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Stress relaxation in high polymers Pattison, James Parker 1952

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STRESS RELAXATION IN HIGH POLYMERS By JAMES PARKER PATTISON A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of CHEMISTRY We accept t h i s thesis as conforming to the standard required from candidates for the degree of MASTER OF ARTS 11? > ft 8T Members of the Department of THE UNIVERSITY OF BRITISH COLUMBIA APRIL 1 9 5 2 11 ABSTRACT The prediction of dynamic energy losses from stress relaxation data, using the equation n\fi- ( slope 6-f rctaxrttW caVy/ e ) I 4.totv r v l ; has received considerable attention recently, but for p o l y i s o -butylene the cor r e l a t i o n between predicted and experimental values of energy loss has been poor. This lack of agreement may have been caused by the f a i l u r e of' the simple "step" d i s t r i b u t i o n function of relaxation times to explain the process adequately. In t h i s work, stress relaxation measurements with the RCA 5 7 3 ^ electronic transducer tube have been made i n polylsobutylene at times as early as ,01 second and the re s u l t s show a very rapid r i s e i n stress as t-*0 rather than the constant value required by the "step" function. Another d i s t r i b u t i o n function may be obtained by measuring the slopes of the log, dynamic modulus vs, l o g , 60 , and the s t a t i c modulus vs, log. time curves. For PIB the absence of data i n the region near 1 second leaves a gap of nearly 2 cycles of log, time for which there are no predicted values of the d i s t r i b u t i o n function and the data obtained i n t h i s research may be used to f i l l t h i s gap. S i m p l i f i e d calculations with t h i s function show q u a l i t a t i v e agreement between predicted and exper-imental energy loss i n polylsobutylene. Dynamic data for Acetate rayon show an Increase i n energy loss with Increased humidity, and stress relaxation experiments were performed on t h i s material at various humidities to see i f the effect was apparent. From equation ( l ) an increase i n slope of the relaxation curve i s expected to occur as the i l l humidity i s increased* No i n d i c a t i o n of t h i s effect was found i n the stress relaxation data, and i n f a c t , the relaxation curves at higher humidity had s l i g h t l y lower slopes. This would; tend to indicate that the prediction of energy loss from equation ( l ) i s not v a l i d i n the case of Acetate rayon e ' . . Values of energy loss for Viscose rayon and Egg Albumin f i b e r at 6 5 $ humidity show moderate agreement with those calculated from stress relaxation experiments* TABLE OF CONTENTS Acknowledgements «••««<> o*a« •«.••»••••» X~ Abstract 1 1 Introduction 1 Experimental Method and Results & Discussion of Results . o«.».. »• i l l -Bibliography • o. » 1 9 ACKNOWLEDGEMENT The author wishes to express his sincere appreciation to Dr. E. A. Dunell f o r the h e l p f u l guidance and assistance which he has so fr e e l y given i n the course of t h i s research. 1 INTRODUCTION In recent years, a considerable amount of work has been done on the mechanical properties of high polymers, under both s t a t i c and dynamic conditions. Various stress-strain-time measurements are commonly made, for example, creep under constant l o a d 1 , stress relaxation at constant elongation 1 and energy loss during harmonic v i b r a t i o n 2 . This thesis w i l l be concerned c h i e f l y with stress relaxation, the experiment i n which the v a r i a t i o n of stress with time at constant s t r a i n i s observed. With the types of apparatus commonly used, few stress r e l a x -ation measurements are taken before several seconds have elapsed aft e r extension of the sample 13»^5»l6,l7. Further, dynamic read-ings are seldom made at vibration frequencies below one cycle per second, a frequency which corresponds approximately to one sec-ond on the stress relaxation time scale. This leaves a gap over which no mechanical data are available and- i t i s the object of t h i s work to make stress relaxation measurements i n the region of very short times af t e r straining the sample, corresponding to v i b r a t i o n a l frequencies greater than one cycle per second. High polymers are unique i n that they are able to undergo large deformations under stress, the deformations being reversible to varying degrees. Two effects are apparent when a polymer i s deformed. The f i r s t i s the completely e l a s t i c part which conforms to Hooke's Law and the