BRELLOUEN SPECTROSCOPY OF MACROMQLECULAR SOLUTIONS Thesis far the Begree of Ph D. MCHIGAN STATE UHWEHSETY fiOUGLAS EDWARB Nomi-MUS i973 1.1312141:ij Michigan State University This is to certify that the thesis entitled BRI LLOU I N SPECTROSCOPY OF MACROMOLECU LAR SOLUT l ONS presented by DOUGLAS EDWARD NORDHAUS has been accepted towards fulfillment of the requirements for Ph .0 degree in CHEMLSIRY. Maior nmfmem / I Date M 0-7639 amomc 3r -. II HUAG & SUNS' j " ‘ BOOK BINDERY INC. 'I ‘. A]- LIBRARY emosns w ~ SPIINGPORT moms: . ABSTRACT BRILLOUIN SPECTROSCOPY OF MACROMOLECULAR SOLUTIONS BY Douglas Edward Nordhaus Brillouin light scattering was evaluated as an experi— mental method for measuring the molecular weights of macro- molecules. The thermodynamic theory of Miller was utilized in the calculations while its limitations were evaluated with the aid of the linearized hydrodynamic equations of Mountain as applied to a thermally relaxing liquid. An instrumental procedure involving parameters associated with the optical quality was develOped to experimentally separate the Rayleigh and Brillouin peaks over a range of scattering angles from 45 to 135 degrees. Brillouin spectra of benzene were taken and the isotropically scattered light identified, separated and measured. A good correspondence was found between the ratio Jv obtained from the experimental measurements and from the calculations using Mountain's theory. This fact indicates that only isotropically scattered light was measured and that the correct spectral base line was selected. An extensive cleaning and filtering procedure was also developed to give consistent scattering measurements from Douglas Edward Nordhaus sample solutions and some parameters of benzene were measured including a relaxation line, the "Mountain" line. Brillouin spectra of macromolecular solutions were taken and it was found that for dilute solutions of polystyrene in benzene the solute did not affect the solvent parameters v v60, and t to any measurable extent. This indicated that 0’ Miller's theory can be effectively applied to this solvent and solute combination. Six macromolecules, used as molecular weight standards and with different molecular weights, were measured with Brillouin scattering and the molecular weights calculated using Miller' theory. The final values compared well with other techniques previously used such as photometric light scattering and viscosity measurements, although the concentra- tions were about one order of magnitude smaller than needed for photometric measurements and the precision slightly lower. A new sample scattering cell was designed for angular measurements from large macromolecules to eliminate reflected light from the back of the cell. This cell was subsequently used in the extrapolation of the scattered intensity measure- ments to zero concentration and zero scattering angle for large macromolecules and also for the measurement of the angu— lar dependence of the Brillouin peak widths for the solvent benzene. BRILLOUIN SPECTROSCOPY OF MACROMOLECULAR SOLUTIONS BY Douglas Edward Nordhaus A THESIS Submitted to Michigan State University in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1973 TO MY PARENTS Dr. and Mrs. E. A. NORDHAUS ii ACKNOWLEDGEMENTS The author wishes to express his deep appreciation to Professor J. B. Kinsinger for his guidance and patience during the preparation of this thesis and to the Chemistry Department of Michigan State University for its financial support. Appreciation is also extended to Henry Yuen and Bill Toth for the many hours of stimulating discussion and their invaluable aid. The author is particularly grateful to his parents for their continued encouragement and understanding. iii TABLE OF CONTENTS I. INTRODUCTION A-Historical Discussion......... .................... B_P11rmse0000000 OOOOOOOOOOOOOO 00...... ....... 0...... II. THEORY A-Light Scattering .......... ... ..................... l-Time Independent Scattering Theory ............ . a-Scattering Processes.............. .......... . b-Fluctuations........... .................... .. c—Thermodynamic Theory........ ............. .... 2-Time Dependent Scattering Theory. ..... . ....... . a-Pure Liquids...................... ..... . ..... (1) Fluctuations....................... ...... (2) Ideal Case ........ ....................... (3) Relaxing Case..... ......... .............. b-Solutions.................................... (1) Mountain's Theory...... ....... ........... (2) Miller's Theory....... ....... . ...... ..... (3) Angular Corrections......... ........... .. 3—Discussion ......... ...... ..... ................. B-Viscosity....... ..... ............... ..... ......... l-Theory............................ ....... ...... 2-Molecular Weight Determinations ................ III. EXPERIMENT A-Light Scattering.... ....... . ..................... . l-Instrument... .......... ....... ...... . ...... .... a-Laser................... ....... ......... ..... b-Interferometer........... .................... c-Aperatures and Optics. ...................... . d—Alignment.................. ...... . ..... ...... e-Detection and Recording........ ............. . 2-Scattering Cell................................ a—Shape........................................ b—Size......................................... C-HOlder....o......ooooo.........o............. iv vi 43 43 43 45 50 52 53 53 63 63 3-Materials and Preparation ........ ..... ..... 65 a-Polystyrenes............................... 65 b—Solvents... .............. . ................. 65 c-Filtration................................. 66 d—Glassware. ..... .. ...... . .................. . 67 e—Solutions................. ..... ............ 59 4-SpeCtra...................... ................ 69 a-Shapes..................................... 69 b-Baseline.. ..... ...... ...... . ............... 71 c-Peak Areas................................. 73 5-Spectra1 Analysis.. .......... ..... ........... 75 a-Solvents......................... ........ .. 75 (1) Mirror Separation.............. ....... . 75 (2) Angular Measurements.. ................. 77 (3) Polarization Measurements. ........ ..... 83 (4) Experimental Values...... ............. . 83 b-Solutions.................................. 87 (1) Peak Widths and Frequency Shifts....... 87 (2) Molecular Weight Determination......... 91 6—Discussion................................... 101 B-Viscosity...................................... 105 l-Constant Temperature Bath.................... 106 2—Viscometer................................... 106 3-Solutions............................ ........ 108 Iv. SUMMARY and CONCLUSIONSOOOOOOOOOOOO00...... ...... O 113 BIBLIOGRAPIiYOOOOOOO......OOOOOOOO......COOOOO0.... 115 APPENDIX........ ..... .................... ....... .. 119 LIST OF TABLES TABLE DESCRIPTION Page 1. Identification of When Brillouin Peak Overlap 0ccurs.................................... 74 2. General Physical Constants of Benzene ........ ..... 88 3. Molecular Weight Measurement of Polystyrenes...... 94 4. Parameters Used in Molecular Weight Measurements.. 95 LIST OF FIGURES FIGURE DESCRIPTION Page 1. Rayleigh-Brillouin Spectrophotometer. ............. 44 2. Brillouin Spectrum of Benzene-Three Consecutive Orders, Intensity vs Frequency........ 49 3. Cylindrical Scattering Cell—Original and Filled... 55 4. Brillouin Spectrum of Benzene—45 Degrees Scattering Angle, Intensity vs Frequency. ...... ... 56 5. Brillouin Spectrum of Benzene—135 Degrees Scattering Angle, Intensity vs Frequency.......... 57 6. Brillouin Spectrum of Benzene-90 Degrees Scattering Angle, Intensity vs Frequency.......... 58 7. Brillouin Frequency Shifts from Benzene-Relative Angular Dependence of Incident and Reflected Light 60 8. Brewster Light Scattering Cell.................... 62 9. Light Scattering Cell Holder...................... 64 10. Refluxing Column for Scattering Cell.............. 68 11. Relative Dependence of Brillouin Peak Separation on Interferometer Mirror Spacing for Benzene...... 76 12. Relative Brillouin Peak Separation for Benzene and a Polystyrene Solution........................ 79 13. Angular Dependence of Brillouin Half—Widths....... 81 14. Angular Dependence of the Scattering Ratio JV for Benzene....................................... 