INVESTIGATEON OF THE LIQUID STSiIICTURE 0F DlMETEYL SULFUXIDE-PYRIDINB MIXTURES WITH BREEQUM SPEETRMCOM’ THESE FOR TEE DEGREE 0? PH. D. EECKEGER STATE UNIVERSITY MARY MARGARET TANNAHILL 1973 lb. 1“" (‘35: k LIBRAR Y ”libiligui State University - ‘ ‘ 5", - ' '9' I :' - ' .. II -0 , _ $ ‘ ‘1 ’7.) V ABSTRACT INVESTIGATION OF THE LIQUID STRUCTURE OF DIMETHYL SULFOXIDE- PYRIDINE MIXTURES WITH BRILLOUIN SPECTROSCOPY BY Mary Margaret Tannahill A Brillouin scattering study of binary solvent mixtures of dimethyl sulfoxide (DMSO) and pyridine was performed in order to gain insight into the structural changes occurring in solution as the composition was varied from pure pyridine to pure DMSO. Thirteen solutions, ranging from neat pyridine to neat DMSO, were examined spectrometrically at eight different temperatures between 20 and 60 oC. The light scattering and acoustical properties determined were the Brillouin shift, Brillouin linewidth, Landau-Placzek ratio, velocity of sound, sonic absorption coefficient and adiabatic compressibility. Density and refractive index data for the mixtures were also recorded. Examination of the refractive index, Brillouin shift, velocity of sound, sonic absorption coefficient and Brillouin linewidth as a function of temperature revealed a linear relationship for each property. The refractive index and Brillouin linewidth were also found to be linear functions of composition in the temperature range from 20 to 60 °c. Fluctuations in the velocity of sound and frequency-corrected absorption coefficient over the entire compositional region from neat pyridine to neat DMSO indicated changes in the degree of association in the liquid. Neat DMSO was ascertained to be a rather highly Mary Margaret Tannahill structured liquid and appeared to possess a greater degree of organiza- tion than either neat pyridine or any of the binary mixtures. Neat pyridine, however, was also noted to possess a considerable degree of ordering. Maximum disorganization of the liquid structure occurred at approximately the equimolar composition for each of the eight tempera- tures, implying that the associated species present in the DMSO—pyridine mixtures are aggregates of DMSO and/or pyridine rather than complexes between DMSO and pyridine. Infrared spectroscopic measurements of the 5-0 stretching frequency for mixtures of dimethyl sulfoxide and pyridine at 22 00 support the concept of self-association of DMSO. The Landau-Placzek ratio was found to be a linear function of the temperature for each of the DMSO-pyridine mixtures investigated. There was a distinct break, however, in the Landau-Placzek ratio- temperature curve for pure DMSO at 45 oC,'indicating reorganization of the liquid structure at this temperature. The overall variations in the Landau-Placzek ratio with compo- sition gave supporting evidence to the interpretations of the velocity of sound and absorption coefficient data. In particular, a vast in- crease in association between 0.80 and 1.00 mole fraction DMSO was heralded by a significant decrease in the Landau-Placzek ratio in this region. INVESTIGATION OF THE LIQUID STRUCTURE OF DIMETHYL SULFOXIDE- PYRIDINE MIXTURES WITH BRILLOUIN SPECTROSCOPY BY Mary Margaret Tannahill A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1973 to the memory of my grandfather, James S. Campbell ii ACKNOWLEDGMENTS The author wishes to express her thanks to the following people, without whose technical and/or intellectual expertise this work could not have been completed: Stuart Gaumer, designer of the Brillouin scattering spectrometer; Charles Hacker and Russell Geyer of the University Machine Shop; Keki Mistry, Jerry DeGroot and Andrew Seer, Jr. of the University Glassblowing Shop; Henny Seeuwen and Kathy Pagan, typists and Jo Llu Long, graphics illustrator, at the University of Texas Medical Branch in Galveston. My sincere appreciation also goes to my mentor, Dr. Jack B. Kinsinger, for his wisdom and humor during the course of the research, and to my parents, Mr. and Mrs. C. F. Tannahill, for their unswerving moral support during this endeavor. iii I. II. III. TABLE OF CONTENTS INTRODUCTION DESCRIPTION OF LIGHT SCATTERING FROM INDEPENDENT PARTICLES A. General Concepts 8. Rayleigh Scattering C. Pine Structure of the Rayleigh Peak: Brillouin Scattering l. Brillouin scattering from a pure fluid 2. Brillouin scattering from a two-component mixture a. Derivation of the spectral distribution of light scattered from a binary liquid system b. Relationship between the Brillouin linewidth, phenomenological coefficients and thermodynamic parameters D. Effects of Optical Properties of Polar Liquids on Light Scattering EXPERIMENTAL INFORMATION A. The Brillouin Spectrometer B. Design and Characterization of the Temperature Control Cell C. Preparation of the Samples D. Measurement of the Refractive Index iv Page 28 28 37 43 53 S3 56 63 64 TABLE or CONTENTS (Cont.) IV. RESULTS AND DISCUSSION A. Variation of the Refractive Index with Temperature and Composition 8. Molar Refractivity at 22.0 °C C. Variation of the Brillouin Shift with Temperature and Composition D. Velocity of Sound and Related Quantities E. Variation of the Brillouin Linewidth with Temperature and Composition F. Sonic Absorption Coefficient and Related Quantities G. Variation of the Landau-Placzek Ratio with Temperature and Composition CONCLUSIONS LIST OF REFERENCES APPENDICES Page 65 65 71 76 91 114 157 181 192 194 199 TABLE 10 11 12 13 14 15 LIST OF TABLES Measured and Corrected Temperatures Obtained During Brillouin Scattering Measurements for Mixtures of DMSO and Pyridine Nominal and Measured Compositions for Mixtures of DMSO and Pyridine ’ Refractiv Index as a Function of Temperature for A = 5890 and 5145 . Molar Refractivities for Mixtures of DMSO and Pyridine at 22.0 °c Brillouin Shift as a Function of Temperature Intercept and Slope for the Brillouin Shift-Temperature Relationship v3 = A + BT Brillouin Shift and Velocity of Sound as a Function of Temperature for Mixtures of DMSO and Pyridine Intercept and Slope for the Sonic Velocity-Temperature Relationship Vs - A + BT Squares of the Ideal and Experimental Velocities of Sound for 21.6 qC Adiabatic Compressibility as a Function of Composition for 2l.6 °C Experimental and Calculated Intensities for the Central Peak of 0.80 Mole Fraction DMSO at 59.0 °C Experimental and Calculated Intensities for the Left Brillouin Peak of 0.80 Mole Fraction DMSO at 59.0 °C Experimental and Calculated Intensities for the Central Peak of Neat nuso at 59.0 °c Experimental and Calculated Intensities for the Left Brillouin Peak of Neat DMSO at 59.0 °C Experimental and Calculated Intensities for the Central Peak of Neat Pyridine at 29.5 0C vi Page 62 66 67 75 84 88 96 100 101 104 122 125 131 133 135 LIST OF TABLES (Cont.) TABLE 16 17 18 19 20 21 22 23 24 25 26 Experimental and Calculated Intensities for the Right Brillouin Peak of Neat Pyridine at 29.5 °C Linewidth of the Central Peak as a Function of Finesse Comparison of the Linewidths of the Central Peaks of Neat DMSO, Neat Pyridine and 0.40 Mole Fraction DMSO at a Given Finesse Brillouin Linewidth as a Function of Temperature for Mixtures of DMSO and Pyridine Absorption Coefficient as a Function of Temperature for Mixtures of DMSO and Pyridine Temperature Derivative of the Sonic Absorption Coefficient Landau-Placzek Ratio as a Function of Temperature for Mixtures of DMSO and Pyridine Temperature Calibration Data Calculation of the Dispersion for Mixtures of DMSO and Pyridine at 26.4 °C Calculation of n51452 for Mixtures of DMSO and Pyridine at 26.4 °c ' Temperature Dependence of the Refractive Index for A . 5890 and 5145 2 vii Page 137 144 151 152 160 167 182 199 203 204 205 FIGURE 10 11 12 13 14 15 16 17 18 19 LIST OF FIGURES The light scattering process The Brillouin spectrometer Sketch of the temperature control cell Cross-sectional view of the temperature control cell Calibration data for the temperature control cell Refractive index versus temperature for mixtures of DMSO and pyridine Refractive index versus temperature for neat DMSO Variation of the temperature derivative of the refractive index with composition for mixtures of DMSO and pyridine Refractive index versus composition for 30.1 0C Refractive index versus composition for 22.0, 26.4 and 30.1 °c Variation of the density with composition for 22.0 0C Molar refractivity versus composition at 22.0 °C Brillouin spectrum.for 0.15 mole fraction DMSO at 34.5 °C Brillouin spectra for neat DMSO a t 29.5 and 59.0 °c Brillouin spectra for 0.60 mole fraction DMSO at 39.4 and 59.0 °C Brillouin spectra for neat pyridine at 21.6 and 59.0 0C Brillouin shift versus temperature for neat DMSO Brillouin shift versus temperatur 0.10, 0.15, 0.30 and 0.50 mole fr variation of the intercept of V8 for mixtures of DMSO and pyridine viii e for neat pyridine, action DMSO - A + BT with composition Page 54 57 58 61 69 70 73 74 77 78 81 82 83 86 87 89 LIST OF FIGURES (Cont.) FIGURE 20 21 22 23 24 25 26 27 28 29 . 30 31 32 33 34 35 36 Variation of the slepe of v3 = A + BT with composition for mixtures of DMSO and pyridine Brillouin shift versus composition at 21.6 0C Brillouin shift versus composition for 21.6, 29.5, 34.5 and 39.4 °c Brillouin shift versus composition for 44.5, 49.3, 54.2 and 59.0 °c Temperature coefficient of the velocity of sound Variation of the velocity of sound with composition for 21.6, 29.5, 34.5, 39.4, 44.5, 49.3, 54.2 and 59.0 °c Squarg of the velocity of sound versus composition for 21.6 C Variation of the adiabatic compressibility with composition for 21.6 00 Variation of the 8-0 stretching frequency with composition at 21.6 °C Square of the velocity of sound versus composition for 21.6, 29.5, 34.5. 39.4, 44.5, 49.3, 54.2 and 59.0 °c Portion of Brillouin spectrum for neat pyridine at 29.5 °C Central peak of Brillouin spectrum for 0.80 mole fraction DMSO at 59.0 9c Left Brillouin peak of Brillouin spectrum for 0.80 mole fraction DMSO at 59.0 °c Centrgl peak of Brillouin spectrum for neat DMSO at 59.0 C Left Brillouin peak of Brillouin spectrum for neat DMSO at 59.0 °c Left-hand side of central peak for 0.80 mole fraction DMSO at 59.0 °C fit to a Lorentzian Rightrhand side of central peak for 0.80 mole fraction DMSO at 59.0 0C fit to a Lorentzian ix _ Page 90 92 93 94 102 103 105 107 109 111 117 118 119 120 121 127 128 LIST OF FIGURES (Cont.) FIGURE 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 Left-hand side of left Brillouin peak for 0.80 mole fraction DMSO at 59.0 °C fit to a Lorentzian Right-hand side of left Brillouin peak for 0.80 mole fraction DMSO at 59.0 °C fit to a Lorentzian Laser-line instrumental profile, spectrum l-b, August 17, 1971 Laser-line instrumental profile, spectrum l-b, August 17, 1971, fit to a Gaussian Left-hand side of laser-line instrumental profile, l-b, August 17, 1971, fit to a Lorentzian Right-hand side of laser-line instrumental profile, 1-b, August 17, 1971, fit to a Lorentzian Linewidth versus finesse for central peak of neat DMSO Linewidth versus finesse for central peak of neat pyridine Linewidth versus finesse for central peak of 0.40 mole fraction DMSO Brillouin linewidth versus temperature for mixtures of DMSO and pyridine Brillouin linewidth versus composition for 34.5 0C Brillouin linewidth versus composition for 21.6, 29.5, 34.5, 39.4, 44.5, 49.3, 54.2 and 59.0 00 Absorption coefficient versus temperature for 0.20 mole fraction DMSO Absorption coefficient versus composition at 21.6 °C Absorption coefficient versus composition at 54.2 0C Frequency-corrected absorption coefficient versus temperature for 0.70 mole fraction DMSO Frequency-corrected absorption coefficient versus temperature for 0.05, 0.10, 0.30 and 0.50 mole fraction DMSO Frequency-corrected absorption coefficient versus composition at 21.6 °C x Page 129 130 139 140 141 142 148 149 150 156 158 159 169 171 172 173 174 176 LIST OF FIGURES (Cont.) FIGURE 55 56 57 58 S9 60 61 62 63 Frequency-corrected absorption coefficient versus composition for 21.6, 29.5, 34.5 and 39.4 °c Frequency-corrected absorption coefficient versus composition for 44.5, 49.3 and 54.2 °c Velocity and absorption of sound for mixtures of DMSO and pyridine at 21.6 00 Absorption of sound and adiabatic compressibility for mixtures of DMSO and pyridine at 21.6 °C Landau-Placzek ratio versus temperature for neat pyridine Landau-Placzek ratio versus temperature for 0.80 mole fraction DMSO Landau-Placzek ratio versus temperature for neat DMSO Variation of the Landau-Placzek ratio with composition for 29.5, 34.5, 39.4, 44.5, 49.3, 54.2 and 59.0 °c Variation of the Landau-Placzek ratio with 85/02 for mixtures of DMSO and pyridine at 21.6 °c Page 177 178 179 180 185 186 187 188 190 I believe that man will not merely endure he will prevail Re is immortal not because he alone among creatures has an inexhaustible voice but because he has a soul a spirit capable of compassion and sacrifice and endurance William Faulkner Nobel Prize Acceptance Speech Stockholm December 1950 I. INTRODUCTION Because of its unique solvation properties, dimethyl sulfoxide (DMSO) has been the subject of wide-ranging scientific investigations in the past twenty years (1). Specific chemical reactions of DMSO have been studied, as well as the physical properties, solvation effects and structure of this aprotic polar solvent. It is generally accepted that liquid and solid DMSO are highly associated substances (cf. the physical properties listed on page 5 of Ref. 1), and that the forces responsible for this high degree of association arise from dipole-dipole interactions between the sulfur and oxygen atoms of adjacent molecules. It is postulated that there are several different structural forms in which molecular aggregates of DMSO exist (1), depending on the mole fraction of DMSO in a particular mixture of the liquid and on the amount of kinetic energy available. These structural forms have been examined previously by a vari- ety of methods: (1) infrared spectroscopy (2-5). (2) thermodynamic studies (6-9). (3) dielectric constant and refractive index measurements (6), (10). (11). (4) nuclear magnetic resonance spectroscopy (12). (5) density and viscosity measurements (6). (6) mass spectroscopy (6) . Many of these investigations were performed in binary solvent mixtures of DMSO, in the hope that the concentration dependence of the prOperty under study would give some clue as to the type of DMSO aggre- gate formed as the mole fraction of DMSO increased. No publications have appeared, however, concerning the use of light scattering spectros- copy to characterize the structural forms of dimethyl sulfoxide in bi- nary solvent mixtures. The purpose of this brillouin scattering study of dimethyl sulf- oxide-pyridine mixtures is to elucidate structural changes which are postulated to occur in the liquid as the composition is varied from pure pyridine to pure DMSO. Specifically, it is the author's intent (l) to present data on the velocity of sound in binary mixtures of DMSO and pyridine, and to attempt to correlate changes in the sonic velocity with changes in the liquid structure. (2) to show that structural changes are also manifest in the concentration and temperature dependence of other light- scattering parameters, namely, the Landau-Placzek ratio and the attentuation coefficient of sound waves in the liquid. (3) to correlate the data obtained from the light scattering experiments with data obtained by other physical-chemical methods. II. DESCRIPTION OF LIGHT SCATTERING FROM INDEPENDENT PARTICLES A. General Concepts Light scattering from a system of independent particles occurs when an incident electromagnetic wave of frequency Vo strikes a scatter- ing particle, inducing an oscillating electric dipole in a direction perpendicular to the electric vector of the incident radiation. This oscillating dipole then emits radiation of frequency vs in a direction perpendicular to the direction of propagation of the incoming radiation. If the frequency of the scattered wave is equal to that of the incoming wave, elastic scattering is said to have taken place. Inelastic scat- tering occurs if the frequency of the scattered wave is not equal to the frequency of the incoming wave; this implies a scattered wave of differ- ent energy. Pictorially, the light scattering process can be viewed in this manner : hvow O Scattering particle hv s 00 = vs, elastic scattering process v0 f vs, inelastic scattering process Figure l. The light scattering process. Rayleigh scattering is the term used to describe an elastic scattering process in which the size of the scattering particle is small compared to the wavelength of the incident radiation. In this process, the scattering particle acts like a radiating linear dipole after im- pact with the incident electromagnetic wave. Mie scattering occurs when radiation of small wavelength (compared to the dimensions of the scat- tering particle) strikes a macromolecular system of large relative index of refraction. Here the scatterer can no longer be considered to be a simple oscillating dipole, and the intensity and distribution of the scattered light assumes a much more complicated form than for small- particle Rayleigh scattering. B. Rayleigh Scattering As outlined in Section A, light scattering occurs when an inci- dent electromagnetic wave interacts with the polarizability of a scat- tering particle. The dipole moment induced in the x direction by an incoming beam traveling in the z direction is given by A “x = axyEy (l) where axy is the x-y component of the polarizability tensor and BY is the magnitude of the electric vector of the incident radiation. In other words, an electric vector in the y direction causes an electric dipole in the x direction because of the distortability of the electron distribution in the molecule. The polarizability component,axy, is a measure of this distortability. In general, the polarizability is a function of direction and must be regarded as a tensor of the second order: a u xx xy axz a = an aYY aYZ - (2) a zx azy azz The subscripts x, y, and 2 denote the Cartesian coordinates of the scattering system under consideration. For isotr0pic molecules, the only terms of importance in the polarizability tensor are axx' ayy' and ozz' The off-diagonal elements are zero in the case of isotropic scattering particles. The intensity of light scattered in the x direction by an in- coming beam of light with the electric vector oscillating in the y di- rection is given by the relation w4 2 2 Ix = —--3- a | E (3) 3C YY Y where w is the angular frequency of the incident radiation and c is the velocity of light in_ygggg. Switching to polar coordinates, we have, for isotrOpic scatter- ing particles of average polarizability, E) the Rayleigh equation (13) for the total intensity of radiation scattered in the direction 0 from an unpolarized incident beam of light: Io BN4 3'2 2 16 = .4 2 (l + cos 8)o (4) A r In equation (4), I0 is the intensity of the incident radiation, a'is the average isotropic polarizability (l4), given by E'= + o (5) 3'[axx + ayy zz ' 6 is the angle between the direction of observation and the direction of propagation of the incident beam, r is the distance between the scattering particles and the observer, and 1' is the wavelength of light in the medium. 1we will assume in the ensuing discussion that we are dealing with isotropic molecules. 0‘ A150, A. = I (6) 5'0» where A is the in vacuo wavelength of the incident light and n is the o ——.———— index of refraction of the medium. Bhagavantam (15) has derived a relationship for the average po- larizability, 0; for an isolated molecule treated as a dielectric sphere: —- At =-—— . 7 a 4fiv ( ) here 08 is the fluctuation in the dielectric constant about the average value, c, in the small spherical scattering volume, v. meloying equation (7) for SB the Rayleigh relation for the scattering of light from an isolated isotrOpic molecule can be written as 2 . . Ion < (110‘5 > (1 + c0526) I = . . (8) 6 2A'4 r2 v2 2 . . . . . where < (As) > is the mean square fluctuation in the dielectric con- stant. 10' it should be recalled, is the total intensity of the light scattered in the directionfi. Specific expressions are derived in sec- tion C for the intensity spectrum of the Rayleigh-scattered light. C. Fine Structure of the Rayleigh Peak: Brillouin Scattering In 1922, almost ten years after initiating a theoretical in- vestigation of the scattering of light from dense media, Leon Brillouin published his theory concerning the fine structure of the Rayleigh peak (16). Brillouin postulated the existence of three peaks for a highly 1Introduction of < (As)2 > is justified by a consideration of the statistical treatment of the fluctuations in the dielectric constant. resolved Rayleigh spectrum: a central peak (the Rayleigh peak itself) and two symmetrical Doppler-shifted side peaks. Beginning with Debye's theory of the specific heat of solids (l7) and Einstein's theory for the scattering of light from density fluctua- tions in an ideal gas (18), Brillouin derived a formula for the scatter- ing of light from density fluctuations in a dense medium. Much to his amazement, he found that "... only one Debye vibration was responsible for the scattering of light at given light frequency v and scattering angle m with hypersonic vibration v0... (19)." These unusual results were published as a note in the Comptes Rendus de 1'Academie des Sciences of Paris in 1914 (20). Before Brillouin had a chance to pursue these light scattering hypotheses to fruition, however, World War I commenced and he left with the French Army for the field. Upon returning to his theoretical papers in 1919, Brillouin decided to attack the problem of light scattering from a different point of view. He considered the passage of ultra- sonic or hypersonic waves in a dense medium as creating "... a succes- sion of planes of higher and lower density, moving along with sound velocity u ... (19)." Since each of the phonon planes of frequency vo reflects light waves of a specific frequency, v 1, there is only one way in which the incident beam can be modified in frequency after an inter- action with one of these planes: it must be Doppler shifted by an amount ivo from the incident frequency, v. In other words, Av : i v-v 2vnu O = i sin w/2 , (9) 1Note that Brillouin's symbols for the frequency of the incident photon and for the scattering phonon are exactly reversed with respect to the symbols which are used in subsequent portions of this work. 8 where Av is the change in frequency exhibited by the Doppler-shifted side peaks, n is the refractive index of the medium and c is the veloci- ty of light, in_yagug. Equation (9) is the principal equation of Brillouin scattering; it describes what happens when an electromagnetic wave of frequency v interacts with a sound wave of velocity u in the liquid. 