second, the i r r e v e r s i b l e part, may follow Newton's v i s c o s i t y r e l a t i o n , such that the rate of extension varies as the stress, or may follow some more complicated non-Newtonian v i s c o s i t y r e l a t i o n ^ . The simplest experiment to i l l u s t r a t e these effects i s the so-called "creep" experiment i n which a load i s attached to one Pig. 1. "Creep" experiment end of a filament, the other end being fixed, and the extension i s measured as a function of time (figure 1). The I n i t i a l extension measures the completely reversible part, while that which occurs as time proceeds measures i r r e v e r s i b l e viscous flow. Sometimes a t h i r d mechanism occurs, a retarded e l a s t i c e f f e c t , such that part of the delayed extension i s re-covered. In other words, i f a sample has been undergoing creep and the load i s suddenly released, the extension w i l l suddenly decrease because of the instantaneous e l a s t i c mechanism, and further decrease w i l l occur as time proceeds, caused by the re-tarded e l a s t i c contribution. F u l l recovery w i l l not be obtained, however, because of the i r r e v e r s i b l e flow mechanism. Mechanical properties are generally described i n terms of mechanical models consisting of arrangements of springs and dash-pots to represent e l a s t i c deformation and i r r e v e r s i b l e flow re-spectively. The simplest arrangement i s perhaps the Maxwell Model consisting of one spring and one dashpot i n series (figure 2). I f the unit i s stretched by a f r a c t i o n of i t s o r i g i n a l length £ and the extension thereafter held constant, the stress w i l l decay logari t h m i c a l l y , governed by the equation S = S 0 6 * , where the rate constant i s equal to-^- -?' , This type of experiment, i n which decay of stress i s observed at constant s t r a i n , i s c a l l e d a stress relaxation experiment. 3 Figure 2. Figure 3. Figure Three Maxwell Model, Voigt Model, Element Model The Maxwell unit, might be used to explain creep also but i t has been found more convenient to employ the Voigt u n i t 1 (figure 3) for t h i s purpose. S i m i l a r l y , the Three-element Model^ (figure k-) i s more convenient for explaining some "dynamic11 e f f e c t s , as for example, the v a r i a t i o n of stress with s t r a i n at constant rate of elongation or of l o a d i n g 1 2 , and the s t r e s s - s t r a i n relationships i n some v i b r a t i o n a l experiments^.. The use of a single Maxwell unit leads to a stress r e l a x -ation curve l i k e that shown i n Figure 5» The t h e o r e t i c a l curve may be compared with experimental curves for Hevea gum stock (natural rubberJ 1^ and polyisobutylene* I t may be seen that while the natural rubber produces a stress relaxation curve which corresponds closely to the th e o r e t i c a l curve, the curve for polyisobutylene decays at a much slower rate. To f i t curves for substances which do not follow the th e o r e t i c a l curve, a number of Maxwell u n i t s , each with diff e r e n t T , may be connected i n p a r a l l e l and, i n the case of polyisobutylene, every stress re-laxation curve may be f i t t e d by using the appropriate d i s t r i b u t i o n p o l y l s o b u t y l e n e , s h o w n w i t h t h e p r e d i c t e d r e l a x a t i o n f o r a s i n g l e M a x w e l l u n i t .  F i g . 6 . P a r a l l e l a r r a y o f M a x w e l l u n i t s w i t h " s t e p " d i s t r i b u t i o n f u n c t i o n . of relaxation times (? values). One very simple d i s t r i b u t i o n to which curves for polylsobutylene have been f i t t e d ^ assumes an I n f i n i t e p a r a l l e l array of Maxwell units (figure 6) such that the number of units with relaxation times between ~ and?*<te i s Nt?)d^ , The modulus associated with these units i s Eftyto; the E r l r ) d r „ The p a r t i c u l a r © d i s t r i b u t i o n function ECrJchoseh by Andrews^ Hofman-Bang and Tobolsky . * • i s a "step" d i s t r i b u t i o n which has a constant value between two l i m i t s (figure 6). I t i s e a s i l y shown that the value E' may be obtained from the slope of the s t r a i g h t - l i n e portion of the stress relaxation curve. This function f i t s stress relaxation data for polylsobutylene f a i r l y well at the long-time end but predicts a plateau at the short-time end beginning from and extending toward t - o 0 Another d i s t r i b u t i o n of relaxation times, of which the above i s a s i m p l i f i c a t i o n , i s that obtained by determining the slope at