that V. OF OF

C H A P T E R 2

STRESS RELAXATION STUDIES O F THE VISCOELASTIC PROPERTIES O F POLYMERS

Arthur V . Tobolsky

I. F o u r R e g i o n s of V i s c o e l a s t i c B e h a v i o r for A m o r p h o u s B e h a v i o r 64

I I . T h e T i m e - T e m p e r a t u r e Superposition P r i n c i p l e 67 I I I . Parameters Defined b y the R e l a x a t i o n M a s t e r Curves 68

I V . Stress Birefringence R a t i o and the M e c h a n i s m of V i s c o e l a s t i c D e f o r m a t i o n 70 V . D i s t r i b u t i o n of R e l a x a t i o n T i m e , F l o w V i s c o s i t y , and D y n a m i c M o d u l u s . . 72 V I . T h e Characteristic R e l a x a t i o n T i m e K(T) as a F u n c t i o n of T e m p e r a t u r e . . 75

V I I . Stress R e l a x a t i o n of P o l y crystalline P o l y m e r s 76 1. D e c a y of Stress of C o n s t a n t E x t e n s i o n Resulting from the G r o w t h of

Oriented Crystallites 78 V I I I . Chemical Stress R e l a x a t i o n ( C h e m o r h e o l o g y ) 79

N o m e n c l a t u r e 81

A n ideally elastic substance subjected t o simple tension o b e y s H o o k e ' s law in the region of small strains:

Stress = Ε X Strain

T h e quantity Ε is a constant k n o w n as the Y o u n g ' s modulus. I t is a v e r y important property of the material in question, being a measure of stiffness or resistance t o deformation. F o r m a n y metallic substances the value of Ε is of the order of 1 0¹¹ d y n e s / c m .². F o r soft vulcanized rubbers the value of Ε is of the order of 1 0⁷ d y n e s / c m .². T h e Y o u n g ' s modulus of a substance generally varies with the temperature.

T h e c o n c e p t of modulus can b e generalized t o nonideally elastic sub- stances b y recognizing that the m o d u l u s is a function of time and also t o some extent a function of the m e t h o d of measurement. F o r example, ex- periments m a y b e c o n d u c t e d in which the substance is subjected t o a fixed extension, and the stress is measured as a function of time. T h e stress per unit strain necessary t o maintain the sample at constant extension will b e defined as the stress relaxation modulus E^r(t). Similarly, the substance m a y b e subjected t o a constant tensile load, and the extension measured as a function of time—a so-called creep experiment. T h e stress per unit strain measured b y this kind of experiment will b e denoted b y E^c(t), the creep modulus. Furthermore vibrational experiments m a y b e carried o u t

63

in which the substance is subjected to vibrations of angular frequency ω.

F r o m these experiments a complex d y n a m i c modulus Ε'(ω) + iE" (ω) m a y be determined. (In actual fact the vibrations used are generally shear vibrations so that the quantity really determined is G'(co) + i(j"(co), where the s y m b o l G denotes shear m o d u l u s ) .

I n most cases the moduli E^r(t), E^c(t) and Ε'(1/ω)—where ω corresponds to 1/t— are approximately equal. F o r substances which o b e y the laws of classical linear viscoelasticity, a determination of any one of these quantities over the entire time scale can be used t o calculate the other t w o time-dependent moduli, or indeed can be used in principle t o predict the results of any viscoelastic experiment.

During the past several years extensive data have been accumulated in our laboratory concerning the stress relaxation modulus E^r(t) of a wide va- riety of polymers over a wide temperature range. T h e practical experi- mental methods for measuring stress at constant extension v a r y with the magnitude of the stiffness m o d u l u s and are described in the original papers.

T h e following pages present a brief summary and generalization of some of the experimental and theoretical findings arising from this work.

Polymers divide v e r y sharply into t w o categories so far as their physical properties are concerned: on one hand there are the amorphous p o l y m e r s whose X - r a y scattering is very similar t o the scattering of liquids; on the other hand there are the p o l y crystalline polymers. T h e viscoelastic proper- ties of amorphous polymers are relatively simpler than those of p o l y crystal- line polymers. In the first place amorphous p o l y m e r s o b e y the classical laws of linear viscoelasticity (at small deformations) whereas crystalline polymers frequently d o not. Secondly amorphous polymers o b e y a time- temperature superposition principle which permits the use of data obtained at different temperatures t o extend the time scale at any temperature.

This is n o t generally true for crystalline polymers. Finally, the amorphous polymers show m a n y features of similarity t o one another in their visco- elastic behavior. F o r these reasons I shall discuss them first.

I. Four R e g i o n s o f Viscoelastic B e h a v i o r f o r A m o r p h o u s P o l y m e r s T h e first complete viscoelastic data for any polymers were obtained b y stress-relaxation studies on p o l y m e t h y l m e t h a c r y l a t e of t w o different m o - lecular w e i g h t s .1 ,2 T h e stress relaxation data at various temperatures are plotted in the form log E^r(t) v s log t for the entire temperature range in Fig. 1. In Fig. 2 are shown similar results for polyisobutylene of molecular weight 1.35 Χ 1 0⁶ for average viscosity. T h e pattern of behavior for these polymers as shown in these figures is v e r y m u c h the same, and this is also

1 J . R . M c L o u g h l i n and Α . V . T o b o l s k y , Colloid Sei. 7, 555 (1952).

2 Α . V . T o b o l s k y and J . R . M c L o u g h l i n , J. Polymer Sei. 8, 543 (1952).

S T R E S S R E L A X A T I O N S T U D I E S 6 5

.001 .01 .1 I 10 100 1000 TIME,HRS.