1. Brillouin scattering from a pure fluid Light is scattered by a pure fluid as a result of local fluctu— ations (inhomogenieties) in the optical dielectric constant, (A62). The fluctuations in the dielectric constant are, in turn, dependent upon several factors: fluctuations in density, orientational fluctuations and fluctuations caused by dipole-induced-dipole interactions and by shear stresses and strains. Fluctuations caused by reorientation of non- spherical molecules, by dipole-induced-dipole interactions and by shear stresses and strains in the liquid are responsible for depolarized light scattering (21). According to Landau and Placzek (22), the density fluctuations that account for polarized scattering are divisible into two major components: isobaric entropy fluctuations, which produce a peak centered at the incident frequency (the Rayleigh peak) and isen- tropic pressure fluctuations, which give rise to two Doppler-shifted side peaks (the Brillouin peaks). The three peaks taken together are called a Brillouin spectrum: this polarized spectrum can also be con— sidered to result from time-dependent fluctuations in the dielectric constant. Several theoretical papers have been published concerning the intensity distribution of light scattered from density fluctuations in a pure fluid (22-29); however, only three of these (25-27) establish a 9 foundation for our later consideration of light scattering in a binary mixture. We begin with the pure thermodynamic fluctuation theory of Cummins and Gammon (25). On the basis of thermodynamic arguments alone, Landau and Placzek (22) derived a theoretical expression for the rates of the in- tensity of the central peak, IC, to the combined intensities of the Brillouin side peaks, ZIB: _E_=_P____‘_’.=_T____§ (10) In equation (10), cp and cv are the specific heats at constant pressure and volume, respectively, B is the isothermal compressibility and 85 T is the adiabatic compressibility. Through the years it was found, how- ever, that the Landau-Placzek equation did not match with experimentally determined values of IC/ZIB (25, 30, 31). By the introduction of a frequency-dispersion correction term, Cummins and Gammon (25) sought to alleviate this difficulty. They examined the classical fluctuation theory of light scattering and then modified the resulting equations to take into account the effects of dispersion (frequency dependence) on the thermodynamic properties of the system. According to Einstein (18), the intensity, 1, of light scat- tered by a small volume element, U, of a pure fluid when a plane po- larized beam of light is incident on it at 90°:Lcan be given by: i Io1T2 2 2 90 = u < (As) > . (11) 2 4 , r A O The scattering angle, 8, is always measured from the direction of propagation of the incident beam to the direction of observation of the scattered light. 10 90 is the intensity of the scattered light, I0 is the intensity of the incident beam, r is the distance from the scatter- In equation (11), i ing element to the point of observation, and < (As)2 > is the mean square fluctuation of the dielectric constant. This equation applies to a system of particles which are small (< one tenth) compared to the wavelength of the incident radiation. Furthermore, it is assumed that the random fluctuations in the local dielectric constant are uncorre- lated from one volume element to another. Einstein expressed the dielectric constant in terms of the in- tensive variables density, p, and temperature, T, such that ._ as .32. At — [30] Ap «9- [8T] AT (12) T 0 p 2 . 2 and <(Ae)2> = 33} <(A0)2> + 32- <(AT)2>.1 <13) 30 T 8T 0 Einstein then neglected the second term in equation (13) on the assump- tion that (Be/3T)p is small compared to (36/3p)T. Coumou, Mackor and Hijmans (32) have verified experimentally that (Be/3T%)<< (35/3p)T so that indeed equation (13) reduces to 2 <(Ae)2> = [32} <(Ap)2>. (14) p T According to statistical mechanical fluctuation theory (33), 2 <(Ap) > = kTBsz/v. (15) By substituting (l4) and (15) into (11), we obtain the following ex- pression for the intensity of light scattered at 900 from the incident beam: 1 The cross terms involving Ap and AT drOp out because of the statistical independence of p and T. 11 3 2 ...__ . 16 pap] ( ) In these equations k is Boltzmann's constant. Another method of evaluating ((A6)2> is to choose entropy and pressure (22) as the independent variables for the dielectric constant, C, and to write the fluctuations in c as Ae = (Be/05)p AS + (3e/3p)s Ap. (17) The cross terms involving AS and Ap are also disregarded because of the statistical independence of p and S, so we are left with 2 2 2 2 2 “(Ad > = (86/35)}? <(AS) 7' + (3t/BP)S <(AP) >- (18) From statistical mechanical fluctuation theory we have <(Ap)2> = kT/vgS and <(AS)2> = kcppv. (19) By using a simple mathematical transformation and thermodynamic iden- tities, it is easy to show that the second term of equation (18) can be given by 2 2 Be 2 _ , 2 as [SPJS <(Ap) > - kFBSp [53} - (20) Employing still another mathematical transformation and more thermody- namic relations one can write the first term of equation (18) as 2 2 2 36 2 _ RT SE. [as] <(As) > - C “(8.1) . (21) P P P After substitution of expressions (20) and (21) into equation (11), the scattering intensity at 900 becomes 2 . I n v 2 2 2 i o kT [as] [ as] 90 = —— — + kTB p— (22) 2 I r A: pep 3T p S 30 S 12 where the first term in the brackets is the Rayleigh component of the scattering intensity and the second term represents the total Brillouin scattering intensity. One final simplification results if we use the thermodynamic transformation 2 (at/Do)s = (ac/3p)T + (T/Cvp )(ap/BTL (BE/8T)p (23) and neglect the second term (see Ref. 25): 2 . 111V 2 2 r 2 1 0 k1‘ dc dc = -—- + ‘ ——- . 24 9° 2 4 pc [3T] kas[°ap] ( ’ er P P T Because of the narrow Doppler shift of the Rayleigh peak observed ex- perimentally, one can assume that the Rayleigh term (first term) re- presents slowly damped fluctuations. The Brillouin (second) term re- presents high frequency non-static pressure fluctuations. The first term can be accurately expressed in terms of the static values of the thermodynamic entities, while the variables in the second (Brillouin) term should be given the hypersonic values associated with the high frequency of the Brillouin shift. Equation (24), with a subscript "static" on the first term in the brackets and a subscript "hypersonic" on the second term, is the dispersion-corrected scattering intensity. The same equation with the subscript "static" on both terms is the original result of the Landau-Placzek theory. As was stated earlier, the original theoretical prediction of the ratio of the intensity of the central component to the combined intensity of the Brillouin components is given by 5?: TI (10) where the values of BT and BS are considered to be the equilibrium 13 (static) values. We have already derived an equation for the total scattering intensity from a pure fluid---equation (16); note that this equation is incorrect only with respect to the contribution of the Brillouin components. Therefore, if we subtract the static-value Brillouin term of equation (24) from the total scattering intensity, we should be left with a correct expression for the intensity of the central component: I 13‘! 2 I = IT - 21B = 02 4 der[g%E] C static r Ab p T I nzv 35 2 . 0 static Combining equation (25) with the second term (hypersonic limit) of (24), we obtain a dispersion-corrected expression for the Landau-Placzek ratio of a pure fluid: [ as 2 as 2 kTB [p-—J — kTB [-——] Ic T 3° T 5 ap T static "“ = (26) 21. 2 . we) 9 T hS 2 36 IC [1 83L, [8T - BSilstatic or ——— = ' 2 [ac—1 . 8p T S hs It should be pointed out that this dispersion correction is contingent on the choice of entropy and pressure as the independent variables. This choice enables us to divide the fluctuations into low- and high- frequency components and use the correct values of the thermodynamic variables in the theoretical expressions for the intensities of the various components . 14 In theory, then, it should be possible to correct the classical Landau-Placzek ratio for the effects of dispersion by introducing a value for the hypersonic adiabatic compressibility in the denominator and multiplying the resultant expression by the correction factor a: 2 38 2 o—- )/ p-- . The hypersonic adiabatic compressibility, 30 . 30 T static T hs hs Bs , is evaluated from the equation 3:5 = 1/[pvis] , (27) where vhs is the hypersonic velocity as determined from the experimen- 2 2 tal Brillouin shift. The correction factor [9&5] ./ [ gEJ p T static p T hs is a difficult quantity to determine experimentally. The applicability of the Cummins-Gammon dispersion-corrected Landau-Placzek ratio is lim- . . 3e ited, therefore, to those liquids for which the values of [036) static as . and [ 5—} are available. p T hs A more preferable method to predict the magnitude and frequency dependence of the various components in a Brillouin spectrum was devel- oped by Mountain using the linearized hydrodynamic equationsof irrevers- ible thermodynamics (24, 27). This theory provides the time dependence of the thermodynamic and hydrodynamic variables. Although Mountain employed this technique initially to derive the intensity distribution of the scattered light from a pure fluid near its critical point (24), he later extended the theory of the spectra of light scattered from pure fluids with a single mode of relaxation (27). In the context of the present paper, we shall examine only the theory in Ref. 27. 15 To facilitate the calculation of the spectrum of the light scattered by a pure fluid, Mountain assumes a model in which the fluid possesses internal degrees of freedom that are weakly coupled to the translational degrees of freedom, thereby affecting the decay of density fluctuations in the liquid. Because density fluctuations are responsible for the observed polarized light scattering spectrum of a liquid, the weak coupling of modes serves to modify spectrum. The modification occurs in such a manner that Landau-Placzek ratio, (cp - cv)/cv, is no longer valid. siders the Specific case of coupling to a single thermal this frequency the classical Mountain con- relaxation mode. He derives formulas for the frequency and intensity distributions of the scattered light, and compares his results to data uids which are assumed to have single thermal relaxation notation used in the following discussion of the problem due to Mountain, with minor changes introduced to aid in of the text. Let us consider the light scattered from density within a small volume element which contains N molecules tering fluid: IN .Jl____ 2 l6n2R A i(R, w) = k: sin2¢<[A€(k, w)]2> . from two liq- times. The is essentially comprehension fluctuations of the scat- (28) In equation (28), I0 is the intensity of the incident plane polarized A wave of vector ki: R is the point of observation of the scattered ..b light intensity, i(R, w), and the scattering is considered to have taken place at the origin. The non-subscripted k in the A equation re- fers to the change in the wave vector ki after scattering. The angle between the electric vector of the incident radiation and the obser- 16 vation point is T, while AE(k, w) is the Fourier component of the fluc- tuation in the dielectric constant; w is the shift in the angular fre- gquency of the scattered light. We can define the magnitude of the change in wave vector as k = (4nn sin 8/2)/Ao rad/cm for calculational purposes, where A0 is the wavelength of the incident radiation, n is the refractive index of the scattering medium, and 6 is the scattering angle. To avoid having to solve directly for Ae(k, m). we invoke sta- tistical mechanical fluctuation theory and express the dielectric con- stant in terms of the density and temperature: AC = (BE/ap)TAp + (OE/8T)pAT. (12) Again, we neglect (ac/8T)p compared to (36/30)T, so that we can re- write equation (28) as I N .-* 4 . 2 2 2 i(R, w) = -.0. 2 2_ ki Sln ¢(3e/8p)T < [p(k, (3)] >. (29) l6n R In this case, p(k, w) is the Fourier component of the density fluctua- tions, and it must be evaluated over the ensemble average of the ini- tial states of the system. For the evaluation of the mean square Fourier component of the density fluctuations, we must go to the lin- earized hydrodynamic equations of irreversible thermodynamics. It is necessary to solve the linearized hydrodynamic equations for p(k, w) in terms of an initial fluctuation, p(k); this can be accomplished by the elegant method of van Hove (34). In the subsequent analysis we assume that the transfer of energy from the internal degrees of freedom to the translational degrees of freedom occurs by a single relaxation process. The linearized hydrodynamic equations which.must be solved are given below; details of the solutions, however, are omitted. 17 The continuity equation for the scattering system under con- sideration is 301/3t + p0 div '{7= 0 , (30) the energy transport equation can be written A 2 oocv 4 .2 i(R, w) = ° 02k. Sln 9 S(k, w) . (33) 2 1 2m S(k, w) being the generalized structure factor. (The reader should be immediately cognizant of the similarity between equations (33) and (29)). The Fourier component of the density fluctuations is related to the generalized structure factor by S(k, w) = < p(k, w) D(-k) > . (34) Integration of S(k, w) over the possible angular frequencies in the liquid yields the ordinary structure factory S(k), which is, in turn, related to the initial density fluctuation, p(k): S(k) = ig'f S(k. w)dw = < p(k) O(-k) > . (35) Obviously, if one can determine either p(k, w) or S(k, w), the other quantity can readily be evaluated. Mountain's ultimate goal in the en- suing analysis is to solve for p(k, m) using the linearized hydrody- namic equations, then to ascertain S(k, w) from the relationship above. Upon solving for p(k, w), Mountain finds that the initial density fluctuations are related to the generalized structure factor by a function 0(k, w), which is the frequency distribution of the scat- tered light. Specifically, 50‘: (D) = < p(k) p(-k) > 00‘: 0)) (36) < p(k. s) p(-k) >] , (37) where 0(k, w) = 2Re I; p(k) p(-k) > saw 3 is the dummy variable utilized in the evaluation of the Laplace and Fourier transforms of the density fluctuations during solution of the linearized hydrodynamic equations. Mountain solves for 0(k, w) approximately by neglecting the 19 small terms in the expression for the inverse Laplace transform of the density fluctuations. He obtains an approximate expression for 0(k, w); a consideration of this equation provides a better understanding of the intensity expressions which are set forth later in this work. Conse— quently, as a close approximation to 0(k, m), we have 2 2Ak /pocP 0(k' (U) 2' (1 - l/Y) (Akz/p c )2 + m2 o p — — N 4 4 2 2 2 - 1)(co/v T + COR (1 - 1/Y)) C2/V4Tz + v2k2 L. 2 2 .L 2co/v 1 4 4 2 2 co/v T + m — —_ (c: - c:)k2 - (vac O X [1 - ci/v2(l - 1/y)][v2k2 + c /v212 - (cf - c:)k2 Cg/v412 + vzk L. __ r r x B + B . (38) 2 2 2 2 PB + (w vk) PB + (w + vk) 2 o 2 F8 is the half-width of a Brillouin peak, v is the phonon speed cal- culated from the Brillouin shift, can is the infinite-frequency phonon speed and T is the relaxation time of the thermal diffusion process responsible for the weak coupling of the internal degrees of freedom to the translational degrees of freedom. As is easily seen from equation (38), all four of the compo- nents are Lorentzian in character. The first term represents the de- cay of a density fluctuation by a thermal diffusion process; this de- cay is recognized as a non—propagating (static) mode. The second 20 term is also a nonpropagating type of decay, and is related to the coupling of the internal degrees of freedom of the molecules. The last term is a propagating decay in the density fluctuation; it represents the phonon modes. Equation (38) for the frequency distribution of the light scattered from a fluid with thermal relaxation is valid onlyr under the condition that the thermal decay process has but a single relaxation time associated with it. ~The intensity of the Rayleigh component of the scattered light and the Brillouin intensities are ascertained by integrating the in- dividual terms of equation (38) with respect to the shift in the angu- lar frequency, m. In general, the intensity ratio of the central peak to the Brillouin components is ——= (1 - l/y) + (c: - c:)k2 - (vz/c: - 1) ZIB 4 4 2 2 2 (Co/V I ) + COR (1 - l/y)] 2 2 l - co/v (l - l/y)Hv2k2 + cg/vzrzl - (c: - c:)k2 (39) For low phonon frequencies (vkr << 1), the relationship simplifies to the classical Landau-Placzek ratio, Ic/ZIB = y — 1. For large phonon frequencies (vkt >> 1), we obtain ..— Ic Y Cw- co -—=(Y-l)1+[Y_1] C2 (40) O Mountain has applied these equations to systemsdof carbon disulfide and carbon tetrachloride, for which much peripheral experimental data is available, and has obtained values for the intensity ratios that are in much better agreement with experimental data than are the clas- sical Landau-Placzek ratios. The most obvious difficulties in utilizing Mountain‘s results 21 for the Landau-Placzek ratio in experimental situations are: (1) one must have previous insight into the relaxational behavior of the liq- uid being studied, so that the single relaxational model can be applied, (2) one must have some method for estimating (vkr) in order to use the simplified expressions for IC/ZI and (3) if one is considering a liq- B uid with more than one mode of relaxation available, a much more com- plicated derivation for the spectral distribution of the scattered light is needed. The final technique to be appraised for the calculation of the intensity and frequency distribution of the light scattered by a pure fluid is a modification of Mountain's approach. Although Mountain treated the case of a singly relaxing liquid in scrutinizing detail, he did not consider the case of a liquid with more than one relaxation time. Montrose, Solovyev and Litovitz (26) have deveIOped a formula for the spectral distribution function, 0(k,w ), for both a nonrelax- ing and relaxing liquid in a manner analogous to Mountain's method of evaluation of 0(k,u)) for a singly relaxing fluid. These authors consider first the case of a nonrelaxing liquid and view the light scattering to be a consequence of fluctuations in the optical dielectric constant. The authors also make the usual as— sumption that the contribution of the thermal fluctuations in dielec— tric constant are small compared to the contribution due to density fluctuations. Montrose, et a1. express the spectral distribution func- tion, < p (k, t) O (-k) > , which is determined by taking the appropriate values of the time derivative of the density. After solving the lin- earized hydrodynamic equations for the Laplace (time) and Fourier 22 . I; 1 (space) transform, (k, s), of the density fluctuation, p(R, t), they evaluate the spectral distribution function, 0(k, w), by noting that _ “we. s) Lek) > o(k. m - 2Re < p(k, pH) > s = w . (41) (The reader will recognize this as being the same equation as (37), Mountain's relation for the frequency distribution of light scattered from a singly-relaxing fluid.) The correlation function ratio of equation (41) is given by [E p(k, 5) p('k) > / < p(k) p(‘k) >:]s = iw . 2 2 2 4 2 = (s2 + [(A/pocv) + (no/00)] 5" + ‘30" (1 ' l/Y) + )‘nok /pocv)/ 3 2 2 2 4 2 (S + [(A/pocv) + (no/p0):ls + [COR + (Anok /pva):]S + Ac2k4/ o Ypocv) . (42) s=1w where A is the thermal conductivity, ”0 is the frequency independent longitudinal viscosity, and _ _ 2 s - Ak /pocp . (43) Equation (42) is equivalent to Mountain's equation (13) in Ref. 24, and Montrose, et a1. remark that up to this point, their treatment has been identical to Mountain's. The authors proceed, however, to decom- pose the correlation function ratio and calculate a spectral distribu- tion function that is markedly different from Mountain's with respect to the Brillouin components. Montrose, et a1. first reduce the complexity of the denominator l -5 . . . p(R, t) is the departure at time t, of the denSity from its equilibrium value. 23 of equation (42) by estimating the relative magnitude of the terms present. Since A/cv << no for most transparent liquids, the terms containing (A/cvno)2 can be ignored. The denominator is then rewritten as n 2 2 o A(l - 1/Y) 2 2 2 2 + -—-+ + . 