Chapter 1 Structure and Physical Properties of Polymers

1

Chapter 1

STRUCTURE AND PHYSICAL PROPERTIES OF POLYMERS 1.1 PHYSICAL STATES OF POLYMERS

Polymers c a n e x i s t i n f o u r p h y s i c a l s t a t e s - - t h e

c r y s t a l l i n e and

t h r e e amorphous s t a t e s ( g l a s s y , r u b b e r y , a n d v i s c o u s f l o w ) .

Polymers

t h a t e x i s t i n t h e g l a s s y o r c r y s t a l l i n e s t a t e are s o m e t i m e s c a l l e d r i g i d polymers.

Each s t a t e h a s i t s own complex o f m e c h a n i c a l

p r o p e r t i e s and i t s own a r e a o f t e c h n i c a l a p p l i c a t i o n . " The p h y s i c a l s t a t e of a polymer i s m o s t o f t e n d e t e r m i n e d by s t u d y i n g i t s mechanical p r o p e r t i e s , f o r example, i t s compliance. One c a n j u d g e t h e b a s i c t e c h n o l o g i c a l p r o p e r t i e s o f a polymer on t h e b a s i s o f i t s c o m p l i a n c e o v e r a w i d e r a n g e o f t e m p e r a t u r e s (10). One c a n d e t e r m i n e t h e d e g r e e o f c o m p l i a n c e by u s e o f thermom e c h a n i c a l method p r o p o s e d by A l e x a n d r o v a n d L a z u r k i n (11) f o r c y c l i c d e f o r m a t i o n and by K a r g i n e t a l . deformation.

(12,13) f o r s t a t i c

F i g u r e 1.1 d e p i c t s a t h e r m o - m e c h a n i c a l c u r v e f o r p o l y m e r s .

A t

temperatures lower than t h e g l a s s t r a n s i t i o n temperature T 9' polymers deform i n t h e manner o f l o w - m o l e c u l a r - w e i g h t g l a s s .

A

s i g n i f i c a n t i n c r e a s e i n r e v e r s i b l e s t r a i n occurs a t temperatures above t h i s , i n d i c a t i n g t h e r u b b e r y s t a t e .

I n t h i s range t h e e l a s t i c

modulus c h a n g e s l i t t l e w i t h t e m p e r a t u r e up t o t h e f l o w t e m p e r a t u r e Tf.

Above t h e f l o w t e m p e r a t u r e , a polymer i s a l i q u i d o f h i g h

viscosity. The r u b b e r y regime f o r a c r y s t a l l i n e polymer l i e s between Tm and

Tf.

Depending o n t h e r a t e o f c o o l i n g , a polymer c a n c r y s t a l l i z e

a t a t e m p e r a t u r e below Tm ( c u r v e DCBF); however, w i t h r a p i d c o o l i n g

a polymer d o e s n o t c r y s t a l l i z e b u t p a s s e s i n t o t h e g l a s s y s t a t e ( c u r v e DCBA) ( F i g . 1.2).

*The r e a d e r who is i n t e r e s t e d i n t h e p r e p a r a t i o n o f polymers c a n f i n d c o n s i d e r a b l e i n f o r m a t i o n i n s o u r c e s (1-6) and modern views on t h e s t r u c t u r e and c l a s s i f i c a t i o n o f p o l y m e r s i n (7-9).

2

'

E

I 0

.+

Fig.

High elastic state (or viscoclastic state)

I

1

1.1 Thermo-mechanical c u r v e f o r p o l y m e r s .

Tempcratu re

Fig. 1 . 2 1.2

Compliance s t r a i n v s . t e m p e r a t u r e f o r v a r i o u s polymers.

PHYSICAL PROPERTIES O F MACROMOLECULES AND POLYMER STRUCTURE

F l e x i b i l i t y is t h e b a s i c property of chain molecules.

A polymer

m o l e c u l e h a s a t r a n s v e r s e dimension of s e v e r a l a n g s t r o m s and a l e n g t h o f s e v e r a l t h o u s a n d a n g s t r o m s , which may f o r m a f i b e r 1 0 m i c r o n s i n diameter and s e v e r a l m i l l i m e t e r s i n length (13).

A macromolecule's

f l e x i b i l i t y i s e f f e c t e d by t h e r o t a t i o n o f i t s l i n k s a r o u n d o r d i n a r y C-C bonds. Through t h e r m a l m o t i o n and t h e i r own f l e x i b i l i t y , macrom o l e c u l a r segments t a k e on v a r i o u s s p a t i a l forms o r c o n f o r m a t i o n s .

I f a macromolecule i s s t r e t c h e d , i t s c o n f o r m a t i o n s w i l l be e l o n g a t e d . I f t h e e x t e r n a l s t r e t c h i n g f o r c e i s removed, t h e n t h r o u g h t h e r m a l motion. t h e macromolecule f o l d s and b e n d s i n t o a random s p a t i a l p a t t e r n , as i n F i g . 1 . 3 . I n a r e a l macromolecule, t h e m o t i o n of m o l e c u l a r segments i s h i n d e r e d by i n t e r n a l a n d i n t e r m o l e c u l a r i n t e r a c t i o n s .

The h i n d r a n c e l e a d s t o

a n i n c r e a s e i n s t i f f n e s s of t h e c h a i n o v e r t h a t o f a c h a i n w i t h f r e e rotation.

The p o t e n t i a l e n e r g y c u r v e f o r i n t e r n a l r o t a t i o n h a s s e v e r a l

3

Fig. 1 . 3

A f l e x i b l e macromolecule c h a i n i n a s p a t i a l l y f o l d e d s t a t e .

minima t h a t d i f f e r i n d e p t h .

Most o f t e n t h e m o l e c u l a r s e g m e n t s o r

l i n k s a r e found i n p l a c e s t h a t c o r r e s p o n d t o t h e e n e r g y minima.

The

s t a b l e c o n f o r m a t i o n s o f a c h a i n t h a t r e s u l t from r o t a t i o n o f i n d i v i d u a l bonds are c a l l e d r o t a t i o n a l i s o m e r s (8,141- On t h e whole, a polymer c a n be viewed, w i t h q u a l i f i c a t i o n s , a s a complex of r o t a t i o n a l i s o m e r s . I n t h e r o t a t i o n a l isomer a p p r o a c h , i n t e r n a l f r i c t i o n , l e a d i n g t o t h e h i n d r a n c e o f r o t a t i o n i n t h e n e t w o r k , r e p r e s e n t s c o u p l i n g s among d i f f e r e n t r o t a t i o n a l isomers. Polymer m a c r o m o l e c u l e s a l w a y s e x i s t i n a condensed p h a s e and i n t e r a c t w i t h t h e e n v i r o n m e n t , which c a n e i t h e r b e a polymer a s w e l l ( a s i n a p u r e polymer) o r b e a l i q u i d ( a s i n s o l u t i o n s o f p o l y m e r s ) . A t a n e a r l i e r s t a g e o f polymer s c i e n c e , a polymer was d e s c r i b e d a s

a complex o f random, t a n g l e d m o l e c u l a r c h a i n s ( 1 5 ) .

Modern t h e o r y ,

however, d e s c r i b e s a polymer as an o r d e r e d m o l e c u l a r s y s t e m .

The

s i m p l e s t k i n d s of o r d e r i n g i n macromolecules ( i . e . , supermolecular s t r u c t u r e s ) a l s o o c c u r i n amorphous p o l y m e r s . The s u p e r m o l e c u l a r s t r u c t u r e s c h a r a c t e r i s t i c a l l y are o f t w o main t y p e s : globules (13,16,17)

.

bundles and

B u n d l e s form i n p o l y m e r s t h a t h a v e s u f f i c i e n t l y

r i g i d c h a i n s ; t h e y are v e r y l o n g c l u s t e r s o f more o r less p a r a l l e l macromolecules.

