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Visco-elastic properties of aluminum soap-hydrocarbon gels Flynn, James Thomas 1951

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VISCO-ILASTIC PROPERTIES OF ALUMINUM SOAP - HYDROCARBON GELS by JAMES • THOMAS FLYNN A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF Applied Science  in the Department 1 of Physics We accept this thesis as conforming to the standard required from candidates for the degree of MASTER OF Applied Science Members of the Department of Physio3 THE UNIVERSITY OF BRITISH COLUMBIA September, 1951 ABSTRACT Measurements o f v e l o c i t y o f p r o p a g a t i o n and damping of t r a n s v e r s e s o n i c waves have been made f o r aluminum soap-h y d r o c a r b o n g e l s . The f r e q u e n c y range c o v e r e d i s 100 t o 1000 c y c l e s p e r second. The e x p e r i m e n t a l r e s u l t s have been f i t t e d t o t h e o r e t i c a l m e c h a n i c a l models. The m e c h a n i c a l * b e h a v i o r o f t h e g e l s i n v e s t i g a t e d can be a p p r o x i m a t e d by a Retarded M a x w e l l Element w i t h a r a t i o o f p a r a l l e l t o s e r i e s v i s c o s i t y o f about 0.01. The r i g i d i t y o f t h e model i s o f t h e o r d e r o f 10 dynes p e r square c e n t i m e t e r and t h e s e r i e s v i s c o u s component o f t h e o r d e r o f 1 t o 10 p o i s e , g i v i n g a r e l a x a t i o n t i m e o f t h e o r d e r o f 10 seconds. I l l ACKNOWLEDGEMENTS The a u t h o r w i s h e s t o e x p r e s s h i s a p p r e c i a t i o n : To t h e Defence R e s e a r c h B o a r d f o r g r a n t i n g l e a v e o f absense and f i n a n c i a l a s s i s t a n c e . To D r. E . J . W i g g i n s , u n d e r whose d i r e c t i o n t h i s work was s t a r t e d a t t h e S u f f i e l d E x p e r i m e n t a l s t a t i o n , f o r h i s c o n t i n u e d a d v i c e and encouragement. To D r. A.M. C r o o k e r , o f t h e U n i v e r s i t y o f B r i t i s h Columbia, f o r t h e l o a n o f o p t i c a l components and f o r much h e l p f u l a d v i c e . To Dr. F.A. K a e m p f f e r , o f t h e U n i v e r s i t y o f B r i t i s h C o l umbia, who a c t e d as s u p e r v i s o r f o r t h i s work a t t h e U n i v e r s i t y . I TABLE OF CONTENTS I INTRODUCTION Page a) General 1 b) Material 1 c) Mechanical Properties of High Polymers The Use of Mechanical Models 3 II PURPOSE AND THEORY OF THE EXPERIMENT 10 III APPARATUS AND PROCEDURE a) General 16 b) Apparatus 18 c) Methods of Measurement Method A 24 Method B 26 d) Material 28 IV EXPERIMENTAL RESULTS a) Digel and Octoic Acid i n Gasoline 30 b) Octal and Octoic Acid i n Benzene 33 c) Octal and Octoic Acid in Gasoline 35 V DISCUSSIORZ 40 VI BIBLIOGRAPHY 44 II LIST OF ILLUSTRATIONS Figures Page 1. Maxwell Element 4 2. Vo&gt Element 5 3. Strain-Time Relations for simple models follows 6 4. Four Parameter Model with Response Curve n 8 5. Theoretical Curves, G/G and ^ / x 0 vs Log " 15 6. Theoretical Curves, G/^ G and y'/y v s Log " 15 7. Photographs of strain patterns " 18 8. Schematic for Optical Components n 18 9 Strobotron Trigger c i rcui t " 23 10 G/G vs Log f for Digel in Gasoline " 31 11 Log G vs Log concentration for Digel in Gasoline Log G vs 1/T " 32 12. Log^ vs Log f for Octal in Benzene " 33 13 Log ^ vs Log f for Octal i n Benzene, variat ion with the temperature " 34 Plate I Apparatus for Studying Propagation of Transverse Sonic Waves fol lows43 1 I. IHTRODUCTIOH a) General Dispersions of aluminum soaps in hydrocarbons such as benzene, gasoline, etc. can exist in a number of different forms ranging from mixtures of discrete swollen lumps of soap i n the solvent through visco-elastic gels( or j e l l i e s ) to completely mobile solutions. The gel state i s of particular interest because of the rather special visco-elastic properties. The physical properties of these gels di f fer markedly from those of ordinary l iquids and render them useful for a number of applications, eg. lubricating greases, pharmaceutical preparations, e t c . . During the last war aluminum soap-gasoline gels were used for incendiary bomb charging and flame thrower fuels A flame thrower fuel should have a high viscosi ty during f l ight to prevent shattering and a low viscos i ty during handling and operation of the flame thrower to prevent f r i c t i ona l losses, permit low f i r i n g pressures e t c . . E l a s t i c i t y i s also believed to be important in preventing shattering during f l i g h t . However, the precise effeot of the properties on the f i r i n g performance of a fuel i s not completely understood at the present time and w i l l not be discussed here. b) Material The physical and chemical properties of aluminum soap -2. hydrocarbon gels have been discussed by Rideal and others • They are quite stable and are characterized by a re la t ive ly high viscos i ty and r i g i d i t y . The actual physical properties 2 obtained depend on the chain length of the soap, the solvent, the presence of peptizing agents, temperature, e tc . . The structure ie thought to be two phase. One phase consists ©f a network of solvated material, the second is a solution of the soap i n the solvent and f i l l s the Interstices of the network. The network may consistjfeither of re la t ive ly r i g i d crystals or of elastic long chain molecules. Junctions i n the network are formed by mechanical interlocking, by hydrogen bonding, true chemical l inks , e t c . . a Sheffer has carried out viscosity and osmotic pressure measurements on dilute benzene solutions of aluminum dicaprylate, dilaurate, dimyristate, dipalmitate, distearate, and monostearate. He concludes that the soaps are polymers of high molecular weight ( 60,000 to 900,000 ) whioh are formed by weak intermolecular l inks , probably hydrogen bonds. When preparing a gel i t i s usual to add a peptizing agent to promote swelling and to increase the so lub i l i ty of the soap. Peptizers are compounds with strong co-ordinating properties eg. amines, alcohols, phenols, fatty acids e tc . , and the ir action i s a breaking of links i n the soap chains by a preferential l inking with the peptizer. After solution i s complete the addition of more peptizer w i l l of course lower the viscos i ty by reducing the chain length. It i s very d i f f i cu l t to exactly duplicate conditions tt manufacture of the soap, so the molecular weight of the soap varies considerably from batch to batch. Water is a strong peptizer so that i t and other impurities have a very marked 3 e f f e c t on t h e g e l l i n g p r o p e r t i e s o f t h e soap and on t h e a g i n g o f t h e r e s u l t i n g g e l . F o r t h e s e r e a s o n s i t i s e x t r e m e l y d i f f i c u l t t o g e t c o n s i s t e n t e x p e r i m e n t a l r e s u l t s even when g e l s a r e p r e p a r e d w i t h t h e g r e a t e s t c a r e i n t h e c h o i c e o f m a t e r i a l s and i n m i x i n g and s t o r a g e . c) The M e c h a n i c a l P r o p e r t i e s o f H i g h Polymers The Use o f M e c h a n i c a l M o d e l s . The use o f m e c h a n i c a l ( o r e l e c t r i c a l ) models t o r e p r e s e n t t h e m e c h a n i c a l b e h a v i o r o f v i s c o - e l a s t i c m a t e r i a l s has been ^ S c o n s i d e r e d i n d e t a i l by A l f r e y , B u r g e r s and o t h e r s . The b e h a v i o r o f p e r f e c t l y e l a s t i c m a t e r i a l s i s d e s c r i b e d by a s e t o f e l a s t i c c o n s t a n t s . I f t h e m a t e r i a l i s i s o t r o p i c t h e number o f c o n s t a n t s r e q u i r e d r educes t o two, Lame's c o n s t a n t s , Young 1 s modulus and t h e B u l k modulus, e t c . . I n t h e i d e a l case t h e e n ergy o f d e f o r m a t i o n i s c o m p l e t e l y r e c o v e r a b l e . A l l t h e work done i n p r o d u c i n g a s t r a i n i s r e c o v e r e d when t h e s t r e s s e s a r e removed. The response o f a Newtonian l i q u i d t o s h e a r i n g s t r e s s i s p u re f l o w , t h e v e l o c i t y g r a d i e n t depending on t h e v i s c o s i t y and t h e a p p l i e d s h e a r i n g s t r e s s . I n t h i s c a s e t h e d e f o r m a t i o n i s n ot r e c o v e r a b l e and a l l t h e work done by t h e a p p l i e d f o r c e s i s d i s s i p a t e d as J o u l e h e a t . Where t h e r a t e o f f l o w i s not a l i n e a r f u n c t i o n o f t h e a p p l i e d s t r e s s , t h e f l u i d i s s a i d t o be non/Newtonian. A l s o i n some c a s e s t h e m e c h a n i o a l p r o p e r t i e s depend upon t h e p r e v i o u s m e c h a n i c a l t r e a t m e n t o f t h e sample; i f f l o w i s accompanied by s t r u c t u r a l changes i t t a k e s a f i n i t e t i m e f o r t h e m a t e r i a l t o r e t u r n t o i t s r e s t e d V//,. 4 s t a t e a f t e r m e c h a n i c a l w o r k i n g . T h i s phenomenon i s c a l l e d t h i x o t r o p y . Many m a t e r i a l s , i n c l u d i n g h y d r o c a r b o n g e l s , e x h i b i t -n e i t h e r p u r e e l a s t i c o r Newtonian b e h a v i o r . T h e i r m e c h a n i c a l b e h a v i o r can o n l y be d e s c r i b e d by c o n s i d e r i n g e l a s t i c and 6 v i s c o u s e f f e c t s a t t h e same t i m e . A l f r e y and Doty have d i s c u s s e d t h e common methods o f d e s c r i b i n g v i s c o - e l a s t i c b e h a v i o r , we w i s h t o c o n s i d e r b r i e f l y t h e use o f m e e h a n i c a l ( o r e l e o t r i c a l ) models. 7 A M a x w e l l element c o n s i s t i n g o f a Hookean s p r i n g i f s e r i e s w i t h a "Newtonian" dashpot d e s c r i b e s t h e b e h a v i o r o f a m a t e r i a l w h i c h undergoes an i n s t a n t a n e o u s e l a s t i c d e f o r m a t i o n and a t t h e same n t i m e f l o w s upon t h e a p p l i c a t i o n o f a s t r e s s , s uch an element i s i l l u s t r a t e d i n f i g u r e (1). S i n c e t h e f l o w deforma-M a x w e l l Element t i o n depends on b o t h t h e magnitude ax& F i g u r e 1 d u r a t i o n o f t h e s t r e s s i t i s n e c e s s a r y t o c o n s i d e r t h e r a t e o f change o f d e f o r m a t i o n o f t h e whole element. I f s h e a r i n g s t r e s s =? S (dynes/cm. 2) modulus o f r i g i d i t y = G (dynes/em 2 ) e l a s t i c d i s p l a c e m e n t =Yt - $/Q f l o w d i s p l a c e m e n t ~ Y3. t o t a l d i s p l a c e m e n t — y~= ^/ + ^2. 