44 (s + Ak /pocp) s p p c k s cok ( ) 0 0V Furthermore. the factor A(l - l/Y)/pocv has been found (36) to be less than one per cent ofno/Oo for a wide variety of liquids, so that ex- pression (44) can be reduced even more. Montrose, et a1. define two I w T : terms , o and B I 2 mo - cok and PB — nok /20O . (45) P8 is the true Brillouin halfewidth at half height, and is similar to Mountain's more complicated expression for TB given on p. 216 of Ref. 27. By eliminating the insignificant terms of equation (42), one arrives at a simplified expression for the correlation function ratio: 2 ' ' 2 < p(k, s) p(-k) > 5 + ZFBS + wo(l - l/Y)+2FBAk /poc < p(k) D(-k) > 2 2 ' 2 (46) s=1w (s + Ak /pocp)(s + ers + mo) Montrose, et a1. separate the right-hand side of equation (46) into partial fractions < Q(kp S) Q(-k) > < p(k) p(-k) > = (l - l/y) 2 +(:) (47) s + (Ak /pocp) 24 and use (47) in equation (41) to calculate the spectral distribution function for a nonrelaxaing liquid: 2 2 k A /pocp 2 2 [AR /pocp] + w 2 I 2c k ] 2TB + * . ' , . (48) J [2wr8]2 + [w2 - wZJZ O The first term denotes the nonpropagating central component of - = [wn ] o o o In equation (49), kB is Boltzmann's constant and 03(k, w) is considered to be the isentropic density fluctuation which is responsible for the phonon modes. The authors continue by making an interesting observa- tion about equation (49): ". . . the form of < p:(k, w) > is the same as that for the spectrum of the displacement correlation function for a harmonic system under the action of a random (white noise) driving force . . ." (Ref. 26, p. 120). What this implies is that the spectrum arising from spontaneous fluctuations is equivalent to the spectrum of a system driven by random forces. The density fluctuation phenomenon 25 is seen, therefore, to be completely analogous to the problem of a damped harmonic oscillator, l with the Brillouin peaks being the re- sults of the "resonant" frequencies of the density fluctuations that are driven by the thermal energy of the system. In this particular situation, the Callen—Welten (37) fluctua- tion dissipation theorem can be applied to the problem to yield further information about the fluctuations in density as they are related to the dissipative (relaxation) processes which occur in the liquid in response to the external driving force (thermal energy). Piercy and Hanes (38) have made this application using the language of electric circuits and Nyquist logic, and have found, among other things, that the traditional neglect of temperature fluctuations of the dielectric constant can lead to serious errors in the analysis of Brillouin spec- tra (Ref. 38, p. 1007). With these facts in mind, Montrose, et al. expand oS(k, w) into four partial fractions: I I F. F F o (k, m) .= l- B + B S Y .2 2 .2 2 PB + [w - (1)4] PB + [w + (1)4] I I — PB w - w4 PB w + N4 - -—- + ——- (50) N4 ,2 2 w4 ,2 2 PB + w — w4 PB + [w + w4]-_ in which w4 is the frequency of free oscillation of the damped system: 2 2 2 4 h = 1 - . “4 wol l‘[mono/poco] (51) The damping or dissipative element can be shown to be the longitudinal viscosity, no (Ref. 26, p. 122). 26 By definition, mo is the product of the low-frequency phonon velocity, Co' and the magnitude of the change in wave vector, k, and w = 0 is the center of the Brillouin spectrum. Equation (50) applies to the specific case of a nonrelaxing fluid. If we replace w4 with mo in the first two terms of equation (50), we obtain the same result as Mountain for a nonrelaxing fluid. The last two terms are present but somewhat obscure, in Mountain's the- ory for a single relaxation when T + 0. As is recognized from equation (50), the first two terms are Lorentzian peaks centered at w = :w4. The last two terms appear to be skewed Lorentzians which have the ef- fects of asymmetrizing and shifting the original Brillouin peaks (the ones defined by the firSt two terms) toward the center of the spectrum (to w = 0). It should be noted, however, that when the individual terms of (50) are integrated in order to determine the intensity ratio of the Rayleigh to the Brillouin peaks, the integrals of the last two terms cancel out, so that they do not contribute to the Landau-Placzek ratio. In the remainder of the theoretical discussion, Montrose, et al. turn their attention to the case of a relaxing liquid, utilizing the same basic method, the linearized hydrodynamic equations and the Piercy- Hanes electrical circuit approach, as they did for the nonrelaxing case. The basic difference in the derivation of the spectral distribu- tion function for the case of a relaxing liquid is the inclusion of the time dependence (frequency dependence) of the various thermodynamic and viscoelastic parameters, such as the specific heat, cv, the thermal expansion coefficient, 8, the shear viscosity, nS and the longitudinal mdmhm,u. By applying the linearized hydrodynamic equations, the authors 27 find for a relaxing liquid that the Spectral distribution function can be given by 2 2Ak /pocp 2 2 2 + Ak /pocp] w 0(k, w) = (l - l/Y) I 2M o[h(w) + “.12 YW [n(w).+n02+]} [OZpr/k -M'2(w)]] where n(w) is the frequency dependent longitudinal viscosity and M'(w) (52) is the real part of the frequency dependent modulus of elasticity. For the special case of a singly relaxing fluid 2 2 2 2 2 2 Coo " Co T Co + C mm T n(w) = po .2 i and M'(w) = p 2 2 , (53) l + w T 1 + w T which leads to the formula 2 2Ak /pocp 0(k, w) = (1 - l/:)[ *{:[:i-[ 2 2 2 Ak /pocp] + m _2] 2 + n l + msz2 o +{pom2/k:{ - po po[c: + ciw212]/[l + m2T2]}2 . (54) This is similar to the expression from Mountain's theory, equation (38). As in the case of a nonrelaxing liquid, Montrose, et a1. then attempt to apply the Piercy-Hanes equivalent electric circuit theory to this problem. The situation for a relaxing liquid is much more com- plicated than for a nonrelaxing liquid, however, so that they can only approximate the form of os(k, m) for the special case of a singly re— laxing liquid exhibiting small dispersion (i.e., for [0: - c:]/c: << 1). 28 (In other situations, the errors inherent in the evaluation of the fre- quency dependent portion of the modulus of elasticity, M'(w), become too great to obtain a meaningful expression for gs(k, m). In spite of these imposing difficulties, one interesting observation is made for the experimental situation in which the liquid possesses a single re- laxation time, small dispersion and small thermal conductivity: a den- sity fluctuation in the liquid can be regarded as being composed of an oscillatory and a nonoscillatory portion, giving rise to the Brillouin peaks and the Rayleigh peak, respectively. The macroscopic relaxation time, T, is assumed to be the sum of the rates at which the oscillatOry and nonoscillatory portions of the density fluctuations are damped. The explicit division of density fluctuations into these component parts is the subject of a second paper by Litovitz and his colleagues (23), with specific application to a relaxing viscous liquid which can form a glass Although it can be seen that several theoretical expressions are available for the intensity distribution of light scattered from a pure fluid, it should be noted that none of these equations has as yet been fully tested. This is because of the lack of information on the various thermodynamic entities in the equations, and because of the high degree of accuracy required for the extraction of the Brillouin scattering par- ameters from experimental spectra. 2. Brillouin scattering from a two-component mixture a. Derivation of the spectral distribution of light scattered from a binary liquid system Because of the complexity of the theoretical analysis of the spectral distribution of light scattered from multicomponent systems, 29 very few papers dealing with this subject have appeared in the litera- ture (39-44). Of the papers which have appeared, only one (44) is of interest to our immediate problem of light scattering from the binary system DMSO and pyridine. Fishman and Mountain (44) consider the spectral distribution of light scattered by a binary mixture with internal degrees of free- dom (i.e., thermal and/or structural relaxation) in much the same man- ner as Mountain and Deutch (43) determined the spectral distribution of light scattered from a binary solution possessing no internal relaxa- tional modes. They use linearized hydrodynamic equations to calculate the relative magnitude of the modes by which the relaxing system re- turns to equilibrium, and thermodynamic fluctuation theory for initial estimates of the correlation functions for the thermodynamic variables. For both relaxing and nonrelaxing fluids, the authors consider the intensity of the scattered light to be related to the Laplace- Fourier transform of the auto-correlation function of the optical di- electric constant. The Laplace-Fourier transform is evaluated by making use of the linearized hydrodynamic equations and thermodynamic fluctua- tion theory in a manner analogous to that described in section C-l for a pure liquid. The basic difference in the method of evaluation of < Ae(k, w) 2 > for a binary mixture is the inclusion of the concentra- tion dependence of the dielectric constant. As was done previously with the dielectric constant for a pure fluid, we express a in terms of certain macroscopic (thermodynamic) variables, and proceed to ana- lyze the fluctuations in e in terms of the fluctuations of these vari- ables. 30 From the Gibbs Postulate (45), we know that any intensive property 1 of the thermodynamic state of a system is a function of K + l intensive prOperties, K being the canonical component number of the system. The dielectric constant is an intensive property, and for a homogenous binary solution, K = 2, so that a can be expressed in terms of three intensive variables. In order to achieve the simplest possible arrangement for calculational purposes, we want to choose three inten- sive variables that are statistically independent, so that their fluc- tuation cross terms go to zero. The obvious choices for the three thermodynamic variables are pressure, temperature and concentration, quantities which are easy to measure experimentally. If we express the dielectric constant c(p,T,c) in terms of space and time (r and t), we can relate the fluctuations in e to the fluctuations in pressure, temp- erature and concentration: 68(r, t) = (Be/Sp)T’C6p(r, t) + (Be/3T)p c6T(r, t) I + (85/8c)p’Tdc(r, t). 2 (55) The linearized hydrodynamic equations are constructed, as usual, but this time they are modified to include a frequency-dependent volume viscosity which relaxes with a single relaxation time, 3 and a diffusion An intensive prOperty is defined at every point in the system (e.g., temperature): it is also the ratio of two extensive properties, such as p = dm/dV. 2 . The symbol 6 is used instead of A for the fluctuation terms to indicate that we are dealing with space-time functions of the thermo- dynamic variables. 3 . . This single relaxation mechanism is usually visualized as thermal relaxation. 31 equation. We have for the continuity equation and the modified Navier- Stokes equation 3Q/3t + 00 div v = 0 (56) and . - t'/T 1%.:2 = — J‘b—Vzp + 3'1on div v + [(330 — c2] re V2 div V(t - t')dt', ° ° (57) respectively. The energy transport equation (58) and diffusion equation (59) can be written 2 ' - ' = T 8 poc (ET/at) pokT(3u/3c)p'T(3c/3t) + poTo(BS/BC)T'C2p/3t AV (5 ) and (ac/3t) = D Vzc + (k /T )VZT - (p /p )2(ap/3c) /(3u/ac) Vzp . T O O O ppT PIT (59) In these equations, p is the chemical potential of the mixture, as de- fined by Landau and Lifshitz (46) to be u = Ul/ml - uZ/m2 (60) where u and u 1 2 are the chemical potentials and m and m are the l 2 masses of the two species in one gram of solution. The concentration, ci, is equal to nimi, where ni is the number of molecules of substance i in one gram of solution. Returning to equation (59), D is the dif- fusion coefficient for the binary mixture, kT is the thermal diffusion ratio, and the subscript 0 indicates an equilibrium value. Examining equations (56-59), we see that there are four equa- tions and five unknowns; this necessitates the assumption that local thermodynamic equilibrium prevails in the system, so that the fluctua- tions in density can be written do = (Sp/3p)T'c6p + (39/3T)p CGT + (ap/ac)p 6c (61) ,T I 32 and equation (61) can be used for 5D in the continuity equation. The four equations are solved by the use of Fourier-Laplace transforms to obtain the initial values (t = 0) of the functions p(r, t), T(r, t) and C(r, t). The generalized structure factor, S(k, m), can then be evalu— ated from the expression for 58(r, t), since S(k, w) is related to 56(r, t) by S(k, w) = 2Re I dt I < 65(r + r',t)6€(0,0) > exp i(kr - wt) drdr'. o v (62) After performing these calculations, Fishman and Mountain arrive at this expression for the generalized structure factor, S(k, 00) 1 Corrections have been made to the original equation to com- pensate for an incorrect thermodynamic expression borrowed by these authors from Miller (39). 33 m m I o . N? I 35 + Nb N3 + 33 + Na vT\ OH ... N335 “no. 3 I xb + 3 + x> I) o o I I o , ... a - .... Ii III? is... - :1: C3 + a. + mp 33 I 3 + m.— —>\oo~ + «53. N N + N N - v N ... mg m." NT\00H ... Lea—00H + NTxao_ I Nara—3 O O c I fl .. . .Ti I .T Ii I .3. . IE I I. . .._ I. VN>\M0N NT§O¢ I VAI>\OUH + Nam—OOH I NRC—80H Okay. N um 3 I a? Q 0 4 D m N3 + 33 + N... I v—>\ u + N333 +_IN§> + 3. + N.— + N33 .3 + Na a 3 + x> O as O a m :N>\Nu I L rN>\Nu L N>\N0Q 53 _I .— a o o o I v?\ 04_ + N333 + N3 + NTN>\NUH L>\ 0‘ + Nth—3 + I O O O 3 O O Nix OH I N333 + ~N>\Nu I LN>\N0 F pN>\N0N Nth—a; I NT“. 0% + N—I>\ OH I v~>\IOHI s s s I n a . ~a+vx~xMuamMuamMmm~u+ommm s3 x3+na+u3+x£+~au as o um an an u-+ aux» 3+3». N Nax N fisxoofl I ..x». .s> I a. + as .3» I at I at m>\0u_ + .pg». 9 N + N N + N N x v NI I I ... N~N>\Mu I L 5? nu ma HN>\.N.0 I L Nth—3 + hN>\ano I Nm—I>\o# a + T (on; Txouw + .53 3 + x 9 ad ad N N N N - I v N N v s 32.5 Tm .. 3 :cm 3 + u~>\mu~ ~—>\oo_ I «Hie; + «.33 I Lino,— uxou cams n 34 In equation (63), c is the concentration of the solute, in moles, 92 is the weight of solute in grams, a is the thermal expansion coefficient of the mixture, v is the small volume element responsible for the scat- tering and the terms subscripted with a zero denote equilibrium quan- tities. Also, cp, Do and BS refer to values for the mixture. The terms in (63) which involve the quantities (vk + w) and (vk - w) are considered to be the Brillouin components; all other terms make up the central peak. By integrating equation (63) with respect to the change in wave vector, k, and the frequency shift, w, one obtains the intensity distribution of the scattered light. Dropping the subscripts for the equilibrium quantities, we find the intensity ratio of the Rayleigh peak to the Brillouin doublet 1 to be P I I [c2 - v2]k2T2 + [c /v]4 - [c /v]2 c m o o --= J 1 + 2 4 B (vk I) + [co/v] u [Ci - c:]k212 + [co/v)4 - [co/v]2 "( (va)2 + [co/v]4 1_ [c: - c3139 + [co/vJ“ - [c.va 2 4 (va) + [co/v] [c2 - VZJRZTZ + [c /v]4 - [c /v]2 - J °° ° ° (64) o 2 4 (VkT) + [co/v] l . . This result differs somewhat from the equation set forth by Fishman and Mountain. 35 » 2 Be 2 (ac/3c) where Jo =([l + 21(1 - In] T [—] + P'T) 92 gch 3T p,c (3p/3C)p’T l. 2 2T 3 a 1 a 2 (”3T ("18%) .+———:;. [:2] [3%] +——. [fl )- gzcv Y p,c_ 92 T v p T,c p,c V S p T,c A tremendous amount of simplification results for small scat- tering angles, since k + 0. For small scattering angles, v + co, also, and we are left with = J I (66) which is essentially the result obtained by Miller (39) using the fluctuation theory approach outlined by Landau and Lifshitz (47). For the case of large scattering angles (i.e., B + 180°), k goes to 2n/Ao, which is very large in comparison with the k values for low-angle scattering. In this situation, we find that equation (64) can again be simplified, if we are willing to assume that k + m as e+lao°I 2 2 C C ——=-3-1+-3J (67) 2 2 O . C C O 0 It is unfortunate that large—and small-angle scattering measurements are difficult to perform experimentally, because the reduction in complexity of the theoretical expressions for the Landau-Placzek ratio is enormous. In spite of the complicated form for the Landau-Placzek ratio, Ic/ZIB, let us examine the situation for a binary mixture at constant temperature. For this type of system we note that (Be/8c)P T f 0; (Bu/ac)“T # 0: (Be/8mmc # Ozand (Be/3T)p C a 0 . I I 36 Equation (64) then becomes VBS (ac/3c): Tlulu/ac) I .£L_~ 2I [B (ac/3p)T p,T + w o o (Vk‘l’ ) + [c /v] 4 [c2 - c2]k 12 + [c /v]4 — [c /v] 1 _ o (ac/3c)2 [c2 - v2]k212 + [c /v]4 - [c /v)2 v gp,T m o o . - V88 2 2 4 (68) (3F—/3£>).r'C .(vkt) + [co/V] J Fishman and Mountain have shown that the expression in the brackets in the denominator is a relatively unimportant term for the case of hyper- sonic waves detected by Brillouin scattering. Furthermore, Miller (39) has shown that the approximation Be (BE/8p)T c = p[—] B (69) I 3p T T,c is valid for a binary solution, so that we can reduce equation (68) to 2 2 2 2 2 4 2 VBs(ae/3c)P'T + [c°° — co]k T + [co/v] - [co/v] 2 2 2 2 4 IC = BTp (an/8c) 'T(ae/3p)m (vkt) + [co/v] . 2I 2 2 B cco - c 12 c /v c /v 1 _ [ w] +[ )4 [ o ] 0 (Wu)2 “(C v]4 Equation (70) is valid for a binary mixture at constant temperature. 37 b. Relationship between the Brillouin linewidth, phenomenological coefficients and thermodynamic parameters The Brillouin linewidth of a pure liquid is directly related to the sonic absorption coefficient of a thermal sound wave in the liquid such that (48, 49) FB = avs . (71) PB is the half width at half height of the Brillouin peak (corrected for the effects of instrumental broadening), and a is the absorption coefficient of the thermally propagated sound wave. V8 is the velocity of the sound wave in the liquid,which is determined by the Brillouin shift, the refractive index of the medium, and the wavelength of the incident light, at a specific scattering angle. The sonic absorption coefficient is describeable in terms of a classical ultrasonic experiment in which a plane sound wave traverses a distance x from the radiation source: I = I e'zux . (72) 0 Here I is the intensity of the sound wave at a distance x from the source, I0 is the intensity at x = O, and a is the absorption coeffi- cient. (The absorption coefficient is known to depend on the physical properties of the medium, on external temperature and pressure, and on the frequency of the propagating wave (50). The frequency-corrected absorption coefficient is denoted by a/vz, where v is the linear fre- quency of the sound wave. Herzfeld and Litovitz (51) derived the classical absorption coefficient, a , for a plane sound wave traveling in a medium class 38 which possesses viscosity and heat conductivity which dissipate the wave. Their result is 2 “’3n+——:‘. (73) C VS 0 a -2. class 3 where w = 2nv is the angular frequency of the sound wave, 0 is the den- sity of the medium, n is the shear viscosity, y is the ratio of the specific heats at constant pressure and volume, respectively, A is the thermal conductivity, and CF is the specific heat at constant pressure. Ultrasonic measurements (52, 53) of the absorption coefficient reveal that many pure liquids possess a values which do not correspond to a . Accordingly, Pinkerton (54) and Herzfeld and Litovitz (51) class divided liquids into groups, depending on the ratio a/aClass and on the sign of the temperature coefficient of absorption, (Ba/3T). The liq- uids investigated fell into three categories (Ref. 51, p. 357): Group I. "Normal liquids," for which a/a = 1.0, and class (Ba/3T) = 0. Group II. Kneser liquids, for which a/aClass > 1.0, and (Ba/3T) is positive. Group III. "Associated liquids," for which a/aclass > 1.0, and (Ba/3T) is negative. It has been suggested by Hall (55) and confirmed experimentally by Rai, Singh, and Awasthi (56), that the abnormally high value of the measured absorption coefficient for associated liquids is due to the presence of a third type of dissipative mechanism, "structural absorp- tion." The essence of the theory of structural absorption is that molecules in an associated liquid can undergo a transition from one type of structure to another under the influence of the passage of a 39 sound wave through the liquid. The total frequency-corrected absorp— tion coefficient for associated liquids can be considered to be composed of two parts: a contribution from the viscosity and thermal relaxation mechanisms ((1C 5) and a term due to structural absorption (a ). las excess Expressing this statement in terms of an equation, we have 2 2 2 = 7 (a/v )total (a/v )class + (u/v )excess ( 4) 2 2n2 4 A 2 or (on/v )total = :l-‘j'; 3") + a; (Y ' l) + (G/V )excess . (75) s Hall (Ref. 55, p. 778) defines the excess absorption from structural relaxation to be (a/vz) a 2n2pV 8 r (76) excess 5 r where Br is the relaxational part of the isothermal compressibility and T is the structural relaxation time. BI is considered to be the difference between the static isothermal compressibility, 8 , and T,O the high frequency limit, 8T a, The structural relaxation time, T, can be calculated from the formula = %g- 1 + exp(AF/RT) (77) where v is the molal volume, AF is the difference in free energy be- tween the two structural configurations of the molecule, and R is the universal gas constant. If we assume that the behavior of binary mixtures of associated liquids is not too dissimilar from the behavior of pure liquids of this type, and that the sonic absorption coefficient for the mixtures can be p given by equation (7S)(using the values for the mixture), then we might expect our DMSO-pyridine solutions to fall into the Group III 40 classification. We might also expect the variation of a with respect to composition and temperature to give us further insight into the structural changes occurring in the DMSO-pyridine solutions. Looking back at equation (71), we see that our primary task in the determination of a is the evaluation of the Brillouin linewidth, 2P8. It is a well-established fact (21, 49, 57-59) that in Brillouin scattering experiments, the linewidths of the spectral components are of the same order of magnitude as the combined linewidths of the inci- dent source and detector, which is called the instrumental profile. The observed spectrum is, therefore, the convolution of the true light scattering spectrum with the instrumental profile of the light detec— tion system. Mathematically speaking, we have (60) 0(v) = JOT(vpv') Ios(v',v")I(v")dv"dv' o (78) with 0(v) being the observed spectrum as a function of frequency, v, T(v,v') being the transmission function of the spectral analyzer (which is usually a Fabry—Perot etalon), S(v',v") being the scattering spec- trum as a function of the incident frequency input, v", and the scat- tering frequency output,v', and I(v") being the intensity spectrum of the incident source, with output frequency v". Leidecker and LaMacchia (60) have given a detailed account of the effects of instrumental profiles on the shape of Brillouin peaks. Following their analysis, we will examine the generalized case of a Gaussian instrumental function convoluted with a Lorentzian scattering Spectrum to produce an observed intensity spectrum which is a combina- tion of the two (namely, a Voigt function). Then we will examine the special case of a Lorentzian instrumental profile convoluted with a 41 Lorentzian scattering spectrum to produce an observed Lorentzian inten- sity spectrum. It has been shown experimentally (61) that most continuous wave lasers possess an intensity profile which can be adequately des- cribed by a Gaussian distribution function, IG(v): 2 oi '—"—I, . . J. v - v (79) IIMZ 15M“ . l 1 6[v - véi]exp ~41n2 In equation (79), N is the number of axial modes lasing, voi is the central frequency of the ith axial mode, and Fiis the full width at half height of the ith spectral line. Many of the recently available, commercial lasers have single mode outputs so that the intensity dis- tribution reduces to V - vo . (80) F The spectral analyzer employed in e majority of Brillouin scattering 0:6 — .. IG(v) [v vOJexp 4ln2 spectrometers (including the one used in this work) is the scanning Fabry-Perot interferometer. A simple theoretical analysis of a Fabry- Perot etalon yields the result that the transmission intensity, T(v), for the etalon is 1 2 . 