F I G. 1. L o g Er(t) v s . l o g ( 0 for p o l y m e t h y l m e t h a c r y l a t e1

true of other amorphous p o l y m e r s . A s was first pointed out b y T o b o l s k y and M c L o u g h l i n² and as can b e clearly seen from the figures, there are four well-defined regions of viscoelastic b e h a v i o r :

1. A low-temperature glassy region in w h i c h the m o d u l u s E^r(t) is of the order of magnitude of 1 0^{1 0 X5} d y n e s / c m .². T h e time effects are n o t v e r y pronounced in this region so that w e m a y speak of a quasi-static glassy modulus Ει. E^r(t) is fairly independent of molecular weight in this region b u t depends v e r y markedly on the rate at which the sample is annealed.³ 2. A transition region of viscoelastic behavior in which E^r(t) changes v e r y rapidly with time and temperature, the values of E^r(t) ranging between 1 0¹⁰ d y n e s / c m .² and 1 0⁷ d y n e s / c m .². T h e p o l y m e r s feel " l e a t h e r y " t o the t o u c h in this region. T h e values of E^r(f) are independent of molecular weight in this region for polymers of sufficiently high molecular weights.⁴ T h e time effects in the transition region are p r o d u c e d b y short-range motions of relatively small segments of the p o l y m e r molecules.

3. A quasi-static " r u b b e r y p l a t e a u " region in which the modulus E^r(t) is of the order of magnitude of 1 0⁷ d y n e s / c m .² and changes quite slowly with

3 J. R . M c L o u g h l i n and Α . V . T o b o l s k y , J. Polymer Sei. 7, 658 (1951).

4 G . M . B r o w n and Α . V . T o b o l s k y , J. Polymer Sei. 6, 165 (1951).

0.01 100

F I G . 2 . 1Q6 2 5 , 26

1.0 TIME, H RS.

L o g Er(t) v s . l o g (t) for p o l y i s o b u t y l e n e of molecular weight Mv = 1.35 X

time. T h e stress in this region appears t o b e maintained b y "entangle- m e n t s " between the long chain molecules as was first pointed o u t in these researches.⁵ F r o m the rubbery plateau region a quasi-static rubbery m o d u - lus E² can be defined, and the molecular weight between "entanglements"

can be calculated.⁵ T h e presence of chemical cross-links will of course increase the value of the rubbery modulus E².

4. A flow region in which E^r(t) changes v e r y rapidly with time from values of 1 0⁷ d y n e s / c m .² t o values of zero. T h e E^r(t) curves in this region are v e r y markedly dependent on the molecular weight and the molecular- weight distribution. F o r samples of the same molecular-weight distribution b u t different average molecular weight M^v, the E^r(t) curves appear t o b e shifted horizontally along the log-time axis, the shifting factor being 3.3 log M^v .⁶ F o r samples of different molecular-weight distribution, b u t

5 H . M a r k and Α . V . T o b o l s k y , " P h y s i c a l Chemistry of H i g h P o l y m e r i c S y s t e m s , "

Chapter X , p . 344. Interscience N e w Y o r k , 1950.

6 R . D . A n d r e w s , N . H o f m a n B a n g , and Α . V . T o b o l s k y , Polymer Sei. 3 , 669 (1948).

STRESS R E L A X A T I O N S T U D I E S 67

the same value of M^v, the shape of the E^r(t) curve vs. log t curve appears t o change, but the relative position along the log-time axis remains the

5, 7

same.

T h e time effects in the rubbery flow region are due t o motions of the mole- cules as a w h o l e or t o v e r y large molecular segments.

W h e n the chemical cross-linking between the linear chains is sufficient t o form a continuous three-dimensional structure, the flow region is c o m - pletely suppressed, t o be replaced at higher temperatures b y a region of chemical flow p r o d u c e d b y the breaking of primary valence b o n d s b y o x y - gen, b y ionic interchanges, or b y other means.

I n the original publication,² the rubbery plateau region and the flow region were considered together as the " r u b b e r y flow" region. It seems desirable t o separate these i n t o t w o regions for purposes of greater clarity.

II. The T i m e - T e m p e r a t u r e S u p e r p o s i t i o n Principle

T h e time-temperature superposition principle w h e r e b y viscoelastic data at one temperature are transformed t o another temperature b y a simple multiplicative transformation of the time s c a l e⁶' ⁸'⁹'^{1 0} was independently proposed b y T o b o l s k y and A n d r e w s⁸ and b y Leaderman.⁹ I n its simplest form this principle means that curves of E^r(t) or log E^r(t) v s . log t that are obtained at different temperatures can b e superimposed b y means of a horizontal shift along the log t axis. Refinements on this procedure are treated elsewhere.¹¹

B y means of this principle it is possible t o extend the values of E^r(t) o b - tained at any temperature t o b o t h shorter and longer times than can b e obtained experimentally. Master curves of log E^r(f) v s . log t can b e con- structed that are applicable for all temperatures and all times. Figure 3 shows a somewhat schematized master curve for unfractionated p o l y i s o - butylene of three different average molecular weights. T h e graph as it stands is valid for 25° C . because the log time origin (log t = 0) is directly under the indicated mark for 25° C . T h e master curve for any temperature shown on the graph can easily b e obtained b y sliding the log time axis t o the new indicated origin. T h e number of decades of time comprised in this master curve is truly staggering. T h e validity of the time-temperature

7 R . D . A n d r e w s , F . H . H o l m e s , and Α . V . T o b o l s k y , unpublished results.

8 Α . V . T o b o l s k y and R . D . A n d r e w s , J. Chem. Phys. 11, 125 (1943); Polymer Sei. 3, 669 (1948).