The l a t e r a l d i m e n s i o n s o f a b u n d l e amount t o s e v e r a l

i n t e r m o l e c u l a r d i s t a n c e s ; i t s l e n g t h f a r exceeds t h e s i z e of elongated m a c r o m o l e c u l e s . G l o b u l e s form i n p o l y m e r s whose m a c r o m o l e c u l e s are v e r y f l e x i b l e and a r e f o l d e d i n t o l i t t l e " c o i l s " . D i s c r e t e segments o f a macromolecule c h a i n w i t h i n a g l o b u l e a r e randomly a r r a n g e d . P o l y m e r i c m a t e r i a l s i n t h e g l o b u l e s t a t e l o s e t h o s e p r o p e r t i e s which a r e a s s o c i a t e d w i t h t h e g r e a t l e n g t h o f m a c r o m o l e c u l e s , a n d behave l i k e s m a l l molecules.

C r y s t a l l i n e p o l y m e r s form s u p e r m o l e c u l a r

s t r u c t u r e s more r e a d i l y t h a n d o amorphous p o l y m e r s .

4

1.3

SPECIAL CHARACTERISTICS OF THERElAL MOTION I N POLYMERS

Thermal m o t i o n o f m a c r o m o l e c u l e s o r t h e i r s e g m e n t s i s q u i t e i m p o r t a n t i n polymer f r i c t i o n p r o c e s s e s .

L e t u s examine t h e s p e c i a l

c h a r a c t e r i s t i c s o f polymer t h e r m a l m o t i o n t h a t b e a r a r e s e m b l a n c e t o thermal motion i n l i q u i d s .

P r e v i o u s l y , u n d e r t h e i n f l u e n c e of

van d e r Waals' i d e a s , l i q u i d s w e r e viewed a s e x t r e m e l y d e n s e g a s e s . Thermal m o t i o n i n l i q u i d s w a s r e d u c e d t o t h e t r a n s l a t i o n a l m o t i o n of p a r t i c l e s .

I n 1 9 2 6 , F r e n k e l (18,191 p r o p o s e d a new view of

thermal motion.

A c c o r d i n g t o F r e n k e l , l i q u i d s , e s p e c i a l l y when

t h e y are n e a r t h e c r y s t a l l i z a t i o n t e m p e r a t u r e , are more l i k e s o l i d s than l i k e dense g a s e s i n s t r u c t u r a l arrangement and i n t h e c h a r a c t e r o f t h e i r thermal motion.

I n s o l i d s and l i q u i d s , a c o n s t a n t m a g n i t u d e

o f k i n e t i c e n e r g y c a n b e a t t r i b u t e d t o t h e t h e r m a l m o t i o n of particles.

Each p a r t i c l e (atom o r m o l e c u l e ) , from t i m e t o t i m e ,

c a n a c q u i r e s u f f i c i e n t k i n e t i c e n e r g y t o surmount t h e p o t e n t i a l b a r r i e r t h a t s e p a r a t e s two a d j a c e n t s t a t e s .

Usually, a p a r t i c l e

i s found i n p l a c e s t h a t h a v e a minimum of p o t e n t i a l e n e r g y a n d i t o s c i l l a t e s a b o u t a n e q u i l i b r i u m p o i n t . As a r e s u l t o f s u c h d i s placements, i n c r y s t a l l i n e s o l i d s v acan cies or " h o les " ( f r e e l a t t i c e p o i n t s ) a n d d i s l o c a t e d a t o m s , l o c a t e d between t h e l a t t i c e p o i n t s , are formed. I n l i q u i d s , such "holes" are microvoids of i n d e f i n i t e s i z e a n d s h a p e d i s t r i b u t e d among m o l e c u l e s .

The l i v e s o f " h o l e s "

are s h o r t , b u t t h e i r number i s s i g n i f i c a n t l y l a r g e r t h a n i n c r y s t a l s . F o r t h e most p a r t , f r e e s p a c e i n l i q u i d s c o n s i s t s o f t h e s e " h o l e s " , which c o n s t a n t l y d i s a p p e a r from o n e p l a c e a n d a p p e a r i n o t h e r p l a c e s . The e x i s t e n c e o f f r e e s p a c e i n l i q u i d s r e s u l t s i n g r e a t m o b i l i t y of the particles. A molecule's duration i n a t r a n s i t i o n a l equilibrium position, c a l l e d " t h e t i m e of a p a r t i c l e a t rest", i s e x p r e s s e d a s a s t a t i s t i c a l mean v a l u e a n d i s d e p e n d e n t upon t e m p e r a t u r e . F r en k el demonstrated t h a t t h e t i m e o f a p a r t i c l e a t rest, t h a t i s , t h e molecular r e l a x a t i o n t i m e T, i s e q u a l t o T

= T 0 e x p (U/kT)

,

(1.1)

where T~ i s t h e p e r i o d o f p a r t i c l e o s c i l l a t i o n a r o u n d e q u i l i b r i u m points ( 9 to second) and U i s t h e corresponding a c t i v a t i o n e n e r g y , e q u a l t o t h e minimum k i n e t i c e n e r g y r e q u i r e d f o r a p a r t i c l e t o surmount t h e e n e r g y b a r r i e r a t a g i v e n t e m p e r a t u r e .

5 The v i s c o s i t y o f a l i q u i d i s d e t e r m i n e d by t h e m o l e c u l a r r e l a x a t i o n t i m e T. The l a r g e r f i s , t h e mcre v i s c o u s i s t h e l i q u i d . According t o t h e Maxwell r e l a t i o n , t h e v i s c o s i t y of a l i q u i d i s g i v e n by q = X T G ~ , where

Go i s t h e " i n s t a n t a n e o u s " s h e a r modulus o f a l i q u i d ,

o b s e r v a b l e d u r i n g h i g h s h e a r r a t e s and x i s t h e c o e f f i c i e n t t h a t r e l a t e s t h e Maxwell r e l a x a t i o n t i m e T~ a n d t h e m o l e c u l a r r e l a x a t i o n time

T.

I n g e n e r a l , from t h e a c t i v a t i o n mechanism, t h e v i s c o s i t y

o f l i q u i d s i s e x p r e s s e d i n t e r m s of t h e well-known Frenkel-Andrade e q u a t i o n f o r Newtonian f l o w : Q = A e x p (U/kT)

.

C o n c e p t s o f t h e mechanism o f t h e r m a l m o t i o n a n d v i s c o u s f l o w i n l i q u i d s h a v e b e e n f u r t h e r d e v e l o p e d i n E y r i n g ' s t h e o r y of anomalously v i sc ous systems ( 2 0 ) .

Modern ideas a b o u t a c t i v a t i o n mechanisms o f

v i s c o u s f l o w a n d d i f f u s i o n are based on c o n c e p t s o f t h e r m a l motion i n l i q u i d s f o r m u l a t e d by F r e n k e l a n d E y r i n g .

According t o Eyring,

v i s c o u s f l o w o c c u r s a s a r e s u l t of t r a n s f e r from a n e q u a l - p r o h a b i l i t y p a t t e r n of a u t o d i f f u s i n g p a s s a g e o f k i n e t i c u n i t s t h r o u g h a s t a t i c l i q u i d i n a l l s p a t i a l d i r e c t i o n s , to an asymmetrical p r o b a b i l i t y d i s t r i b u t i o n o f p a r t i c l e t r a n s f e r i n a v i s c o u s s t r e a m , where w i t h t h e h i g h e s t p r o b a b i l i t y p a r t i c l e s are t r a n s f e r r e d i n t h e d i r e c t i o n o f a t a n g e n t i a l force.

E y r i n g i s o n e o f t h e s c i e n t i s t s who have

made a m o r e p r e c i s e d e s c r i p t i o n o f v i s c o s i t y ( 2 1 ) .

W e have s i n c e

learned t h a t t h e passage o f p a r t i c l e s occurs i n a l l s p a t i a l d i r e c t i o n s , n o t only i n t h e d i r e c t i o n of t h e o p e r a t i n g t a n g e n t i a l force.