82 15. Concentration Dependence of Rayleigh and Brillouin Peak Half-Widths for Polystyrene in Benzene-(A) Brillouin, (B) Rayleigh, (C) True Brillouin Half—Width..................... 90 16. Molecular Weight Measurements of Polystyrene in Benzene (A) MW = 48,000, (B) MW = 78,400, 93 (c) E, = 124,000, (D) En = 194,000, (E) 171'“I = 260,000. vi 17. 18. 19. 20. 21. 22. 23. 24. Molecular Weight Measurement of Polystyrene in Benzene. M“ = 1.85 x 106........................ Brillouin Spectrum of Polystyrene in Benzene at 45 Degrees Scattering Angle, o -— _ 6 C2—O0O38A,MW—I0BSXIO 0.00 ooooooooo 00...... Brillouin Spectrum of Polystyrene in Benzene at 90 Degrees Scattering Angle, o _ 6 C2-00038A,MW—1085X10 00.000000000000000... Brillouin Spectrum of Polystyrene in Benzene at 135 Degrees Scattering Angle, ... O — 6 C2-0.038/°,MW_1.BSXIO 00000000000000.000000 Constant Temperature water Bath.................. Viscosity of Polystyrene in Benzene at 30.0°C.... Viscosity of Polystyrene in Benzene at 30.0°C.... Viscosity of Polystyrene in Benzene at 30.0°C.... vii Page 97 98 99 100 107 110 111 112 I.Introduction A—Historical Discussion Scattered light has excited man's imagination for thousands of years as seen by his frequent attempts to ex- plain common phenomena such as the origin of the blue sky and the red sunset. Experimentally Tyndall [1] produced the scattering effect of a "blue sky" by passing a beam of white light through a tube of small suspended particles from the mixed vapors of butyl nitrate and hydrochloric acid. Unfor— tunately his attempts to explain this scattering process were unsuccessful as were the attempts of many others before him, but in 1871 a theoretical paper based on his work was written by Lord Rayleigh [2] which correctly identified the process as one of diffraction. Using this theory, one could predict other properties from scattering such as the angular dependence of the scattered intensity from horizontally polarized incident light and the dependence of the intensity on the fourth power of the wave—length. Thus Lord Rayleigh initiated the science of light scattering as it is presently known. Since Rayleigh's original theory [3,4] dealt only with small, simple, and noninteracting particles, others made an effort to extend his theory to solids and liquids. These attempts met with many difficulties since in a liquid the positions and interactions of individual particles are inter— dependent and thus can not be simply summed as had previously been done; thus Lord Rayleigh's theory had to be modified or 1 completely changed. In a liquid, the physical properties at equilibrium are practically always equal to their mean values with great accuracy, although small deviations from these mean values do occur and can be described by fluctuation theory. Einstein [5] utilized the idea of fluctuations in a liquid to modify Rayleigh's earlier theory and calculated light scattering intensities for liquids and solutions. All the theoretical work on light scattering processes up to this time, including Einstein's theory, dealt only with the intensity of the scattered light and entirely ne- glected any frequency dependence. This deficiency was finally remedied in a paper by Brillouin [6] who predicted that light is actually inelastically scattered from a liquid because of the presence of thermally excited acoustic or pressure waves. The incident light shifted in frequency due to this inelastic scattering is related directly to the velocity of the thermal waves in the liquid. Gross[7] experimentally verified this theory a short time later, but the experimental techniques were so difficult and the accuracy so poor that few eXperi— ments were ever attempted [8]. Most researchers instead directed their attention either to Raman scattering or to elastic light scattering. The study of elastic light scattering developed rapidly and led to solution studies of polymers in which Debye [9] made an important contribution. By measuring the extra scattering from a solution over that of the same pure liquid, he could count the number of molecules per unit volume and so determine the molecular weight of a macromolecule. But with macromolecules, the intensity of vertically polarized scattered light varies with the scattering angle, whereas for small molecules no variation is noticeable. To correct for this angular dependence and another effect due to the non— ideal behavior of solutions, Zimm [10] and his co—workers devised a double extrapolation plot to zero concentration and zero scattering angle in order to obtain a correct molec- ular weight for large macromolecules. Since then the theoret- ical development and experimental techniques have changed only slightly [11]. In the early 1960's the development of a powerful new optical source, the laser [12], now made Brillouin scatter— ing a viable technique [13,14] in the study of liquids and solutions. The major advantages that this new source offered included both a very high intensity of polarized light and a very narrow frequency distribution. Those two properties were the main deterrents to obtaining good Brillouin spectra earlier, since it was necessary to separate three closely spaced peaks. When intensity is plotted against frequency, the Brillouin spectra of liquids are very simple consisting of a central peak resulting from elastically scattered light, the Rayleigh peak, and two symmetrical frequency shifted peaks on either side of the Rayleigh peak resulting from inelastic— ally scattered light. It was observed that a macromolecule added in small quantities to a pure liquid greatly increased the total amount of scattering from a liquid but this change was noticeable only in the intensity of the central peak [15,16]. The two frequency shifted side peaks related ex— clusively to the solvent and were unaffected by the solute. Thus, the increased scattering of a solution could be measur— ed by relating the increased intensity of the central peak to the constant intensity of the two shifted peaks which could thereby act as internal standards. A theory for the ratio of the intensity of the central peak to the frequency shifted peaks or Brillouin peaks, IC/éI , was developed by Miller [17] using strictly thermo- dynamic arguments. Subsequently it was applied to determine the molecular weight of a macromolecule in the same manner as Debye previously had done with elastic light scattering, but the primary assumptions made were never experimentally verified. Simultaneously Mountain [18,19] developed a theory which cal- culated the total frequency spectra of scattered light in the Brillouin—Rayleigh region in liquid mixtures. This theory included relaxation effects due to the solvent which can modify the Brillouin spectra sufficiently to render Miller's theory valid only under restricted circumstances. B-Purpgse This thesis will examine experimentally the theory of Miller [17] for determining the molecular weights of macromolecules in solution, using Brillouin light scatter- ing. The limitations which must be observed will be analyzed in the context of Mountain's theory [19] for solutions in thermally relaxing liquids, and experimentally verified where possible. A comparison between molecular weights obtained from Brillouin light scattering and other methods for a series of high molecular weight polystyrenes will be made along with a general comparison of this method with regular elastic light scattering [11] to evaluate major advantages and disadvantages. II—Theory A-Light Scattering 1-Time Independent Scatterinngheory a-Scatterinqurocesses Light scattering is a general phenomenon which occurs whenever electromagnetic radiation in the visible region interacts with matter. If a particle is small compared with the wave-length of light and sufficiently isotropic to be polarized in the direction of the incident electric vector, the exciting field will induce a dipole moment in the particle as it interacts with the field [20]. This dipole oscillates in phase with the original radiation and becomes a secondary source of energy, which emits light in all directions with the same frequency as that of the incident radiation. This induced dipole moment vector pi, is a function of the polarizability tensor ai,j of a particle since the induced dipole is dependent on the shape of the particle and on the amplitude of the exciting radiation vector Ei‘ *0 II Q m Il-" This is an expression for