each point along the stress relaxation curve and p l o t t i n g the neg-ative slope against log X •*•« This d i s t r i b u t i o n may be extended to shorter V values by considering dynamic data^. The actual molecular mechanisms involved i n the processes of e l a s t i c deformation and flow have not been completely worked out and, indeed, vary with the type of polymer. For example, poly-sulfide rubber undergo stress relaxation i n accordance with the _ t simple exponential function S = S 0e 7, in d i c a t i n g that a single type of bond i s responsible for the relaxation, while other polymers require a d i s t r i b u t i o n of r v a l u e s to explain the effect. Relax-ation i n polysulfide rubbers has been found to be due to oxidative scission along the molecular chain, f or when the relaxation i s performed i n highly p u r i f i e d nitrogen the relaxation rate i s 5 reduced 1000-fold 1 1. The highly e l a s t i c deformation occuring i n rubber-like high polymers i s ce r t a i n l y due to uncoiling of polymer segments from a random orientation between two Juncture points to a more order-ed one under the action of the stress. Upon release of the tens* ion the segments may go back to t h e i r I n i t i a l random orientation, the process being completely re v e r s i b l e . The loss of "randomness" i s associated with an entropy change and the process i s sometimes referred to as entropy e l a s t i c i t y ^ The flow mechanism, which i s the i r r e v e r s i b l e part of the relaxation,'must involve rearrange-ment of the molecules as a whole with respect to each other. This may be caused by rupture of primary bonds or secondary i n t e r -actions between molecules or by slippage of whole molecules impeded by an in t e r n a l f r i c t i o n . Stress relaxation measurements have, been made with a wide variety of polymers^' 1 0 » 1 5» 1 5»17 a n ^ the general c h a r a c t e r i s t i c s of the stress relaxation curve have been w e l l established. Polyisobutylene has received a great deal of attention i n t h i s respect 7>S»10,lS,19 s i n c e i t i s non-polar and since i t has a very- simple structure, being believed to be completely l i n e a r . Tobolsky and Brown 1 0 have made an intensive study of the elasto-viscous properties of polyisobutylene showing the effect of molec-ular weight, temperature, and elongation. They found that the shape of the stress relaxation curve- (stress/elongation vs. log time) was independent of temperature and elongation, and that a change i n temperature served only to move the curve along the time axis. They also found that a c t i v a t i o n energies for the stress relaxation process were e s s e n t i a l l y constant from 30-100° C. but that at lower temperatures, especially at -35°C. and lower the 6 stress magnitudes increase greatly and curves become non-super-lmposible. At room temperature, then, we would expect a great increase i n the stress which we might observe at very short times aft e r the i n i t i a l s t r a i n , which i s the time o r i g i n for the stress relaxation experiment. Such an expectation i s based on the qual-i t a t i v e equivalence between cooling the sample and observing stress values at shorter times i n a stress relaxation experiments When a v i s c o - e l a s t i c material i s subjected to a sinusoidal stress S = Sbto*(u»t) , the material responds with a sinusoidal v a r i a t i o n i n s t r a i n €= t0cos(u>t£y, In the apparatus used by Dunell p and D i l l o n , a c o i l to which f i b e r s are attached at each end i s suspended between the poles of a magnet and voltage from a low frequency o s c i l l a t o r i s applied to the c o i l . The current pass-ing through the c o i l i s a measure of the stress applied to the system and the maximum elongation 6 0 i s measured by cathetometer e The dynamic properties of a material may be described by a Voigt unit (figure 3) l n which the parameters ^ and E are-fre-quency-dependent. The motion of the system under.sinusoidal f o r c e 2 f r f0tos(u>t)is given by M 1 t ? - " t ? » r & ¥ f - - f . " " * ...:.....u) where X i s the displacement of-the vibrator u n i t , t i s the time, M the mass vibrating,!*) the radian frequency, A the cross-sectional area of the f i b e r and £ the length of each of the two fi b e r s . Equation ( l ) i s solved for X , and from the data at,^mechan-i c a l resonance • — the condition where -X- / f i s maximum with frequency E and ¥| can be c a l c u l a t e d 2 . The assumptions are made that bu> and E are constant with change i n frequency. The value rju> Is proportional to the energy loss per cycle per unit volume of material and may be c a l l e d the hysteresis index. I t has been found possible to predict the value of i^ u) from stress relaxation data . Using the "step" d i s t r i b u t i o n of relax-ation times (figure 6) the value of rjui i s given by where 2.