2 l + [E-F Sin 6] T(v)¢ (81) where F is the effective finesse of the etalon, e = 2ndnv/c, and T(v) has the form of an Airy function. In the expression for e, n is the refractive index of the medium between the Fabry-Perot mirrors and d is the mirror separation. For cases in which the effective finesse is greater than 30, the Airy function reduces to Lorentzian form: 42 l l + [2[v - VOJ/FFP Here v0 is again the central frequency of the transmitted beam, and T (v)¢ (82) L 2 ° PFP is the full width at half height of the Fabry-Perot spectral peak. The restrictions on the validity of equation (82) are the conditions of absolute parallelism and flatness of the mirrors (62). The linewidth of the instrumental profile is a combination of the linewidths of the laser beam and Fabry-Perot transmission function. Therefore, it can be seen a_priori that in a situation in which the linewidth of the Fabry-Perot transmission function is very much larger than the linewidth of the light source, the resulting instrumental profile will possess the shape of the Fabry-Perot function; i.e., it will be a Lorentzian. In the less common situation in which the line- width of the incident source is comparable in size to the linewidth of the transmission function, the resulting instrumental profile will be a Voigt function. From the theoretical discussion of the intensity profiles which are predicted for pure liquids (pp. 19, 27) and for bi- nary mixtures (p. 33), and from reported experimental linewidth meas- urements for Brillouin spectra (49, 57-59, 63-65), one has every reason to expect the true shape of a Brillouin peak from a DMSO-pyridine spectrum to be Lorentzian. For the remainder of this discussion, we assume that this conclusion is correct, 1 and proceed to an important discovery about the shape of an observed Brillouin peak when the in- strumental function is a Lorentzian. For the special case of the instrumental profile being l . . . . . It will be demonstrated in Part IV-E that this assumption is indeed correct. 43 Lorentzian in nature, it can be shown (Ref. 60, p. 145) that the ob- served intensity distribution for a Brillouin peak in the light scat- tering spectrum is a Lorentzian with full width at half height given by 2 2r + P . (83) rB/obs = B inst Here ZFB is the true width of the Lorentzian-shaped Brillouin peak, and rinst is the full width at half height of the Lorentzian-shaped instrumental profile (viz., the Fabry-Perot transmission function). In conclusion of this discussion of Brillouin linewidths, it shOuld be noted that the above analysis is not valid if other relaxa- tional modes are observed in the experimental spectra (i.e., if the Rayleigh and/or Brillouin peaks are composed of more than one Lorentzian function). D. Effects of Optical Properties of Polar Liquids on Light Scattering From equation (9) on p. 7, it is seen that the velocity of hypersound in a fluid is dependent upon both the observed Brillouin shift and the refractive index, measured at a given wavelength. Con- sequently, in order to obtain accurate values for the hypersonic veloc- ities of the DMSO-pyridine solutions, it was necessary to determine the refractive index of each solution under the apprOpriate thermal conditions. During the course of the data analysis on the refractive index measurements, it was discovered that there is a dearth of material available on the optical properties of mixtures of polar compounds. 44 Furthermore, it was recognized that the Clausius-Mosotti-Lorentz- Lorenz equation 1 describing the relationship between the dielectric constant, refractive index, and polarizability of a nonpolar compound, would in no way be valid for the polar substances DMSO and pyridine. Since the additivity of molar refractivities of mixtures of compounds is based on the Clausius-Mosotti-Lorentz-Lorenz equation, we felt that there was also doubt concerning the linearity of the molar refractivity of the DMSO-pyridine solutions with mole fraction DMSO. We were cog- nizant of the fact that any abnormality in the values of the refractive index would be transmitted directly to the hypersonic velocity in such a manner as to render this data invalid. Therefore, we felt that it would be advisable to confirm the additivity of the molar refractivities of DMSO and pyridine, both experimentally and theoretically. The following is a brief theoretical discussion concerning the optical properties of polar and nonpolar liquids which we hope will lend significance to the refractive index data presented in section IV. For a uniform isotropic medium, the refractive index, n, is defined as (84) 5 ll < IO where c is the velocity of light i2_vacuo and v is the velocity of light in the medium. 1From Born and Wolf (66), p. 87, the Clausius-Mosotti-Lorentz- Lorenz equation is given by .e-lszi—iNd e + 2 n2 + 2 3 where e is the dielectric constant, n is the refractive index, N is the number of molecules per unit volume, and a is the polarizability of the molecules. 45' Born and Wolf (Ref. 66, p. 11) discuss the relationship between the velocity of light in a medium, the dielectric constant, 5, and mag- netic permeability, u. They state that the solution of Maxwell's wave equations, 24 22‘ 2 32E A mes—3...; “3.2%.; (85) c at c at "... suggest the existence of electromagnetic waves propagated with a velocity 1 v .. CNS: <86) .3 A In equation (74), H and E are the magnetic and electric field vectors, , 2a 2 2a 2 . . respectively, 3 H/at and 3 E/at are second derivatives with respect to time, and 2 2 2 ‘ 2A 3 H 8 H 8 Hz v H = -——- + 2 + 2 L ax By 82 j 2 2 2 ‘ 2a (a E a E a B2 v E = + + . (87) 3 2 3 2 2 k x y 32 J From the definition of n, we have n = (em). (88) The form of this equation presented in most optics texts, n2 = e, arises from the fact that p = l for "nonmagnetic” substances. Although the relationship n2 = s has proven valid for experi- mental observations on gases and many liquid hydrocarbons, it has been shown to be incorrect for certain other liquids (namely, highly associated ones). Values of n and JE'given by Born and Wolf (Ref. 66, p. 14) for methyl alcohol, ethyl alcohol, and water are illustrative 1Ref. 66, p. 11. 46 of the strong deviation from the n2 = a relation exhibited by highly polar compounds: n(yellow light) 3g; c9305 1.34 s 7 - .36 5.0 CZHSOH 1 H o 1.33 9.0 2 The temperature for these measurements is assumed to be ambient, since no information was provided. The values of n and V; cited in Szmant's work (Ref. 1, p. 11) for DMSO exhibit a similar behavior: n30 = 1.4783 «2': “8.9 = 6.99 25 —- nD = 1.47674 Vt = “46.4 = 6.81 Experimental determinations of n and e are made under quite different conditions. Refractive index measurements are made with electromagnetic 15 radiation of rather high frequency ( 10 sec-1), whereas dielectric constant measurements take place with electric fields in the frequency range 5 x 105 to S x 106 sec-1. Because of the extreme differences in frequency in the two cases, certain criteria must be met in order for the relation n2 = e to be valid (67): (1) the molecules of the dielectric medium must possess no dipole moment (2) the measurement of refractive index should be made with long-wavelength infrared radiation so that both the electrons and the nuclei are affected by the electromagnetic wave (3) the wavelength of the infrared radiation should be far enough from an absorption band to insure that anomalous dispersion is not occurring. 47 The first restriction arises from the frequency dependence of the dielectric constant, and from the contribution of the permanent dipole moment, u, to the dielectric constant at low frequencies. Lipson and Lipson (68) derive the frequency dependence of the dielectric con- stant for a medium consisting of polar molecules: a brief Summary of their discussion follows. The frequency dependent dielectric constant, £(w), is related to the electric susceptibility, x(w), by the equation 8 (w) = 60 + 411x80) (89) where 80 is the value of the frequency independent portion of the di- electric constant and x(w) is found to be a complex function of the form _ X(O) x0») ‘—"_1+im . (90) Here m is the frequency of the applied electric field, x(0) is the zero-frequency electric susceptibility and T is the relaxation time, or the time required for the electric dipoles to orient themselves in a state of minim. potential energy while under the influence of the external electric field. Equations (89) and (90) lead to the result a = e ' (w) l + 162 , (91) where n - cl = 60 + ngla. and £2 = 113—13422:— ° (92) 1 + w T 1 + w T In the limit of low frequencies, cl-+ so + 4nx(0) e + 0 (93) 2 '48 so that C(w) + e = 50 + 4"X(0)v which is the value of the static di- electric'constant. At high frequencies the molecules do not have time to change with the rapidly oscillating field so that and C(w) + E = so . (94) It can be seen from the expressions for the dielectric constant at the ' low and high frequency limits that experimental determinations of the dielectric constant must be made at low electromagnetic frequencies in order to obtain meaningful values of e. Debye (69) has shown, however, that in this low frequency region the permanent dipole moment of a polar molecule can contribute to the total polarization of the medium. Debye calculated the contribution of the average value of the permanent dipole moment, u, to the dielectric constant for a polar liquid. He showed that the total molar polarization, P , in the region M of low electromagnetic frequencies is due to the contributions from both the induced and permanent dipole moments: 4nN 2 e - 1 MW 3 L n PM [a + 2]p 3 [u + 3kT] ' (95) In equation (95) MW is the molecular weight of the liquid, p is the density, a is the polarizability, NL is the Avogadro-Loschmidt constant, and kT is the thermal energy of the system at temperature T. What all of this means, of course, is that for polar molecules, the value of the low-frequency dielectric constant will contain a con- tribution due to the permanent dipole moment of the molecule. In the 49. high frequency region of the electromagnetic spectrum, where refractive index measurements are made, the dielectric constant will be much lower, due to the fact that at high frequencies the permanent dipole moment cannot contribute to the dielectric constant. Therefore, for polar molecules, the relation 2 a n2 is not valid. In order to obtain a suitable relationship between the dielectric constant and the refractive index for a polar fluid, Onsager (70) reviewed Mosotti's internal field theory of the polarization of dielectric media and found it to be un- applicable to polar molecules. From his subsequent analysis of the effect of the environment of a polar molecule on the permanent and in- duced dipole moments of that molecule during the presence of an external electric field, Onsager derived the following equation for a pure polar fluid: (96) £[n2 + 2]2 p 9kT The first step in solving (96) for the refractive index is to [e - n2][2e + n2] [143] = M rewrite the equation in expanded form: . 2 41'"Lu _2_ = 2:2 - en2 - n4 (97) MW 0 9kT e[n4 + 4n2 + 4] Let c' = 4nNLp/(9kMW) so that 2 2 2 4 I 2 _ _ cgp g [ e an n ] . (98) T e[n4 + 4n2 + 4] After cross multiplication and suitable algebraic manipulations, we find that 2 2 [l + ec'u /T]n4 + [e + 4ec'u2/TJn - 252 + 4ec'u2/T = O . (99) Solving for the refractive index, n, we have 50 115 + 4€Cifl?/T]2 - 4(1 + ec'pZ/T)(-252 + 4ec'p2/Til:_ 5 (100) 2 n . 2[l + ec'u /T] for a polar liquid. Applying equation (100) to the case of pure DMSO at 25°C 1, we find that n = 1.12. The agreement between this value and the experimen- tal value of n = 1.4767 cited by Szmant (Ref. 1, p. 5) is quite good, considering the errors inherent in the experimental measurements of e and u. Recall that the value of n calculated from the equation 6 = n is 6.81. The value of n calculated for pure pyridine from equation (100) is 1.16 at 22°C, as opposed to the value 3.54 obtained from the rela- tion 5 = n2. The values of c and p for pure pyridine were taken from the International Critical Tables. The results for DMSO and pyridine tend to support Onsager's theoretical relationship between the refractive index of a polar liquid and its static dielectric constant. It has been found empirically that the refractive index be- havior of mixtures of molecules is directly related to the refractive index characteristics of the constituent molecules. For pure nonpolar liquids, a quantity known as the molar re- fractivity, A, can be defined by 2 A=£W______n"1 , (101) p n2 + 2 2 2 . . where the factor (n - 1)/(n + 2) comes from the ClauSius-Mosotti— Lorentz-Lorenz equation (see footnote, p.44) 1The values of e and u for DMSO used in equation (100) were taken from Ref. 1, p. 5. " 51 2 (5-1 n-1 4 = =-—a. e + 2 2 3TrN (102) n+2 The total molar refractivity, AT, for mixtures of nonpolar compounds is linearly dependent on the mole fraction of each component of the mixture. 1 Therefore, for a binary mixture of nonpolar sub- stances, AT = xlnl + x2112 , (103) where X1 and X2 are the mole fractions of substances 1 and 2, respec- tively, and A and A? are the molar refractivities. 1 In view of the complicated functional relationship between E and n for polar liquids, it is not intuitively obvious that the total molar refractivity of a binary mixture of two polar compounds should be linearly dependent on the mole fraction of component 1. Onsager (Ref. 70, p. 1490), however, hinted at a linear dependence in his derivation of the equation relating C and n for a mixture of polar liquids. Onsager defined the effective polarizability, ai, of species i in a mixture of polar fluids to be , (104) where ai is the radius of the i spherical molecular species. In light of equation (104), we may define the molar refractivity, A; of polar species i to be ‘-——‘—"—’ p (105) 1 . . . . This linear relationship is known as the law of Gladstone and Dale. 52 where ni is the refractive index for the polar species i, and is given by equation (100). Note that equation (105) is exactly analogous to the molar refractivity equation for nonpolar substances. The total molar refractivity for a binary mixture of polar species can then be expected to be linearly dependent on the mole frac- tion of component 1 such that A = xlAl + X2A2 . (106) Summarizing, we have the following equations relating the dielectric constant, refractive index, and molar refractivity for nonpolar and polar substances: pure nonpolar fluids: n = C II X 3’ 4. binary mixtures of nonpolar fluids: A X A pure polar fluids: 2 2 2 + 8c - lGec'u /T 5 2 C + 4cqip /T :1 | NIH 2 l + ec'u /T binary mixtures of polar fluids: A'=XA'+ ' 1 1 X2A2 III. EXPERIMENTAL INFORMATION A. The Brillouin Spectrometer The Brillouin spectrometer employed in these light scattering studies was designed and constructed by S. J. Gaumer. A detailed description of the apparatus is contained in his Ph.D. thesis (71). For the sake of completeness, however, and in order to be able to discuss separate portions of the apparatus, a schematic diagram of the spectrometer is shown in Figure 2. Omitted from the diagram is the temperature control cell which houses the glass sample tubes. The temperature control cell is placed on the rotating table and is constructed to allow the incident and scattered beams into and out of the sample tube while maintaining the temperature to + 0.1 °C of a prescribed setting. The design, construction and calibration of the temperature control cell are the subjects of section III B. The incident, vertically polarized light of wavelength 5145 g originates from a single mode, frequency stabilized Spectra Physics Model 165 Argon Ion laser. The incident beam travels to a front surface mirror mounted on a triangular optical rail, is reflected 90°, and passes into the sample on the rotating table. Light scattered at 90° from the incident beam in the horizontal plane is collected for analysis. Optics housing I is a light-tight metal box which sits at the head of the detection train: it contains two adjustable iris diaphragms and an achromatic lens. The iris diaphragms are located on either end of a threaded aluminum pipe which is positioned in the front portion of the box. Scattered light passes through the ~ 1 mm diameter 53 54 acmemwsvo Hmuoomwuom nua3 manna mcaumuom mono anon assessaa Hoowoomm acumecuomuousw nouumuaunmm How was: Houusoo euaumwogfima .uouufiouuoemm swsoaawum was .N onsmflm HeueaEMOOfim vogue 92m How mamas Hosea mammfim emmudo> : H000 -l(,.I.IHI(Ia liq!) fl “Hmmunfl mum I! / noses Hmowumo manuaon moaned % House: .1.) 1.: I I.:.I...|.r..:nWMVLII. Henna Aflllll|)l)l: 55 diaphragms and is collimated by a 500 mm focal length achromatic lens. The collimated light then impinges on a Fabry-Perot interferometer to be resolved. The Fabry-Perot etalon possesses 1 inch diameter mirrors which have inside surfaces polished to 1/100 flatness: the etalon is scanned piezoelectrically. The resolved light travels to a second light-tight metal box, optics housing II, to pass through (1) a third aperture (~ 5 mm diameter) (2) another achromatic lens (1000 mm focal length) (3) a final aperture (~ 1 mm diameter) and (4) a disc of polarizing material (HGCP-Zl, Eastman Kodak) set in the vertical position. The vertically polarized, resolved, scattered light then travels to a photomultiplier tube to be detected. The signal from the photomultiplier is amplified by a Keithley picoammeter and is fed into a Sargent recorder. The spectrum of the scattered light is traced by the recorder as the linear ramp voltage to the piezoelectric device is increased. Five spectral orders are obtained from a single O-l700-volt scan of the interferometer. The finesse for the spectra obtained in this project ranged from 30 to 45. In order to obtain accurate data from a Brillouin scattering experiment, it is necessary to remove the depolarized (horizontal) contribution to the total light scattering spectrum. The vertical polarizer included in the detection train is used to accomplish this task. The horizontal component appears to have been of little consequence, however, since the depolarization ratio for the two neat liquids was found to be < 0.02. 56 B. Design and Characterization of the Temperature Control Cell In order to maintain a constant, controlled temperature within the DMSO—pyridine solutions and yet enable light to pass through, a special temperature control cell had to be constructed for use with the Brillouin spectrometer. An artist's sketch of the unit is shown in Figure 3, while a cut-away representation is given in Figure 4. The basic component of the temperature control cell is a solid copper cylinder 4 1/2 inches in height. The diameter of the cylinder in the upper and lower flanges is 3 3/8 inches, while the diameter of the smaller central section is 2 1/2 inches. A slit 1/2 inch in width is positioned 1 3/4 inches from the bottom of the cylinder and extends 1 7/16 inches (horizontal depth) into the cylinder. The slit allows light to pass through a sample tube which sits in the center of the copper cylinder. A piece of copper tubing 3/16 inch O.D. is wound around and soldered to the outside surface of the copper cylinder to serve as a cooling coil. The ends of the tubing are soldered to hollow posts on the top of the cylinder to provide an inlet and outlet, respectively, for cooling solutions. A layer of molded asbestos with 12 feet of #30 nichrome wire embedded in it forms the outermost layer of the central section of the copper cylinder. Each end of the nichrome wire is wound around a drilled and tapped, electrically insulated brass post which emerges on the top of the cylinder. Voltage applied to the nidhrome wire via the brass posts serves to heat the copper cylinder. The outer insulation material for the temperature control cell consists of a Transits board on the bottom, on which the copper cylinder nugutuluuuuudh(UHLu(H;3F Hllhtlluwfium Figure 3. Sketch of the temperature control cell. 58 \ r ' I / 1 m N\/ (I\ \ //////“‘ ite "J. h~T~J§E§E:::::////// / \l \ .