ACOUSTIC BEREFRINGENCE 0F LIQUID POLYSTYRENE Thesis for the Degree of M. S. MICHIGAN STATE COLLEGE Donald Alan Hall .1954 J WWW! WU] 11111111117111 L g_.1 t 0 1:23;"- 3 1293 01693 8791 ’rene PL LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINE return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 1!” WWW.“ ACOUSTIC BIREFRINGENCE 0F LIQUID POLYSTYRENE by Donald Alan Hall A.Thesis Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Applied Science in partial fulfillment of the requirements for the \ degree of MASTER OF SCIENCE Department of Physics 199+ ACKN OWLEDGEIVEN T I wish to express my sincere thanks to G. S. Bennett for introducing me to the field of Ultrasonics and to. Professor E. A. Hi‘edemann for his encouragement and assist; ance toward the completion of this problem. I wish also to thank W. Gessert in particular for the use. of' equipment and the others in the ultrasonic laboratory for their many (B'OULW 0;. M timely suggestions. 331299 TABLE OF CONTENTS I. INTRODUCTION . . . . . . . . . . . . A. historical . . . . . . . . . . . B. Theory . . . . . . . . . . . . C. Earlier Experimental Studies . . . . . II. EXPERIMENTAL METHODS USED IN THE PRESENT STUDY A. Method Using Wollaston Prism . . . . . B. MfitnOd USing P hatomQt” e e e o o e I I I . EXPERIMTAL DAT‘ O O O O O O O O O 0 IV. DISCUSSION OF RESULTS . e . . o ». o . . V C 8.01me O O O O O O O O O O O O O ‘ VI. LITERATURE CITED . e e . o . . . o e O\I\)HH I. INTRODUCTION A. Historical Brewsterl was the first to investigate the effect of a transparent solid on polarized light. It was shown later by Fresnel? that the observed effects could be attributed to double refraction, or, as it is also known, birefringence. This double refraction was found to be due to strains in the solid. In 1866, James c. Maxwell3 attempted to: determine if a state of strain existed in a viscous liquid that was in motion by observing the effect on polarized light when passed through the liquid. He as unable in the first attempts to. detect an occurrence of double refraction. At a later date the experiment was repeated and Maxwell” observed a relation between the indicces of refraction n, the viscosity" , and the velocity gradient. This relation is es .D . AM=“¢-V\o M 5‘ ”L (1:) 2 the Maxwell constant of the liquid 3 the velocity gradient. where W J? :< The phenomena of double refraction has also been observed when a liquid is subjected to sound energy. The frequency of the sound is in the ultrasonic range. The effect was observed in solutions of iron oxide and vanadium pentoxide by Cookson and Osterberg5 and by Lucas6 in liquids such as linseed oil and castor oil. Several theories of acoustic double refraction have been proposed, each differing significantly from.the others. Lucas6 bases his macroscopic theory on the stress in the . liquid and relates the acoustic double refraction to the dynamic or flow double refraction of Maxwell. Peterlin7 givesa microscopeic theory based on particle orientation. The particles are oriented by the velocity gradient giving rise to the whole liquid becoming anisotropic. A.third theory is given by Oka8. This is also a microscopic theory which attributes the double refraction to an orientation of colloidal suspensions by the sound pressure in the liquid. Due to the nature of the liquid used in this experimental work, the theory given by Lucas, which does not depend upon colloidal suspensions or particles, will be Outlined e B. Theory Lucas6 makes use of the normal and tangential stress relations for viscous liquids as given by the NaviernStokes hydrodynamic equations. These-are: N= "A6+2\‘%—-‘;-P T':‘L%‘T’1i+%é N1: A9*1‘\—"P 71.103: + 95'; a) N5: Ae‘vb‘fi" T331 31* Six = the Normal stresses, T 5 the Tangential stresses, “8 the pressure, at Vt are two viscosity coefficients from.Stokes theo 9: its 4 9.: ins; , the divergence of the particle °" &“\ 3". velocity. Applying these equations to a plane progressive longitudinal wave with velocity v traveling in the 0)! direction, the expressions for the normal and tangential stresses become: = )vI-i All. .. N1=Nt="§-‘—:‘P (3) where the velocity components in the OT and 02 direction are zero. This is then equivalent to a liquid which is under an isotropic pressure given by P — A 3;: - (to The liquid is also: experiencing a normal force acting in the OX direction as shown in Figure l. $9»- rs: P "x‘t-P “343-? 