5 d r s i s +±<Ui F o r t h e case where t h e sample i s f o r c e d t o s u d d e n l y undergo a g i v e n d e f o r m a t i o n and i s t h e n h e l d a t t h i s c o n s t a n t s t r a i n The s t r e s s decays e x p o n e n t i a l l y w i t h t i m e and t h e r a t i o VIQ. i s c a l l e d t h e r e l a x a t i o n t i m e o f t h e m a t e r i a l . The v o i g t o r r e t a r d e d e l a s t i c element, f i g u r e 2 . , i s used t o r e p r e s e n t t h e b e h a v i o r o f t h o s e m a t e r i a l s w h i c h undergo an e l a s t i c d e f o r m a t i o n upon t h e a p p l i c a t i o n o f s t r e s s b u t r e q u i r e s a c e r t a i n t i m e i n w h i c h t o t a k e up t h e new e q u i l i b r i u m p o s i t i o n . Such a damped o r r e t a r d e d e l a s t i c r e s ponse i s l i k e t h a t o f a s p r i n g s u r r o u n d e d by a v i s c o u s medium. The d i s p l a c e m e n t o f t h e two components must be t h e same and t h e I I V o i g t Element F i g u r e 3 d i f f e r e n t i a l e q u a t i o n d e s c r i b i n g t h e d i s p l a c e m e n t i s F o r a g i v e n s t r e s s , S , t h e d e f o r m a t i o n t i m e r e l a t i o n i s g i v e n by t h e e q u a t i o n & The q u a n t i t y V/Q i s c a l l e d t h e r e t a r d a t i o n t i m e o f t h e element. Upon t h e removal o f s t r e s s t h e sample w i l l r e t u r n t o i t s o r i g i n a l shape a c c o r d i n g t o t h e e q u a t i o n 6 The r e s p o n s e t o a c o n s t a n t s t r e s s a p p l i e d f o r a t i m e i n t e r v a l t± t o t 2 i s i l l u s t r a t e d i n f i g u r e 3. f o r t h e f o u r t y p e s of m a t e r i a l d i s c u s s e d above. Y E l a s t i c Newtonian M a x w e l l V o i g t F i g u r e 5. The e f f e c t o f t h e mass o f t h e m a t e r i a l has been n e g l e c t e d so f a r . T h i s may o n l y be so f o r t h e s t a t i c c a s e , f o r e l a s t i c r e s p o n s e , and f o r s t e a d y s t a t e c o n d i t i o n s i n c o n s i d e r i n g f l o w . Where f o r c e s v a r y w i t h r e s p e c t t o t i m e t h e i n e r t i a l f o r c e s of/ihe m a t e r i a l must be c o n s i d e r e d . E s s e n t i a l l y t h e mass o f t h e elements o f a body d e l a y s t h e r e s p o n s e o f p a r t s remote f r o m t h e p o i n t s o f a p p l i c a t i o n o f t h e s t r e s s e s . D i s t u r b a n c e s produced by t h e a p p l i c a t i o n o f f o r c e s a t p o i n t s o f t h e body a r e p r o p a g a t e d t h r o u g h o u t t h e body by e l a s t i c waves. To c o n s t r u c t a model t o r e p r e s e n t t h e dynamic b e h a v i o r p o i n t o f a m a t e r i a l i t i s n e c e s s a r y t o i n c l u d e / m a s s e s a t a p p r o p r i a t e p l a c e s i n t h e s y s t e m o f s p r i n g s and d a s h p o t s . The use o f m e c h a n i c a l ( o r e l e c t r i c a l ) models t o r e -p r e s e n t t h e m e c h a n i c a l p r o p e r t i e s o f v i s c o - e l a s t i c systems s e r v e s two main p u r p o s e s . The f i r s t , and most o b v i o u s , i s t h a t t h e y p r o v i d e a c o n v e n i e n t and r e l a t i v e l y s i m p l e means o f r e c o r d i n g and u s i n g d a t a o b t a i n e d e x p e r i m e n t a l l y , 7 e s p e s c i a l l y i n t h o s e c a s e s where t h e r e i s no adequate t h e o r e t i c a l b a s i s f o r r e f e r e n c e , s e c o n d l y , i t i s sometimes p o s s i b l e t o a s s o c i a t e t h e v a r i o u s components of t h e model w i t h more o r l e s s e l e m e n t a r y p r o c e s s e s o f t h e m a t e r i a l i t s e l f . There i s obiriLously no r e a s o n t o e x p e c t o n l y one M a x w e l l element o r V o i g t model t o r e p r e s e n t t h e e n t i r e m e c h a n i c a l b e h a v i o r o f a m a t e r i a l . Many d i f f e r e n t i n t e r n a l mechanisms determine t h e m e c h a n i c a l r e s p o n s e , each w i t h i t s own d i s t r i b u t i o n o f p a r a m e t e r s . However, i t i s u s u a l l y p o s s i b l e t o use a d i s c r e t e number o f p a r a m e t e r s , M a x w e l l elements i n p a r a l l e l , o r V o i g t elements i n s e r i e s . The t i m e s c a l e o f an experiment i s i m p o r t a n t i n d e t e r m i n i n g a s u i t a b l e model. By t i m e s c a l e i s meant t h e e f f e c t i v e l e n g t h o f t i m e t a k e n t o make a measurement o f response t o s t r e s s . C o n s i d e r a s y s t e m made up o f M a x w e l l e l e m e n t s i n p a r a l l e l , w i t h t h e c o r r e s p o n d i n g r e l a x a t i o n t i m e s d e c r e a s i n g i n g o i n g f r o m element 1 t o element 2 ete.,^« 2i., ^ z / ^ ^ . Upon t h e a p p l i c a t i o n o f a s t r e s s a l l elements w i l l undergo e x t e n s i o n f r o m b o t h e l a s t i c d e f o r m a t i o n and f l o w . I f t h e d u r a t i o n o f t h e s t r e s s i s l o n g i n comparison w i t h % but s h o r t i n c o m p a r i s o n w i t h 2^ , t h e e f f e c t o f t h a a p p l i e d s t r e s s w i l l be e s s e n t i a l l y d i f f e r e n t on t h e c o r r e s p o n d i n g elements t h o u g h t h e i r t o t a l d i s p l a c e m e n t s a r e t h e same. The response o f element 1 w i l l be m o s t l y due t o f l o w w h i l e t h a t o f element 3 w i l l be m o s t l y e l a s t i c d e f o r m a t i o n . I n o t h e r words, i f t h i s were t h e o n l y o b s e r v a t i o n made, t h e e l a s t i c p a r t o f element 1 and t h e v i s c o u s p a r t o f element 3 would 8 p r a c t i c a l l y n o t be o b s e r v e d . I n o r d e r t o s e p a r a t e t h e e f f e c t s o f a l l s i x parameters i n t h i s model i t w o u l d be n e c e s s a r y t o p e r f o r m a t l e a s t t h r e e e x p e r i m e n t s w i t h t i m e s c a l e s c o r r e s p -o n d i n g t o t h e t h r e e r e l a x a t i o n t i m e s , The e x t e n s i o n t o any number o f elements i s o b v i o u s . I n d e a l i n g w i t h m o l e c u l a r mechanisms t h e r e w i l l n o t be a d i s c r e t e d i s t r i b u t i o n o f r e l a x a t i o n t i m e s f o r each i n d i v i d u a l meohanism but a c o n t i n u o u s d i s t r i b u t i o n about some v a l u e . However, i t i s u s u a l l y p o s s i b l e t o lump them i n t o a s i n g l e element, a t l e a s t a s a f i r s t a p p r o x i m a t i o n . A l f r e y t a k e s t h e f o u r p a r a m e t e r model shown i n f i g u r e 4 as t h e s i m p l e s t r e p r e s e n t a t i o n o f t h e b e h a v i o r o f a polymer i n s h e a r . p r o v i d e s t h e i n s t a n t a n e o u s e l a s t i c r e s p o n s e , G 2 a n d ^ 2 t h e r e t a r d e d e l a s t i c response and 17 3 t r u e f l o w . The s t r a i n t i m e r e l a t i o n s h i p f o r such a s y s t e m i s i l l u s t r a t e d i n f i g u r e 4. The i n s t a n t a n e o u s e l a s t i c response r e p r e s e n t e d b y G^ c o r r e s p o n d s t o t h e i n s t a n t a n e o u s d e f o r m a t i o n o f t h e whole s t r u c t u r e upon t h e a p p l i c a t i o n o f s t r e s s . Suchk. d e f o r m a t i o n would i n v o l v e changes o f d i s t a n c e between n e i g h b o r i n g m o l e c u l e s , s m a l l shape changes of m o l e c u l e s , e t c . . The modulus Gg i s t a k e n t o r e p r e s e n t t h e s o - c a l l e d c o n f i g u r a t i o n a l e l a s t i c i t y . I n t h e u n s t r e s s e d s t a t e t h e macromolecules a r e c o n s t a n t l y c h a n g i n g i n shape but obey a d e f i n i t e d i s t r i b u t i o n l a w. Upon t h e a p p l i c a t i o n o f s t r e s s t h e i r s h apes, on t h e a v e r a g e , w i l l be changed f r o m t h e e q u i l i b r i u m p o s i t i o n . The e f f e c t i s c o m p l e t e l y r e v e r s i b l e t o f o l l o w page 8 9 sinoe the molecules w i l l revert to the ir or ig inal equilibrium position upon the removal of stress, sinoe i t i s reversible i t may be considered as an elast ic effect. The true flow represented b y ^ 3 and the viscous component connected with the oonf igurational e l a s t i c i ty , rj 2 , are governed by the same mechanisms. The biased thermal diffusion of molecular segments . 10 II Purpose and Theory of the Experiment The investigation d^esoribed here i s part of a broader program of experiments planned by the Defence Research Board Suffield Experimental Station. As explained i n Part I, i t i s not expected that the measurements over the re la t ive ly narrow frequenoy range used here w i l l describe the behavior of the material adequately. Other investigators are working of the measurement of properties over time intervals of the order of several seconds down to intervals of the order of 1/50 th of a second. The measuremants to be described here extend the time interval down to about 0.001 seconds. Except for the work of Van Wazer, Goldberg and Sandvik measurements of the physical properties of hydrocarbon gels has been confined to re la t ive ly long time intervals . The Q resonance elastometer developed by Van Wazer et a l permits measuremants to about l/50th of a seoond. A few investigations have been made; by suitable techniques on other materials, but so far as i s known this i s a new region of investigation for flame fuels . It i s of interest not only beoause i t w i l l extend the data available for consideration of moleoular effects but also the time periods involved here are of the the measurements are possibly of direct emperical use i n assessing fuels . The behavior of viseo-elastio systems over short time intervals i s much easier to study by means of periodical ly varying stresses rather than attempting to study very rapid those involved in f i r i n g . Thus 11 transient phenomena. Ferry has described a method of measuring the physical properties of concentrated polymer solutions by means of transverse sonic waves. This method depends upon the material being optical ly clear and stra in birefringent. It was known that hydrocarbon gels are strain ftf birefringent and since i t was thought that s l i p at polymer-instrument interfaces was a possible source of error in direct measurements of stress and strain this method was chosen to begin the investigation of physical properties i n the higher frequency range. It i s hoped that i t w i l l be poss-ible to make measurements in the same frequency range by other methods and investigate the effect of s l i p at some la ter date. It i s known that very pronounced s l ip can be obtained in rotating cylinder viscometers with rapidly increasing rate of shear. However, i t i s not known whether this effect can be produced or observed at higher frequencies. When the mass of the material can be neglected the response to a shearing stress can be completely described to either by a complex viscosity , ?