [26, 28, 29] indeed show a small deviation from a straight line close to the critical point, consistent with the predicted behaviour. Also, in agreement with the theory of Goldstein et al. [26], the diameter slope A\ was found to vary linearly with appc in these nonpolar substances. For polar fluids, the situation is somewhat more complex [28, 30]. Even though dipolar couplings between the molecules do not change the universality class of the system, they do influence the values of the critical amplitudes. In particular, the theory of Goldstein et al. predicts that the coexistence curve diameter amplitude A\ is expected to be proportional to the quantity ctppc rather than ctppc, where the "effective " polarizability dp of the polar fluid includes the effects of dipole-dipole interactions and is given by 1/2 (1.7) p 3 / 9IkBTc^ Here, p0 is the molecular electric dipole moment of the molecule, I is its dissociation energy and Tc is the critical temperature. In order to compare the behaviour of a strongly polar fluid close to its critical point to that of weakly polar and nonpolar fluids, we have carried out experiments to measure the order parameter and coexistence curve diameter of the pure fluids CHF 3, CC1F3 and Chapter 1. Introduction 9 Xe. CHF3 and CCIF3 have fairly similar critical temperatures, pressures and densities, but very different dipole moments: CHF3 is strongly polar (po — 1.65 D [31]), whereas CCIF3 is only weakly polar (/i0 = 0-50 D [32]). Also, CHF 3 molecules, which contain electronegative atoms together with hydrogen atoms, will form hydrogen bonds, which are absent in CCIF3. In contrast, the interactions between the nonpolar Xe atoms are purely via van der Waals interactions. Thus, by examining the critical behaviour of these three fluids, one can obtain important information about the relevance of dipolar interactions and hydrogen bonds close to the critical point. We find that in all three fluids, the relation between A\ and Bo is linear, in accordance with the data taken on nonpolar fluids [28], This indicates that three-body interactions indeed play a role, as proposed by the theory of Goldstein et al. The expected proportionality between A\ and ctppc is well fulfilled for CCIF3; however, Xe and CHF3 exhibit deviations, suggesting that three-body interactions other than the Axilrod-Teller forces play a dominant role. 1.5 Diffusivities Up to now we have only considered static critical phenomena. Their universality class was determined by the dimensionality of the system, the number of components of the order parameter and the range of the interactions. When discussing dynamic critical phenomena, the universality class is also determined by the number of relevant hydro-dynamic modes [33]. In fluids and fluid mixtures, there are two relevant (and coupled) modes: the diffusive decay of the order parameter fluctuations and that of the transverse momentum fluctuations. Thus we expect pure fluids and binary fluids to belong to the same dynamic universality class and to exhibit the same critical exponents and scaling functions for dynamic as well as static properties [33]. We have performed experiments on the binary liquid system nicotine + water and Chapter 1. Introduction 10 measured the generalized diffusivity D, which is proportional to the decay rate of order parameter fluctuations in the fluid mixture. According to the fluctuation-dissipation the-orem [34], D can be written as the ratio of a transport coefficient £ and a generalized static susceptibility xt- D = £/xt [33]. In the case of a binary mixture, D is the concentration diffusivity, £ the mass conductivity and xr = (dx/dfj,)r the osmotic compressibility [35]. Xt diverges strongly at the critical point; £ also diverges, but more weakly than xt, and therefore the diffusivity D goes to zero as the critical point is approached. This phenomenon, known as "critical slowing down", reduces the speed of equilibration in the critical region and leads to long time constants close to Tc. The conventional method for measuring diffusivities is by the method of light scat-tering [36, 37, 35, 38]. The light is scattered by fluctuations of the refractive index of the medium, which are related to order parameter fluctuations. The range of these fluc-tuations is the correlation length £ which diverges at the critical point as £ oc t~". The exponent v has a value of approximately 0.63 in pure fluids and binary mixtures. In the light scattering experiments, one keeps the fluid at a given fixed temperature in the one-phase region and probes the equilibrium density fluctuations; their characteristic decay time gives information about the equilibrium diffusivity D and is obtained by measuring the density-density autocorrelation function in the fluid. A second method of obtaining information about the diffusivity consists of putting the system into a nonequilibrium situation and observing its decay to equilibrium. This corresponds to quenching the fluid under investigation from the two-phase region into the one-phase region and watching the subsequent relaxation of the density profile of the fluid in the cell under the influence of gravity. From the relaxation one can again obtain a diffusivity, which now, however, is a nonequilibrium diffusivity. We performed experiments on the binary liquid mixture nicotine-water using both methods. This system is well suited for the study of critical phenomena, because the Chapter 1. Introduction 11 densities of the two constituents are very closely matched and thus gravity effects, which otherwise limit the accuracy of light scattering data, play a minor role. In binary liquids, the order parameter is proportional to the difference in concentration Ax* = xrA — xlAl of one of the constituents (say A) in the two phases I and II. We show that due to the density-matching of the constituents, gravitational rounding effects due to the divergence of the osmotic compressibility x r are negligible on the time scale of this experiment. From the light-scattering experiment, we measure the critical exponent of the viscosity and find good agreement with results of other experiments and with theory. The diffusivity measured by the interferometric method is found to be a function of the order parameter Ax* in the two-phase region before the quench. The diffusivity is thus concentration-dependent. In the limit Ax* —• 0, corresponding to a quench from the critical point into the one-phase region, the diffusivity data from the interferometric method are found to be consistently larger than the data from the light scattering experiment. This is the first time a comparison like this has been performed. 1.6 "Gas-Gas Equilibrium" Finally, a set of experiments were carried out on a system that exhibits features of both a binary liquid and a gas-liquid system. Whereas in pure fluids only gas-liquid equilibria exist, three different types of two-phase equilibria have to be considered in fluid mixtures: liquid-gas, liquid-liquid, and the so-called "gas-gas" equilibria [39, 40]. They all consist of two coexisting fluid phases of different densities separated by a meniscus. At a critical point of the mixture the intensive properties of the two phases in equilibrium become identical. Whereas pure substances are characterized by a critical point for the liquid-gas equilibrium, binary systems exhibit a critical line in three-dimensional T — p — x space (where x is the concentration). The various phase behaviours of binary fluids can Chapter 1. Introduction 12 X Figure 1.3: Phase diagram of gas-gas equilibrium of the first kind: line AB corresponds to the liquid-gas coexistence curve of the less volatile conponent and ends at the critical point B. From there, the critical line of second order phase transition points starts (dashed line). be classified by looking at the p(T) projections of these critical curves. An example of a system with liquid-liquid equilibrium was encountered above, in the binary liquid nicotine-water, under its equilibrium vapour pressure. "Gas-gas" equilibria can occur in systems of binary fluids at pressures and temper-atures above the critical temperature of the less volatile substance. The critical curve is interrupted and consists of two branches. The branch starting form the critical point of the more volatile component (/) ends at the so-called critical end point. The other branch begins at the critical point of the less volatile component (II) and either im-mediately tends to higher temperatures and pressures ("gas-gas equilibrium of the first kind", see figure 1.3) or goes through a temperature minimum first and then runs to increasing pressures and temperatures ("gas-gas equilibrium of the second kind"). The Chapter 1. Introduction 13 term "gas-gas equilibrium" is somewhat confusing for these systems, because in fact both substances are, at these temperatures and pressures, beyond the critical point , and so the distinction between "gas" and "liquid" does not exist any more. Rather, the coexistence is between two supercritical fluids of differing compositions. These phase separation effects were predicted by Van der Waals [41] and have been found in a wide range of systems (for a review see [42]). More and more sophisticated cal-culations of equations of state have enabled theorists to predict with reasonable accuracy which type of "gas-gas" equilibrium will be found in a given system [43], and to calculate the phase diagram. However, no investigations have been made up to now on the critical behaviour of the system along the second-order phase transition line. As the universality class of the system is the same as for binary liquids and pure fluids, we expect to observe the same critical exponents as for the three-dimensional Ising model. However, whereas the nonequilibrium density profiles in cells with pure fluids are known to relax quite fast, this process is much slower in binary liquids. We thus expect to observe a crossover from gas-liquid equilibrium at the end point of the second order critical line (pure component 77) to the behaviour of a binary fluid as the concentration of component J is increased. In the vicinity of the critical line, we expect both the compressibility and the osmotic compressibility to diverge. We have investigated the system Helium-Xenon which exhibits a "gas-gas equilibrium curve of the first kind". The phase diagram of this system has been studied by de Swaan-Arons and Diepen [44] for pressures up to 2000 atm and temperatures up to 60° C. We performed experiments on a sample of critical density and concentration of this system, containing « 5% He, and measured the coexistence curve, compressibility and diffusivities close to the critical point. We found no appreciable difference between the behaviour of this binary fluid sample and a pure Xenon sample, suggesting that the phase transition in this binary fluid is more of the gas-liquid type, rather than the binary liquid type. The Chapter 1. Introduction 14 only effect of the presence of He in the mixture seems to be an increase in the critical temperature and the pressure of the system. This is in agreement with a recent neutron scattering experiment on a noncritical He-Xe mixture [45] which finds that the structure factor of the Xe-Xe pairs is unchanged by the presence of He-atoms. In the two-phase region, our optical technique provided a novel way of measuring the surface tension: It enabled us to directly measure the width of the meniscus between the two coexisting phases in the cell, which is a measure of the rise height of a wetting layer on the cell windows. We used this method to extract the critical exponent of the surface tension and find it to be in good general agreement with other experiments. 1.7 Outline of the Thesis The remainder of this thesis is organized as follows: Chapter 2 gives an overview of the theory of critical phenomena and defines the quantities which we have measured. Chapter 3 describes the optics and the data extraction from the measurement. In Chapter 4 the experimental setups of the various experiments are described. Chapter 5, 6 and 7 present the results of the freon, nicotine + water and He-Xe experiments respectively, and Chapter 8 contains the conclusions. Chapter 2 Theory 2.1 Critical Phenomena in Pure Fluids 2.1.1 Critical Exponents As was discussed in the Introduction, the phase diagram of a pure fluid contains a region in which the liquid and vapour phases coexist (see figure 1.2). In the P-T plane, this region is represented by the liquid-vapour coexistence curve, which ends in the critical point as shown in figure 1.1. For many fluids it is found experimentally that the variation of the liquid and the vapour densities pi(T) and pv(T) with temperature is identical for different fluids when density and temperature are rescaled by their critical values pc and Tc. Since the difference in density between the phases goes to zero as the critical point is approached, the width of the coexistence curve A,'= (2.1) 2/>c is called the order parameter of the gas-liquid phase transition. Setting t = (T — Tc)/Tc, the width of the coexistence curve in many fluids is measured to behave approximately as [1] Ap* oc M 1 / 3 . More generally, one can write Ap* = B0\tf (2.2) The exponent /3 is found experimentally to be the same for all fluids close to the critical point. It is thus a universal quantity. 15 Chapter 2. Theory 16 Similarly, other thermodynamic quantities in fluids near their critical points are ob-served to obey universal power laws, for example: • the specific heat above (+) and below (—) the critical point CV(T) = A±\t\-a± (2.3) • the compressibility above (+) and below (—) the critical point *£CO = r J l * r * (2.4) • the chemical potential along the critical isotherm \n(p,Tc) - pc)\ Pc • the correlation length * = (2.6) A satisfactory theory of critical phenomena is expected to explain why the exponents a, /3,7, 6 and v are universal, and to produce numerical values for them which agree with experiment. 2.1.2 Van der Waals Equation of State Many attempts have been made in the past to find an analytical formula describing the equation of state of a pure fluid. The earliest attempt at a quantitative description is due to van der Waals; his equation of state is (P + ap2)(- - 6) = RT (2.7) P~ Pc (2.5) Chapter 2. Theory 17 where R is the gas constant and a and b are system-dependent parameters. The critical point is determined by the simultaneous conditions The solution is pc = 1/36, Tc = Sa/27bR and Pc = a/2762. Introducing the reduced variables t = (T - Tc)/Tc, Ap = (p - pc)/pc and p = (P - Pc)/Pc, the van der Waals equation can be written: p(2 - Ap) = 8t(Ap + 1) + 3A/>3 (2.9) This equation is independent of the system-dependent parameters a and b and thus has the same functional form for all gases, in agreement with the principle of corresponding states [1] which postulates that for a group of similar fluids the equation of state can be written in the form p = *(*,Ap) (2.10) where $ is the same function for all substances in the group. For t < 0, there is a segment along the isotherms with (dp/dAp)x < 0 (drawn as a dotted line in figure 1.2b) which corresponds to a region of mechanically unstable thermodynamic states. This is clearly not a physical solution. Instead, in this region the system demixes spontaneously into two coexisting phases, the equilibrium densities of which can be found from the van der Waals equation using the Maxwell equal-area construction. This is based on the condition that in equilibrium the chemical potential, pressure and temperature of the two phases have to be the same, and corresponds to replacing the van der Waals equation in the two-phase region by a line of constant pressure such that the areas A + and A - (see figure 1.2b) are the same [34]. Eq. (2.9) can be used to extract values for the critical exponents and amplitudes in Chapter 2. Theory 18 exponent van der Waals Experimental RG value a 0 0.08 - 0.13 0.110 P 1/2 0.3 - 0.4 0.325-0.327 7 ± 1 1.2 - 1.3 1.237 - 1.241 6 3 4 - 5 4.82 Table 2.1: Values of some critical exponents, as obtained from the van der Waals equation of state, experiments on pure fluids and renormalization group theoretical calculations. the van der Waals model: Along the critical isotherm (t — 0) one obtains from which one deduces, by comparison with definition (2.5), that in the van der Waals model, 6 = 3 and D0 = 3/2. Similarly, one can calculate the isothermal compressibility Along the critical isochore (Ap = 0) in the one-phase region, we thus obtain Kj = l/6t, from which we deduce, by comparison with eq. (2.4), that in the van der Waals equation 7+ = 1 and = 1/6. The extraction of the exponent 0 is much more difficult, since in the two-phase region, for any given temperature and pressure, there exist three solutions for the density, from which only two are physical. It can be shown however [34, 2] that 0 = 1/2 and B0 = 2. Experimentally, one finds very different values of the critical exponents in real fluids. Table 2.1 gives an overview of some experimental results in comparison with values obtained from the van der Waals equation of state. The discrepancy between the values is due to the fact that the van der Waals theory is a mean field theory, which means that it can be derived by assuming that each particle moves in the mean field of all the other particles. Mean field theories do not give the correct values for the critical p = 3A/>3(2 - Ap)-1 = (3/2)A/>3(l + Ap/2 ± ...) (2.11) (2.12) Chapter 2. Theory 19 exponents, because they ignore fluctuations and thus do not correctly take into account correlations which are important near the critical point. Thus, even though the van der Waals equation gives a qualitatively correct picture of the behaviour in the critical region, correlations have to be considered explicitly to obtain quantitatively correct results. 2.1.3 Scaling Mean field theories assume that the free energy at the critical point can be expanded as a power series in integral powers of the order parameter Ap*. The fact that the exponents which are found experimentally deviate strongly from the mean-field values indicates that this is not a good assumption. Rather, the free energy in the critical region will contain a singular part fs(T, p) which contains the leading critical behaviour. The scaling assumption postulates that /„ is a generalized homogeneous function [2], i.e., that A / .(M) = / , ( A ° % A > ) (2.13) for any value of the number A. By differentiating the free energy / , with respect to its variables one obtains the various thermodynamic quantities, and one can thus express the critical exponents in terms of the exponents a< and aM. One obtains 0=i^L, 6 = - ^ - , 7 ± = ^ l I Q+ = 2 - i (2.14) at a — dp at at Since in this scaling form the free energy contains only two exponents, aM and a*, it follows that the critical exponents are not independent, but related by so-called scaling relations: 7+ = 7" = 0(6 - 1) and a+ = a~ = 2-0(6+1) (2.15) Choosing A = i - 1 / 0 ' , one can then write singular part of the free energy as f3=t*-°f8(l,p/\AP\s) (2.16) Chapter 2. Theory 20 More scaling relations can be obtained from the conjecture that the long range corre-lations of density fluctuations near Tc are responsible for all singular behaviour [3]. The correlation length £ is a measure of the range of the density fluctuations. The singularities in various physical quantities at Tc can thus be understood as a result of the divergence of £ at Tc. Empirical data show that £ obeys a power law close to Tc with an exponent v (see eq. (2.6)). As £ diverges at the critical point, close to Tc it becomes much larger than any other length scale in the system and therefore is the only relevant length scale as far as the singular behaviour of the free energy is concerned. The density-density correlation function is a quantity that is experimentally accessi-ble and gives information about the fluctuations in the system. The static correlation function G(r) between the densities in infinitesimal volume elements spaced a distance r apart can be written as [2] G(r) =< p(r)p(0) > - < p >2 (2.17) where < p > is the average density of the sample and is assumed to be independent of position. Away from the critical point, the correlation function is found to fall off exponentially with r for large distances. The general behaviour is therefore given by G(r) oc exp(—r/£). At the critical point, the correlation length £ becomes infinite; as the correlation function experimentally is still measured to decay to zero as r —• oo, one expects an inverse power law in r for the correlation function close to the critical point: Gc(r) oc r-<d-a+*> for r - » o o (2.18) where d is the dimensionality of the system and rj is another critical exponent. This correlation function can be measured by quasielastic light scattering. The inten-sity I(q) scattered at wave number q is, from the Wiener-Khinchin theorem, proportional Chapter 2. Theory 21 to the Fourier transform of the correlation function: I(q) oc J d3r exp(-iq • r)G(r) (2.19) On the other hand, from the fluctuation-dissipation theorem [34] it follows that the scattering intensity in the forward direction (q = 0) is proportional to the compressibility: 1(0) oc J d3rG(r) = 4 (2.20) This indicates that the phenomena observed close to the critical point, • the increase in the size of the density fluctuations • the increase in the range of the density correlation function • the increase in the compressibility are all interrelated phenomena. As a consequence one expects the exponents 77 and v to be related to the thermodynamic exponents a, /?,7 and 6. Indeed from the fluctuation-dissipation theorem it follows that 7 = (2 - r,)v (2.21) Furthermore, a dimensional analysis of the free energy yields the "hyperscaling relation" dv = 2 - a. (2.22) We thus have a scaling theory which determines all exponents from a knowledge of two of them. While the scaling relations hold for all dimensions d, the hyperscaling relation fails for dimensions d > 4, where the classical, mean field exponents are valid. Chapter 2. Theory 22 Uij = < 2.1.4 Lattice Gas, Ising Model and Series Expansions In order to calculate the critical exponents, one needs a microscopic model which contains the important features of the physical system under consideration while at the same time being simple enough to be treated theoretically. For pure fluids, such a model is the so-called lattice gas model [5], which consists of a system of "occupied" and "unoccupied" sites on a lattice. In order to model the repulsive cores of the fluid molecules, we assume that each site can be occupied by at most one molecule. If two adjacent lattice sites are occupied, their interaction energy is —eo. The potential between two molecules i,j is thus: oo if i = j —to if i,j are nearest neighbours (2.23) 0 otherwise It can be shown [5] that the lattice gas is mathematically equivalent to the Ising model, which is the simplest model for treating magnetic phase transitions and has been widely analysed in the literature. Again, the space is divided up into a lattice, and each lattice point i is occupied by a spin s< which can point "up" (s,- = +1) or "down" (s,- = —1). The Ising spin is thus a scalar. The simplest Hamiltonian for the interaction between Ising spins on a lattice with T V sites in an external magnetic field H can be written as HN({si}) = -H £ ^ - J £ Si • Sj. (2.24) <=i <«\j> When the strength of the interaction J > 0, this describes a ferromagnet. < i,j > indicates summation between nearest neighbours. In order to extract thermodynamic quantities from this microscopic model, one has to calculate the partition function ZN = T r i i = ± 1 exp(-HN/kBT) (2.25) Chapter 2. Theory 23 from which the total free energy (i.e., the sum of the singular part and the background term) can be obtained in the thermodynamic limit: F = -kBT lim \nZN (2.26) N—*oo Analytic solutions exist for the one-dimensional Ising model and for the two-dimensional Ising model in zero field, but the three-dimensional model can only be solved approxi-mately. One approximate method, called high-temperature expansion, consists in expanding the partition function in powers of the reduced coupling strength J/kBT [46]. In the limit of high temperatures, J/kBT is small, and the exponential in eq. (2.25) can be approx-imated by a power series in J/kBT. The various thermodynamic quantities correspond to the derivatives of the free energy and are thus also represented by a power series in J/kBT. From the ratios of successive coefficients in these power series expansions the critical exponents can be obtained. Through tedious and time-consuming calculations, the values of the exponents are found to be [7, 47]: a = 0.105(10), /? = 0.328(8), 7 = 1.239(2), v = 0.632(2) 2.1.5 Renormalization Group The idea giving the deepest insight into the physics of critical phenomena is the appli-cation of the renormalization group (RG) transformation to systems close to the critical point [48, 49]. The RG is capable of explaining scaling, universality and where the critical power laws come from. The effective Hamiltonian H = H/kBT is a function of a set of parameters {if, h,...}, with K = J/kBT, h = H/kBT, etc. The set of all Hamiltonians H(K,h,...) forms a space of Hamiltonians in which every Hamiltonian H(K, ft,...) is represented by a point. Chapter 2. Theory 24 The dimensionality of this space is, in general, infinite. The renormalization group is a set of transformations on the set of parameters {K, h,...} of the system's Hamiltonian. In practical terms, the RG transformation consists in partially carrying out the trace in the partition function, thus summing over selected spins while leaving others unaf-fected. By carrying out this partial sum, one decimates the number of spins in the trace, while replacing the parameters {K,h,...} in the Hamiltonian by new, effective, "renormalized" parameters {K',h',...}. Thus the RG procedure can be written formally as H'(K\h',...) = K(H(K,h,...)) (2.27) A fixed point {K*>h*,...} of the renormalization group satisfies the equation {K*,h\...} = K{K\h*,...} (2.28) This fixed point can be shown to coincide with a critical point: if we rescale the whole lattice by a factor b, i.e., substitute each "block" of bd lattice sites (b > 1) by a single new renormalized spin, then in the new lattice the correlation length £' has the length £' = £/b. The RG transformation thus has the effect of reducing the correlation length, which is equivalent to driving the system away from criticality. As £ is a function of the parameters {K, h,...}, the flow of the correlation length under the RG transformation can be written as £(K,h,...) —• £(K',h',...) = ((K,h,...)/b. At the fixed point of the transformation this implies that ((K*,h*,...) = £{K*, h*, ...)/&, which, since b > 1, has the solutions £* = oo (critical point) or £* = 0 (trivial fixed point). The divergence of the correlation length was recognized earlier as the crucial feature of a critical point, and we conclude that the fixed point of the RG transformation is a critical point. How is all of this related to "real" systems? Figure 2.1 illustrates the consequences of an RG transformation on the Hamiltonian of a real system. In the space spanned by the parameters of the system, the Hamiltonian corresponding to the neighbourhood of the Chapter 2. Theory 25 Figure 2.1: Schematic illustration of the motion of the Hamiltonian of a real physical system under the influence of the RG transformation, in parameter space. Chapter 2. Theory 26 "physical critical point" of, say, Xenon, is expressable as a function of the parameters t and fi as 7i°(*,/z). At the critical point (t = 0,/x = 0), we have f = oo; however, TQ = 7t°(0,0) is in general not a fixed point. After an RG transformation, we obtain a renormalized set of Hamiltonians Ti'(t', fi') into which is embedded the renormalized critical Hamiltonian Ti'e = Tl(Ti%). Thus, under repeated RG transformations, a line of critical points is generated. The Hamiltonian wandering along this line may eventually end up at a fixed point Ti*, at which the RG does not cause any further motion. That means that the initial Hamiltonian Ti? lies on the stable critical manifold of the fixed point Ti*. All systems lying on the critical manifold of the same fixed point will display identical critical behaviour because they flow to the same point under RG transforma-tions. This explains the concept of universality classes. If the initial Hamiltonian is perturbed in such a way that it lies in a manifold flowing into a different fixed point, then this perturbation is relevant. For our example of the critical point of a fluid, this is true for the parameters t and fi. They are called relevant scaling fields (i.e., in order for the system to be at the critical point, they have to be zero), whereas other parameters of the system (like the shape of the molecules or short-range interactions between them) are irrelevant. Close to the critical point, the RG transformation, which in general is nonlinear, can be linearized. As a consequence of the semi-group property of the RG, the eigenvalues of the linearized transformation can be expressed as Ai = bXl, A 2 = b*2,.... Small deviations from the critical point can then be written as linear combinations of the eigenvectors be-longing to these eigenvalues. If A,- > 0, then A,- > 1 for b > 1, and successive RG transformations carry the system away from criticality; the conjugate scaling fields are then called relevant. If A,- < 0, then A,- < 1, and successive RG transformations eventu-ally make these terms imperceptibly small; the conjugate scaling fields are irrelevant. The fixed point is thus insensitive to irrelevant variables, but depends strongly on the Chapter 2. Theory 27 relevant variables. In the vicinity of the fixed point, where the RG transformation can be linearized, the linear scaling fields behave like g'i=giAi = bXigi (2.29) under the RG transformation. Since each RG iteration changes the length scale of the system by a factor of 6, the free energy per unit volume transforms as f[H'] = bdf[H\. (2.30) Expressed in the set of scaling fields t, p, gi,the flow equation for the free energy then takes the asymptotic form /(*, a«. ft, •••) = **i*> - . **au.