^ 03 S° i s the slope of the stress relaxation curve. Dynamic experiments with Acetate Rayon 2 0 show an increase i n energy loss (v^a) with Increased humidity. As a check of the formula >jw= , stress relaxation experiments were performed on Acetate Rayon. 6 ' EXPERIMENTAL METHOD Many types of apparatus with which to measure stress r e l a x -ation are described i n the l i t e r a t u r e The majority of these 13,21,2k, adapt the beam or platform balance to t h i s use but other types use calibrated metal springs1**" > k y m o g r a p h s 2 2 , and p i e z o e l e c t r i c c r y s t a l s 2 3 0 One method measures the resonance frequency of l a t e r a l vibrations i n a stretched sample to ob-t a i n stress measurement1?. When i t i s considered that measurements as early as .01 second a f t e r extension of the f i b e r must be made, i"t becomes apparent that most of the methods l i s t e d above cannot be used because of high I n e r t i a . However, special transducers with very low i n e r t i a are av a i l a b l e , for example, the mechano-electronic transducer tube RCA 573^. I t was decided to investigate the use-fulness of t h i s vacuum tube i n stress measurement. The RCA 573*4- (Plates l a , lb) i s a small trlode measuring an inch i n length and 3/S inch i n diameter. I t i s constructed of metal such that there i s at one end a diaphragm through which a very t h i n rod of metal protrudes by l / g inch. This rod serves as anode and when a force i s applied to the external end of the rod, the diaphragm bends s l i g h t l y So that the i n t e r i o r part of the anode i s moved r e l a t i v e to the cathode, r e s u l t i n g i n a change i n current through the tube. This change can then be used as a measure of the force applied to the protruding end of the rod. A maximum load of 20 grams can be applied at the end of the rod and at t h i s load the current i s changed by roughly 20$ from that at zero load. The s e n s i t i v i t y of the tube was found to drop as the tube aged, but t h i s i s probably due to excess loads which were inadvertently applied. PLATE I ( c ) D i s t o r t e d S q u a r e W a v e . ( d ) T r u e S q u a r e W a v e 9 Various c i r c u i t s were t r i e d i n order to a r r i v e at the best method of interpreting the signal from the tube, though i t was decided from the start that a cathode ray oscilloscope would be the only suitable Instrument for recording the short-time r e l a x -ation. At longer times, however, the oscilloscope would be rather inconvenient because of d r i f t , and a way of measuring the long-time relaxation had to be devised. As i l l u s t r a t e d i n f i g -ure 7, one possible method was to inse r t a milliammeter i n the plate lead to record the plate current v a r i a t i o n . This method was unsuitable since the change i n current was small i n compar-ison with the normal current* The si m i l a r objection of i n s u f f -iciency i n the change of the position of. the trace on the o s c i l l o -scope with change i n stress also rendered t h i s method unfeasible. Next, the c i r c u i t of figure S was t r i e d . Here, the high D.C. voltage occuring at the anode i s biased to that with the tube i n an unstrained condition, the point A i s at zero v o l t s . In addit-ion, a vacuum-tube voltmeter was connected i n p a r a l l e l with the oscilloscope to measure the long-time relaxation. This c i r c u i t i s v a l i d only because the very high input resistance of both the oscilloscope (2.2 Megohms) and the voltmeter (10 Megohms) allow only an i n f i n i t e s i m a l current i n the plate take-off lead© The instruments used were a Sylvania Model 132 oscilloscope and a Hickok Vacuum-tube Voltmeter. Both were rather unsatisfactory i n that the GRO needed an external DC l i n e a r a mplifier and also had a non-linear sweep while the VTVM d r i f t e d badly. A DuMont Model 3C4 CRO and a Heath VTVM were then obtained. These were both found to be e n t i r e l y s a t i s f a c t o r y . . A Wheatstone Bridge c i r c u i t was considered but since the c i r c u i t described i n the previous paragraph was more d i r e c t i n 13oo v. ( 7 to -aS \ i 2.2 * \o^a r . 