For

s m a l l s h e a r stresses t h e p r o b a b i l i t y d i s t r i b u t i o n a p p e a r s a s a l i n e a r f u n c t i o n o f t h e s h e a r stress. C o n s e q u e n t l y , t h e s h e a r r a t e i s p r o p o r t i o n a l t o t h e s h e a r stress; t h a t i s , o n e o b s e r v e s Newtonian f l o w of c o n s t a n t v e l o c i t y . Under h i g h stresses, which a r e r e a l i z e d i n h i g h l y v i s c o u s l i q u i d s t h a t h a v e complex s t r u c t u r e s ( p o l y m e r s , d i s p e r s e d systems and o t h e r s ) ( 2 2 - 2 6 ) , t h e !linear approximation i s i m p o s s i b l e b e c a u s e v i s c o s i t y d i m i n i s h e s w i t h i n c r e a s e i n stress or s h e a r r a t e . Thermal m o t i o n o f m a c r o m o l e c u l e s i s d i s t i n g u i s h e d from t h a t o f s u b m o l e c u l a r matter i n a number o f ways.

I n a n i s o l a t e d macromolecule,

t h e r m a l m o t i o n i s c h a r a c t e r i z e d by f o r w a r d Brownian m o t i o n o f t h e macromolecule a s a whole, by r o t a t i n g m o t i o n o f t h e macromolecule c h a i n s , by o s c i l l a t i o n o f t h e macromolecule s e g m e n t s a b o u t o n e a n o t h e r , a n d by r o t a t i n g i n t e r m o l e c u l a r Brownian motion of

macromolecule s e g m e n t s a b o u t o n e a n o t h e r .

Brownian m o t i o n of polymer

c h a i n s o r d i s c r e t e m a c r o m o l e c u l e s i s u s u a l l y c a l l e d macro-Brownian motion.

Brownian m o t i o n o f d i s c r e t e s e g m e n t s o f a c h a i n i s c a l l e d

micro-Brownian. As a r e s u l t o f i n t r a m o l e c u l a r i n t e r a c t i o n s i n a polymer c h a i n , ( c o n s i s t i n g of i n t e r a c t i o n s o f s e p a r a t e s e g m e n t s of t h e c h a i n o r i t s lateral groups), r e p u l s i v e f o r c e s arise t h a t reduce molecular m o b i l i t y , p a r t i c u l a r l y where t h e r e a r e s t a b l e v a l e n c e a n g l e s . R e s t r i c t i o n s o n t h e r m a l motion i n polymer c h a i n s a r e d e s c r i b e d i n

terms o f p o t e n t i a l b a r r i e r s . A segment i s t h e k i n e t i c u n i t t h a t c h a r a c t e r i z e s t h e m o b i l i t y o f a polymer c h a i n . The s i z e o f a segment c h a r a c t e r i z e s t h e r i g i d i t y of a polymer c h a i n .

The c l o s e r

t h e s i z e of t h e k i n e t i c u n i t (segment) i s t o t h e chemical u n i t ( l i n k ) , t h e more f l e x i b l e t h e polymer c h a i n i s . i n elastomers ( e . g . , l a r g e r than t h e link.

Thus, f o r example,

n a t u r a l r u b b e r ) t h e segment i s s e v e r a l t i m e s The r i g i d i t y o f a c h a i n a l s o d e p e n d s i n

l a r g e m e a s u r e upon b o t h t h e s t r u c t u r e o f l a t e r a l g r o u p s a n d i n t e r a c t i o n s among them.

The p r e s e n c e o f d o u b l e a n d t r i p l e bonds

i n a polymer c h a i n a l s o i n c r e a s e s t h e c h a i n ' s r i g i d i t y . A s p e c i a l c h a r a c t e r i s t i c o f thermal motion i n c r y s t a l l i n e polymers

i s r e l a t e d t o t h e f a c t t h a t s u c h p o l y m e r s c o n s i s t o f amorphous a n d I n c r y s t a l l i n e polymers molecular motion i s more d i f f i c u l t t h a n i n amorphous p o l y m e r s . An i n c r e a s e i n t h e degree of c r y s t a l l i n i t y decreases mobility not only i n t h e k i n e t i c u n i t s o f t h e c h a i n s ( s e g m e n t s ) , b u t a l s o i n t h e l a t e r a l g r o u p s (27). The v i s c o u s - f l o w s t a t e o f a p o l y m e r , w h i c h i s o b s e r v e d a b o v e t h e c r y s t a l l i n e regions.

g l a s s t e m p e r a t u r e i n amorphous p o l y m e r s or above t h e m e l t i n g p o i n t i n c r y s t a l l i n e polymers, i s due t o g r e a t molecular m o b i l i t y .

In

l i n e a r p o l y m e r s , t h e v i s c o u s - f l o w s t a t e i s c h a r a c t e r i z e d by v i s c o s i t y i n t h e r a n g e l o 3 t o 1 0 l 2 p o i s e . V i s c o u s f l o w i s accompanied by t h e development o f viscoelastic deformation, and r e l a t e d t o t h i s t h e r e i s r e c t i f i c a t i o n and o r i e n t a t i o n o f t h e macromolecules.

The

b a s i c mechanism of polymer v i s c o u s f l o w i s u n c l e a r , b u t i t may be d e s c r i b e d as a micro-Brownian

t r a n s f e r of m a c r o m o l e c u l a r s e g m e n t s ,

l e a d i n g t o g e n e r a l t r a n s f e r of t h e macromolecules.

However, it i s

n e c e s s a r y t o k e e p i n mind t h a t d u r i n g v i s c o u s f l o w t h e macromolecule o r i e n t a t i o n i n f l u e n c e s v i s c o s i t y (28-32).

7 1.4

THE RUBBERY STATE I N POLYMERS

For r u b b e r y p o l y m e r s , o n e o b s e r v e s a s p e c i a l k i n d o f f r i c t i o n . The p r e d o m i n a n t a s p e c t o f d e f o r m a t i o n i n t h i s case i s v i s c o e l a s t i c (13,15,33-37).

The t e m p e r a t u r e s i n which a polymer e x i s t s i n t h e

r u b b e r y s t a t e i s above t h e g l a s s t r a n s i t i o n t e m p e r a t u r e T

9'

However,

t h e r e g i o n where r u b b e r y p r o p e r t i e s d e v e l o p d e p e n d s upon t h e s t r a i n frequency and t h e o b s e r v a t i o n t i m e . I n o t h e r words, t h e p o s i t i o n o f t h e r u b b e r y r e g i o n i s r e l a t e d t o polymer r e l a x a t i o n phenomena; i t s lower l i m i t o c c u r s a t t h e m e c h a n i c a l t r a n s i t i o n t e m p e r a t u r e , Tm, which u s u a l l y l i e s above T

9' The m o s t d i s t i n c t r u b b e r e l a s t i c i t y d e v e l o p s i n c r o s s l i n k e d

rubbers (vulcanized rubbers).

I n l i n e a r amorphous p o l y m e r s , d u r i n g

a l o n g enough p e r i o d of o b s e r v a t i o n , v i s c o u s f l o w i s s u p e r i m p o s e d upon r u b b e r y d e f o r m a t i o n . D e f o r m a t i o n p r o p e r t i e s o f c r y s t a l l i n e

p o l y m e r s depend upon b o t h t h e r u b b e r y p r o p e r t i e s of amorphous m i c r o - r e g i o n s a n d t h e more r i q i d n a t u r e o f c r y s t a l l i n e m i c r o - r e g i o n s , i n p r o p o r t i o n t o t h e d e g r e e o f c r y s t a l l i n i t y i n t h e polymer a s a whole

.

N a t u r a l r u b b e r s , v u l c a n i z e d r u b b e r s , some r u b b e r l i k e p o l y m e r s , and even swollen r i g i d - c h a i n e d polymers a r e t y p i o a l rubbery

m a t e r i a l s , e a c h i n i t s own t e m p e r a t u r e r a n g e s .