2 Q“ g. 1‘53? 0 tag“ P X a- Fig. 1. Pressure and Force acting on Liquid. The normal force can be: expressed as e M. = 2 fig; (5) where ‘l is the usual value for shear viscosity. This force gives rise to an anisotropic condition in the liquid and can be found if‘ the velocity gradient, 3—: , in the direction of propagation can be determined. It will be possible to obtain an expression for the velocity gradient from the. viscous decay of the wave along the 0X direction. From the formula for the displacement u we have u=ue e’“"smw(T-63 (6) or an __ “W 2-2 5; - «(co-aw 1'43) fi_u.we e men ’53 < 1 (7) In general 2_ 7.1! a | . v " TC >>°‘ warmer wanton To "6 (8) Therefore the velocity gradient as a function of the maximum displacement will be in the first but sufficient approximation d“ _. w‘ -‘X . A) w“ . 6% -u, 7 e SIN ”(I 5) p u. V (9) The sound intensity can also“ be expressed as a function of the maximum displacement and can be written as I: 719‘! u.“ 0.0“ where (3 is the density. Combining equation (9) and equation (10) yields the amplitude of the particle velocity (lo) gradient which is g. gun», a...» 9.1 (é$)M‘* T \7 (—5- . (11) By substituting this expression for the velocity gradient into the formula for the normal force we obtain the amplitude of the normal staresses as 1.! ( ) su‘naag 1‘5? =1\€' 9—5 12 which is the anisotropic- force present in the progressive wave. From this w. see that for a liquid of negligible viscosity or for elastic waves at low frequencies, the anisotropic stresses disappear. Lucas shows that the double refraction that is caused by a laminar flow of the liquid can also be analyzed on the basis of an anisotropic normal force. He then compares the double refraction thus obtained with that from the acoustic case. From Stokes, the tangential stress on a liquid moving with laminar flow is T = 1 an; , ‘5 a1 (13) According to elastic theory, this tangential stress can be replaced by two perpendicular normal stresses which act along OE and 02. This is shown in Figures 2. u. 5... 5L {éS—t‘ 131‘ Fig. 2. Forces Acting on 1.1qu Moving with Laminar Flow. These forces are of magnitude ‘31.: . The anisotropy that results- from these two. forces can be replaced by a single normal stress acting at #50 to the velocity of the flow u, d is of magnitude an $5145:ng ' (It?) Refering back to Maxwell's relation for the double refraction arising from laminar flow of a liquid, equation I, we can obtain an expression for the double refraction in terms of the normal force givm in equation (1%). 0r- AVL-‘KO'V‘O '3 %- SN4‘ e (15) By analogy, then, the acoustic double refraction can be related to the acoustic normal force of equation (12). The acoustic double refraction then becomes _. — as I (16) (AkSMM " % (“l-3m — \M V EFT where 3 viscosity, 3 Maxwell's constant of flow double refraction, §~angular»frequency 3 21rf, a frequency of the sound wave, e'wave velocity, §-acoustic intensity, The optical axis is given by the plane of the wave front, i.e., it coincides with the velocity of the liquid whereas in the case of laminar flow, as in Maxwell's measurements, it is at 15° to the direction of the velocity of the liquid. Summing up the predictions that can be made on the ‘bHdHEE’A basis of the Lucas theory expressed in equation (16), we should find that the double refraction should be propor4 tional to the following: 1. the frequency of the sound wave; 2. the Maxwell constant; a. the viscosity of the_liquid; . the reciprocal of the temperature; 5. the square root of the sound intensity which is proportional to-the sound amplitude. C. Earlier Experimental Studies' Lucas pointed out that the usual methods for the measurement of the birefringence cannot be applied in the Special case of acoustic birefringence. Measurements by means of compensators or quarteréwave plates are based on the comparisons of beams of elliptically polarized light 'which have the same degree of'ellipticity. Therefore ILucas used another method. The liquid is brought between 'two crossed Nicols so that no light emerges from the analyzer. If an ultrasonic field is applied to the liquid, light again emerges. Lucas derived that the intensity of the emergent light is directly proportional to the sound intensity and to the square of the ultrasonic frequency. Although Lucas reports on some experimental observations, he did not make any quantitative measurements. Some