-.) = <2"a/(i, M/^.W** .-•) (2-31) choosing 6Al = 1/t and setting Ax = d/(2 — a), A2 = 8fid/(2 — a) and A,- = fad/(2 — a), where fa is a so-called correction to scaling exponent. Bearing in mind that Ap oc t13, this equation is seen to be identical to eq. (2.16). The scaling form of the free energy thus follows in a natural way from the RG transformation. 2.1.6 Implementation of the Renormalization Group In order to calculate numerical values for the exponents, the RG has to be implemented explicitly, starting from a microscopic model of the system under consideration. This can be accomplished either in real space [50] or in momentum space [48, 49]. Once the RG transformation has been constructed, fixed points can be found and tested for stability. For Ising models of dimensionality d < 4, the classical (mean field) fixed point is found to be unstable, and the system flows instead into the so-called Ising fixed point under RG transformations. The critical exponents are obtained from the eigenvalues Chapter 2. Theory 28 of the linearized RG transformation around that fixed point (see eq. (2.31)). For the three-dimensional Ising model one obtains [51, 52]: a = 0.116, 0 = 0.325, 7 = 1.238, v = 0.628. 2.1.7 Monte Carlo "Experiments" One way to check the theoretical results and to develop a quantitative understanding of the local microscopic correlations in a system (which are not accessible to experiment) is to carry out computer simulations on a lattice, producing various microstates of the system and summing over them with their respective thermodynamic weights. These "Monte Carlo Experiments" give some insight into the thermodynamic states that a system goes through under thermodynamic equilibrium conditions. In particular, the evolution of fluctuations can be followed explicitly. Monte Carlo simulations also give values of the critical exponents and critical amplitudes. However, due to limited computer power and time, the "thermodynamic limit" N —• oo is not directly accessible but has to be obtained by extrapolation of data obtained for finite (and actually quite small) N. Since the correlation length £ —• oo as the critical point is approached, the size of the lattice used in the simulation limits the degree to which the critical point can be approached. This problem can be avoided by combining Monte Carlo (MC) simulations with RG analysis of the critical properties [53, 10]. The MC simulation on a system at its critical point produces a sequence of microscopic configurations, and the RG is applied directly to these individual configurations, yielding.a sequence of configurations for the "blocks". As the original Hamilitonian is critical, the renormalized Hamiltonian flows towards the fixed point. This model yields accurate values for the exponents [54]: a = 0.113, 0 - 0.324, 7 = 1.238, v = 0.629. Chapter 2. Theory 29 2.2 Subtleties and Applications In this section we look at some consequences of the theory developed above, and their applications to real fluid systems. 2.2.1 Corrections to Scaling The terms in the free energy (see eq. (2.31)) containing the scaling fields </,- deserve some consideration, because they are of importance in experimental situations where data are taken not only in the critical region, but also further away from the critical point. If the exponent fa < 0, then </, is an irrelevant variable, and the free energy close to Tc can be expanded in powers of this leads to the so-called corrections to scaling [55]. The exponents fa can be calculated from the RG equations. In Ising-like systems, the most important correction exponent A is due to the scaling field u in the Landau-Ginzburg-Wilson expansion [4] and is irrelevant for dimensions d < 4. It has been calculated to be A w 0.5 [6, 7, 8, 56]. In addition to these singular corrections to the asymptotic scaling form, there are also analytic corrections due to the fact that at appreciable distances from the critical point, the linearization of the RG transformation is no longer valid and terms nonlinear in the scaling fields have to be included. This leads to added terms of the form i,i 2 , . . . . Since A is found to be so close to 0.5 for the three-dimensional Ising model, these terms can be assumed to have exponents of 2A,4A etc. [11]. Taking these corrections to scaling into account, the order parameter then can be written A / = BQt\l + BitA + B2t2A + ...) (2.32) Other thermodynamic quantities can be expanded in a similar way into power series in r A . For large reduced temperatures, the correction to scaling expansion requires too many Chapter 2. Theory 30 correction terms and therefore too many parameters to be useful. Here, an equation of state incorporating the crossover to classical mean field behaviour can be introduced [57]. In the experiments of this thesis, corrections to scaling play a role in all the pure fluids considered and the exponent A can be determined with considerable accuracy. 2.2.2 Universal Amplitude Ratios From the free energy in eq. (2.16) follows an equation of state [58, 59] Ap* = (p-pc)/pc = Ap\Ap\s-1h(x), with x = i/\Ap\1/0 (2.33) where h is a nonuniversal function. By rescaling the function h and the variable x by two (nonuniversal) constants ho and xo, this equation of state can be made universal; i.e., the rescaled function h(x) = h(x/x0)/h0 is the same for all systems in a given universality class. Thus, from the knowledge of the two system-dependent scale factors ho and x0 all critical amplitudes can be calculated. This leads to universal relations between the critical amplitudes [12], analogous to the scaling laws for the exponents. For example [12], B0 = Xq13, D0 = hQ and Tq = Xoh'1 nm*—O0[x'y/h(x)}. Using the scaling relation 7 = /3(6 — 1), it follows that h\ = T+DqB*-1 = Jim [xy~h(x)] (2.34) X—+00 is independent of ho and xq and is thus universal. Similarly, i ? r = r+/ro (2.35) can be shown to be universal [12]. Rx and Rr have been calculated by high-temperature series expansion and by implementation of the renormalization group (e-expansion [4]) and have been found to have values Rr « 4.8 — 5.07 and Rx « 1.6 — 1.75 [12]. Similarly, certain ratios of the correction to scaling amplitudes are expected to be universal [60, 61]. Chapter 2. Theory 31 We have measured the quantities i?x and Rr in the polar fluid C H F 3 . Our experiments give the most accurate experimental values to date for these amplitude ratios. The results are discussed in Chapter 5. 2.2.3 Long Range Forces If there are long-range forces present between the molecules of the fluid, for example dipolar forces varying as 1/r3 between molecules with permanent electric dipole moments, will they affect the universality class? In ferromagnets, dipolar interactions can be shown to be relevant and thus change the universality class of the system [62]. In fluids, however, the dipolar interaction does not couple to the order parameter, and therefore, when calculating the partition function in eq. (2.25), the trace over the orientations of the dipoles can be performed independently of the configurational trace [63]. As the angle integral over dipolar interactions is zero, the lowest contribution is proportional to 1/r6 and is irrelevant to the Ising fixed point [64]. The dipolar coupling therefore does not change the universality class of the fluid, and we expect polar fluids to exhibit the same critical exponents and amplitude ratios as nonpolar ones. This hypothesis is tested in this thesis by comparing results of experiments on CHF 3, a strongly polar fluid, with those on CC1F3, a weakly polar fluid, and Xe, a nonpolar fluid. These experiments are described in Chapters 5 and 7. 2.2.4 Asymmetric Lattice Gas and Deviation from Rectilinear Diameter In section 2.1.4 we found that the lattice gas model is useful because it maps onto the Ising model, and because it correctly describes many features of real fluid systems. The lattice gas has the property that the sum of the liquid and vapour densities is constant [5], which closely resembles the "law of rectilinear diameter" [24], found experimentally to Chapter 2. Theory 32 be quite well obeyed for fluids. The lattice gas model is artificial, however, in assuming symmetrical behaviour of occupied lattice sites (particles) and empty ones (holes). Some models have been invented that circumvent this shortcoming [65, 66] by treating the fluid particles as interpenetrable spheres rather than point particles on a lattice. In these models, the law of rectilinear diameter is no longer obeyed. Rather, close to the critical point, the diameter acquires the leading temperature dependence Pd = Pi±±L = i + A l _ a t i - + ... (2.36) lPc The exponent a is identical to the specific-heat exponent. An alternate way of obtain-ing the same result [67] is by extending the scaled equation of state, valid in the close proximity of the critical point, by a parametric representation, thereby introducing cor-rections to scaling. Even though the law of rectilinear diameter is violated, the scaling relations between the exponents can be shown to be still valid in these models [68], and the universality class of the system is still the same as that of the Ising model [69]. Recently, a microscopic theory [26] has interpreted this deviation of the diameter from a straight line as resulting from many-body interactions between the fluid molecules. The strength and direction of the deviation from rectilinear diameter thus give information about the importance of many-body interactions. We have carried out experiments to measure the critical behaviour of the coexistence curve diameter in CHF 3, CC1F3 and Xe. Both CHF 3 and CC1F3 exhibit a diameter singularity close to the critical point, in accordance with the microscopic theory. This singularity is absent in Xe, indicating that interactions not included in the microscopic theory play a role in this system. Chapter 2. Theory 33 2.3 Dynamic Critical Phenomena: Diffusivities By measuring the spectrum of light scattered from a fluid sample close to the critical point, one can obtain information about the order parameter fluctuations and their decay times. The two relevant hydrodynamic modes in a fluid correspond to two types of decay processes and give rise to two types of lines in the spectrum of the scattered radiation: The line due to the thermal fluctuations is centered at the incident frequency, and its width is proportional to the the diffusivity D\ mechanical fluctuations, or equivalently phonons, produce lines shifted from the incident frequency by an amount ±Aw, proportional to the sound velocity in the medium [70]. To obtain information about the dynamics of the system, one wants to measure time-dependent correlation functions. The quantity of interest is the order-parameter correlation function, obtained from measurement of the autocorrelation function < £n*(r,r)6n(r,0) > of refractive index fluctuations at times T and 0 in an infinitesimal volume of fluid centered at position r. Equivalently, one can investigate its Fourier transform < 6n*(q, r)6n(q, 0) >. By assuming a linear relation between £n(q, T) and the scattered electric field Es(q, T), one can relate the order parameter correlation function to the electric field correlation function g{1) =< £*(q, r )£ (q , 0) > / < E*E > . (2.37) In practice, the electric field itself cannot be detected, but only its intensity. Thus, the simplest correlation function that can be determined experimentally is the intensity correlation function 0(2>=</(q,T)/(q,O)>/</>2, (2.38) where < / >=< E*E >. If the scattered field is Gaussian distributed (which is normally the case when the number of scattering centers is high, i.e., when the correlation length Chapter 2. Theory 34 is much smaller than the linear dimension of the scattering volume), it can be shown that [71] 5(2)(q,T) = l-fT|^1)(q,T)|2, (2.39) where T is a factor of order unity depending on the scattering geometry. The order parameter correlation function, to a good approximation, decays exponen-tially [35]: MqTF " </(q)> " eXp(-r(q)r>' (2-4°) with T(q) = D(q)q2. Here, D(q) is the wavenumber dependent diffusivity. The half-width T(q) measured in the autocorrelation measurements is identical to the linewidth of the Rayleigh line, which is the central line in the spectrum of the scattered radiation [70]. In the hydrodynamic limit C 1, when the characteristic length of the order pa-rameter fluctuations is much smaller than the wavelength of light, the diffusivity can be written as the ratio of a generalized conductivity £ and a generalized susceptibility x as limiZ>(q) = Ijx. (2.41) X diverges strongly at the critical point, with an exponent 7 « 1.24, and £ is also expected to exhibit a singular behaviour close to Tc. The main contribution to the anomaly in £ comes from couplings between the hydrodynamic modes [72, 73]. In the framework of the mode-coupling theories, the conductivity can be expressed in terms of the generalized susceptibility x and the shear viscosity fj as [33] £fj = RkBTXe-d- (2.42) Similar results have been obtained by renormalization group calculations [74]. Using this the singular part of the diffusivity can be written as [75]: D°(q) = D(q) -D= n ( r f ) , (2.43) Chapter 2. Theory 35 where D is the nonsingular background contribution to the diffusivity. Experimentally, D can be obtained by extrapolating diffusivity data taken far from critical into the critical region [76]. It can be written as D = o^/x(?)> where Iq is the background contribution to the conductivity. J? is a universal constant and is expected to have a value of R = 1/67T (from mode coupling calculations), or R = l/5w (from renormalization group calculations). The shear viscosity fj consists of a background term rj° and a singular term T)a: fj = rf -f 77*. Close to Tc, the dynamic renormalization group predicts 77s to diverge as ( Z n [33, 75], whereas in mode-coupling theory the viscosity, to a first approximation, can be written as 77 = 77°[1 + (8/157r2)ln(QO]) with Q a system-dependent amplitude [33]. The two predictions agree if one considers fj to obey a power law fl = v°{Qt)*- (2-44) The viscosity exponent zv = 8/157T2 = 0.054 [77] is related to the static exponent 77 introduced in eq. (2.18) by a scaling law [33]. The universal dynamic scaling function ft has the form [78] ft(x) = ft*(x)[l + (x/2) 2]^ / 2 (2.45) with the Kawasaki function [73] CtK{x) = (3/4x2)[l + x 2 + (x3 - 1/x) arctan x] (2.46) According to eq. (2.43), the linewidth of the decay curve can be written as a power law in the scattering vector q: T(q) = q*n(qO. (2.47) The dynamic scaling function satisfies the boundary conditions: Jim ft(x) = Coo*" ( 1 +* , , ) and limft(x) = C 0 (2.48) Chapter 2. Theory 36 where Co and Coo are constants. The dynamic scaling exponent z and the viscosity exponent zv should obey the scaling relation z = 3 + z„. (2.49) This can be used to obtain an estimate of the viscosity exponent from autocorrelation measurements, even if the viscosity itself is not being measured. In Chapter 6 of this thesis, experiments to measure the diffusivity in the binary liquid system nicotine + water are described. Two different experimental techniques were used, and their results are compared. From the light-scattering data, the exponent zn can be determined. Chapter 3 General Experimental Considerations: Temperature Control and Optics This chapter describes the general experimental techniques used. In section 3.1 the temperature control system is discussed. In section 3.2 the optical methods are introduced and the physical quantities of interest are derived. 3.1 Temperature Control In order to obtain data on the asymptotic power laws close to the critical point, mea-surements have to be extended into the scaling region which, for pure fluids, is limited to reduced temperatures t < 10~4. Approaching the critical point so closely requires very precise temperature control: For the fluids investigated in this thesis, which have Tc w 300K, reduced temperatures of the order of t « 10"5 corresponds to stabilizing the system at 3mK from the critical point. Thus we needed a temperature control system that controlled the cell temperature to within at least lmK over several hours. This was accomplished by placing the sample cell in the center of a two-stage thermo-stat (see Figure 3.1). The inner heating stage consisted of a copper or aluminium block wrapped with heating wire or heating foil. A thermistor embedded in this block sensed its temperature and formed one arm of a Wheatstone bridge. The remaining arms were formed by a decade resistance box and, in series with standard resistors, a potentiometer which could be turned by switching on a toy motor. The error signal of the Wheatstone bridge was fed into an HP nullmeter (model #419A DC) which amplified it; the output of the HP served as the input of a Kepco operational power supply (model OPS 7-2). 37 Chapter 3. General Experimental Considerations: Temperature Control and Optics 38 Figure 3.1: Thermal control system for conducting optical experiments on fluids close to the critical point. Chapter 3. General Experimental Considerations: Temperature Control and Optics 39 The output of the OPS drives a heater on the metal block encasing the sample cell. The OPS was supplemented by an "amplification box" which enabled us to vary the time response and the amplification of the heater circuit. The switching of the toy motor driving the Wheatstone bridge was controlled by a Commodore PET computer: A "Sweeping Program" written in BASIC contained infor-mation on the number of sweeps, the sweep times (i.e., the "on" times of the motor) and the waiting times between individual sweeps. It also contained commands to light LEDs which could be used as time markers on the film used to record the data. The output of the PET was fed into an interface box, which contained a solid-state switch for turning the motor on and off. Details of the computer-motor interface are found in Ref. [79]. The innermost heating stage was surrounded by a layer of styrofoam 2.5 cm thick, fol-lowed by an outer heating stage which consisted of a cylindrical shell of copper to which 3/8" copper tubing was soldered. For the CHF 3, CC1F3 and He-Xe-experiments water from a temperature regulated bath (FORMA model # 2095) was circulated through the copper tubing and kept the cylinder at a chosen temperature, always between 0.5° and 1.0° below the cell temperature. In the nicotine + water experiment the critical tem-perature was about 61°C, where water evaporates at a fairly high rate, so the outermost heating stage was heated electrically. In the nicotine + water and the He-Xe setups, a passive heat shield, consisting of a cylindrical shell of 1/16" thickness copper sheet, was inserted into the insulation between the inner and outer shells. Windows in the metal thermal stages and the styrofoam allowed an expanded laser beam to be fed through the cell. Chapter 3. General Experimental Considerations: Temperature Control and Optics 40 Z ( a ) P ( b ) ( c ) Figure 3.2: Refractive index n as a function of height z in a cell of critical overall density, (a) Two-phase region, (b) one-phase region close to Tc and (c) one-phase region far from Te. 3.2 Optics In this section we discuss the optical setups for the experiments and derive formulas for the quantities that can be obtained from the experiments. Figure 3.2 shows the refractive index profiles in a cell of critical overall density as the temperature is changed. Far below the critical point (T « Tc), two phases of different refractive index coexist in the cell, separated by a sharp transition line called meniscus (see figure 3.2a). Far inside the one-phase region, the refractive index is homogeneous over the whole cell, as shown in figure 3.2c. Close to the critical point, in a pure fluid the compressibility (dp/dP)T diverges, leading to a strongly curved refractive index profile inside the cell (figure 3.2b). In binary liquids, the corresponding osmotic susceptibil-ity (dxjdp)T diverges, but in a much smaller temperature region close to the critical point. Due to the very similar densities of the constituents, the divergence of the osmotic susceptibility is not observable in the nicotine + water experiment. Chapter 3. General Experimental Considerations: Temperature Control and Optics 41 The refractive index profile contains a wealth of information about various thermody-namic properties of the system. In order to utilize it, though, in pure fluids the refractive index has to be related to the fluid's density, and in binary liquids to the composition. An experimental method which provides the link between the refractive index and the density in pure fluids, is described in section 3.2.1. An estimate of the concentration dependence of the refractive index in nicotine + water is presented in chapter 6. The interference methods used in this thesis are geared to mapping out the refrac-tive index profile as a function of temperature. Sections 3.2.2 and 3.2.3 describe these methods and outline the data analysis procedures. Limitations of the methods due to gravity effects are discussed briefly in section 3.2.4. Finally, Section 3.2.5 describes how diffusivities can be measured by light scattering. 3.2.1 Prism Cell Experiments The prism cell experiment can be used to measure the density and the refractive index of the fluid simultaneously. Thus it gives information on the Lorentz-Lorenz function C(p) which characterizes the density dependence of the refractive index n: The sample cell used for these experiments [80] had an aluminium body with a prism-shaped head. Two sapphire windows mounted on the prism faces allowed a laser beam to be shone through the cell's head. By measuring the deflection angle of this laser beam from the incident direction one can deduce the refractive index of the medium inside the cell. This calculation is carried out in Appendix A. Figure 3.3 shows a schematic of the optical setup: The expanded beam from a He-Ne laser passes through the prism and hits a micrometer driven mirror (Lansing Research Corp. model 10.253) which reflects it into an autocollimating telescope (Davison model Chapter 3. General Experimental Considerations: Temperature Control and Optics 42 BEAM EXPANDER THERMOSTAT *<0 MICROMETER DRIVEN MIRROR HE-NE LASER IRIS 7 TELESCOPE e Figure 3.3: Optical setup of the prism cell experiment. D275). By adjusting the angle of the mirror, the deflection angle due to refraction by the prism and thus the refractive index can be measured. A reference beam passing outside the cell is used to monitor the stability of the alignment. In order to obtain precise measurements, a series of calibrations had to be carried out. Appendix A describes these calibrations and shows the effects of the various errors on the results. The measurements of the Lorentz-Lorenz function proceeded as follows: The cell, containing a measured mass of fluid, was cooled into the two-phase region where the fluid phase separates into liquid and vapour phases, in each of which the refractive index is temperature-dependent. The temperature of the cell was then, raised until the system passed from the two-phase region into the one-phase region where the refractive index becomes essentially independent of temperature. The refractive index was measured just above the coexistence curve in the one-phase region. The mass of the fluid in the cell Chapter 3. General Experimental Considerations: Temperature Control and Optics 43 and thus its density was determined by weighing the cell on a chemical balance. By repeating this procedure for different overall densities, the Lorentz-Lorenz coefficient £ was measured and thus the relationship between density and refractive index on the coexistence curve obtained. Details of the calibrations are given in Appendix A. For the measurement of the coexistence curve and its diameter, the prism cell was filled with the fluid at its critical density. In the two-phase region, the refractivities of both liquid and vapour phases were measured as a function of temperature and converted to densities using the Lorentz-Lorenz relation. After changing the cell temperature, the system was allowed to equilibrate for at least two hours. This experiment thus yielded pi(T) and pv{T) along the coexistence curve, from which the order parameter and the diameter could be calculated. 3.2.2 Focal Plane Interference Technique Figure 3.4 shows the optical setup of the focal plane experiments. The sample was contained in a flat cell of length L with parallel windows, which was placed in a ther-mostatic block. A laser beam from a He-Ne laser, expanded by a pinhole filter and then collimated into a parallel beam, traversed the sample cell. The light was then collected by a focussing lens which was positioned as closely behind the cell as physically feasable to collect as much of the refracted radiation as possible. In the focal plane of this lens, a Fraunhofer interference pattern was formed which contains information on the refractive index profile in the cell. A slit camera with continuous film transport was used to record the interference pattern as a function of time on film. We now derive the formulae necessary for the interpretation of the data. The Fraun-hofer diffraction pattern from a flat cell far from the critical point (see Fig. 3.2a and 3.2c) is just a point. As the cell temperature is changed from a temperature T, in the two-phase region to a temperature Tf in the one-phase region, the refractive index profile inside the Chapter 3. General Experimental Considerations: Temperature Control and Optics 44 Focal Plane H e - N e Laser Beam Expander Thermostat with Cell Camera with Continuous Rim Transport Figure 3.4: Optics of the focal plane interference technique. cell evolves from the profile given in figure 3.2a to the one in 3.2c. The construction of the interference pattern due to an intermediate density profile is shown in figure 3.5. A ray travelling in the y-direction through a medium with varying refractive index n(z) is being bent by an angle 0 given by f ' ~ T (3'2> dy n dz For a thin cell, where the total bending is small, eq. 3.2 can be integrated to give the total deflection angle 6{ inside the cell n dz (3.3) where L is the cell thickness. Upon emerging from the cell, the ray is refracted according to Snell's law: n sin 0,- = na,-r sin 0, where 0 is the angle at which the ray leaves the cell. One thus obtains 0 = L dn «„«> dz (3.4) Chapter 3. General Experimental Considerations: Temperature Control and Optics 45 incoming plane waves ceil - y focal plane Figure 3.5: Formation of the Fraunhofer diffraction pattern in the focal plane. for the total bending angle of a ray passing through a cell of length L at a height z where the refractive index profile exhibits a slope (dn/dz). Two rays entering the cell at heights z\ and z2 where the gradient of the refractive index profile at z\ and z2 is the same are bent by the same angle 6 and thus are mapped onto the same spot in the focal plane. They will interfere constructively if the total optical path difference between them is an integral multiple of the laser wavelength. The multitude of rays passing through the cell in the height interval [^ m ,zmax] around the meniscus thus gives rise to a pattern of light and dark interference spots in the focal plane. This pattern, which changes with time after a change of the cell temperature, is recorded on film. This experimental method was used to measure the order parameter and the diffusivity of a binary liquid mixture. In a binary liquid of two constituents A and B, the order parameter Ax* is proportional to the difference in concentration of one species, say A, in the two coexisting phases I and II. Thus Ax* oc xf — xfj. The relation between the Chapter 3. General Experimental Considerations: Temperature Control and Optics 46 Temperature T \ , An(T») J \s- A n C n " ' ) — , \ " — !  