Polymers t h a t

e x i s t i n . t h e r u b b e r y s t a t e are w i d e l y u s e d i n e n g i n e e r i n g , c h i e f l y i n t h e form of v a r i o u s v u l c a n i z e d r u b b e r p r o d u c t s ( p a c k i n g s , v a l v e s , v i b r a t i o n dampers, e t c . ) , a u t o m o b i l e a n d a i r p l a n e t i r e s , and s o f o r t h , where t h e i r f r i c t i o n a l p r o p e r t i e s are i m p o r t a n t . t e c h n i c a l p r o p e r t i e s of r u b b e r y m a t e r i a l s a r e :

The u s e f u l

l o w moduli o f

e l a s t i c i t y , h i g h f r i c t i o n c o e f f i c i e n t s , a n d good s h o c k a b s o r p t i o n capacity. The n e e d f o r s t a b i l i t y o f t h e s e p r o p e r t i e s compels t h e u s e of v u l c a n i z e d r u b b e r s i n t h o s e t h e r m a l r e g i o n s , and restricts s t r a i n r a t e s and s t r a i n s t o t h e l i n e a r r a n g e o f b e h a v i o r . C r o s s l i n k e d polymers ( v u l c a n i z e d r u b b e r s ) can completely recover t h e i r shape a f t e r unloading, j u s t l i k e r e s i l i e n t s o l i d s .

But i n

o t h e r c h a r a c t e r i s t i c s t h e y a r e more l i k e l i q u i d s . Submolecular l i q u i d s a n d c r o s s l i n k e d p o l y m e r s a r e more a l i k e i n t h e i r c o e f f i c i e n t s o f t h e r m a l e x p a n s i o n a n d c o m p r e s s i b i l i t y , which are much l a r g e r t h a n those of s o l i d s .

T h e s e a n d o t h e r p r o p e r t i e s conform t o P a s c a l ' s law.

Thus, t h e b u l k t h e r m a l e x p a n s i o n c o e f f i c i e n t s f o r g a s e s are 3.6 x degree-': but f o r organic degree-'; f o r metals, 6 x l i q u i d s and polymers t h e c o e f f i c i e n t s are c l o s e t o g e t h e r : 10 x -1 and 3.6 x degree The i s o t h e r m a l c o m p r e s s i b i l i t y c o e f f i c i e n t s

.

8 are a p p r o x i m a t e l y e q u a l f o r a i r , 1 cmL/kq ( a t a p r e s s u r e o f 1 a t m ) 2 and f o r metals, c m / k q , b u t f o r o r g a n i c l i q u i d s and p o l y m e r s t h e y a r e n e a r l y t h e same and d i f f e r from t h e v a l u e f o r m e t a l s by 2 t w o o r d e r s of t e n ( and 0 . 5 x cm /kg). By i t s n a t u r e , r u b b e r y d e f o r m a t i o n d i f f e r s from d e f o r m a t i o n of hard c r y s t a l l i n e and g l a s s y b o d i e s , b u t i s analogous t o moleculark i n e t i c ( e n t r o p i c ) e l a s t i c i t y i n g a s e s . F o r example, t h e e q u i l i b r i u m stress i n a deformed v u l c a n i z e d r u b b e r , l i k e t h e p r e s s u r e i n a compressed g a s , i s p r o p o r t i o n a l a t c o n s t a n t volume t o t h e a b s o l u t e temperature. Such a c o n j u n c t i o n o f p h y s i c a l p r o p e r t i e s , c h a r a c t e r i s t i c o f a l l t h r e e s t a t e s o f a g g r e g a t i o n , d i s t i n g u i s h e s r u b b e r y materials from o t h e r s . Rubbery d e f o r m a t i o n o c c u r s by t h e o r i e n t a t i o n and t r a n s f e r o f l i n k s i n f l e x i b l e c h a i n s , t h a t i s , t h e t r a n s i t i o n o f c h a i n s from folded to elongated conformations.

The c l a s s i c a l s t a t i s t i c a l t h e o r y

o f r u b b e r y d e f o r m a t i o n w a s d e v e l o p e d by Wohl, Kuhn, James and Good, T r e l o a r , and o t h e r s (5,7-9,151. 1.5

RELAXATION PROPERTIES OF POLYMERS

M o l e c u l a r - k i n e t i c p r o c e s s e s i n p o l y m e r s i n c l u d e d i f f u s i o n and s e l f - d i f f u s i o n , c r y s t a l l i z a t i o n and f u s i o n , v a p o r i z a t i o n a n d d i s s o l u t i o n , g l a s s t r a n s i t i o n a n d s o f t e n i n g , d e f o r m a t i o n and d i s i n t e g r a t i o n , m e c h a n i c a l and d i e l e c t r i c losses, v i s c o u s f l o w , a n d many o t h e r s . Some o f t h e s e p r o c e s s e s ( i n p a r t i c u l a r , r e l a x a t i o n ) are c h a r a c t e r i s t i c o f p o l y m e r s . Among them a r e t h e r e l a x a t i o n of m e c h a n i c a l stresses, m e c h a n i c a l losses d u r i n g r e p e a t e d s t r a i n , v i s c o u s flow, t h e mechanical and s t r u c t u r a l g l a s s t r a n s i t i o n , and d i e l e c t r i c and m a g n e t i c r e l a x a t i o n . The f o r m a t i o n o f t h e c o n t a c t area b e t w e e n a polymer a n d h a r d s u r f a c e s and t h e f r i c t i o n p r o c e s s a r e a l s o i n c l u d e d among r e l a x a t i o n phenomena. The s t u d y o f m e c h a n i c a l r e l a x a t i o n i n p o l y m e r s ( c r e e p , stress r e l a x a t i o n , and t h e temperature-frequency

dependence o f dynamic

c h a r a c t e r i s t i c s ) allows n o t only t h e evaluation of e x p l o i t a b l e p r o p e r t i e s i n polymer m a t e r i a l s b u t a l s o t h e d e t e r m i n a t i o n o f t h e r e l a t i o n s h i p between t h e c h e m i c a l and p h y s i c a l s t r u c t u r e o f p o l y m e r s , and t h e n a t u r e o f t h e i r m o l e c u l a r m o b i l i t y a n d m a c r o s c o p i c r e l a x a t i o n p r o p e r t i e s (13,36-38)

.

Polymer r e l a x a t i o n p r o c e s s e s a r e d i v i d e d i n t o l i n e a r a n d n o n l i n e a r ( 2 7 ) . L i n e a r r e l a x a t i o n i s d e s c r i b e d by t h e l i n e a r t h e o r y o f visco-elasticity

(39-41).

I t i s c h a r a c t e r i z e d by t h e r e a r r a n g e m e n t

9

o f m a c r o m o l e c u l a r s e g m e n t s d u r i n g t h e f o r m a t i o n o f e q u i l i b r i u m stress w i t h o u t a l t e r a t i o n i n t h e polymer s t r u c t u r e . (E =

For c o n s t a n t s t r a i n

c o n s t ) , l i n e a r s t r e s s r e l a x a t i o n f o l l o w s t h e r e l a x a t i o n law

u = u 0 f ( t ), where u o i s t h e i n i t i a l s t r e s s a t t h e moment t = 0. N o n l i n e a r r e l a x a t i o n p r o c e s s e s a r e o b s e r v e d d u r i n g l a r g e stresses and d e f o r m a t i o n s and a r e g e n e r a l l y r e l a t e d t o s t r u c t u r a l c h a n g e s o f p o l y m e r s i n t h e p r o c e s s of d e f o r m a t i o n .

The n o n l i n e a r stress

r e l a x a t i o n f o r a given deformation corresponds to t h e r e l a x a t i o n l a w u = uof ( t , ~ )

.