quan- titative studies have been reported by Zvetkov, Mindlina, and Makarov9. The results of these authors appear to prove the Lucas theory. Lucas developed his theory for the case of progressive waves. The experimental set-up of the Russian authors did not exclude the possibility of stationary waves. Zvetkov, Mindlina, and Makarov used the following set-up. For a sound source they used a piezoquartz fastened on a brass plate which served as a holder and at the same. time, as one of the electrodes. Aluminum foil pasted on the other face of the quartz acted as the second electrode. The quartz was placed directly into the liquid to be- investigated. The cell to hold the liquid was made of glass. Windows for the light to pass through the liquid were of strain-free cover glass. The quartz was fed from an oscillator with a frequency range of 600kc/s to lIOOOkc/ s. Zvetkov, Mindlina, and Makarov measure the birefringence of the liquid by measuring the intensity of the; emergent light after it has passed through a polarizing Nicol prism, the liquid cell, and a second Nicol prism that has been crossed with the polarizer. A schematic drawing of their apparatus is shown in Figure 3. o [TE-II- Fig. 3. Apparatus used by 2vetkov, Mindlina, and Makarov. In Figure 3, S is the light source wnich was the filament of an incandescent lamp; 0 is a collimating lens; B the glass cell which contained the liquid under investigation and the quartz K; P is the polarizing Nicol whose principle section was set at an angle of 45° with the ultrasonic wave front; W is a Wollaston prism; and 4.13 the analyzing Nicol. The scale of angular displacement on the analyzing Nicol was divided into 0.01° divisions. The image of the filament was observed through the telescope F. The wollaston prism was used in the following manner. It was mounted so that its principle section was in the same plane as the polarizer P. This would cause-one of the two images seen in the telescope to be extinguished. The analyzer was then rotated until the second of the images was extinguished. The analyzer is then said to be in the zero position. When an ultrasonic field was introduced into the liquid from the piezoquartz, the liquid became optically anisotropic. This caused the first image, the one that had been extinguished by placing the principle section of the wollaston parallel to the plane of polarization from P, to reappear. The brightness of this restored image could then serve as a measure of the birefringence present. The use of the Wollaston prism to measure the birefrin-L- gence is based on the assumption that for small values of An the intensity of the light beam giving the second image, the one extinguished by the analyzer, is not changed when the sound is present in the liquid. Let the intensity of this beam be; Io. When birefringence is present and the first Wollaston image has reappeared with intensity I, the: analyzer can be turned until the second Wollaston image is of equal intensity, or, I 3 10. The angular displacement of the analyzer from the zero position to the point where the images are of equal brightness is designated by o< . From the relationship I g tan20( (17) I0 it is possible to obtain a working formula that, will relate the angular displacement e( and the birefringence An. The instantaneous brightness of the first image is: equal to {sheaf-(1‘5) s 1, am E50; (K9 0‘0) (18) where 3 is the optical: path difference of the rays, l is the length of the light beam in the acoustic field, ?\ is the light wave length, and (116 -'~ no): is given by the 10 following equation Vie _“°= afl‘fi;AM‘\., SIN z-,-,§("'— '3’) (19) where c is the velocity of the acoustic wave, a the fre- quency of the wave, and A the maximum amplitude of the wave. Since the observed path difference does not exceed a few per cent, one can make the approximation that s-in2(v{)e(«£\"' in equation (18) so that the mean value of the brightness I is found tO‘-_b6 I'I.(1'.i'-:)‘Au" (20) where An is the mean quadratic birefringence. Combining equation (17) and equation (20} we obtain An. = 3%.. +0.04 0( . (21) The birefringence-can thus be determined by measuring the: angular displacement of the analyzer. 