quench n|" ne n<|> refractive index n Figure 3.6: Schematic of the nicotine + water phase diagram. Temperature is plotted as a function of refractive index which is proportional to the concentration of nicotine [81]. The order parameter is proportional to the refractive index discontinuity in the two-phase region. refractive index An and the order parameter Ax* for the binary liquid nicotine + water can be approximated by assuming a linear relationship between An and Ax* [81]. The interferometric data can then be used to measure the order parameter and the diffusivity. Order Parameter The refractive index discontinuity ni — n2, of the initial refractivity profile at a tem-perature T = Ti in the one-phase region can be measured by quenching the system into the one-phase region and counting the total number of interference minima Nm(Ti). Figure 3.6 illustrates the various quantities in an inverted phase diagram of the type encountered in the nicotine + water system. The difference ni — n2 is related to the Chapter 3. General Experimental Considerations: Temperature Control and Optics 47 number of minima Nm(Ti) by m - n 2 = An(TJ) = (Nm(Ti) - 1/2)X/L (3.5) where A is the laser wavelength and L the thickness of the cell. The temperature de-pendence of the order parameter was obtained by repeated quenches from the two-phase region into the one-phase region for different initial temperature T,-. A set of experi-ments then proceeded in the following way [82]: The sample cell was removed from the thermostat, mixed by shaking, and then replaced and heated to a temperature T/1^ in the two-phase region, where demixing into two liquid phases occurs. After an equili-bration period of several hours, the cell was quenched into the one-phase region in one temperature step which was quick on the time scale of the diffusivity, but slow enough to avoid convection in the cell. The interference pattern due to the relaxation of the concentration profile was recorded on film in the focal plane. The number of interference minima was used to calculate An(T/1^). Subsequently, the cell was heated to a temper-ature deeper inside the two-phase region, equilibrated, and the interference pattern was recorded as the cell was cooled into the one-phase region again. The number of interference minima gives An(7f }). The purpose of this experimental procedure is to obtain an equilibrium profile in the cell by increasing the composition discontinuity in each successive step. This question will be discussed in detail below (see section 6.3.2). Diffusivity After a temperature step taking the system from the two-phase to the one-phase region, an initial refractive index profile n(z) = n c - An/2 for z < 0 (3.6) nc + An/2 for z > 0 Chapter 3. General Experimental Considerations: Temperature Control and Optics 48 will "relax" according to the diffusion equation An(z,r) = D^„(z,r) .. (3.7) Here, r is the time and z is the height in the cell, as measured from the meniscus, and D is assumed to be concentration independent. n c is the critical refractive index and, for a critically filled cell, equal to the refractive index in the one-phase region. The solution to eq. (3.7) is [83] „(z,T) = „c-^ $(_iL=), (3.8) where <&(*) = 4= IXeM-P2)dp (3.9) V7T JO is the error function. Close to the meniscus, i.e., for small z, the error function can be approximated by $(r) « 2r/-v/7r, so that U M ~ n c = -UWr ( 3 ' 1 0 ) The maximum refraction angle 6max in the Fraunhofer pattern is due to rays refracted in the center of the cell where the slope of the density profile is highest. The time evolution of 6max after a quench from Ti in the two-phase region to Tf in the one-phase region is related to the diffusivity D(Tf) by = L[nc-n(z + 6z,r)} = { L / 2 ) A n / J ^ T y ; ( 3 n ) Thus, the diffusion constant D(Tf) can be determined from a measurement of the maxi-mal refraction angle as a function of time. 3.2.3 Image Plane Interference Technique The image plane technique uses a Mach-Zehnder interferometer which forms an inter-ference pattern between an expanded laser beam passing through the sample confined Chapter 3. General Experimental Considerations: Temperature Control and Optics 49 Beam Thermostat Figure 3.7: Schematic of the optical setup for image plane interference experiments. in a flat cell and a reference beam. Figure 3.7 shows a schematic of the experimental setup. A laser beam from a He-Ne laser enters a spatial filter which expands it into a collimated beam of about 1" diameter. It is subsequently split by a beam splitter into a "cell beam'' which passes through the sample, and a reference beam. A second beam splitter recombines the two beams, thus producing an interference pattern between them. A lens focusses this interference pattern onto the image plane, where it is recorded on film by a slit camera with continuous film transport. Each part of the cell maps directly onto a unique point in the image plane. Thus, by measuring the interference pattern, one obtains direct information about the refractive index as a function of height in the cell. The distance sq between the sample cell and the focussing lens determines the magnifi-cation factor of the image of the cell: the magnification M is given by M = f/(s0 — /), Chapter 3. General Experimental Considerations: Temperature Control and Optics 50 where / is the focal length of the lens. In order to resolve the interference pattern around the meniscus as well as possible, a rather large magnification was chosen. The length of the optical table and the power of the laser place a limit on M; in our experiments, we had M « 3 — 5. M is determined by measuring the height of the image of the cell in the image plane hi and dividing it by the original height of the cell h0: M = hi/h0. By mixing a plane wave (corresponding to the reference beam) with the wave which has interacted with and has thus been distorted by the refractive index profile in the cell, one obtains an interference pattern consisting of horizontal fringes. The change in refractive index An between two neighbouring fringes is An = X/L, where A is the wavelength of the laser light (in our case, 6328A), and L is the cell thickness. Far away from the critical point, the compressibility is small and thus the densities of each phase are almost height-independent. In the one-phase region, the whole cell is filled by the sample at (almost) uniform density (see Figure 3.2c), leading to a field of uniform intensity in the image plane. In the two-phase region, far from Tc, the density at any given temperature is uniform in both liquid and vapour phases (Figure 3.2a). Thus, the cell's image contains no interference fringes, but is only a field of uniform intensity. As the temperature is changed, the change in densities of the coexisting phases leads to a phase change between the cell beam and the reference beam. This appears as a periodic change in intensity in the image plane. Since the whole field corresponding to one phase is of uniform intensity, however, it is difficult to see whether the refractive index difference between the two phases is actually decreasing or increasing. The interpretation of the data can be greatly facilitated by slightly tilting the reference beam wave fronts with respect to the cell beam, thereby "misaligning" the beams with respect to each other (see Figure 3.8). Then even far from the critical point, one obtains interference fringes in the image plane, the spacing of these fringes depending on the tilt angle of the mirror. For our experiments, the tilt angle was between 0.01° and 0.02°, corresponding Chapter 3. General Experimental Considerations: Temperature Control and Optics 51 incident plane wc n(z) Figure 3.8: Effect of a tilted reference beam on the interference pattern. to 2 - 4 fringes per cm of cell height. On passing through the cell, the plane wavefront of the incident beam is distorted into a shape which mirrors the density profile in the cell. The propagation speed v of a light ray through a medium of refractive index n is v = c/n, so that the rays passing through the bottom of the cell (where n is larger) are retarded relative to the rays passing higher in the cell (where n is smaller). An interference maximum between the reference and the cell beams occurs whenever there is a path difference of 2nN\ between them (where JV is an integer). In order to obtain the refractive index profile from the interference pattern, one has to deconvolve the interference pattern by subtracting the effect of the tilted reference beam, which distorts the image of the cell as indicated in Figure 3.8. The total path Chapter 3. General Experimental Considerations: Temperature Control and Optics 52 around the interferometer, Ltot, can be written as Ltot = Lo + Ln(z = 0) where Lq is the path through air, L is the length of the cell and n(z = 0) is the refractive index at height z = 0, in the middle of the cell. A ray passing through the cell at height z interferes with the tilted reference beam with a phase <f>(z) given by Ln(z) + L0 + z tan a = \<j>(z)/2ir (3.12) Here, a is the tilt angle of the mirror, and A is the laser wavelength. Two rays passing at heights z and z' will therefore interfere with a phase 8(j> — <f>(z) — <j>(z'), when L(n(z) - n(z')) + (z- z') tan a = A S<f>/2n (3.13) These rays will form adjacent interference fringes in the image plane if 8<f) = 2TT, i.e., £An + Aztana = A (3.14) Thus, if a ray passing through the cell at height z\ interferes with the reference beam to give an interference maximum, then a ray passing at height z2 will also interfere to produce a maximum if the refractive index difference at the two heights z\ and z2 is n(z\) — n(z2) = [A — (z\ — z2) tan ct]/L. The height difference z\ — z2 = Az is related to the height difference Az as measured on film by Az = Az/M, where M is the magnification factor of the lens. The refractive index profile in the cell can thus be directly inferred from a measurement of the interference pattern as a function of height, provided the tilt angle a and the magnification factor M axe known, a can be obtained from the interference pattern of the cell far from T c, where any curvature of the refractive index profile is negligible. Order Parameter and Coexistence Curve Diameter The image plane interference technique can be used to measure the order parameter and the coexistence curve diameter of the fluid. As the temperature of the cell is lowered Chapter 3. General Experimental Considerations: Temperature Control and Optics 53 below Tg, the fluid phase separates. As T is lowered further, the liquid density increases, whereas the vapour density decreases. The density discontinuity at the meniscus in the middle of the cell (i.e., the order parameter) becomes larger, which causes interference fringes to "disappear" into the image of the meniscus. The number of missing fringes is related through the Lorentz-Lorenz function to the density differences pi — pc and pc — pv of the liquid and vapour phases from their critical value. At each temperature, the order parameter is obtained from the sum of the number of fringes missing from each phase, while the diameter is obtained from the difference. Before each experimental run, the cell was cooled deep into the two-phase region. Then it was warmed up in small steps (« 0.1° far from Tc, and as small as 0.001° close to Tc) and the evolution of the interference pattern was recorded on film in the image plane. Between steps, the system's temperature was held constant for 1-2 hours to allow the cell to reach equilibrium. After the waiting period, the fringe pattern near the meniscus was found to be stationary, which lead us to believe that thermal equilibrium had been reached (at least locally). Compressibility To evaluate the isothermal compressibility kt, the slope of the density profile in the middle of the cell had to be measured. Due to critical slowing down, even for pure fluids the equilibrium slope takes hours to establish itself in the critical region. Therefore, when taking compressibility data, the sample was allowed to equilibrate for « 12 — 24 hours before a data point was taken. A difference in chemical potential Ap is related to the corresponding pressure differ-ence AP and the corresponding height difference Az by the equation Ap = AP/p = -gMAz (3.15) Chapter 3. General Experimental Considerations: Temperature Control and Optics 54 where p is the density in moles/cm3, Ai is the molar mass of the fluid, and g = 9.81 m/s2. For T > T c, one can approximate p « pc. For T < Tc, in the two-phase region, corrections have to be taken into account. However, these corrections are small (of the order of < 8% for our experiments, see section 5.4), and are much smaller than the experimental scatter of the compressibility data in the two-phase region. The compressibility can thus be expressed as: Pc (dp Pc (Ap} (3.16) Pi \dpJT Pl9M \&zJt We approximate the slope of the density profile by the density difference corresponding to one interference fringe, divided by the vertical distance in the cell over which this difference occurs. Using the Lorentz-Lorenz relation (see eq. (3.1)), Ap can be expressed approximately as A n 6 n c A n ° (3.17) *P =  Pc (n 2-l)(n c 2 + 2) where An 0 = X/L is the refractive index difference correponding to one fringe. From this relation, kj can be evaluated, because Az can be directly inferred from the measurements of the fringe spacing on film (after correcting for the magnification factor of the lens and the tilt of the reference beam). Chemical Potential Profile along the Critical Isotherm The reduced chemical potential Ap* = P(p) ~ Pc Do P~ Pc (3.18) is evaluated using eq. (3.15). Over the height of the image, the difference Ap = p — pc is very small, so that we can approximate: P~ Pc _ pcgM . •1**1 (3.19) Chapter 3. General Experimental Considerations: Temperature Control and Optics 55 where 6z is the height in the cell as measured from the meniscus, and M. is the molar mass of the fluid. The critical amplitude Do is thus determined by -s Do = —— \Az\ P- Pc (3.20) Diffusivity The diffusivity D is obtained from the image plane interferometric data in a similar fashion as from the focal plane experiment. After a quench from the two-phase region into the one-phase region of the phase diagram, the initial refractive index discontinuity decays as described in eq. (3.8). Near the meniscus the decay can be approximated by eq. (3.10). Thus the time evolution of the fringe spacing close to the middle of the cell can be used to estimate the diffusivity: AnAz = n(z, T) - n(z + Az, r) = X/L (3.21) 2VTTDT Here, An is the initial refractive index discontinuity, Az is the fringe spacing in the middle of the cell, and T is the time. Calling N the number of interference fringes (corresponding to the initial discontinuity), one gets: Dr = (7VAz)2/47r. (3.22) 3.2.4 Gravitational Rounding A possible source of error in all optical experiments is the effect of gravitational round-ing [84, 22]. It limits the accuracy of data in the critical region and determines the minimum distance in temperature to which the critical point can be approached. To see how this effect comes about and how it influences the data, recall eq. (3.4) which describes the amount by which a ray traversing the cell is bent. According to this equation, a ray entering the cell at height z = H will emerge from the cell at the height Chapter 3. General Experimental Considerations: Temperature Control and Optics 56 z = H — h, where h « LO. This ray will therefore probe a slab of fluid of height h; it will not only contain information on the refractive index at height H, but will average over the refractive indices between heights H — h and H. Gravitational effects are important when the change in refractive index over the height h becomes significant. The height averaging becomes larger as the density profile in the cell becomes more strongly curved. It is most noticeable close to the critical point, where the compressibility diverges and thus dn/dz —* oo at the meniscus. The gravitational rounding effect can be made small by choosing a very thin cell, so that L is small. This was the case in our interference measurements: The cell lengths in all experiments were < 2 mm, leading to negligible rounding effects even as close to the critical point as t « 5 x 10~6. Gravitational rounding thus does not play an appreciable role in the interference experiments described in this thesis. The situation is different in the case of the prism experiment, however [80]. Here, the optical path through the cell is of the order of « 0.5 cm, so that gravitational rounding comes into play for the fluids considered here at reduced temperatures of the order t < 10~4. Data taken closer to the critical point are strongly smeared and difficult to correct for gravitational rounding effects. 3.2.5 Light Scattering Experiments Light scattering experiments were performed on the nicotine-water system in order to compare diffusivities obtained by the interference method to ones obtained in a more conventional way. For these experiments, the same optical cell was used as for the fringe method. Figure 3.9 shows a schematic of the optical setup. The light source was a 10 mW He-Ne laser whose intensity was for the most part further reduced to minimize optical heating. The beam was expanded and rendered uniphase by a spatial filter. Lens LI Chapter 3. General Experimental Considerations: Temperature Control and Optics 57 He-Ne Laser Autocorrelator Figure 3.9: Optical setup for light scattering experiments. collimated the emerging beam, and L2 (with a long focal length « 400 mm) focussed the beam to a diameter of « 0.1 mm inside the scattering volume. Polarizers P l and P2 in front and behind the cell defined the polarization of the incident and scattered radiation. In our experiment, the polarization direction was chosen perpendicular to the plane of scattering. The photomultiplier tube was mounted on a turntable which could be rotated around the scattering cell. The scattered light entered the photomultiplier through an iris the diameter of which was variable from 0.5 mm to 6 mm. Far from Tc, where the scattered intensity is low for large scattering angles, the iris was opened up to 2 mm. Close to Tc, it was reduced to 0.5 mm. The photomultiplier tube contained a built-in adjustable lens which forms an image of the scattering volume on the pinhole filter inside the tube. To align the photomultiplier optics, and to assure that the fluid cell was properly positioned at the centre of the turntable, the cell was replaced by a hollow pipette (diameter w 1 Chapter 3. General Experimental Considerations: Temperature Control and Optics 58 mm) containing a strongly scattering substance (latex spheres). The spot illuminated by the laser beam in the pipette provided a bright target for alignment of the photomultiplier lens. Further checks on the correct alignment of the optics were carried out as in Ref. [85]. 3.2.6 The Correlator Our correlator was a Malvern K7023 digital correlator. Its operation and the method of photocount autocorrelation have been discussed in detail in the literature [37, 71, 86], Here, only the features of relevance for the experiments in this thesis are presented. Prior to processing by the correlator, the photon signal was digitized: in each time interval Ta, the number of photons detected by the photomultipier was counted. At any instant, the probability (per unit time) of detecting a photon was proportional to the intensity. The photomultiplier, discriminator and amplifier assembly used was an EMI D307K, which produced pulses of height —1.2V and width 30ns, with a uniform rise and fall time, each pulse corresponding to a single photon. The method of integrating the total signal arriving between two sample time clock pulses had the advantage that no photocounts were lost due to dead time between sample times. The number of photons detected in the sample time Ta was processed by the autocor-relator, which operated in the time domain and used a set of M parallel channels. The operation of the autocorrelator is shown schematically in Figure 3.10. The principle of operation of the correlator is, by taking N samples, to construct the sum J V Cm = J2n(Ti)n(Ti-m), (3.23) t'=l where T,+I — r; = Ta and m lies in the interval 1 < m < M. This is achieved in the following way: The signal accumulated during the time Ta is fed into the first channel, while the Chapter 3. General Experimental Considerations: Temperature Control and Optics 59 M Channels r r Clipping ^ j 0 i « • • i 0 9 9 c, c , Storage C M , CM Autocorrelator Figure 3.10: Schematic diagram of the autocorrelator. Chapter 3. General Experimental Considerations: Temperature Control and Optics 60 contents of the remaining M — 1 channels are shifted down by one position each, the last one being discarded. The signal in the first channel is multiplied with the contents of the other channels and stored, thus building up the autocorrelation function Cm. The frequency attained by the instrument is determined by the total sample length T = MTa, chosen to be 2 — 3 times the correlation time f, which is of the order of microseconds or less. In order to make the multiplications of the correlation function as fast as possible, the signal, before being stored in the shift register, is "clipped", i.e., it is compared to a preset threshold k; if n(0) > k, n*(0) is set to 1, otherwise njt(0) is set to zero. The shift register then contains a set of 0s and Is, and the multiplication operations can be replaced by simple "AND" gates, controlled by the bit in the appropriate shift register: If nfc(mT8) = 1 (with 1 < m < M), then n(0) is added to the storage position m, and if nk(mTa) = 0, nothing is added. It can be shown [71] that the "single-clipping" operation here described does not distort the time dependence of the intensity autocorrelation function. The Malvern K7023 autocorrelator has a store of M = 24, and the sample time Ta can be varied from 50 ns to 1 s. Three additional channels collect the total number of photon counts, the total number of clipped counts and the number of sample times N. The autocorrelator was interfaced to the UBC Amdahl 5850 mainframe via a Z-80 microcomputer (built by the UBC electronics shop). Details of this interface are described in reference [87]. Chapter 4 Experiments This chapter describes the experimental details of the apparatus and materials used in this thesis. The experimental setups and sample cells of the Freon experiments and the nicotine + water experiments are similar to those used in previous experiments [80, 82, 87, 88, 89] and are therefore not discussed in detail here. Only information specific to the individual experiments is given (sections 4.1 and 4.2). The high-pressure cell for the He-Xe experiment was designed and built especially for this thesis and is therefore described in detail (see section 4.3). 4.1 Freon Experiments The CHF 3 ("Freon 23") and the CC1F3 ("Freon 13") used in these experiments were obtained from Matheson Gas Products. The CHF 3 was rated to be 98% pure, and the CC1F3 was rated to be 99% pure. Prism cell experiments were performed on both CHF 3 and CC1F3, using the technique and the equipment described in section 3.2.1. In order to get very accurate data close to the critical point, an image plane interfer-ence experiment was carried out on CHF 3. The sample cell consisted of an aluminium body with two sapphire windows (diameter 1", thickness 1/4") spaced 1.86±0.01 mm apart [88]. Before filling, the cell was evacuated to minimize contamination of the sam-ple. Subsequently the cell was filled with CHF 3 at the critical density. The deviation from critical filling can be estimated by observing the rise or fall of the meniscus between the liquid and vapour phases as the critical point is approached. Our cell was slightly 61 Chapter 4. Experiments 62 overfilled, the deviation from critical density being less than 0.1%. The cell was placed into a two-stage thermostat which controlled the temperature to an accuracy of ±0.2 mK. Its temperature was measured by an HP 2804A quartz thermometer, the probe of which was embedded in the innermost heating stage of the thermostat. The interference pattern is extremely sensitive to changes in ambient room temper-ature, which change the optical path length of the reference arm of the interferometer, but not that of the temperature-controlled sample arm. In order to minimize this source of error, the whole interferometer was contained in a temperature-stabilized box which controlled temperature to better than 10 mK over a day and largely eliminated air tem-perature fluctuations. Still, changes in humidity and barometric pressure lead to a scatter in the diameter data which is somewhat larger than that for the prism cell data. 4.2 Nicotine 4- Water Experiment For this experiment, a flat spectrophotometer cell (Hellma QS 282) of inside thickness 2 mm was filled with a critical mixture of nicotine and water, corresponding to about 40 wt% nicotine [81], and fire-sealed. Nicotine is a clear liquid of similar appearance to water, but with a higher viscosity. It is very hygroscopic and reacts with air, thereby acquiring a brownish tinge. In an effort to have as clean a sample as possible, the nicotine, purchased from Kodak, was purified by distillation under a nitrogen atmosphere at a pressure of « 5 mm Hg, where the boiling point occurs at « 95°C. Immediately after distillation, the nicotine was sealed into the cell together with the appropriate amount of deionized water. From observing the meniscus close to the critical point, we deduce that the cell was critically filled to about ±0.2% of the critical composition. The phase diagram of nicotine + water in the T — x plane is a "closed loop" [81]. Chapter 4. Experiments 63 Phase separation occurs only in the temperature interval from « 60° C to « 210° C. Our experiments were carried out at the lower critical point, where the phase diagram is "inverted", i.e., cooling rather than heating takes the system from the two-phase into the one-phase region (see figure 3.6). Prior to the measurement of order parameter and diffusivity, the critical refractive index had to be determined. Due to the high refractive index of the mixture, our prism cell setup could not be used for this measurement. Instead, the Hellma cell containing the mixture was slowly rotated around its vertical axis in one arm of a Mach-Zehnder interferometer, and the resulting interference pattern was used to calculate the critical refractive index. The experimental method and the results are discussed in Appendix B. In an effort to determine the critical density and the densities of the two coexisting phases close to the critical point, precision densitometry was carried out. The densito-meter consisted of a bulb-shaped glass bubble of volume « 63 ml connected to a pipette. Nicotine-water mixtures in the composition range 30 wt% < xnic0 < 49 wt% were inves-tigated, at temperatures ranging from 55°C to 67°C. The results of this measurement are also presented in Appendix B. Having measured the density and refractive index of the mixture, focal plane interfer-ence experiments and light scattering experiments were carried out. The same sample cell was used for all these experiments. The cell temperature was measured by thermistors embedded in the innermost heating stage of the thermostat. 4.3 High-pressure experiment The He-Xe system was studied through two experiments: in one of them we measured the order parameter and diameter as a function of density in Xe to as high an accuracy as possible, using the prism cell setup. Our objective was to obtain an estimate of the Chapter 4. Experiments 64 critical density and of the variation of pc for samples from different sources. Two different samples were studied: One of them had been purchased from Matheson Gas Products just prior to the experiments and was rated to be 99.995% pure. The other one was obtained from Professor R. Gammon at the University of Maryland. It is currently being analysed for impurities, but the result is not available at the time of this writing. The other experiment used the image plane interference technique, and was carried out in a high-pressure cell designed for pressures up to « 400 atm. The cell design is described in detail in the following section. Two samples were investigated: One of them consisting of pure Xe, the other one of a He-Xe mixture containing « 5% He. Both gases for these experiments were purchased from Matheson Gas products (rated 99.995% pure for both Xe and He). Because of the high pressures involved in the He-Xe experiment, special care had to be taken in designing and dimensioning the equipment that was to come in contact with the gas under high pressure. The gas-handling system was made up of two connected parts: One of them was designed for low pressures only and consisted of the containers of the uncompressed gases and the overflow volumes. The other one was designed for pressures up to 500 atm and contained the gases after compression. 