* v/~ <• 7" (l) or a complex r i g i d i t y " g:= Qt i &" (S) With high polymers P o i s s o n « s Ratio i s usually very nearly one half and so E = 3Q- , where E i s Youngs Modulus and Gr i s the shear modulus as before. Since behavior in shear i s generally easier to study experimentally we consider only behavior in shear. The real part of the complex r i g id i ty G* i s equal to the componant of the stress i n phase with the IE strain divided by the s t ra in . The rea l part of the complex viscosi ty JJM i s the component of stress in phase with the rate of strain divided by the rate of s t ra in . I f i s 2ir times the frequency then G^coy 7 , -y"^ etc. The values of G* and G w obtained experimentally generally vary with frequency and mechanical models are made up by combining springs and dashpots to duplicate the experimental results . For a Mastwell element with a spring of r i g i d i t y G and dashpot of viscosity37 and relaxation time t ~ y J G-G <#x**~ (3a) W = yj ! (5b) 7 ( J + u ^ 1 ? 2-For a Voigt element G (4a) V- * (tt) the values being independent of frequency in th i s case. For a retarded Maxwell element, that i s a spring with r i g i d i t y G i n series with a dashpot of viscos i ty yj s and i n para l l e l with a dashpot of viscosity ?7 : / p «S - ( s a ) 13 », D+^Crs+tfi] (5d) In our case we wish to determine the mechanical behavior by studying the propagation of transverse sound waves, we assume that the elastic wave may be described by the following: . _. 7 7 r , ( U ) t _ j (- =^-~ i | A U r U . t € ^ * ° J (6) where u i s the displacement. By analogy with the theory for a perfectly elastic solid one can define a new r i g i d i t y £T for a visco-elastic material such that the differential equation describing the motion i s / , A = &y<* (7) and ^ „ x (8) where v is the measured Telocity of propagation of the wave. The two quantities a and A / x c may then be used to describe the physical properties of the material at the frequency at which they were measured. The relation between G and A / x # , and G« and 77» , may be found by substituting from (8) into the d i f f e r e n t i a l equation 2 ^ /O d u, ^ fQ+ c ooy'j d_a, M*- a x -then putting ^ 14 from equation (8), giving Equating rea l and imaginary parts tS^Q )£<- fro**/' and £' , g W » f i ^ - f l f e ) ' J (9a) 7 ~ ~ [*wU The measured values may then be compared with those for theoretical models by substituting from relations sueh as (3) , (4) or (5) into equations (9a) and (b). For the three models considered this gives: (9b) Maxwell Element Q = Q ^ " i l l - (10a) ^ - * f „ (10b) Yoigt Element G - G + ( l la) X/ - 2 7 T a . y (l ib) 15 Retarded Maxwell Element The above relations are shown graphically i n figure 5. It i s perhaps easier to visualize the quantities G* and *j* also^it i s necessary to convert to these to make comparisons with other types of measurements. For the retarded Maxwell element we define ?^ * VJ® a n d ^ = 9p/* and the relations are: These are i l lus trated graphically in figure 6. . For the Maxwell element the value of G approaches a l imit ing value at high frequencies and the damping i s severe at low frequencies and f a l l s to zero as the frequency i s increased. For the voigt element both G and the damping increase indef ini te ly with frequency. For the retarded Maxwell element the behavior i s l ike that of a Maxwell element at low frequencies and of a Voigt element at high frequencies. to follow page 15 2 • EH / / / 1 rs 1 1 I SL G 1 voia^T^LgyE^ r RETARDED MAXWELL ELEMENT8 / J £ = o.o/ sy y MAXWELL ELEMENT %=0- 005 \j \ 0 LOG «•>*" 2 2 \ i 1 «H / / / RETARDED / VOIOT CLEMCMT \ \ MAXWELL ELEMENT / % « > * I 9 O O 1 I / * i O / M A X W E L L \ ^ ELEMENT \>^~ / / • 0 L O G W f r F I G U R E 5 1 2 t o f o l l o w page 15 16 I I I APPARATUS AND EXPERIMENTAL PROCEDURE a) General To o b t a i n t h e v a l u e s o f G and A./x Q f o r a g i v e n , sample i t i s n e c e s s a r y t o measure t h e v e l o c i t y o f p r o p a g a t i o n of a p l a n e s h e a r wave and a l s o , t o measure t h e damping o f t h e wave. I f t h i s can be done o v e r a w i d e enough f r e q u e n c y range t h e n i t i s p o s s i b l e t o f i t t h e e x p e r i m e n t a l d a t a t o t h e o r e t i c a l c u r v e s and so f i n d a model t o r e p r e s e n t t h e m e c h a n i c a l b e h a v i o r o f t h e m a t e r i a l u n d e r t h e c o n d i t i o n s o f t h e e x p e r i m e n t . The e x p e r i m e n t a l method used was e s s e n t i a l l y as d e s c r i b e d b y F e r r y • T r a n s v e r s e sound waves were produced i n a sample o f g e l by a t h i n vane v i b r a t i n g s i n u s o i d a l l y i n i t s own p l a n e . The d i r e c t i o n o f p r o p a g a t i o n was h o r i z o n t a l and a t r i g h t a n g l e s t o t h e d i r e c t i o n o f o b s e r v a t i o n . Under t h e a c t i o n o f t h e s t r e s s e s produced by t h e s h e a r waves t h e g e l becomes b i r e f r i n g e n t w i t h axes a t f o r t y f i v e d e g r e e s t o th e d i r e c t i o n of p r o p a g a t i o n o f t h e sound wave. I f t h e g e l i s o b s e r v e d between c r o s s e d p o l a r o i d s ' , w i t h t h e a x i s o f t h e p o l a r i z e r v e r t i c a l and t h a t o f t h e a n a l y z e r h o r i z o n t a l , t h e f i e l d w i l l be da r k i n t h e absense o f s i g n a l and w i l l be c r o s s e d by a l t e r n a t e l i g h t and dark bands i n t h e p r e s e n c e o f a s h e a r wave. I n o r d e r t o obs e r v e t h e s e l i g h t and d a r k bands i t i s n a c e s s a r y t o , e i t h e r produce s t a n d i n g waves i n t h e g e l , o r , t o use t r a v e l l i n g waves and s y n c h r o n i z e t h e so u r c e o f l i g h t w i t h t h e s o u r c e o f t h e sound waves. The l a t t e r i s much s i m p l e r and i s t h e method used h e r e . The 17 s p a c i n g between a d j a e e n t l i g h t and dark bands c o r r e s p o n d s t o one h a l f w a v e l e n g t h and t h u s m e asuring t m i s s p a c i n g w i l l g i v e t h e w a v e l e n g h t o f sound i n t h e g e l . However, s i n c e the i n t e n s i t y o f l i g h t v a r i e s a c c o r d i n g t o sin 2§ where S i s t h e r e l a t i v e o p t i c a l r e t a r d a t i o n i n t h e g e l , i t i s d i f f i c u l t t o l o c a t e t h e mid p o i n t o f t h e bands a c c u r a t e l y . An a l t e r n a t e method o f o b s e r v i n g t h e b i r e f r i n g e n c e i s t o i n s e t t a compensator between t h e g e l and t h e a n a l y z e r . I f a B a b i n e t compensator i s used i n t h e u s u a l way w i t h i t s o p t i c a x i s a t f o r t y f i v e degrees t o t h e axes o f t h e p o l a r i z e r and a n a l y z e r , t h e f i e l d i s c r o s s e d by l i g h t and darft bands p e r p e n d i c u l a r t o i t s o p t i c a x i s . I n t h i s case/the r e t a r d a t i o n p roduced by t h e compensator i s e q u a l t o t h e amount o f double r e f r a c t i o n i n t r o d u c e d b e f o r e i t . However, i f t h e compensator i s used w i t h i t s o p t i c a x i s a t a s m a l l a n g l e , «< , f r o m t h e v e r t i c a l t h e n t h e f i e l d i s f a i n t l y i l l u m i n a t e d a l l o v e r and a l t e r n a t e l y c r o s s e d by l i g h t and d a r k bands as b e f o r e . But t h e d i s p l a c e m e n t produced by a s m a l l amount o f double r e f r a c t i o n i n t r o d u c e d b e f o r e i t i s m a g n i f i e d a c c o r d i n g t o t h e r e l a t i o n tan ~ tan A Sin 2o< where $ i s t h e amount o f s t r a i n d ouble r e f r a c t i o n and A i s t h e e q u i v a l e n t r e l a t i v e r e t a r d a t i o n i n t r o d u c e d by t h e compensator . The e f f e c t o f s t r o b o s c o p i c a l l y i l l u m i n a t i n g t h e s h e a r wave i s t h e n t o produce wavy l i n e s c r o s s i n g t h e f i e l d o f v i e w . The d i s t a n c e between s u c c e s s i v e peaks p. c o r r e s p o n d s t o t h e w a v e l e n g t h o f around i n t h e g e l . T h i s method has t h e f u r t h e r advantage t h a t i f i t can be assumed 18 t h a t t h e amount o f s t r a i n double r e f r a c t i o n i s p r o p o r t i o n a l t o t h e amount o f s t r a i n , t h e n m e a s u r i n g t h e d e c a y o f d i s p l a c e m e n t o f t h e l i n e s c r o s s i n g t h e compensator w i l l g i v e t h e damping f a c t o r f o r s h e a r waves i n t h e g e l . photographs o f a 10% g e l a t i n e g e l a t 500 c y c l e s p e r second w i t h no compensator and a t 1000 c y c l e s p e r second w i t h t h e compensator a x i s a t 5° t o t h e v e r t i c a l a r e shown i n f i g u r e 7.. b) A p p a r a t u s P l a t e I shows t h e a p p a r a t u s as used w i t h t h e compensator. The s u p p o r t s t a n d c o n s i s t e d o f t h r e e 1 i n c h d i a m e t e