9 H . L e a d e r m a n , " E l a s t i c and C r e e p Properties of F i l a m e n t o u s M a t e r i a l s . " T e x - tile F o u n d a t i o n , W a s h i n g t o n D . C . 1943.

1 0 J. D . Ferry, J. Am. Chem. Soc. 72, 3746 (1950).

1 1 J. D . F e r r y and E . R . Fitzgerald, J. Colloid Sei. 8, 224 (1953); see also E . Catsiff and Α . V . T o b o l s k y , J. Colloid Sei. 10, 375 (1955).

68 A R T H U R V . T O B O L S K Y

Π ι ι ι ι ι ι ι I I 1 I I ! I ! I ! I L

Log time, hours

F I G. 3. Idealized master relaxation curve l o g Er(t) v s . l o g (t) for p o l y i s o b u t y l e n e of three different average molecular weights (1) M^v = 1.36 Χ 1 0⁶; (2) M^v = 2.80 X 1 06; (3) Μυ = 6.60 Χ 1 06.1

superposition principle will be p r o v e d later when relaxation modulus and d y n a m i c modulus are compared.

III. P a r a m e t e r s D e f i n e d b y the R e l a x a t i o n M a s t e r C u r v e s

Stress-relaxation data, such as are e m b o d i e d in Figs. 1, 2, and 3 can b e used to define certain parameters that are characteristic of the polymers being studied.12 In the first place w e have Ex and E2, the quasi-static glassy modulus and the quasi-static rubbery modulus. A t any temperature Τ the time required for the logarithm E^r(t) t o reach the value (log Ε^λ + \ogE²)/2 was called the characteristic relaxation time Κ.¹² Κ is o b v i o u s l y a function of temperature K(T). A n attempt was also m a d e t o define a distinctive temperature Td from the Er(t) curves alone which w o u l d correspond in some w a y t o the glass transition temperature T^g obtained from v o l u m e - temperature curves. A s is well k n o w n , evidence exists which indicates that the T^g value obtained from V-T curves depends on the rate at which specimen is cooled (or heated). I t was h o p e d that the log Er(t) vs. log t curves would provide a value of Td that did not depend u p o n time effects.

Td was defined as the temperature at which K(T) had a m a x i m u m value of apparent activation energy. T h e apparent activation energy is defined as Rd In K/d(\/T). A difficulty that arises with this definition is that the time-temperature superposition principle v e r y p r o b a b l y becomes only partially valid at temperatures close t o Tg and lower. T h e horizontal shifts along the log time scale which produce superposition of the log Er(t) curves

1 2 J. Bischoff, E . Catsiff, and Α . V . T o b o l s k y , Am. Chem, Soc. 74, 3378 (1952).

S T R E S S R E L A X A T I O N S T U D I E S 6 9

for temperatures below T^g m a y well be valid only for the time scale of the relaxation experiments. Nevertheless, the values of T^d obtained b y the m e t h o d described a b o v e seems t o be v e r y close t o the T^g values obtained from V-T curves, at least for the p o l y m e r s w e have studied thus far. T h e determination of T^d appears t o have experimental-operational validity.

T h e value of T^g or T^d is perhaps the most important characteristic of the p o l y m e r . Should our experimental-operational definition of T^d fail for certain polymers, our discussion of these polymers could be couched in terms of T^g rather than of T^d .

T h e characteristic relaxation time Κ at the temperature T^d is denoted h^YK^d.

Another important parameter which characterizes the master relaxation curves is the quantity n, which is the negative slope of the master curve log E^r(t) v s . log t at the point at which log E^r(t) = (log Ε^λ + log E²)/2.

A n o t h e r characteristic parameter is the quantity h, related t o n, whose quantitative definition will be discussed subsequently.

T a b l e I contains a compilation of the parameters Ει, E², h, n, T^d , and K^d for the polymers whose viscoelastic properties have been studied b y stress relaxation.

It is, of course, of great interest t o inquire h o w these parameters depend on the molecular structure of the p o l y m e r s . A m o n g the polymers studied there does not appear t o be a tremendous variation in the values of E\ and E². T h e glassy modulus Ei m a y well depend u p o n the strength of the secondary valence forces between atomic groups and u p o n the a m o u n t of

"free v o l u m e " in the glass. Nevertheless the value of E\ obtained thus far for various polymers appears t o lie in the range 1 0¹⁰ t o 1 0^{1 0 X5} d y n e s / c m .². E² will depend on the chain stiffness in the case of linear polymers and u p o n

T A B L E I

P A R A M E T E R S CHARACTERIZING S T R E S S R E L A X A T I O N OF AMORPHOUS P O L Y M E R S

Polymer Tdi°K log Ei log E² η h log Kd

P o l y m e t h y l m e t h - 3 8 4 1 0 . 3 5 7 . 3 5 0 . 5 25 0 . 3 1 - 1 . . 5 acrylate

Paracril 2 6 2 4 1 . 0 1 0 . 1 0 7 . 4 0 0 . 6 3 0 . 4 15 - 1 . . 5 4

G R - S 2 2 0 1 0 . . 2 4 7 . 4 4 0 . . 7 1 0 . 4 5 - 1 . 5 0

6 0 / 4 0

b u t a / s t y 2 3 7 . 1 1 0 . 2 7 8 . 0 3 0 , 5 05 0 . 4 0 - 1 . 9 4 5 0 / 5 0

b u t a / s t y 2 5 0 . 8 1 0 . , 2 1 2 7 . 5 5 8 0 . . 5 4 5 0 . 3 6 4 - 1 . 1 2 3 0 / 7 0

b u t a / s t y 2 8 5 . 1 9 , . 9 5 5 7 . 2 5 5 0 , . 5 5 0 . 3 6 - 1 . 1 3 P o l y i s o b u t y l e n e 1 9 7 . 0 1 0 , . 4 8 6 . 8 8 0 . 7 4 5 0 . 3 6 7 + 1 . 0 4

( D y n a m i c )

—

70

the amount of cross linkage in the case of cross-linked p o l y m e r s . T h e quantity η and the related quantity h m a y depend u p o n such factors as chain configuration and the heterogeneity of composition along the chain.