4.3.1 Cell Design The cell was designed for pressures up to « 400 atm, allowing for a safety factor of 5 x. The body was made out of 316 Stainless Steel, and the cell valve was directly worked into the cell body, in an effort to minimize the cell's "dead volume". Several tapped holes (1/2" deep, 1/4-20 thread) in the outside wall of the cell allowed thermistors to be implanted for temperature measurements. Figure 4.1 shows a technical drawing of the cell and Figure 4.2 a photograph. The window construction was based on the design by Poulter [90] and uses the principle of "unsupported area" which employs the high Chapter 4. Experiments Figure 4.1: Technical drawing of the high-pressure cell. Chapter 4. Experiments 66 Figure 4.2: High-pressure stainless steel cell. In the actual experiments, the fill line points vertically, with the valve being in the horizontal. The windows are shown with retaining rings and sealing O-rings in position. Chapter 4. Experiments 67 pressure to force the window against its support (see also [91, 92, 93]). For windows we used sapphire single crystal cylinders (1" in diameter and 1/2" thick), flat to 2A and c-cut in order minimize effects due to the polarization of the laser beam. Their surfaces were coated with a thin film (1/4 A thickness of MgF2) to reduce stray reflections at the windows. The windows were then mounted onto SS support plugs. Two types of plug-window assemblies were made: One set, using a higher plug stem, made the optical path length in the cell 0.195 ± 0.002 cm long, whereas the other, with a shorter stem, made the cell length 1.167 ± 0.002 cm. In order to provide a good seal between the sapphire window and the steel support plug, the plug surface had to be perfectly flat. The plug was lapped on a polishing machine with diamond paste down to 1/4 pm grain size. During the polishing process, the plug was fastened to a 4" diameter SS lapping block in order to prevent curvature of the surface. The smoothness and flatness of the surface were checked optically by observing Newton's rings between the plug surface and a high-quality glass plate, using a sodium lamp. All the plugs were lapped until the surfaces were very shiny, no scratches were apparent, and at most 2 Newton's rings were visible over the whole 1" diameter surface. The windows were then glued to the support plugs using low-vapour-pressure epoxy as a sealant. Care was taken to squeeze out any air trapped in the epoxy. After setting of the glue, the assembly was cured under w 10 atm for about 24 hours. A retaining ring held the window and the plug together in good alignment during the glueing and curing process. No leaks were encountered at the windows when this procedure was carefully implemented. The seal between the window support and the body of the cell was made with a rubber O-ring. The support plug was forced into position by a SS sealing bolt. In order Chapter 4. Experiments 68 Figure 4.3: Components of the windows for the high-pressure cell to ensure good slippage between the bolt and the plug, a thin disk of softer steel was interleaved between the two. This thin disk also contained weepholes for pressure relief. The various components of the windows are shown in Figure 4.3. The force exerted on the bolts during the tightening process deformed the O-rings, so that, even when the tension of the sealing bolts was released, the window plugs could not be easily removed. To facilitate the extraction of the the windows, the inside of the SS support plugs was threaded, so that an extraction bolt could be inserted to pull the windows out. 4.3.2 Pressure Handling System Aside from safety, the main consideration in designing the high-pressure gas handling system was the cost of Xe. As 1 mole of Xe costs about $700, we tried to minimize the Chapter 4. Experiments 69 amount of Xe needed, and built the gas handling system in such a way that all the Xe used for the experiments could be recovered in storage cylinders. To reach pressures of the He-Xe mixture up to 500 atm, two procedures could be used: 1. Mix He and Xe in the correct stoichiometric ratios at a lower pressure and then use a compressor to compress them up to the desired pressure. 2. Compress the two components separately without use of a compressor and then mix them at high pressure. The first procedure has the disadvantage that it is not certain that He and Xe are compressed equally efficiently by the compressor, i.e., that the mixture at the discharge end has the same stoichiometry as at the inlet. Moreover, the compressor cannot be operated efficiently for suction pressures below 500 psi; however, due to the high cost of Xe, only a small gas bottle would be available in which the pressure would quickly drop below the required suction pressure. Thus, an initial precompression stage would be necessary. As the gas out of the precompression stage is compressed into the high-pressure part, the amount of gas in the precompression volume decreases, leading to a pressure drop which will make the compressor more and more inefficient, thus preventing a continuous operation. Our decision was therefore made in favour of the second procedure, which seemed more straightforward and easier to realize. The strategy was to freeze out a well-defined quantity of Xe in a cylinder submerged into liquid air (at which temperature Xe is solid), then add an appropriate quantity of He such that the desired concentration ratio would be reached. Upon warming up of the Xe, the mixture has the correct ratio and (hopefully) the correct pressure to make it critical. Chapter 4. Experiments 70 He-supply Xe-supply to pump <-to air (1000 atm) €H Cell j Vc 0 (35 atm) Figure 4.4: High pressure gas handling system. Components shown in red, yellow and blue are dimensioned for pressures up to 500 atm or more, components in black and green are for low pressures. For sample pressures up to about 300 atm, the (unregulated) pressure of our He bottle (1800 psi) was enough to produce a sufficiently high partial He pressure. For mixtures of higher He concentration, the He would have to be precompressed. This compression could be effected by cooling the He down to liquid nitrogen temperatures, where its density is 3 times as high as at room temperature; in this manner a threefold compression could be obtained. Our gas-handling system consisted of several separate sections, as indicated by the different colours in Figure 4.4. The heart of the system was the "high-pressure part" shown in red (from now on called Vred) containing the cell and two pressure gauges, one Chapter 4. Experiments 71 which went up to 200 atm, the other to 1000 atm. Connected to it were the compression stages for the gases: the Xe freeze-out cylinder Cy e (shown in yellow) and the He com-pression stage C//e (shown in blue). Both freeze-out cylinders were mounted in such a way that they could be immersed into dewars with LN 2 for cooldown. The cylinders were equipped with relief valves (from supplier HIP), set at 7000 psi (« 500 atm). Whereas the outlet of the relief valve on the He cylinder went straight into the room (He being cheap), the outlet of the valve on the Xe cylinder was connected to the storage volume Cs, so that in case of the bursting of the valve the Xe would be instantly recovered. All the components described so far were designed for pressures up to 500 atm. All tubing, valves and containers were made out of 316 SS, and the connections were made with high-pressure connectors. The valves were rotating-stem valves (Autoclave), and the tubing had 1/4" o.d. and 0.083" i.d. We picked the heavy wall tubing in order to keep the gas volume inside the tubing as small as possible. The physical dimensions of the apparatus were, among other things, determined by the large minimum bend radius of the heavy wall tubing (1.25" [94]). The connectors were purchased from Autoclave, and the tubing was prepared in our lab: the joints were coned and threaded, and the connections were tightened at 40 ft lbs [94] in order to make a good seal. The connections to Autoclave components (valves and tees) seemed to cause less trouble than the ones to HIP components (safety valves). Also connected to the high-pressure volume V r ed were the Xe supply and the Xe storage cylinder Cs. Moreover, there was a low-pressure "measurement circuit" (shown in green) which permitted precision measurements of the He-Xe concentrations: It contained a small cylinder (volume » 16 cm3) for freezing out Xe and a large cylinder of w 1000 cm3. This part of the system contained the gases only at low pressures (below 5 atm), and therefore the ideal gas law could be used to estimate densities from pressures. A precision gauge (Heise, for pressures < 35 atm) was used for measuring the partial pressures of He Chapter 4. Experiments 72 Pc Tc Pxe Phe (atm) [•C] [%) [moles/cm3] [moles/cm3] [moles/cm3] 75.7 16.25 95.1 0.00876 0.00833 (~ 0.98/9C) 0.0004 (~ lOatm) 102.8 16.9 89.5 0.00941 0.00842 (~ 0.99/JC) 0.0010 (~ 20atm) 132.0 18.4 84.65 0.01016 0.00860 (~ l.Olpc) 0.0016 (~ 35atm) 184.9 19.3 76.15 0.01138 0.00866 (~ 1.02pc) 0.0027 (~ 60atm) 198.0 19.65 74.56 0.01166 0.00869 (~ \mPc) 0.0030 (~ 65atm) 239.3 20.35 69.64 0.01226 0.00853 (~ 1.00pc) 0.0037 (~ 84atm) 298.3 22.55 64.67 0.01372 0.00887 (~ 1.04/JC) 0.0049 (~ 114atm) 375.7 23.75 59.47 0.01520 0.00904 (~ 1.06pc) 0.0062 (~ 144atm) Table 4.1: Critical parameters of He-Xe mixtures [44]. Here, Pc is the critical pressure, Tc the critical temperature, p*ot the total critical density and xcxe the critical Xe-fraction of the mixture. Phe and pxe are the partial He and Xe densities. and Xe. 4.3.3 Dimensioning of the Gas Handling System In order to calculate the sizes of the freeze-out volumes, we needed an estimate of the pressures, densities and critical temperatures involved. Table 4.1 gives an overview of approximate values of the critical parameters [44] and the He and Xe densities that are needed to reach them. The Xe densities for all mixtures up to 400 atm are only slightly larger than the critical density (and thus the partial pressures of Xe are < 70 atm at room temperature). The required partial pressures of He rise quickly with increasing He concentration. As we owned about 2 moles of Xe, we chose Cxe large enough so that our total amount of Xe could be frozen out and thus compressed. Since the solid density of Xe is p*xe = 3.52 g/cm3, this meant that Cxe « 74 cm3. The total volume of Vr e (*, including the 200atm pressure gauge and the cell (with window spacing = 2mm) was about 33 cm3. The vessel used to compress He had a volume of Vj/e = 115 cm3, the tubing in that Chapter 4. Experiments 73 part of the system being negligible compared to the freeze-out cylinder. The volumes were chosen in such a way as to minimize Cxe while still enabling us to reach mixture pressures of « 400 atm and compositions of « 50% Xe in the cell. 4.3.4 Filling the Cell In order to reach the appropriate composition of the gases corresponding to the desired pressure, the necessary densities of He and Xe were estimated using Table 4.1. By referring to the literature values for isotherms of the pure gases at room temperature [95, 96], the pressures corresponding to these densities could be determined. Before the first filling of the cell, the gas handling system was evacuated for about 2 weeks with a diffusion pump. The required amount of Xe was tben taken out of the storage cylinder by freezing it out into Cxc The partial pressure of Xe corresponding to the amount of Xe frozen out could be checked by warming up Cxe and letting the fluid fill V r e ( f . If the pressure was too high, the gas could be bled back into the storage vessel. As the pressure around Pc is a very weak function of density, it is difficult to estimate the actual Xe density by measuring its pressure. So, in order to be on the safe side, we retained more Xe than necessary. This Xe was then frozen back into Cxe, filling up about 35cm3 of its volume. Subsequently, V r ed and the empty portion of Cx> were filled with He at the desired pressure. If the He bottle pressure was not high enough (as is the case for the larger He concentrations), the He could be precompressed by cooling it to 77K in C// e-When the Xe frozen out in Cxe was allowed to warm up, it mixed with the He, giving the desired mixture. As long as the room temperature was kept well above T c , the gas mixture was homogeneous. In the cell, which was kept at a temperature below T c , demixing into two phases (one He-rich and the other Xe-rich) occurred. Whereas there exists a well-tested procedure for critically filling a pure fluid cell (see section 4.1 and Ref. [88]), filling the He-Xe cell is more complicated. Here, at a given Chapter 4. Experiments 74 temperature, one has to adjust two variables, e.g. density p and Xe concentration xxe to their appropriate values (the value of the third one, pressure, then being determined by the equation of state). As it is extremely unlikely that the correct values of these two variables would be reached at a first attempt, the following procedure was applied to fill the cell iteratively: The cell was filled, at a certain temperature T* < Tc, with fluid at a total density and Xe concentration slightly higher than the desired one. As the cell was overfilled, the Xe-rich phase occupied a larger fraction of the volume. Since the cell valve was located horizontally exactly in the middle of the cell, any bleed swept out fluid from the Xe-rich (heavier) phase, and reduced both the overall density and the overall Xe concentration. So, by iteratively bleeding the cell, the fluid mixture in the cell travelled along the dotted line in the phase diagram indicated in Figure 4.5. Eventually, one reached "the line onto which the manifold of critical points maps for the given temperature T*. At this intersection the cell was critically filled, the meniscus was exactly in the middle of the cell, and both density and concentration were adjusted to their critical value. In order to check that the cell was really critically filled, it was warmed up to a temperature T « Tc, and the position of the meniscus was observed. For example (see the schematic in Figure 4.5), if we start at an overall Xe concen-tration of 91.6% and total density 0.0117 moles/cm3, this fluid mixture decomposes, at 14.5°C, into one phase with xxe = 76% and density pl = 0.0067 moles/cm3 and another phase with xx[ = 93% and density pu = 0.0128 moles/cm3. The bleed then takes the system along the dotted line to an overall density = 0.0093 moles/cm3, overall com-position xcxt = 90%, and pressure Pc = 102 atm. On warming, the system thus prepared passes through a second order phase transition point into the one-phase region. Chapter 4. Experiments 75 Figure 4.5: Phase diagram of the He-Xe system at 14.5°C (semiquantitative schematic): The dashed lines correspond to lines of constant composition. The set of critical points maps onto the bold line. Chapter 4. Experiments 76 4.3.5 Thermal Control of the Cell The cell itself was used as the innermost stage of the thermal regulating system. Because the thermal conductivity of steel is so small, the cell was wrapped with a layer of copper, to which the heater was attached. The cylindrical copper shells were bent out of 1/2" thick copper sheet and then machined to the correct diameter on a lathe. Copper end plates were screwed to it, so that the whole cell was covered with copper sheet (at least 1/4" thick), except for holes for the windows, valve and thermistor screws. The copper cylinder was wrapped with heating foil (MINCO, four sheets, each 2"x9", with 10.6 each). The cell was heated using a feedback circuit as described in section 3.1. A thermistor embedded directly inside the copper sheet served as a "control" thermistor. As the thermistor measured the temperature of the copper sheet, its feedback was fast and temperature oscillations did not pose a problem. The thermal response of the cell as a whole was slow: After a change of temperature, the cell took about 1 hour to reach thermal equilibrium. The cell, consisting of such a large mass of stainless steel, was very slow to respond to heating. The 4 heating foils were connected in parallel. As the total current supplied by the OPS was « 2A, the total heating power available was « 5W. In normal operation, the cell was heated to about half a degree above the temperature of the outer temperature stage. In this case, the heating power was about 0.5 - 1 W. Three thermistors used for temperature monitoring were embedded in different places in the stainless steel body of the cell and measured using Wheatstone bridges. The imbalance of the Wheatstone bridges were amplified by HP nullmeters and monitored on an x-t chart recorder. The inner heating stage of the cell was surrounded by a layer of styrofoam as thermal Chapter 4. Experiments 77 insulator, then by a thin copper cylinder as a heat shield, followed by another layer of styrofoam and the outer temperature control cylinder. The outer cylinder had 3/8" o.d. copper tubing soldered to its outside in a spiral, and water flowing through it from a temperature regulated water bath (FORMA) kept it at a chosen temperature (always between 0.5° and 1°C below the cell temperature). The cell was tested for temperature gradients by attaching one calibrated thermistor to the inside top and another one to the inside bottom of the cell, and feeding the leads out through a styrofoam plug replacing one of the windows. In one experiment, heating the cell to a temperature of about 1°C above the surroundings, we measured the bottom of the cell to be warmer than the top by 2-3 mK. In another experiment, heating the cell to higher temperatures in small steps (imitating the actual experimental conditions), we discovered no systematic temperature difference between top and bottom. The consequences of thermal gradients on the experiment are discussed in Appendix C. Chapter 5 Freon Experiments: Results and Discussion This chapter presents the results of experiments on C H F 3 , a strongly polar fluid, and CCIF3, a weakly polar fluid. We measured the Lorentz-Lorenz function (section 5.1), the order parameter (section 5.2) and the coexistence curve diameter (section 5.3) of these two substances. We found them to exhibit considerable differences, suggesting that dipo-lar interactions play a major role in determining the values of the critical amplitudes and the Lorentz-Lorenz function. On the other hand, the critical exponents were observed to be the same, in accordance with the principle of universality. Sections 5.2.2, 5.4 and 5.5 present measurements on the order parameter, compressibility and critical isotherm of C H F 3 , from which universal critical amplitude ratios were obtained with very high accu-racy. They were found to be in excellent agreement with values obtained from nonpolar fluids and with theoretical results. Finally, section 5.6 describes the determination of the critical temperature from the experiments, and section 5.7 contains a discussion of the results. All experiments on CCIF3 were carried out using the prism cell method; on C H F 3 , data were taken both by prism cell and image plane interference method [23, 97]. 5.1 Lorentz-Lorenz Data Information on the density dependence of the refractive index was obtained from the prism cell experiment, in which the refractive index and density are measured simulta-neously in the one-phase region. The Lorentz-Lorenz function C, as defined by eq. (3.1), 78 Chapter 5. Freon Experiments: Results and Discussion 79 Substance £o(cm3/mole) £i(cm 6/mole 2) £ 2(cm 9/mole 3) £ c(cm 3/mole) CHF 3 6.905 37.6 -2.8 x 103 7.03±0.02 CCIF3 11.735 41.3 -3.7 x 103 11.85±0.02 Table 5.1: Results of a quadratic fit to the Lorentz-Lorenz data of CHF3 and CCIF3. exhibits a weak density dependence along the coexistence curve, and can be expanded as a power series in p: C = C0-rClP + C2p2 + ... (5.1) Figures 5.1 and 5.2 show the Lorentz-Lorenz data for CHF 3 and CCIF3 respectively, and the lines are a quadratic fit to them. The fit parameters are given in Table 5.1, together with the critical value £ c = C(pc). The Lorentz-Lorenz results for CHF3 can be compared to the results of the experiment of Buckingham and Graham [98] who measured the Lorentz-Lorenz function of CHF 3 as a function of density for pressures up to 5 atm. They fit £ to a straight line and found £0 = 7.05 cm3/mole and C\ = 3.4 cm6/mole2. By extrapolation they found for the critical value £ c = C(pc) = 7.08 cm3/mole. Our measurements, which extend up to 50 atm, yield £ c = 7.03±0.02 cm3/mole, in reasonable agreement with their result. From the Lorentz-Lorenz data, the electronic (optical) polarizability ctp can be deter-mined using [99] \imC(p) = ^ N A , (5.2) where NA is Avogadro's number. For CHF3, we obtained ctp to be 2.74±0.02A3, which agrees with the value ctp = 2.8A3 cited in the literature [98, 100]. The polarizability of CC1F3, as measured in our experiment, is ap = 4.65 ± 0.02A3, and thus considerably larger than the value for CHF3. For a polar fluid, the molecular polarizability ap calculated from eq. (5.2) difffers Chapter 5. Freon Experiments: Results and Discussion 80 0.000 0.002 0.004 0.006 0.008 Density (moles/cc) 0.010 Figure 5.1: Lorentz-Lorenz function of CHF 3 as a function of density. The line is a quadratic fit to the data. Chapter 5. Freon Experiments: Results and Discussion 81 oo o co O O ^ o 3 CM +-> O c LL O CO CD CO CD <s> o \ 0 o O \ 0_ P u o o % o o c?°Oo CD CO CD 0.000 0.003 0.006 0.009 Density (moles/cc) 0.012 Figure 5.2: Lorentz-Lorenz function of CC1F3 as a function of density. The line is a quadratic fit to the data. Chapter 5. Freon Experiments: Results and Discussion 82 from the effective polarizability ctp due to dipole-dipole and dipole-induced-dipole in-teractions [28, 30] according to eq. (1.7). For the case of CHF 3, with dipole mo-ment p 0 = 1.65 x 10"18esu [31], polarizability ctp = 2.74A3 and dissociation energy J « 12.5eV [101], these effects enhance the polarizability ap, resulting in the so-called effective polarizability [28, 30] (corresponding to a low-frequency polarizability) which then has the value dp « 3.5A3. For CC1F3, with p 0 « 0.50 x 10"18 esu [32], ap = 4.65A3 and / w 12.5eV [101], the effective polarizability becomes ap —• o7p w 4.67A3. The prism cell experiment also yields the critical density: A fit to the coexistence curve diameter gives the refractive index at the critical temperature, which can be converted into the critical density using the Lorentz-Lorenz relation. For CHF 3, we measured nc = 1.0806(2), from which we obtained pc = 0.5272 ± 0.0015 g/cm3, in close agreement with other experiments [102, 103], which find pc in the range 0.525 — 0.526 g/cm3. For CC1F3, we measured the refractive index to be nc = 1.1009(3), slightly higher than cited in the literature (nc = 1.0996, see Ref [104]). Our measurement thus gives a critical density of pc = 0.582±0.002 g/cm3 and agrees well with the values pc = 0.578 g/cm3 [102] and pc = 0.581 g/cm3 [105] reported in the literature. 5.2 Order Parameter Measurements 5.2.1 Order Parameter of CC1F3 The coexistence curve data obtained in the prism cell experiment on CC1F3 were fitted to the expression Ap* = Bot0(l + Bxt* + B2t2A + B3t3A). (5.3) Some results are shown in Table 5.2. Data were taken in the temperature interval 10 -4 < t < 6.5 x 10-2, and all the fits were performed over the whole temperature interval. For some of the fits the critical exponent ft and the correction exponent A were held fixed Chapter 5. Freon Experiments: Results and Discussion 83 /? A B 0 Bi B 2 B 3 (0.327) (0.5) 1.645 0.654 -1.00 (0.0) (0.325) (0.5) 1.619 0.733 -1.14 (0.0) 0.3298 (0.5) 1.681 0.554 -0.80 (0.0) (0.327) 0.43 1.632 0.555 -0.63 (0.0) (0.327) (0.5) 1.639 0.754 -1.74 1.65 0.3257 (0.5) 1.621 0.838 -2.08 2.20 Table 5.2: Results of the coexistence curve fit of CCIF3. Parameters shown in parentheses were kept fixed for the fit. at the theoretically expected values, 0.325 < /? < 0.327 and A = 0.5. When /? is treated as a free parameter, and two correction terms are taken into account, the value of /? is slightly larger than theoretically expected. However, when one more correction term is taken into account, the value of /? obtained by the fit agrees very well with theory. The correction to scaling exponent A, when fitted as a free parameter, favours a value of 0.43±0.01, which is somewhat smaller than the theoretically predicted value. Similar values have been reported for ethylene and hydrogen [89, 106]. A plot of the data from two independent runs (total of 204 data points) is shown in Figure 5.3, where logi0(A/)*/r/3) is plotted versus the reduced temperature t. The line corresponds to a fit which treats /? as a free parameter with a fitted value of 0.3257, A fixed at 0.5, and includes three correction to scaling terms. The fit is seen to describe the data quite well. 5.2.2 Order Parameter of CHF 3 The order parameter data obtained from both the fringe and the prism cell experiment were fitted to the expression (5.3). Table 5.3 shows several fits to this equation, with various combinations of the amplitudes and exponents as free parameters. Fits were performed for both experiments separately, and then for all the data taken together. For Chapter 5. Freon Experiments: Results and Discussion 84 Figure 5.3: log-log plot of coexistence curve data of CC1F3. The curve corresponds to a fit with three correction to scaling terms and with A = 0.5 held fixed and /3 = 0.3257 a free parameter. Chapter 5. Freon Experiments: Results and Discussion 85 A Bo B2 B3 Image plane interference data 0.326 - 1.75 - - -(0.325) (0.50) 1.717 0.96 -1.85 -(0.327) (0.50) 1.747 0.85 -1.59 -0.331 (0.50) 1.806 0.64 -1.08 -(0.327) 0.42 1.748 0.65 -0.77 -(0.327) (0.50) 1.739 1.03 -3.39 5.3 0.3287 (0.50) 1.770 0.85 -2.34 2.9 (0.327) 0.42 1.731 0.68 -1.00 0.5 Prism cell data (0.325) (0.50) 1.723 0.87 -1.41 -(0.327) (0.50) 1.748 0.80 -1.31 -0.332 (0.50) 1.813 0.60 -0.70 -(0.327) 0.43 1.742 0.60 -0.60 -All data (0.327) (0.50) 1.747 0.85 -1.55 -0.331 (0.50) 1.810 0.62 -1.01 -(0.327) 0.40 1.727 0.60 -0.60 -0.326 0.37 1.703 0.60 -0.55 -(0.327) (0.50) 1.739 1.03 -3.44 5.5 0.3287 (0.50) 1.768 0.86 -2.45 3.3 Table 5.3: Coexistence curve fits for C H F 3 . The first fit was done in the tempera-ture range 10~6 < t < 10~4. All other fits were done over the temperature range 10 -6 < t < 4 x 10~2. Parameters in parentheses were held fixed for the fit. Chapter 5. Freon Experiments: Results and Discussion 86 the fringe data, a fit to a pure power law was performed over an "inner" temperature range 10~6 < t < 10~4, where the corrections to scaling in eq. (5.3) are expected to contribute on the order of 1% and are therefore smaller than the statistical scatter of the data. (At t=10~4, the number of missing fringes is about 50, and the resolution is of the order of half a fringe). The value of ft = 0.326 ± 0.001 found in this temperature interval is in excellent agreement with the theoretical value 0.325 < ft < 0.327 (Ref. [11]) and with other experimental data [89, 17]. The prism cell data do not extend close enough to Tc to allow a similar fit to that data. Fits over the whole temperature interval 10~6 < t < 4 x 10 -2 consistently yield a higher value for ft. When two correction to scaling terms are included in the fit, we obtain ft = 0.331 ± 0.001 for the fringe and ft = 0.332 ± 0.003 for the prism data; both of these values are slightly larger than the theoretical value. Adding a third correction term results in ft = 0.329 ± 0.001 for the combined data, which is close to the theoretically predicted value. Figure 5.4 shows the coexistence curve data from both experiments. The fringe data consist of 604 points from four individual runs, and the prism data consist of 70 points from two runs. As in figure 5.3, the leading t13 temperature dependence has been divided out, and logi0(A/)*/<'J) is plotted versus t. For reduced temperatures t > 10"4, the data exhibit a clear deviation from a horizontal straight line, indicating that a pure power law is insufficient to describe the data and demonstrating the importance of the correction to scaling terms. The curves in the figure correspond to fits with the exponents fixed at ft = 0.327 and A = 0.5, and with the amplitudes as free parameters. The dashed curve is a fit with two, and the solid curve with three correction terms. These fits are seen to describe the data very well; however, the tendency of the data to favour values of ft larger than the theoretical value, even when more correction to scaling terms are included, suggest that the correction to scaling series for Ap* is not adequate over the Chapter 5. Freon Experiments: Results and Discussion 87 reduced temperature Figure 5.4: Coexistence curve data of CHF 3, from interference experiment (•) and prism cell experiment (o). The curves correpond to fits with /3 = 0.327 and A = 0.5. The dashed curve is a fit with two, the solid curve with three correction to scaling terms. Chapter 5. Freon Experiments: Results and Discussion 88 large temperature range covered in this work. In order to obtain a more satisfactory description of the coexistence curve in the whole temperature interval, the crossover from the asymptotic scaling behaviour near the critical point to the regular behaviour away from the critical point should be taken into account [57, 107]. As was the case for CCIF3, the correction to scaling exponent A for C H F 3 also exhibits a trend to values smaller than theoretically expected. We find A = 0.42 ± 0.02 for CHF 3. This may be another indication that the correction to scaling series is not a good approximation for the large temperature intervals covered in our experiments. It may also mean that A is indeed smaller than 0.5, as suggested by other experiments [106]. In addition to the statistical error in Bo due to the scatter in the data, there is an additional systematic uncertainty in the fringe data due to the uncertainty in the cell thickness. To make the data of the prism and the interference experiments coincide, Lceu was fixed at 1.865 mm, which is equal to the measured thickness within our experimental error. The resulting values of the critical amplitude Bo depend on the value of /? and to a lesser extent on how many correction to scaling terms are taken into account. The inter-ference data, extending further into the critical region, furnish a more reliable estimate of the critical amplitude Bo. Averaging over the results of various fits of the CHF 3 fringe data, we obtain B0 = 1.743±0.003 when keeping /3 = 0.327 fixed, and B0 = 1.722±0.003 when keeping /3 = 0.325 fixed. The error on Bo is thus very small for fixed 0. These values are in good agreement with previous measurements [102], which were fitted using 0 = 0.324 and found BQ = 1.74 [108]. Chapter 5. Freon Experiments: Results and Discussion 89 0.000 0.004 0.008 0.012 reduced temperature 0.016 Figure 5.5: Coexistence curve diameter pd as a function of t for C H F 3 . Data from interference experiment (•) and prism cell experiment (0). The dashed line corresponds to a straight-line fit to data with <>8x 10"3. 