11 II. EXPERIMENTAL METHODS USED IN THE PRESENT STUDY For the investigations reported in this thesis, two different methods were used. The first method was a varia-1 tion of the method used by the Russian authors; the second method was based” on photometric measurements. A. Method Using Wollaston Prism As the experiment of Zvetkov, Mindlina, and Makarovg did not exclude the possibility of stationary waves, it seemed interesting to investigate the case of stationary waves. In this experimental arrangement use was made of Barium Titinate: ceramics: as a source of ultrasonic energy instead of piezoquartz. The choice was made so] that the output characteristics of the oscillator used for the electromotive driving voltage and current could be utilized. This oscillator is a U.S. Navy radio frequency transmitter, Model ATD, with a frequency range of fill-Okc/s- to: 15,000kc/s. The output circuit of the transmitter was modified so. as to conform to the Pi-lsection matching circuit as described? in the dRRL Handbooklo. This change was necessitated due to the low impedance of the BaTi transducers and with this circuit it is possible to obtain maximum transfer of rf energy from the oscillator to the transducer. The voltage necessary to operate the transmitter is obtained from the dynamotor that was supplied with the transmitter. Fig. 4. Tank to Hold Liquid and Transducer. 12 The dynamotor in turn obtained its operating voltage from the 28 volt DC supply in the building. The cell or tank which was used to hold the.liquid and the transducer is shown in Figure #. The body of the Vitank.is made from a section of‘3 cm. copper wave guide. Its inside dimensions are; I cm. 2:2.3 cm. 1.6.5 cm. long. Two openings, I cm. x:# cm. are cut in the sides of the wave guide. These are covered'with optically' strainéfree glass which act as windows for the light beam. A piece of brass, 5 cm. x'S cm. x:0.3 cm., is soldered on the back of the tank. A.hole, 1.6 cm. in diameter, is drilled into the brass plate to: a- depth of 0.15 cm. A second hole is started at this depth of 1.3 cm. diameter which continues into the plate for 0.075 cm. A.third hole of 1.0 cm. diameter-starts at this point and continues through into the tank. Figure 5’is a crossbsection through the brass plate and the wall of the copper wave guide which shows the details of the holes just described. , l.6 cm. .4 l.3 cm. l.0 cm. Brass Plate / ' / /9¢¢¢éé Copper Wave Guide Fig. 5. Section Through Tank for Liquid and Transducer. 13 The BaTi ceramics used as transducers are flat circular disks, 1.28 cm. in diameter and 0.25 cm. thick. These dimen- sions are for the ceramicszith a fundamental frequency of lOOOkc/s. The ceramic disk fits into-the 1.3 cm. diameter' hole in the brass plate-and is held in place by a Neoprene rubber "0* ring which is placed in the 1.6 cm. diameter hole. ‘A second brass plate, with the same dimensions as the plate soldered to the copper wave guide, is then placed on top of the "O" ring. It is held to the first plate by means of four' machine screws, one in each corner. A.l.2h cm. diameter hole is drilled through the.second plate at its center. Through this hole is passed a spring loaded brass contact that bears against the silver plated surface of the BaTi ceramic. This acts as one of the electrical connections to the transducer. The reverse face of the ceramic is also silver plated, and, since it is in direct contact with the first brass plate, the whole metallic tank acts as the second contact. Two "bananna" plug terminals are mounted.on.the second brass plate. The output of the oscillator is fed to the BaTi ceramic through these terminals. Measurement of the birefringence of the-liquid was made in the same manner as described by avetkov, Mindlina, and Makarov9. Figure 6 is a schematic drawing of the optical arrangement that was employed. 11+ «om 0: ONE-JEN m S L' Al F LZAZ L3 NI To W N2 Te Fig. 6. Optical Arrangement for the. Measurement of Acoustical Birefringence. In Figure 6, S is the light source, a high pressure mercury arc lamp or a tungsten filament lamp. used for a white light source; L1 is a focusing lens; ‘1 is a limiting aperture; F is an optical filter which passes the mercury green line of $60.7 8 which is not used with the white light source; L2 is a second focusing lens; 5.2.13 a circular aperture which acts as the effective source; 1.3 is a collimating- lens which renders the light parallel; N1 is: the polarizing Nicol prism; Ta is the tank containing the liquid and the trans-Q d‘ucer; w is the Wollaston prism; N2 is the analyzing Nicol prism; and Te is the viewing