T h u s far the variation in these quantities as between different polymers has n o t p r o v e d t o b e v e r y great, although larger variations have been found in the case of other polymers n o w under study.

T h e parameter which is of greatest importance in determining the p r o p - erties of amorphous p o l y m e r s is Td (or T^g ) . T o a v e r y crude approximation one can state that the viscoelastic properties of m o s t amorphous p o l y m e r s are roughly similar for the same values of T/T^du or Τ - T^d . T h e qualita- tive validity of this principle m a y be verified b y inspecting Figs. 1 and 2 . T h e value of T^d (or T^g) for any given p o l y m e r is determined b y the cohesive energy density of the p o l y m e r (measured b y swelling or solubility) and the stiffness of the chains (which can be measured b y light scattering in p o l y m e r solutions). T h e higher the cohesive energy density, the higher is the value of T^d ; the stiffer the chain, the higher the value of Td . T h e latter point is nicely demonstrated b y the fact that polyacenaphthalene has a m u c h higher T^g value than polystyrene.

A l t h o u g h all amorphous h o m o p o l y m e r s and h o m o g e n e o u s c o p o l y m e r s studied thus far show the qualitative features of viscoelastic behavior de- scribed a b o v e , p o l y m e r alloys or p o l y b l e n d s , which are finely dispersed mechanical mixtures of t w o p o l y m e r s just barely on the borderline of c o m - patibility, show a splitting of the transition region into t w o discrete parts,¹⁴ as if the p o l y b l e n d had preserved the t w o Td values of each individual c o m - ponent.

I V . Stress Birefringence R a t i o a n d the M e c h a n i s m o f V i s c o e l a s t i c D e f o r m a t i o n

F o r an ideal cross-linked rubber (in the rubbery plateau region of visco- elastic b e h a v i o r ) , the stress at constant extension is proportional t o the absolute temperature. T h i s means that the entropy of the rubber sample is decreased when the rubber is stretched isothermally, and the stress in the stretched rubber sample actually arises from this decrease in entropy.

T h e decrease in entropy occurring when a rubber sample is stretched is, in turn, due t o a decrease in the long-range configuratiönal possibilities of the chain molecules. Since birefringence is also a measure of the long-range chain configurations, it is not surprising that the following relation can be derived.15

Stress

™ τ ^w ρ - ^£ . = constant

l e m p e r a t u r e X Birefringence

1 3 Α . V . T o b o l s k y and E . Catsiff, J. Am. Chem. Soc. 76, 4204 (1954).

1 4 R . B u c h d a h l and L . E . Nielsen, J. Polymer Set. 1 5 ,1 (1055).

STRESS RELAXATION STUDIES 71 T h e value of the constant in the above equation can be used to deter- mine the size of the "freely rotating segment" of the polymer c h a i n s .1 5 , 16

It was further shown that the quantity stress/(temperature X birefrin- gence) is a constant independent of temperature and time during stress-re- laxation experiments, provided that the experiments were carried out in the rubbery flow region, the rubbery plateau region, or the low modulus por- tion of the transition r e g i o n .1 7 , 18 This indicates that in these regions of visco- elastic behavior even during a stress-relaxation experiment, where stress is changing rapidly with time, the stress must be considered as an "entropy stress," arising from long-range configurational changes. On the other hand, as one enters the high modulus portion of the transition region and the glassy region, the ratio stress/(temperature X birefringence) changes its magni- tude and often changes its s i g n .1 7 , 19 This indicates that the molecular mechanism of stretching in the glassy region must be different from that operating in the rubbery region. V e r y probably the stress in the glassy region arises from internal energy changes corresponding to distortion of van der Waals' bonds and bending of valence angles, rather than from configurational entropy changes.

This fundamental change in the mechanism giving rise to stress should be taken as an indication that the time-temperature superposition principle may become invalid (or only partially valid) at temperatures close to and below the glass transition temperature.

1. M A T H E M A T I C A L FUNCTIONS D E S C R I B I N G Er(i)

T h e function Er(t) has been fitted quite well b y empirical functions in both the transition region and the rubbery flow region. In the transition region the following empirical function holds very w e l l :1 2 , 20

log

(t/K) =

+

- ( t o g ^ - t o g ^

where

erf χ = 2τ γ "^{1 /2} Γ e~^u* du.

Jo

Erf is an abbreviation of the term error function, and A is the adjustable parameter of the Gauss error c u r v e ) . T h e q u a n t i t y A, related t o n, has

1 5 W . K u h n and F. Grun, Kolloid-Z. 101, 248 (1942); L . R . G . Treloar, Trans. Fara- day Soc. 43, 277, 289 (1947).

1 6 R . S. Stein and Α . V . T o b o l s k y , Polymer Sei. 11, 285 (1953).

1 7 R . S. Stein, S. K r i m m , and Α . V . T o b o l s k y , Textile Research J. 19, 8 (1949).