N o n l i n e a r r e l a x a t i o n s i n amorphous and c r y s t a l l i n e p o l y m e r s have much i n common, which allows u s t o a s s o c i a t e them w i t h t h e p r e s e n c e o f supermolecular formations i n r i g i d polymers (13,421,

The

d i s i n t e g r a t i o n o f supermo l e c u l a r s t r u c t u r e s d u r i n g d e f o r m a t i o n leads to t h e beginning of nonlinear r e l a x a t i o n p r o p e r t i e s . M e c h a n i c a l losses p l a y a n i m p o r t a n t r o l e i n f r i c t i o n o f d i f f e r e n t materials. The maximum f r i c t i o n c o e f f i c i e n t s are o f t e n a s s o c i a t e d w i t h maximum m e c h a n i c a l l o s s e s ( w h i c h a r e examined b r i e f l y b e l o w ) . The modulus of e l a s t i c i t y i n t h e case o f c y c l i c d e f o r m a t i o n s h a s a complex form, f o r example E* = E '

+

iE",

where s t o r a g e modulus

o f e l a s t i c i t y E l a n d t h e loss modulus E" a r e e x p r e s s e d by t h e formulas

Here, n i s t h e number o f e l e m e n t a r y r e l a x a t i o n mechanisms; E

are j t h e c o e f f i c i e n t s t h a t i n d i c a t e t h e c o n t r i b u t i o n of each r e l a x a t i o n mechanism. F o r t h e m e c h a n i c a l loss c o e f f i c i e n t s w e h a v e

(2a/E')

(

El

WT 1

1

+

w2'12 1

@'I

+

E2 1

+

2 W*T2

+

... +

E

4...).

j l + w . r

+

(1.2)

4

I f w e a s c r i b e t o e a c h m o l e c u l a r r e l a x a t i o n p r o c e s s i t s own

-

characteristic relaxation t i m e w-rl

'I then as t h e c o n d i t i o n s j' 1, w-r2 * 1 e t c . , are s u c c e s s i v e l y a t t a i n e d , t h e m e c h a n i c a l

l o s s e s s u c c e s s i v e l y p a s s t h r o u g h maxima.

These losses o f t e n are

10 n o t c h a r a c t e r i z e d by t h e m e c h a n i c a l loss c o e f f i c i e n t , b u t by a q u a n t i t y p r o p o r t i o n a l t o i t , namely t h e loss t a n g e n t , t a n 6 . E x p e r i m e n t a l d a t a on c h a n g e s o f m e c h a n i c a l l o s s e s u s u a l l y are p l o t t e d tan 6 v s t h e c y c l i c frequency w or t a n 6 v s temperature T. The r e l a x a t i o n t i m e s T T

j

have a t e m p e r a t u r e r e l a t i o n

= T~ e x p ( U . / k T ) , where U i s t h e a c t i v a t i o n e n e r g y f o r t h e j t h j 7 j

i s a c o e f f i c i e n t whose v a l u e c a n v a r y from o n e r e l a x a t i o n mechanism t o a n o t h e r .

molecular p r o c e s s , and

T~

Each w e l l - d e f i n e d r e l a x a t i o n p r o c e s s t h a t i s c h a r a c t e r i z e d by

a f i x e d v a l u e o f a c t i v a t i o n e n e r g y U and a r e l a x a t i o n t i m e T j j a p p e a r s a s a s h a r p o r d i f f u s e maximum i n t h e t e m p e r a t u r e dependence o f t h e m e c h a n i c a l loss. The t e m p e r a t u r e , h e i g h t , and w i d t h o f m e c h a n i c a l - l o s s maxima f o r polymers o f d i f f e r e n t s t r u c t u r e s have been i d e n t i f i e d ( 4 2 ) .

F i g u r e s 1 . 4 a n d 1 . 5 show t h e t e m p e r a t u r e

dependence o f t a n 6 , b a s e d upon a n a n a l y s i s o f e x p e r i m e n t a l d a t a f o r amorphous l i n e a r a n d c r y s t a l l i n e p o l y m e r s ( 4 3 - 4 5 ) .

Fig. 1 . 4 Temperature a is l i n e a r polymer: p r o c e s s ; maxima y ' , y , i n t h e g l a s s s t a t e : ri

dependence o f loss t a n g e n t f o r a n amorphous t h e maximum c o r r e s p o n d i n g t o t h e g l a s s t r a n s i t i o n and 8 are r e l a t e d t o t h e m o l e c u l a r m o b i l i t y i s t h e t r a n s i t i o n t o v i s c o u s flow.

The dynamic modulus and m e c h a n i c a l losses o f p o l y m e r s depend upon t h e r a t e o f d e f o r m a t i o n and t h e f r e q u e n c y o f a p p l i e d

stress, because

i n dynamic s y s t e m s work i s done n o t o n l y a g a i n s t l a r g e e l a s t i c f o r c e s , b u t a l s o a g a i n s t i n t e r n a l f r i c t i o n , d e p e n d i n g upon t h e rate o f t h e process.

F o r example, t h e modulus o f e l a s t i c i t y i n v u l c a n i z e d r u b b e r s

c o n s i s t s of t w o p a r t s : E = Em + El, where Em i s t h e r e l a x e d modulus, which c o r r e s p o n d s t o t h e s t a t e i n which r e l a x a t i o n i s a l m o s t c o m p l e t e and i s g e n e r a l l y d e p e n d e n t upon t h e e x t e n t o f t r a n s v e r s e c r o s s l i n k i n g . The n o n e q u i l i b r i u m p a r t El, due t o i n t e r n a l f r i c t i o n , i s e s s e n t i a l l y d e p e n d e n t upon t h e number o f p o l a r g r o u p s i n t h e polymer c h a i n and t h e q u a n t i t y o f a c t i v e f i l l e r , t h a t i s , upon t h e n a t u r e and i n t e n s i t y

11

I

T

-

F i g . 1 . 5 T e m p e r a t u r e dependence of t h e loss t a n g e n t f o r a c r y s t a l l i n e polymer: a i s t h e maximum c o r r e s p o n d i n g t o t h e g l a s s t r a n s i t i o n f o r t h e amorphous component; y ' , y , B are r e l a t e d t o m o l e c u l a r m o b i l i t y i n t h e g l a s s y s t a t e ; c1 a n d a' a r e r e l a t e d t o t h e m o b i l i t y of s u p e r m o l e c u l a r s t r u c t u r e s ; 17 i s t h e m e l t i n g transition. of i n t e r m o l e c u l a r i n t e r a c t i o n s . El and t h e mechanical l o s s e s i n c r e a s e w i t h d e c r e a s i n g t e m p e r a t u r e and w i t h i n c r e a s i n g f r e q u e n c y and r a t e of deformation. 1.6

THE PRINCIPLE O F TIME-TEMPERATURE EQUIVALENCE

The p r i n c i p l e o f t i m e - t e m p e r a t u r e

e q u i v a l e n c e has been a p p l i e d

i n t h e c a l c u l a t i o n o f many r e l a x a t i o n p r o c e s s e s a n d s e v e r a l polymer f r i c t i o n p r o p e r t i e s . I t i s known t h a t a l l m e c h a n i c a l , e l e c t r i c a l , and o t h e r polymer r e l a x a t i o n p r o c e s s e s are c h a r a c t e r i z e d b y c o r r e s p o n d i n g r e l a x a t i o n

times. The t e m p e r a t u r e dependence o f polymer m e c h a n i c a l a n d electrical p r o p e r t i e s i s d i r e c t l y r e l a t e d t o t h e e f f e c t of temperature on r e l a x a t i o n t i m e s . With a d e c r e a s i n g t e m p e r a t u r e , r e l a x a t i o n t i m e s i n c r e a s e s i g n i f i c a n t l y and i n t e r a c t i o n t i m e s d e c r e a s e .

This indicates

t h a t r e l a x a t i o n p r o c e s s e s are r e t a r d e d so much a t l o w t e m p e r a t u r e s t h a t they almost disappear.