1 input <r««i<t<i«t« F i g u r e 7. F i r s t e x p e r -i m e n t a l c i r c u i t . + 300 V. 5 < 10A 1 4 0 V . C*0 VTVM 1 1 F i g u r e 8. S e c o n d e x p e r -i me n t a.1 c i r c u i t . + 300 v. I On 1 F i g u r e 9 . T h i r d e x p e r -i m e n t a l c i r c u i t . F i g u r e 1 0 . F i n a l e x p e r -i m e n t a l c i r c u i t . 1 1 0 application i t was adopted i n preference to the Bridge. On testing the c i r c u i t with a square-wave generator, i t was found that d i s t o r t i o n occurred at short times (Plate I c ) . The trouble was eventually located and found to be caused by capacit-ance i n the bias batteries. The effect interfered only with read-ings up to about \ second, that i s , on those obtained from the CRO. I t was considered that I f the CRO were attached d i r e c t l y to the tube plate through a high-capacity condenser the trouble might be eliminated. The time constant for the c i r c u i t - — time for the voltage to drop to a f r a c t i o n l/e of i t s i n i t i a l value - — might be adjusted to a value greater than a minute so that the condenser i s charged very l i t t l e i n the i n i t i a l second and during that time acts as an ordinary conductor. In figure 9, consider a voltage to appear suddenly at the plate of the tube; the condenser w i l l charge i n accordance with the well-known equation N^cv^O"^^ 8"). I f C R i s large enough, the voltage drop across the condenser during the f i r s t second of relaxation w i l l be very small while the remaining voltage w i l l be across the CRO and w i l l appear on the screen. In these experiments, C I * was set at 1 5 0 , using a 3 0 mfd ^ 5 0 V. ^ondenser and a 2.S Megohm r e s i s t o r i n series with the oscilloscope (2.2 Megohms), at which value the charging during the i n i t i a l second was about 1 ^ . The f i n a l arrangement i s shown i n figure 1 0 . During a run, a ll+ second time exposure photograph of the CRO screen, recording the stress relaxation up to one second, i s taken the instant the f i b e r i s stretched. Then at two seconds, a Gra-Lab e l e c t r i c timer energizes the relay (R i n figure 1 0 ) so that the plate of the 5 7 3 ^ i s disconnected from the CRO c i r c u i t . , 11 arm and connected to the VTVM c i r c u i t arm. Subsequent readings of stress are taken from the VTVM. Instantaneous stretching of the filament i s accomplished by a spring triggered by an electromagnet. The filament clamp (figure 11) i s notched to f i t an ir o n bar which holds the clamp against the action of the spring. When the electromagnet i s switched on, the iron bar i s pulled out of the notch, whereupon the spring stretches the filament. The plate-supply voltage for the 573^ tube i s obtained from a conventional full-wave r e c t i f i e r , voltage regulated to 300 v o l t s D.C. and fed at 115 v o l t s A.C. from a Sola Constant-Voltage trans-former which also supplied both the CRO and VTVM. The r e c t i f i e r c i r c u i t i s shown i n figure 12. The apparatus i s calibrated by measuring the voltage produced when the 573^ tube i s under the tension of the calibrated spring whose length i s measured by a cathetometer. The CRO and VTVM are correlated by means of a dry c e l l and tapping key i n the follow-ing manner. The c i r c u i t shown i n figure 13 i s inserted at Point B i n figure 10 and when the tapping key i s depressed, the voltage change which occurs i s read from the VTVM. The relay i s then switched to the CRO side and the key pressed again, The r e s u l t i n g change i n trace height corresponds to the voltage change read from the VTVM. Since the s e n s i t i v i t y of the tube changed as i t aged, i t was thought advisable to ca l i b r a t e before each run. Calculations have been made as to what i s the e a r l i e s t read-ing which the apparatus may be expected to record r e l i a b l y . The most important factor involved i s the time required for the spring to move the f i b e r clamp weighing i2.7 grams. The only other variable i s the reaction of the tube diaphragm and t h i s has been ] ? i g « 1 1 . D e t a i l o f f i l a m e n t c l a m p . . F i g . 1 3 . C i r c u i t t o "be i n s e r t e d w h e n c o r r e l a t i n g o s c i l l o s c o p e a n d v o l t m e t e r , £o v. «BB 2 5 0 0 "* O D -r8. ^ 1 (2) | | 0 V . A . C TifJWS-fcorner F i g . 1 2 . C i r c u i t d i a g r a m f o r p o w e r s u p p l y . electron betun .> error 0 yew tube F i g . 1 4 . D i a g r a m i l l u s t r a t i n g p a r a l l a x e r r o r w h i c h o c c u r s w h e n CRO i s p h o t o g r a p h e d . 