1 8 R . S. Stein and Α . V . T o b o l s k y , J. Polymer Sei. 14, 443 (1954).

1 9 J . R . M c L o u g h l i n , P h . D . thesis, Princeton University (1950).

2 0 E . Catsiff and Α . V . T o b o l s k y , J. Appl. Phys. 25, 1092 (1954).

72 A R T H U R V . T O B O L S K Y

already been considered as one of the characteristic parameters of the master relaxation curve E^r(t).

I n the region of rubbery flow the relaxation curve E^r(t) for p o l y i s o b u t y l - ene at 298° K . is fitted quite well b y the following empirical f u n c t i o n :^{6 , 21} '^{2 2} E^r(t) = E⁰[Ei( — t/n) — Ei( — t/r^m)] where Ei(—x) is the exponential integral function, r³ and r^m are parameters that depend on the 3.3 p o w e r of the average molecular weight and E⁰ is a parameter that appears t o depend u p o n the molecular-weight distribution (see ref. 5, p . 3 4 3 ) .

F o r a class of amorphous polymers, n o t a b l y the butadiene-styrene c o - polymers, the empirical relation hT^d = 100 appears t o be valid. W h e n this is true, expressions for E^r(t), which are essentially universal functions of T/Td, can b e d e r i v e d .¹³ T h i s reduced equation of state for viscoelastic behavior is not valid for polymers like polyisobutylene where hTd 9^e 100.

V . Distribution o f R e l a x a t i o n Times, Flow Viscosity, a n d D y n a m i c M o d u l u s

A distribution function of relaxation times H (log r ) can be defined in terms of E^r(t) b y means of the following integral equation:

E^r(t) = Γ H(logr)e-^tlTd(\ogr)

J—oo

T h e distribution function Η (log r ) is v e r y useful since other viscoelastic properties such as the flow viscosity and the real and imaginary c o m - ponents of the d y n a m i c modulus can be calculated therefrom.

τΗ (log r ) d(\og τ)

" O O

E

" = Γ

1\

rf(log

Formulas relating E^f and E" t o the distribution of relaxation times were first presented b y T o b o l s k y and E y r i n g .^{2 2 , 23} A l l of these formulas presup- pose the validity of the laws of linear viscoelastic behavior. T h e n laws h a v e been p r o v e d valid for amorphous p o l y m e r s b u t are invalid for crystalline polymers under certain conditions.

A n idealized distribution of relaxation times at 298° K . had previously been presented (in ref. 22) which had the simple graphical aspects of a

" b o x " for the rubbery flow region and a " w e d g e " for the transition region.

2 1 R . D . Andrews and Α . V . T o b o l s k y , J. Polymer Sei. 6, 221 (1951).

2 2 Α . V . T o b o l s k y , / . Am. Chem. Soc. 74, 3786 (1952).

2 3 Α . V . T o b o l s k y and H . E y r i n g , / . Chem. Phys. 11, 125 (1943).

S T R E S S R E L A X A T I O N S T U D I E S 73

14 -12 - Ι Ο - 8 - 6 - 4 - 2 - L O G u > X^{2 9 8 l}( H R S . )

F I G. 4. D y n a m i c shear m o d u l u s G' v s . n e g a t i v e l o g frequency for p o l y i s o b u t y l e n e at 25° C . C o m p a r i s o n of experimental points w i t h solid c u r v e p r e d i c t e d from stress relaxation r e s u l t s .^{2 5, 26}

T h e " w e d g e " is unaffected b y the molecular weight of the sample (for sufficiently high molecular weights), whereas the " b o x " is shifted along the log time axis with a shift factor 3.3 log M^v . A l t h o u g h this idealized function is o n l y an approximation of the true values of Η (log τ), it has the virtue that exact mathematical expressions for E^r(t) and for E' and E" can be derived from it, so that n o mathematical approximations are necessary for converting relaxation m o d u l u s t o d y n a m i c m o d u l u s in this case.

A v e r y accurate determination of Η (log r) can be obtained from the ex- perimental value of E^r(t) b y the use of first- and second-order approxima- t i o n s .2 4, 2 5 , 26 T h e values of Η (log r) at 298° K . as a function of log r are tabulated in ref. 26.

F r o m the distribution function Η (log τ), obtained entirely from stress- relaxation data and the stress-relaxation data master curve E^r(t), one can c o m p u t e and predict the values of Ε'(ω) and Ε"(ω). Such c o m p u t a t i o n s (and tabulations) h a v e been m a d e in refs. 25 and 26. Figures 4 and 5 show the comparison between predicted values and the experimental values

2 4 J. D . F e r r y , E . R . F i t z g e r a l d , L . D . G r a n d i n e , Jr., and M . L . Williams, Ind. Eng.

Chem. 44, 703 (1952).

2 5 E . Catsiff and Α . V . T o b o l s k y , J. Colloid Sei. 10, 375 (1955). See also J. Appl.

Phys. 25, 145 (1954).

2 6 Α . V . T o b o l s k y and E . Catsiff, J. Polymer Sei. 19, 111 (1956).

A R T H U R V . T O B O L S K Y

obtained elsewhere.²⁷ T h e extremely g o o d agreement provides another verification of the validity of the theory of linear viscoelastic behavior in the case of p o l y i s o b u t y l e n e . I t also provides a verification of the time- temperature superposition principle, because stress-relaxation data and dynamic-modulus data are obtained in v e r y different time intervals. Figure 6 shows stress-relaxation data and d y n a m i c data for polyisobutylene o b - tained at — 4 4 ° C . T h e solid curve represents the master curve E^r(t) o b - tained from stress-relaxation data at various temperatures b y the time- temperature superposition principle. T h e agreement between the t w o types of data is striking.