By C o n t r a s t , a t h i g h t e m p e r a t u r e s

r e l a x a t i o n o c c u r s so q u i c k l y t h a t s u c h a change a s r u b b e r y d e f o r m a t i o n instantaneously a t t a i n s its limiting value. The s t u d y o f r e l a x a t i o n i s o f t e n done by o n e o f two e q u i v a l e n t e i t h e r by a l t e r i n g i n t e r a c t i o n t i m e w i t h i n b r o a d l i m i t s o r by v a r y i n g t h e t e m p e r a t u r e . I n t h e f i r s t method, t h e i n t e r a c t i o n

methods:

t i m e e x c e e d s a l l r e l a x a t i o n t i m e s , a n d i n t h e s e c o n d method a l l r e l a x a t i o n t i m e s are a l t e r e d a n d v a r y a t a f i x e d i n t e r a c t i o n t i m e . E x p e r i m e n t a l l y t h e s e c o n d method i s s i m p l e r , a n d t h e r e f o r e i t h a s been used widely. A l e k s a n d r o v and L a z u r k i n (11) f i r s t a p p l i e d t h e f r e q u e n c y method t o t h e i n v e s t i g a t i o n of r e l a x a t i o n e f f e c t s i n r u b b e r l i k e polymers during c y c l i d deformations. They d e m o n s t r a t e d t h a t a n i n c r e a s e i n

i n t e r a c t i o n f r e q u e n c y e x e r t s a n e f f e c t s i m i l a r t o t h a t p r o d u c e d by an e q u i v a l e n t d e c r e a s e i n temper?.ture.

Similar r e s u l t s i n s t a t i s t i c a l

t e s t s have been o b t a i n e d by Kobeko, K u v s h i n s k y , a n d G u r e v i c h ( 4 6 ) a n d by Leadermann ( 4 7 1 , a n d worked o u t i n more d e t a i l by Tobolsky ( 3 3 ) . These r e s e a r c h e r s have d e t e r m i n e d t h a t c u r v e s o f c r e e p a n d stress r e l a x a t i o n , obtained a t d i f f e r e n t temperatures

by a t r a n s l a t i o n a l o n g t h e l o g a r i t h m - o f - t i m e

,

can b e s u p e r p o s e d

axis.

F i g u r e 1 . 6 s c h e m a t i c a l l y shows r e s u l t s f o r c r e e p ( o r s t r e s s r e l a x a t i o n ) obtained a t d i f f e r e n t temperatures.

As i n d i c a t e d , t h e

c u r v e s c a n b e b r o u g h t i n t o c o i n c i d e n c e by t r a n s l a t i o n t o t h e r i g h t o r left.

T h i s r u l e o f t r a n s l a t i o n i s a p p l i c a b l e t o many polymer

r e l a x a t i o n p r o p e r t i e s b o t h s t a t i c and dynamic.

For dynamic p r o p e r t i e s ,

which i n c l u d e dynamic moduli a n d m e c h a n i c a l losses, i n p l a c e o f t i m e

(i.e.,

log t ) a x i s w e have t h e f r e q u e n c y ( i . e . ,

l o g w)

axis.

For

m e c h a n i c a l losses, i n s t e a d o f monotonic waves on t h e g r a p h , t h e r e

w i l l b e c u r v e s w i t h maxima.

h

c

Y

I

8 CI

c

Y

W

-t?, A

B

log t

Fig. 1 . 6 R e l a t i o n o f t h e l o g a r i t h m o f t h e modulus o f e l a s t i c i t y [or function H ( t ) 1 of a vulcanized rubber to the logarithm of T3 < T 4 < T 5 ) . t i m e , f o r d i f f e r e n t t e m p e r a t u r e s (T1 < T2 T h i s r u l e o f c u r v e s h i f t i n g , which s e r v e s a s a b a s i s f o r f o r mulation of t h e time-temperature

equivalence principle, is equivalent

t o t h e a s s e r t i o n t h a t t h e e f f e c t o f t e m p e r a t u r e T o n polymer stresss t r a i n p r o p e r t i e s o c c u r s independently of t i m e t and t h e polymer's p h y s i c a l p r o p e r t i e s ( f o r example, t h e modulus o f e l a s t i c i t y ) a n d c a n b e e x p r e s s e d by t h e f u n c t i o n

13

where t h e c o e f f i c i e n t a ( T ) i s t h e f u n c t i o n o f t e m p e r a t u r e o n l y and a t a c e r t a i n s t a n d a r d t e m p e r a t u r e Tst i.e.,

a(Tst)

= 1.

i s chosen e q u a l t o o n e ,

The t i m e dependence o f t h e modulus a t t h i s

t e m p e r a t u r e i s w r i t t e n i n t h e form E s t ( t ) .

Knowing t h i s r e l a t i o n

from e x p e r i m e n t , i t i s p o s s i b l e t o c a l c u l a t e t h e dependence of t h e modulus upon t i m e a t any t e m p e r a t u r e w i t h t h e a i d o f t h e f u n c t i o n The d e t e r m i n a t i o n o f t h i s f u n c t i o n i s t h e c h i e f m e t h o d o l o g i c a l

a(T).

t a s k i n p r e d i c t i n g polymer r e l a x a t i o n p r o p e r t i e s a t d i f f e r e n t temperatures. To c l a r i f y t h e s i g n i f i c a n c e o f t h e v a l u e a ( T ) , l e t u s examine

t h e s i m p l e s t case o f stress r e l a x a t i o n a c c o r d i n g t o Maxwell’s law:

where t h e Maxwellian r e l a x a t i o n t i m e (E =

‘ I depends ~

upon t e m p e r a t u r e

const).

The e l a s t i c modulus f o r t h e s t a n d a r d t e m p e r a t u r e Tst

and f o r any

temperature T i s e q u a l to

I f it i s supposed t h a t

( t ) ,one c a n c a l c u l a t e E ( t ) , h a v i n g s u b s t i t u t e d st t i n t / a ( T ) , t h a t is, E ( t ) = E s t ( t / a ) . G e n e r a l l y , a n y p h y s i c a l q u a n t i t y @ ( tt)h a t depends upon a r e l a x a t i o n p r o c e s s c h a r a c t e r i z e d by a r e l a x a t i o n t i m e rM, i s some f u n c t i o n f o f t h e r a t i o t / T N ( f o r s t a t i c p r o c e s s e s ) or o f t h e p r o d u c t u-rM ( f o r c y c l i c p r o c e s s e s ) : t h e n , knowing E

I f , a s b e f o r e , a ( T ) i s d e t e r m i n e d by t h e r a t i o o f r e l a x a t i o n t i m e s , t h e n f o r s t a t i c p r o c e s s e s , by c a r r y i n g o u t i d e n t i c a l o p e r a t i o n s , w e obtain

14

a n d f o r c y c l i c modes w e o b t a i n

where O s t ( t )

and O s t ( w )

a r e known r e l a t i o n s a t t h e t e m p e r a t u r e T s t '

The f o l l o w i n g " r e v e r s e " r e l a t i o n s are a l s o v a l i d

where @ ( ta)n d @ ( w ) a r e t h e r e l a t i o n s f o r s o m e t e m p e r a t u r e T.

It

i s obvious t h a t t h e c o e f f i c i e n t a ( T ) has d i f f e r e n t values f o r various p o l y m e r s a n d f o r d i f f e r e n t r e l a x a t i o n p r o c e s s e s i n t h e s a m e polymer. L e t us a p p l y t h e p r i n c i p l e of time-temperature e q u i v a l e n c e t o e x p e r i m e n t a l r e s u l t s s u c h t h a t t h e same r e l a x a t i o n t i m e s a t a g i v e n t e m p e r a t u r e do n o t c h a n g e w i t h t h e p a s s a g e o f t i m e .