5.3 Coexistence Curve Diameter for C H F 3 and CC1F 3 The coexistence curve diameter of CHF 3 from the prism and fringe experiment are shown in Figure 5.5. Each point of the fringe data in the figure is the average of from 5 to 30 individual data points. In this way, the scatter of the original data, due to fringe counting uncertainties and variations in room pressure and humidity, was averaged out. (The fits, however, were performed on the raw, unaveraged, data.) The dashed line is a straight-line fit to data with reduced temperatures t > 8 x 10-3. Both data sets agree very well for reduced temperatures t > 10-3. For small reduced Chapter 5. Freon Experiments: Results and Discussion 90 temperatures, they do not quite coincide, but show the same general behaviour: Close to Tc both data sets bend downward from a straight-line fit to the data far from Tc. This behaviour is consistent with the expected <1_Q anomaly [68]; according to the model of Goldstein et al, the downward bend is indicative of repulsive three-body forces between the molecules [26, 28]. Figure 5.6 shows a plot of the coexistence curve diameter of CC1F3. The dashed line in the figure corresponds to a straight-line fit to the data for reduced temperatures t > 8 x 10~3. The straight line seems to describe the data very well. However, as is obvious from the more detailed graph of data with t < 2 x 10~2, the CC1F3 data bend away from the straight line close to Tc. Even though this effect is not as pronounced in CC1F3 as it was in CHF 3, there is still evidence for the critical diameter anomaly, and, as in CHF 3, the downward bend of the data from the straight line can be interpreted, in the framework of the Goldstein model [26], as due to repulsive three-body interactions between the fluid molecules. Table 5.4 shows fit results of the diameter data of CHF 3 and CC1F3 to expres-sion (2.36). To get a value of the amplitude Ai, the data were fitted to a straight line in the outer temperature interval t> 8 x 10-3. The fits give Ax = 1.31 ±0.01 and 1.33±0.06 for the CHF 3 fringe and prism cell experiments respectively, and A\ = 0.860 ± 0.002 for the CC1F3 prism experiment. The remainder of the table contains fits over the total tem-perature range of the experiments. When both A\ and A\-a are fitted as free parameters to the CHF 3 data, the prism data give a much smaller error in the amplitudes than the fringe data. Due to the scatter in the CHF 3 data, fits with additional terms in eq. (2.36), corresponding to corrections to scaling, do not yield any further useful information. Chapter 5. Freon Experiments: Results and Discussion 91 LO CN O LO O -CD +-> CD E eg O LO CD CD 6 "l 1 r 0.000 0.015 0.030 0.045 0060 0.075 reduced temperature (b) P o 0.000 0.005 0.010 0.015 reduced temperature 0.020 Figure 5.6: Coexistence curve diameter pd of CC1F3. (a) whole temperature range, (b) temperature range t < 2 x 10"2. The dashed line corresponds to a straight-line fit to the data with t > 8 x 10"3. Chapter 5. Freon Experiments: Results and Discussion 92 Ai Ai_ Q Ao C H F 3 data outer temperature range (t> 8 x lO"3) Interference data: 1.31 ±0.01 (0.0) 1.0022 ± 0.0002 Prism cell data : 1.31 ± 0.06 (0.0) 1.002 ± 0.01 total temperature range (10~6 < t < 1.8 x lO"2) Interference data: 1.40 (0.0) 1.001 (0.0) 0.94 1.001 -1.4 ±1.0 1.8 ±1.0 1.001 ±0.001 Prism cell data: 1.46 (0.0) 1.006 (0.0) 0.95 1.0002 0.32 ± 0.04 0.69 ± 0.02 1.00014 ± 0.00002 CC1F3 data outer temperature range (t > 8 x lO"3) total temperature range 10"4 < t < 6.5 x 10"2 0.860 ± 0.002 (0.0) 1.0010 ± 0.0002 0.869 (0.0) 1.0005 (0.0) 0.631 0.9992 0.482 0.261 1.00005 Table 5.4: Fit results of the coexistence curve diameter of C H F 3 and CCIF3. Parameters shown in parentheses were kept fixed for the fit. Chapter 5. Freon Experiments: Results and Discussion 93 5.4 Compressibility of C H F 3 The isothermal compressibility /ey, as given by eq. (2.4), is measured from our data by measuring the fringe spacing in the middle of the cell (i.e., close to the meniscus in the two-phase region), where the gradient of the density is largest and where the average density equals the critical density (see eq. (3.16)). Figure 5.7 shows a plot of the compressibilities of CHF 3 in both the one and the two phase regions. In the one phase region, a least squares fit of the logarithm of /cJ as a function of the logarithm of t (shown as a solid line in the figure) yields an exponent 7 + = 1.230(8) and a critical amplitude TQ — 0.058(3). The critical temperature was determined as a free parameter from the compressibility data and was found to agree with the value determined in the coexistence curve fit to within 0.2 mK. The figure shows that the straight line is sufficient to describe the data. Correction to scaling terms in this temperature range are expected to be of the order of 1% and cannot be distinguished from the statistical scatter of the data. In the two-phase region, data extraction is much more difficult, due to a lower cur-vature of the interference pattern (i.e., a smaller compressibility), and thus the data in this region are somewhat more scattered. A two parameter fit for T < TC, for which the critical temperature was kept fixed at the value found in the one-phase region, yields 7~ = L18(3), which is substantially lower than the theoretical value ( 7 " = 7+ = 1.24). In order to determine the critical amplitudes TQ consistently, data in both regions were fitted by power laws with 7 = 1.23 (corresponding to the exponent in the one-phase region) and 7 = 1.24 (the theoretical value). For'these fits, the critical temperature was held fixed at the value found in the coexistence curve fit. The results are summarized in Table 5.5. Chapter 5. Freon Experiments: Results and Discussion 94 1 1 I I I I I l | 1 1 I I I I I l | 1 1 I I I I I I J 10'6 10"5 10~4 10"3 reduced tempera tu re Figure 5.7: Compressibility of CHF 3. The data for T < Tc (two-phase region) are represented by open circles (vapour phase) and open triangles (liquid phase). Data in the one-phase region (T > Tc) are represented by solid circles, and the curve is a power-law fit to them, with exponent 7 = 1.230. Chapter 5. Freon Experiments: Results and Discussion 95 7 + r+ T> Tc 1.230 ± 0.008 0.058 ±0.005 (one-phase region) (1.23) 0.058 ± 0.003 (1.24) 0.052 ± 0.002 7 " r-T <TC 1.18 ±0.03 0.019 ± 0.008 (two-phase region) (1.23) 0.012 ± 0.002 (1.24) 0.011 ±0.001 Table 5.5: Compressibility fits of C H F 3 data. Parameters in parentheses were held fixed for the fit. 5.5 Critical Isotherm of C H F 3 In order to extract the critical amplitude Do of the critical isotherm for C H F 3 (see eq. (3.20)), we follow the approach suggested by Pestak and Chan [17]: At temperatures close to Tc the distance Az of each of several interference fringes from the meniscus was measured. Then, for each temperature, the quantity D t ~ RjApy ~ -pI(Apy ( 5 - 4 ) was calculated for the reduced density Ap=\p(Az)-pc\/pc (5.5) corresponding to this fringe. The value of 8 was determined using the scaling relation 7 = 0(8 — 1) and the values of 7 and 0 obtained earlier in the same experiment. On the critical isotherm, all the values of Dt thus calculated should coincide (this will give the value of the critical amplitude Do). However, for data taken at temperatures away from T c, the functional relation Ap = Do\(p — pc)/Pc\S is not correct: below T c, the density profile is "steeper" than on the critical isotherm and thus the values of Dt, as calculated Chapter 5. Freon Experiments: Results and Discussion 96 o CO - 0 . 0 1 2 - 0 . 0 0 9 - 0 . 0 0 6 - 0 . 0 0 3 0 . 0 0 0 0 . 0 0 3 0 . 0 0 6 0 . 0 0 9 temperature (T-T c) Figure 5.8: Example of the graphical method used for extracting D0. The lines correspond to interference fringes with a fixed density difference Ap from the critical density. The lines intersect at the critical temperature, determining the critical amplitude D0. from formula (5.4), are smaller than the critical value D0. At any given temperature T < Tc, the values of Dt decrease with increasing distance Az from the meniscus. Above Tc, the density profile is "less steep", and therefore the Dt values are larger than D0. In order to assure that gravitational rounding does not play an appreciable role, only data with Ap > 4 x 10-2 were taken into account, for which beam-bending errors are less than 0.1% [22]. An antisymmetric density profile was assumed, and the Az used for the evaluation was the mean of the vapour and liquid values. The data of one run (with 6 = 4.80) are shown in Figure 5.8. As can be seen, lines corresponding to various values of Ap indeed do intersect at the critical temperature, and this furnishes the amplitude Chapter 5. Freon Experiments: Results and Discussion 97 D 0 . Using 6 = 4.76 (corresponding to ft = 0.327,7 = 1.23), one obtains D0 = 3.5 ± 0.2, whereas for 8 = 4.80 (corresponding to ft = 0.327,7 = 1.24), one obtains D0 = 3.85±0.3. 5.6 Critical Temperature of CC1F3 and C H F 3 In the data evaluation of the prism cell experiments of CC1F3 and CHF 3, the critical temperature was determined as a free parameter of the coexistence curve fits. In the CC1F3 experiment, in which two coexistence curve runs were performed, Tc was found to be 28.104 ± 0.001°C, considerably lower than the values cited by other researchers (Tc w 28.400°C [102], Tc « 28.715°C [104]). The reason for this difference might be attributed to the presence of different impurities in the system [109]. In the CHF 3 prism cell experiment, in which three runs were performed, we found Tc = 26.008 ±0.007°C, higher than the value cited in the literature (Tc « 25.8FC [102]). For the evaluation of the diameter data, the critical temperature was held fixed at the value obtained from the coexistence curve fit for each particular run. In the interference experiment on CHF 3, the critical temperature in each run was determined in three different ways: from a least-squares fit to the coexistence curve data, from a power-law fit to the one-phase compressibility, and from the intersection of Dt—lines belonging to different densities near the critical isotherm. The three values of Tc thus obtained agree to within 0.2 mK, which indicates that the results for the critical amplitudes and exponents are quite consistent. This suggests also that the system was in thermal equilibrium. Interference data on C H F 3 were taken over a period of about 8 months in 11 consec-utive runs. During this time, we observerd a steady drift of the critical temperature at a rate of approximately 4mK/month, from 26.01°C initially to 26.04°C. A run performed one year before this set of experiments, using the same sample, found Tc « 25.80°C. Chapter 5. Freon Experiments: Results and Discussion 98 0.0 2.0 4.0 6.0 T i m e (months) Figure 5.9: Drift of the critical temperature of CHF 3 as a function of time, as measured in the interference experiment. Figure 5.9 shows a graph of the critical temperature as a function of time in the CHF3 interference experiment. The temperature stability was regularly checked by a triple point cell and was found to be better than 2 mK/year. This temperature drift seems to have no observable effect on the critical amplitudes. Whenever several data sets were evaluated together, their difference in critical temperature was corrected for. We suspect that chemical reactions of the CHF 3 or impurities coming out of the cell walls or the windows may be responsible for the drift in critical temperature, and also for the difference between our values of Tc and literature values for CC1F3 and CHF 3. 5.7 Discussion The prism cell experiment and the interference experiment are two equivalent methods of measuring the order parameter and the coexistence curve diameter of a fluid. The prism Chapter 5. Freon Experiments: Results and Discussion 99 5.7 Discussion The prism cell experiment and the interference experiment are two equivalent methods of measuring the order parameter and the coexistence curve diameter of a fluid. The prism cell method has the advantage of measuring directly the densities of liquid and vapour phase as a function of temperature. Close to the critical point, however, there is some uncertainty in the data due to gravitational rounding [22]: As the laser samples regions of strongly varying density in the cell, one observes a continuous distribution of deflection angles. This makes the determination of the liquid and vapour densities at the meniscus difficult and limits the temperature range accessible to this experiment to t > 3 x 10 -5. In contrast, the interference method is much less affected by gravitational rounding [22]. Due to the short path length of the C H F 3 sample cell, the error introduced by beam bending is negligible in comparison to the statistical scatter: At reduced temperatures as small as t = 2 x 10~6, gravitational rounding introduces an error of only about 1/5 fringe close to the meniscus, which is smaller than the accuracy with which fringes can be measured close to the critical point. For the C H F 3 experiment, in which order parameter and coexistence curve diameter were measured both by the fringe and the prism cell method, the two experiments give consistent results. From the results of the interference experiment of C H F 3 , some amplitude ratios can be calculated. Table 5.6 presents a collection of the various critical amplitudes obtained in fits with different critical exponents, and Table 5.7 shows the corresponding values of the critical amplitude ratios. The experimental values are in good agreement with theoretical predictions [12], independent of the precise values of exponents one chooses for the fits. There is excellent agreement between the amplitude ratios obtained by different choices of the critical exponents. Our results are not accurate enough, however, to decide whether high-temperature series expansions or explicit implementations of the Chapter 5. Freon Experiments: Results and Discussion 100 p 7 6 Bo r + A 0 r0- Do (0.327) (0.327) (0.325) (0.325) (1.23) (1.24) (1.23) (1.24) (4.76) (4.80) (4.79) (4.82) 1.743 ± 0.003 1.743 ± 0.003 1.722 ± 0.003 1.722 ± 0.003 0.058 ± 0.003 0.052 ± 0.002 0.058 ± 0.003 0.052 ± 0.002 0.012 ± 0.002 0.011 ±0.001 0.012 ± 0.002 0.011 ±0.001 3.5 ±0.2 3.85 ± 0.30 3.75 ± 0.30 4.1 ±0.3 Table 5.6: Critical amplitudes of C H F 3 . Exponent values in parentheses were kept fixed for the fit. r^ /Tp- DQT^B^ Our data (0 = 0.327,7 = 1-23) 4.8 ±0.6 1.64 ±0.12 (0 = 0.327,7 = 1.24) 4.8 ± 0.6 1.61 ± 0.14 (0 = 0.325,7 = 1.23) 4.8 ± 0.6 1.70 ± 0.14 (0 = 0.325,7 = 1.24) 4.8 ± 0.6 1.69 ± 0.14 Pestak et al. N 2 4.8 ±0.6 1.71 ±0.5 Ne 4.8 ± 0.8 2.05 ± 0.8 Theory high-temperature series 5.07 1.75 e-expansion 4.80 1.6 Table 5.7: Critical amplitude ratios Chapter 5. Freon Experiments: Results and Discussion 101 renormalization group (e-expansions [12]) yield the better values of the ratios. We also find good agreement with the results of Pestak and Chan [17] from mea-surements on N 2 and Ne. For the product DQTQBQ"1 the uncertainty in our results is lower than theirs due to the fact that close to the critical point, their data are strongly influenced by gravitational rounding which causes a large error in the amplitude DQ. Our experiment, being less affected by gravitational rounding errors, determines DQ to much higher accuracy, resulting in a more precise value of the amplitude ratio. In order to check the self-consistency of our CHF 3 interference data, the critical tem-perature was determined separately from fits to the coexistence curve and compressibility data and from the intersection of the lines of constant density as used for the evaluation of the critical isotherm. The values of TC obtained by these three different methods agree to within 0.2 mK. A discrepancy of the TC values can be an indication of gravitational rounding or insufficient equilibration time between temperature steps. We thus conclude that our data are to a large extent free of these errors. Also, capillary effects do not play an appreciable role in CHF3; they would manifest themselves as a smearing out of the meniscus separating the liquid and the vapour phase. On our films, the meniscus is always very narrow, which indicates that there is negligible wetting of the sapphire windows, even far from the critical point. The compressibility data of CHF 3 do not extend far enough away from TC to detect any deviation from pure power-law behaviour, but the evaluation of the coexistence curve clearly shows the importance of correction to scaling terms. Even though three correction terms are sufficient to describe the data in the temperature interval 10 -6 < t < 2 x 10"2, the large variations in the amplitudes B2 and B3 from fit to fit indicate that using a power law series for the evaluation is not satisfactory any more, and that the data should instead be evaluated using crossover theories [107, 57]. However, when both /3 and A are kept fixed at their theoretically expected value, the amplitude B0 changes very little (by Chapter 5. Freon Experiments: Results and Discussion 102 less than 0.5%) when more correction to scaling terms are included. Thus the leading critical amplitude can be reliably extracted independent of the exact behaviour of the coexistence curve far from critical. The absolute value of the critical amplitude Bo in CHF3 is considerably higher than the ones measured for nonpolar gases [17, 89, 110, 111], whereas Do is appreciably lower [17]. This is probably due to the fact that in the polar gas fluoroform the in-termolecular interactions are of a different nature than in nonpolar fluids, and thus fluo-roform is not expected to obey the "principle of corresponding states" as well as nonpolar fluids do [1]. The same trend has also been observed for H2O and D2O [107]. However, even though the absolute values of the critical amplitudes deviate from the ones found in nonpolar fluids, polar fluids still furnish the same amplitude ratios as nonpolar ones, in accord with the universality principle. The evaluation of the coexistence curve clearly shows the importance of corrections to scaling, for both CHF3 and CC1F3. Even though three correction terms are sufficient to describe the data over the whole temperature interval, the correction amplitudes B2 and Bz vary greatly with ft and A. Also, the value of ft depends on the temperature interval considered and on the number of correction terms. This suggests that the correction to scaling series is not sufficient over the large range of reduced temperatures studied in these experiments. For both fluids, the diameter data close to Tc exhibit a deviation from the law of rectilinear diameter towards smaller densities, in agreement with the model of Goldstein et al [28, 26] for systems with repulsive many-body interactions. The diameter slope of CC1F3, as obtained from a straight-line fit to the data with t > 8 x 10 -3, gives A\ = 0.86 and has thus a value very similar to those observed in nonpolar fluids, which typically have A\ values between 0.5 and 1.0 [28]. In contrast, CHF 3 has a diameter slope of Ai = 1.32 ± 0.01, considerably larger than those for nonpolar gases. According to the Chapter 5. Freon Experiments: Results and Discussion 103 theory of Goldstein et a/., A\ should increase linearly with the molecular polarizability ctp. Whereas this prediction holds well in the case of CCIF3, the polarizability of C H F 3 is in fact smaller than that of most nonpolar fluids [99]. Due to its larger dipole moment, the polarizability of C H F 3 becomes considerably enhanced by dipole-dipole and dipole-induced-dipole interactions. (In contrast, taking the dipole-dipole and dipole-induced-dipole interactions into account has a negligible effect on the polarizability of CCIF3.). Even when the enhancement of the polarizability due to dipole-dipole and dipole-induced-dipole interactions is taken into consideration, C H F 3 still does not exhibit the same relationship between dppc and A\ as one would expect from Goldstein's theory. A similar discrepancy is observed for H 20 which also has a high value of A\ [112] accompanied by a small polarizability. This deviation may be due to hydrogen bonding, which is known to play an important role in systems containing electronegative atoms (like 0, N, F, CI) together with hydrogen atoms [30]. These hydrogen bonds strongly influence the intermolecular potentials. A more subtle point is the behaviour of the Lorentz-Lorenz function close to the critical point. Both diameter and refractive index have been predicted to have a critical anomaly [113, 114], both with the same exponent 1 — a, which makes it difficult experi-mentally to distinguish between the two phenomena. We observe a weak critical anomaly in the diameter close to Tc for both CHF 3 and CC1F3, of the same order of magnitude as observed in other experiments on nonpolar substances [28]. This effect, which seems to be more pronounced in CHF 3 than in CCIF3, may result from the fact that C H F 3 is more polar than CCIF3. No definite experimental evidence exists for the anomaly of the refrac-tive index at optical frequencies [115]. Our method of determining the Lorentz-Lorenz function does not allow us to perform the precise measurements close to Tc which would detect such an anomaly. The Lorentz-Lorenz data of neither C H F 3 nor CC1F3 show such an anomaly with our experimental accuracy. Even though the Lorentz-Lorenz data of Chapter 5. Freon Experiments: Results and Discussion 104 C H F 3 show a slight increase close to the critical point (see figure 5.5), it is doubtful whether this is a real effect or just a statistical scatter in the data. We therefore assume C to have the form of eq. (5.1) and not to exhibit any singularity in the critical region. Chapter 6 Nicotine + Water Experiment: Results and Discussion On the binary liquid system nicotine + water, focal plane interference measurements and light scattering experiments were carried out. In this way, the order parameter and the diffusivity were measured. Section 6.1 presents the order parameter measurements. The data suggest that corrections to scaling are of relatively minor importance in this system. Section 6.2 compares the results of the diffusivity experiments using the interfer-ence method and the light scattering method. In the interference method, a systematic dependence of the diffusivity on concentration was observed. The light scattering data were used to extract the critical exponent of the viscosity. In section 6.3 potential errors due to gravitational effects and nonequilibrium effects are discussed. Finally, section 6.4 presents a discussion of the results. 6.1 Order Parameter For binary liquids, the order parameter Ax* is defined as the difference in concentration x of one of species (say nicotine) between the two coexisting phases I and II: Ax* = x[ico - xnL = B^{1 + Bxt* + ...) (6.1) There is no definite agreement in the literature on whether x should be picked as the mole fraction, mass fraction or volume fraction [116]. Usually, one chooses the one that produces the most symmetric phase diagram. Since our experiments did not measure xi\ic0 and xnfco separately, but only their difference, we could not make a decision based 105 Chapter 6. Nicotine -f Water Experiment: Results and Discussion 106 on our data; but since the nicotine + water phase diagram looks very symmetric when expressed in terms of mass fraction [81], we decided to define our order parameter in wt%. This choice has no influence on the exponent ft [116], but affects the values of the amplitudes. The order parameter Ax* is related to the refractive index difference between the coexisting phases, which is the quantity measured in our experiments. According to the data of Campbell et al. [81], the refractive index n varies linearly with composition xm c o. in a region x^ico — 0.3 < x m c o < xcnico + 0.3 around the critical composition, so that we can write A**=(9,A" »ith (!)c=4-72- <6-2> An is related to the the number of recorded interference minima Nm by An = (JV m — l/2)\/L (see section 3.2.2), where A = 6328Ais the laser wavelength and L = 0.2 cm the cell thickness. On the nicotine + water system, data were collected during a period of over two years. The order parameter data could be grouped into six data sets. Within each set the temperature was measured using the same thermistor, but because different thermistors (or different Wheatstone bridges or batteries) were used in different sets, the critical temperatures varied somewhat, between 61.37°C and 61.43°C. Within each data set, the critical temperature was determined by a power law fit to the order parameter data, and when fitting several data sets together, corrections were made for their differences in (apparent) critical temperature. Figure 6.1 shows a log-log plot of Ax* as a function of reduced temperature, in the reduced temperature interval 10"6 < t < 2 x 10-2. Ax* was fitted to the expression 6.1 with various amplitudes and exponents as free parameters. The results are shown in Table 6.1. In the first two fits to a pure power law (i.e., Bi = 0 in eq. (6.1)) the Chapter 6. Nicotine + Water Experiment: Results and Discussion 107 • D w r B OS o B • • I I i I 111 l l | I I I I In i j i I I I MTTf '™l I I I IM!|"""I T T T m i 10"6 10"6 10"4 10'3 10"2 10"1 reduced temperature Figure 6.1: log-log plot of the order parameter Ax* of nicotine + water as a function of reduced temperature (six data sets). 0 Bo A Bi (0.325) 1.644 - -(0.330). 1.691 - -0.3153 1.557 - -(0.325) 1.692 (0.5) -0.403 0.3292 1.750 (0.5) -0.535 Table 6.1: Results of fits of the order parameter of the nicotine + water system. Param-eters shown in parentheses were held fixed for the fit. Chapter 6. Nicotine + Water Experiment: Results and Discussion 108 <0. X < o ti) o ID CN O ID CN O ID CM CN cr- O O o CN O ID o I D ID CN 10 -6 P - %S>5 £ = 0.3I5 • D c S D V D » ° d S d | 5 S D D 0 ° (P Q D I I I l l | 1—I M I I l l | 1—I I I I 1111 1—I I I I 11| | 1—I I I I 11| | f 10"6 10'4 10"3 10"2 10"1 reduced temperature Figure 6.2: log-log plot of Ax*/t^ with ft = 0.3153, as a function of reduced temperature. exponent ft was held fixed at values obtained in other experiments [82, 117] or expected theoretically. In the third fit ft was treated as a free parameter and was found to be ft = 0.3153, considerably lower than values obtained in other experiments on binary liquids [116]. Figure 6.2 shows a plot of Ax*ft1* as a function of reduced temperature, with ft = 0.3153. The lines drawn into the graph indicate slopes that the data would exhibit for exponent values ft = 0.325 and ft = 0.330. Clearly, when fitting the data by a pure power law, ft = 0.315 gives the most satisfactory result. Introducing one correction term B\t0+A leads to a value of ft = 0.329 in much better agreement with the results of Chapter 6. Nicotine + Water Experiment: Results and Discussion 109 other researchers. The amplitude B\ is then found to be negative. Including one more correction term may make B\ positive, but the statistics of our data do not justify fitting so many free parameters. 6.2 Diffusivites 6.2.1 Fringe Data As discussed in section 3.2.2, the diffusivity can be obtained from a measurement of the maximum deflection angle 6max as the system is quenched from a temperature T,- in the two-phase region to a temperature Tf in the one-phase region (see also figure 3.6). If one plots K6^ax (where K = (Z/An)2/47r, and An is the initial refractive index discontinuity) as a function of time, the data points in a given run are expected to follow a straight line. The slope of this line determines the diffusivity D(Tj). Figure 6.3 shows examples for three different runs, all with tf = (Tc - Tf)/Tj « 1 . 