the dipole induced in a particle by a radiation field and can be used in Maxwell's electro- magnetic equations to obtain a scattering equation for independent particles which are small compared to the wave-length of light. unfortunately this theory specifically 6 applies to a dilute gas since in a liquid a particle is not independent of other particles in its position or properties. A different and more practical approach to calculating the scattered intensity from a liquid is to consider it as a continuum instead of a collection of individual atoms. If the liquid is dense and homogeneous, the scattered intensity should be zero because the phases of the radiation scatter- ed by each particle destructively interfere with each other. H0wever, a pure liquid does scatter light. This is the re— sult of optical inhomogeneities or fluctuations within the liquid which are due to the random thermal motions of the molecules [5]. b-Fluctuations A statistical thermodynamic approach is taken to de— scribe the fluctuations present in a liquid but the models require the system to be both isotropic and continuous. These conditions imply that the polarization vector of the incident light is not rotated during the scattering process (isotropic media) and the mean free path of the molecules in a liquid is much shorter than the wavelength of light in the media. In a pure liquid, fluctuations are very small changes in the average value of some property of a liquid, for ex— ample fluctuations in the density. If these fluctuations affect the dielectric constant, light passing through the liquid will be scattered and the intensity will depend on the extent to which the fluctuations are coupled to the dielectric constant. Because of the fluctuations, the dielectric constant can be separated into an average value, (0, and a fluctuating part A61 j' [20] This latter term 3 accounts for the total optical inhomogeneities in a liquid and the total light scattered. We can now write the dielec- tric constant as follows: 6. . ==£018. . + AEC. 1:] The second term can be further divided into two terms, the I isotropic part, AC , and the anisotropic part A£i j' ’ AC. .==A€°811j+-A£’ 3 1)] igj H Ir\’1w H D I“ H-‘- L O The first term on the right side is due to isotropic fluctuations and is simply described by a set of statisti- cally independent thermal variables such as density and temperature or entropy and pressure. The anisotropic part ‘56:,j’ has not been fully described by theory but appears to be partly due to fluctuations in the orientation of anistropic molecules. Since there is no agreement on its I evaluation [21], we set ACi j equal to zero for our 3 theoretical and experimental development. [M Iw c-Thermodynamic Theory The intensity of the scattered light can be obtained by beginning with Maxwell's electromagnetic equations [19] and using the assumption that only first order terms are neces— sary to describe the scattering. The total scattering volume in a liquid V, is divided into a number of small parts V*, in such a way that each of the parts contains a sufficient number of atoms to be independent of the others, yet small compared to the wave—length of light. The fluctuations in the dielectric constant of each volume AE are related to AC V*’ for the total scattering volume since both have the same density dependence. Maxwell's equation for the scattered intensity is 2 _ ‘fl' v . 2 2 lb The components of this equation are V, the scattering volume, L, the distance between the scattering volume and the detector, A , the wave—length of the incident light in the medium, 8, the viewing angle for the scattered light, and <(A£)2>, the average value of the square of the fluctuations in the dielectric constant. This is the time independent scattering equation for which the values of all the terms can be measured except for the last term <(A£)2> . Einstein derived the value of this last term using statistical thermodynamics by separating the fluctuations in the dielectric constant into two statistically 10 independent parts. A6: (13%) AS + (-%%) AP P S lm In Brillouin spectroscopy, it is possible to separate ex— perimentally these two contributions [7] since the part of the scattered light due to the pressure fluctuations AP, is shifted in frequency sufficiently to separate it from the entropy fluctuations, AS. Consequently an intensity ratio of the nonfrequency shifted light to that of the shifted components (there is both a positive and a negative fre- quency shift) is related to the ratio of the entropy and pressure fluctuations. 36 2 2 IC (3‘5")13 < (AC) >P = 6 21 2 - B 36 2 (31,) <(A€) >S S This relation has been reduced by Landau and Placzek [22] to a very simple form by using the assumption that a ‘52 36 a?) = a ( T) -7— T a P The Landau—Placzek ratio is defined as, I C c _ _ P B 21 ' 7' 1 ’ Y" c 11 However, in the event that the thermodynamic parameters do vary with the measurement frequency, this thermodynamic derivation is not sufficiently accurate [23]. A variation with frequency often appears to be present in liquids because the measured Landau—Placzek ratio is usually larger than calculated and another parameter, namely the velocity of sound, which is related to the adiabatic compressibility, also in- creases when measured at increasing frequencies [22]. Thus a different theory is necessary which will lead to a better correspondence between theoretical and experimental results when a dispersion in these terms is present. 2—Time Dependent Scattering Theory a-Pure Liquids (l) Fluctuations The frequency shifted light due to pressure fluctuations is caused by molecular motions which, according to ' Debye's theory of heat in a solid [24], are a superposition of longitudinal and transverse waves of different frequencies. Liquids also contain these waves except that the transverse waves are almost nonexistent since the viscosity of a liquid is usually very low. Energy to excite the longitudinal waves is readily available in the liquid since the energy per phonon is much less than the energy available as heat. hi) << k T Io 12 Therefore at room temperature the frequency of the phonon, (Imay have any value up to about 1012 Hz. Each propagating longitudinal wave is a periodic varia— tion of the density and so will scatter light in a manner similar to the scattering of x-rays from the planes in a crystal in that both types of scattering must follow Bragg's law. This requires that the sum of the vectors for the l incident wave ki’ the scattered wave ki , and the sound wave qi is equal to zero, or ki = k1 + qi. 19. Graphically, this is The value of qi is approximated by assuming that lkil = lk;| , since the scattered wave differs from the incident wave only by a very small amount. This assumption makes the vector triangle isosceles and therefore determines the value of the sound wave qi when the scattering angle 8 is selected. qi = 2ki sin(9/2) 11 The wave vectors for light and sound are respectively ki = Zw/A and qi = 27r/AS = 2” vs/vs , with A and A's 13 the wave-lengths of light and sound in the media, 'Vg the sound frequency, and vS the sound velocity. Substituting in the above equation, we obtain equations for the character— istic properties of the sound wave. Letting A=AO/n, with n the refractive index of the scattering media, v8 = vs AO/[Zn sin (9/2)] 1_2 A's =AO/[2n sin (8/2)] _1_3_ V Vs = T: [2n sin (9/2)] 1% The velocity of sound in a liquid can be calculated according to equation 12 by measuring the frequency shift I; of the Brillouin peaks, and substituting in the known values of the scattering angle, refractive index of the liquid and the wave-length of light in vacuo. In equation 1;, the wave—length of the sound waves is dependent on the scattering angle and is of the same order of magnitude as the wave—length of the incident light waves, while the ultra high acoustic frequency of the waves can be calculated either from equation 14_or directly measured from the spectra. Examining equation 12 more closely and recognizing that at increasing scattering angles the scattered light is from sound waves of increasing frequency, we can measure the dispersion in the velocity of sound by simply changing the scattering angle. It will be noted here that the theoretical velocity dependence of the sound wave on the increasing scattering vector exhibits 14 a slight negative