r b r a s s rods screwed i n t o a 1x12x12 i n c h b r a s s p l a t e . Mounted on t h e r o d s were two 3/8 i n c h t h i c k aluminum p l a t e s w h i c h c o u l d be clamped a t any d e s i r e d p o i n t on t h e r o d s . The t o p p l a t e c a r r i e d t h e e l e c t r o m a g n e t i c d r i v e r and vane. The bottom p l a t e c a r r i e d t h e m a t e r i a l t o be t e s t e d i n a r e c t a n g u l a r g l a s s c e l l w h i c h i n t u r n was i n s i d e a t e m p e r a t u r e c o n t r o l b a t h . A l s o mounted on t h e bottom p l a t e was a p i e c e o f s t e e l c h a n n e l on w h i c h t h e o p t i c a l components c o u l d be mounted. The whole assembly r e s t e d on a t h i c k p i e c e o f foam r u b b e r t o damp out e x t r a n e o u s v i b r a t i o n s . The p r i n c i p l e p a r t s o f t h e o p t i c a l s y s t e m a r e shown i n P l a t e I and a r e s k e t c h e d i n f i g u r e 8. The l i g h t s o u r c e was a 631-P s t r o b o t r o n s y n c h r o n i z e d w i t h t h e o s c i l l a t i o n s o f t h e vane. Next was a t h r e e i n c h d i a m e t e r c o n d e n s i n g l e n s t o d i r e c t t h e l i g h t t h r o u g h t h e g e l . The l i g h t was p o l a r i z e d i n t h e v e r t i c a l d i r e c t i o n b e f o r e e n t e r i n g t h e g e l by t h e P o l a r o i d P. The p o r t i o n o f t h e g e l i n t h e p l a n e L-l B FIGURE 8 L-2 O M> O H H a 09 • 19 c o n t a i n i n g t h e g l a s s vane and a t r i g h t a n g l e s t o t h e d i r e c t i o n o f o b s e r v a t i o n was f o c / u s s e d onto t h e compensator by a two i n c h ( f 1.5) camera l e n s . Then t h e image o f t h e s t r a i n e d g e l and B a b i n e t compensator was p r o j e c t e d t h r o u g h t h e P o l a r o i d a n a l y z e r A onto t h e f o c a l p l a n e o f a f i l a r m i c r o m e t e r eye-p i e c e by a sma 11 t e l e s c o p e l e n s . W i t h t h i s o p t i c a l arrangement i t was p o s s i b l e t o make o b s e r v a t i o n s e i t h e r w i t h o r w i t h o u t t h e compensator and t o make w a v e l e n g t h measurements e i t h e r d i r e c t l y w i t h t h e c r o s s h a i r and mi c r o m e t e r movement o r w i t h a p h o t o g r a p h . The p o l a r i z e r , a n a l y z e r and compensator c o u l d be r o t a t e d i n d e p e n d e n t l y . A l l t h e components except t h e l i g h t s o u r c e , c o n d e n s i n g l e n s and p o l a r i z e r were mounted on a 1 1/4 i n c h s t e e l c h a n n e l . T h i s c h a n n e l had a 1/4 i n c h s l o t m i l l e d out f o r most o f i t s l e n g t h and was f a s t e n e d t o t h e bottom o f t h e l o w e r aluminum p l a t e by means o f two 1/4 i n c h s c r e w s . The e y e p i e c e , a n a l y z e r and second l e n s were mounted per m e n a n t l y onto t h e end o f the c h a n n e l . A b r a s s c a r r i a g e w h i c h s l i d a l o n g t h e t o p o f t h e c h a n n e l c a r r i e d t h e compensator mounting and t h e o b j e c t i v e l e n s . Thus i t was p o s s i b l e t o f o c u s on t h e p l a n e o f t h e B a b i n e t by s l i d i n g t h e c a r r i a g e a l o n g t h e c h a n n e l . F o c u s s i n g o f t h e p l a n e i n t h e g e l onto t h e compensator was a c c o m p l i s h e d by moving t h e o b j e c t i v e l e n s i n i t s mount, e i t h e r b y means fif a cou r s e s l i d i n g a djustment o r , by means o f t h e r e g u l a r screw f o c u s s i n g a d j u s t m e n t . Adjustment o f t h e w v e r - a l l m a g n i f i c a t i o n was made by moving t h e c h a n n e l on i t s s u p p o r t i n g s c r e w s . 20 The g e l sample was h e l d i n a r e c t a n g u l a r g l a s s c e l l . Two s i z e s were u s e d , t h e f i r s t was a s m a l l a b s o r p t i o n c e l l 2x3 c e n t i m e t e r s i n c r o s s . s e c t i o n and 5 c e n t i m e t e r s h i g h . The second was 2x10 c e n t i m e t e r s i n c r o s s s e c t i o n and 5 c e n t i m e t e r s h i g h amd t h e d i r e c t i o n o f p r o p a g a t i o n was i n t h e l e n g t h w i s e d i r e c t i o n i n b o t h c a s e s . The use o f c e l l s o f f i n i t e s i z e has 13 been i n v e s t i g a t e d and i t has been shown t h a t w i t h r e a s o n a b l e c a r e t h e e r r o r i n t r o d u c e d by assuming p l a n e waves i n a medium o f i n f i n i t e expent i s s m a l l compared w i t h t h e e x p e r i m e n t a l e r r o r s . T h i s i s p a r t i c u l a r l y so w i t h measurements o f wave-l e n g t h . Measurements o f damping were n o t made w i t h t h e s h o r t o e l l s . The l a r g e c e l l s were c o n s t r u c t e d o f o r d i n a r y window g l a s s , c u t and ground t o s i z e and cemented t o g e t h e r w i t h D e K o h t i n s k y Cement. No d i f f i c u l t y has been encountered w i t h t h e s e c e l l s , e i t h e r f rom m e c h a n i c a l f a i l u r e o r f r o m s p u r i o u s d ouble r e f r a c t i o n . The t e m p e r a t u r e o f t h e sample was c o n t r o l l e d by h o l d i n g t h e g l a s s c e l l i n s i d e a L u c i t e box 8x15x5 c e n t i m e t e r s , i n s f l . d e d i m e n s i o n s , t h r o u g h w h i c h w a t e r o f t h e d e s i r e d t e m p e r a t u r e was c i r c u l a t e d . The L u c i t e box was h e l d s e c u r e l y t o t h e l o w e r aluminum p l a t e and s i n c e t h e g l a s s c e l l was clamped between t h e bottom and t o p c o v e r o f t h e box t h e sample was h e l d s e c u r e l y i n p l a c e . V i b r a t i o n s were sep up i n t h e sample by means o f a g l a s s vane w h i c h was o s c i l l a t e d v e r t i c a l l y i n i t s own p l a n e by a h o r n l o u d s p e a k e r d r i v e r ( A t l a s Sound C o r p . ) . The vane was a p p r o x i m a t e l y 0.1x0.6x8 c e n t i m e t e r s : t made by c u t t i n g a m i c r o s c o p e s l i d e l e n g h t w i s e and p o l i s h i n g down t h e edges. 21 I t i n t u r n was cemented i n t o a s l o t i n a d u r a l u m i n a d a p t e r w h i c h was t h r e a d e d on i t s o p p o s i t e end. The a d a p t e r screwed i n t o a s m a l l L u c i t e c y l i n d e r w h i c h was cemented d i r e c t l y t o th e t h i n b a k e l i g h t diaphram o f t h e d r i v e r . The L u c i t e c y l i n d e r was cemented c e n t r a l l y and p a r a l l e l t o t h e a x i s o f t h e d r i v e r by f i r s t mounting on a cap w h i c h t h r e a d e d onto t h e mounting t h r e a d o f t h e d r i v e r . I t was made t o match t h e s p h e r i c a l c o n t o u r o f t h e diaphram by s o f t e n i n g t h e end w i t h g l a c i a l a c e t i c a c i d and t h e n a p p l y i n g s u f f i c i e n t f o r c e d u r i n g t h e cementing t o make t h e m a t e r i a l f l o w . The t e m p e r a t u r e was m a i n t a i n e d by a c i r c u l a t i n g w a t e r system. The t e m p e r a t u r e o f a b a t h o f w a t e r was h e l d t o ± 0 . 5 degrees c e n t i g r a d e by means o f a b i m e t a l l i c t h e r m o - r e g u l a t o r (American I n s t r u m e n t Go.) and a 250 w a t t k n i f e h e a t e r (Cenco). Temperatures below room t e m p e r a t u r e were o b t a i n e d by c i r c u l a t i n g c o l d t a p w a t e r o r by s u r r o u n d i n g t h e b a t h w i t h i c e . Water f r o m t h e b a t h was pumped t h r o u g h t h e L u c i t e b a t h p a s t t h e c e l l by a s m a l l c e n t r i f u g a l pump ( E a s t e r n I n d u s t r i e s Model B - l ) a t a bout 5 l i t e r s p»r m i n u t e . A l i g n m e n t o f t h e a p p a r a t u s was checked by f i r s t l i n i n g up t h e v e r t i c a l c r o s s h a i r o f t h e m i c r o m e t e r e y e p i e c e w i t h a han g i n g plumb l i n e . The v e r t i c a l c r o s s h a i r was t h e n used as a r e f e r e n c e f o r t h e r e s t o f t h e components. The p o l a r i z e r was s e t w i t h i t s a x i s p a r a l l e l t o t h i s c r o s s h a i r . The B a b i n e t compensator has a l i n e on t h e s t a t i o n a r y wedge and a t r i g h t a n g l e s t o t h e o p t i B a x i s t h u s t h e a n g u l a r s c a l e and v e r n i e r c o u l d be u s e d t o measure a n g l e s f r o m t h e v e r t i c i l . I 22 The a n a l y z i n g P o l a r o i d was s e t w i t h i t s a x i s h o r i z o n t a l b y f i n d i n g t h e p o s i t i o n f o r minimum t r a n s m i s s i o n w i t h b o t h t h e p o l a r i z e r and compensator axes v e r t i c a l . S t a t i c a l i g n m e n t o f t h e g l a s s vane was s i m p l y compared w i t h t h e v e r t i c a l c r o s s h a i r . Whether o r not i t s m o t i o n was t r u l y i n i t s own p l a n e was a l s o checked v i s u a l l y . T h i s c o u l d be done by r e p l a c i n g t h e r e g u l a r s y n c h r o n i z e d d r i v i n g c i r c u i t o f t h e s t r o b o t r o n w i t h a G e n e r a l R a d i o c o . s t r o b o t a c Type 6S1-B I f t h e f r e q u e n c y o f t h e l i g h t s o u r c e was a d j u s t e d m a n u a l l y u n t i l i t d i f f e r e d f r o m t h a t o f t h e v i b r a t i n g vane by a f r a c t i o n o f a c y c l e a second t h e vane c o u l d be seen t o move S l o w l y t h r o u g h o u t I t s complete c y c l e . The o v e r - a l l m a g n i f i c a t i o n -was about f o r t y t i m e s and u n d e r t h e s e c o n d i t i o n s no a p p r e c i a b l e l a t e r a l m o t i o n o f t h e vane was o b s e r v e d . The e l e c t r o - m a g n e t i c t r a n s d u c e r u s e d t o d r i v e t h e vane was a commercial h o r n l o u d s p e a k e r d r i v e r , r a t e d a t 25 w a t t s , 16 ohms. The d r i v i n g s i g n a l was s u p p l i e d b y an o r d i n a r y a u d i o o s c i l l a t o r c a l i b r a t e d f o r f r e q u e n c y a g a i n s t t h e 60 c y c l e house s u p p l y ; The s i g n a l was a m p l i f i e d and f e d t o t h e l o u d -s p e a k e r by a t h r e e t u b e a u d i o o s c i l l a t o r . The a m p l i f i e r was o f c o n v