1 2 calculated from tube data to be leas than 1 0 " ^ seconds At a spring tension equal to 1 0 0 grams and for ,1 cm. elongation the response time should be about ©005 second and to allow f or poss-i b l e error, no stress readings corresponding to times less than • 0 1 second were plotted^ from the photographic tracer Values of stress at . 0 1 second are considered to be quite r e l i a b l e , and indeed, on one run stress was measureable at .005 second. The most convenient method for setting the sweep frequency accurately to e s t a b l i s h a time basis for the f i r s t part of the experiment, was to set i t at 1 0 sweeps per second and synchronize t h i s with the 60 cycles per second l i n e frequency. A small parallax error occurs when photographing the face of the GRO tube. This w i l l be apparent from figure 1*4-. A correction curve has been calculated. For runs at temperatures below room temperature, a cold Jacket was constructed. I t consisted of a metal box measuring 2 X 2 X 4 - Inches f i t t i n g t i g h t l y around the clamp which held the 573^ tube, one side of the box being hinged to form a door and the top being l e f t open. A thick layer of powdered asbestos lagging was applied around the entire e x t e r i o r . Low temper-atures were obtained by bubbling nitrogen through a mixture of dry ice and acetone at such a rate as to obtain the required temperature and then forcing the cool gas into the cold Jacket. In t r i a l s with fibrous materials such as rayon, which require regulated humidity, the apparatus was enclosed i n a cardboard box and humid a i r forced into i t (Plates IB, H a ) . A 65$ r e l a t i v e humidity was obtained by bubbling a i r through a saturated solution of Magnesium Acetate at room temperature 2^. In t h i s investigation, stress relaxation experiments were performed on 6 . 6 m i l l i o n molecular weight polyisobutylene (sample W-7-7 of those/polyisobutylene samples Investigated by Andrews and Tobolsky^), 2 5 denier Acetate and 3 ° denier Viscose rayons, and on a sample of egg albumin f i b e r ( 5 * $ steam elong-ated)*. The curves obtained with these materials are shown i n Figures 1 5 , 1 9 * 2 0 , 2 1 . , 22 and i n each case the slopes at both long and short times are Indicated. For comparison, the predicted value of energy los s , fjtA> , calculated from the slope of the pa r t i c u l a r curve and the corresponding value of l^ u) determined experimentally from dynamic data are shown i n Table T.i% * Prepared by A.D. Maclntyre i n t h i s laboratory. Id 16 i 1 4 '12 to 8 VP o 10 .1 10 TiMt 10 Sec. 10 10 4. to Figure 15. Stress Ifclaftcation Paly isobuty Lene . (sample W - 7 - 7 . 6 . 6 m i l l i o n M . W.) 24- °C Figure 16. Composite Curve shovnn with curves of Tobolsky awd Andrews^. (Sample W-7-7) \ \ \ \ 7 \ o \ \ \ \ \ \ • trio** Composite Cwr/e Fi<j /4 . ©.fcMeb M. {royn dynamic data. ) 26 1 c 2 6 , t o * <Ut« . 1 /.j . ,„J6 M ^ i n j u r e 17. D i s t r i l n t i o . i function i n Polylsobutylene. Figure 18 tnor^v loss in Poly i so butyl cnewith points calculated f^ om Function in Figure 17. Figure 20. Stress "Relaxation in Acetate Kayon (2.4°C) • o 8 o 7 o # • °. 5_ 4 o o o o o .Ol t i I me 10 Sec too ,o4 F i g u r e 2 1 . S t r e s s r e l a x a t i o n i n V i s c o s e r a y o n ( 2 4 ° C ) . F i g u r e 2 2 . S t r e s s r e l a x a t i o n i n E g g a l b u m i n f i b e r ( 2 4 ° c ) . PLATE II (a) View of entire apparatus. (b) Typical trace obtained using polyisobutylene • (c) Typical trace obta using rayon. Ik DISCUSSION OF RESULTS The r e s u l t s for polyisobutylene ( 6 . 6 m i l l i o n M.-W.) shown in Figure 1 5 show a rather poor degree of r e p r o d u c i b i l i t y , the extreme high and low values d i f f e r i n g by a factor of about 1 . 5 . However, when mul t i p l i e d by a factor, any curve may be almost exactly superimposed upon another. This suggests that the trouble may l i e i n the measurement of tension. Clearly, i f the point of application of the tension on the protruding end of the transducer tube i s altered, the torque applied to the diaphragm w i l l vary, and since s t r i p s from a large sheet of polyisobutylene were cut for samples,.it i s quite probable that variations i n thickness would occur. In runs using rayon, which has a more uniform cross-section, the r e p r o d u c i b i l i t y was much better. In i. addition, since a l l samples were attached to the transducer tube by polystyrene cement, the p o s s i b i l i t y of having layers of cement of d i f f e r e n t thicknesses cannot be excluded. The v a r i a t i o n i n height between the several curves may be attributed