5 x 10"4, but with different initial temperatures T,- and consequently with different initial refractive index discontinuities An and order parameters Ax*. The data are seen to be well described by straight lines in the time interval considered (< 15 hours after the quench), indicating that the approx-imations made in section 3.2.2 are indeed valid. Thus, the slopes are well defined and D(tf) can be extracted quite accurately. However, as is seen from figure 6.3, the slopes of the runs with different initial temperature T,- (or equivalently, different An,) do not agree, indicating that D(tj) depends also on the the initial composition discontinuity Ax* across the meniscus in the two-phase region. Thus D is in fact concentration dependent (in contrast to the assumption in eq. (3.7), and this manifests itself in our experiment as a dependence of D on the order parameter, so that D = D(Ax*,tj). Figure 6.4 illustrates the dependence of D on the order parameter Ax* for the data taken at tf = 1.5 x 10-4: D is seen to increase with decreasing composition discontinuity Ax* across the meniscus. Chapter 6. Nicotine -f Water Experiment: Results and Discussion 110 0.0 5.0 10.0 t ime (hours) 15.0 Figure 6.3: Examples for the evaluation of the diffusivity D. Plot of K9~?ax as a function of time for three runs with the same value of the final quench temperature Ty, but different initial refractive index discontinuities An, corresponding to different initial order parameters Ax*: • - An = 0.020 (Ax* = 0.094), A - An = 0.025 (Ax* = 0.118), O - An = 0.032 (Ax* = 0.151). Figure 6.4: Plot of the diffusivity D(tj, Ax*) as a function of composition discontinuity Ax*. Chapter 6. Nicotine -f Water Experiment: Results and Discussion 111 This dependence of D on Ax* is strongest for small tf, and less pronounced further from the critical point. In order to compare the diffusivities measured by the interferometric method with the results of the light scattering experiment (which measures D in the one-phase region where Ax* = 0), D(Ax*,tf) has to be extrapolated to Ax* —* 0, and we get D(tf)= Um D(Ax*,tf) (6.3) Ax*—»0 For lack of a better estimate, the logarithm of D as a function of Ax* was fitted by a straight line, and the zero intercept was taken as the value for D(tf). Figure 6.5 shows a log-log plot of all diffusivity data D(Ax*,tf) as a function of reduced temperature tf. The limiting values D(tf) are marked by full circles. A power law fit of the form D(tf) = Do{tf)K (see eq. (2.43)) yields n = 0.61 ± 0.05. 6.2.2 Light Scattering Data The autocorrelation data accumulated in the storage of the Malvern autocorrelator were transferred into the UBC Amdahl 5850 and fitted to a relation of the form G%\r) = A[\ + Cexp(-2rY + /z2r2)] (6.4) where r stands for time. A =< n >< >, where < n > is the total number of undipped counts and < ra* > is the number of clipped counts. C is a parameter depending on the experimental geometry, < n >, and the sampling time, and is a constant for a given experimental run. The diffusivity D is given by the equation r = 1/f = Dq2 (6.5) where f is the correlation time and q is the scattering vector, q can be expressed in terms of the refractive index n of the medium, the wavelength A of the laser light and Chapter 6. Nicotine + Water Experiment: Results and Discussion 112 O-I > 13 O -LO + ++ * • *+ + + + + + • + + + + I I 1111] 10": 1 T T 10' 10 .-4 10" 10" reduced temperature t f Figure 6.5: log-log plot of diffusivity data as a function of reduced final quench temper-ature tj. Data of D(Ax*,t/) are indicated by crosses (+), and the limiting values D(t/), with their corresponding error bars, by the symbol •. Chapter 6. Nicotine + Water Experiment: Results and Discussion 113 the scattering angle 6 as g =^sin(0/2) (6.6) A Since the refractive index n enters the diffusivity D quadratically, a precise measurement of D requires a precise knowledge of n. Our experiments were carried out on a critically filled cell in the one-phase region, and thus we had n = n(T) = nc. The critical refractive index was determined in a separate experiment (see Appendix B) and found to be nc = 1.3811 ±0.0004. The parameter p2 in eq. (6.4) is a measure of the variance of the distribution of diffusivities given by [118]: M 2 =<(r-<r>) 2 >=g<r> 2 where Q= < ( r ~ p ^ 2 > ) 2 > (6.7) Thus, Q is a measure of the polydispersivity of the sample. For our experiments, only data with Q < 0.02 were taken into account for the evaluation. Whereas Q is usually quite small far from the critical point, it increases as Tc is approached. This is due to the fact that in the critical region the fluctuations become so important that the approximation of single scattering of the incident light is not valid any more. Rather, multiple scattering comes into play, distorting the spectrum of scattered radiation. A large Q can also be an indication of insufficient equilibration time. Therefore, in our experiments data were only taken after waiting long enough so that Q had reached a constant value. We have taken data of D in the one-phase region at four different scattering angles (9 = 30°, 60°, 80° and 90°), the uncertainty in angle being about 0.5°. At angles 30° and 80°, the diffusivities were measured out to a reduced temperature of « 10 -2, and at angles 60° and 90° in a somewhat smaller temperature interval. Measurements further from Tc are difficult, because the intensity of the scattered radiation is very low. A transition into the two-phase region is accompanied by the appearance of a meniscus. We defined the Chapter 6. Nicotine + Water Experiment: Results and Discussion 114 .CO T-cs E > CO 3 A A • £ A O to 4& O o • • <t> o i—11111HI—i—111111n— 10*5 10"4 10 i i i 1111| 1—i i i 11111 10"3 10": reduced temperature Figure 6.6: Diffusivities of nicotine + water, as obtained from light scattering experiments at different angles 9: o = 30°, • = 60°, A = 80° and O = 90°. Error bars are indicated for each data point. critical temperature as the temperature beyond which a meniscus was observed. This allowed us to determine Tc with an accuracy of 2 — 3 mK. Figure 6.6 shows a log-log plot of the diffusivity data as a function of reduced tem-perature. Equation (2.43) describes the behaviour of the diffusivity D(q) = T(q)/q2 as a function of reduced temperature and wavevector. In the hydrodynamic limit q£ <C 1 the dynamic scaling function Q(q£) becomes a constant (see eq. (2.48)), so in this limit the Chapter 6. Nicotine + Water Experiment: Results and Discussion 115 diffusivity fe^-^iF <6-s> is expected to be independent of the scattering vector q. Indeed, our data show that for large reduced temperatures, where £ is much smaller than the wavelength of the laser light, the data obtained at different scattering angles agree. The variation of D with temperature in the hydrodynamic limit can be used to obtain an estimate of the viscosity exponent zv: Since fj oc £2", the diffusivity in the hydrodynamic limit behaves as D oc £-(1+*i) oc t« with n = v(\ + z„). Fitting the diffusivity data for t > 10 -4 and 0 = 30° by a power law in reduced temperature, we obtain D0 = 4.4 x 10~6 and K = 0.658 ± 0.005. Assuming v = 0.630, this implies z„ = 0.044 ± 0.008, slightly lower but in good general agreement with other experiments and theory. Diffusivity data in the critical region can be used to obtain an estimate of z, the exponent characterizing the wavevector dependence of the Rayleigh linewidth in the crit-ical limit q£ >^ 1 (see eq. (2.47)). Since we have not measured the bare correlation length £o and thus cannot calculate how far the critical region (q( >^ 1) extends, we use the method suggested by Chu and Lin [119] to evaluate z: We calculate an "effec-tive" exponent zef/(t) as a function of reduced temperature t by fitting our data to the expression T(q,t) = C^q'-HM (6.9) (see eq. (2.47 and eq. (2.48)). For large t (hydrodynamic limit q£ <C 1), ze/f should approach the value 2, whereas for < -> 0, ze/f tends to the critical value z. Figure 6.7 shows a plot of the zeff data. A straight line fit of ze// as a function of t, for z e// > 2.6, t < 5 x 10_s, gives z = zeff(t = 0) = 3.104 ± 0.026, slightly higher than the value z = 3.063 ±0.024 found by Burstyn and Sengers [76]. From the relation between the dynamic scaling exponent z and the viscosity exponent zv (see eq. (2.49)) we calculate that zv = Chapter 6. Nicotine -f Water Experiment: Results and Discussion 116 CN CO N O CO 0 0 C N • C N CN ~l C N C N C N o o o 0.0 1.0 2.0 3.0 -3 reduced temperature *10 Figure 6.7: Plot of the effective exponent zejf(t) as a function of reduced temperature. The exponent z is obtained as the limit of ze/f(t —* 0). Chapter 6. Nicotine + Water Experiment: Results and Discussion 117 0.104 ± 0.026, somewhat higher than the value calculated above from considering data in the hydrodynamic limit. 6.3 Potential Sources of Error Two potential sources of error in these experiments will be discussed in some detail: The question of gravity effects and the problem of equilibration times. 6.3.1 Gravitational Concentration Gradients Concentration gradients are known to develop in binary liquid systems due to the diver-gence of the osmotic susceptibility close to the critical point. However, because of the small diffusivities in the critical region, they take a long time (>• 10 days) to develop and reach their equilibrium value [120, 121]. The effects of gravity close to the liquid-liquid critical point in nicotine + water can be calculated following Ref. [122]. The variation of equilibrium concentration x with height z in thermal equilibrium can be written as point with an exponent 7 . Since the exact values for the nicotine + water system are not known, we use the result of Ref. [122] as an estimate: (dx/dp)pj « 5.75 x 10_7(T — Te)~12. In the nicotine + water system, around the critical composition, (dp/dx)ptx « 0.064 g/cc [81], and pc « 0.995 g/cm3. Thus we obtain dxnico/dz « l O ^ T c - r ^ / c m . Hence, the equilibrium composition profile in the one-phase region is not uniform, as assumed in section 3.2.2, but has a finite slope at the site of the meniscus of order of magnitude (dxnico/dz). This leads to a finite equilibrium refraction angle 0max, so that (6.10) Here, (dx/dp)pj is the osmotic susceptibility and expected to diverge close to the critical Chapter 6. Nicotine -f Water Experiment: Results and Discussion 118 for times T —• oo one expects that emax(r - » « > ) = LIA dxf° » 5 x i o - 7 ( r c - T)- 1- 2 (6.11) (ax/dn)c az with (dx/dn)c = 4.72 (see eq. (6.2)) and L = 0.2 cm = cell length. This effect is thus just barely detectable close to the critical point: at Tc — T = 0.001 K, 0MAX(T —• 00) w 0.1°. However, for temperatures so close to T c, the diffusivities are so small that this state would only be observed after a very long time (^ > 10 days [123]), much longer than any of our experiments. Thus we conclude that, due to the close density matching, in the nicotine + water system gravity effects play a negligible role. 6.3.2 Equilibration Times The long equilibration times, due to the critical slowing down in the critical region, raise the question of the validity of experiments performed close to the critical point. We present here an estimate of equilibration times in the two-phase region of the nicotine -f water system. If the critically filled cell, before it is heated from the one-phase into the two-phase region, is thoroughly mixed (by removing it from the thermostat block and shaking it), we can assume that it is at a homogeneous composition xc. After heating to a reduced temperature U = (Ti — Tc)/Tc in the two-phase region (see Fig. 6.8), phase separation occurs via spinodal decomposition [124]. This means that one phase will form a network of interconnected domains on all length scales in the other phase. The dominant length scale of the domains, r, increases with time r as a power law, r oc r~*, where <f> depends on the reduced temperature U and on the time regime. In the time interval of interest to us, corresponding to the "late stages of spinodal decomposition", <j> lies between —0.6 and —0.8 [124, 125]. To simplify calculations, we assume that we are dealing with droplets of radius r(r) of the "wrong" composition X\ (and thus the "wrong" density P\) embedded Chapter 6. Nicotine + Water Experiment: Results and Discussion 119 Figure 6.8: Phase diagram of a binary liquid of the type of nicotine + water (with an inverted phase diagram). We assume that the system, after a temperature step from Tf to Tf, has not reached thermal equilibrium, so that concentrations in the range x\ < x < x° are present in the cell. A subsequent temperature step to temperature T( in the two-phase region then takes only part of the liquid mixture in the cell into the spinodal region: fluid in the composition range x{ < x < x\ will demix via spinodal decomposition, whereas fluid in the ranges x\< x <x{ and will demix via nucleation. Chapter 6. Nicotine + Water Experiment: Results and Discussion 120 into a fluid of the "right" density p2, which will thus sink down or float up under the influence of gravity. The time it takes these droplets to reach the meniscus can be used to determine the relaxation time f. The droplets' velocity is given by Stokes' law, . „ i _ a r ' t o - f t ) ( 6 . 1 2 ) yr/ where 7/ is the viscosity of the medium and will be approximated as r/ « 0.8 cP (using the weighted mean with r}water = 0.45 cP and T/„,c0 = 1.7 cP [126]). In the case of nicotine + water, the density difference p\ — p2 in the region around the critical composition is proportional to the composition difference [81], i.e., Ap = cAx*, with c w 0.06 g/cc. In order to approximate r, we use results of a light-scattering experiment on isobutyric acid + water [124] which give Here, qo is a dimensionless scattering vector which can be estimated to be roughly equal to 4 [124]. For an estimate of r in the nicotine + water system, we use the same value. We also assume that the bare correlation length £o « 3A [123]. In section 6.2.2 we found that D = Dot* with D0 w 4.4 x 10_6cm2/s and K « 0.66; also, from our order parameter measurements in section 6.1 we obtained Ax* = B0t^ with B0 = 1.6 and ft = 0.315. Finally, using £ = £o*,~" w ^h v w 0.63 we can write v.(<T) = v0t-*"a-2'l> (6.14) with v0 = (2gn2cBo$2*+1))/(9jiqlD&) and eeff = ft-2u-2<f>(2u + K) W -0.95-3.85c6. From this equation the distance d(f) travelled in time f can be calculated as 2(f)' The time f taken by a droplet to travel from the top (or bottom) of the cell to the meniscus can then be calculated using d(f) = 1.6 cm, corresponding to half of the fill d(f) = j\a(T)dr = -f^t-W (6.15) Chapter 6. Nicotine + Water Experiment: Results and Discussion 121 u f 10"4 -0.6 « 6000 s spinodal quench lO"4 -0.8 « 1000 s lO"3 -0.8 « 150 s lO"2 -0.8 « 20 s 10"4 -0.2 « 107 8 quench into lO"3 -0.2 « 107 8 nucleation region lO"2 -0.2 « 107 s Table 6.2: Estimated equilibration times f for quenches to reduced temperature t,- in the two-phase region, into the spinodal and the nucleation region of the nicotine + water system height of the liquid in the cell. Table 6.2 gives an estimate of the equilibration time f as a function of reduced temperature t,-. For most reduced temperatures t,-, the equilibration times are on the order of a few hours or less for a spinodal quench. A more complicated situation arises when the cell is not shaken between individual runs (see figures 6.8 and 6.9). After a quench from the two-phase region at temperature where two compositions x° and x° coexist, into the one-phase region, the composition profile relaxes as described by the diffusion equation. Since the diffusivity is so small, the system has not reached equilibrium after a time « 24 h after the quench, and exhibits a composition profile as shown in figure 6.9b. Thus, the top and the bottom of the cell contain liquid of different compositions x° and x° before the heatup into the two-phase region. When heating to a temperature T( in the nucleation region of the phase diagram, <f> is much smaller (<f> « —0.2) [125] than for quenches into the spinodal.region (<j> « —0.8), leading to much longer equilibration times (see Table 6.2). Thus we face the following situation for a cell heated into the nucleation region of the phase diagram: the liquid close to the meniscus, in the composition regime x[ < x < x\, demixes rapidly via spinodal decomposition, whereas the rest of the cell demixes much more slowly via nucleation. This Chapter 6. Nicotine + Water Experiment: Results and Discussion 122 Figure 6.9: Composition profile as a function of height in the cell (compare to figure 6.8). (a): in equilibrium at T = Tf, (b): after a quench into the one-phase region, (c): after a heat-up into the two-phase region to a temperature T( > Tf. In the density regime x[ < x < x'2, phase separation occurs via spinodal decomposition Chapter 6. Nicotine -f Water Experiment: Results and Discussion 123 may lead to a "warped" composition profile in the cell, as shown in figure 6.9c, where the composition and thus the refractive index are no longer monotonic functions of height. Since only the volume very close to the meniscus is important for the order parameter measurements, this effect is not expected to affect the order parameter data. However, as the relaxation of the composition profile close to the meniscus depends crucially on the composition profile in other parts of the cell, it may have severe consequences for the diffusivity measurements. 6.4 Discussion The fits of the nicotine + water order parameter indicate that over the temperature interval t < 2 x 10-2 there are deviations from a pure power law. They cause the exponent /? to differ from the theoretically expected value, but are less noticeable than in pure fluids, where corrections to scaling start playing a role at reduced temperatures t > 10~4 (see figures 5.3 and 5.4). Similar results have been obtained in many other binary liquids [116] which leads to the conclusion that in binary liquids the scaling region extends to larger reduced temperatures t « 5 x 10-3. The inclusion of one correction to scaling term results in a negative value of the amplitude B\, which at first sight is suspicious. However, the values of amplitudes de-pend strongly on the temperature range of the data: A similar result was obtained in experiments on n-heptane + acetic anhydride [127] and on carbon disulfide + ni-tromethane [128] where B\ was also found to be negative when data in a reduced tem-perature interval t < 2 x 10~2 were fitted with one correction to scaling term. For a comparison of the results of the diffusivity measurements obtained by light scattering and by the interference method, the diffusivities from both experiments are shown in the same log-log plot in figure 6.10. From the interference experiment, only Chapter 6. Nicotine + Water Experiment: Results and Discussion 124 O -CM . E ' o > CO D T 5 • 6 I s ° o s o a AA A A AA ^ _ o • U 0 • • O <ED OD 10" "i—i i i 11 ii| 1—i i i 1111| 1—i i i 11 ii| r v-3 i i 111 i I i M i 11 10 10 10" 10" reduced temperature t 10 Figure 6.10: Comparison of the diffusivities obtained from light scattering experiments (open symbols) and interference measurements (with Ax* —• 0, solid circles), as a func-tion of temperature in the nicotine + water system. the data D(tj) (i.e., with Ax* —+ 0) are included. Both data sets show the same quali-tative trend: the diffusivity decreases as the critical point is approached, with the same exponent within error. However, there are some quantitative discrepancies: The diffu-sivities measured by the interference method are consistently larger than the diffusivities measured by light scattering. This discrepancy is not due to nonequilibrium .effects in the cell during the interference measurements: In the evaluation only those diffusivity data were considered for which the waiting time before the quench into the one-phase region was more than five equilibration times f (as calculated in section 6.3.2 for quenches into the spinodal region). This was Chapter 6. Nicotine + Water Experiment: Results and Discussion 125 especially true for runs with small Ax*, i.e., runs in which the cell was shaken before heating it into the two-phase region. The diffusivities D(Ax*,t/) found for these runs lie close to the limiting values D(tf) and are thus consistently larger than the light-scattering data. Even if for some runs with large initial concentration discontinuities Ax* the demixing occurred via nucleation, this has only very small influence on the limiting values D[tj) as Ax* —• 0. We thus conclude that the larger values of the diffusivity in the interference experiment are to high probability not due to nonequilibrium concentration profiles at the outset of a run. The reason of the discrepancy may lie in the fact that the interference method is inherently a nonequilibrium method, whereas in the light scattering experiment equilib-rium concentration fluctuations are being probed. The diffusion process can be looked upon as a diffusion of little "droplets" of the size of the correlation length in a viscous medium (see eq. (2.43)). At a given temperature, the correlation length in a system of critical composition is larger than in a system whose composition is off-critical. Thus it is plausible that in the interference experiment, where diffusion occurs in two phases of off-critical composition, the size of the diffusing droplets is smaller than in the light scattering experiment, where the system is kept at critical composition. To our knowl-edge, the experiment presented here is the first measurement in which microscopic and macroscopic diffusivities are directly compared. Further studies on different systems are necessary to illuminate the discrepancy in more detail. The light scattering diffusivity data, being less affected by statistical variations than the interference data, were used to calculate the viscosity exponent zn. From our mea-surements we obtain that ze// = 3 + zn = 3.104 ± 0.026. This value is slightly higher than theoretically expected, in contrast to some other experiments [129] which find z = 2.992 ± 0.014, but they agree within error with the result obtained by Burstyn and Sengers [76] who found z = 3.063 ± 0.024. We thus measure a viscosity exponent Chapter 6. Nicotine + Water Experiment: Results and Discussion 126 of zv = 0.104 ± 0.026, higher than the theoretical value which is z„ = 0.054 [77] from mode-coupling calculations and zv = 0.065 [130] from RG. The reason for the deviation may be found in our evaluation method: The quality of the result in applying the method of Ref. [119] hinges on having as many precise data close to Tc as possible. We may not have approached Tc closely enough to observe the limiting behaviour of zn. Moreover, a small change in the value used for the critical temperature will influence the value of zv drastically. A change of lmK in Tc changes the value of zv by « 0.05. If our critical temperature was not correct, this may account for the discrepancy. Another way of determining zv is a fit of the diffusivity to a power law in the hydrody-namic limit. The exponent « is then related to the viscosity exponent by K = + zv). From the light scattering data at scattering angle 0 = 30° at reduced temperatures t > 10~4, we deduce that z^ — 0.044 ± 0.008, in close agreement with other experiments and with theory. According to eq. (2.43), the diffusivity has a nonsingular contribution which we have not yet substracted from the total measured diffusivity as plotted in Fig-ure 6.6. For some systems, the correction due to the background terms has been found to be as large as 1% at q£ = 1 and several percent for smaller q( [76]. Background terms in D lead to higher values of the exponent K and thus z„ than expected theoret-ically [82, 131, 132]. Since the exponents found for nicotine + water in our experiment are of the same order of magnitude as the theoretical values, we conclude that back-ground terms play a relatively small role in nicotine + water in the temperature regime covered in our experiment, so that their neglect has a minor impact on the diffusivity measurements. Because the densities of the two constituents are so closely matched in the nicotine + water system, gravitational rounding of the composition profile plays a negligible role. Depending on the temperature, it is either too small an effect to be observed, or it takes too long to develop on the scale of the experiments to be noticeable. Chapter 6. Nicotine + Water Experiment: Results and Discussion 127 An interesting aspect is the possible observation of nonequilibrium density profiles. As we have shown, spinodal quenches lead to equilibrium after a relatively short time (a few hours or less), so that a wait of « 12 h between runs is sufficient to assure equilibrium in most runs. If the cell is not thoroughly mixed between runs, however, there is a chance of quenching part of the cell into the nucleation region, even if in a series of runs the distance from the critical point, T,- — Tc, is increased with each run. One piece of evidence for this would be observed in the process of heating the cell from the one-phase to the two-phase region: Areas of the cell undergoing spinodal decomposition would exhibit critical opalescence, whereas regions of nucleation would remain clear. Indeed, we have observed in some of our runs that only the region of the cell close to the meniscus becomes milky close to the critical point. The hypothesis of a "warped" concentration profile in the cell could explain the occurence of so-called "ghost fringes" which have have frequently been observed in our lab (and not just in the nicotine -f water experiment): If the refractive index profile is not a monotonic function of height, but exhibits "bumps", these will lead to interference fringes at angles 0 < 0 and appear as ghost fringes on the films. Chapter 7 He-Xe Experiments: Results and Discussion This chapter discusses the results of the experiments on pure xenon and on a helium-xenon mixture containing 5% He. Section 7.1 presents the results of measurements of the Lorentz-Lorenz function, coexistence curve and diameter. The critical density of Xe is determined to very high accuracy and found to be insensitive to impurities in the sample. The coexistence curve diameter for Xe does not deviate appreciably from a rectilinear diameter, in apparent contradiction of the theory of Goldstein et al [26]. Section 7.2 gives the results of the experiments on the He-Xe mixture. The order parameter and compressibility of a He-Xe mixture are measured to be very similar to those of pure Xe, indicating that the addition of He does not change the liquid-gas character of the phase transition appreciably. We present a novel way of estimating the effect of wetting, by measuring indirectly the rise height of the wetting phase on the sapphire windows of the cell. Finally, section 7.3 contains a discussion of the results. 7.1 Experiments on Pure X e In an effort to measure the critical density of xenon with as high precision as possible, experiments were performed using the prism cell setup. In order to test the consistency of the results for different samples, we performed prism cell experiments on two Xe samples (Sample #1 and Sample #2, described in Section 4.3) obtained from different suppliers. Subsequently, data were taken using the image plane interferomentric technique, as a 128 Chapter 7. He-Xe Experiments: Results and Discussion 129 Co [cm3/mole] Ci [cm6/mole2] c2 [cm9/mole3] rc(fit) [cm3/mole] <CC> [cm3/mole] Sample #1 10.413 25.28 -1569. 10.515 ±0.001 10.510 ±0.008 Sample #2 10.382 26.19 -1377. 