e n t i o n a l d e s i g n . I t c o n s i s t e d o f a r e s i s t a n c e c a p a c i t a n c e c o u p l e d phase i n v e r t e r and d r i v e r and a p a i r o f p u s h - p u l l output t u b e s t r a n s f o r m e r c o u p l e d t o t h e , l o a d . The t u b e s were one 6S3LJ7GT and two 6L6jGs. The o u t p u t t r a n s f o r m e r was a Hammond t y p e 1637, The o u t p u t was a l s o used t o s u p p l y t h e t r i g g e r i n g i m p u l s e s f o r t h e s y n c h r o n i z e d l i g h t s o u r c e . The c i r c u i t f o r t h e s t r o b o t r o n l i g h t s o u r c e i s shown £3 i n f i g u r e 9,. I t has two main s e c t i o n s , t h e s t r o b o t r o n and i t s a s s o c i a t e d d i s c h a r g e c i r c u i t , and, t h e p u l s e s h a p i n g and t r i g g e r i n g c i r c u i t . The l e n g t h o f t h e d i s c h a r g e i s d e t e r m i n e d by t h e 1 m i c r o f a r a d condenser and t h e tub e i n t e r n a l and l e a d r e s i s t a n c e . W i t h a s i m i l a r c i r c u i t and a 4 m i c r o f a r a d condenser a d u r a t i o n o f l e s s t h a n 5 mic r o s e c o n d s i s c l a i m e d f o r t h e l i g h t f l a s h • The d i s c h a r g e i s i n i t i a t e d by c a u s i n g a glow d i s c h a r g e between t h e two g r i d s . T h i s i s done by a v o l t a g e p u l s e t o e i t h e r one o f t h e g r i d s ; t h e magnitude aled s i g n b e i n g d e t e r m i n e d by t h e s t a t i c p o t e n t i a l s o f b o t h g r i d s r. To o b t a i n good s y n c h r o n i z a t i o n t h e t r i g g e r i n g p u l s e s s h o u l d have a s h a r p f r o n t and s h o u l d be o f s h o r t d u r a t i o n compared t o t h e i n t e r v a l between p u l s e s . Sharp p i p s were o b t a i n e d w i t h a p u l s e s h a p i n g c i r c u i t . A 6N7 d o u b l e t r i o d e was used as a s q u a r e wave g e n e r a t o r by a p p l y i n g a v e r y l a r g e i n p u t s i g n a l and t h u s o v e r - d r i v i n g i n b o t h g r i d and p l a t e c i r c u i t s . The r e s u l t i n g a p p r o x i m a t e l y square wave was t h e n changed t o a s e r i e s o f s h a r p p i p s by an RC d i f f e r e n t i a t i n g c i r c u i t . The t r i g g e r i n g p i p s were a p p l i e d ; t o t h e i n n e r g r i d a n d , s i n c e t h e second g r i d was n o r m a l l y 100 v o l t s p o s i t i v e , , o n l y n e g a t i v e p u l s e s produced a glow d i s c h a r g e . Thus t h e t u b e was t r i g g e r e d once e v e r y c y c l e , p r o v i d e d p l a t e and second g r i d v o l t a g e s were n e a r n o r m a l . The 631-P i s r a t e d a t a maximum o f 250 f l a s h e s p e r second so t h e c i r c u i t c o n s t a n t s a r e su c h t h a t at around 300 c y e l e s p e r second t h e r e c h a r g e t i m e o f t h e d i s c h a r g e condenser i s o f t h e same o r d e r as one p e r i o d o f t h e s i g n a l . From t h i s p o i n t a d i s c h a r g e o c c u r s o n l y when t h e t o f o l l o w page 23 24 second g r i d r e a c h e s some l i m i t i n g v a l u e ; s y n c h r o n i z a t i o n i s m a i n t a i n e d hut t h e f l a s h i n g r a t e does not exceed t h e r a t e d maximum. T h i s method o f s y n c h r o n i z i n g was found t o he s u p e r i o r t o t h a t o f a p p l y i n g a s y n c h r o n i z i n g s i g n a l t o t h e m u l t i v i b r a t o r c i r c u i t i n t h e G e n e r a l P a d i o S t r o b o t a e as used by F e r r y e t a l . e) Methods o f M e a s u r i n g Wavelength and Damping Two methods o f making measurements were u s e d , one p e r m i t t e d measurement o f w a v e l e n g t h o n l y , t h e second p e r m i t t e d measurement o f w a v e l e n g t h and damping, we w i l l r e f e r t o them as Method A and Method B r e s p e c t i v e l y ; Method A A compensator was not a v a i l a b l e when t h e s e experiments were s t a r t e d so i t was n e c e s s a r y t o f i n d some o t h e r method o f m e a s u r i n g t h e w a v e l e n g t h o f sound i n t h e g e l . A s mentioned b e f o r e , t h e i n t e n s i t y o f l i g h t t r a n s m i t t e d by a d o u b l y r e f r a c t i n g medium ( i n t h i s case a s t r a i n e d g e l ) between c r o s s e d P o l a r o i d s , o p t i c axes o f medium a t f o r t y f i v e degrees t o t h o s e o f t h e P o l a r o i d s , i s p r o p o r t i o n a l t o s i n 2 6 , where S i s t h e r e l a t i v e r e t a r d a t i o n . I t has been shown t h a t i f a second d o u b l y r e f r a c t i n g medium, w i t h r e l a t i v e r e t a r d a t i o n A, i s p l a c e d b e f o r e t h e a n a l y z e r w i t h i t s o p t i c a x i s a t an a n g l e <•< t o t h a t o f t h e p o l a r i z e r , t h e i n t e n s i t y i s governed by t h e f o l l o w i n g : j o < sivf z£- + i - stoAsm2*csi*£ + 5-in_4 sml* cosS Suppose A =-A/4 and t h a t «< i s a s m a l l a n g l e . Then s i n c e 8 i s always s m a l l i n t h e p r e s e n t case ( « i r ) , cosS w i l l a l w a y s 25 be p o s i t i v e and i t s e f f e c t w i l l be t o i n c r e a s e t h e o v e r a l l i l l u m i n a t i o n . However, s i n S w i l l be p o s i t i v e f o r p o s i t i v e S and n e g a t i v e f o r n e g a t i v e § . Thus i t s s i g n w i l l r e v e r s e e v e r y h a l f w a v e l e n g t h i n t h e g e l . The t r a n s m i t t e d l i g h t w i l l be i n c r e a s e d f o r p o s i t i v e r e t a r d a t i o n $ and d e c r e a s e d f o r n e g a t i v e S . A m i c a p l a t e was used t o produce t h e d e s i r e d A • I t was o b t a i n e d by s p l i t t i n g m i c a sheet and s e l e c t i n g by t r i a l f o r t h e d e s i r e d e f f e c t , t h e t h i c k n e s s was about 0.027 mm.. By a d j u s t i n g t h e a n g l e o f t h e p l a t e f o r optimum e f f e c t i t was p o s s i b l e t o p r a c t i c a l l y e l i m i n a t e e v e r y o t h e r l i n e and a t t h e same t i m e g r e a t l y i n c r e a s e t h e apparent s h a r p n e s s o f t h e r e m a i n i n g l i n e s . Measuremnets of w a v e l e n g t h were made w i t h t h e m i c r o m e t e r e y e p i e c e . The p o s i t i o n o f p o i n t s o f e q u a l i n t e n s i t y on e i t h e r s i d e o f a l i n e o f minimum o r imximum i n t e n s i t y were measured and t h e a v e r a g e t a k e n as t h e p o s i t i o n o f t h e m i d - p o i n t r e f e r r e d t o some a r b i t r a r y datum l i n e . T h i s was done f o r a l l t h e l i n e s v i s i b l e on each s i d e o f t h e vane, u s u a l l y about t h r e e o r f o u r . D i f f e r e n c e s between a d j a c e n t l i n e s gave t h e w a v e l e n g t h . Measurements were not made c l o s e r t h a n a h a l f w a v e l e n g t h f r o m t h e vane i n o r d e r t o a v o i d p o s s i b l e e r r o r s due t o s t r u c t u r a l changes e t c . a t t h e vane s u r f a c e . No s y s t e m a t i c v a r i a t i o n i n w a v e l e n g t h was observed a t v a r y i n g d i s t a n c e s f r o m t h e vane. The l o w e s t measureable f r e q u e n c y was d e t e r m i n e d by the d i f f i c u l t y o f s e t t i n g t h e c r o s s h a i r a c c u r a t e l y on t h e r e l a t i v e l y b r o a d l i n e s . The upper end o f t h e f r e q u e n o y range was d e t e r m i n e d by t h e d i s a p p e a r a n c e o f d o u b l e r e f r a c t i o n . 26 Method B This method used the Babinet compensator set up as described above, A few measurements, only, were taken with the micrometer eyepiece. Measurements of wavelength could be made re lat ively easi ly but i t was not possible to measure the decay of l ine displacement by this method. The l a t te r measurement was attempted by rotating the eyepiece 90 degrees so that the motion of the cross hair was v e r t i c a l . The variation of intensity i s sinusoidal,for monochromatic l i g h t , and with the compensator at a small angle ( 5 ° ) small so that i t i s impossible to hake accurate settings of the cross ha i r . This d i f f i cu l ty was overcome by photographing the image i n the eyepiece and printing on high contrast paper. High speed panchromatic f i lm was used (Kodak Super XX r o l l film) • By suitable exposure and development on high contrast enlarging paper i t was possible to get quite sharp variation between the l ight and dark l ines and i t was then possible to make measurements of decay of amplitude. Measurements of wave-length were also made on the photographs, the image of the vane being used as a ca l ibrat ion. As i n Method A, readings were not taken closer than a half wavelength from the vane. Measurements of wavelength could be made over the same frequency range as with Method A, but measurements of decay of l ine displacement could only be made over a much smaller range due to the rapid decay of birefringence with increasing frequenoy. 