almost completely to t h i s difference i n torque applied to the tube at given f i b e r tension, although some sample-to-sample v a r i a t i o n may be expected to occur from differences i n thickness along the sample. The samples were cut from a sheet of polyisobutylene with a photographic cutter and the cross-sectional area was calculated from the length of the sample, i t s weight and i t s density rather than by di r e c t measurement, because the c r i n k l y nature of the surfaces of the o r i g i n a l sheet made measurement of the thickness tedious and inaccurate. Curve ( l ) i n Figure 1 6 Is a composite curve i n which each point i s the average of the corresponding points on the four experimental curves of Figure 1 5 , and from t h i s composite curve calculations of slope were taken and comparisons were made with the r e s u l t s obtained from the same sample of material at various temperatures by Andrews and Tobolsky^.. In the composite curve, the high i n i t i a l values of stress.predicted to occur at very early times are c l e a r l y indicated. However,, the slopes of the stress relaxation curves obtained i n t h i s research are generally s l i g h t l y higher than those of Andrews and Tobolsky, although the stress values themselves are of s i m i l a r magnitude. Moderate success i s obtained i n superimposing the curves for low temper-'s atures (Figure 16) upon the composite curve at early times, although the low temperature curves contain an almost f l a t portion at times just past those corresponding to the rapid drop i n stress while such a plateau i s found i n only one of the curves obtained here. ' , • ?6 Ferry et a l . have constructed a graph showing the relaxation d i s t r i b u t i o n function for polyisobutylene calculated from stress relaxation and v i b r a t i o n a l data (Figure 17). The curve i s necess-a r i l y broken because of the absence of data to bridge the gap of, nearly two cycles of log time between dynamic data, corresponding to relaxation times less than about a second, and stress relaxation data, corresponding to relaxation times greater than a second. Although the sample used here had molecular weight d i f f e r e n t from that of the sample used by Ferry, the stress relaxation data obtained here at very early times may be used to calculate the values of the d i s t r i b u t i o n function at times not covered by Ferry's data. The d i s t r i b u t i o n function i s obtained by measur-ing the slope at points along the stress relaxation curve and at times corresponding to the relaxation time desired. p o r example TABLE I MATERIAL HUMIDITY SLOPE CALCULATED EXPERIMENTAL VjU(Ref. 20) ACETATE 38^ (1) 7.87 10 9 5.37 10 9 .72 10 9 RAYOM 38:; (2) 4.77 " 3.26 » .72 " 24dC. 58;?' 3.57 2.44 « 1.45 65% 3.46 " 2.36 " 2.24 " 16% 5.37 » 3.66 »• 2.15 «• 82% 3.43 " 2.34 »• 2.15 " VISCOSE 65%, 24*C 6.86 11 4.68 " 4.8 11 * RAYOH ii 6.12 " 4.17 » 4.8 " * • EGG ALBUMIH ii 5.5 " : 3.7 " 1.2 •• "FIBER 24°C • POLYISO-BUTYLEKE SEE FIGURE 18 * Hel 8 _2*\o dynes/cm the height of the d i s t r i b u t i o n function corres-ponding to ^nostc i s given by The values of the function obtained i n t h i s way from the compos-i t e curve (Figure l 6 ) are shown with Ferry's data i n Figure 17 • The values of the d i s t r i b u t i o n function obtained through the com-posite curve for 6.6 m i l l i o n M.W. are higher than those for the not e n t i r e l y due to the difference i n molecular weight, for when the data obtained by Andrews and Tobolsky^ for l.k m i l l i o n M.W. i s used the values for the d i s t r i b u t i o n function are just as high. The reason for t h i s v a r i a t i o n i s not clear for the re s u l t s of Andrews and Tobolsky indicate that the slope of the st r a i g h t -l i n e portion of the.stress relaxation curve i s nearly independent of molecular weight. Hence, the height of the d i s t r i b u t i o n function over the l e v e l portion of the curve should be the same for a l l molecular weights. Stress relaxation data vary Consid-er erf of. erably, however, with d i f f e r e n t observers- 7' ' . The curve of the relaxation d i s t r i b u t i o n function i s observed to pass through a minimum near relaxation times of one second. This i s not apparent from previous d i s t r i b u t i o n functions and indicates that the relaxation mechanisms are divided into two broad groups of relaxation times, one with very short relaxation times and another with long ones. Energy losses i n polyisobutylene have been found to increase sharply with frequency^'^, a fact which