Because

t h e r e l a x a t i o n t i m e d e p e n d s n o t o n l y upon t e m p e r a t u r e b u t a l s o T~! w i t h t i m e i n d i c a t e s a change i n t h e polymer's s t r u c t u r e d u r i n g prolonged o b s e r v a t i o n .

upon t h e p o l y m e r ' s s t r u c t u r e , t h e c h a n g e i n

For a given r e l a x a t i o n p r o c e s s , t h e e q u iv alen ce p r i n c i p l e i s f u l f i l l e d i f t h e r e l a x a t i o n t i m e d o e s n o t depend upon t h e o b s e r v a t i o n t i m e . I n many cases t h i s c o n d i t i o n i s f u l f i l l e d t o a good a p p r o x i m a t i o n . For p r a c t i c a l a p p l i c a t i o n o f t h e equivalence p r i n c i p l e , i n r e l a t i o n t o polymer f r i c t i o n p r o p e r t i e s ,

it i s n e c e s s a r y t o s e l e c t

a s t a n d a r d temperature and t o f i n d t h e c o e f f i c i e n t a ( T ) and t h e f o r t h e standard temperature. Then, i n

p r o p e r t y Q s t ( to)r O s t ( w )

o r d e r t o f i n d t h e t i m e a n d f r e q u e n c y d e p e n d e n c e s @ ( ta)n d @ ( w ) f o r

i t i s n e c e s s a r y t o s u b s t i t u t e t f o r t / a and Ost(wa), where d en o tes each f u n c t i o n t h a t i s d e t e r m i n e d by e x p e r i m e n t s a t t h e s t a n d a r d t e m p e r a t u r e .

any o t h e r temperature, w f o r w/a:

O ( t ) = aSt(t/a);

L e t u s examine some p r o c e d u r e s f o r t h e c a l c u l a t i o n o f a s t ( t ) a n d a ( T ) . The f u n c t i o n a s t ( t ) c a n b e d e t e r m i n e d e x p e r i m e n t a l l y ,

w i t h s o m e ' d i f f i c u l t y , f o r a wide range o f t i m e s .

However, i n

p r a c t i c a l experiments a r a t h e r narrow observation-time r e a l i z e d ( f o r example, from o n e s e c o n d t o o n e d a y ) .

interval is

The l i m i t s

o f t h i s i n t e r v a l are p r o v i s i o n a l l y p l o t t e d as p o i n t s A a n d B i n Fig. 1.6.

I n s u c h a case, a n e f f o r t i s made t o o b t a i n a s many

15

d e p e n d e n c e s as p o s s i b l e f o r d i f f e r e n t t e m p e r a t u r e s b e t w e e n p o i n t s

A a n d B.

Suppose, as a n example, i n F i g . 1 . 6 f o r t h e s t a n d a r d

t e m p e r a t u r e w e t a k e T3.

Segment 3 between p o i n t s C a n d D ,

e x p e r i m e n t a l l y o b s e r v a b l e f o r t h i s t e m p e r a t u r e , i s chosen as a base. I n o r d e r t o o b t a i n a c o m p l e t e c u r v e beyond t h e l i m i t s o f zone AB, s e g m e n t 2 i s s h i f t e d t o t h e l e f t p a r a l l e l t o t h e t i m e a x e s s o a s t o c o i n c i d e w i t h s e g m e n t 3, a n d t h e n s e g m e n t 1 i s s h i f t e d t o c o i n c i d e w i t h t h e new l o c a t i o n o f s e g m e n t 2 . b r a n c h of t h e b a s i c c u r v e CK i s o b t a i n e d .

As a r e s u l t ,

the l e f t

F u r t h e r , segment 4 i s

s h i f t e d t o t h e r i g h t so a s t o c o i n c i d e w i t h segment 3 , a n d t h e n s e g m e n t 5 i s s h i f t e d t o c o i n c i d e w i t h t h e new l o c a t i o n of segment

4.

A s a r e s u l t , t h e r i g h t b r a n c h o f t h e b a s i c c u r v e DM i s o b t a i n e d .

Thus, w e d e t e r m i n e t h e whole r e l a t i o n

acT(t)

graphically.

The

c o e f f i c i e n t a ( T ) i s d e t e r m i n e d f o r d i f f e r e n t t e m p e r a t u r e s by t h e m a g n i t u d e o f t h e n e c e s s a r y s h i f t s and i s e x p r e s s e d a s a f u n c t i o n of temperature i n a graph o r table.

,

we propose t h e following a n a l y t i c procedures F i r s t o f a l l , w e l o o k a t t h e s i m p l e s t , i n which t h e r e l a x a t i o n t i m e i s e x p r e s s e d by t h e f o r m u l a For d e t e r m i n i n g a (T)

T~ = b e x p

(U/kT)

,

(1.9)

where T~ i s t h e Maxwellian r e l a x a t i o n t i m e , a n d b i s t h e c o n s t a n t . From t h i s f o r m u l a i t f o l l o w s t h i t log a (T) =

- (U/2.

.

3kTst) (T-Tst/T)

Usually t h e g l a s s temperature T 109 a ( T ) = -(U/2.3kTq) (T-T /T)

s

9

i s s e l e c t e d f o r Tst;

therefore,

.

F o r many p o l y m e r s t h i s e x p r e s s i o n p r o v e s t o b e t o o i m p r e c i s e . For t h i s r e a s o n , F e r r y (38) proposed t h e i n t r o d u c t i o n o f c o e f f i c i e n t a(T) a s i n Eq.

(1.31,

a n d s u g g e s t e d a more p r e c i s e e m p i r i c a l f o r m u l a

w h e r e A = 17.4, B = 51.6O. F e r r y also i n d i c a t e d t h a t a ( T ) = q ( T ) / n ( T s t ) , where n ( T ) and n(Tst)

a r e t h e v i s c o s i t i e s a t t e m p e r a t u r e s T and Tst

> T

s

.

This

16

c a l c u l a t i o n f o l l o w s from t h e Naxwell e q u a t i o n 17 = T ~ , G ~ w , here Go i s t h e s h e a r modulus and d e p e n d s w e a k l y upon t e m p e r a t u r e . Fulcher ( 4 8 ) had a l r e a d y proposed t h e f o l l o w i n g e q u a t i o n f o r t h e viscosity

,

= C exp(a/T-To)

T-

(1.11)

which p r o v e d t o b e more p r e c i s e t h a n F r e n k e l a n d A n d r a d e ' s e q u a t i o n . Because t h e v i s c o s i t y i s p r o p o r t i o n a l t o t h e r e l a x a t i o n t i m e ,

it

f o l l o w s t h a t t h e r e is a need f o r a m o r e p r e c i s e formula t h a n ( 1 . 9 ) f o r the r e l a x a t i o n t i m e

.

= b exp(a/T-To)

T ! ,

(1.12)

T h i s f o r m u l a m a t c h e s t h e u s u a l f o r m u l a (1.9), i f w e assume t h a t t h e a c t i v a t i o n e n e r g y d e p e n d s upon t e m p e r a t u r e by t h e l a w U = ka/(l

T

-+

-

(TO/T) )

,

where k a i s t h e a c t i v a t i o n e n e r g y Uco f o r

T h i s a g r e e s w i t h t h e well-known e x p e r i m e n t a l t e m p e r a t u r e

m.

d e p e n d e n c e o f a p o l y m e r ' s a c t i v a t i o n e n e r g y a t a n d below t h e g l a s s temperature.

I f w e now assume t h a t Tst

= T

9'

then w e obtain

leading t o the expression A = a / 2 . 3 ( T -T ) 9 0

,

B = T 9 - T O '

from which w e o b t a i n Eq.

(1.10).

From t h e above n u m e r i c a l v a l u e s o f A a n d B , i t f o l l o w s t h a t To = T

9

-

51.6'

( t h a t i s , T o i s a p p r o x i m a t e l y 50'

below t h e g l a s s

t e m p e r a t u r e ) : a = 2.04 ,x lo3 d e g r e e s : Urn = 4 k c a l / m o l e . The p r i n c i p l e o f t i m e - t e m p e r a t u r e e q u i v a l e n c e i s a p p l i e d i n t h e p r o c e s s i n g o f d a t a on t h e f r i c t i o n o f r u b b e r y materials: t h e r e f o r e ,

i t w i l l b.e u s e d f r e q u e n t l y i n t h e f o l l o w i n g s e c t i o n s . 1.7

POLYMER GLASS-TRANSITION PROCESSES

W e commonly d i s t i n g u i s h s e v e r a l p o l y m e r g l a s s - t r a n s i t i o n p r o c e s s e s .

The most e s s e n t i a l f o r polymer f r i c t i o n p r o p e r t i e s a r e m e c h a n i c a l

17 and g l a s s t r a n s i t i o n s (11,13,37,50-55). examined t h i s problem i n d e t a i l (52).