27 The loudspeaker driver was driven at maximum rated input for a l l measurements. Measurements were made on a 6% Digel-3%0otoie Acid in Gasoline gel to determine the variation in amplitude of m§tion of the vane with frequency and the effect of input power on measured wavelength. The results are shown i n Table 1. for amplitude frequency measurements. Table 1. Amplitude of vane, speaker input 20 watts Frequency Amplitude cycles/sec. millimeters 100 0.5 200 0.8 300 0.4 400 0.15 500 0.05 The maximum around 200 cycles/second was caused by a mechanical resonance i n the system. The variat ion in wavelength was less than 2.5% for driving powers between 10$ and 140% of the rated power for the same ge l at 340 cycles/second, 5 0 ° C . Because of these results no attempt has been made to l imi t the vane amplitude to any part icular value, to attempt to have a constant strain e t c . . strains and rates of s tra in are apparently small enough to avoid thixotropic effects, e t c . . No systematic variat ion of wavelength with distance from the vane was ever noted. The measurement of damping i s d i f f i cu l t to do accurately, part icular ly in the case of the experiments reported here. The method of measurement was as follows. The positions of peaks and troughs of the nevy l ines were 28 measured r e l a t i v e t o an a r b i t r a r y base l i n e , i n t h i s case t h e l i n e on t h e B a b i n e t . D i f f e r e n c e s were t a k e n between s u c c e s s i v e measurements and r a t i o s t a k e n o f t h e s u c c e s s i v e d i f f e r e n c e s . I f t h e de£ay o f double r e f r a c t i o n S i s -e x p o n e n t i a l t h e s e r a t i o s w i l l be c o n s t a n t and e q u a l t o e ° s i n c e r e a d i n g s were t a k e n e v e r y h a l f w a v e l e n g t h . The q u a n t i t y A / x was t h e n f o u n d by t a k i n g t h e l o g _ o f t h e average r a t i o . The r a t i o s found i n t h e s e e x p e r i m e n t s were between 1.1 and 1.5 and were g e n e r a l l y measured w i t h a v a r i a t i o n o f ~£o.l, From t h i s i t was assumed t h a t t h e decay o f s t r a i n d o u b l e r e f r a c t i o n and t h e r e f o r e t h e decay o f s t r a i n was e x p o n e n t i a l . U n f o r t u n a t e l y w i t h r a t i o s o f t h i s magnitude t h e v a r i a t i o n i n t h e c a l c u l a t e d A / x Q i s o f t h e o r d e r o f 100% f o r a v a r i a t i o n o f 0.1 i n t h e argument, so t h e v a l u e s o f A / x 0 a r e s u b j e c t t o c o n s i d e r a b l e e r r o r . d) M a t e r i a l The aluminum soaps used i n t h e two s e r i e s o f measure-ments r e p o r t e d here a r e s u p p o s e d l y o f t h e same t y p e . However s i n c e t h e b e h a v i o r appears t o be q u i t e d i f f e r e n t i # t h e two c a s e s , we l a b e l t h e f i r s t soap used by t h e o r i g i n a l Canadian name D i g e l and t h e second by t h e l a t e r name O c t a l . The o r i g i n o f t h e D i g e l powder i s n o t known a t p r e s e n t and an attempt i s b e i n g made t o t r a c e i t . The o r i g i n and p a r t i c u l a r s o f manufacture e t c . a r e known f o r t h e O c t a l powder u s e d . The g a s o l i n e used i s o r d i n a r y c o m m e r c i a l grade but 29 requires some comment. An attempt was made to make gels i n Vancouver with looal ly ohtaine^gasoline and considerable d i f f i c u l t y was encountered in making observations due to the small amount of strain double refraction. This could possibly have been due to a different miTing procedure or to different soaps. However, i t was learned that other d i f f i cu l t i e s have been experienced with gels during the past year and that much of the d i f f i cu l ty was caused by the quality of the gasolines used. Samples of gasoline have heen checked by Defence Research chemical Laboratories, Ottawa^and the only acceptable gasoline at the present time appears to be straight run from Alberta crude.. For this reason a l l the gels used were obtained from the Defence Research Board Suffield Experimental Station and were prepared with "acceptable" gasoline• The benzene used was reagent grade. Mixing was done at 2 5 ° C±1 ° C , in a large " f i e l d mixer" for the f i r s t series and i n a small scale laboratory mixer for the Octal series. The small mixer i s designed to give results s imilar to those obtained with operational mixers but this i s not necessarily so with the properties investigated here. 30 IV" EXPERIMENTAL RESULTS Measurements have been made on two groups o f g e l s . A s e r i e s o f mixes o f D i g e l (soap) w i t h o c t o i c a o i d i n g a s o l i n e have been i n v e s t i g a t e d u s i n g Method A. Method B has been used on a s e r i e s o f O c t a l (soap) w i t h o c t o i c a c i d i n g a s o l i n e and i n benzene. The r e s u l t s o f t h e s e two groups o f measurements a r e q u i t e d i f f e r e n t . I t was p o s s i b l e t o ap p r o x i m a t e t h e b e h a v i o r o f t h e f i r s t group w i t h a s i m p l e M a x w e l l element but t h i s c o u l d n ot be done f o r t h e O c t a l mixes i n e i t h e r s o l v e n t , a) D i g e l and O c t o i c A c i d i n G a s o l i n e Method A I t was found p o s s i b l e t o f i t t h e r e s u l t s o f t h e s e measurements t o M a x w e l l models. The proce d u r e was as f o l l o w s . V a l u e s o f G were c a l c u l a t e d f r o m t h e measured wa v e l e n g t h s a c c o r d i n g t o t h e e q u a t i o n G - f2«Vy<? • The d e n s i t y / * ? was t a k e n t o be 0.75 gm/cm3 i n a l l o a s e s ; e r r o r s i n t r o d u c e d by assuming c o n s t a n t a r e n e g l i g i b l e compared t o e x p e r i m e n t a l e r r o r s . The l i m i t i n g v a l u e o f G was found by p l o t t i n g G v s G / f 2 , t h e i n t e r c e p t on G / f 2 s 0 b e i n g t a k e n a s G, t h e r i g i d i t y of t h e e q u i v a l e n t M a x w e l l M o d e l , w i t h t h i s r e s u l t v a l u e s of G/G were c a l c u l a t e d and p l o t t e d a g a i n s t l o g f . The p l o t t e d p o i n t s were superimposed on a s t a n d a r d p l o t f o r a M a x w e l l element f o r b e s t f i t . The v a l u e o f f f o r = 1 was t a k e n f r o m t h i s c u r v e g i v i n g t h e r e l a x a t i o n t i m e o f t h e model. T a b l e 2.summarizes t h e r e s u l t s o f t h e s e measurements. Typical"* c u r v e s a r e i l l u s t r a t e d in„figure ' i b . 31 Tafcle 2 Constants f j^? Maxwell Model Digel and Octoi© Acid in Gasoline Concentration Temperature Rigidity Relaxation Time viscosity ^DigeHOotoic 3 3/4 1 1/4 1 1/2 2 1/2 6 °c dynes/cm2 10""3sec. poise 2.5 1200 0.6 0.7 11.2 750 0.7 0.4 20.5 650 >1.6 >1. 31.2 700 >1.6 >1.1 1.5 1600 0.5 0.8 11.2 1600 0.5 0.8 21.5 1500 0.7 1. 30.7 1300 0.8 1. 41.2 1270 1.0 1.5. 50.5 1160 >2.0 >2. 0.7 3900 0.2 0.8 12.3 3400 0.3 1. 21.7 2800 0.4 1.1 32.2 2200 >1.1 >2. 42.0 1900 — — — 51.3 1700 1.2 780 0.5 0.4 7.0 640 0.8 0.5 14.5 600 0.9 0.5 25.3 520 1.3 0.7 36.7 500 1.1 0.6 49.0 470 1.8 0.8 0.5 1220 0.6 0.7 11.7 1120 0.6 0.7 21.0 1020 0.8 0.8 31.3 940 0.9 0.8 41.3 940 0.9 0.8 51.3 900 1.2 1.1 1.0 4600 0.3 1.4 11.7 4100 0.3 1.2 20.0 3400 0.5 1.7 31.3 3100 0.6 1.9 41.7 2900 0.6 1.7 51.3 2600 0.9 2.3 1.1 3900 0.4 1.6 10.5 4900 0.3 1.5 20.5 4100 0.4 1.6 30.6 3500 0.5 1.8 40.7 2800 >1 >2.8 50.7 2800 ^1 t o f o l l o w page 31 32 The r i g i d i t y , G, i n c r e a s e s w i t h c o n c e n t r a t i o n . R e s u l t s f o r b o t h soap t o p e p t i z e r r a t i o s a r e shown i n dfigure 1<I>, where l o g . G i s p l o t t e d a g a i n s t l o g . (soap c o n c e n t r a t i o n , w e i g h t ^ ) . The r e l a t i o n i s a p p r o x i m a t e l y l i n e a r w i t h a s l o p e o f about t h r e e i n b o t h c a s e s . Thus t h e r i g i d i t y i s a p p r o x i m a t e l y p r o p o r t i o n a l t o t h e cube of t h e soap c o n c e n t r a t i o n . R i g i d i t y d e c r e a s e s w i t h i n c r e a s i n g t e m p e r a t u r e . Log.G v s 1/T, where T i s t h e a b s o l u t e t e m p e r a t u r e , g i v e s a s t r a i g h t l i n e f o r any one mix but t h e s l o p e v a r i e s somewhat w i t h soap c o n c e n t r a t i o n and w i t h soap t o p e p t i z e r r a t i o . These curves f o r t h e 8:1 soap t o p e p t i z e r s e r i e s a r e shown i n f i g u r e 11. and t h o s e f o r t h e 4:1 s e r i e s were s i m i l a r b u t w i t h s l i g h t l y h i g h e r s l o p e . I f we assume t h a t t h e r i g i d i t y - t e m p e r a t u a r e dependence can be d e s c r i b e d by an e q u a t i o n s u c h as - Oc t h e n t h e a c t i v a t i o n energy Q, f o r t h e r i g i d i t y i s between 1 and 2 K e a l . p e r mole. S i n c e t h e v a l u e s f o r t h e r e l a x a t i o n t i m e s a r e found by a r a t h e r i n d i r e c t p r o c e d u r e h e r e , t h e y can o n l y be t c o n s i d e r e d as v e r y a p p r o x i m a t e . However, i n a l l c ases i t has been found t h a t t h e r e l a x a t i o n t i m e t e n d e d t o d e c r e a s e w i t h i n c r e a s i n g soap c o n c e n t r a t i o n and t o i n c r e a s e w i t h i n c r e a s i n g t e m p e r a t u r e . The magnitude o f t h e v i s c o u s 1 component o f t h e M a x w e l l model was c a l c u l a t e d f rom t h e v a l u e s o f G and t {yj - G>^), and found t o i n c r e a s e w i t h i n c r e a s i n g soap c o n c e n t r a t i o n and w i t h i n c r e a s i n g t e m p e r a t u r e and t o d e a r e a s e w i t h i n c r e a s i n g p e p t i z e r . V i s c o s i t i e s were o f t h e o r d e r o f 1 p o i s e and t h e v a r i a t i o n was about 33 100 <f0 i n t h e t e m p e r a t u r e i n t e r v a l 0° t o 50°G. b) O e t a l - O c t o i c A c i d i n Benzene Two mixes were i n v e s t i g a t e d , 5% O c t a l 1.25% O c t o i c A c i d and 5% O c t a l 1.7% O c t o i c A c i d . B o t h were measured a t 25°C, t h e r e s u l t i n g v a l u e s f o r to G a r e shown i n T a b l e 3. below. T a b l e 5. R i g i d i t y , G*, f o r O c t a l - O c t o i c A c i d i n Benzene a t 25°C 5ft 4:1 Frequency, c/s 126 I§1§ 199 251 316 397 500 6T, Bynes/cm 2 2350 2320 2170 2240 2460 2400 2320 5% 5:1 Freq u e n c y , c/s 100 T26" 158 224 355 500 630 792 dynes/cm 2 2340 2620 2520 2280 2500 2730 2900 3010 The 4:1 g e l gave v a l u e s o f r i g i d i t y a p p r o x i m a t e l y independent o f f r e q u e n c y , t h e v a l u e s f o r t h e 3:1 g e l show a s l i g h t i n c r e a s e a t t h e h i g h f r e q u e n c y end o f t h e measure-ments. F i g u r e 12. shows l o g . A p l o t t e d a g a i n s t l o g . f r e q . , t h e c u r v e s a r e s t r a i g h t l i n e s drawn a t f o r t y f i v e degrees i n d i c a t i n g t h a t i n b o t h cases l o g . X - f - l o g . f may be app r o x i m a t e d by a c o n s t a n t o v e r t h e f r e q u e n c y range c o v e r d d . The v a l u e s o f f A c a l c u l a t e d f r o m t h e i n t e r c e p t s o f t h e s e l i n e s w i t h e i t h e r a x i s a r e t a k e n as t h e ave r a g e v e l o c i t y o f p r o p a g a t i o n and used t o c a l c u l a t e t h e average v a l u e o f t h e r i g i d i t y , G & y . The average v a l u e s found were: 5 % 0 c t a l 1 . 