T h e relaxation curves of polyisobutylene in the rubbery flow region have been used t o predict the bulk viscosity, using the " b o x " distribution as the approximation for the relaxation time distribution in that region.²⁸ V e r y satisfactory agreement was obtained between predicted values and experi- mental values of viscosity.

2 7 J. D . Ferry, L . D . Grandine, Jr., and E . R . Fitzgerald, J. Appl. Phys. 24, 911 (1953).

2 8 R . D . Andrews and Α . V . T o b o l s k y , Polymer Sei. 7, 221 (1951).

S T R E S S R E L A X A T I O N S T U D I E S 75

N.as. PO -YISOBUTYL EN Ε

• OYNAMIC SH ο S T R E S S - R E

— S T R E S S - R E

EAR MODULUS (X LAXATION MODU LAXATION MAST

3 ) AT - 4 4eC LUS AT - 4 4 * C

ER CURVE AT 4 - C \

LOG TIME ( HRS. )

— LOG FREQUENCY (RAD./HR.)

F I G . 6. D y n a m i c shear modulus (times three) v s . negative l o g frequency and stress relaxation m o d u l u s v s . l o g time for p o l y i s o b u t y l e n e at 4 4 ° C . Solid c u r v e repre- sents the stress relaxation master c u r v e . Shear m o d u l u s Gf multiplied b y three t o c o n v e r t t o a tension m o d u l u s .

V I . The Characteristic R e l a x a t i o n Time K(T) a s a Function o f T e m p e r a t u r e

It is clearly v e r y desirable t o express the temperature variation of K(T) in a manner that w o u l d b e applicable t o all linear p o l y m e r s . Catsiff and T o b o l s k y had suggested that h log K(T)/(K^d) v s . T/T^d should be the same for all amorphous p o l y m e r s .^{1 3 , 12} R e c e n t l y W i l l i a m s²⁹ has proposed that log A^T V S . Τ — T⁸ is a universal function for all polymers, where T⁸ is a reference temperature selected for each p o l y m e r . T h e function A ^T is linearly proportional t o K(T). T h e t w o theories discussed a b o v e are in agreement over the limited temperature interval over which they were c o m m o n l y tested. Williams, Landel, and Ferry h a v e ascribed a functional form for the variation of log A ^T with Τ — T^s .^{3 0} Expressed in terms of the notation and the parameters used in this article this new theory has the following f o r m :

, K(T)

. Τ — Td

l 0 g- K T ⁼ -^{1 6}-^{1 4}5 6 + T - T ^d

All of our values for K(T) for the p o l y m e r s studied b y stress relaxation fit this formula v e r y w e l l .^{2 5 , 26} T h e agreement obtained with this formula and the experimental data is shown in Fig. 7.

2 9 M . L . W i l l i a m s , J. Phys. Chem. 59, 95 (1955).

3 0 M . L . W i l l i a m s , R . F . Landel, and J. D . F e r r y , J . Am. Chem. Soc. 77, 3701 (1955).

76

- 2 0 -10 0 10 20

FIG. 10 T-T

F I G . 7 . L o g K(T)/Kd v s . Τ - Td for several p o l y m e r s2 6' 26

V I I . Stress R e l a x a t i o n o f Polycrystalline P o l y m e r s

In the previous discussion of the viscoelastic properties of linear amor- phous polymers it was shown that certain features are c o m m o n t o the b e - havior of all of these p o l y m e r s . I n particular it was shown that there are four characteristic regions of viscoelastic b e h a v i o r : a glassy region, a transi- tion region, a rubbery plateau region, and a rubbery flow region. In Fig. 2 which presents log E^r(t) v s . log t for polyisobutylene the behavior in the transition region is t o be especially noticed. In the time scale of the relaxa- tion experiments (.01 to 10 hr.) the transition region occurs in the tempera- ture interval between —80° C . and —40° C , a matter of 40 degrees. In this interval the modulus changes v e r y rapidly with time and temperature be- tween values of 1 0¹⁰ d y n e s / c m .² and 1 0⁷ d y n e s / c m .² T h e behavior of par- tially crystalline polymers is rather different and shall b e discussed, using polytrifluorochloroethylene as a well studied example.

Polytrifluorochloroethylene

Polytrifluorochloroethylene is a polycrystalline p o l y m e r whose melting temperature T^m is 215° C . Its glass transition temperature has not been

S T R E S S R E L A X A T I O N S T U D I E S 77

accurately determined b u t should b e somewhere in the neighborhood of r o o m temperature if the approximate relation T^g = % T^m is valid. Figure 8 shows stress relaxation d a t a³¹ for polytrifluorochloroethylene in the tem- perature range between 30 and 193° C . I n this interval the modulus varies from 1 0^{1 0 1}t o 1 0⁷'² d y n e s / c m .² I t is particularly interesting t o contrast Figs.

2 and 8. I n Fig. 8 the log E^r(t) v s . log t curves between 30° and 144° C . are relatively flat, i.e., the modulus change with time in the ''transition r e g i o n " p o l y m e r is m u c h smaller for the polycrystalline p o l y m e r as c o m - pared t o the amorphous p o l y m e r s . A l s o in Fig. 8 the value of log E^r(t) at t = 0.01 hr. changes from a value of 1 0^{1 0}'¹ d y n e s / c m .² at 30° t o a value of 1 0⁸'³⁵ at 144° C , a v e r y gradual change. T h e transition region, if such it can b e called, for a polycrystalline p o l y m e r extends over a m u c h wider temperature range than for an amorphous p o l y m e r . T h e "transition r e g i o n "

blends into a high-modulus " r u b b e r y region'' at high temperatures, crys- tallites playing the same role in the polycrystalline p o l y m e r that entangle- ments or cross links d o in the amorphous p o l y m e r s .