dispersion in light scattering and must be interpreted differently from acoustic propagation exper— iments [25]. The scattered light is considered to be shifted in frequency, either increasing or decreasing the original energy of the radiation, corresponding to the annihilation or creation of a phonon in the liquid. Since the phonon population is large at room temperature, the equilibrium population is maintained if energies of the incident light are less than 105 milliwatts per square centimeter [26]. This is still well in excess of the energies we use in laser scattering, indicating we are not disturbing the system during our measurements and hence are measuring liquids in thermal equilibrium. The shortest pressure fluctuation that can be measured by Brillouin scattering is dependent on the incident wave— length of light. (As)minimum = AO/2n l2 This fluctuation is about 10-5 cm. in length and has a frequency of 1010 Hz. In terms of molecular sizes, these are very long fluctuations, so the scattered light can be interpreted as coming from coherent fluctuations in the pressure. Associated with these pressure fluctuations are in— dependent fluctuations in the entropy which cannot be included in Debye's purely mechanical description of 15 heat motion because they do not propagate in the form of waves. In comparison to the coherent pressure fluctuations, entropy fluctuations occur only over a short range, and are related to the short range motions of the thermal diffusion process. Hence these are considered as incoherent processes. Thus Brillouin spectroscopy can be used to measure these two types of fluctuations but requires the scattered light to be sufficiently separated into two parts due to (1) frequency shifted light from pressure fluctuations and (2) light not shifted in frequency from entropy fluctuations. For an iso— tropic liquid in which the scattered light does not depend on the direction of the incident light vector ki, it has been established using Onsager's symmetry relations that the spectrum contains only those two frequency shifted lines [27]. (2)-Ideal Case Since the pressure and entropy fluctuations in a liquid are time dependent, these fluctuations can be calcu— lated by adapting thermodynamic and hydrodynamic equations to the calculations. An approximate theory of the fluctua— tions employing both of these processes to account for the temporal dependence has been developed by Mountain to ex- plain the frequency shifted light in the Rayleigh-Brillouin region [28]. Since this is a phenomenological theory which treats a liquid as a continuum, it should be strictly valid only for measurements at long wave—lengths and low frequen— cies. The fluctuations that scatter light are long, 10—5cm, compared with molecular sizes although the frequency, 1010 Hz, 16 is much higher than usually encountered in hydrodynamics. This theory should still apply up to frequencies close to the reciprocal of the collision time,1012 to 1014 Hz [29,30,31]. This implies that some properties may have a significant frequency dependence, and for those cases a modified or completely different approach must be used. Mountain's theory begins with the neutron scattering equation of van Hove [32] which has been modified for light scattering [33]. 2 I(r.,t) = I 7’ V . 2 1 0 4 2 Sin 9 ff A L iw(t —t) - ik. (r —r) e 2 l l 2 1 dr dr dt dt _];_§_ 2 l 2 1 The fluctuations in the dielectric constant are a function of the two statistically independent variables,density and temperature. Since the temperature fluctuations are small compared to those of the density [34], we can neglect the temperature fluctuations and obtain the resulting equation 2 2 _ 17 V ' 2 g \. I(ri,t) — I0 731-2- Sin 9 (OP) ff 3.9. The first term in the braces in equation 2§_represents the unshifted or Rayleigh peak while the other two terms represent the Brillouin doublet which is shifted in fre— quency by the amount vok. It is evident from equation 25 20 that a ratio of the unshifted to the shifted components gives the Landau—Placzek ratio of vertically polarized light originally derived using only equilibrium thermodynam— ics and which we have mentioned earlier in equation 8, (a Mountain's theory gives results consistent with the thermo- ICD dynamic treatment, since the final Landau—Placzek ratios are identical, although it does not account for any frequency dispersion in the experimental measurements. 3) Relaxing Case If we assume that the velocity dispersion is entirely due to a coupling between the molecular motions already described by Debye's theory and the internal degrees of freedom of the particles such as vibrational modes in a thermal relaxation process,as opposed to a structural relax— ation process [35], the frequency spectrum of the scattered light can then be derived by using either an additional hydrodynamic equation for the internal degrees of freedom or a frequency dependent bulk viscosity [36]. The second alternative is more easily followed in the mathematical derivation although both modify the frequency shift and the width of the Brillouin peaks and also add a new unshifted mode in the spectrum [37] which we will refer to as the Mountain line after his theory. 21 To effect this change in the theory, the Navier—Stokes equation is modified by inserting an extra term to allow for a frequency dependent bulk viscosity; (the last term on the left side). 2 2 avi V0 V0 a po pO—3t_ +-——- grad pl + grad T1 t 2. _ I _ I ' I I _ - 3 ”S + 17B)grad grad vi / 7)B(t t )grad div vi (t )dt — O O 27 The bulk viscosity thus has a frequency independent part 17B and a frequency dependent part 1"3 . The analysis proceeds in the same manner as in the ideal case except that the final equation is too detailed and the terms too interre- lated to be able to identify specific expressions with properties of the material. Because of this complexity, a simplified approach is taken, even though the exact solution to the hydrodynamic equations is available for the fluctua— tions in the density. This simplification process is explained well by Mountain, who gives the final frequency dependent scattering equation from a relaxing liquid as follows: 22 I r' A k2 2 2 c v . 2 be 1 p0 P mum = I 7’ sm e(——) T a (1— —) OA4R2 5p 0 T y Alkz 2 2 c +0) . po P 4 2 2 v 2v .. 2 _ ,. + (v - v2)k2—(!——-- l) O + v2k2(l— l) O ] oo 0 2 4 2 0 ‘Y 2 V0 V7 . V ‘r v 4 ] v2 2 L O 2 2 _ O 2 - 4 2 + V k < 2 :) +Ia’ v v V2 V2 2.2: _ o l 2 2 0 2 2 2 _ (l--—§-(l- —~)) °(v k + —§—)- (v — v )k + v 7! v 1, oo 0 0 v 4 _ 2 + V2k2 .1 V 1' F F ___ B + B [‘2 2 2 7- 2 2 B+((0+vk) PB+(m-vk) The first term corresponds to the Rayleigh peak and is identical to those arising in the nonrelaxing case. This is the non-propagating decay of a fluctuation by thermal dif- fusion with the peak half-width at half-height. I R A 2 =———— k 29 1/2 pOCP —— 23 Because of the dependence on the scattering vector k, this half—width should decrease with decreasing scattering angle, since k2 = §l§_23.(1 - cos 8). This has been shown to be the 0 correct angular dependence by self-beating spectroscopy [38]. The second term corresponds to a nonpropagating decay that is coupled to the internal degrees of freedom of the molecules. This is a new peak, which is called the Mountain line, and it has a half-width The half—width is not dependent on the scattering vector as is the Rayleigh peak, but it does depend on the disper— sion of the velocity of sound and (H: a single relaxation time (1) of the liquid. The third and fourth terms correspond to the propa— gating modes, the Brillouin peaks, and result from the moving sound waves. The frequency shift from the incident frequency (I) is [10) = v-k and depends on the scattering vector k = 2”“ sin (9/2) 31 *0 Thus for a decreasing scattering angle the frequency shift will show a decrease. The half-width of this shifted term due to pressure fluctuations is 24 2 B ' v _.1__£ .. _._Q_ 2 I‘ ’p03n3+173+ (7 2 k 1/2 2 2 v - O 1 f 2 2 + 222 - c k 3.2. l + V'1'k p0 v This half-width is also dependent on the scattering vector k2, and decreases with a decrease in scattering angle, but the change is not linear as it is for the Rayleigh peak. The ratio of the intensities of the central components to the shifted components is termed J instead of the Landau- V Placzek ratio because of the relaxation processes present, and is obtained from equation 28. 