telescope. Measurements of the angular displacement of the analyzer Nicol and the corresponding transducer current are given in Table I. B. Method Using Photometer For further verification of the Lucas theory, use was made of an experimental set-up:- of W. Gessert, working in our own laboratories. In addition to a similar optical arrange-e ment, with exception of the Wollaston prism, as shown in 15 Figure 6, there was included an American Instrument Company photometer in place of the viewing telescope. The photometer used a very sensitive photo-multiplier tube. The polarizing Nicol was set at #50 to the ultrasonic wave front. Then the analyzer Nicol was rotated until a minimum reading was obtained on the photometer. This reading was then used as a “zero” point. When the ultrasound was turned on, the light reading on the photometer increased due to the birefringence in the liquid. The difference between the "zero" reading and the reading with the sound turned on is then recorded. At the same time the quartz current is measured. These readings are given in Table II. Of note is the fact that this experimental setéup used a piezoquartz as a source of ultrasonic energy, Operating at lOOOkc/s, as compared to the BaTi ceramics that are used in the former method. Use of the piezoquartz compares to the experimental method of Zvetkov, Mindlina, and Makarov. As a point of interest, a.trial was made using the method of the Russian authors and the quartz source of ultrasonic energy. The measurements thus obtained are given in Table III. is a qualitative check on the Lucas theory, Measurements cxf acoustic birefringence were attempted by the Wollaston prism method at four frequencies; lOOOkc/s, 2000kc/s, 2500kc/ s, and 5000kc/ s on solutions of 100%, 80%, and 60% concentra-1 tion. The polystyrene was diluted, by volume, with carbon- tetrachloride in order to reduce the dynamic viscosity. 16 The results of these observations are given in Table IV. The liquid that was investigated is poly-alpha?- methyl-styrme. The sample was obtained from the Dow Chemical Company and is designated by them as Dow Resin 276-1112. This resin is primarly composed of tri—polmner alphaémethyl-Lstyrene. 17 III. EXPERIMENTAL DATA Table I BIREFRINGENCE MEASURED WITH WOLLASTON PRISM ULTRASONIC SOURCE, BaTi TRANSDUCER Frequency : 1000kc/s Sourcel Current Rotation (rf amps) (degrees) white 0.% 1.0 white 0.6 #.0 white 1.5 6.25 white 1.3 10.0 white 0.3 2.5 white 0.3 2.0 White 0. 3 2.0 White 003 5O 5 green 1. 11.25 green 1.# 16.0 green , 1.h 11.25 green 0.2 1.0 green 0.2 1.0 green 0.2 1.0 grm O. 2 100 green 0.2 1.0 green 1.8 2.0 green 1.8 2.0 L'White" indicates that a white light source was used easel "green" indicates that a filtered mercury source was tuseni. 18 Table II BIREFRINGENCE MEASURED WITH PHOTOMETER ULTRASONIC SOURCE, PIEZOQUARTZ Frequency: TOOOkc/s Cede Light, Crystal Current Current Squared (rf me) (:10'” amps) G-l 2.0 75 56 1.0 95 90 3.0 125 156 5.0 Th5 ' 209 5.0 165 271 6.0 175 305 G-2 1.0 80 6% 2.0 110 121 3.0 1&0 195 .0 170 288 6.0 195 8 3.0 220 2 .0 2H5 598 8.0 250 612 2.5 1t6 213 3'0 £36” 2%? .0 ' t ' 6.? 210 $13.3 7. 2 2 9.0 237 606 9 . 0 21+? 606 13.0 265 700 15.0 280 782 17.0 300 900 19.0 310 960 G-#a 1.5 100 100 5.0 150 225 7.0 180 322 9.0 180 £32 9.5 200 0 11.5 220 #82 11.0 255 62k 19 Table II continued Code Light Crystal Current Current Squared (rf ma) (3:10")+ amps) G-ha 7 - 8 200 #00 10 - 12 235 £30 11.5 233 2 16.5 265 700 G-JH: 10.0 120 11+» 12.0 192 202 29.0 190 3,860 6.0 220 ‘ 5 .0 2+0 575 51+.0 267 713 11+.0 160 256 20.0 160 256 33.0 208 L106 31.0 215 l+55 5.0 215 L155 2.0 250 625 no.0 265 700 I+9.0 265 700 G‘-5 2.0 115 210 3.0 170 288 .0 195 80 5.5 210 0 6.0 228 5140 7.5 2% 575 10.0 265 700 Table III BIREFRINGENCE MEASURED'WITH WOLLASTON PRISM ULTRASONIC SOURCE, PIEZOQUIRTZ Frequency: lOOOkc/s Light Source: High Pressure Mercury Arc Filtered for 5‘+60.7 A Current Rotation (rf ma) (degrees) 1201 0.75 139 1. 50 152 1.50 182 1.75 230 1.75 Table IV QUILITLTIVE MEASUREMENTS 0F BIREFRINGENCE IT VARIOUS FREQUENCIES AND CONCENTRATIONS ULTRASONIC SOURCE, BaTi TRANSDUCER Light Source: High Pressure Arc Filtered for $460.74 Frequency Concentration Birefringence (kc/s) (per cent) Present 2000 100 Yes 2500 100 Yes 50001 100 No 1000 80 Yes 2000 80 Faint 5000 80 Na 1000 60 ‘Yes 2000 60 no. 5000 60 No 3an .2. 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