Because there are changes with temperature in the microcrystalline struc- ture and in the stress-bearing mechanisms, it is certain that the simple time- temperature superposition that is valid for amorphous p o l y m e r s in the tran-

3 1 Α . V . T o b o l s k y and J . R . M c L o u g h l i n , J. Chem. Phys. 59, 989 (1955).

78

sition region is not valid for p o l y crystalline polymers. T h e r e is n o t o n l y a horizontal displacement along the log-time axis due t o changing rate of molecular motions with temperature, b u t also an even more important verti- cal shift along the log E^r(t) axis resulting from the changing structure and other factors.

I t is interesting t o note that the polycrystallinity of plasticized p o l y v i n y l chloride was first discovered b y T o b o l s k y and Stein from an examination of the E^r(t) c u r v e s .³² These rubbery samples behaved like cross-linked rub- bers in that E^r(t) was relatively independent of time. A t progressively higher temperatures the values of E^r(t) b e c a m e smaller, although the time depend- ence of E^r(t) always was slight. I t was inferred that crystallites were acting as temperature-sensitive b u t n o t time-sensitive cross links. T h e presence of crystallites was also p r o v e d b y X - r a y , birefringence, and other tech-

32

niques.

1. D E C A Y O F S T R E S S A T C O N S T A N T E X T E N S I O N R E S U L T I N G F R O M T H E G R O W T H O F O R I E N T E D C R Y S T A L L I T E S

T h e first observations on the d e c a y of stress at constant extension due t o the growth of oriented crystalline material w^Tas made in this laboratory in 1 9 4 6 .^{3 3 , 3 4}'^{3 0} I t was found that vulcanized Neoprene showed v e r y little re- laxation of stress after 1 0 0 hr. at 3 5 ° C . and 5 0 % extension. H o w e v e r , a complete d e c a y of stress at zero stress was observed after only 5 0 hr. at 0 ° C . and 5 0 % extension—a rude shock t o any simple-minded belief that rate of stress relaxation should follow the Arhennius law in all cases. A c t u - ally the Neoprene sample, after d e c a y i n g t o zero stress, began t o increase in length b e y o n d the original 5 0 % stretched value, a p h e n o m e n o n k n o w n as spontaneous elongation. This p h e n o m e n o n was attributed b y us t o crystal- lization of the N e o p r e n e . Observations of spontaneous elongation, b u t n o stress d e c a y measurements, w^rere previously reported for natural rubber and for ether polysulfide r u b b e r .^{3 6 , 37}

An extensive study was made ³⁴ '^{3 5} on the effect of temperature and elon- gation on the stress d e c a y curves of unvulcanized natural rubber (cast latex sheet) in the temperature region of crystallization. In Fig. 9 stress d e c a y curves are shown for samples maintained at 5 0 % elongation at temperatures of 0 , - 1 0 , - 2 0 , - 2 5 , - 3 0 , - 4 0 , and - 5 0 ° C . T a k i n g the time required t o attain zero stress as an index of the rate of crystallization, it is clear that the results of Figure 9 are in accord with the results of Bekkedahl, w h o

3 2 R . S. Stein and Α . V . T o b o l s k y , Textile Research J. 18, 302 (1948).

3 3 R . D . A n d r e w s , P h . D . thesis, P r i n c e t o n U n i v e r s i t y (1948).

3 4 G . M . B r o w n , P h . D . thesis, P r i n c e t o n U n i v e r s i t y (1948).

3 5 Α . V . T o b o l s k y and G . M . B r o w n , J. Polymer Sei. 17, 547 (1955).

3 6 C . Park, Rubber Chem. Technol. 12, 778 (1939).

3 7 W . H . Smith and C . P . Saylor, Rubber Chem. Technol. 12, 18 (1939).

STRESS R E L A X A T I O N S T U D I E S 7 9

studied the rates of crystallization in unvulcanized natural rubber b y a dilatometric p r o c e d u r e .³⁸ T h e stress-relaxation data show a m a x i m u m rate of stress d e c a y at —20° C . A t —50° C . the rate of crystallization is v e r y slow.

I n m a n y crystalline p o l y m e r s for w h i c h n o p h e n o m e n o n of spontaneous elongation was observed, the d e c a y of stress was nevertheless a c c o m p a n i e d b y an increase of birefringence, indicating that even in these cases stress d e c a y m a y b e caused in the part b y a g r o w t h of oriented crystalline ma- terial, p r o b a b l y around pre-existing crystal n u c l e i .³⁹

V I I I . Chemical Stress R e l a x a t i o n ( C h e m o r h e o l o g y )

I t might b e expected that for chemically cross-linked rubbers, in the region of the rubbery plateau, the stress-relaxation curves E^r(t) should re- main constant with time, inasmuch as thermoplastic flow is suppressed b y the cross links. In actual fact it was discovered that all rubbers show E^r(t) curves that d e c a y t o zero stress at sufficiently high temperatures, and w e attributed this stress d e c a y t o chemical reactions such as chain scission b y oxidative cleavage or reorganization of the network structure b y ionic i n t e r c h a n g e s .^{4 0 , 41}

3 8 N . Bekkedahl, J. Research Natl. Bur. Standards 13, 411 (1934).

3 9 R . S. Stein and Α . V . T o b o l s k y , Textile Research J. 18, 201, 302 (1948).