\ ////// . ////// Sleeve .Air , / ace ° I / Copper 20 0: / Cooling Ceil 1//J : hole ; ////// Role for Soldered to 2.9 O. Ax” Thermistor outside of ::;//f 2 for ; ,z” PrObe Copper Block /////A fig ’ O: r/::::: Nichrome / : Sam- 2+— 7 A fleeting wire a 1 f :fi 03 / Embedded in o e or ./A 4_v ple t //////’ Asbestos Thermometer ///’ 4 z ; ' ¢ :0 -— Cell 05 ¢ // : Copper ' .4 O: - so 0 Air ? E BlOCk :S+93C% I \ < ‘T““~ ‘““‘~~ /(1/:4 Air Space szj/9/Transite pad f Aluminum Plate J I Figure 4. Cross-sectional view of the temperature control cell. 59 rests, a Transite sleeve around the periphery of the cylinder and a Transite inset on tep. The Transite pipe and board were gifts of the Johns-Manville Corporation. In order to provide a firm mounting for the heavy unit, the Transite board was cemented with epoxy glue to three 1/4 inch thick rectangular pieces of Transite, which were in turn epoxied to a 1/4 inch thick circular aluminum plate. An air space the thickness of the Transite pads helped to insulate the main portion of the temperature control cell from the aluminum base plate. Holes for a thermometer, thermistor probe and sample tube were drilled through the vertical portion of the copper cylinder in suitable positions. The 1/2 inch diameter hole for the sample tube was placed in the center of the upper face of the cylinder, while the holes for the thermometer and thermistor probe were located one quarter inch to either side of the central hole. Calibration of the temperature control cell was performed using glycerin as the sample and two accurate etched-stem mercury thermometers as the temperature monitors. An Owens-Illinois mercury thermometer, designated A, reading from -1 to 101 0C in divisions of 0.1 0C, was placed inside a glass sample tube1 in the copper cylinder and was immersed in glycerin to the proper depth. A Nurnbero mercury thermomr eter, designated C, reading from -10 to 80 0C in divisions of 0.2 0C, was positioned in the thermometer hole in the copper cylinder. A thermistor probe from a YSI Model 72 temperature controller occupied l The glass sample tube was a 9 mm diameter Fischer-Porter joint sealed at the bottom. 60 the second small hole in the top of the copper cylinder. This partic- ular arrangement, with thermometer C in the thermometer hole and the thermistor probe in the other small hole in the copper cylinder, con- stituted the experimental arrangement by which all of the temperatures for the DMSO-pyridine solutions were measured and controlled during the Brillouin scattering experiments. In response to a signal from the YSI temperature controller, voltage was applied to the nichrome wire around the copper cylinder, causing the cylinder to be heated. The band width of the controller was adjusted to 0.1 0C, and temperatures could be set on the controller to the nearest tenth of a degree. It was found during the calibration procedure that the temperatures set on the controller agreed very closely with the temperatures read from thermometer C. The calibration procedure was begun by immersing both of the thermometers in an ice bath to determine a correct zero point for each. After the zero points had been obtained, the temperature controller was set to + 70.0 0C, and the control cell was allowed to come to thermal equilibrium. The set-point was then lowered by five- degree increments down to 40 oC: below 40 0C, the controller set-point was lowered one degree at a time. At least 20 minutes was allowed for thermal equilibrium after each lowering of the set-point, even for the one degree increments. A summary of the data obtained is given in Appendix A; only the pertinent results are included here. Figure 5 illustrates some of the data gathered from the temperature calibration experiments. The temperatures from the two thermometers are designated TC and TA, respectively. TC refers to the temperature of the thermometer in the hole in the copper cylinder, 61 .Hdoo Honusoo mhsumnogadu on» How mono sowumnnwamo .m owsmwm m.o + anm.0 u .< h as no do an on as ov on on as ou _ _ a L) .‘ ()4 _ )2 a a _ d m.o : cahoo.a u a “00 co 30Hem 1 o a a "00 co e>oo¢ 1 dado madame cw oesowa mo ousumuomfieu use i xuoan Hoodoo mo ousueuomaou "09 an mm on an ow mm on no 00 no 62 while TA refers to the temperature of the thermometer in the sample liquid. It was found that the data above 40 oC gave rise to a straight line of slightly different slope than the data between 20 and 40 oC. The linear relationships for each temperature region are shown on the respective portions of the graph. Not surprisingly, the similarity of the two temperatures (TC and TA) decreases at readings above 40 0C. From the data obtained in this series of experiments, it was a simple matter to adjust the temperatures taken in the DMSO-pyridine light scattering experiments to the correct values established by the calibration curve in Figure 5. Typical temperature readings from thermometer C taken during light scattering measurements are given in Table 1, along with the corrected values. The precision of the measured and corrected temperatures is i 0.1 0C. TABLE 1 Measured and Corrected Temperatures Obtained During Brillouin Scattering Measurements for Mixtures of DMSO and Pyridine Measured Corrected Temperature Temperature TC(°C) TA(°C) 64.7 63.9 59.8 59.0 54.6 54.2 49.6 49.3 44.7 44.5 39.6 39.4 34.8 34.5 29.6 29.5 26.8 26.7 21.6 21.6 It should.be noted at this point that all the temperatures presented in other sections of this work are corrected values. 63 C. Preparation of the Samples Purification of the two solvents, DMSO and pyridine, and prep- aration of eleven of their mixtures were carried out by M. S. Greenberg; details of these procedures are given in Ref. 72. It was found from Karl Fischer titrations that the residual water in the two pure fluids amounted to < 0.077% by weight. This corresponds to a maximum contri- bution of 0.33 mole per cent water to the DMSO-pyridine mixtures. To insure that the samples were as free of dust as poSSible for the light scattering measurements, the solutions were passed repeatedly through an ultrafine-Millipore filter apparatus designed by H. K. Yuen (73). This filtering device was a closed-system type whereby the solution was forced under nitrogen pressure through an ultrafine filter, through a Millipore filter, and finally into a collection tube. The collection tubes consisted of 9 mm diameter Fischer-Porter joints which had been sealed at the bottom. Filtration and collection of the samples proceeded according to the steps outlined below, beginning with neat pyridine and ending with neat DMSO. Filtration Procedure 1. Flushed apparatus twice with 2 m1 of the new sample. 2. Passed five more 2-ml aliquots of sample through the filtration apparatus. Each sample was collected in a Fisdher-Porter collec- tion tube, was rolled around in the tube and discarded. 3. After the rinsing process was completed, 12 ml of solution was passed repeatedly through the filtering apparatus and collected until, upon examination with a laser beam, the resultant liquid appeared to be dust-free. (The small dust particles were visible 64 as tiny scintillators when the beam.was passed through a dirty sample.) After a dust-free solution had been obtained, the sample tube was sealed to prevent contamination from dust and moisture. Duplicate or triplicate samples were prepared for each composition for examination with the Brillouin spectrometer. This was done to eliminate the possi- bility of a sample being prepared with an incorrect composition. In all cases it was found that the light scattering data from the duplicate or triplicate samples agreed within the limits of experimental error. D. Measurement of the Refractive Index Refractive indices for the DMSO-pyridine mixtures were obtained on a Bausch and Lomb Abbe 3-L Refractometer at a.wavelength of 5890 g (the sodium D line). The refractive index for eadh of the thirteen solutions was measured at five different temperatures. Temperature control for the prisms of the refractometer was provided by a Haake circulating-bath temperature control unit. The refractive indices for A = 5145 g were calculated from information given in the dispersion table provided by the Bausch and Lomb Company, and from compensator readings obtained on the refracto- 0 meter for each refractive index measurement made at 5890 A. IV. RESULTS AND DISCUSSION A. Variation of the Refractive Index with Temperature and Composition Refractive indices for the DMSO-pyridine solutions were re- quired at each temperature to calculate the velocities of sound. Since the light scattering observations were made using an incident wave- length of 5145 g, and the refractive index measurements were made employing a wavelength of 5890 X (the sodium D line), corrections to convert the refractive index data to obtain n5145 g for the DMSO- pyridine mixtures were necessary. Details of the correction procedure are given in Appendix B; only the refractive indices meaSured at A a 5890 g and calculated for A = 5145 g are reported here. Preceding the refractive index data is a table of actual and approximate compositions for the DMSO-pyridine solutions (Table 2). The rounded (nominal) values of the mole fraction are used throughout the remainder of this monograph to conserve space. The molecular weights of the two solvents are listed to illustrate their remarkable similarity. The refractive indices of the DMSO-pyridine solutions at five different temperatures are given in Table 3. Typical graphs of the measured refractive index, n5890 2, versus temperature are shown in Figure 6, while data for n5890 X and n5145 g versus temperature for pure DMSO are presented in Figure 7. Graphs of the refractive index versus temperature give rise to no discontinuities in the temperature range 22.0 to 39.0 0C, indicating that there are no anomalies in the refractive index behavior of the individual mixtures in this tempera- ture region. The temperature coefficient of the refractive index, 65 66 Table 2 Nominal and Measured Compositions for Mixtures of DMSO and Pyridine mole fraction mole fraction DMSO DMSO (nominal) (measured) 0.00 0.0000 0.05 0.0495 0.10 0.0993 0.15 0.1506 0.20 0.1983 0.30 0.2997 0.40 0.3995 0.50 0.5005 0.60 0.6002 0.70 0.7001 0.80 0.8004 0.95 0.9453 1.00 1.0000 Molecular weight pyridine = 79.10 Molecular weight DMSO = 78.13 The average standard error in these values is r 0.0003. The DMSO-pyridine mixtures were made up by M. S. Greenberg (72). 67 Table 3 , o Refractive Index as a Function of Temperature for A a 5890 and 5145 A O o O T(OC) “53902 n5145A T(OC) n5890A n5145A pure pyridine 0.05 mole fraction DMSO 22.0 1.5090 1.5164 22.0 1.5073 1.5146 26.4 1.5064 1.5138 26.4 1.5053 1.5126 30.1 1.5046 1.5120 30.1 1.5031 1.5104 30.3 1.5042 1.5116 30.3 1.5028 1.5101 39.0 1.4988 1.5062 0.10 mole fraction DMSO 0.15 mole fraction DMSO 22.0 1.5056 1.5128 22.0 1.5052 1.5123 26.4 1.5035 1.5107 26.4 1.5020 1.5091 30.1 1.5017 1.5089 30.1 1.5003 1.5074 30.3 1.5008 1.5080 30.3 1.5000 1.5071 39.0 1.4957 1.5029 39.0 1.4942 1.5013 0.20 mole fraction DMSO 0.30 mole fraction DMSO 22.0 1.5044 1.5113 22.0 1.4979 1.5045 26.4 1.5010 1.5079 26.4 1.4962 1.5028 30.1 1.4993 1.5062 30.1 1.4941 1.5007 30.3 1.4988 1.5057 30.3 1.4950 1.5016 39.0 1.4931 1.5000 The refractive index measurements were a joint effort of M. S. Greenberg and the author. The average error for “58902 is 1 0.0002. According to the Bausch and Lomb reference manual, values of the refrac- tive index calculated for a wavelength other than 5890A are subject to a standard error of 1 0.0005. Table 3 (cont.) T(OC) O n5890A O n5145A 0.40 mole fraction DMSO 22.0 1.4962 1.5025 26.4 1.4942 1.5005 30.1 1.4923 1.4986 30.3 1.4925 1.4988 0.60 mole fraction DMSO 22.0 1.4915 1.4973 26.4 1.4894 1.4952 30.1 1.4877 1.4935 30.3 1.4877 1.4935 39.0 1.4827 1.4885 0.80 mole fraction DMSO 22.0 1.4852 1.4909 26.4 1.4831 1.4888 30.1 1.4809 1.4866 pure DMSO 22.0 1.4784 1.4836 26.4 1.4763 1.4815 30.1 1.4748 1.4800 30.3 1.4748 1.4800 39.0 1.4709 1.4761 68 33:3) “58902 n51453 0.50 mole fraction DMSO 22.0 1.4942 1.5004 26.4 1.4917 1.4979 30.1 1.4901 1.4963 30.3 1.4901 1.4963 39.0 1.4858 1.4920 0.70 mole fraction DMSO 22.0 1.4880 1.4937 26.4 1.4863 1.4920 30.1 1.4846 1.4903 30.3 1.4838 1.4895 0.95 mole fraction DMSO 22.0 1.4802 1.4855 26.4 1.4785 1.4838 30.1 1.4768 1.4821 30.3 1.4766 1.4819 69 .msfioauhm one omzo mo mowsuxflfi you musumnwmamu msmno> xmocw m>wuomwmom .m mummwm Auov m~3p Noose m>wuomummm .m onpmflh 0min. 20:U<~: m._0<< 0°.— 00.0 00.0 05.0 00.0 00.0 01.0 00.0 0nd 0—.0 00.0 a s _ _ i a 33.. l 000‘.— l 000‘.— l. 000V.— once.— «mini 0000.— 0n0n.— .... 00.“.— 74 00; .oo H.om one «on .o.NN How sowuwmomaoo moms? x88." gwuomumum Om<UZmDOm~I “w..- ...—H .3 083m ‘——— AllS N31N| .Uo o.¢m can m.m~ um omzn ummc How muuommm chOHHHHm ‘Illu >UZmDOm~E m NIOhvdu a 81 N I 022 um; 00 Wan n a . 3 magma a——-— AllSNElNI 82 ”8:5.75 G H Z INTENSITY -—0 VB=5.48C3 H z FREQUENCY ——-O Figure 15. Brillouin Spectra for 0.60 mole fraction DMSO at 39.4 and 59.0 °c. 83 T = 21.6 °c I "fill =6.AIGHz-‘ / W -.MJ TLW/ T = 59.0 °c INTENSITY ——O A '— ”8:5.87GH2" FREQUENCY—'4 Figure 16. Brillouin spectra for neat pyridine at 21.6 and 59.0 0C. 84 Table 5 Brillouin Shift as a Function of Temperature T(°c) vB(GHz) T(OC) vB(GHz) T(OC) VB(GHz) T(OC) vB(GHz) purejygidine 0.05 m.f. DMSO 0.10 m.f. DMSO 0.15 m.f. DMSO 21.6 6.39 21.6 6.29 21.6 6.27 21.6 6.28 29.5 6.27 29.5 6.17 29.5 6.12 29.5 6.14 34.5 6.19 34.5 6.08 34.5 6.02 34.5 6.05 39.4 6.11 39.4 6.01 39.4 5.92 39.4 5.97 44.5 6.03 44.5 5.92 44.5 5.82 44.5 5.88 49.3 5.95 49.3 5.85 49.3 5.73 49.3 5.80 54.2 5.87 54.2 5.77 54.2 5.63 54.2 5.71 59.0 5.79 59.0 5.69 59.0 5.54 59.0 5.63 0.20 m.f. DMSO 0.30 m.f. DMSO 0.40 m.f. DMSO 0.50 m.f. DMSO 21.6 6.17 21.6 6.22 21.6 6.11 21.6 6.02 29.5 6.05 29.5 6.08 29.5 5.99 29.5 5.90 34.5 5.97 34.5 5.98 34.5 5.90 34.5 5.82 39.4 5.89 39.4 5.89 39.4 5.82 39.4 5.75 44.5 5.81 44.5 5.79 44.5 5.73 44.5 5.67 49.3 5.74 49.3 5.70 49.3 5.65 49.3 5.59 54.2 5.66 54.2 5.61 54.2 5.57 54.2 5.51 59.0 5.59 59.0 5.52 59.0 5.49 59.0 5.44 85 Table 5 (cont .) T(OC) vB(GHz) T(OC) VB(GHz) T(OC) VB(GHz) T(OC) vB(GHz) 0.60 m.f. DMSO 0.70 m.f. DMSO 0.80 m.f. DMSO 0.95 m.f. DMSO 21.6 6.00 21.6 6.05 g 21.6 6.04 21.6 6.13 29.5 5.91 29.5 5.92 29.5 5.95 29.5 6.00 34.5 5.84 34.5 5.83 34.5 5.88 34.5 5.92 39.4 5.78 39.4 5.75 39.4 5.82 39.4 5.83 44.5 5.72 44.5 5.66 44.5 5.75 44.5 5.75 49.3 5.66 49.3 5.57 49.3 5.69 49.3 5.67 54.2 5.60 54.2 5.49 54.2 5.63 54.2 5.59 59.0 5.54 59.0 5.41 59.0 5.57 59.0 5.51 EDIE DMSO 34.5 5.91 39.4 5.84 44.5 5.75 49.3 5.68 54.2 5.60 59.0 5.53 Average standard deviation for T is 0T = i 0.1 0C. Average standard deviation for VB is 0v = i 0.05 GHz. 86 mm .OWZQ “MUG HON mgflflHGQEB MUWHO> HMHQM C; 53:22: :flsoflaaum .na «Human on n* 0‘ mm on nN ON 1 A 4 q —‘ 1 — _ .ncifi .1606 L006 ..OOd .10N6 (2H9) a 87 .095 :oflomum waoa om.o can. om.o .mH.o .36 .wawpwuhm was: How wH:#mnmemu mnmnm> umfifiw agoaawhm .mH mupmflh Au; $523.22: (2H9) a 00 mm on mv ov mm on mm on n.. . .... 1‘ _ _ _ _ a _ _ _ o n n I o md 1 end a 1 00.0 093 Gawuomuw 0H9: I o ...o 095 530mb 39: 8.0 0 85 5383 32 3.0 I 1 and 8.8 c9303“ 205 36 b unwmflgm umoz O ... 0 n6 Tfifle6 88 Intercept and Slope for the Brillouin Shift- mole fraction DMSO 0.0000 0.0495 0.0993 0.1506 0.1983 0.2997 0.3995 0.5005 0.6002 0.7001 0.8004 0.9453 1.0000 Temperature Relationship A (GHz) 0A 6.74 $0.06 6.63 0.05 6.70 0.09 6.65 0.02 6.50 0.03 6.62 0.04 6.47 0.03 6.36 0.06 6.27 0.02 6.42 0.05 6.31 0.14 6.49 0.09 6.45 0.05 VB=A+BT -B (GHz/OC) 0.0160 0.0159 0.0197 0.0174 0.0155 0.0186 0.0166 0.0156 0.0124 0.0172 0.0126 0.0166 0.0157 OB i0.0014 0.0013 0.0024 0.0005 0.0006 0.0011 0.0008 0.0014 0.0005 0.0010 0.0030 0.0021 0.0011 89 .0825th can omza mo moguxd: How soauflmomfioo 5H3 9m + .a u .uO.— Onic o-.o 05.0 0220 ZOZ.U “masm sflsoaawum .HN oHDmHm Owio ZOCU<~I m ..OE 00.. 00.0 00.0 050 00.0 0n0 0V0 0nd 0nd 0-0 00.0 n 1 3 4 fl 1 _ 7 s _ W F Le 0 0.0 0 0.0 0 ..0 0 «.0 0 0.0 0 0.0 0 0.0 (Inmga .00 3mm 98 mém .....mm 6.3 you 80339.8 mama»... unfit 59335 .2 983m 0min ZO_»U<~: m..0<< 93 00.— 0. 0nd 0 0.0 00.0 00.0 0 9.0 0«.0 0 —.0 00.0 05.0 0 0.0 00.0 00.0 0«.0 09.0 0 v.0 (2H9)8d 94 00.— maOE .o 05.0 00.0 00.0 0 v.0 0nd 0 «.0 1 — a fi 1 .oo o.mm can ~.em .m.m¢ .m.e¢ you coauamomsoo msmump umwsm nanoaaanm 0min ZO_~U<~I 00 00.0 . mm shaman 0 —.0 0 0.0 0 V...“ 0 0.0 00.0 0nd (2H9)aa 95 An advantage of Brillouin scattering over ultrasonic measure- ments is that no transducer is required; the sound velocities detected are the products of thermal motion in the liquid. Velocity of sound data gathered by light scattering and by ultrasonics for a variety of liquids have been found to be in good agreement (29,49,82). A logical consequence would be to assume that one can use documented ultrasonic findings on mixtures to aid in the interpretation of data obtained from Brillouin scattering measurements. As seen in the remaining portion of this chapter, the author has drawn extensively from ultra- sonic research to aid in the interpretation of the data from this study. If there were no structural changes occurring in proceeding from pure pyridine to pure DMSO, one would expect two phenomena: (1) the change in the velocity of sound in proceeding from one pure fluid to the other would be a smooth, continuous function (81)1 (2) the change in st/dT with composition would also be a smooth, continuous function. In Table 7 are displayed the frequency shifts, refractive indices and velocities of sound for the DMSO-pyridine mixtures as a function of temperature. By numerical analysis the sonic velocity- temperature curves for each composition are fit by a linear relation- ship, Vs = A + ET. The parameters A and B obtained for each composi- tion and their standard deviations are exhibited in Table 8. The 1From Ridhardson, p. 178, "If two liquids of different density and elasticity are mixed in various proportions, one would expect the velocity to pass gradually from the value corresponding to one pure fluid to that of the other." This type of behavior has been observed for many liquid mixtures; of. Ref. 81, pp. 178-181 and Ref. 77. 96 Table 7 Brillouin Shift and Velocity of Sound as a Function of Temperature for Mixtures of DMSO and Pyridine T(OC) vB(GHz) n5145§ Vs(m/sec) T(OC) VB(GHz) n51452 VS(m/sec) pure CSHSN 05 mole % DMSO 21.6 6.39 1.5168 1530 21.6 6.29 1.5149 1510 29.5 6.27 1.5121 1510 29.5 6.17 1.5107 1490 34.5 6.19 1.5091 1490 34.5 6.08 1.5080 1470 39.4 6.11 1.5061 1480 39.4 6.01 1.5053 1450 44.5 6.03 1.5031 1460 44.5 5.92 1.5026 1430 49.3 5.95 1.5002 1440 49.3 5.85 1.5000 1420 54.2 5.87 1.4974 1430 54.2 5.77 1.4974 1400 59.0 5.79 1.4944 1410 59.0 5.69 1.4948 1380 10 mole % DMSO 15 mole % DMSO 21.6 6.27 1.5134 1510 21.6 6.28 1.5125 1510 29.5 6.12 1.5087 1480 29.5 6.14 1.5075 1480 34.5 6.02 1.5058 1460 34.5 6.05 1.5043 1460 39.4 5.92 1.5029 1430 39.4 5.97 1.5012 1450 44.5 5.82 1.5000 1410 44.5 5.88 1.4979 1430 49.3 5.73 1.4971 1390 49.3 5.80 1.4948 1410 54.2 5.63 1.4942 1370 54.2 5.71 1.4917 1390 59.0 5.54 1.4914 1350 59.0 5.63 1.4887 1380 Values for the average standard deviation for each of the entities listed in the table are as follows: temperature CT = t 0.1 0C refractive index on Brillouin shift 0v i 0.05 GHz velocity of sound CV 0.0005 10 m/sec H- H- 97 Table 7 (cont.) T(OC) vB(GHz) n5145§ Vs(m/sec) T(OC) vB(GHz) n5145§ VS(m/sec) 20 mole % DMSO 30 mole % DMSO 21.6 6.17 1.5115 1490 21.6 6.22 1.5047 1500 29.5 6.05 1.5063 1460 29.5 6.08 1.5015 1470 34.5 5.97 1.5030 1450 34.5 5.98 1.4994 1450 39.4 5.89 1.4998 1430 39.4 5.89 1.4974 1430 44.5 5.81 1.4965 1410 44.5 5.79 1.4953 1410 49.3 5.74 1.4933 1400 49.3 5.70 1.4934 1390 54.2 5.66 1.4901 1380 54.2 5.61 1.4914 1370 59.0 5.59 1.4870 1370 59.0 5.52 1.4894 1350 40 mole % DMSO 50 mole % DMSO 21.6 6.11 1.5027 1480 21.6 6.02 1.5005 1460 29.5 5.99 1.4990 1450 29.5 5.90 1.4966 1430 34.5 5.90 1.4967 1430 34.5 5.82 1.4942 1420 39.4 5.82 1.4944 1420 39.4 5.75 1.4918 1400 44.5 5.73 1.4921 1400 44.5 5.67 1.4893 1390 49.3 5.65 1.4899 1380 49.3 5.59 1.4869 1370 54.2 5.57 1.4876 1360 54.2 5.51 1.4846 1350 59.0 5.49 1.4854 1340 59.0 5.44 1.4822 1330 98 Table 7 (cont.) T(OC) vB(GHz) n5145§ VS(m/sec) T(OC) vB(GHz) n5145§ VS(m/sec) 60 mole % DMSO 70 mole % DMSO 21.6 6.00 1.4977 1460 21.6 6.05 1.4940 1470 29.5 5.91 1.4936 1440 29.5 5.92 1.4903 1450 34.5 5.84 1.4910 1430 34.5 5.83 1.4880 1430 39.4 5.78 1.4885 1410 39.4 5.75 1.4856 1410 44.5 5.72 1.4859 1400 44.5 5.66 1.4832 1390 49.3 5.66 1.4834 1390 49.3 5.57 1.4810 1370 54.2 5.60 1.4809 1380 54.2 5.49 1.4787 1350 59.0 5.54 1.4784 1360 59.0 5.41 1.4764 1330 80 mole % DMSO 95 mole % DMSO 21.6 6.04 I 1.4912 1470 21.6 6.13 1.4857 1500 29.5 5.95 1.4870 1460 29.5 6.00 1.4823 1470 34.5 5.88 1.4843 1440 34.5 5.92 1.4802 1460 39.4 5.82 1.4818 1430 39.4 5.83 1.4781 1440 44.5 5.75 1.4791 1410 44.5 5.75 1.4759 1420 49.3 5.69 1.4765 1400 49.3 5.67 1.4738 1400 54.2 5.63 1.4739 1390 54.2 5.59 1.4717 1380 59.0 5.57 1.4714 1380 59.0 5.51 1.4696 1360 99 Table 7 (cont.) T(OC) vB(GHz) n5145§ Vs(m/sec) pure DMSO 21.6 6.11 1.4837 1500 29.5 5.99 1.4803 1470 34.5 5.91 1.4781 1460 39.4 5.84 1.4760 1440 44.5 5.75 1.4737 1420 49.3 5.68 1.4716 1400 54.2 5.60 1.4695 1390 59.0 5.53 1.4674 1370 100 TafleB Intercept and Slope for the Sonic Velocity- Temperature Relationship Vs = A + BT mole fraction A 8 DMSO (mésec) (misec/OC) 0.0000 160314 -3.23i0.09 0.0495 158914 -3.51i0.09 0.0993 160614 -4.36i0.09 0.1506 158514 -3.5210.09 0.1983 155914 -3.2510.09 0.2997 158811 -4.02t0.02 0.3995 1560i4 -3.68:0.09 0.5005 153415 -3.3810.12 0.6002 1516i4 -2.60i0.09 0.7001 1560i5 -3.84iO.11 0.8004 152915 -2.57i0.12 0.9453 1584:5 -3.74i0.11 1.0000 1575:4 -3.47:0.09 The errors listed are the standard deviations for each value. 