2 V4 1 2 2 v V0 2 ..(7 v. V? J V V2 V2 O 1 2 2 0 2 2 2 l — ——- (l- —) + ——-— - (v - v )k 2 ‘Y V212 oo 0 33 At low phonon frequencies this reduces to the Landau—Placzek ratio, so for the case vk1’ << 1 JV=Y-l 2‘1 At very large phonon frequencies equation 33 reduces to a slightly different but simpler form for the case vk1>> l. J=—%°—1-i 35 25 Between these two extremes the more complex equation 33 must be used to determine the ratio J although it should v) be noted that the value JV can be equal to or larger than the Landau-Placzek ratio but not smaller. This is true for all liquids examined to date. The relaxation time 1', can be calculated from the velocity dispersion equation in Mountain's theory. _1_ 2 2 -. (vk‘n =:,_l- [(mGT)2-l]+ -21—[ ((vmk‘r)2 — 1)2 + 4(v0k‘r)2] 36 0’ is measured acoustically while the velocity of sound at infinite The velocity of sound extrapolated to zero frequency v frequencies is approximated by using acoustic theory and assuming that only one relaxation mechanism with a single relaxation time contributes to the velocity dispersion [39], giving the equation 2 :99_=E_E__:_SI_.E_Y. 37 v3 CV‘CI CP —_ The internal specific heat CI, is calculated from vibration— a1 energy levels and their known degeneracies [39] from the equation N' 2 hlw. _.:2: 2 X _ “X , —.__4L CI — i=1 giR x / e (l e ) , x — kT I__ The measured velocity v, with the corresponding incident vector k, is then substituted in the dispersion equation 26 to obtain 1', the relaxation time. b—Solutions l) Mountain's Theory The frequency dependence of light scattered from a solution [19] with the solvent a relaxing liquid, can be treated in a manner similar to the case of a pure relaxing liquid by including another equation to describe the mass diffusion due to the solute. k 2 2T + (—£) v p 39 0 p0 302 2 kT = _ V The components of this equation are the binary diffusion coefficient D, the thermal diffusion ratio kT, and a term kP, containing different thermodynamic quantities, k _ _§g (59/692193 P- '0 (all-1 ) — /5C2 PT ) The solution of these equations has been approximated, as was done previously, even though an exact solution again is available. Nevertheless it is extremely complex, requir- ing at least a full page to reproduce in its entirety. Since we are principally interested in the intensity ratio JV, the complex set is reduced to a much simpler set based on a T35°1n° derived by Miller [40]. We thermodynamic ratio Jv must first assume that the relaxation peak is narrower in width than the frequency shift of the Brillouin peaks, so 27 vk1"2 1. This is true experimentally since for benzene we find vk1’ = 5.44. The complex equation is then reduced to the following form based on the ratio JVT’SOIn' T soln. soln. JV ’ [l + A(k)] + B(k) JV 2 T soln 3; l — B(k) - Jv’ ‘[A(k)] The two composite terms A(k) and B(k) are 2 v 4 2 2 2 2 V (voo- v )k 'r — (39) + (39-) MM = 42 (vk‘r) + (—) v 2 v 4 2 2 V (Vio- V0)k2‘r - (59') 4439') B(k) = 43 2 v0 4 '— (vk1') + (——) v These two terms are equal to zero for a very small wave vector k,(A(k) = B(k) = O) and equation 3; reduces to the following for ka << 1 Jsoln _ T,soln .gg V V At the other limit of a very large vector k, A(k) = O 2 v and B(k) = 1 _(_;Q) and equation 31: reduces to the 00 following for ka >> 1 28 v 2 v soln. T,soln. -—£L + —99-— 45 J = J v 2 -—- v v 0 v Using our experimental spectra of benzene at a 90° scatter- ing angle and the known values of k, and v00, the value v0, of A(k) = 0.00253 is small and relatively unimportant, although the second term B(k) = 0.2769 is too large to be neglected in our calculations. Thus the thermodynamic deri— vation of the ratio JVT cannot be used directly for benzene and its solutions but must be modified to account for the internal relaxation present. 2) Miller's Theory A light scattering theory of solutions has been de— rived by Miller [40] which separates pressure fluctuations from entropy and concentration fluctuations. The derivation is similar to that described in the section on thermodynamic theory, but it also includes fluctuations in the concentra- tion in deriving the average value of the fluctuation in the square of the dielectric constant <(A 6 )2) . Using the variables temperature T, pressure P, and number of moles of solute n2, the scattering equation §_is 29 _ 2 2 Iscatt. _ I0 ”4‘72 sinze (35%) <(AT)2> 46 A R P,n2 2 as 66 + 2 6T) (36%) <(ATAP)> + (j) <(AP)2> P,n2 T,n2 T,n2 2 + (37%) <(An2)2>] The cross terms, <(AT°AT)>and<(AP-AT)> are zero, since the variables anus statistically independent. Statistical thermodynamics is used to obtain the fluctuation averages and gives the final scattering equation 2 TT2V 2 313 (BC) P n Iscatt. = Io A4112 cV '57.? ’ 2 31 2 ' 2 + 2RT a. (fig) (fig) RT (36) fl'I‘Cv T P,n2 DP T,n2 Vfis 3P T,n2 2 mac/a n2) ( af‘Z/Dnz) T,P 30 The Brillouin peaks are due only to adiabatic pressure fluctuations, therefore, this contribution to the fluctua- tions in the dielectric constant is given as follows 2 36 2 2 <(A£) > = (————) <(AP) > 34.9 P,n2 3P S,n2 From thermodynamics we obtain the equation Bi) = (LG 129. + (3.3T 49 (‘31)sz bT)P,n2 CP DP ,n2 — From statistical thermodynamics the pressure fluctuations are given as follows <(AP)2> = RT/vfi 5.9. S Combining the above to obtain the contribution due to the adiabatic pressure fluctuations, the final equation obtained is 2 135 2 . 2 RT 1 <(A ) > = —"- (1 - —) T e C3v Y ( T)P,n2 2 2RT a BS DE RT (36) + —— + B C (3T)P,n2 ( 5P)T,n2 V55 DP T,n2 31 Since this part is included in equation 41, subtrac-— tion gives the fluctuations contributing to the central peak. A ratio of the fluctuations contributing to the central peak to the fluctuations contributing to the Brillouin peaks is found as follows: 2 I <(AE) >S’n2 = 2 B ((A6) >1”).2 52 2 (36/3 lflfii(.%¥§) ‘+ RT n T CV P,n2 (3"2 / 3112).]? P 2 2 7- ‘ RL _ l 35 2M a 36 6 RT a6 2 CV (1 Y) (fi)P,n + f3TCV (“>335 (%E)T:n + VB: .5?) 2 2 T,n 2 zflmp 2 Values of eggp) are usually not available in the litera— T n 2 ture, but by assuming that 5: n2, and rearranging the denominator using a term X , where z: = 1'+ 0‘ (an/51%, 2; for which excellent values are available from the literature [34], equation g; reduces to 32 2 2 (bn/bnz)?’ RT 3n) + RT T,P cP TPn (BIAZ/bnz) JT,soln. = ’ 2 _ T,P 54 V RT2/ 1 )(3n)2 1 +Y(-—--2X + —————YX2 ) CP ‘7- 1 8T P,n2 l-x (l-x)2 Simplifying this equation further by letting 2x ),x2 f =-——— + l—x (l-x)2 and writing the solvent chemical potential in terms of the concentration of a macromolecule C2, [11—fl0=-RTV1C2(—:’T+A2C2+A3C§+---) 56 The ratio JVT for a real polymer solution is J C 2 2 T,soln.=()’-1)+()’.1 \ CP . (an/“913p v 1+)’f l+)’f/ RTZ (an/3min ’ 2 —l l 2 .(——— + 2A2C2 + 3A3C2 + ...) “w This is an exact thermodynamic ratio without approximations or relaxation effects adding to the central peak. This equation can be reduced to a simpler and more easily handled form by setting the exact thermodynamic ratio for the pure solvent equal to JVT. 33 Grouping other terms into a constant K, dependent on the solvent and solute system, we have 2 C (on/8C ) 2 P T,P K: 2 2 E2 RT (an/amp ,n 2 Now we can write equation 21 in a simplified form. JT’E301m =JT+JTKC (——1+2AC 3 c2 ‘1 6 V V V 2 M 22+A32+'°°) _9 W If no relaxation or dispersion effects are present to form a significant Mountain line, the experimentally measured ratio Jv for the solvent is equal to the thermodynamic ratio JVT. The presence of thermal relaxation effects can be detected by an increased velocity of sound in the pure liquid at increas— ing frequencies and a larger experimental intensity ratio JV, than calculated from thermodynamics as JVT. When these effects arepresent, equation 99 must be modified to use the experimen- tal ratio Jv‘ This is accomplished by multiplying each term due to the pure liquid by the quantity (JV/JVT) so that equation gg may now be written as soln. 1 2 —l J = J + J KC 1:— If no relaxation or dispersion effects are present JV = JVT and the original equation is recovered. Unfortunately 34 equation 6; is not entirely correct when dispersion is present, since to evaluate the molecular weight we need to measure the relative amount that the central peak is increased due to concentration fluctuations. This increase is dependent on the original amount of scattering present which is due only to entropy fluctuations and so should not include other