10.505 ± 0.003 10.504 ± 0.006 Table 7.1: Results of a quadratic fit to the Lorentz-Lorenz data of two Xe samples. £ c(fit) is the critical value of C, obtained from quadratic fits. < Cc > is the average of C measurements in the density interval 0.88/>c < p < 1.12pc. test of the high-pressure setup and in order to compare coexistence curve measurements in the high-pressure cell with those performed in the prism cell. 7.1.1 Lorentz-Lorenz Function The Lorentz-Lorenz function, as defined in eq. (3.1), provides a link between the refractive index and the density of a fluid. It is measured by the prism cell method. The Lorentz-Lorenz function was fitted to a second order polynomial in the density p (see eq. (5.1)). Table 7.1 shows results of fits obtained in the density interval 0.3pc < p < 1.4/t»c, and results of the critical value Cc = C(pc). Due to the scatter of the data points, the error in Cc obtained from various fits, £ c(fit) is considerably smaller than < Cc >, calculated from averaging the Lorentz-Lorenz values over the interval 0.88/>c < p < 1.12/>c. Figure 7.1 shows the Lorentz-Lorenz data of both samples, together with curves corresponding to the fit parameters given in Table 7.1. Note that the discrepancy between the £c-values of the two data sets is less than 0.1%. Our results for C are in good agreement with other experiments which find the Lorentz-Lorenz function to have the value C = 10.52 ± 0.02 cm3/mole [133] and C = 10.53 ± 0.07cm3/mole [134] on average in the density region 0.6pc < p < \.lpc. Within the accuracy of our measurements, we observe no anomaly in C close to the critical point, in agreement with the results of other researchers [133,134]. Using eq. (5.2) Chapter 7. He-Xe Experiments: Results and Discussion 130 0.000 0.003 0.006 0.009 0.012 Density (moles/cc) 0.015 Figure 7.1: Lorentz-Lorenz data of Sample #1 (o) and Sample #2 (•). The curves correspond to quadratic fits. Chapter 7. He-Xe Experiments: Results and Discussion 131 Ax A\-a pc [mole/cm3 outer temperature range (i > 8 x lO"3) Sample #1 0.73 ± 0.04 Sample #2 0.65 ± 0.04 (0.0) 0.008495(4) (0.0) 0.008481(8) total temperature range (lO"5 < t < 2 x 10"2) Sample #1 0.73 ±0.02 (0.0) 0.008497(2) (0.0) 0.47 ± 0.02 0.008492(1) 1.3 ±1.0 -0.5 ±0.3 0.008502(2) Sample #2 0.65 ± 0.01 (0.0) 0.008484(7) (0.0) 0.43 ±0.02 0.008481(6) 1.4 ±1.0 -0.5 ±0.3 0.008488(8) Table 7.2: Fit results of the coexistence curve diameter of Xe. Parameters in parentheses were kept fixed for the fit. we can calculate the electronic polarizability otp from the limiting value of C(p) as p —* 0. We find a p = 4.12 ± O.OlA. 7.1.2 Coexistence Curve Diameter and Critical Density The prism cell data can be used to evaluate the coexistence curve diameter, pd = (pi + pv)/2pc, of Xe. For all fits of pd the critical temperature was held fixed at the value obtained in the coexistence curve fits (see section 7.1.3). Figure 7.2 shows a plot of the diameter data of Sample #1 and Sample #2 as a function of reduced temperature. In all cases, the data exhibit no significant deviations from straight lines. We thus do not observe any singularity of the coexistence curve diameter. Table 7.2 gives the diameter fit results. The first part of the table shows the parameters obtained from a straight line fit to the data with reduced temperatures Chapter 7. He-Xe Experiments: Results and Discussion 132 1.003 £ l o o o - ~ Q) 0.997 -E .S2 1003 1.000 - i r 0.997 J 1.003 1.000 0.997 ,.0*3 O J O O -o 0 O -0.00 0.01 0.02 reduced temperature Figure 7.2: Coexistence curve diameter of the four experimental runs in the prism cell experiment. Open symbols correspond to the data of Sample #1, filled ones to the data of Sample #2. Chapter 7. He-Xe Experiments: Results and Discussion 133 t > 8 x 10 -3. For each sample, the data of the two runs were fitted together. The error is a measure of the difference between the two runs. The second part of the table gives results of fits treating A\ and A\.a as free parameters. The exponent a was kept constant at OL = 0.11. When only a linear term is fitted (Ai-a = 0), the values found for A\ agree very well with the slopes of the straight-line fits in the outer temperature range t > 8 x 10~3. This indicates that there are indeed no systematic deviations from rectilinear diameter close to the critical point. Fitting both A\ and Ai-a as free parameters does not give useful results, as indicated by the large errors. The limiting value of the diameter p& as t —• 0 determines the critical density pc. We calculated the value of pc for our samples by averaging the pc-values obtained in the various diameter fits. The results are: Sample #1: pc = 0.008496(4)mole/cm3 = 1.1156(5)g/cm3 Sample #2: pc = 0.008489(4)mole/cm3 = 1.1139(5)g/cm3 For an evaluation of the uncertainty in the value of pc due to systematic errors, it turns out to be easier to extract the limiting value n c as t —> 0 of the refractive index diameter (n/ + n„)/2 which is directly accessible experimentally and to calculate pc using the Lorentz-Lorenz relation. The error calculation is presented in Appendix A. The results are: Sample #1: nc = 1.1377 ± 0.0001 pc = 1.1160 ± 0.0017g/cm3 Sample #2: nc = 1.1374 ± 0.0001 pc = 1.1147 ± 0.0017g/cm3 The small differences of the critical densities found in the two evaluations are well within error. Table 7.3 compares our values of the critical density with literature values. Our values are in good agreement with the literature values, but their error is considerably smaller. Chapter 7. He-Xe Experiments: Results and Discussion 134 Experimented Pc Habgood et al. [135] 1.099±? g/cma Levelt [136] 1.091±? g/cm3 Chapman et al. [134] Garside et al. [137] Baidakov et al. [138] Cornfeld and Carr [139] 1.106 ±0.004 g/cm3 1.119 ±0.011 g/cm3 1.1128±? g/cm3 1.1113 ±0.0017 g/cm3 1.1128 ±0.0003 g/cm3 this w o r k : Sample #1 Sample #2 1.1160 ±0.0017 g/cm3  1.1147 ±0.0017 g/cm3 Table 7.3: Critical density of Xe Also, the values of Sample #1 and Sample #2 agree very well, indicating that whatever impurities they may have contained, these impurities have a minor impact on the critical density. 7.1.3 Coexistence Curve Coexistence curve data were obtained both by the prism cell method (samples #1 and #2) and by the interferometric technique. The data were fitted to a power law in the reduced temperature t> with corrections to scaling as given in eq. (5.3). In the prism cell experiment, two runs were performed on each of the two samples. Each data set was fitted separately. In order to be able to compare the results of the different samples and the different techniques, each data set was evaluated keeping the exponents fixed at their theoretically expected values (ft = 0.327, A = 0.5) and fitting the critical temperature and two amplitudes as free parameters. The fit results are given in Table 7.4. The values cited for each prism cell sample is the average of the results of the two runs. The fit results of the amplitudes of the two samples are seen to agree within error. Figure 7.3 shows a plot of Ap*/tp as a function of reduced temperature t for Chapter 7. He-Xe Experiments: Results and Discussion 135 Sample Bo Br B2 T C (K) Prism cell data (f < 3 x 10"2) Sample #1 1.479 ±0.011 1.15 ±0.19 -2.6 ±1.0 289.752 ± 0.001 Sample #2 1.470 ± 0.010 1.20 ±0.17 -2.8 ±1.1 289.789 ± 0.002 Image plane interference data (5 x lO"5 < t < 2 x 10"2) Sample #3 (long cell) 1.400 2.00 -5.5 289.920 ± 0.002 Sample #4 (short cell) 1.412 1.71 -4.1 289.882 Focal plane interference data (5 x 10"5 < t < 4 x 10"2) 1.444 1.41 -2-4 289.807 Table 7.4: Results of fits to the coexistence curve of Xe. For these fits, the exponents /? = 0.327 and A = 0.5 were held fixed. the prism cell data of Samples #1 and #2. As is obvious from the figure, individual runs on the same sample show excellent agreement. There is a slight systematic difference between Sample #1 and Sample #2. In order to avoid overcrowding of the figure, the curves corresponding to the individual fits were omitted. The curves correspond to fits of all data collected on each sample with /? = 0.327 and A = 0.5 held fixed. Interference experiments were carried out on Xe samples in cells of different thickness. By changing the window support plugs in our high-pressure cell, the cell length could be changed from L = 0.195(2) cm to L = 1.176(2) cm. As the high pressure cell and its temperature control system were so bulky, the second beam splitter in the optical setup could not be moved very close to the cell. Therefore light refracted from the neighbourhood of the meniscus, which is strongly bent close to the critical point, was lost from the image. This limited the reduced temperature regime accessible in this experiment to t > 2 x 10~5. In order to compare the amplitudes obtained in the different experiments, the data Chapter 7. He-Xe Experiments: Results and Discussion 136 Figure 7.3: log-log plot of coexistence curve data of Xe, as measured in the prism cell, o - Sample #1, • - Sample #2. The dashed curves correspond to fits to data of Sample #1 and Sample #2 separately, with two correction to scaling terms and keeping ft = 0.327 and A = 0.5 fixed. Chapter 7. He-Xe Experiments: Results and Discussion 137 were fitted over the same reduced temperature range. Therefore, for the interference ex-periments only runs in the temperature interval 5 x 10"5 < t < 2 x 10~2 were taken into account. This leaves one set for each long cell (Sample #3) and short cell (Sample #4). The fit results on these data sets (again, with ft = 0.327 and A = 0.5 held fixed) are included in Table 7.4. The critical temperatures of the samples in the interference ex-periments are substantially higher than the ones measured in the prism cell experiments. Also, the critical amplitudes exhibit a marked deviation from the results obtained on Sample #1 and Sample #2. As a comparison, Table 7.4 also contains results of a focal plane interference exper-iment [111], reevaluated using our measurements of the Lorentz-Lorenz function. Here, data were taken in the reduced temperature interval 8 x 10 -5 < t < 4 x 10-2. The values of Tc and Bo are between the values found in the prism cell experiment and those found in the image plane interference experiment. Figure 7.4 shows a log-log plot of Ap*/t^ as a function of t for the interferometric data. The image plane data are seen to agree very well. There is some disagreement, however, with the focal plane experiment [111], both as far as the amplitudes and the critical temperature is concerned. The interferometric data contain a curious feature which is absent in the prism cell data: Around a reduced temperature of t < 3 x 10~4, the data exhibit a "bump" which is the more pronounced the thinner the cell. As careful reevaluation of the data shows, this is not due to a miscount in interference fringes. Since the size of the effect decreases with increasing sample thickness, we assume that it has to do with an interaction of the fluid with the cell windows. We will come back to this feature later (see section 7.3). Chapter 7. He-Xe Experiments: Results and Discussion 138 Figure 7.4: log-log plot of the coexistence curve data of Xe from the interference experi-ments. ® - image plane interference, short cell, x - image plane interference, long cell, o - focal plane interference [111]. Chapter 7. He-Xe Experiments: Results and Discussion 139 7.2 He-Xe Mixtures After the interferometric experiments on pure Xe, the cell was filled with a He-Xe mixture containing w 5 mole% He. For this mixture, order parameter and compressibilities were measured. Since our prism cell was designed for pressures up to 60 atm, the Lorentz-Lorenz function in the critical region of this mixture (with a critical pressure Pc fa 90 atm) could not be measured. We used the Lorentz-Lorenz data of pure Xe to convert refractivity data to densities in the He-Xe mixture. 7.2.1 Coexistence Curve Two coexistence curve runs were performed on the He-Xe mixture. The data were fitted to a power law equation with corrections to scaling, as given in eq. (5.3). The critical temperature was found to be T c = 291.252(2) K for the first run and Tc = 291.228(2) K for the second run. For the coexistence curve fits, in which both data sets were taken together, this difference in critical temperature was corrected for. Data were taken in taken in the reduced temperature interval 3 x 10~5 < t < 2 x 10-2. Table 7.5 shows the fit results. When the data were fitted with two correction to scaling terms, a fit treating the exponent ft as a free parameter gave /3 = 0.3249, in excellent agreement with theory. Also, keeping /? = 0.325 fixed and fitting A as a free parameter yielded A = 0.498, again very close to what is expected theoretically. Fitting the data with one more correction to scaling term has a negligible influence on Bo and B\, and changes the value of B2 only slightly. We thus conclude two corrections to scaling are sufficient for describing our data. Figure 7.5 shows a plot of Ap*/^ as a function of reduced temperature for two experimental runs. The curve corresponds to a fit keeping A = 0.5 fixed and treating /? as a free parameter, with two correction to scaling terms. Both data sets taken on the mixture exhibit a "bump" of the coexistence curve in the Chapter 7. He-Xe Experiments: Results and Discussion 140 reduced temperature Figure 7.5: log-log plot of Ap*/t^ as a function of reduced temperature t for a He-Xe mixture containing « 5 mole% He. The curve corresponds to a fit with two correction to scaling terms, holding A = 0.5 fixed and fitting /? = 0.3249 as a free parameter. Chapter 7. He-Xe Experiments: Results and Discussion 141 A B0 Br B2 B3 total temperature interval 2 x lO"5 < t < 2 x 10"2 (0.327) (0.325) 0.3249 (0.325) (0.327) (0.325) 0.498 (0.5) (0.5) (0.5) (0.5) (0.5) 1.370 1.96 -4.3 1.344 2.12 -4.7 1.343 2.13 -4.8 1.344 2.11 -4.7 1.370 1.93 -3.8 1.343 2.17 -5.4 (0.0) (0.0) (0.0) (0.0) -1.8 2.7 outer temperature range 5 x 10"4 < t < 2 x 10"2 (0.327) (0.325) (0.5) (0.5) 1.373 1.93 -4.2 1.349 2.06 -4.5 (0.0) (0.0) Table 7.5: Results of coexistence curve fits of a He-Xe mixture containing « 5 mole% He. Parameters in parentheses were held fixed for the fit. vicinity of t < 3 x 10~4 as observed in pure Xe. This feature causes deviations of the data from the assumed power law close to Tc. In order to make sure that this feature does not bias the fit results, fits were performed on the reduced temperature interval t > 5 x 10-4, where the effects of the bump are not noticeable. The results of these fits are also given in Table 7.5. Clearly, the bump has only a small effect on the amplitudes. 7.2.2 Compressibilities The compressibilities can be obtained from the fringe spacing near the meniscus, in the manner described in reference to eq. (3.16). For the He-Xe mixture containing » 5 mole% He, the critical pressure Pc was measured to be Pc = 90±1 atm and pc w 1.11 g/cm3. The compressibilities were measured in the one-phase and the two-phase region. Figure 7.6 shows a log-log plot of the compressibility as a function of reduced temperature. The data were fitted to a power law Kj = rot|r|~')'. Table 7.6 presents the results of this fit. In the one-phase region, a power law fit to the data leaving 7 a free parameter Chapter 7. He-Xe Experiments: Results and Discussion 142 reduced temperature Figure 7 . 6 : Compressibilities of a He-Xe mixture containing 5 mole% He. • - data in the one-phase region, o - data in the two-phase region, + - data point of pure Xe in the one-phase region. Chapter 7. He-Xe Experiments: Results and Discussion 143 7 One-phase region 1.07 ±0.03 1.07 ±1.0 (1.241) 0.21 ± 0.04 Two-phase region 0.94 ± 0.02 1.05 ±1.0 (1.241) 0.07 ± 0.03 Table 7.6: Compressibility fits of a He-Xe mixture. Parameters in parentheses were held fixed for the fit. yields 7 + = 1.07 ± 0.03, considerably lower than the theoretically expected value. In the two-phase region, a power law fit gives 7 = 0.942 ± 0.018, also substantially lower than the theoretical value. Fitting the compressibility data in the one-phase region with 7 + = 1.241, we obtain TQ = 0.212 ± 0.041. This value is substantially higher than the one measured by Giittinger and Cannell [13], who found that for pure Xe TQ « 0.0575. One reason for the discrepancy might be that in our experiments the system was not in thermal equilibrium. Close to the critical point, the equilibration times become very long, due to the small diffusivities in the critical region. We found, however, that data taken after an equilibration period of 3-4 days do not exhibit significantly lower values of TQ than data taken after « 12 hours, and therefore we conclude that the discrepancy is not due to insufficient equilibration time. Another possible explanation is the presence of temperature gradients in the cell [79]. The consequences of a thermal gradient on the compressibility are discussed in Ap-pendix C. We find that the discrepancy in the compressibility can indeed be ascribed to a nonuniform temperature in the cell, with a gradient of w 2 mK/cm. The deviation of our measured value of 7 from the theoretically expected value there-fore could be attributed to a thermal gradient which depends on the temperature dif-ference between cell and surroundings. Since the room temperature varied by several degrees, this also may explain the comparatively large scatter in the data. Chapter 7. He-Xe Experiments: Results and Discussion 144 Figure 7.6 contains one data point of the compressibility of pure Xe in the one-phase region, which is seen to agree very well with the He-Xe data. Thus, there is no substantial difference in the compressibility of the mixture and the compressibility of pure Xe, and the large value of the compressibility of the mixture close to the critical point indicates that the phase transition in the binary fluid has very similar features to the liquid-gas phase transition. 7.2.3 Wetting In the Xe and the He-Xe experiments, the image of the meniscus far from Tc is not seen as a narrow line (as in the C H F 3 experiment), but as a broad streak the width of which decreases as Tc is approached. This behaviour of the meniscus can be related to wetting of the walls by the fluid and suggests a novel experimental method for measuring the rise of a wetting fluid on the wall of the container. Figure 7.7 shows a log-log plot of the meniscus width as a function of reduced temperature for the two He-Xe runs and the Xe run using the thin cell. Far from Tc the the meniscus width is seen to approximately obey a power law in the reduced temperature, as indicated by the dotted lines which represent power-law fits to the Xe and He-Xe data in the temperature regime 4 x 10 -3 < t. Around t w 4 x 10~3 deviations from this power law occur towards larger meniscus values. The data for pure Xe exhibit a similar exponent to the data for the He-Xe mixture, but with a slightly different amplitude. The deviation between the two data sets may be due to a small difference in magnification factor from the cell to the image which was not properly taken into account. The results for the meniscus width can be understood in the framework of wetting: Classically, a liquid in equilibrium with its vapour partially wets a confining wall. The rise height adjusts itself in such a way that the pressure drop across the meniscus due to Chapter 7. He-Xe Experiments: Results and Discussion 145 T | 1 1—I I I I I I | 1 1—I I I I I I | 10"4 10' 3 10*2 reduced temperature t i — r Figure 7.7: log-log plot of the meniscus width as a function of reduced temperature, o, •— He-Xe experiment (two data sets), and •— Xe experiment (one data set). The dashed (dotted) line corresponds to a power law fit to the He-Xe (Xe) data in the temperature range t > 4 x 10-3. Chapter 7. He-Xe Experiments: Results and Discussion 146 surface tension is balanced by the buoyancy force. The equilibrium rise height is [140] 2E h = J——Vl-sm0 (7.1) where E is the surface tension, Ap = p\ — pv is the density difference between liquid and vapour and 0 is the contact angle at the wall, determined by cos 6 = (crav — crai)/cr, where o~av (<rai) are the surface free energy between the vapour (liquid) and the solid wall and tr is the surface free energy of the liquid-vapour interface. As the contact angle is small [141, 142], we obtain that h w yjTE/gAp [143]. The density difference varies with reduced temperature as Ap oc t&. The surface tension E obeys a power law E oc ((T 0-T)/r 0) n with n > 0, where T0 is a temperature close to Tc [144]. Thus the rise height h decreases as the critical point is approached following a power law h oc r(n-^)/2. Wetting therefore leads to a curved meniscus, with fluid near the windows creeping up to wet them. Incident rays hitting the cell in the region of this meniscus are bent or scattered and lost in forming the image of that region of the cell. Identifying the meniscus width measured in the experiment with the rise height h, one can thus extract the exponent n. From a power law fit to our meniscus data, M(t) oc r^ , for reduced temperatures t > 10-3, we get C = 0.441 ± 0.020 for the He-Xe runs and C = 0.473 ± 0.018 for pure Xe. From this, the surface tension exponent can be calculated. We obtain n = 1.21 ± 0.04 for He-Xe and n = 1.27 ± 0.04 for Xe. These values are in good general agreement with the one cited by Cahn [142] who finds n = 1.3. The deviation of the meniscus widths from a power law for small reduced temperatures may be interpreted as an indication of total wetting close to the critical point [143, 142, 145]. This effect distorts the meniscus close to Tc from its classically expected shape, causing the denser phase to creep up very high ("infinitely high") along the walls. Thus a broader region around the meniscus, in which light rays are scattered and thus lost in the formation of the image, can be understood in terms of a critically expanded wetting height Chapter 7. He-Xe Experiments: Results and Discussion 147 in the proximity of the critical point. The determination of the exact meniscus width is made difficult by the presence of interference fringes inside the image of the meniscus. Very close to Tc, the meniscus looks totally sharp, indicating that the meniscus height vanishes as the critical point is approached. 7.3 Discussion Within the accuracy of our experiment, we observe no critical anomaly in the Lorentz-Lorenz function of Xe. The Lorentz-Lorenz data measured in two different samples of Xe agree to within 0.1% and agree with results obtained by other researchers. They can be adequately described by a second order polynomial in the density. In our prism cell data on pure Xe, we observe no critical deviation from a rectilinear diameter. This is in contradiction with the theory of Goldstein et al. [26] according to which we would expect an anomaly whose magnitude should be proportional to appc. For Xe, aapc = 0.021, larger than in any of the nonpolar gases studied in Ref. [28] which all display critical singularities in the diameter. Thus, we would expect the anomaly in Xe to be larger than observed in other fluids. However, there are indications [146] that in the Xe system the Axilrod-Teller interactions are not the most important three-body forces. Rather, exchange interactions play a dominant role. These, however, are not taken into account in the theory of Goldstein et al. This is a possible explanation of why the theory does not correctly predict the diameter anomaly of Xe. From the diameter data, critical densities can be calculated. The values of pc obtained from Sample #1 and Sample #2 agree to within 0.1%, demonstrating the reproducibility of the method and the insensitivity of the result to impurities. The coexistence curve amplitudes, as obtained from fits on Sample #1 and Sample #2 Chapter 7. He-Xe Experiments: Results and Discussion 148 agree very well. The critical amplitude B0 is slightly lower in Sample #2 than in Sam-ple #1. Also, the critical temperatures of the two samples differ by about 40 mK. Since the same experimental method and the same cell was used in the two measurements, this discrepancy must be due to differing impurities in the two samples. It is well known that the presence of impurities can strongly influence the critical temperature of a pure fluid sample [109]. As Tc in Sample #1 is so close to the literature value (Tc = 289.74 K), we assume that it is cleaner than Sample #2. In contrast to this good agreement, the amplitudes obtained from fits to the inter-ferometric data are markedly different: Even though the values of Bo obtained from the image plane experiments on Xe agree well, they are substantially lower than the values obtained in the prism cell experiments. Also, the critical temperatures measured in the image plane experiment are considerably higher than in the prism celll experiments. This is probably due to He impurities in the high pressure sample cell: Since the setup was pressure-tested using He, and since the high-pressure tubing has such a small inner di-ameter, it is quite possible that not all the He was removed from the cell before filling it with Xe. A He contamination has the effect of increasing the critical temperature [44], as indeed observed in our experiment. The focal plane interference data exhibit values of Bo and Tc between the image plane interferometric experiment and the prism cell experiment, indicating that the sample was probably also contaminated. A He-Xe mixture containing 5 mole% He has a lower critical amplitude Bo than pure Xe. Simultaneously, the critical temperature is substantially higher. The compressibility measurements were complicated by the presence of thermal gradients in the cell. This caused some scatter in the data, and led to distorted values of the critical amplitudes and exponents. In the image plane interference experiments on Xe and the He-Xe mixture, and area in the vicinity of the meniscus appears as smeared; the streak is the broader, the further Chapter 7. He-Xe Experiments: Results and Discussion 149 the system is from Tc. Identifying the width of this streak with the rise height of a wetting fluid, we measure an exponent of the surface tension which is in good general agreement with the value cited in the literature. This agreement indicates that it is indeed reasonable to assume that the finite meniscus width as seen in the image plane is due to wetting. The meniscus width data deviate from the power law at reduced temperatures f « 4 x 10-3. In the interference measurements on the order parameter of Xe and He-Xe, we observed a wiggle on the data in the reduced temperature regime of t « 4 x 10~4. Since this feature is the more pronounced for the thinner cell, we assume that it is also somehow related to wetting. The exact relation, however, is not clear at the moment. Comparison of the coexistence curve data and compressibilities of Xe with those of the He-Xe mixture shows that there is no substantial difference between the two systems. This fact may be explained by the conjecture that the interference experiments cannot "see" the He, since He has a refractive index so close to unity. Moreover, the He atoms are small compared to the Xe atoms and could therefore easily slip through voids between the Xe atoms. In this case, they would not displace any Xe atoms, and would not have any effect on the density and refractive index of the xenon. An investigation of a He-Xe mixture using neutron scattering [45] found that the presence of He had a negligible effect on the Xe-Xe correlation function, so that the main effect of the presence of He is a shift of the critical temperature and pressure. This indicates that the phase transition has characteristics of the gas-liquid type rather than the binary liquid type. Chapter 8 Conclusions In this chapter, the results of the experiments performed in this thesis will be briefly reviewed and compared. In order to test the influence of dipolar interactions on the order parameter and the coexistence curve diameter in pure fluids, experiments were carried out on CHF3 (a strongly dipolar fluid), CCIF3 (a weakly dipolar fluid) and Xe (a nonpolar fluid). Table 8.1 gives an overview of the results. Here, ctp is the effective polarizability (as defined by eqn. (1.7)), A\ is the linear term in the coexistence curve diameter (fitted for reduced temperatures t > 8 x 10~3), and B0 is the critical amplitude of the order parameter. Our results for Bo and A\ can be compared to the theory of Goldstein et al. [26, 28], which, starting from a microscopic model of three-body interactions between the particles in a fluid, calculates the impact of these three-body forces on the critical amplitudes BQ and A\. It is found that the amplitudes depend on a parameter x = q/ab, where q is the integrated strength of the three-body potential and a and b are the van der Waals parameters. In the limit of small x one obtains that A\ = 2/5 + 22x/15 + ... Substance Pc [g/cc cyA3] <XpPc Ai B0 CHF 3 0.527 3.5 0.016 1.31 ±0.04 1.75 ±0.02 CCIF3 0.582 4.67 0.016 0.86 ± 0.02 1.64 ± 0.02 Xe 1.115 4.12 0.021 0.69 ± 0.04 1.45 ± 0.03 Table 8.1: Comparison of critical polarizability product, order parameter amplitude and diameter slope for substances with different strength of dipolar interactions. 150 Chapter 8. Conclusions 151 Bo O 1.7 " CHF 3 1.6" 1.5 1.4 0.75 A 0.50 1.00 1.25 Figure 8.1: Plot of the order parameter amplitude Bo as a function of the diameter slope Ai for a variety of pure fluids. The line corresponds to a straight-line fit to the data. and Bo = 2 + 2x/3 + .... This implies that the order parameter amplitude Bo increases linearly with the diameter slope Ai like Figure 8.1 shows a plot of Ai as a function of Bo. The data of all substances investigated fall on a straight line, which supports the hypothesis of three-body interactions as pos-tulated in the theory of Goldstein et al. Even though C H F 3 , being a polar fluid, exhibits a value of Ai which is markedly larger than for nonpolar fluids, its value of Bo is also larger, so that the proportionality of Ai and BQ is still approximately satisfied for C H F 3 . If the dominant three-body forces are triple-dipole forces of the Axilrod-Teller type [27], (8.1) Chapter 8. Conclusions 152 A, 1,0 0.8 0.6 o C H F . 