101 quantity B, lst/dTI, as a function of composition is shown in Figure 24. The variation in the velocity of sound with composition for eadh temperature is displayed in Figure 25. Within experimental precision in the measurements, neither the Vs versus x1 nor IdVS/dTI versus x1 curve is a smooth, continuous function over the entire temperature range, indicating that there are structural changes in the mixture as one progresses from pure pyridine to pure DMSO. According to kudriavtsev (83), the square of the velocity of sound for an ideal binary solution at a given temperature should vary linearly with composition: ' 2 2 2 _. M (v ) — xA _5. vA + xB E§_ VB . (108) Mmix Mmix In equation (108), XA and XB are the mole fractions of components A and B, MA and “B are the molecular weights of pure A and B, and VA and VB are the sonic velocities of the pure species, respectively. “mix is the molecular weight of the mixture. Deviations from this linear relationship occur when association takes place. Values of (V.)2 for mixtures of DMSO and pyridine at 21.6 0C are given in Table 9, along with values for the square of the experi— mentally determined velocity of sound, V52. Table 9 Squares of the Ideal and Experimental Velocities of Sound for 21.6 °C Ideal _ Experimental mole fraction (V2)2X1g Vs x10"6 DMSO, xDMSO (m /sec ) (mZ/Secz) 0.0000 2.341 2.34 0.0495 2.338 2.28 0.0993 2.332 2.28 0.1506 2,326 2.28 0.1983 2.323 2.22 102 .055m «0 huwooag 05 mo ucowowmmmou wgumuwmfimfi .gn 05.5.3 0329 20:93: 302 69. and cod and cod. and 3.6 and one 3.0 cod ,7 q 4 T 1 d a 1 _ _ \ PH M m '9...» Go.“ on." o V.N oofi 00$ 0 o." 0'.” cod 00.0 00.1 on; octv oo.‘ 1P SAP 103 .00 0.0m can Ném 5.3 6.3 .¢.mm .mém .m.m~ 6:3 . Hon ”3.330930 5.; 050m «0 huwooag 05 mo nowumflnflw .mN «Human 0min ZOZ.U<~: 302 00.. 09.0 00.0 omo oo.o . 0nd 016 end o~.o 0.6 00.0 a q _ 1 . j _ _ _ _ onu— con- 1 Ohm— cov— onw— con. 00 o.mmue 4 oo Tom "96 00 23.1.9 I oo mémuen uo Wmeue > .oo m.m~ue> of... co mévue 0 0o mé~uao (DOS/w) SA 104 Table 9 (cont.) Ideal Experimental mole fraction (V')2x10"6 Vsleo‘6 ouso, xDMSO (mz/secz) (mz/secz) 0.2997 2.313 2.25 0.3995 2.304 2.19 0.5005 2.295 2.13 0.6002 2.286 2.13 0.7001 2.277 2.16 0.8004 2.268 2.16 0.9453 2.256 .2.25 1.0000 -2.250 2.25 Average standard deviation for V' and V5 is 0v = 1 10m/sec. Average standard deviation for (V')2 and Vs2 is (N2 = 1 0.03x106 mz/secz. 2 with composition at 21.6 °C, plus Variations of (v')2 and VS the standard deviations, are shown in Figure 26. The DMSO—pyridine solutions exhibit extremely large deviations from ideality, indicating that a considerable amount of association is present in these mixtures at 21.6 0c. The adiabatic compressibility, BS, is another quantity of interest and can be calculated from velocity of sound and density data according to 8 = 1 (109) S "‘1Z OVS ’ Table 10 contains values of the adiabatic compressibility for the DMSO-pyridine mixtures at 21.6 0C. Table 10 Adiabatic Compressibility as a Function of Composition for 21.6 °C XDMSO (cmZ/dyne) 0.0000 4.38 0.0499 4.46 105 .Uo mém How COHUflmomfioo 3925 @50m mo fiwooag. 05 mo mhmawm .om gunman 0920 ZOCU<~E 302 00.. 00.0 om.o oud 00.0 0nd ov.o om.o o~.o 0.6 00.0 .1 _ 1 a ,._ . _ i a _ oo: 4.- B non_.~ 4 -.F Hmucwfiwummxm J] loomfl .1. I 4| ll Jo. lomNN a ... . _ 4| Lu . w 31 H _ x J 182 .r ” 1.. .i L‘ . IV [V Hmong” 1r 39m Ir oowfi (zaas/sz-le zA 106 Table 10 (cont.) 85 x 1011 XDMSO (cmZ/dyne) 0.1005 4.44 0.1500 4.42 0.2000 4.50 0.2996 4.40 0.4016 4.48 0.4993 4.55 0.6002 4.49 0.7005 4.38 0.8004 4.34 0.9453 4.09 1.0000 4.06 Average standard deviation for 85 is 0BS = 1 0.05 x 10"11 cmz/dyne. The apparent adiabatic compressibility as a function of composition is plotted in Figure 27. Several interesting observations can be made about Figure 27 and part b of Figure 26. The most noticeable one is that the two functions are almost mirror images of one another. This indicates that the structural changes which produce the characteristic behavior in the BS versus Xl curve at 21.6 0C produce a similar but opposite effect in the plot of V82 with X1. The compressibility of a liquid is inversely proportional to the strength of the intermolecular forces or degree of association in the liquid. The velocity of sound is directly proportional to the strength of attraction between the molecules. By examining changes in adiabatic compressibility, sonic velocity and dVS/dT with composi- tion for a series of binary fluids, one can gain insight into the local structural modifications occurring over the full range from one pure fluid to the other. IO? 00.— .00 012.. How cowufimoguoo $5.5 huwawflmmoumeoo owumnmwpm 05 no coaumaumxw KN 055.3 00.0 0min. ZO3U<~I m._O<< 00.0 05.0 00.0 0nd 0V0 0n.0 00A. 0—.0 00.0 « 1 i 1 a a i a a no; .. ou.v + a a .... . -5. A. t.. . [I G“ 0 L on... P L 006 (auAp/zun) u 01 x 9g 108 Referring to Figures 26-b and 27, one sees that the addition of dimethyl sulfoxide to pyridine up to ~ 0.55 mole fraction DMSO causes a continuing decrease in the square of the velocity of sound and a slight increase in the adiabatic compressibility. This type of behavior, in general, is associated with a decrease in order or a disorganization of local structure in a liquid. The point of maximum disorganization 2 reaches a appears to occur at ~ 0.55 mole fraction DMSO, where VS minimum and BS attains a maximum. These observations are interesting in view of the fact that the S-O stretching frequency (Figure 28) declines steadily in the region from pure pyridine to 0.60 mole fraction DMSO at 21.6 0C, indicating increased dipole-dipole interaction and alignment of the S-O dipoles (72). Evidently the microscopic changes in the bond order of the S-O bond are not reflected by the light scattering behavior of the fluid, which is dependent upon dimethyl sulfoxide §2§_pyridine. In the compositional region from 0.00 to 0.50 mole fraction DMSO, in which there are more pyridine molecules present than DMSO molecules, it is entirely possible that the attempts of the dimethyl sulfoxide molecules to form aggregated species (as indicated by the behavior of the S-O stretch frequency) are masked by the pyridine- pyridine, pyridine-DMSO interactions, which also influence the light scattering behavior of the liquid.1 The dramatic fall in the adiabatic compressibility and rise in VS2 between 0.50 and 1.00 mole fraction DMSO are accompanied by a 1 It should be noted that these observations pertain to one temperature only, 21.6 °C, and that the behavior of the molecules at higher temperatures may be entirely different. 109 .U 0.HN um coauamomfioo €53 hocmsvoum mcwsoumhum Cum 9.3 «0 coaumwflg .mm 050.: o . . Om< 055m mo bacon“; 9.3 «0 005m .3 dawn 0min ZO_._U<~_ u. m.— 02 00.— 00.0 090 Okd 00.0 00.0 04.0 on.o ON.O O—.O 00.0 4 q a q a fi M 1 4 q ON.— \7 Low.— - J 00.— O 4 O 4 D > I oo.~ .1 ..v 4 U I Q .I p D b D b D C . 4 U n p » 46: I D < e O 4 I D 4 a O o 4 D o . 4 . o: O 4 I o O 0 co camae) Uo «.mmueb . o .62. oo mémuel oo mimosa 00m.mefln.4 uo n.0muad o 00 934.9. ooméwuao (gm/aw) 9_0t x §A 112 DMSO at the higher temperatures. The temperature coefficient of the hypersonic velocity, IdVS/dTI, is seen to fluctuate dramatically in these compositional regions also. From the ultrasonic work of Lutskii and Solon'ko (79,80) concerning variations of V5 and IdVS/dleith association for aliphatic carboxylic acids, substituted phenols and anisole derivatives, one can make the following observations: (1) Association causes a significant increase of VS and a sharp lowering of ldVS/dTI. (2) In passing from a chain-like structure to cyclic dimer, there is a sharp decrease in V5 and a sharp increase in Ide/d'rl.1 (3) Branched chain complexes exhibit an even larger increase in V5 and decrease in Ist/dTl than do linear complexes. From these facts one can draw several reasonable conclusions concerning the behavior of the DMSO-pyridine mixtures. In particular, the significant increase in the velocity of sound (at the higher tem- peratures) and sharp decrease in the temperature coefficient between 0.10 and 0.15 mole fraction DMSO signify the formation of a more associated species from a less associated one. This could mean the formation of DMSO dimers, trimers or more highly associated chain-like aggregates in this region. Infrared spectroscopy measurements (72) of the S-0 stretching frequency tend to support this interpretation, as does the cryoscopic and dipole moment data of Lindberg, et al. for The chain-like association of formic acid has been confirmed by infrared studies, while the cyclic dimerization of acetic acid has been studied by proton magnetic resonance (80). 113 mixtures of DMSO and benzene (9,10). Infrared spectroscopic data for binary mixtures of DMSO and benzene have been interpreted by Szmant and his coworkers (2) to be indicative of the formation of cyclic dimers of DMSO in the composition range from 0.002 to 0.029 mole fraction DMSO. The difference in composi- tional region required for the formation of the DMSO dimers can be attributed to the difference in interaction between the S-O dipole and the pi electrons of the benzene ring, on one hand, and the polarizable electron cloud of the spherical carbon tetrachloride molecule, on the other. The minimum in the velocity of sound recorded for the 0.50-0.50 mole fraction mixture at each temperature signifies a maximum in dis- organization of the liquid structure at this composition. These minima in the velocity of sound are indirect evidence for the presence of more substantial attractive forces between DMSO species than between DMSO and pyridine molecules. If the association noted in this study were caused primarily by the interaction of DMSO with pyridine, one would expect to see a maximum in organization of the liquid structure at the equimolar composition. This is exactly opposite of the behavior observed experi- mentally. One can conclude, then, with a reasonable degree of certainty, that the associated species present in binary mixtures of dimethyl sulfoxide and pyridine are aggregates of DMSO, not DMSO-pyridine species. The complex pattern of variations in V3 and Ist/dTI between 0.50 and 0.80 mole fraction DMSO could be due to the formation and rearrangement of more highly associated species in the liquid, possibly chain-like and/or ring-like aggregates of dimethyl sulfoxide. The sequence of formation, breakdown and recombination of the species, however, remains obscure. 114 In the composition region between 0.80 and 1.00 mole fraction DMSO the velocity of sound increases at temperatures below 40 oC, signifying an increase in association. Presumably, more and more DMSO molecules are rearranging into the long-chain ring structure which has been proposed for the neat liquid at temperatures below 40 °C (84,85). The types of associated species present at temperatures above 40 °C are difficult to approximate because of the apparent breakdown in association _that occurs in the pure liquid at these higher temperatures.1 8. Variation of the Brillouin Linewidth with Temperature and Composition It appears to be a straightforward process to determine the linewidth of a Brillouin peak in a light scattering spectrum; however, this information has been obtained for relatively few liquids (49, 57- 59, 63-65). The reason for the lack of information on Brillouin line- widths is that it is exceptionally difficult to obtain a liquid which is free enough from dust so that accurate peak heights (and widths) can be measured. This difficulty holds true for both pure liquids and multicomponent fluids and explains in large measure why there is only slightly more linewidth data available for pure liquids (49, 57-59, 65) than for binary mixtures (63,64). The dearth of information on experimental Brillouin linewidths, therefore, renders this work the first major attempt to correlate effects of temperature and composition on the linewidths of Brillouin peaks for binary solvent systems. 1 This breakdown in association is evidenced by the change in slope of the Landau-Placzek ratio-temperature curve at ~ 45 °C (see section III G). 115 One realizes from the theoretical discussion of Brillouin linewidths on pp. 40-43, that a vast simplification of the data analy- sis occurs if the instrumental profiles and the observed Brillouin peaks can be shown to be Lorentzian in nature. Specifically, one finds that 2TB = 2FB/obs - I‘inst , (110) where 2FB is the true linewidth of the Brillouin peak, er/obs is the observed linewidth of the Brillouin peak and rinst is the linewidth of the instrumental function. In this particular situation, the true Brillouin linewidths can be obtained without the use of complicated computer methods for the deconvolution of the instrumental profile. Furthermore, if one shows that the instrumental profile is approximated by the central peak of the Brillouin spectrum, the instrumental line- width, rinst' can be replaced by the linewidth of the central peak, PC. In order to justify the use of such a simplification in our Brillouin linewidth analysis, we felt that we must demonstrate the fOIIOWing: l. The central and Brillouin peaks for the spectra of the DMSO-pyridine solutions are Lorentzian in shape. 2. The instrumental profile of the detection system is equivalent to the profile of the central peak of a Brillouin spectrum, via the fact that the linewidths of the central peaks for pure DMSO, pure pyridine and one of their mixtures are equal at a given value of finesse. 3. The linewidth of the central peak for a given solution is a linear function of the experimental finesse. 116 Portions of experimental Brillouin spectra which were used in testing the Lorentzian nature of the component peaks are shown in Figures 30-34. The central peak and one Brillouin peak from each spectrum were used in this analysis. Each peak was divided at the center so that the_left and right portions could be fit to appropriate Lorentzian functions.1 Numerical values for the intensities and graphical comparisons of the experimental and calculated intensities for the peaks of 0.80 mole fraction DMSO are presented in Tables 11 and 12 and Figures 35-38, respectively. As one can see from the computer plots of the 0.80 mole fraction DMSO data,2 each side of each peak fits a Lorentzian function, especially in the tail regions of the peaks, where one would expect the greatest discrepancies to occur for an incorrect choice of mathe- matical function. Experimental and calculated intensities for analo- gous peaks of the two neat liquids are given in Tables 13-16. To illustrate the difference in the appearance of experi- mental and calculated peaks when one attempts to fit a Lorentzian profile to a Gaussian mathematical function, one finds an experimental profile in Figure 39, and the respective Gaussian and Lorentzian computer fits to this profile in Figures 40-42. A beam of monochro- matic (5145 X), vertically polarized radiation was passed through the detection system to obtain the experimental spectrum. The natural profile of the laser is superimposed on the transmission function of the Fabry-Perot interferometer to produce an instrumental profile which 1This was done for computational simplicity. 2Similar computer plots were obtained for the central and Brillouin peaks of pure DMSO and pure pyridine, but were omitted to conserve space. 7 l 1 .oo Tam um 053.43 sum: .80 5.300% 58:05 no 5.3.84 .8 8:62 I >UZmDOw~I \2/ 5341/, 2.. ,1 1/ N 1 w _. 1 ‘—— AllSNalNl 118 .00 0.0m no 093 sowuomnm oaofi 00.0 How gamma cgoadflum no 030m HmHusoU 0|.Il >UZmDOm~I . Hm 0.33m 4"— AllSNBlNI 119 .00 0.0m um omzo cofiuomuw oHoE 00.0 How Ebupoomm sfldoaaflnm mo xmom sflsoHHflHm puma OIII >UZm30w~I . mm mash *— AllSNEIlNI 120 4> < R INTENSITY ——9 FREQUENCY—4 Figure 33. Central peak of Brillouin spectrum for neat DMSO at 59.0 °C. 121 .00 0.0m um omzn um0s Mom 5.30000 “~33...”me no 030% 5.50.3me pus .wm gunman OIIII >UZmDOm~E ‘—‘—— AllSNSiNI 122 Table 11 Experimental and Calculated Intensities for the Central Peak of 0.80 Mole Fraction DMSO at 59.0 0C 0.80 mole fraction DMSO; March 18, 1972, No. 6; Central peak, left hand side; fit to Lorentzian 0.80 mole fraction DMSO; March 18, 1972, No. 6; Central_peak, right hand side; fit to Lorentzian Point Intensity Intensity* Point Intensity Intensity Number (exptl.) (calcd.) Number (exptl.) (calcd.) 1 0.0 1.095 1 289.5 259.1 2 0.0 1.148 2 280.0 284.1 3 0.0 1.204 3 240.0 243.8 4 0.0 1.265 4 190.0 179.4 5 0.0 1.331 5 118.0 124.8 6 0.0 1.402 6 83.5 87.79 7 0.0 1.478 7 64.5 64.12 8 0.0 1.561 8 51.0 48.01 9 0.0 1.652 9 42.5 37.25 10 1.0 1.750 10 33.0 29.47 11 1.0 1.857 11 25.0 23.94 12 1.5 1.974 12 21.0 19.72 13 2.0 2.104 13 17.0 16.57 14 2.0 2.245 14 14.0 14.05 15 2.5 2.402 15 10.5 12.10 16 2.5 2.576 16 9.0 10.50 17 3.0 2.769 17 8.0 9.184 18 3.5 2.984 18 7.0 8.125 19 4.0 3.225 19 . 7.0 7.219 20, 4.0 3.498 20 6.0 6.471 *Intensities are given in arbitrary units. lllllllllllllllllll’lll.l. {.1 .\[ Table 11 (cont.) 0.80 mole fraction DMSO; March 18, 1972, No. 6; Central_peak, left hand side; fit to Lorentzian Point Intensity Intensity Number (exptl.) (calcd.) 21 4.5 3.806 22 4.5 4.155 23 5.0 4.555 24 5.0 35.014 25 5.0 5.553 26 6.0 6.168 27 6.5 6.909 28 7.0 7.767 29 8.0 8.821 30 9.0 10.07 31 10.0 11.64 32 12.0 13.60 33 15.5 16.02 34 19.5 19.22 35 24.5 23.33 36 31.5 29.01 37 39.5 36.69 38 54.0 47.94 39 68.0 64.17 40 98.5 89.43 41 137.0 128.4 42 184.0 183.5 123 0.80 mole fraction DMSO; March 18, 1972, No. 6; Central peak, right fit to Lorentzian hand side; Point Intensity Number (exptl.) 21 5.0 22 5.0 23 4.0 24 4.0 25 4.0 26 3.0 27 3.0 28 3.0 29 3.0 30 3.0 31 2.5 32 2.5 33 2.5 34 2.0 35 1.5 36 1.5 37 1.5 38 1.0 39 1.0 40 0.5 41 0.0 42 0.0 Intensity (calcd.) 5.819 5.273 4.789 4.377 4.008 3.691 3.404 3.148 2.926 2.721 2.542 2.375 2.228 2.091 1.969 1.855 1.753 1.657 1.568 1.488 1.413 1.345 124 Table 11 (cont.) 0.80 mole fraction DMSO; March 18, 0.80 mole fraction DMSO; March 18, 1972, No. 6; Centralypeak, left 1972, No. 6; Central peak, right hand side; fit to Lorentzian hand side; fit to Lorentzian Point Intensity Intensity Point Intensity Intensity Number (exptl.) (calcd.) Number (exptl.) (calcd.) 43 241.5 246.9 43 0.0 1.280 44 270.5 273.3 44 0.0 1.221 45 289.5 235.0 45 0.0 1.164 46 0.0 1.112 125 Table 12 Experimental and Calculated Intensities for the Left Brillouin Peak of 0.80 Mole Fraction DMSO at 59.0 0C 0.80 mole fraction DMSO; March 18, 0.80 mole fraction DMSO; March 18, 1972, No. 6; Left Brillouin peak, 1972, No. 6; Left Brillouin peak, left hand side; fit to Lorentzian right hand side; fit to Lorentzian Point Intensity Intensity Point Intensity Intensity Number (exptl.) (calcd.) Number (exptl.) (calcd.) 1 0.0 1.716 1 110.0 109.8 2 1.5 1.834 2 100.5 102.0 3 2.0 1.966 3 81.5 82.98 4 2.0 2.111 4 64.0 63.11 S 3.0 2.274 5 48.5 ‘ 47.20 6 4.0 2.455 6 35.0 35.51 7 3.5 2.659 7 26.0 27.31 8 4.0 2.890 8 21.0 21.45 9 3.5 3.151 9 20.0 17.19 10 3.0 3.449 10 16.0 14.03 11 3.5 3.791 11 11.0 11.64 12 4.0 4.185 12 10.0 9.794 13 5.0 4.643 13 8.0 8.344 14 5.5 5.179 14 7.0 7.188 15 6.0 5.811 15 5.5 6.251 16 7.0 6-574 16 5.0 5.476 17 8.0 7.482 17 5.0 4.841 18 9.0 8.586 18 4.5 4.309 19 9.5 9.944 19 4.0 3.859 20 10.0 11.64 20 3.0 3.476 126 Table 12 (cont.) 0.80 mole fraction DMSO; March 18, 0.80 mole fraction DMSO; Mardh 18, 1972, No. 6; Left Brillouin peak, 1972, No. 6; Left Brillouin peak, left hand side; fit to Lorentzian right hand side; fit to Lorentzian Point Intensity Intensity Point Intensity Intensity Number (exptl.) (calcd.) Number (exptl.) 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I I I I o I I uIIHonI 00u0asoa0o 0 0G00E 0 I m m I I M ”I560 H0uc0fifiu09n0 G0 0:008 x u I I m I m I I I I I I I I I Im---am--now---unuuuumuuunn-cuonouuun-nu-m-nu-nanuumu-nnm-uuumuuuu0unuuwuuu-:ununms---wunnnmuuuumu---I 130 .cmfluucouoa m on uflm uo o.mm um omza :owuomum mIoe om.o you xmom afiaoaflflum puma no mean cannuunmfim 0.02 ~IoI.oI ruacx omxc IZuu «u; uqox on com new; 2I2:44_m¢ Iszu Imnuuamuuo-mu---m----m-s--m-u--m----m----m----m----m----m----m---um----r.---m----r----w----m----n----I mIIIIIIIIx I CIIIIIO “III II In c a x o x I x c I muamn In x muamn QEMm may cw mum 03» any madmfi unwom ovuMHIUHmo m mamms o unflom Imucoefiuomxo :m mammfi x CI uuunmunuumI—II-umus-II..-fipc—uuw—umuwbouuud‘uuu—ImunnumI-I-I-Infi I I I m I I I I m I I I I m I I I I m I I I I m I I I I m I I I I m I I I I m I I I I m I I I I I m----“I---m----m-"-m----m'--'m-'--n'---m'---m----m"I'mI---fi-'-'fi--|-n--'-fi--'-flvll--.fi--"”---~9----~ .mm onsmflm Experimental and Calculated Intensities for the 131 Table 13 Central Peak of Neat DMSO at 59.0 °c Pure DMSO; December 15, 1971, No. 6; Central Beak, left hand side; Point Intensity Number (expt1.) 1 0.0 2 0.0 3 0.5 4 1.0 5 2.0 6 2.2 7 2.5 8 3.5 9 4.0 10 4.5 11 5.0 12 5.5 13 6.0 14 7.5 15 9.0 16 11.5 17 14.0 18 15.0 19 20.5 20 33.7 fit to Lorentzian Intensity (calcd.) 2.021 2.191 2.383 2.602 2.852 3.140 3.474 3.864 4.322 4.868 5.524 6.319 7.303 8.523 10.10 12.12 14.80 18.52 23.73 31.42 Pure DMSO; December 15, 1971, No. 6; Central peak, right fit to Lorentzian hand side; Point Number 1 2 10 11 12 13 14 15 16 17 18 19 20 Intensity (e52t1.) 485.0 427.0 261.0 148.7 84.5 44.0 34.0 22.0 17.0 12.3 10.0 7.5 6.0 5.5 5.5 5.5 5.5 Intensity (calcd.) 456.4 476.6 251.0 128.2 73.77 46.81 32.23 23.37 17.74 13.91 11.16 9.175 7.671 6.495 5.579 4.835 4.237 3.744 3.326 2.979 Table 13 (cont.) Pure DMSO; December 15, 1971, No. 6; Central peak, left hand side; Point Intensity Number (e§2t1.) 21 22 23 24 25 26 27 fit to Lorentzian 48.5 80.0 124.0 199.0 330.5 427.5 485.0 Intensity (calcd.) 43.57 63.52 99.23 169.2 298.8 430.6 349.2 132 Pure DMSO; December 15, 1971, No. 6; Central peak, right hand side; fit to Lorentzian Point Number 21 22 23 24 25 26 27 28 Intensity (e52t1.) 4.0 4.0 4.0 3.0 2.0 2.0 1.0 Intensity (Calcd.) 