effects such as the Mountain line. Since Miller's calculations are based on the complete separation of fluctuations due to pressure from those due to entropy and concentration effects, the relative intensity of the central peak must be reduced to the thermodynamic value. This is accomplished by multiplying the measured ratio Jv by the term which is derived as follows: the thermodynamic ratio is a ratio of the scattered intensity from entropy and concentration fluctuations Ic, to the intensity due to pressure fluctuations I whereas the measured ratio I J‘—ST'th tthtI' I dI\I V 21 15 8 same excep a C > C an B , B B) due to an extra relaxation peak, the Mountain line. Assum- ing that the total scattered intensity in each case is equal, the intensity of the Rayleigh and Brillouin peaks in terms of the ratios and the thermodynamic value from Miller's 35 equation is Therefore by multiplying the intensity of the central Rayleigh peak as measured in a spectra by the ratio T JV(Jv + l) we obtain the correct thermodynamic value and T Jv (Jv + 1) equation §l_becomes T -l J (J + l) Jsoln. = J + J V V KC l—— + 2A C +3A C2+--- v v v J (JT + 1) 2 —- 2 2 3 2 v v Mw 9:1. Simplifying this expression by letting B (Jv+l) (Z—l) 65 Y(l+f) — and rearranging, the final working equation becomes BKC 2 _ l 2 v ~7va If there is no dispersion present in the constants which characterize the solvent, then the measured ratios and thermodynamic ratios are equal and we find for the solvent _ T_ _ Y-l Jv-Jv-B-TTTE 9—7- 36 Therefore with no dispersion,equation 66 reduces to the thermodynamic expression 69. 3) Angular Correlations Equation 66 is often modified by inserting an extra term P(9), the internal interference factor. BKC 2 = 1 +2AC+3AC2+--- — 2 2 3 2 Jv Mw P(G) This term P(9), is defined as [41] sin(K-r ) P(9) = l3 2 Z K-r X1Y N X Y Xsy where 4wn . K =-———— Sin(9 2). A / 0 The relative amount of destructive interference which decreases the scattered intensity can be determined from the scattering at N different points which are pairwise separated by the distance rx,y' If a solution is dilute, this destructive interference is calculated only from the separate scattering points within each molecule, since rx,y is very large for widely separate random points and thus produces a very small interference effect at these distances. Therefore P(9) becomes a factor internal to the molecule, although only becoming a significant factor in reducing the scattered intensity if the size of the macro— 37 molecule exceeds about l/20 of the wavelength of the light used. Consequently the molecular weight should be greater than about 106 g/mole for P(9) to be smaller than unity and significantly affect the molecular weight measurements. 3-Discussion An analysis of the light scattered from a solution of macromolecules or even from a pure liquid is difficult to interpret completely without a precise liquid theory. Con— sequently we can only approximate the scattering with simplified models [42]. The two different theories we have used to evaluate Brillouin light scattering from solutions and pure liquids are (l) Miller's thermodynamic theory, and (2) Mountain's thermodynamic-hydrodynamic theory. Other theories similar in nature are available [42,43,44,45,46,47]. Both theories attempt to explain only the light scatter- ed from isotropic fluctuations; so only the vertically polarized light VV or Jv can be evaluated. In many liquids, a significant amount of the scattered intensity is not vertically polarized and therefore must be separated and eliminated from the measured value. Also, part of the re— maining vertically polarized light is due to anisotropic fluctuations which may have broad frequency shifts such as is exhibited in the Rayleigh "wing" in benzene and also must be separated and eliminated from the analysis. Because much of the scattered light is not included, both of these theories are only approximations to the total scattering and should 38 only be treated as such. Both theories also often contain simplifying assumptions to allow a clearer mathematical representation of the final result. Mountain's theory necessarily employs more approx- imations because of its complexity. The most significant approximations are in (l) describing extremely rapid fluctuations on the order of 1010 Hz by hydrodynamic equa— tions, (2) evaluating the fluctuations in the dielectric constant with only density fluctuations, and (3) using a single frequency dependent bulk viscosity to describe dis- persion effects [46]. The first approximation mentioned above is Mountain's basic assumption and appears to result in a reasonable correspondence with experimental measure- ments. The second approximation neglects temperature fluc- tuations since they are small when compared with density fluctuations. Since the temperature fluctuations are about 0.2%.of the density fluctuations for many liquids [34], this approximation is often used in light scattering theory. The third approximation results in an entirely new peak which appears to be present. The theory by Miller makes a basic assumption that fluctuations in entropy and concentration can be completely separated from fluctuations in the pressure in order to derive a ratio. In some solutions this separation is not possible if the solute affects the values of v voo and 1', o) as has been seen in gas mixtures [31,48]. When this happens, a coupling between pressure and concentration fluctuations 39 occurs requiring a very complex expression such as Mountain has derived. It is also assumed in Miller's theory that the Brillouin peaks scatter an identical amount of light even after large macromolecules are added to the solvent. This assumption also appears to be valid for very dilute solu-— tions in solvents where significant structural relaxation is not present. An evaluation of these assumptions with experimental data will be made when possible along with the ability of Brillouin scattering to determine accurate molecular weights of large macromolecules. B—Viscosity l-Theory A liquid which is under small shear stress will easily distort and flow, whereas a solid will show a fixed shear strain. This distortion and flow in a liquid is a result of adjacent molecules or molecular layers moving relative to each other, while the frictional forces between them reduces this relative motion. The rate that mechanical energy from a stress is transformed into heat J, by friction— al resistance is [49] J =1]qu 70 The uniform velocity gradient is q and the viscosity of a pure liquid 7'0 . 40 The addition of a solute to a liquid will modify J by an amount AJ. J+AJ=nq2 71 The relative change or specific viscosity, ”sp’ is written as =Efle €19: = nsp ’10 _- If the solute consists of large rigid spheres, the specific viscosity depends on the volume of these spheres plus second order effects, and can be expressed as [49] :1. Va 2 2 flsp=2.5 M C2+12.6 31-— c2 _7__3_ 2 2 2-Molecular Weight Determinations Experimentally we use an empirical equation which has the same form as above even though macromolecules in solution do not act exactly as large rigid spheres. $2 = . 