4 0 Α . V . T o b o l s k y , I . B . P r e t t y m a n , and J . H . Dillon, J. Appl. Phys. 15, 309 (1944).

4 1 M . D . Stern and Α . V . T o b o l s k y , Chem. Phys. 14, 93 (1946).

I.POLYSULPHIDE RUBBER 2.NE0PRENE

3.HEVEA 4. B U N A - Ν 5. BUTYL 6. B U N A - S

7 P 0 L Y F S T E R R I I R R F R

8.P0LYETHY LACRYLATE R U B B E R 8.P0LYETHY

§

\ 3 4 V \

• Ol O.l 1.0 10 100 1000

T I M E , ( HOURS)

F I G. 10. C h e m i c a l stress relaxation for v a r i o u s v u l c a n i z e d rubbers at 130° C⁶

I n m a n y cases the d e c a y curves p r o d u c e d b y chemical stress relaxation follow the d e c a y law e x p ( — /Λ) where A:1 is a rate constant that o b e y s the Arhennius law for temperature dependence. T h e number of chain scissions per gram of rubber occurring per unit time can be calculated from the stress relaxation c u r v e s .⁴²

I n m a n y of the h y d r o c a r b o n rubbers (natural rubber, G R - S , B u t y l , and so o n ) the chemical stress relaxation is b r o u g h t a b o u t b y a cleavage induced b y molecular o x y g e n . A t the same time there are simultaneously occuring cross-linking reactions. Fortunately these t w o reactions can b e b o t h ex- amined b y physical techniques. D e c a y of stress at constant extension meas- ures the scission reaction alone, even t h o u g h cross linking is occurring simultaneously. On the other hand, if the sample is k e p t in a relaxed condi- tion at the elevated temperature and the m o d u l u s measured ' 'intermit- t e n t l y / ' the net effect of cross linking and scission is measured. I n rubbers for which cross linking predominates o v e r scission, the " i n t e r m i t t e n t "

modulus increases with time, in rubbers in which scission predominates the intermittent m o d u l u s decreases with t i m e .^{8 , 40} T h e c o n c e p t of simultaneous scission and cross-linking originally introduced in these studies t o explain the physical changes p r o d u c e d b y high temperatures and o x y g e n n o w p l a y s a vital role in the interpretation of the changes p r o d u c e d in p o l y m e r s b y high energy radiation.

T h i n samples of rubber maintained in a stretched condition at elevated

4 2 Α . V . T o b o l s k y , D . M e t z , and R . B . M e s r o b i a n , J. Am. Chem. Soc. 72,1942 (1950).

S T R E S S R E L A X A T I O N S T U D I E S 81

temperatures d e v e l o p an irrecoverable permanent set because of the simul- taneous cross linking and scission reactions. T h e permanent set can be predicted from the " c o n t i n u o u s " relaxation curves and the " i n t e r m i t t e n t "

measurements of the m o d u l u s .⁴³

T h e polysulfide- or T h i o k o l - t y p e rubbers show a chemical stress relaxa- tion resulting from catalyzed interchanges i n v o l v i n g the disulfide link- a g e .^{4 1, 44} In this case the " i n t e r m i t t e n t l y " measured m o d u l u s shows n o change with time, inasmuch as the interchanges leave the over-all structure of the network unaffected. Various types of ionic reagents catalyze the inter- c h a n g e .⁴⁴ I n 1947 our studies on the stress relaxation of vulcanized silicone rubbers c o n v i n c e d us that the relaxation process was due t o an ionic inter- change of S; Ο linkages catalyzed b y some u n k n o w n reagent.³⁴ In 1950 w e prepared cross-linked silicone rubbers in a special w a y which u n a v o i d a b l y contained traces 0ÎH2SO4 as a residual c a t a l y s t .1 9 , 45 These rubbers showed complete decay to zero stress at room temperature in a matter of several minutes unless the residual catalyst was stabilized b y water or pyridine.

Once stabilized, these specially prepared silicone rubbers showed practically no stress decay even at elevated temperatures.

A recent paper has identified the trace catalysts that cause stress relaxa- tion in commercially prepared silicones.46

In Fig. 10 chemical stress relaxation is shown for a number of synthetic crossed linked rubbers at 130° C. T h e date is plotted in the form stress/ini- tial stress vs. log time. T h e tremendous variation that can be obtained in the rate of chemical stress relaxation as befoveen several different synthetic rubbers is clearly seen from the graph. Although the stability to reactions at high temperatures is partly inherent in the structure of the polymers, very great variations in stability can be achieved—and have been achieved

— b y suitable purification, incorporation of stabilizers anti-oxidants etc.

Nomenclature

Εr{t) R e l a x a t i o n m o d u l u s ; m o d - E' R e a l part of c o m p l e x d y - ulus measured with sam- namic modulus

pie at constant extension E" Imaginary part of c o m p l e x Ee(t) C r e e p m o d u l u s ; m o d u l u s d y n a m i c modulus

measured w i t h sample ω Angular frequency of v i ' under c o n s t a n t stress bration

4 3 R . D . A n d r e w s , Α . V . T o b o l s k y , and Ε . E . H a n s o n , / . Appl. Phys. 1 7 , 352 (1946).

4 4 M . M o c h u l s k y and Α . V . T o b o l s k y , Ind. Eng. Chem. 40, 2155 (1948).

4 5 D . H . J o h n s o n , J. R . M c L o u g h l i n , and Α . V . T o b o l s k y , J. Phys. Chem. 58, 1073 (1954).

4 6 R . C . Osthoff, A . M . B u e c h e , and W . T . G r u b b , J. Am. Chem. Soc. 7 6 , 4659 (1954).