2.684 2.427 2.208 2.017 1.848 1.701 1.570 1.454 133 Table 14 Experimental and Calculated Intensities for the Left Brillouin Peak of Neat DMSO at 59.0 0C Pure DMSO; December 15, 1971, No. 6; Left Brillouin peak, left hand side; fit to Lorentzian Pure DMSO; December 15, 1971, No. 6; Left Brillouin peak, right hand side; fit to Lorentzian Point Intensity Intensity Point Intensity Intensity Number (exgt1.) (calcd.) Number (e§Et1.) _1§21gghl_ 1 0.0 6.343 1 429.0 .422.2 2 1.0 6.711 2 421.8 421.5 3 2.0 7.112 3 372.7 385.9 4 3.0 7.550 4 318.2 330.5 5 5.0 8.029 5 272.7 272.1 6 6.0 8.555 6 233.6 220.3 7 6.0 9.134 7 187.3 177.9 8 8.0 9.774 8 153.6 144.6 9 10.0 10-48 9 127.3 118.7 10 10.0 11-27 10 107.3 98.52 11 12.0 12.15 11 86.4 82.70 12 14.0 13.13 12 70.0 . 70.19 13 14.0 14.24 13 57.3 60.18 14 16.0 15.50 14 45.4 52.07 15 17.0 16.92 15 36.4 45.44 16 17.0 18-54 16 30.0 39.96 17 19.0 20.41 17 24.5 35.39 18 20.0 22.57 18 23.6 31.54 19 ' 22.7 25.08 19 22.7 28.29 20 28.2 28.03 20 20.0 25.48 '— 134 Table 14 (cont.) Pure DMSO; December 15, 1971, No. 6; Left Brillouin peak, left hand side; fit to Lorentzian Pure DMSO; December 15, 1971, No. 6; Left Brillouin peak, right hand side; fit to Lorentzian Point Intensity Intensity Point Intensity Intensity Number (egptl.) (calcd.) Number (exptl.) (calcd.) 21 32.7 31.51 21 17.3 23.08 22 34.5 35.66 22 17.3 20.99 23 39.1 40.65 23 13.6 19.17 24 43.6 46.73 24 10.0 17.57 25 50.0 54.20 25 10.0 16.17 26 57.3 63.51 26 9.1 14.92 27 68.2 75.24 27 8.2 13.81 28 89.1 90.23 28 6.4 12.82 29 112.7 109.6 29 6.4 11.93 30 141.8 134.9 30 5.4 11.13 31 172.7 168.2 31 5.4 10.41 32 216.3 211.4 32 4.5 92751 33 264.5 265.8 33 3.6 9.155 34 335.4 328.8 34 2.7 8.612 35 383.6 389.4 35 2.7 8.115 36 420.9 426.8 36 1.8 7.660 37 429.0 422.7 37 1.4 7.242 38 0.0 6.857 135 Table 15 Experimental and Calculated Intensities for the Central Peak of Neat Pyridine at 29.5 0C Pure pyridine, December 20, 1971, Pure pyridine, December 20, 1971, No. 4; Central peak, left hand No. 4; Centralgpeak, right hand side; fit to Lorentzian side; fit to Lorentzian Point Intensity Intensity Point Intensity Intensity Number (egpt1.) (calcd.) Number (expt1.) (calcd.) 1 0.0 0.8891 1 337.5 337.12 2 0.0 0.9783 2 273.5 282.60 3 0.0 1.0817 3 185.0 196.83 4 0.0 1.2023 4 132.0 131.30 5 0.0 1.3443 5 90.0 89.790 6 0.0 1.5130 6 65.5 63.920 7 0.0 1.7155 7 48.0 47.415 8 0.0 1.9623 8 38.0 36.283 9 0.0 2.2635 9 29.2 28.557 10 0.0 2-6429 10 24.5 23.047 11 1.0 3.1259 11 20.5 18.936 12 1.5 3-7485 12 17.0 15.819 13 2.2 4.5840 13 15.0 13.404 14 3.5 5.7318 14 12.2 11.510 15 5.5 7.3679 15 10.5 9.9765 16 8.5 9.7877 16 8.0 8.7279 17 12.5 13.648 17 6.0 7.7055 18 20.2 20.276 18 4.5 6.8455 19 37.5 32.855 19 2.8 6.1211 20 78.0 61.243 20 1.2 5.5097 . /_I 136 Table 15 (cont.) Pure pyridine, December 20, 1971, Pure pyridine, December 20, 1971, No. 4; Central peak, left hand No. 4; Central peak, right hand side; fit to Lorentzian side; fit to Lorentzian‘ Point Intensity Intensity Point Intensity Intensity Number (exptl.) (calcd.) Number (egpt1.) '(calcd.) 21 188.0 139.62 21 0.0 4.9813 22 288.0 275.36 22 0.0 4.5251 23 337.5 338.74 23 0.0 4.1315 24 0.0 3.7844 Pure pyridine, December 20, 1971, No. 4; Right Brillouin peak, left hand side; fit to Lorentzian Point Number 1 2 10 11 12 13 14 15 16 17 18 19 20 137 Table 16 Experimental and Calculated Intensities for the Right Brillouin Peak of Neat Pyridine at 29.5 °C Intensity (expt1.) 0.0 0.0 10.5 12.2 13.5 16.5 19.2 24.0 30.0 39.0 Intensity (calcd.) 3.1568 3.4179 3.7125 4.0466 4.4273 4.8638 5.3674 5.9541 6.6346 7.4434 8.4065 9.5546 Pure pyridine, December 20, 1971, No. 4; Right Brillouingpeak, right hand side; fit to Lorentzian 10.960 12.690 14.847 17.552 21.056 25.641 31.695 39.960 Point Intensity Intensity Number (expt1.) (calcd.) 1 152.0 149.49 2 134.0 143.91 3 109.5 117.74 4 89.5 88.130 5 70.0 64.505 6 54.0 47.817 7 42.0 36.149 8 32.0 27.999 9 23.0 22.188 10 17.5 17.973 11 13.2 14.796 12 10.5 12.370 13 7.5 10.496 14 5.5 8.9995 15 4.0 7.7964 16 3.8 6.8229 17 3.3 6.0130 18 2.5 5.3376 19 2.0 4.7729 20 1.5 4.2890 138 Table 16 (cont.) Pure pyridine, December 20, 1971, Pure Pyridine, December 20, 1971, No. 4; Right Brillouin peak, left No. 4; Right Brillouin peak, right hand side; fit to Lorentzian hand side; fit to Lorentzian Point Intensity Intensity Point Intensity Intensity Number (expt1.) (calcd.) Number (efiptl.) (calcd.) 21 59.0 51.315 21 1.5 3.8745 22 76.0 66.803 22 1.0 3.5169 23 95.0 87.724 23 0.5 3.2086 24 116.5 113.42 24 0.0 2.9369 25 132.0 138.30 25 0.0 2.6981 26 149.0 151.15 26 0.0 2.4886 27 152.0 143.09 27 0.0 2.3012 28 0.0 2.1340 139 12.3 .ha #395 .AIH 5900mm .oawmoum 3:35.33 ofidnummmn 0|III >UZmDGm~I . mm 8:63 *——- AllSNalNl 1140 .GMHmmnmo m on uwm .Hhma .ha umomam .QII ESHDUmmm .mawmoum Hmucmfisuumcfi wcwalhommq .o¢ memflm IhaIehI ImDODI OI taahuuam act Educ Gum<4 u“ mu “ IN: . u I“. u N N U I U M n- Ul H “- M I m----m"|'~ mI I II I II II I II I IO 00 O oo o co I II II I II Inm I x xx x Io co x x xx I I K K x K I I o x o x I I Ox 0 x I m x x m I o I I x u I I I I x x I m o o m I x I I I I o x I I x o I m m I I I x I I o . o I I I m x m I x I I I I o o I I I m x m I I I I I I I o o I m m I x K I I I I I I o o I m n I I I I I x I I o o I m x m I I I I I o o I I x I m x m I o o I I I I I x I I I Imu---m-uo-muuummuccumn-namauunm---um----m----m----mu---m----muuu-m----m-uu-m--u-mu-u-m----mnu--munuuI l4]. .GMINucoHoq m on paw .Hbma .54 umswam .QIH .maflwoum Hmucmanhumcfl mcflaiummma mo mcwm ocmnluqu .Hv musmflh IsoIohI ImDIID< I: {Ebowam to; ¥3 x muamw wEMm may CI mum 03“ man mnmmfi u M I _ I K I I unwom cwHMHIUImo m mauve o _ m 0 T I I n o ucflom Imucosflummxw an mamas x _ I I I I m : I I I n I I I I I I Im--u-m--.-mauu-m----m--oum----m----mc---m----m----m-u--m---an----m---um---an--u-m----:----¢----mu---I 142 .QMfiNucuHoq m 0p paw ~Hhma .hH umamam .QTH .uafimonm Hmucwssuumcfi ucwdlhmeH mo wwwm cqmgnunmam .po—.~. bmac2< 2. :zrpuumm 1cm xn x mufloc M m mEdm wnu cw mum o3u wgu mcmoE u w _ _ . . ” unwom ovuMHsoamo m mammfi o . c _ m u r “ unwom HmucmEHHmmxo cm mauve x u . . u _ n o . . _ . . 9 ~ . m----m----m----m----m----m----m----m----m----m----m----m----n----n----r----n----n----L----:----r---._ .mv unamfim III|||I|||||||III 143 we call a "laser-line instrumental profile." This instrumental pro- file is for pure coherent light, whereas the instrumental profile of a scattering spectrum is the frequency distribution of incoherent scattered light. As one can see from the graphical comparisons of the experi- mental profile to Gaussian and Lorentzian functions, the laser-line instrumental profile is Lorentzian in nature. The great difference between the experimental profile and the Gaussian distribution function helps to reinforce the conclusion that the experimental peaks for the 0.80 mole fraCtion DMSO mixture and for the two neat liquids are true Lorentzians. Having proven that the experimental light scattering peaks are Lorentzians functions, the next step in the endeavor to simplify the analysis of the Brillouin linewidth data is to prove that the central peak of a Brillouin scattering spectrum is equivalent to the instrumental profile of the incoherent scattered light. One knows from statements in the Brillouin scattering literature (21), that the true Rayleigh linewidth is often a factor of 102-103 smaller than the observed linewidth of the central peak. In order to test the validity of this assumption for our particular case of the light scattering from binary mixtures of DMSO and pyridine, we compared the linewidths of the central peaks of several different solutions at a given finesse. we were cognizant of the fact that if the central peak were a true representation of the instrumental profile of the scattered light, we would obtain similar linewidth values for the various solutions. The linewidths of the central peaks for three solutions are listed in Table 17, and individual graphs of PC versus finesse for each of the 144 Table 17 Linewidth of the Central Peak as a Function of Finesse Pure DMSO spectrum Mirror Date Number Finesse FQ(MHz) Spacing, d(cm) December 15, 1971 1-b 40.1 415 0.879 December 17, 1971 9 41.3 428 0.875 December 17, 1971 9 41.3 435 0.875 December 17, 1971 10 27.6 620 0.875 December 18, 1971 5 46.5 383 0.875 December 18, 1971 S 46.5 376 0.875 December 16, 1971 1 46.5 383 0.874 December 15, 1971 2 41.2 414 0.879 December 19, 1971 2 33.6 512 0.872 December 19, 1971 2 34.4 499 0.872 December 19, 1971 1 45.7 376 0.872 December 19, 1971 1 43.7 394 0.872 145 Table 17 (cont.) Pure Pyridine Spectrum Mirror Date Number Finesse FC(MHz) §pacing, d(cm) December 20, 1971 4 29.8 575 0.872 December 20, 1971 3 39.8 432 0.872 February 18, 1972 3 36.7 467 0.874 December 19, 1971 3 41.4 416 0.872 December 19, 1971 3 34.0 505 0.872 December 18, 1971 28 33.9 505 0.875 December 18, 1971 28 27.3 628 0.875 December 14, 1971 14 49.0 348 0.879 December 14, 1971 14 45.7 373 0.879 December 18, 1971 17 36.3 473 0.875 December 18, 1971 16 47.4 362 0.875 December 18, 1971 16 42.7 401 0.875 December 17, 1971 1 32.7 524 0.875 December 17, 1971 1 29.7 577 0.875 146 Table 17 (cont.) 0.40 Mole Fraction DMSO Spectrum Mirror Date Number Finesse [glMHz) Spacing, d(cm) January 3. 1972 7 27.3 622 0.882 January 3, 1972 6 37.8 450 0.882 January 10, 1972 19 38.2 449 0.875 January 10, 1972 20 ' 28.8 595 0.875 January 10, 1972 7 32.8 522 0.875 January 10, 1972 8 26.8 639 0.875 January 8, 1972 9 32.0 533 0.878 January 8, 1972 8 40.2 425 0.878 January 9, 1972 14 33.6 502 0.875 January 9, 1972 13 41.1 417 0.875 January 9, 1972 7 44.3 387 0.875 January 9, 1972 8 31.8 539 0.875 January 3, 1972 11 35.2 483 0.882 January 2, 1972 6 34.4 493 0.882 147 three solutions are given in Figures 43—45. A comparison is given in Table 18 of the full width at half height, PC, at a given finesse, for pure DMSO, pure pyridine, and 0.40 mole fraction DMSO. Considering that the standard deviation in a typical linewidth measurement is i 8 MHz, the values of PC are in excellent agreement. We may conclude that the central peaks of our Brillouin scattering spectra are equivalent to the instrumental pro- files of the scattered light at specific values of the finesse. In addition, we can couple this knowledge with the conclusion readhed earlier that the peaks in a Brillouin scattering spectrum are Lorent- zian, to arrive at the following equation for the true Brillouin linewidth, ZFB: ZFB = 21‘B/obs ' I‘C . (111) In equation (111), er/obs is the observed full width at half height of the Brillouin peak while PC is the full width at half height of the central peak. The results of the calculations of the Brillouin linewidths for each solution at the various temperatures are tabulated in Table 19. Typical plots of Brillouin linewidth versus temperature at a specific composition are shown in Figure 46. The Brillouin linewidth is found to decrease linearly with temperature for each composition. These results represent the first deliberate effort at correlating the variation in Brillouin linewidth with temperature for either pure or multicomponent fluids. The only other reported study concerning the change in Brillouin linewidth with temperature (64) was published in terms of the decrease in the frequency-corrected absorption co- 2 efficient, a/v , with temperature. No Brillouin linewidths were 148 .0915 use: no xoom Hmnucoo How unmocfiu msmhm> ficfisgq .me Gunman 3.ch oun oov ovv oov 000 q a q _ a ON 149 .ocwcflumm anon mo xmom Hmuucoo How ommoswm msmuo> fiwflzocflq 3: mafia U :15: K on» our. ORV Ont o—v on" Onn a ._ a q q _ ON .omzn sawuomnm mace ov.o mo xmom Hmuucoo How ommocwm n5mHm> fipazoswq .mv mun—53E u A :3: ouo our. own com 001 . ONV can 150 Comparison of the Linewidths of the Central Peaks of Neat DMSO, Neat Pyridine and 0.40 Mole Fraction DMSO at a Given Finesse Sample neat 0.40 neat neat 0.40 neat neat 0.40 neat neat _ 0.40 neat neat 0.40 neat neat 0.40 neat neat 0.40 neat neat 0.40 neat neat 0.40 neat neat 0.40 pyridine mole fraction DMSO pyridine mole fraction DMSO pyridine mole fraction DMSO pyridine mole fraction DMSO pyridine mole fraction DMSO pyridine mole fraction DMSO pyridine mole fraction DMSO pyridine mole fraction DMSO pyridine mole fraction DMSO pyridine mole fraction DMSO DMSO DMSO DMSO DMSO DMSO DMSO DMSO DMSO DMSO 151 Table 18 Finesse 45 45 45 43 43 43 41 41 41 39 39 39 36 36 36 35 35 35 34 34 34 33 33 33 32 32 32 30 30 Full Width at Half Height of Central Peak (6c) 382 i 8 MHZ 380 387 400 400 410 419 419 430 436 439 451 477 468 482 491 484 493 506 502 503 521 520 514 536 537 525 565 572 152 Table 19 Brillouin Linewidth as a Function of Temperature for Mixtures of DMSO and Pyridine neat pyridine 0.05 mole fraction DMSO 21.6 29.5 34.5 39.4 44.5 49.3 54.2 0.10 mole fraction DMSO 459,469,459, 456 469,442,420, 439 439,423 426,424 411,407 379,379,403, 410 379,388,377 461 442 431 425 409 393 381 temp.(°C) 2FB(MHz) ZPBavg(MHz) temp.(°C) 2FB(MHz) ZPBavg(MHz) 21.6 454,498,448 467 21.6 472,459 466 29.5 436,466,449 450 29.5 445,450 448 34.5 441,437,439 439 34.5 432,435,435 434 39.4 430,430,431 430 39.4 421,422 422 44.5 415,404 410 44.5 396,407,422 408 49.3 388,411 400 49.3 397,397 397 54.2 406,374 390 54.2 397,390,381 389 59.0 369,376,404 383 0.15 mole fraction DMSO 21.6 29.5 34.5 39.4 44.5 49.3 54.2 471,449,442, 468 401,450,475, 439 426,424,441 418,418 401,407 406,384,390 382,382 458 441 430 418 404 393 382 153 Table 19 (cont.) 0.20 mole fraction DMSO 0.30 mole fraction DMSO temp.(°C) 2FB(MHz) ZFBavg(MHz) temp.(°C) 2FB(MHz) 2FBavg(MHz) 21.6 434,470,470 458 21.6 446,446 446 29.5 441,436,436 438 29.5 432,435,417 428 34.5 431,431,426 429 34.5 418,418,418, 418 418 39.4 421,416 418 39.4 419,400,396 405 44.5 396,412 404 44.5 402,386 394 49.3 387,387 387 49.3 356,409 382 54.2 394,365,383, 380 54.2 398,349 373 377 59.0 364,358 361 0.40 mole fraction DMSO 0.50 mole fraction DMSO 21.6 429,460,436 442 21.6 436,440,443, 438 . 433 29.5 429,403,435, 423 29.5 426,423,411 420 425 34.5 411,418 414 34.5 396,417 406 39.4 408,395 402 39.4 413,365,396, 394 400 44.5 406,367,384 386 44.5 399,383,371 384 49.3 398,355 376 49.3 394,348 371 54.2 344,344,388, 368 54.2 359,356 358 398 59.0 334,376 355 59.0 363,332 348 154 Table 19 (cont.) 0.60 mole fraction DMSO 0.70 mole fraction DMSO temp.(°C) 2FB(MH2) ZFBavg(MHz) temp.(°C) 2PB(MHz) ZPBavg(MHz) 29.5 393,414,422, 411 21.6 435,418,423, 425 405,416,416 423 34.5 412,398 405 29.5 406,406 406 39.4 372,372,412, 390 34.5 388,410,410, 396 402 388,382 44.5 374,370,388, 378 39.4 388,372,379, 384 385,375 391,387,387 49.3 385,385,349, 365 44.5 370,374,368 369 343,361 379,355 54.2 361,361,356 359 49.3 371,370,354, 358 338 59.0 355,352,318, 342 54.2 332,355,351, 347 355,353,320 349 59.0 335,335 335 0.80 mole fraction DMSO 0.95 mole fraction DMSO 29.5 430,388,404, 399 21.6 408,425,401 411 396,355,422 400,421 34.5 389,389,392, 392 29.5 394,394,401, 393 396 384 39.4 392,422,363, 378 34.5 380,370,402 383 337 379 44.5 349,367,370, 368 39.4 373,373 373 367,385 49.3 366,363,363, 357 44.5 366,318,395 357 335 ., 349 54.2 318,348,348, 342 49.3 338,352,352 347 355 59.0 363,325,309 328 54.2 348,364,400, 336 231 59.0 318,329 324 155 Table 19 (cont.) 9951523192 temp.(°C) 2FB(MHz) ZFBavg(MHz) 21.6 407,407,390,4l7 405 29.5 384,395 390 34.5 394,350,382 375 39.4 353,379,373 368 44.5 363,346 354 49.3 334,343 338 54.2 337,323 330 59.0 318,318 318 156 coauuaum 366 no. . o coauumuw wHOE 05.0 cOfiDUMHw oHOE Om.o cowuomnw wHOE ov.o .maflpwumm cam omzn Mo mmuduxwfi How muflumnomawu mdmhm> nupflzmcwa nfldoaawhm co OmSD umwz G noduomuw maoa :oHuomuw oHOE cowuumuw oHoE cowuomuw oHoE GOHUUMHM 0H0:— mcwcfiu>m an on 3. av _ V 0 V mm3h nupflzmafla cwfloaaflhm .hv oHfimflm Omio 20:91: mAOE 8.. oa.o 8.6 Ed 668 one 3.8 38 on... 2.6 38 4 _ "1 2 2 _ J 2 T a can 1 can 0 1 1 can 17. 7v 4.. 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I I s I O > b p r-.VV O a o o o 664 I .. onw 160 Table 20 Absorption Coefficient as a Function of Temperature for Mixtures of DMSO and Pyridine freq-correc. absorption Brill. velocity Brillouin absorp. coefficient, temp. shift,v of sound, half-width coeff., 01/9le018 (0c) (GHz) Vs(m/sec) FB(MHz) d(cm’l) (seCZ/cm) pure C5H5N 21.6 6.39 1530 234 1530 37.5 29.5 6.27 1510 224 1480 37.7 34.5 6.19 1490 219 1470 38.4 39.4 6.11 1480 213 1440 38.6 44.5 6.03 1460 207 1420 39.0 49.3 5.95 1440 202 1400 39.6 54.2 5.87 1430‘ 196 1370 39.8 59.0 5.79 1410 190 1350 140.2 0.05 mole fraction DMSO 21.6 6.29 1510 233 1540 39.0 29.5 6.17 1490 224 1500 39.5 34.5 6.08 1470 217 1480 39.9 39.4 6.01 1450 211 1460 40.3 44.5 5.92 1430 205 1430 40.8 49.3 5.85 1420 199 1400 40.9 54.2 5.77 1400 193 1380 41.4 59.0 5.69 1380 161 Table 20 (cont.) freq-correc. absorption Brill. velocity Brillouin absorp. coegficignt, temp. shift,v of sound, half-Width coeff., a/v x10 (°C) (GHz) VS(m/sec) FB(MHz) d(cmfl) (secz/cm) 0.10 mole fraction DMSO 21.6 6.27 1510 232 1540 39.1 29.5 6.12 1480 222 1500 40.0 34.5 6.02 1460 216 1480 40.8 39.4 5.92 1430 210 1470 41.9 44.5 5.82 1410 204 1450 42.8 49.3 5.73 1390 198 1420 43.4 54.2 5.63 1370 192 1400 44.2 59.0 5.54 1350 0.15 mole fraction DMSO 21.6 6.28 1510 230 1520 38.6 29.5 6.14 1480 220 1490 39.4 34.5 6.05 1460 214 1470 40.0 39.4 5.97 1450 208 1430 40.2 44.5 5.88 1430 202 1410 40.9 49.3 5.80 1410 197 1400 41.6 54.2 5.71 1390 191 1370 42.2 59.0 5.63 1380 162 Table 20 (cont.) freq-correc. absorption Brill. velocity Brillouin absorp. coefficient: temp. shift,v of sound, half-width coeff., d/vleole (°C) (GHz) Vs(m/sec) FB(MHz) d(cmfl) (secz/cm) 0.20 mole fraction DMSO 21.6 6.17 1490 230 1540 40.4 29.5 6.05 1460 220 1510 41.2 34.5 5.97 1450 214 1480 41.5 39.4 5.89 1430 208 1450 41.8 44.5 5.81 1410 202 1430 42.4 49.3 5.74 1400 196 1400 42.5 54.2 5.66 1380 190 1380 43.1 59.0 5.59 1370 0.30 mole fraction DMSO 21.6 6.22 1500 223 1490 38.5 29.5 6.08 1470 214 1460 39.5 34.5 5.98 1450 208 1430 40.0 39.4 5.89 1430 203 1420 40.9 44.5 5.79 1410 197 1400 41.8 49.3 5.70 1390 192 1380 42.5 54.2 5.61 1370 186 1360 43.2 59.0 5.52 1350 180 1330 43.6 163 Table 20 (cont.) freq-correc. absorption Brill. velocity Brillouin absorp. coefficient, temp. shift,v of sound, half-width coeff. , 111/1122:1018 (°C) (GHz) V3(m/sec) FB(MH2) d(cm‘l) (secz/cm) 0.40 mole fraction DMSO 21.6 6.11 1480 221 1490 39.9 29.5 5.99 1450 212 1460 40.7 34.5 5.90 1430 206 1440 41.4 39.4 5.82 1420 200 1410 41.6 44.5 5.73 1400 194 1390 42.3 49.3 5.65 1380 189 1370 42.9 54.2 5.57 1360 183 1350 43.5 59.0 5.49 1340 178 1330 44.1 0.50 mole fraction DMSO 21.6 6.02 1460 219 1500 41.4 29.5 5.90 1430 210 1470 42.2 34.5 5.82 1420 204 1440 42.5 39.4 5.75 1400 198 1410 42.6 44.5 5.67 1390 192 1380 42.9 49.3 5.59 1370 186 1360 43.5 54.2 5.51 1350 180 1330 43.8 59.0 5.44 1330 174 1310 44.3 164 Table 20 (cont.) freq-correc. absorption Brill. velocity Brillouin absorp. coefficient, temp. shift,v of sound, half-width coeff., u/v2x1018 (°C) (GHz) Vs(m/sec) FB(MHz) d(cm'l) (sec2/cm) 0.60 mole fraction DMSO 21.6 6.00 1460 216 1480 41.1 29.5 5.91 1440 206 1430 40.9 34.5 5.84 1430 201 1410 41.3 39.4 5.78 1410 195 1380 41.3 44.5 5.72 1400 189 1350 41.3 49.3 5.66 1390 184 1320 41.2 54.2 5.60 1380 178 1290 41.1 59.0 5.54 1360 172 1260 41.1 0.70 mole fraction DMSO 21.6 6.05 1470 213 1450 39.6 29.5 5.92 1450 204 1410 40.2 34.5 5.83 1430 197 1380 40.6 39.4 5.75 1410 192 1360 41.1 44.5 5.66 1390 185 1330 41.5 49.3 5.57 1370 180 1310 42.2 54.2 5.49 1350 174 1290 42.8 59.0 5.41 1330 168 1260 43.0 165 Table 20 (cont.) freq-correc. absorption Brill. velocity Brillouin absorp. coefficient, temp. shift,v of sound half-width coeff., a/v2x1018 (°C) (GHz) Vs(m/sec) FB(MHz) d(cmfl) (secz/cm) 0.80 mole fraction DMSO 21.6 6.04 1470 210 1430 39.2 29.5 5.95 1460 200 1370 38.7 34.5 5.88 1440 195 1350 39.0 39.4 5.82 1430 189 1320 39.0 44.5 5.75 1410 184 1300 39.3 49.3 5.69 1400 178 1270 39.2 54.2 5.63 1390 172 ‘ 1240 39.1 59.0 5.57 1380 166 1200 38.7 0.95 mole fraction DMSO 21.6 6.13 1500 206 1370 36.5 29.5 6.00 1470 197 1340 37.2 34.5 5.92 1460 191 1310 37.4 39.4 5.83 1440 185 1280 37.7 44.5 5.75 1420 179 1260 38.1 49.3 5.67 1400 174 1240 38.6 54.2 5.59 1380 168 1220 39.0 59.0 5.51 1360 , 162 1190 39.2 Table 20 (cont.) temp. (°C) 21.6 29.5 34.5 39.4 44.5 49.3 54.2 59.0 166 freq-correc. absorption Brill. velocity Brillouin absorp. coefficient, shift,v of sound half-width coeff., a/v2x1018 (GHz) VS(m/sec) PB(MHz) d(cmfl) (secz/cm) pure DMSO 6.11 1500 204 1360 36.4 5.99 1470 194 1320 36.8 5.91 1460 188 1290 36.9 5.84 1440 182 1260 36.9 5.75 1420 176 1240 37.5 5.68 1400 170 1210 37.5 5.60 1390 165 1190 37.9 5.53 1370 159 1160 37.9 167 energy to translational energy in a liquid. More specifically, it is believed that a greater amount of absorption will occur when collisions between the molecules in a liquid are inefficient in producing a transfer of energy. A high efficiency of transfer of internal vibra- tional energy to translational energy will result in a sustained pro- pagation of the sound wave, or decreased absorption. The effectiveness of the molecular collisions, of course, is directly related to the degree of interaction between the molecules. As the temperature of a liquid is changed the degree of order or structure of the liquid changes. Variation in the sonic absorption coefficient with temperature can be expected to reflect these altera- tions in structure, via alterations in the interactions between molecules. Values for the temperature derivative of the sonic absorption coefficient were calculated for the thirteen DMSO-pyridine mixtures and are given in Table 21. Table 21 Temperature Derivative of the Sonic Absorption Coefficient mole fraction (ad/3T) DMSO (cm-l °C‘1) 0.0000 -4.81 0.0495 -4.91 0.0993 ' -4.29 0.1506 -4.60 0.1983 -4.91 0.2997 -4.28 0.3995 -4.28 0.5005 -5.08 0.6002 -5.88 0.7001 -5.08 0.8004 -6.15 0.9453 -4.81 1.0000 -5.35 168 The negative values of aa/BT render the mixtures "associated fluids" according to the classification scheme of Herzfeld and Litovitz (see p. 38). The sonic absorption coefficient as a function of temperature for a typical DMSO-pyridine mixture is illustrated in Figure 49. The absorption coefficient decreases linearly with an increase in tempera- ture for all the solutions studied. The principal contribution to the error in these values is the error in measurement of the Brillouin linewidth. A comparison of the measured sonic absorption coefficient and the calculated "classical" absorption coefficient (see p. 34) for DMSO at 25 0C reveals an unexpected result. The classical absorption coefficient can be evaluated from the equation 2 2 , 811’ V3 3). Class 3‘7??