3 / o C , H C 2 H 6 0.006 0010 0.014 0.018 0.022 Figure 8.2: Plot of the diameter slope A\ as a function "of the critical polarizability product for a number of fluids. The line corresponds to a straight line fit to the data (without Xe and CHF3). then x is found to be proportional to the dimensionless quantity dppc, called the "critical polarizability product" [28]. Figure 8.2 shows a plot of A\ as a function of dppc for a number of nonpolar fluids investigated before [28], and the three fluids of this thesis. The agreement with the theoretically expected curve is good for CCIF3, but the values for Xe and C H F 3 deviate strongly from this curve. This suggests that in the case of CHF 3 and Xe the three-body interactions are not Axilrod-Teller dipole-induced-dipole forces. In the case of C H F 3 , the reason may be due to the strong molecular dipole moment of the molecules. For Xe, there are indications [146] that the modification of the two-body Chapter 8. Conclusions 153 potential due to three-atom exchange interactions is more important than the one due to Axilrod-Teller interactions. This may explain why Xe exhibits a different relationship of Ai to ctppc than other nonpolar fluids [28]. Since hydrogen bonds between the molecules play an important role in C H F 3 , it is not easy a priori to decide whether the difference in the critical amplitudes of C H F 3 and nonpolar fluids are due to the dipolar coupling between the molecules or due to hydrogen bonds. This question must be addressed in a separate investigation. A deviation from rectilinear diameter is observed in C H F 3 and CCIF3, but not in Xe. This is in contradiction to the theory of Goldstein et al, according to which the strength of the diameter singularity is expected to be proportional to the critical polarizability product dppc. Xe, having a large value of ctppc, should therefore exhibit an especially strong critical singularity. The reason for this discrepancy may again be the fact that the Axilrod-Teller forces, in the case of Xe, are less significant than the three-body exchange forces [146]. Precision measurements of the critical density of Xe give values that are in close agreement with other data, but are more accurate. Impurities are found to have a negligible influence on the value of the critical density. The universality conjecture states that the critical exponents and certain amplitude ratios should be the same for all systems in a given universality class. Table 8.2 gives a comparison of the values of the order parameter exponent /? and the correction to scaling exponent A found in the various systems studied in this thesis. The values of /? found in these vastly different fluids and liquid mixtures agree within error, in agreement with the principle of universality. They are also in close agreement with values found in other experiments and expected theoretically. For C H F 3 , CC1F3 and nicotine + water, the value of A is smaller than expected theoretically (A « 0.5 [11]). This same trend has been previously observed in other experiments [89]. For He-Xe, A is significantly larger Chapter 8. Conclusions 154 Substance 0 A CCIF3 0.328 ± 0.002 0.43 ± 0.02 CHF 3 0.328 ± 0.002 0.41 ± 0.02 nicotine + water 0.329 ± 0.003 Xe 0.328 ± 0.003 0.44 ± 0.02 He-Xe 0.325 ± 0.002 0.50 ± 0.02 Table 8.2: Comparison of the order parameter exponent 0 and the correction to scaling exponent A for the substances studied in this thesis. than in the other substances investigated in this thesis and very close to theoretically calculated value. From experiments on CHF 3, the universal amplitude ratios TQ /T~ and DQTQBQ'1 have been determined (see table 5.7). They are in excellent agreement with theoretical predictions and with values found for nonpolar fluids, indicating that, as expected, the dipolar interaction between the molecules does not influence the universality class. In the experiments on the binary liquid system nicotine -f water, diffusivities were measured both by light scattering, an equilibrium method, and by focal plane inter-ference, a nonequilibrium method. In the nonequilibrium method, the diffusivity data depend on the initial concentration difference between the phases, and the values of the diffusivity to be compared with the light scattering data are obtained in the limit as the initial concentration discontinuity goes to zero. In each experimental run, lasting up to 15 hours, the diffusivity was found to be constant in time for the duration of the exper-iment, which indicates that boundary effects did not play a role. The data from light scattering and interference experiments follow the same trend, but exhibit quantitative differences. From the light scattering diffusivity data the critical exponent of the viscosity was extracted to yield zn = 0.05 ± 0.01, in good agreement with other experiments [76] and theory [77, 130]. Chapter 8. Conclusions 155 In the binary fluid system He-Xe, an example of "gas-gas equilibrium", the critical amplitude of the order parameter, Bo, and the compressibility are close to those of pure Xe. The fact that the isothermal compressibility of the mixture is as high as that of the pure substance leads to the conclusion that the He-Xe mixture behaves more like a fluid than a binary liquid. Thus, it seems as if the presence of He has mostly the effect of increasing the critical temperature and the critical pressure [109]. Even though this effect may be explained by our experimental method being unable to "see" the He, there is strong evidence [45] that in fact the He-Xe phase transition is of the liquid-gas type. In order to obtain a conclusive answer, binary fluid systems like Xe-Ar, in which both constituents have refractive indices markedly different from 1, and in which the sizes of the atoms are less dissimilar, should be investigated. Appendix A Error Analysis for Prism Cell Measurements This appendix describes how the refractive index of a fluid inside the prism cell can be obtained from the refraction angle, and presents an error evaluation for the critical density obtained from a prism cell measurement. A . l Calculation of refractive index from refraction angle We assume that the incident beam hits the first (straight) window of the prism at right angles. The refracted beam leaves the prism at an angle K with respect to the incident beam. Figure A.l shows the geometry and defines the angles. All dotted lines ( ) correspond to directions parallel to the incident beam. All dashed lines ( ) indicate normals to the window faces. na, na and the refractive indices of sapphire, air and xenon respectively. 8s(6t) is the wedge angle of the straight (tilted) window, and 6 is the prism angle. Angles a; are between the beam direction and a surface normal, are angles between the beam and the direction of the incident beam, and K is the total (measured) refraction angle. For obtaining an expression of the refractive index nxe of the substance in the cell as a function of the deflection angle K, we use the following equations: • Snell's law n„ sin ai = nxe sin a2 nxe sin a 3 = na sin a 4 na sin a 5 = n„ sin a6 156 Appendix A. Error Analysis for Prism Cell Measurements 157 Figure A.l: Refraction geometry of the prism cell experiment. For an explanation of the symbols, see text. Appendix A. Error Analysis for Prism Cell Measurements 158 • Window wedges cti = 6S and 0:5 — 0:4 = 0t • Direction of beam with respect to incident direction: « 2 + 0 — <*4 OLQ — K + a 5 — K2 As 0„ and 0t are small, we can Taylor-expand to obtain: sin 6a « 9, smOt ~ 0, cos « cos0a « 1 One thus obtains, to first order in 0a and tV-n = (n./ sin 0) [sin(K + 5) + • 0« (COS(K + 0) - ^ / n * - sin2( + 0,(sin(/c -f-0) cot 0 — n, cos (A.l) This relates the refractive index of the medium nxe to the refraction angle K. A. 1.1 Error Estimates Careful calibrations and an estimate of errors was essential for judging the accuracy of the critical densities determined in the prism-cell experiments. The following quantities were measured with the given accuracy: • Systematic errors: 1. Prism angle 0 = 20.525°, with 60 = 0.010°. 2. Sapphire window wedges: Straight window: 0a = -0.00018, 60a = 0.00002 Tilted window: 0t = -0.00041, 60t = 0.00003 3. Volume of prism cell V = 12.066cm3, 8V = 0.003cm3. Appendix A. Error Analysis for Prism Cell Measurements 159 4. Mass of empty prism cell: M0 = 179.3995g, 8M0 = 0.0030g. Items 1 and 2 limit the accuracy of refractive index measurements. They lead to errors 8npr, 6ns and Snt due to the error in the prism angle 6 and the tilt angles 0a and 6t respectively. Items 3 and 4 limit the accuracy of the density p — M/V. • Random errors 1. Mass readings 2. Micrometer reading 3. Temperature reading These errors produce a random error 8CC in the value of the Lorentz-Lorenz function at the critical point and a statistical error 8nc in the critical refractive index. In our experiments we found that The parameter C is obtained from a Taylor expansion of the Lorentz-Lorenz relation around the critical point, and is given by The total error in the critical density can then be evaluated to be (A.2) where C = 6nl (n2 + 2)( i .a- l)" Appendix B Critical Refractive Index and Critical Density of Nicotine + Water This appendix descibes the experimental methods used for measuring the critical refrac-tive index and the density of the nicotine + water mixture, and gives the results of the experiments. B . l Critical Refractive Index Measurement As is clear from equations (6.5) and (6.6), the scattering vector q and therefore the diffusivity D depend on the exact knowledge of the refractive index of the liquid mixture. In order to determine the refractive index nc of the critical mixture of nicotine + water as accurately as possible, we used an interferometric method. The mixture of critical composition was contained in a Hellma cell with parallel windows which in turn was held in a cylindrical cell holder to whose end a gear was attached. A geared-down toy motor turned the cell slowly about its vertical axis at a rate of about 1/24 rph. The cell-motor assembly was placed inside the thermostat in one arm of a Mach-Zehnder interferometer (see figure 3.7). As the cell was rotated, the optical path through it changed, leading to a change in the interference pattern between the cell beam and the reference beam. This interference pattern was detected in one spot in the image plane by a light sensitive diode (Hamamatsu). The diode output was monitored on an x — t recorder. The number of interference fringes is simply related to the optical path through the cell which depends on the refractive index of the medium inside. Thus, a measurement of the interference minima and maxima as a function of time (and thus of the rotation angle of the cell) 160 Appendix B. Critical Refractive Index and Critical Density of Nicotine + Water 161 could be used to extract the refractive index of the liquid. As the cell is rotated around its vertical axis, the angle <f> between the cell normal and the incident beam changes as a function of time r. The number of interference fringes N(T), as measured from normal incidence (<j> = 0) is N(T) = ^ (yfnl — sin2 <j> — cos <f> — n0 + 1^  + (y/n2 — sin2 <f> — cos <j) — n + lj (B.l) where d is the thickness of the quartz windows of the cell, no = 1.45709 is their refractive index, D is the sample thickness in the cell, n is the refractive index of the sample, and Ao is the laser wavelength. As the rotation frequency of the cell is known, one can relate the rotation angle c6(r) at any time r to the number of interference maxima recorded on the chart, and thus N(T). In order to determine the time To at which </>(r0) = 0, the run was started at <f> w —10° and TQ was taken such that the interference pattern was symmetric around To, i.e., that N(r + r0) = N(T — To). A calibration run with an empty cell (n = 1) gave 2d = 0.246 ± 0.001 cm, and measurement of the total cell thickness yields (D + 2d) = 0.445 ± 0.001 cm. Thus the window spacing was measured to be D = 0.199 ± 0.001cm. For the run with a filled cell, the interference pattern was recorded from <f> w —10° to <j> « 50°, where the number of fringes is 940. A least squares fit to expression (B.l) yielded n c = 1.3811 ±0.0004 (B.2) Figure B.l shows a plot of the number of interference fringes as a function of angle <f>. B.2 Densities of Nicotine + Water Mixtures The densitometer used for the measuring the densities of nicotine + water mixtures -consisted of a pyrex bulb with a 0.1 ml pipet of resolution 1/1000 ml attached to it. The Figure B.l: Plot of the number of interference fringes as a function of rotation angle <f>. The line corresponds to a fit giving a refractive index of nc = 1.3811. Appendix B. Critical Refractive Index and Critical Density of Nicotine + Water 163 volume of the bulb was calibrated by filling the bulb with deionized water at 20° C and weighing it on a chemical balance. We found Vo(20°C) = 63.274 ± 0.002cm3. Due to the thermal expansion of pyrex [147], the bulb's volume changes in the temperature range 20° C < T < 80° C approximately as V(T) = Vro(20°C) [l + (T - 20) * 12 x 10~6], (B.3) with T given in °C. The pipet was also calibrated with dionized water. The empty densitometer weighed 22.664 ± 0.001 g. For a measurement of the temperature dependence of the density p in a nicotine + water mixture of a given composition, the densitometer was filled with the mixture and weighed. The bulb was subsequently immersed into a carefully temperature regulated (to 0.1°C) stirred water bath, and the rise height of the liquid in the pipet and equivalently the volume occupied by the liquid, was determined. Since the density was measured over a large temperature interval, it was necessary to "bleed" liquid from the densitometer at regular intervals as the temperature was increased, in order to keep the liquid from overflowing, and to reweigh the densitometer. Figure B.2 shows a plot of the mixture densities as a function of temperature for pure nicotine, pure water, and a variety of mixtures around the critical composition. Note that for T > T c, a near-critical mixture is in the two-phase region, which means that the densitometer contains two phases in equilibrium. This may account for the fact that in the mixtures, the p vs. T curves exhibit a slight change in slope in the neighbourhood of the critical temperature, w 61.4°C. The concentration dependence of the density around the critical temperature is mea-sured from our experiments to be 0.273 ±0.044 mg/%cm3. This information can be used to "density-match" the liquid mixture close to the critical point, where the nicotine con-centration is « 40% [81]: An increase in the water density by about 0.66% will make the Appendix B. Critical Refractive Index and Critical Density of Nicotine + Water 164 o O iri CJ o \ + 4-2 o D O CO l _ 0) CL O io in o 100% nicotine 0.97 100% water /\ \\\\ 30.5 \? 7 X \ \ v 4 8 , 7 \ \ \ \ M5.9 40.3 32.2 36.7 % nicotine 0.98 0.99 density p (g/cm3) 1.00 1.01 Figure B.2: Temperature dependence of the density p of nicotine + water mixtures of a variety of compositions. The numbers indicate nicotine concentration in weight %. The uncertainty in the compositions is « 0.2%. Appendix B. Critical Refractive Index and Critical Density of Nicotine -f Water 165 coexisting phases in the vicinity of the critical point have the same density. To accom-plish this, 0.7 wt% heavy water D2O have to be admixed to the "normal" dionized water. Under these circumstances all gravity effects, which are already small in the nicotine -f water system (see section 6.3.1), can be made even smaller. Appendix C Thermal Gradients in the Sample Cell In this appendix we study the influence of a thermal gradient in the sample cell on compressibility measurements. A thermal gradient has the effect that the measured variation of density with height z has two contributions, one isothermal and the other due to the temperature variation [79]: For the extraction of compressibility data, we measure the spacing of the interference fringes in the vicinity of the meniscus (z = 0) where the pressure and density are at their critical value. There, one can write [2] (frL " (§)„, = " (IP)t (ff)„ " ik Or ©„' (C'2) Using the law of corresponding states [1], (dP/dT)Pc can be estimated to yield where R is the gas constant and b is one of the parameters in the van der Waals equation. The compressibility as measured in the experiment can then be written as 4 " = / C T [ i + -L. JL (£.)] (CA) T [ 0A95gM \dz)\ v ; with M. denoting the molar mass of the fluid. If the temperature dependence of the second term is weak, then Kj * has the same temperature dependence as KJ, and we can write: 4 " = o r (c.5) 166 Appendix C. Thermal Gradients in the Sample Cell 167 A comparison of the compressibility amplitude in the one-phase region of pure Xe, Y*jj = 0.21, (corresponding to the data point marked by "+" in figure 7.6) with the literature value TQ = 0.0575 [13] permits a calculation of the temperature gradient in our experiment. We obtain: This corresponds to an overall temperature difference of 7.5 mK over the total height of the cell. This seems like a very large gradient. It must be due to some heat leak from the cell's environment, which for the actual running conditions was 2-4 degrees warmer than the cell. This result is in general agreement with the test for thermal gradients performed prior to the experiments (see section 4.3), in which we found the cell bottom to be < 3 mK warmer than the top when the cell was 0.1-1 degrees warmer than the environment. In order to test for possible influences of the temperature of the outer bath on the compressiblity, the bath temperature was varied while keeping the cell temperature constant. This variation had little effect on the compressiblity, well within statistical scatter, so that we conclude that a thermal gradient like this does not stem from a heat leak from the cell to the bath. Rather, it is probably due to the stainless steel gas fill line. (C.6) • Bibliography [1] E. A. Guggenheim, Thermodynamics. North Holland, Amsterdam (1967). [2] H. E. Stanley, Introduction to Phase Transition and Critical Phenomena. Oxford University Press, New York (1971). [3] S.-K. Ma, Modern Theory of Critical Phenomena. Benjamin/Cummings Publish-ing Co., Reading, Massachussetts (1976). [4] M. E. Fisher in Critical Phenomena: Proceedings of the Summer School held in Stellenbosch, South Africa, ed. F. J. W. Hahne, Springer Verlag, New York (1983). [5] F. D. Lee and C. N. Yang, Phys. Rev. 87, 410 (1952). [6] M. J. George and J. J. Rehr, Phys. Rev. 53, 2061 (1977). [7] J. H. Chen, M. E. Fisher, and B. G. Nickel, Phys. Rev. Lett. 48, 630 (1982). [8] J. C. LeGuillou and J. Zinn-Justin, Phys. Rev. Lett. 39, 95 (1977). [9] B. Nickel and M. Dixon, Phys. Rev. B26, 3965 (1982). [10] S. Ma, Phys. Rev. Lett. 37, 461 (1976). [11] K. E. Newman and E. K. Riedel, Phys. Rev. B30, 6615 (1984). [12] A. Aharony and P. C. Hohenberg, Phys. Rev. B13, 3081 (1976). [13] H. Guttinger and D. S. Cannell, Phys. Rev. A24, 3188 (1981). [14] M. Barmatz, P. C. Hohenberg, and A. Kornblit, Phys. Rev. B12, 1947 (1975). 168 Bibliography 169 [15] L. A. Weber, Phys. Rev. A2, 2379 (1970). [16] M. W. Pestak, Ph.D. thesis. Pennsylvania State University (1983). [17] M. W. Pestak and M. H. W. Chan, Phys. Rev. B30, 274 (1984). [18] P. Schofield, Phys. Rev. Lett. 22, 606 (1969). [19] J. T. Ho and J. D. Litster, Phys. Rev. B2, 4523 (1970). [20] D. A. Balzarini, Can. J. Phys. 50, 2194 (1972). [21] L. R. Wilcox and D. Balzarini, J. Chem. Phys. 4 8 , 753 (1968). [22] M. R. Moldover, J. V. Sengers, R. W. Gammon, and R. J. Hocken, Rev. Mod. Phys. 51, 79 (1979). [23] U. Narger and D. A. Balzarini, Phys. Rev. B39, 9330 (1989). [24] L. Cailletet and E. C. Matthias, R. Hebd. Seanc. Acad. Sci. (Paris) 102, 1202 (1886). [25] M. Ley-Koo and M. S. Green, Phys. Rev. A16, 2483 (1977). [26] R. E. Goldstein, A. Parola, N. W. Ashcroft, M. W. Pestak, M. H. W. Chan, J. R. de Bruyn, and D. A. Balzarini, Phys. Rev. Lett. 58, 41 (1987). [27] B. M. Axilrod and E. Teller, J. Chem. Phys. 11, 299 (1943). [28] M. W. Pestak, R. E. Goldstein, M. H. W. Chan, J. R. de Bruyn, D. A. Balzarini, and N. W. Ashcroft, Phys. Rev. B36, 599 (1987). [29] D. Balzarini, J. R. de Bruyn, U. Narger, and K. T. Pang, Int. J. Thermophysics -9, 739 (1989). Bibliography 170 [30] J. N. Israelachvili, Intermolecular and Surface Forces. Academic, New York (1985). [31] W. L. Meerts and I. Ozier, J. Chem. Phys. 75, 596 (1981). [32] P. C. Johnson, T. L. Weatherby, and Q. Williams, J. Chem. Phys. 35, 2261 (1961). [33] P. C. Hohenberg and B. I. Halperin, Rev. Mod. Phys. 49, 435 (1977). [34] L. E. Reichl, A Modern Course in Statistical Physics. University of Texas Press, Austin (1980). [35] W. I. Goldburg in Light Scattering near Phase Transitions, eds. H. Z. Cummins and A. P. Levanyuk, North-Holland, Amsterdam (1983). [36] B. J. Berne and R. Pecora, Dynamic Light Scattering. J. Wiley & Sons Inc., New York (1976). [37] H. Z. Cummins and H. L. Swinney in Progress in Optics VIII, p. 133, ed. E. Wolf, North Holland, Amsterdam (1970). [38] H. L. Swinney in Photon Correlation and Light Beating Spectroscopy, eds. H. Z. Cummins and F. R. Pike, Plenum, New York (1974). [39] M. L. McGlashan, J. Chem. Thermodynamics 17, 301 (1985). [40] W. B. Streett, Can. J. Chem. Eng. 52, 92 (1974). [41] J. D. van der Waals, Zittingsv. K. Akad. Wet., p. 133. Amsterdam (1894). [42] G. M. Schneider, Adv. Chem. Phys. XVII. Wiley, New York (1970). [43] R. L. Hurle, F. Jones, and C. L. Young, J. Faraday Trans. II7Z, 613 (1977). [44] J. de Swaan Arons and G. A. M. Diepen, J. Chem. Phys. 44, 2322 (1966). Bibliography 171 [45] M. C. Bellissent-Funel, U. Buontempo, C. Petrillo, and F. P. Ricci, Phys. Rev. A40, 7346 (1989). [46] C. Domb, Adv. Phys. 9, 149 (1960). [47] J. Zinn-Justin, J. Physique 42, 783 (1981). [48] K. G. Wilson, Phys. Rev. B4, 3174 and 3184 (1971). [49] K. G. Wilson and J. Kogut, Phys. Rep. 12, 75 (1974). [50] L. P. Kadanoff, Physics 2, 263 (1966). [51] E. Brezin, J. C. LeGuillou, J. Zinn-Justin, and B. G. Nickel, Phys. Lett. 44A, 227 (1973). [52] A. A. Vladimirov, D. I. Kazakov, and 0. Tarasov, JETP 50, 521 (1979). [53] R. H. Swendsen, Phys. Rev. Lett. 42, 859 (1979). [54] G. S. Pawley, R. H. Swendsen, D. J. Wallace, and K. G. Wilson, Phys. Rev. B29, 4030 (1984). [55] F. J. Wegner, Phys. Rev. B5, 4529 (1972). [56] J. Adler, M. Moshe, and V. Privman, Phys. Rev. B26, 3958 (1982). [57] P. C. Albright, J. V. Sengers, J. F. Nicoll, and M. Ley-Koo, Int. J. Thermophys. 7, 75 (1986). [58] R. B. Griffiths, Phys. Rev. 158,176 (1967). [59] J. Rudnick and D. R. Nelson, Phys. Rev. B13, 2208 (1976). [60] A. Aharony and G. Ahlers, Phys. Rev. Lett. 44, 782 (1980). Bi bliography 172 [61] M.-C. Chang and A. Houghton, Phys. Rev. Lett. 44, 785 (1980). [62] M. E. Fisher and A. Aharony, Phys. Rev. Lett. 30, 559 (1973). [63] J. S. Rowlinson, Liquids and Liquid Mixtures. Butterworth, London (1969). [64] A. Aharony in Phase Transitions and Critical Phenomena, Vol. 6. Eds. C. Domb and M. S. Green, Academic Press, London (1977). [65] B. Widom and J. S. Rowlinson, J. Chem. Phys. 52, 1670 (1970). [66] B. Widom and F. H. Stillinger, J. Chem. Phys. 58, 616 (1973). [67] M. S. Green, M. J. Cooper, and J. M. H. Levelt Sengers, Phys. Rev. Lett. 26, 492 (1971). [68] J. J. Rehr and N. D Mermin, Phys. Rev. A8 , 472 (1973). [69] J. F. Nicoll and R. K. P. Zia, Phys. Rev. B23, 6157 (1981). [70] R. D. Mountain, Rev. Mod. Phys. 38, 205 (1966). [71] E. Jakeman in Photon Correlation and Light Beating Spectroscopy, eds. H. Z. Cummins and F. R. Pike, Plenum, New York (1974). [72] L. P. Kadanoff and J. Swift, Phys. Rev. 166, 89 (1968). [73] K. Kawasaki, Ann. Phys. (New York) 61 , 1 (1970). [74] E. D. Siggia, Phys. Rev. B13, 3218 (1976). [75] J. V. Sengers, Int. J. Thermophysics 6, 203 (1985). [76] H. C. Burstyn and J. V. Sengers, Phys. Rev. A25, 448 (1982). Bibliography 173 [77] T. Ohta and K. Kawasaki, Progr. Theor. Phys. 55, 1384 (1976). [78] H. C. Burstyn, J. V. Sengers, J. K. Bhattacharjee, and R. A. Ferrell, Phys. Rev. A28, 1567 (1983). [79] J. R. de Bruyn, Ph.D. thesis. University of British Columbia, Vancouver (1987). [80] D. Balzarini and P. Palffy, Can. J. Phys. 52, 2007 (1974). [81] A. N. Campbell, E. M. Kartzmark, and W. E. Falconer, Can. J. Chem. 36, 1475 (1958). [82] D. A. Balzarini, Can. J. Phys. 52, 499 (1974). [83] W. Jost, Diffusion. D. Steinkopff, Darmstadt (1957). [84] P. C. Hohenberg and M. Barmatz, Phys. Rev. A6, 289 (1972). [85] Operating and Installation Manual, Type K7023. Malvern Digital Correlator. [86] R. Foord, E. Jakeman, C. J. Oliver, E. R. Pike, and R. J. Blagrave, Nature 227, 242 (1970). [87] L. Williams, M.Sc. thesis. University of British Columbia, Vancouver (1986). [88] M. P. Stein, M.Sc. thesis. University of British Columbia, Vancouver (1986). [89] J. R. de Bruyn and D. A. Balzarini, Phys. Rev. A36, 5677 (1987). [90] T. C. Poulter, Phys. Rev. 35, 297 (1930). [91] E. W. Comings, High Pressure Technology. McGraw Hill (1956). [92] I. L. Spain and J. Paauwe, High Pressure Technology. Marcel Dekker Inc., New York (1977). Bibliography 174 [93] p. 37, Aminco Superpressure Catalogue, #466D. American Instrument Co. Silver Spring, MD (1974). [94] Autoclave Engineers, Catalog 6000. Autoclave Engineers Inc., Pennsylvania, 16512. [95] A. Michels, T. Wassenaar and P. Louwerse, Physica 20, 99 (1954). [96] US Bureau of Mines Report Invest. 7639 (1972). [97] U. Narger, J. R. de Bruyn, M. Stein, and D. A. Balzarini, Phys. Rev. B39, 11914 (1989). [98] A. D. Buckingham and C. Graham, Proc. R. Soc. London A336, 275 (1974). [99] H. H. Landolt and R. Bornstein, Zahlenwerte und Tabellen 1/3, pages 509ff. Springer Verlag, Berlin, (1951). [100] F. Baas and K. D. Van Den Hout, Physica A95, 597 (1979). [101] Estimated from comparison with ionization potentials of similar molecules in Lan-dolt & Bornstein, Zahlenwerte und Tabellen, Springer Verlag, Berlin (1951). [102] S. M. Khodeeva and I. V. Gubochkina, Zh. Fiz. Khim. 51, 1708 (1977). [103] T. E. Morsy, Kaltetechnik 18, 203 (1966). [104] J. M. H. Levelt Sengers, J. Straub, and M. Vincentini-Missoni, J. Chem. Phys. 54, 5034 (1971). [105] H. H. Landolt and R. Bornstein, Zahlenwerte und Tabellen II/l, page 346. Springer Verlag, Berlin (1951). Bibliography 175 [106] J. R. de Bruyn and D. A. Balzarini, Phys. Rev. B39, 9243 (1989). [107] J. V. Sengers and J. M. H. Levelt Sengers, Ann. Rev. Phys. Chem. 37, 189 (1986). [108] Yu. E. Sheludyak and V. A. Rabinovich, High Temp. Sci. 21 , 48 (1983). [109] J. R. Hastings, J. M. H. Levelt Sengers, and F. W. Balfour, / . Chem. Thermody-namics 12, 1009 (1980). [110] 0 . G. Mouritsen, D. Balzarini, and P. Palffy, Can. J. Phys. 61 , 1301 (1983). [Ill] D. Balzarini and 0 . G. Mouritsen, Phys. Rev. A28, 3515 (1983). [112] J. M. H. Levelt Sengers and S. C. Greer, Int. J. Heat Mass Transfer 15, 1865 (1972). [113] S. Y. Larson, R. D. Mountain, and R. Zwanzig, J. Chem. Phys. 42, 2187 (1965). [114] R. Hocken and G. Stell, Phys. Rev. A8 , 887 (1973). [115] J. V. Sengers, D. Bedeeaux, P. Mazur, and S. C. Greer, Physica 104A, 573 (1980). [116] A. Kumar, H. R. Krishnamurti, and E. S. R. Gopal, Phys. Reports 98, 58 (1983). [117] J. Shelton and D. Balzarini, Can. J. Phys. 59, 934 (1981). [118] D. E. Koppel, J. Chem. Phys. 57, 4814 (1972). [119] B. Chu and F. J. Lin, J. Chem. Phys. 61 , 5132 (1974). [120] S. C. Greer, T. E. Block, and C. M. Knobler, Phys. Rev. Lett. 34, 250 (1975). [121] E. Dickinson, C. M. Knobler, V. N. Schumaker, and R. L. Scott, Phys. Rev. Lett. 33, 180 (1975). Bibliography 176 [122] M. Giglio and A. Vendramini, Phys. Rev. Lett. 35, 168 (1975). [123] D. Beysens, J. Chem. Phys. 71, 2557 (1979). [124] W. I. Goldburg in NATO Advanced Study Series B73, Scattering Techniques Ap-plied to Supramolecular and Nonequilibrium Systems, eds. S.H. Chen, B. Chu and R. Nossal. Plenum Press, New York (1981). [125] M. Takahashi, H. Horiuchi, S. Kinoshita, Y. Ohyama, and T. Nose, J. Phys. Soc. Japan 55, 2687 (1986). [126] H. H. Landolt and R. Bornstein, Zahlenwerte und Tabellen 11/5. Springer Verlag, Berlin (1951). [127] N. Nagarajan, A. Kumar, E. S. R. Gopal, and S. C. Greer, J. Phys. Chem. 84, 2883 (1980). [128] S. C. Greer, Phys. Rev. A14, 1770 (1976). [129] C. M. Sorensen, R. C. Mockler, and W. J. O'Sullivan, Phys. Rev. Lett. 40, 777 (1978). [130] E. D. Siggia, B. I. Halperin, and P. C. Hohenberg, Phys. Rev. B13, 2110 (1976). [131] D. L. Henry, H. L. Swinney, and H. Z. Cummins, Phys. Rev. Lett. 25, 1170 (1970). [132] H. L. Swinney and H. Z. Cummins, Phys. Rev. 171, 152 (1968). [133] D. Y. Parpia and B. L. Smith, J. Phys. C4, 2254 (1971). [134] J. A. Chapman, P. C. Finnimore, and B. L. Smith, Phys. Rev. Lett. 21 , 1306 (1968). Bibliography 177 [135] H. W. Habgood and W. G. Schneider, Can. J. Chem. 32, 98 (1954). [136] J. M. H. Levelt, Physica 26, 361 (1960). [137] D. H. Garside, H. V. Molgaard, and B. L. Smith, J. Phys. Bl , 449 (1968). [138] V. G. Baidakov, A. M. Rubshtein, V. R. Pomortsev, and I. J. Sulla, Phys. Lett. A131, 119 (1988). [139] A. B. Cornfeld and H. Y. Carr, Phys. Rev. Lett. 29, 28 (1972). [140] L. D. Landau and E. M. Lifshitz, Fluid Mechanics. Addison-Wesley, Reading (1959). [141] L. V. Mikheev, Phys. Lett. A129, 245 (1988). [142] J. W. Cahn, J. Chem. Phys., 66, 3667 (1977). [143] R. F. Kayser, J. W. Schmidt, and M. R. Moldover, Phys. Rev. Lett. 54, 707 (1985). [144] M. W. Zemansky and R. H. Dittman, Heat and Thermodynamics. McGraw Hill, New York (1981). [145] M. R. Moldover and R. W. Gammon, J. Chem. Phys. 80, 528 (1984). [146] E. E. Polymeropoulos, J. Brickmann, L. Jansen, and R. Block, Phys. Rev. A30, 1593 (1984). [147] R. J. Corrucini and J. J. Gniewek, Thermal Expansion Coefficients of Technical Materials. NBS Monograph 29, US Department of Commerce, Washington (1961). 

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