2 CZ [7’]+k[‘))]C2 74 The limiting viscosity number [1,], is related to the hydrodynamic volume Ve’ for a mole of particles but varies with the size and shape of the particles, the solvent, solute and temperature. Also and most importantly, the hydrodynamic volume varies with the molecular weight 41 according to the well established theoretical equation of Flory [50]. _3 2 2 l '- _ 0 2 3 _ [17]-(1)0(‘73— M a 1- The components of this equation are the molecular weight M, the average value of the square of the end-to-end distance < r02 > , the expansion factor a, and a universal constant QC . When a macromolecule is in a solvent at the theta temperature, 9, as defined by Flory, equation 1§_reduces to a very simple form, where the constant K depends only on the specific solvent and solute [")]e=KMa 5 a=l/2 76 This simple form can be used at temperatures hdgher than the theta temperature but then the constants K and a must be experimentally determined, since they are not simply derived from theory. If we define a viscosity average molecular weight as N 2 b E = W.Mb 77 n i=1 1 i ’ -—— we find that when b = 1.0, the viscosity average molecular weight equals the weight average molecular weight M“ . Since Mn is often a closer approximation to M; than the number average molecular weight Mn, the constants K and a are usually determined by an experimental method such as elastic Photometric light scattering which permits determination of the 42 weight average molecular weight. In the experimental section on viscosity we have used light scattering data from the literature to determine the constants K and a so as to experimentally evaluate the weight average molecular weights for a series of polystyrenes. These then are compared with experimental values calculated from Brillouin scattering experiments. III Experiment A-Light Scattering l-Instrument An instrument to measure Brillouin light scattering, shown in Figure l, was designed and constructed in the chemistry laboratory at Michigan State University [51]. Basically it consists of a laser, collecting and resolving optics, plus a detector and recorder which are constructed along two parallel tracks. The optical system, which is very sensitive to vibrations, is mounted entirely on a large, flat, acoustically isolated table to eliminate building vibrations which otherwise would force the system out of alignment. One track of this system holds an argon ion laser and beam directing mirror, while the other holds the scattering cell and angular adjustment system, collecting lens, inter- ferometer, resolving lens and detector. Each of these com- ponents will be described separately, although only briefly, since a more complete description can be found in the thesis of S. Gaumer. a-Lsse: The primary light source is a commercial Spectra Physics argon ion laser, model 165-03. Due to its high intensity and single frequency, the detection of Brillouin spectra is quite simple. Also because of stable output, vertical polar— ization and a single spacial mode configuration of the light 43 44 umquOponmonpowmm sasoaafinmnnmfloammm 30:23.11 55.212 39.08”. szL NEE I m emmzz,‘ $3.95 I8... 6238““. mozemzmo .35. «$2: 5&3 5&8 mo<50> :9: $30.. $30.. $9: .35 ... EB 9555. s e m e ezmzzoza. l _ _ _ u 2: _ _ _ mum‘s 200m< was. 70..me 45 beam, the spectra can be quantitatively analyzed. Besides these necessary features, there are also eight separate wave-lengths which can be used for light scattering, although only five (514.5, 496.5, 488.0, 476.5, 457.9 nm) have suffi- cient intensity to provide good spectra. The most commonly used wave-length is 514.5 nm with a corresponding single mode intensity of 400 mw. This intensity, lower than the maximum 800 mw, allows both a good stability of the light output and a longer usable laser lifetime. The light beam is usually used with a vertical polar— ization vector although infrequently the vector is rotated horizontally for depolarization measurements. Consequently a good vertical alignment is necessary and can be quickly established by reflecting the "vertically" polarized beam from a glass plate located at the critical angle and adjust— ing the direction of the incident polarization to obtain a minimum reflection. Specific details of the stabil- ity of the light intensity, frequency drift, single mode selection and construction of internal laser components are readily found in the literature [51] and from the manufact- urer . b-Interferometer The frequency dependence of the scattered light in the range of i_l.0 cm_1 from the incident light frequency, the Rayleigh-Brillouin spectral region, is obtained with a commercial Fabry-Perot scanning interferometer, Lansing 46 Research model 30.205. Although this interferometer is a fine interference filter and allows only a very narrow frequency distribution of light to pass at one time, it still can be scanned over a small range of frequencies by varying the optical path length between two very flat mirrors (JR/100). The interferometer is the most sensitive component of the entire instrument and its alignment and operation are critical to the quality of the final Brillouin spectra. Thus, its operation and associated terminology will be discussed in more detail. Before a specific frequency is passed through the two parallel interferometer mirrors, it must meet the inter— ference condition given by an Airy function [52]: I =1. . (i2- 78 trans inc1d l—R 4R 2 8 ——- 1 + —— sin —-— (l—R)2 " where 8 = 21m£ sin 9. The light intensity transmitted through the mirrors depends on the incident intensity, the transmission T, and reflec— tion R, coefficients of the mirrors, R = 98.5%.for our mirrors, and on the optical path length 8 . When the inter— ferometer is scanned to obtain a Brillouin spectrum ,the optical path length 8 can be changed by varying either the refractive index n, between the mirrors, the angle 9, between the mirrors and the incident light, or the separa— tion I , between the mirrors. 47 A piezoelectric transducer is used to move one mirror towards the other. This transducer is an anisotropic crystal which expands when a large voltage difference is applied across two opposite crystal faces. If this applied voltage is gradually increased, a mirror attached to one crystal face is slowly moved toward the other mirror and so allows the separation between the two parallel mirrors to decrease. Because this movement is very small, it cannot be mechan— ically changed with a micrometer screw with good precision, which is why a piezoelectric transducer is employed. Applying from zero - 1600 volts across the crystal faces will expand a four layered crystal 1.2]! and scan almost five spectral orders. This voltage is linearly increased with time with a Lansing Research model 80.010 power supply. Unfortunately neither the lower voltage range nor the largest crystal ex- pansion gives a linear expansion with increasing voltage. A linear range is only over a voltage range of -250 to -1500 volts. While this still allows four spectral orders to be scanned, we maintain a safety factor by selecting only the three central orders. The measured values from three orders are averaged in the final analysis for this gives a more consistent value than if only one order is selected. Continuously increasing the applied voltage results in a series of repeating spectral orders in which one order immediately follows another. An example of three consecutive orders for benzene with the dark current baseline also 48 included is shown in Figure 2. The peaks are the time de— pendent intensity changes for an interference pattern originally in the form of a concentric series of rings. Reducing the mirror separation forces the rings to converge toward a center until the interference condition given by the Airy function is met, then all the rings brighten simul— taneously. Only the small central spot of this ring pattern is detected for reasons which will be explained in the section on Aperatures and Optics. Specific"terminology :regarding the interferometer, which will frequently be used is described in detail below [52]. Free Spectral Range; f = C . 79 Zhul '—— This is the frequency range that is scanned in one spectral order. It is inversely proportional to the mirror separation l , the index of refraction n, of the media between the mirrors, and the velocity of light C. Choosing a specific mirror separation for a spectrum must be done carefully so as to eliminate overlap of Brillouin peaks from adjacent orders yet identify the correct central peak associated with each Brillouin peak. Our method for meeting these two objectives is given in the section on Spectral Analysis. 49 wosmsvmum m> muwmsmusH .mnmpno m>flpsommsou mongeamsmusmm mo Esnuommm GHSOHHHHm .N wusmflm 4m... . --. .. .-.-_w ....