cannot be explained i f the simple ''step" d i s t r i b u t i o n function (Figure 6) i s used. With 8 1.2 m i l l i o n M.W. sample used by Ferry, but t h i s i s apparently the function of Figure 17, however, better correl a t i o n between theoreti c a l and experimental values of energy loss w i l l probably be obtained. As a preliminary examination, we may use the formula Y)ixi = - (_^ °Pe) to calculate energy lo s s , taking the slope of the stress relaxation curve at times equal to — . As the slope varies, then, values of energy loss w i l l also vary. While t h i s method i s not intended to give r e s u l t s which are very accurate, i t w i l l indicate whether the more complex d i s t r i b u t i o n function predicts energy loss values which Increase with frequency. A comparison between energy loss values' predicted i n t h i s way and g those determined experimentally i s given i n Figure lS-> Results with fibrous materials show.rather better agreement with theory than result s using polyisobutylene do. Stress relax-a t i o n curves for Acetate and Viscose rayon and egg albumin f i b e r are shown i n Figures 19-22. Experiments with.Acetate rayon were performed at several humidities while the experiments with the other two materials were done at a.constant humidity (65^)o 1 V i b r a t i o n a l experiments with Acetate r a y o n 2 0 indicate an increase in energy loss with increased humidity, and from the stress relaxation theory of energy losses t h i s effect should show up i n a stress relaxation experiment as a more; rapid relax-ation at higher humidity. However, no ind i c a t i o n of t h i s tend-ency is.found i n stress relaxation experiments with Acetate rayon (Figures 19,20). Calculated and experimental values are shown for comparison i n Table I. At no point along the stress relaxation curves for low humidity i s the slope less than that at higher humidity for corresponding time values. This evidence puts the whole theory of prediction of energy loss from stress relaxation slopes i n a rather__.unfavorable l i g h t s 18 The energy loss calculated from stress relaxation data f o r the egg albumin f i b e r i s larger than the measured value of energy loss by a factor of more than 3 &t the lowest value of slope along the curve. In conclusion, the r e s u l t s of these experiments indicate that while the predicted values of energy loss i n polyisobutylene calculated from stress relaxation data compare favorably with experimental values, the values calculated for fibrous materials give rather poor agreement. BIBLIOGRAPHY 1. T. A l f r e y , High polymers Vol. 6, Appendix I I , ;p. 537. Interscience Publishers Inc. (1948). 2. B. A. Dunell and H. D i l l o n , Text. Res. Journ. 21, 393',: (1951). •3. R. D. Andrews and A. V. Tobolsky, J . Polymer Sci» 7, 224, (1951). 4. A. V. Tobolsky and H. Eyring, J . Chem. Phys. 11, 125, (1943). 5. J . C. Maxwell, P h i l . Mag. (IV) 35, 134, (186-8). 6. ¥. Kuhn, Ze i t . physik. Chem. B42, 1, (1939). 7. R. D. Andrews, N. Hofman-Bang and A.. V. Tobolsky, J . Polymer S c i . 3, 669, (1948). 8. B. A. Dunell, PhD Thesis, Princeton Un i v e r s i t y , (1949). 9. R. S. Marvin, S. R. F i t z g e r a l d and J . D. Ferry, J . App. Phys. 21, 513," (1950). 10. A. V. Tobolsky and G. K. Brown, J . Polymer S c i . 6, 165, (1951). 11. M. D. Stern and A. V. Tobolsky, J . Chem. Phys. 14, 93, (1946). 12. R. H. Carey, E.G. Schulz and G. J . Dienes, Ind. Eng. Chem. 42, 842, (1950); .13. A. V. Tobolsky, I. B. Prettyman and J . H. D i l l o n , J . App. Phys.. 15, 380, (1944). 14. R. Meredith, J . Text. Inst. 39, P245, (1948). 15. E. G. Burleigh and H. Wakeham, Text. Res. Journ. 17, 245, (1947). 16. L. R. G. Treloar, Trans. Faraday Soc. 36, 538, (1940). 17. M. Mooney, ¥. E. \7olstenholne and D. S. V i l l a r s , J . App. Phys. 15, 324, (1944). 18. R. Sips, J . Polymer S c i . 5, 69, (1950). 19. J . D. Ferry, \I. !€• Sawyer, G. V. Browning and A. H. Groth, J . App. Phys. 21 9 197, (1950). 20 20. A. D. Haclntyre, K. A. Thesis, University of B r i t i s h Columbia, 1952. 21. L. E. Peterson, R. L. Anthony and E. Guth, Ind. Eng. Chem. 34, 1349, (1942). 22. H. Walceham and E. Honold, J . App. Phys. 17, 698, (1946). 23. H. Tohara, J . Phys. Soc. Japan 4, 139, (1950). 24. R. L• Anthony, R. H. Gaston and E. Guth, J . Phys. Chem. 45, 826, (1942). 25. Handbook of Chemistry and Physics, p. 1925. Chemical Rubber publishing Co. (1947). 26. J . D. Perry, E. R. F i t z g e r a l d , K. F. Johnson and L. D. Grandine J . App. Phys. 22, 717, (1951). 

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