— 4cp . where the first term in the brackets represents the shear viscosity of the medium and the second term involves thermal conduction parameters. For all liquids except liquid metals the heat conduction term is negligible (51,86), so that the classical absorption coefficient can be approximated by 2 ~ 8712 VB 7] a .. class —3V;31;—— (113) For dimethyl sulfoxide at 25 °C , one Obtains a value of “class = 5320 cm'l. The experimentally determined value of the sonic absorption coefficient is a = 1350 i 40 cm'l. The ratio “meas/ meas “class is 0.254, which is much lower than would.be expected from ultrasonic measurements for analogous associated fluids in lower frequency regions (51). This abnormally low value of “meas/“class 169 .093 .5305 can! on . o Mom 050% Bung uaodudumooo sojmhonnm mm AU; mafi~<¢wa2mp 6n n! 01 an on 31 .,_ q "a nu ON I 1 l , oun— oom— cow— 0“— 00V. can— con— . 2. «names (“may 170 suggests that the relaxation mechanisms responsible for the absorption of ultrasonic waves in a liquid have response times that are slower than the period of the high frequency (~6 GI-Iz) waves detected by Brillouin scattering. Variation in the sonic absorption coefficient with composi- tion for two different temperatures is plotted in Figures 50 and 51. Within experimental error, the absorption coefficient is found to be a linear function of composition. As would be expected, the absorp- tion coefficient reaches a minimum.value for pure DMSO at each tempera? ture. Highly associated neat DMSO is expected to possess a maximal amount of intermolecular interaction, thereby causing a minimal amount of absorption to take place. Because the absorption coefficient usually varies with fre- quency, ultrasonics researchers prefer to use the term e/vz, the frequency-corrected absorption coefficient, when reporting acoustical data. Typical plots of c/v2 versus temperature are exhibited in Figures 52 and 53. The change in the frequency-corrected absorption coefficient.with temperature is found to be a monotonically increasing function for each of the solutions studied. The linearity of these plots again suggests that eadh of the DMSO-pyridine mixtures is be- having like a pure fluid with regard to temperature variations in the acoustical properties. As in the case of the sonic absorption coefficient, the frequency—corrected absorption coefficient that is determined experi- mentally for pure DMSO at 25 °C is found to be ~1/4 the value calculat- ed using the "classical" absorption coefficient, “class! of equation (113). Instead of there being an ”excess" absorption present in pure 171 .00 wt: um soflamomfioo 9..an ucofioammuoo soflmuonnm .om 080?.— Om<< zit/D 177 .00 0.00 was mém .m.0~ .013 How coflanomfioo 090.»? ugfiowumgu 00.308096. psoowwglhosgvam .mm 080mm Om< b 01 C O I I o.o~ «.1 I I O C 38 v I a/ I m I l o —V l\ C - I 00 0 an n .H. I I 00 m fin I H. . I 1 o.NV 00 m @N I .H. . 178 .00 szm 0:0 900 3.3. now :oHuauofluoo 090.85 acounowmm0oo nodumuomam ©300HH818§030 .mm 090.."0 Omin— ZO_._.U<~_“_ m._O<< 3.. use one on... 36 one 36 one one o..o oo.o . - I 1 d _ 9 H I u p I 0.: 0.3 D I a t X 0.: nIu. u .....I a 3 z e: W 1 w 00 «#6 u a I I 1 6.: oo 29. 1 a. b I . UOmQfiflB. I 106‘ 179 V3 (tn/sec) .N?\e ug0m0um0u 008an 20> 3800800..“ 0000.900 .uo 0.18 no «.5388 use 093 no measures now 058 mo 838.8% 0.8 5832, .3 6930 Om<<.a/v 180 .0550 no 00008308 #:000ng 008.30 .huwawflmm0wmaoo 0300.30 0.000.000.» 00.00.00 .00 0. HN 00 00.3.."qu 9.0 0043 «0 00883.... How hueawfi000uguoo 0300030 000 05.00 .00 09.398000 .mm 050.3 0920 ZOCU<~Z m._O<< 0 0.. no.0 0 4.0 as... 04.0 04.0 04.0 on... on... 0 ..0 0 0.0 00.4 . o . . 4 _ .1 n _ 0.3 1/I °—.I .I J $.80 p 11 / 0 At a n I 1m. x 2/ 3.4 I 1 0.: O m u c ) n I I u 0 38 ‘l 006 I. . l 0.00 / X I I 3 I B. I m\ 04.4 1 I 4 0.04 I I I I on 4 1 -. I 1 o .4 ooéfi 1 0.: 181 Examining Figure 57, one sees that the maximum of the q/vz curve correspond to the minimum.of the velocity of sound curve. This observation fits quite well with the concept of increased absorption being the result of inefficient transfer of internal vibrational energy to translational energy in the liquid, and decreased absorption being the result of efficient transfer of energy of sustained propagation of the sound wave between highly interacting molecular species. Variations in c/v2 and 88 with composition at 21.6 °C follow the same trends, with the maximum in the c/v2 curve falling at the same x1 value as the maximum for the as curve. These results indicate that changes in the adiabatic compressibility with intermolecular forces in an associated binary fluid are reflected accurately by changes in the frequency-corrected absorption coefficient. The structural changes which are presumed to account for the variations in both properties with composition have been elaborated in Section IV D. G. variation of the Landau-Placzek Ratio with Temperature and composition Although it had been hoped that a direct comparison could be made-between the theoretical relationship for the Landau-Placzek ratio (equation (70)) and the values Obtained experimentally for mixtures of dimethyl sulfoxide and pyridine, lack of information on quantities in the theoretical expression rendered this impossible. Consequently, only a semi-quantitative explanation for the change in Landau-Placzek ratio with temperature and composition is presented here. Table 22 contains values for the Landau-Placzek ratio of DMSO- pyridine solutions at seven temperatures. The effect of temperature 182 Table 22 Landau-Placzek Ratio as a Function of Temperature for Mixtures of DMSO and Pyridine T = 29.5 °c T = 34.5 °c mole mole fraction fraction DMSO RLP DMSO RLP 0.00 0.56 0.00 0.60 0.10 0.59 0.10 0.62 0.15 0.65 0.15 0.67 0.30 0.57 0.30 0.60 0.40 0.61 0.40 0.66 0.60 0.64 0.60 0.63 0.80 0.72 0.80 0.71 0.95 0.70 0.95 0.69 1.00 0.45 1.00 0.46 r = 39.4 °c T = 44.5 °c 0.00 0.63 0.00 0.67 0.10 0.65 0.10 0.67 0.15 0.69 0.15 0.71 0.30 0.62 0.30 0.65 0.40 0.71 0.40 0.77 0.60 0.63 0.60 0.63 0.80 0.70 0.80 0.69 0.95 0.67 0.95 0.66 1.00 0.46 1.00 0.47 Table 22 (cont.) T = 49.3 °c mole fraction 0050 RM: 0.00 0.70 0.10 0.71 0.15 0.73 0.30 0.67 0.40 0.82 0.60 0.62 0.80 0.68 0.95 0.65 1.00 0.46 'r .. 59.0 °c 0.00 0.77 0.10 0.76 0.15 0.77 0.30 0.72 0.40 0.93 0.60 0.62 0.80 0.67 0.95 0.63 1.00 0.43 183 -r = 54.2 °c mole fraction DMSO RLP 0.00 0.74 0.10 0.73 0.15 0.75 0.30 0.70 0.40 0.87 0.60 0.62 0.80 0.67 0.95 0.64 1.00 0.44 184 on the Landau-Placzek ratio for the two neat liquids and one of their mixtures is illustrated in Figures 59-61. The most surprising of these graphs is the one for pure DMSO. There is a distinct break in the curve at 45 oC, indicating a change in the degree of association in the liquid at this temperature. An analogous discontinuity observed by Schlafer and Schaffernicht (6) for refractive index data of pure DMSO between 40 and 50 °C was interpreted to signify a breakup in association of the DMSO molecules. 6 It should be noted in Figures 59-61 that although the Landau- Placzek ratio increases with temperature for the two pure fluids (up to 45 oC), IC/ZIB decreases with temperature for the 0.80 mole fraction DMSO mixture. The compositional dependence of the Landau-Placzek ratio for temperatures in the range 29-59 0C is shown in Figure 62. The Landau- Placzek ratio is seen to decrease between 0.00 and 0.60 mole fraction DMSO, increase slightly between 0.60 and ~ 0.90 and finally plummet between 0.90 and 1.00 mole fraction DMSO to a minimum value for pure DMSO at all temperatures. In an attempt to explain the significance of the compositional dependence of the Landau-Placzek ratio, we turn to equation (70), which was derived for the specific case of a binary mixture at constant temperature. According to equation (70), the Landau-Placzek ratio is dependent upon several physical and thermodynamic properties, including 9: BS and BT. It is also dependent upon the infinite-and zero-frequency sound velocities, C0 and co, respectively, for which we have no data. At 21.6 °C we have values for the density and adiabatic compressibility, so that a semi-quantitative comparison can be made 185 . 05.0% 000: How 000.90.39.00» 0:0.H0b AU; waah<¢mm< .Nm 0005.3 0min ZO_._.U<~I maOE 00.0 and 00.0 and 0‘6 and 0.4.6 O—.o cod [ V.° — fl _ 1 A _ _ '— o la and Ce .006 cc A...44 n a D 4 2.. I 0o 4.00 ... 0. o 4 00 m.vm u .H. 4 e 06 004 u a o 1 co— 189 between theory and experiment for the change in IC/2IB with BS/pz at this temperature. From equation (70) we would expect a linear relation- ship between the Landau-Placzek ratio and 85/02 at constant temperature; however, we should remember that equation (70) was derived under the tacit assumption that the binary mixture is ideal in nature. We know from the results of sections III A-F, that DMSO-pyridine mixtures exhibit distinctly non-ideal behavior, so that the variation in Landau- Placzek ratio with adiabatic compressibility and density can be expect- ed to be linear only so long as the degree of association in the liquid affects IC/ZIB and ss/pz to the same extent. Figure 63 demonstrates the variation in the Landau-Placzek ratio with 83/02 for DMSO—pyridine mixtures at 21.6 0C. The values of IC/ZIB were obtained from the respective Landau-Placzek ratio- temperature curves for each of the mixtures. The variation of IC/2IB with BS/pz in the range from pure pyridine to 0.80 mole fraction DMSO is seen to be a linear function, which is the type of relationship predicted from equation (70). The fact that the Landau-Placzek ratio increases with decreasing values of 85/02 indicates that the change in chemical potential with concentration, (3U/3C)P'T, is negative as one progresses from.pure pyridine to 0.80 mole fraction DMSO (see equation (70)). At ~0.90 mole fraction DMSO, the change in Landau-Placzek ratio with 85/92 reverses sign and the Landau-Placzek ratio falls precipitously to a minimum for pure DMSO. A qualitative explanation of this abrupt drop in the Landau—Placzek ratio involves the assump- tion of a decrease in the damping forces in the liquid between 0.90 and 1.00 mole fraction DMSO due to an increase in association of the 190 .0003 coflo0um 0.38 c... 50.30.009.50 00000.00.“ 000.40% 0000 05. 000000 0HOQ§z .00 023 #0 003.303 0.0 09.3 no 000503.0— HOu ~0\mm 5M3 owu0H x0000amls0pc0n 05 mo 00.300000, .mm gunman O a . 0.x.& 0 m .— NUO NED cod n06. on.w haw oo.w nu.” ond mm.” _ J 4 1 . . . owd 00.... 1 and 0.. I cod 1 00.0 80.8 4 00.0 191 DMSO molecules. A decrease in the damping forces in a liquid causes an increase in the magnitude of the pressure fluctuations whidh are responsible for the Brillouin peaks, thereby decreasing the Landau- Placzek ratio. The decrease in damping forces is also reflected by a concomitant decrease in the frequency-corrected absorption co- efficient. A glance at Figure S4 reveals that the frequency-corrected absorption coefficient decreases significantly between 0.80 and 1.00 mole fraction DMSO at 21.6 °C, lending credence to the qualitative explanation given above for the decrease in Landau-Placzek ratio. The fact that the change in Landau-Placzek ratio with 85/02 takes on a positive slope between ~0.90 and 1.00 mole fraction DMSO indicates that the change in chemical potential with concentration reverses sign and becomes positive in this region. The changes in sign are undoubted- ly due to the influence of increased association of the DMSO molecules in the compositional region from 0.80 to 1.00 mole fraction DMSO. Variations in the Landau-Placzek ratio with 85/02 for tempera. tures above 21.6 0C can be expected to follow the same general trends as those observed for IC/ZIB at 21.6 0C, although these variations cannot be monitored because of lack of density data. V- CONCLUSIONS Brillouin scattering parameters and acoustical properties have been used to monitor structural changes occurring in dimethyl sulfoxide- pyridine mixtures in the temperature range from 20 to 60 oC. Linear variation of the refractive index, Brillouin shift, velocity of sound and sonic absorption coefficient with temperature indicated that each of the mixtures was behaving like a pure fluid with respect to these physicochemical properties. It was also noted that the refractive index and Brillouin linewidth were linear functions of composition in this temperature range. The Brillouin linewidth data represent the first reported measurements of the variation of Brillouin linewidth with temperature or composition for pure or multi- component liquids. Fluctuations in the velocity of sound and frequency-corrected absorption coefficient with composition denoted changes in the degree of association in the liquid. These changes were observed for each of the eight temperatures in the range from 20 to 60 °C. Neat DMSO and neat pyridine were both found to be rather highly structured liquids, although it appeared that neat DMSO was the more ordered of the two at any given temperature. The binary mixtures were found to be less structured than the two pure solvents, with maximum disorganization of the liquid structure appearing to occur at approximately the equimolar composition. This behavior indicated, of course, that the species responsible for the highly associated nature of the mixtures of DMSO and pyridine were not complexes between DMSO and pyridine, but were homomolecular aggregates of DMSO and/or pyridine. Infrared spectrosc0pic data for the S-O stretching frequency for 192 193 mixtures of DMSO and pyridine at 22 oC support the concept of self- association of DMSO. A linear relationship was observed between the Landau-Placzek ratio and the temperature for each of the DMSO-pyridine mixtures. There was a distinct break, however, in the Landau-Placzek ratio- temperature curve for pure DMSO at 45 oC, indicating a rearrangement of the liquid structure at this temperature. The overall changes in the Landau-Placzek ratio with composi- tion gave supporting evidence to the interpretation of the velocity of sound and absorption coefficient data. In particular, a vast increase in association between 0.80 and 1.00 mole fraction was indicated by a precipitous drop in the Landau-Placzek ratio in this region. The linear variation of the Landau-Placzek ratio with Bs/v2 between 0.00 and 0.80 mole fraction DMSO at 21.6 0C lent validity to the theoretical expression derived for ideal binary mixtures at con- stant temperature (equation (70)). The deviation from linearity observed for the mixtures between 0.80 to 1.00 mole fraction DMSO at 21.6 °C was interpreted to be a consequence of the substantial increase in association presumed to be taking place in this compositional region. 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Taylor and H. wynberg,eds., Wiley Interscience, New York, 1965). Cited in Ref. 1. R. Thomas, C. B. Shoemaker and K. Eriks, Acta Cryst. JJ, 12 (1966). M. Greenspan, in American Institute 35 Physics Handbook, 3rd ed. (0. B. Gray, coordinating ed., McGraw-Hill Book Company, New York, 1972). APPENDICES Appendix A Temperature Calibration Data for the Thermal Characterization of the Control Cell Employed in the Brillouin Scattering Measurements TC - temperature of thermometer C in the copper block TA - actual temperature of the glycerin sample used for the calibration procedure Tc and TA are both equilibrium temperatures Table 23 Temperature Calibration Data TC(°C) “(°C) 69.9 68.8 64.7 63.9 59.8 58.9 54.6 53.9 49.6 49.3 44.7 44.5 39.6 39.3 37.8 37.4 36.8 36.5 35.7 35.4 34.8 34.5 33.7 33.5 32.8 32.6 31.7 31.5 30.6 30.5 199 200 Table 23 (cont.) TC(°C) TA(°C) 29.6 29.5 28.6 28.6 27.8 27.7 26.8 26.7 24.9 24.9 21.6 21.6 Appendix B 0 Calculation of the Refractive Index at 5145A Following the equations given in the Bausch and Lomb Abbe 3-L o Refractometer instrument manual, the refractive index for A - 5145A was calculated for each of the DMSO-pyridine solutions at five differ- ent temperatures. The equations used in the calculations are I nT 0:15. 4* 8 5145A (514.5nm)z A. = nDT - 2.8796 x 10-63. ' 6 B — 0.52364 x 10 (nF-nc) where A, B, and C are empirical constants for the system under investi- gation. The dispersion, (nF-nc) , is the difference in the refractive index of the liquid at two known wavelengths, 656 and 486 nm, respectively. Because of the construction of the compensators in the Bausch and Lamb Abbe 3-L Refractometer, the dispersion can be accurately determined from values of A, B, and C, which are given in the Bausch and Leah Dispersion Table. A, B, and C depend on the refractive index of the sample and on the drum readings of the re- fractometer obtained during measurements of nD. In Table 24 are given values for nD, the average drum reading, A, B, and C, and the calculated dispersion, (nF-nc), for the DMSO- pyridine mixtures at 26.4 °C. Calculations of the dispersion at other tenperatures followed the same format. Table 25 presents the 201 202 results for A', B. and n51“: for the DMSO-pyridine mixtures at 26.4 °C. The accuracy in the calculation of n51“): using this dis- persion correction procedure is 1 0.0005. After the refractive index at A =- 5145101 had been calculated in a similar manner for the DMSO-pyridine mixtures at the four other tenperatures, it was determined that the refractive index at 5145: was a linear function of the refractive index at 58903.. For each mixture it was found that “5145:. a “3 + k' (mere 1: is a constant independent of temperature. The values of “51452' "D and k are presented in Table 26 for the DbBO-pyridine mixtures . mole frac. DMSO 1.000 0.9453 0.8004 0.7005 0.6002 0.4993 0.4016 0.2996 0.2000 0.1500 0.1005 0.0499 0.0000 203 Table 24 Calculation of the Dispersion for Mixtures of 1.4763 1.4785 1.4831 1.4863 1.4894 1.4917 1.4942 1.4962 1.5010 1.5020 1.5035 1.5053 1.5064 DMSO and Pyridine at 26.4 0C average drum reading 21.0 21.5 21.8 21.8 22.5 23.0 23.4 23.4 23.4 24.0 0.02417 0.02417 0.02415 0.02415 0.02414 0.02414 0.02413 0.02413 0.02412 0.02412 0.02411 0.02411 0.02411 (nF-nc)=a+sc 0.02704 0.02694 0.02676 0.02662 0.02650 0.02641 0.02630 0.02621 0.02600 0.02595 0.02588 0.02582 0.02576 -0.431 -0.416 -0.416 -0.383 -0.358 -0.339 -0.339 -0.339 0.01114 0.01132 0.01200 0.01206 0.01272 0.01315 0.01319 0.01409 0.01481 0.01532 0.01534 0.01536 0.01615 mole frac. DMSO 1.000 0.9453 0.8004 0.7005 0.6002 0.4993 0.4016 0.2996 0.2000 0.1500 0.1005 0.0499 0.0000 204 Table 25 Calculation of n5145§ for Mixtures of DMSO and Pyridine at 26.4 0C .... (514.5nm)2 n51453 = A A. 8 1.4595 5.833 1.4614 5.928 1.4650 6.284 1.4681 6.315 1.4702 6.661 1.4719 6.886 1.4743 6.907 1.4750 7.378 1.4787 7.755 1.4789 8.022 1.4804 8.033 1.4821 8.043 1.4820 8.457 n51453 1.4815 1 0.0005 1.4838 1.4887 1.4920 1.4954 1.4979 1.5004 1.5029 1.5080 1.5092 1.5108 1.5125 1.5140 205 Table 26 0 Temperature Dependence of the Refractive Index for A = 5890 and 5145A pure CSHSN 05 mole % DMSO k - 0.0074 k . 0.0073 T‘OC) “5890: n51453 T‘°C’ n58903 n5145: 59.0 1.4870 1.4944 59.0 1.4875 1.4948 54.2 1.4900 1.4974 54.2 1.4901 1.4974 49.3 1.4928 1.5002 49.3 1.4927 1.5000 44.5 1.4957 1.5031 44.5 1.4953 1.5026 39.4 1.4987 1.5061 39.4 1.4980 1.5053 34.5 1.5017 1.5091 34.5 1.5007 1.5080 29.5 1.5047 1.5121 29.5 1.5034 1.5107 21.6 1.5094 1.5168 21.6 1.5076 1.5149 10 mole 4 DMSO 15 mole % 0160 k - 0.0072 k a 0.0071 59.0 1.4842 1.4914 59.0 1.4816 1.4887 54.2 1.4870 1.4942 54.2 1.4846 1.4917 49.3 1.4899 1.4971 49.3 1.4877 1.4948 44.5 1.4927 1.5000 44.5 1.4908 1.4979 39.4 1.4957 1.5029 39.4 1.4941 1.5012 34.5 1.4986 1.5058 34.5 1.4972 1.5043 29.5 1.5015 1.5087 29.5 1.5004 1.5075 21.6 1.5062 1.5134 21.6 1.5054 1.5125 Table 26 (cont .) 20 mole % DMSO k - 0.0069 T(°C’ n58903 n5145: 59.0 1.4801 1.4870 54.2 1.4832 1.4901 49.3 1.4864 1.4933 44.5 1.4896 1.4965 39.4 1.4929 1.4998 34.5 1.4961 1.5030 29.5 1.4994 1.5063 21.6 1.5046 1.5115 40 mole % DMSO k = 0.0063 59.0 1.4791 1.4854 54.2 1.4813 1.4876 49.3 1.4836 1.4899 44.5 1.4858 1.4921 39.4 1.4881 1.4944 34.5 1.4904 1.4967 29.5 1.4927 1.4990 21.6 1.4964 1.5027 206 30 mole % D160 x = 0.0066 T(°C’ n5890101 n51453 59.0 1.4828 1.4894 54.2 1.4848 1.4914 49.3 1.4868 1.4934 44.5 1.4887 1.4953 39.4 ' 1.4908 1.4974 34.5 1.4928 1.4994 29.5 1.4949 1.5015 21.6 1.4981 1.5047 50 mole % DMSO 59.0 54.2 49.3 44.5 39.4 34.5 29.5 21.6 k 8 0.0062 1.4760 1.4784 1.4807 1.4831 1.4856 1.4880 1.4904 1.4943 1.4822 1.4846 1.4869 1.4893 1.4918 1.4942 1.4966 1.5005 Table 26 (cont.) 60 mole % DMSO k ' 0.0058 T<°C) “58902 n51453 59.0 1.4726 1.4784 54.2 1.4751 1.4809 49.3 1.4776 1.4834 44.5 1.4801 1.4859 39.4 1.4827 1.4885 34.5 1.4852 1.4910 29.5 1.4878 1.4936 21.6 1.4919 1.4977 80 mole % DMSO k = 0.0057 59.0 1.4657 1.4714 54.2 1.4682 1.4739 49.3 1.4708 1.4765 44.5 1.4734 1.4791 39.4 1.4761 1.4818 34.5 1.4786 1.4843 29.5 1.4813 1.4870 21.6 1.4855 1.4912 207 70 mole % DMSO k = 0.0057 T<°C) n58902 n51453 59.0 1.4707 1.4764 54.2 1.4730 1.4787 49.3 1.4753 1.4810 44.5 1.4775 1.4832 39.4 1.4799 1.4856 34.5 1.4823 1.4880 29.5 1.4846 1.4903 21.6 1.4883 1.4940 95 mole % DMSO 59.0 54.2 49.3 44.5 39.4 34.5 29.5 21.6 k 3 0.0053 1.4643 1.4664 1.4685 1.4706 1.4728 1.4749 1.4770 1.4804 1.4696 1.4717 1.4738 1.4759 1.4781 1.4802 1.4823 1.4857 208 Table 26 (cont.) pure DMSO k = 0.0052 31:31. n5890101 n5145: 59.0 1.4622 1.4674 54.2 1.4643 1.4695 49.3 1.4664 1.4716 44.5 1.4685 1.4737 39.4 1.4708 1.4760 34.5 1.4729 1.4781 29.5 1.4751 1.4803 21.6 1.4785 1.4837