— 50 When a careful determination of Brillouin peak separa— tion is necessary, as in sound velocity measurements, the small linear increase in the free spectral range due to a decreasing mirror separation must be corrected. This is done mathematically by determining the increasing distance between consecutive Rayleigh peaks and applying this correction to the Brillouin separations. Instrumental Band Width; 8V3 W . This is the narrowest full peak width at half height passed by the interferometer when a perfectly monochromatic light beam is used as a source. As seen in the expression for the Airy function, it is dependent on the T and R coefficients for the mirror and is an instrumental constant of about 293 MHz for our particular instrument. The eXper— imental value of the instrumental band width is also dependent on the alignment and reaches a minimum value with perfect alignment of the interferometer mirrors, parallel incident light, etc. c-Aperatures and Optics The Brillouin scattering instrument contains two achromatic lenses and three pinholes to collect the scattered light and define its direction during analyzing and detection. Most of these components are aligned on precision optical rails inside two light tight boxes which help eliminate extraneous light from adding to the scattered 51 beam and also to keep the lenses as clean as possible. In the front of the first box closest to the scattering cell is a collimating tube which is blackened inside and supports a variable aperature at each end. The first aperature de— termines the acceptance angle of the diverging scattered light and the second determines the cone angle. We have found the best finesse (an eXperimental ratio of the width of the central peak to the separation between central peaks of consecutive orders) is obtained when the first aperature is 1.0 mm and the second 2.0 mm. After passing through the two aperatures, the scattered light is gathered by the first lens and made parallel before entering the interferometer. The first lens (f = 50.0 cm) is positioned so its focal point is at the center of the scattering volume which is within the sample cell. An exact positioning of this lens is critical only to within 1_2 cm of the focal length because the scattered beam has a small divergence angle. A large aperature, 2.0 cm in diameter, is placed between the first lens and the front interferom— eter mirror to reduce light reflected off the front surface of the front mirror and prevent it from re-entering the beam path of the instrument. A second lens (f = 100.0 cm) is positioned behind the interferometer and its focal point is placed on a small pinhole 2.0 mm in diameter, directly in front of the photo— multiplier tube. Thus, only a central spot of the complete 52 ring pattern is detected. This will give a finesse at least two times better than if the complete pattern is used. The main reason for the increased finesse is that the two mirrors are not perfectly flat over their complete surface nor per— fectly parallel. If only a small section of the surface of the mirrors is used to determine the interference pattern, as when a small spot is isolated from the ring pattern, a closer approximation to the ideal case of perfectly flat and parallel mirrors is possible, thus allowing a better finesse. d-Alignment A complete experimental alignment procedure is provided in the appendix and is based on an extra stationary laser which is used only for alignment of the optical components. We use a Spectra-Physics model 125 helium neon laser which has a well defined, moderately intense beam to initially allow an interlocking set of pinholes to be accurately aligned. Then, sequentially, the mirrors and lenses are placed in the Optical train and finely adjusted by matching the path of the reflected light from their front surfaces with the path of the incident beam. Only by a very careful alignment are good reproducable spectra possible and then only when realignments are made between each run. f—Detection and Recording To detect the radiation, an E.M.I. 9558B photomultiplier 53 tube with a S-20 frequency response is supplied with —l,200 volts by a regulated high voltage power supply from KEPCO. A low dark current is maintained by exposing the de— tector only to the low intensities of the scattered light, not the room light, and continuously cooling the cathode to only —10° C with a Products for Research, Model TE-lO4TS, refrigerated chamber. When lower cathode temperatures are used, moisture condenses on the front window of the detector and decreases the signal intensity obtained. The signal from the detector is then fed into a preamplifier with a variable current range, usually set at 0-10-9 amps, and damping con- trol set at 30%.of the maximum value. The signal is then fed into a Keithly Model 417 picoamplifier, and finally to a Sargent Model SRC strip chart recorder where the final spectra is recorded. Specifications for each component are obtained from the manufacturers and from the literature [51]. 2-Scattering5Cell a-Shape A regular, rectangular light scattering cell with parallel entrance and exit windows is excellent for Brillouin scattering at 90 degrees to the incident beam since this shape allows the scattered light beam to be observed with a minimum of extraneous reflections from the surfaces of the cell. A cylindrical cell at least one inch in diameter is also satisfactory and much less expensive if the solutions are to be degassed and sealed. A design we incorporated 54 consists of a precision bore tube one inch in diameter and sealed to a 15 mm Fisher—Porter joint. A neck was formed in the tube about two inches below the joint while the bottom, about five inches below the joint was sealed and flattened for a base as seen in Figure 3. This cell was attached to a similar Fisher—Porter joint on the filter with a teflon gasket making an air tight system. However, for scattering angles other than 90 degrees, an entirely different cell design is required. We have noted that for a scattering angle other than 90 degrees with a standard cylindrical light scattering cell, a small amount of light from the incident beam is reflected back from the glass—air interface at the point where the incident beam exits the cell. This reflected beam produces an extra set of small Brillouin peaks in the spectra with a frequency shift corresponding to the complementary scattering angle, while the Brillouin shift A” , is dependent on the scattering angle as seen from the previous equation 14, These small peaks can be seen very faintly in a vertically polarized, VV, benzene spectrum at 45 degrees, Figure 4, and at 135 degrees, Figure 5, where the main Brillouin peaks and the reflected peaks exchange positions. In comparison with these two spectra, a Brillouin spectrum of benzene at 90 degrees, Figure 6, does not contain these separate reflected peaks. When the frequency shift of the small peaks is plotted with an equation for the complementary angle, 55 omaaflm 6cm Hmcflmflnolaamu mcflnmpumom Havauocflawo .m mHsmHm TI 14] _. :plllv_ J U \\ s5 56 mocmsqwhm m> m a H m I m a a a umam a q a a u m mam mcflumuumo mmmumwn mg mcmucmm mo ESHDU m m a :0 H Hm.¢ musm - p .m gig? I§¥§§i§i 57 wosmsqmum m> huflmcmusH .mamc¢ mcflnmpumum mmmnmma mmalmcmucmm mo Esnuommm QHSOHHHMm .m musmflm ii 58 wocmskum m> muflmcmucH .wamcfi mcflumpumom mmmummm Colmcmuamm m0 Esugowmm QHDOHHHHm .w musmflm JIil€fskT2$i$l§$f{%{lé+?}<}5$++}2I5é§1iS43J34§?9iii3453iK{5IP5JkiSéiififlfilliii§ififléJ€$ 59 AV reflected: 2:; .- sin( (180 " 9)/2 E39. the points correspond very well with the regular angular dependence as seen in Figure 7, indicating these peaks are from the same scattering volume. The intensity of small reflected Brillouin peaks are approximately 5%.of the inten— sity of the main Brillouin peaks. This value corresponds well with a reflected intensity of 4%, calculated from the index of refraction for the interfaces benzene-glass plus glass-air perpendicular to the incident beam. Irregularities in the surfaces of the scattering cell and difficulties in accurately measuring the intensities of the small peaks easily account for the small difference between these two values. These small peaks, which are due to the reflected beam, affect the use of the spectra by modifying the experimental ratio Jv in two ways. First, these peaks can be separated from the regular Brillouin peaks by selecting a scattering angle other than 90 degrees. This separation is impossible for the central peak, therefore an accurate Jv ratio must include the intensity from all four Brillouin peaks plus the central peak. Second, the location of the base line is difficult to determine since the extra two peaks generate a large uncertainty in its position because a consistent flat portion cannot be obtained. For these reasons we were obliged to design a different type of cell to eliminate the problems mentioned. 6O Dnmflq Umuomammm tam ucmpHUCH mo wocmosmmma “wasmcd m>HumHTMuwcmNcmm Eoum mDMHnm woswswmum :flsoHHHum .n wusmflm 82:25 3 ed no No 3 no to no «o ..o o q u . m u u u q q A O .. 2.0 \© © .. 93 ©\ \ AU\\\mW 1 OMAvmflNII ©\ Ad \ © @\ AV\\ l OVAV .9 © 20.53:; xo