2 5 % 0 c t o i c A c i d G & v= 2700 dynes/em 2 5 % 0 o t a l 1.7 % 0 c t o i c A c i d G a v= 2400 dynes/cm 2 t o f o l l o w page 33 34 The e f f e c t o f t e m p e r a t u r e was a l s o d e t e r m i n e d f o r t h e 3:1 mix. The r e s u l t s a r e shown i n f i g u r e 14, below. R i g i d i t y , f o r 5 % 0 e t a l 1.7#0etoic A c i d i n Benzene. L o g . f 2.0 2.1 2 i 2 2*3 2.4 2.45 2.6 2.7 2.75 2.8 Temp. 4 °C 2800 2900 2850 2850 2750 3050 17 2750 2900 2800 2750 2750 3400 3270 35 2800 2800 2800 2650 2600 3200 3500 3400 47.3 2600 2650 2400 2400 2500 2550 2650 2600 As found i n t h e measurements a t 25°C, t h e r i g i d i t y G i s independent o f f r e q u e n c y a t l o w f r e q u e n c i e s and tj'ends t o h i g h e r v a l u e s a t t h e h i g h f r e q u e n c y end o f t h e range measured. T h i s i s i l l u s t r a t e d g r a p h i c a l l y i n f i g u r e 14... l o g ^ + l o g . f 2* 2.78 and G a v ^ 2850 dynes/cm 2. U n l i k e t h e mixes o f D i g e l i n g a s o l i n e i n v e s t i g a t e d by Method A t h e r i g i d i t y o f t h i s mix appears t o be alm o s t independent o f t e m p e r a t u r e o v e r t h e range c o v e r e d . I n f a v o r a b l e c a s e s , i t i s p o s s i b l e t o make measurements of t h e decay o f b i r e f r i n g e n c e . The te m p e r a t u r e and f r e q u e n c y dependence o f ^ / x 0 i s i l l u s t r a t e d i n Uable 5. below. I t i s no t p o s s i b l e t o make a c c u r a t e measurements o f damping w i t h t h i s method and i n many cases i t was n o t p o s s i b l e t o make any measurement a t a l l . However, t h e v a l u e s shown i n T a b l e 5. i n d i c a t e a minimum i n t h e v a l u e o f A / x Q as a f u n c t i o n o f f r e q u e n c y and t h a t t h i s minimum t e n d s t o move t o h i g h e r f r e q u e n c i e s as t h e t e m p e r a t u r e i s i n c r e a s e d . 35 T a b l e 5. <Vx0 f o r 5 % 0 c t a l 1 . 7 % 0 c t o i c A o i d i n Benzene Temp.°C 4 17 35 47.5 L o g . f £.0 .5 .6 .8 2.1 .4 .4 .7 2 ,2i .3 .4 n.5 .4 2.3 .3 .3 .5 2.45 .5 .4 .3 .3 2.6 — .5 — — 2.7 — .7 Comparison w i t h t h e t h e o r e t i c a l c u r v e s o f f i g u r e 5. shows t h a t t h e b e h a v i o r cannot be a p p r o x i m a t e d by a s i m p l e M a x w e l l element i n t h i s c a s e . Not o n l y because o f t h e f a c t t h a t t h e r i g i d i t y remains independent o f f r e q u e n c y and t h e n t e n d s t o r i s e a t h i g h f r e q u e n c i e s , b u t a l s o because o f t h e minimum f o r ^ / x Q v s f r e q u e n c y . The v a r i a t i o n o f b o t h R e t a r d e d r i g i d i t y and damping may be a p p r o x i m a t e d by a/Maxwell element w i t h t h e r a t i o ^ j j ^ = 0.005. I t i s not p o s s i b l e t o d e t e r m i n e t h e p o s i t i o n o f t h e minimum p r e c i s e l y so a d e t e r m i n a t i o n o f r e l a x a t i o n t i m e has n o t been a t t e m p t e d . The o r d e r o f magnitude f o r r e l a x a t i o n t i m e i s about l O x l O " 8 s e c o n d s . c) O c t a l - O c t o i c A c i d i n G a s o l i n e Method B The r e s u l t s o f t h i s s e r i e s o f measurements a r e e s s e n t i a l l y t h e same as f o r t h e O c t a l - O c t o i c - B e n z e n e 36 measured by t h e same method and, t h e r e f o r e , d i f f e r c o n s i d e r a b l y f r o m t h o s e o f t h e D i g e l mixes i n g a s o l i n e . Measurements have been made n e a r room t e m p e r a t u r e on g e l s r a n g i n g i n soap c o n c e n t r a t i o n f rom 3% t o 6^ and w i t h soap t o p e p t i z e r r a t i o s o f £:1, 3:1 and 4:1. Some d i f f i c u l t y has been o b t a i n e d i n o b t a i n i n g c o n s i s t e n t r e l a t i o n s h i p s between g e l s o f d i f f e r i n g c o n c e n t r a t i o n s . T h i s i s p o s s i b l y due t o d i f f e r e n c e s i n a g i n g r a t e s , p r e s e n c e o f i m p u r i t i e s , e t c . . I n c e r t a i n - o a s e s i t has been found t h a t t h e r i g i d i t y o£ a p a r t i c u l a r g e l i s l e s s t h a n g e l s of^Lower soap c o n c e n t r a t i o n . A> The v a r i a t i o n o f r i g i d i t y , G, w i t h f r e q u e n c y i s i l l u s t r a t e d i n T a b l e 6 f o r some o f t h e samples t e s t e d . T a b l e 6 R i g i d i t y G o f O e t a l - O c t o i c A c i d i n G a s o l i n e a t 25°G (age 2 weeks) C o n c e n t r a t i o n Frequency  c y c l e s / s e c o n d # O o t a l ^ O c t o i c 100 126 158 199 251 316 597 500 4 1 850 950 950 1000 950 950 1150 5 1.25 1500 1550 1500 1450 1600 1650 1660 3 1 1050 1000 1000 1250 1150 1200 1250o 4 1.33 900 900 900 900 850 850 1200 1300 5 1.7 700 700 750 800 800 950 4 2 400 500 450 450 550 650 750 5 2.5 1400 1300 1400 1350 1400 1400 1600 AS w i t h t h e benzene m i x e s , t h e r i g i d i t y t e n d s t o be independent o f f r e q u e n c y a t l o w f r e q u e n c i e s and t o b e g i n t o i n c r e a s e r a t h e r s h a r p l y a t t h e h i g h f r e q u e n c y end o f t h e sp e c t r u m i n v e s t i g a t e d . 37 The b e h a v i o r o f t h e q u a n t i t y A / x Q i s a l s o s i m i l a r t o t h a t f o r t h e benzene m i x e s . The minimum i n t h e ease o f t h e g a s o l i n e g e l s was n o t q u i t e as l o w as f o r t h e benzene g e l s , i n t h e m a j o r i t y o f cases i t was around .5. T h i s t o g e t h e r w i t h thm f r e q u e n c y c h a r a c t e r i s t i c f o r t h e r i g i d i t y s u g g e s t s t h a t t h e m e c h a n i c a l b e h a v i o r may be app r o x i m a t e d by a r e t a r d e d M a x w e l l element w i t h a r a t i o o f p a r a l l e l t o s e r i e s v i s c o s i t y o f t h e o r d e r o f 0.01. The dependence o f r i g i d i t y on t e m p e r a t u r e was det e r m i n e d f o r t h e 5 $ 0 c t a l 1 . 7 % 0 c t o i e A c i d m i x . Measurements were made a t 2.5, 21.2 and 42.5 °G. The r i g i d i t y , G, was found t o be s u b s t a n t i a l l y independent o f t e m p e r a t u r e o v e r t h i s s m a l l range and as b e f o r e was a p p r o x i m a t e l y independent o f f r e q u e n c y f r o m 100 t o 500 c y c l e s p e r second. The average v a l u e o f t h e r i g i d i t y was a p p r o x i m a t e l y 750 dynes/cm 2. T h i s i s t h e same mix as i n T a b l e 6. and t h i s v a l u e compares w e l l w i t h t h e v a l u e s shown t h e r e f o r 25°C. As a cheek on t h e f a l l o f r i g i d i t y w i t h i n c r e a s i n g c o n c e n t r a t i o n a second measurement was made on t h e 4% O c t a l m i x w i t h t h e same p e p t i z e r r a t i o . An average r i g i d i t y o f a p p r o x i m a t e l y 900 dynes/cm 2 was f o u n d . T h i s confiBmed t h e r e a d i n g s o f T a h l e 6. and a g a i n i n d i c a t e d a de c r e a s e o f r i g i d i t y w i t h c o n c e n t r a t i o n . The above two groups o f t e s t s were made a t ages o f two and t h r e e weeks r e s p e c t i v e l y . The two g e l s r e c h e c k e d appear t o be q u i t e s t a b l e and t h i s would seem t o e l i m i n a t e d i f f e r i n g a g i n g r a t e s as a p o s s i b l e e x p l a n a t i o n o f t h e anomaly. 3 8 The e f f e c t s o f a g i n g p r e s e n t a d i f f i c u l t p r o b l e m i n ex p e r i m e n t s o f t h i s t y p e w i t h g e l s . I t i s v e r y d i f f i c u l t t o d e t e r m i n e t h e e f f e c t o f soap c o n c e n t r a t i o n , soap t o p e p t i z e r r a t i o , e t c . , u n l e s s t h e g e l s a r e . r e l a t i v e l y s t a b l e o r a v e r y d e t a i l e d knowledge o f t h e i r a g i n g c h a r a c t e r i s t i c s i s known. The p r a c t i c e has been t o make a l l measurements on a p a r t i c u l a r s e r i e s a t t h e same age, i f p o s s i b l e . T h i s i s p o s s i b l y n o t t h e b e s t p r o c e d u r e , s i n c e , i n g e n e r a l , d e t e r i o r a t i o n w i t h age i s l i k e l y t o be more r a p i d f o r lo w soap c o n c e n t r a t i o n s o r f o r l o w soap t o p e p t i z e r r a t i o s . V e r y marked d i f f e r e n c e s may be caused by i m p u r i t i e s , eg. minu t e t r a c e s of a c i d on t h e w a l l s o f c o n t a i n e r s can cause v e r y r a p i d b r e a k down o f t h e g e l . The d i f f e r e n t a g i n g r a t e s o f two g e l s o f t h e same soap c o n c e n t r a t i o n but d i f f e r e n t p e p t i z e r c o n c e n t r a t i o n a r e i l l u s t r a t e d i n T a b l e 7 below. T a b l e < 7. E f f e c t o f A g i n g on O c t a l - O c t o i c A c i d i n G a s o l i n e ( a t 21°C) C o n c e n t r a t i o n Age Average R i g i d i t y . G . ^ O c t a l ^ O c t o i c A c i d Days dynes/cm' 13 1500 5 2.5 42 750 5 1.25 8 48 1850 1500 39 I t i s not p o s s i b l e t o make an a c c u r a t e d e t e r m i n a t i o n o f t h e values o f A / x Q and so l o c a t e t h e p o s i t i o n o f t h e minimum and f i n d a v a l u e o f f a . A l s o measurements do n o t extend f a r enough i n t o t h e d i s p e r s i o n r e g i o n s t o f i x t h e v a l u e o f "2? f r o m r i g i d i t y measurements a l o n e . I t i s p o s s i b l e t o make an e s t i m a t e of o r d e r . S i n c e t h e minimum always o c c u r s i n t h e f r e q u e n c y range 100 t o 500 c y c l e s / s e c o n d , u>lra z 10 i n t h i s f r e q u e n c y range. T h e r e f o r e 'tf i s of s & —3 t h e o r d e r o f 10 x 10 seconds and t h e c o r r e s p o n d i n g v a l u e o f *77 _ o f t h e o r d e r o f 10 p o i s e . 40 Y DISCUSSION Measurements have been c a r r i e d out on aluminum soap h y d r o c a r b o n systems a t low s o n i c f r e q u e n c i e s . The p r i m a r y a i m o f t h e p r e s e n t i n v e s t i g a t i o n has been t o d e t e r m i n e t h e p r a c t i c a b i l i t y o f u s i n g t h e t r a n s m i s s i o n o f t r a n s v e r s e s o n i c waves as a method o f d e t e r m i n i n g t h e m e c h a n i c a l p r o p e r t i e s . W i t h c e r t a i n l i m i t a t i o n s t h i s has been found t o be a u s e f u l method, p a r t i c u l a r l y f o r t h e measurement o f r i g i d i t y . The f r e q u e n c y range o v e r w h i c h measurements can be made i s r a t h e r l i m i t e d . A l s o t h e measurement o f damping o r v i s c o u s f o r c e s i s d i f f i c u l t because measurements must be made i n a f r e q u e n c y range where damping, i n space, i s s m a l l ; a t l e a s t f o r t h e cases i n v e s t i g a t e d . An attempt has been made t o app r o x i m a t e t h e m e c h a n i c a l b e h a v i o r w i t h m e c h a n i c a l models. T h i s i n one way o f comparO i n g r e s u l t s o f e x p e r i m e n t s u s i n g d i f f e r e n t t e c h n i q u e s , i n d i f f e r e n t f r e q u e n c y ranges e t c . . I t has been found t h a t t h i s may be done w i t h r e l a t i v e l y s i m p l e models, m c e r t a i n cases M a x w e l l elements a r e s u f f i c i e n t , i n o t h e r s , R e t a r d e d Maxwelih elments a r e n e c e s s a r y . The d a t a a v a i l a b l e , i n most c a s e s , i s not s u f f i c i e n t t o make a r e a l l y a c c u r a t e f i t t o a model. When more d a t a f r o m o t h e r e x p e r i m e n t s a t d i f f e r e n t f r e q u e n c i e s a r e a v a i l a b l e i t w i l l be p o s s i b l e t o choose much more u s e f u l models. By u s i n g e l e c t r i c a l a n a l o g s , a n d a d j u s t i n g e x p e r i m e n t -a l l y f o r t h e c o r r e c t impedance c h a r a c t e r i s t i c , i t i s p o s s i b l e ( i n p r i n c i p l e ^ t o f i n d models t o match b e h a v i o r o f any c o m p l e x i t y . 41 The r e s u l t s f o r mixes made w i t h two s u p p o s e d l y i d e n t i c a l soaps g i v e d i f f e r e n t r e s u l t s , t h o u g h o n l y i n d e g r e e . I n one ca s e a s i m p l e M a x w e l l element i s s u f f i c i e n t o f d e s c r i b e t h e b e h a v i o r , i n t h e second a more c o m p l i c a t e d model i s n e c e s s a r y . W i t h i n t h e a c c u r a c y o f t h e measurements and f r e q u e n c y range c o v e r e d a r e t a r d e d M a x w e l l element w i t h a v e r y s m a l l p a r a l l e l v i s c o u s component i s needed i n t h e second c a s e . The r i g i d i t i e s , G-, n e c e s s a r y f o r t h e r e s p e c t i v e models a r e of t h e same o r d e r but t h e s e r i e s v i s c o u s elements d i f f e r by a f a c t o r o f about t e n , b e i n g h i g h e r i n t h e case w i t h t h e r e t a r d e d element. l b i s p r o b a b l e t h a t a r e t a r d e d M a x w e l l element w o u l d be n e c e s s a r y i n t h e f i r s t c a s e a l s o i f measurements c o u l d be made a t h i g h enough f r e q u e n c i e s . Because o f t h e low v i s c o s i t y and r e l a x a -t i o n t i m e measurements have been made i n t h e r e g i o n where a M a x w e l l element i s s u f f i c i e n t i f ^ f/trs i s s m a l l w h i c h i t c e r t a i n l y i s i n t h i s c a s e . A f u r t h e r d i f f e r e n c e has been o b s e r v e d . T h i s i s t h e d i f f e r e n c e i n t h e range o f f r e q u e n c i e s o v e r w h i c h o b s e r v a t i o n s may be made w i t h t h e two s o a p s . I n o n l y one case w i t h t h e D i g e l mixes was i t not p o s s i b l e t o make measurements up t o 1000 c y c l e s p e r second, i n t h e m a j o r i t y o f ca s e s measurements were made a t f r e q u e n c i e s o v e r 1000 c y c l e s p e r second. W i t h O c t a l m i x e s , b o t h i n g a s o l i n e and benzene t h e r a p i d decay of s t r a i n d o u b l e r e f r a c t i o n w i t h i n c r e a s i n g f r e q u e n c y made i t i m p o s s i b l e t o make measurements p a s t 500 c y c l e s p e r second i n most c a s e s . T h i s may be due t o d i f f e r e n c e s i n s t r a i n o p t i c a l c o e f f i c i e n t s o r t o d i f f e r e n c e s i n t h e damping o f t h e ssonic 4 2 waves . I n t h e D i g e l mixes measurements have been made i n t h e r e g i o n where <*r«l and ^/Xo decreases w i t h i n c r e a s i n g f r e q u e n c y . A r e t a r d e d H a r w e l l element i s n e c e s s a r y f o r t h e O c t a l m i i e s , A / X 6 has a minimum and t h e n i n c r e a s e s w i t h i n -c r e a s i n g f r e q u e n c y . At t h e h i g h e r f r e q u e n c i e s t h e damping i s more i n t h e second case and t h e r e f o r e l i m i t o f o b s e r v a b i l i t y of t h e d o u b l e r e f r a c t i o n i s l o w e r e d . 3 . The r i g i d i t i e s f o u n d a r e o f t h e o r d e r o f 10 dynes per square c e n t i m e t e r i n a l l e a s e s . These a r e o f t h e same o r d e r 1ST as r i g i d i t i e s d e t e r m i n e d by Gunn f o r aluminum soaps i n . b benzene a t f r e q u e n c i e s o f a few c y c l e s p e r second i n a c o n c e n t r i c c y l i n d e r a p p a r a t u s . I t must be s t r e s s e d h e r e t h a t t h i s r i g i d i t y i s n e i t h e r o f t h e r i g i d i t i e s i n t h e model, d i s e u s s e d i n t h e I n t r o d u c t i o n but t h e c u m u l a t i v e e f f e c t o f s e v e r a l mechanisms. We f i n d t h a t t h e r i g i d i t y t e n d s t o deorease w i t h i n c r e a s i n g t e m p e r a t u r e b u t t h a t t h e e f f e c t i s s m a l l . Now, t h e r e t a r d e d o r c o n f i g u r a t i o n a l . e l a s t i c i t y , w h i c h a r i s e s f r o m t h e change of e n t r o p y due t o m o l e c u l e s u n c u r l i n g f r o m t h e i r most p r o b a b l e c o n f i g u r a t i o n s , i s p r o p -yl o r t i o n a l t o t h e a b s o l u t e t e m p e r a t u r e . On t h e o t h e r hand, i n t e r m o i i e c u l a r f o r c e s g i v e r i s e t o a r i g i d i t y w h i c h d e c r e a s e s w i t h i n c r e a s i n g t e m p e r a t u r e . A l s o i f we c o n s i d e r t h e system t o be two phase t h e r e i s t h e a d d i t i o n a l c o m p l i c a t i o n o f i n t e r c h a n g e o f soap between t h e network and t h e s u r r o u n d i n g s o l u t i o n . T h i s would p r o b a b l y temd t o reduce t h e r i g i d i t y w i t h i n c r e a s i n g t e m p e r a t u r e by removing m a t e r i a l f r o m t h e network. The o b s e r v e d t e m p e r a t u r e dependence i s t h e sum o f t h e above e f f e c t s . 43 I t i s not p o s s i b l e t o c o n c l u d e a n y t h i n g about t h e dependence o f r i g i d i t y on soap c o n c e n t r a t i o n . One would e x p e c t t h a t t h e c o n c e n t r a t i o n would have a v e r y marked e f f e c t on t h e r i g i d i t y due t o t h e r a p i d i n c r e a s e i n t h e number o f mesh . p o i n t s w i t h i n c r e a s i n g c o n c e n t r a t i o n ? The r e s u l t s a r e s o n t r a d i e t o r y h e r e , i n some cases t h e r i g i d i t y does v a r y r a p i d l y w i t h c o n c e n t r a t i o n and i n o t h e r s i t appears t o d e c r e a s e w i t h i n c r e a s i n g c o n c e n t r a t i o n . I t i s f e l t t h a t t h e p r e s e n c e o f i m p u r i t i e s i s t h e main cause o f t h e s e d i f f e r e n c e s . The v i s c o s i t y o f t h e s e r i e s damping element i s o f t h e o r d e r o f 1 t o 10 p o i s e . T h i s i s 10 3 t o 10 5 t i m e s s m a l l e r t h a n t h e v i s c o s i t y as measured i n c o n v e n t i o n a l v i s c o m e t e r s . T h i s i s t o be e x p e c t e d s i n o e t h e mechanisms i n v o l v e d i n t h e two c a s e s a r e d i f f e r e n t . I n t h e case o f t r u e f l o w one must c o n s i d e r t h e b r e a k i n g and r e f o r m i n g o f bonds o f t h e network. I n t h e p r e s e n t case s t r a i n s a r e s m a l l , t h e network i s deformed but bonds a r e a p p a r e n t l y not b r o l e n o r t h i x o t r o p i c e f f e c t s would have been o b s e r v e d . The v i s c o s i t y o b s e r v e d s h e r e i s p r o b a b l y e n t i r e l y due t o t h e movement o f segments o f macro-m o l e c u l e s . Measurements o f damping a r e not e x t e n s i v e enough a t p r e s e n t t o d e t e r m i n e t h e dependence on t e m p e r a t u r e o r c o n c e n t r a t i o n e s p e c i a l l y siince i t i s a l s o c o m p l i c a t e d by the two phase s t r u c t u r e o f t h e system. From the r e s u l t s a v a i l a b l e t h e e f f e c t o f t e m p e r a t u r e seems t o be s m a l l . To f o l l o w page 43 PLATE I V I BIBLIOGRAPHY 1. F i e s e r , L.F., H a m s , S.C., H s r s h b e r g , E.B., Morgana, M., N o v e l l a , F.C.,and Putman, S.T. I n d . Eng. Chem.,38, 788 (1946) 2 R i d e a l , E.K., and o t h e r s P r o o . Roy. S o c , A 200, 135 (1950) 3 S h e f f e r , H. Can. J o u r . Res., B 26, 481 (1948) 4 A l f r e y , T. M e c h a n i c a l B e h a v i o r o f H i g h P o l y m e r s , I n t e r s c i e n c e P u h l i s h e r s I n c . , N.Y., 1948. 5 B u r g e r s , F i r s t Report on V i s c o s i t y and P l a s t i c i t y R o y a l N e t h e r l a n d s Acamemy o f S c i e n c e s , N o o r d - H o l l a n d s c h e , Amsterdam, 1935. 6 A l f r e y , T. and Doty, P.M. J o u r . App. Phys., 16, 700 (1945) 7 M a x w e l l , J.C. P h i l . T r a n s . Roy. Soc. London, 157, 49, 1867. 8 Van Wazer, J.R. and G o l d b e r , H. J o u r . App. Phys., 18, 207 (1947) G o l d b e r g , H. a#d S a n d v i k , 0. A n a l . Chem,, 19, 123 (1947) 9 F e r r y , J.D. Rev. S c i . I n s t . , I S , 79, (1941) 10 Gemant, A. T r a n s . F a r aday S o c , 31, 1582 (1935) 11 Wegel, R.L. and W a l t h e r , H. P h y s i c s , 6 , 141 (1935) 12 F e r r y , J.D., Sawyer, W.M., and Ashworth, J.W. J o u r . Polymer S c i . , 2, 593 (1947) 13 A d l e r , F.T., Sawyer, W.M., and F e r y y , J.D. J o u r . A p p . S c i . , 20, 1036 (1949) 14 Gray, V.R., A l e x a n d e r , A.E. J o u r . Phys. and C o l l . Ohem., 53, (1949) 15 Gunn, G<.B'. M o G i l l Ph.D. T h e s i s , A p r i l , 1950 16 Mark, H. J o u r . App. P h y s . , 12, 41 (1941) 17 F e r r y , J.D. J o u r . Am. Chem. S o c , 64, 1323 (1942) 

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