One o f t h e a u t h o r s h a s Some d a t a on g l a s s t r a n s i t i o n s

n e c e s s a r y f o r a n u n d e r s t a n d i n g o f polymer f r i c t i o n p r o c e s s e s i n d i f f e r e n t p h y s i c a l s t a t e s a r e g i v e n below. The g l a s s t r a n s i t i o n i s t h e change from t h e l i q u i d s t a t e , which h a s c o n s t a n t l y c h a n g i n g s t r u c t u r e , t o t h e s o l i d s t a t e , which h a s fixed structure. The g l a s s t r a n s i t i o n a p p e a r s a s a change i n t h e t e m p e r a t u r e dependence o f a s u b s t a n c e ' s p h y s i c a l p r o p e r t i e s ( t h e r m a l , e x p a n s i o n , h e a t c a p a c i t y , v i s c o s i t y , and s i m i l a r c h a n g e s ) i n t h e a b s e n c e of c y c l i c m e c h a n i c a l i n t e r a c t i o n s .

The g l a s s t r a n s i t i o n

f o r a l i q u i d o r a f u s e d s o l i d may b e d e f i n e d a s a l o s s o f m o b i l i t y i n b a s i c u n i t s o f m a t t e r ( i o n s and m o l e c u l e s ) as a r e s u l t o f The g l a s s y s t a t e decrease i n temperature or i n c r e a s e i n pressure. i s thermodynamically u n s t a b l e , b u t i t is k i n e t i c a l l y s t a b l e because o f t h e e x t r e m e l y s l o w r e a r r a n g e m e n t o f s t r u c t u r e and c r y s t a l l i z a t i o n

a t l o w temperatures. One c h a r a c t e r i s t i c o f t h e g l a s s y s t a t e i s i r r e v e r s i b i l i t y of s t r u c t u r e under changes of p r e s s u r e even a t a

.

L i q u i d s , a n d e v e n r u b b e r y polymers, by t e m p e r a t u r e f a r from T g c o n t r a s t , a r e c h a r a c t e r i z e d by s t r u c t u r a l c h a n g e s t h a t o c c u r q u i c k l y w i t h a change of t e m p e r a t u r e o r p r e s s u r e . With a d e c r e a s e i n t e m p e r a t u r e , t h e s t r u c t u r e o f a l i q u i d o r amorphous polymer g r a d u a l l y and c o n t i n u o u s l y c h a n g e s b e c a u s e o f t h e r e a r r a n g e m e n t o f k i n e t i c u n i t s , which l e a d s d i r e c t l y t o o t h e r c h a n g e s , s u c h a s a d e g r e e o f m i c r o s t r a t i f i c a t i o n , and o t h e r s p e c i a l s t r u c t u r a l c h a r a c t e r i s t i c s of t h e l i q u i d .

The r a t e of r e a r r a n g e m e n t d e c r e a s e s w i t h d e c r e a s i n g

t e m p e r a t u r e , so t h a t i n t h e r a n g e of t h e g l a s s t e m p e r a t u r e a short-range-order

e q u i l i b r i u m c a n n o l o n g e r b e m a i n t a i n e d and t h e

l i q u i d ' s s t r u c t u r e i s immobilized.

During h e a t i n g up, s o f t e n i n g - - t h a t

i s , a t r a n s i t i o n from g l a s s t o l i q u i d - - i s

observed.

This softening

p r o c e s s d o e s n o t g r e a t l y d i f f e r from t h e g l a s s t r a n s i t i o n p r o c e s s , i f t h e r a t e o f h e a t i n g up i s t h e same a s t h e r a t e o f c o o l i n g t h a t produces t h e g l a s s . Glasses t h a t a r e c r e a t e d a t d i f f e r e n t r a t e s o f c o o l i n g have d i f f e r e n t s t r u c t u r e s and d i f f e r e n t s o f t e n i n g points. The g l a s s t r a n s i t i o n i s n o t n e c e s s a r i l y c o n n e c t e d w i t h a n y mechanical i n t e r a c t i o n s .

I n c o n t r a s t , t h e s o - c a l l e d "mechanical

t r a n s i t i o n " develops under t h e a c t i o n o f e x t e r n a l f o r c e s .

At

h i g h s t r a i n r a t e s , a n y l i q u i d ( o r any amorphous polymer) i n some r a n g e above T b e h a v e s a s a n e l a s t i c s o l i d , b e c a u s e w i t h r e d u c t i o n g i n o p e r a t i o n t i m e o r p e r i o d o f f o r c e d o s c i l l a t i o n , a l i q u i d (polymer) g r a d u a l l y loses i t s f l o w c a p a c i t y , s o t h a t i t deforms v i s c o e l a s t i c a l l y

18

and p a s s e s t o t h e e l a s t i c s t a t e .

T h i s t r a n s i t i o n i s o b s e r v e d when

t h e m o l e c u l a r r e l a x a t i o n t i m e , r e l a t e d t o m a c r o m o l e c u l a r segment mobility, is equal to t h e loading t i m e or e x t e r n a l f o r c e t i m e . For s u b m o l e c u l a r l i q u i d s , it i s v e r y d i f f i c u l t t o r e a l i z e a t r a n s i t i o n t o t h e e l a s t i c s t a t e , inasmuch a s t h e m o l e c u l a r r e l a x a t i o n t i m e f o r a l l t e m p e r a t u r e s above t h e m e l t i n g p o i n t i s e x t r e m e l y

short.

An a l t o g e t h e r d i f f e r e n t s i t u a t i o n a r i s e s w i t h v i s c o e l a s t i c

s u p e r c o o l i n g o f h i g h l y v i s c o u s l i q u i d s and e s p e c i a l l y w i t h p o l y m e r s , whose r e l a x a t i o n t i m e i s many o r d e r s of m a g n i t u d e l o n g e r t h a n t h a t of simple l i q u i d s . The t r a n s i t i o n - t e m p e r a t u r e r a n g e where r u b b e r y d e f o r m a t i o n c a n n o t o c c u r , b u t where m e c h a n i c a l l o s s e s p a s s t h r o u g h a maximum, i s c h a r a c t e r i z e d by t h e m e c h a n i c a l t r a n s i t i o n t e m p e r a t u r e TM. T h i s t e m p e r a t u r e i s d e f i n e d a s t h e t e m p e r a t u r e a t which t h e maximum mechanical losses a r e observed (Fig. 1 . 7 ) .

I n a c t u a l experimental

conditions rubbery deformation does n o t occur a t a l l temperatures above T a' b u t o n l y i n t h e r a n g e o f t e m p e r a t u r e s above TM. During a n upward t r a n s i t i o n a c r o s s TM, a p o l y m e r ' s f l e x i b i l i t y rises a b r u p t l y and t h e modulus o f e l a s t i c i t y d r o p s a b r u p t l y ( F i g . 1 . 7 ) . The m e c h a n i c a l t r a n s i t i o n t e m p e r a t u r e rises w i t h d e c r e a s i n g d u r a t i o n of f o r c e o r w i t h i n c r e a s i n g d u r a t i o n of f o r c e or w i t h i n c r e a s i n g deformation frequency

.

F i g . 1 . 7 Temperature dependence o f t h e l o g a r i t h m o f t h e e l a s t i c modulus E a n d loss t a n g e n t t a n 6 . T = g l a s s temperature; TM = m e c h a n i c a l t r a n s i t i o n t e m p e r a t u g e .

19 I t f o l l o w s t h a t a p o l y m e r ' s t r a n s i t i o n from t h e r u b b e r y t o t h e

e l a s t i c - r i g i d s t a t e ( a s t h e t e m p e r a t u r e decreases) i s o f a molecular-kinetic nature, l i k e t h e g l a s s transition. However, t h e t r a n s i t i o n t o t h e e l a s t i c - r i g i d s t a t e i s n o t r e l a t e d t o t h e cong e a l i n g o f s t r u c t u r e b u t a l w a y s t a k e s p l a c e above T

9'

though below

Therefore, t h i s t r a n s i t i o n i s n o t associated with the glass transition. However, i n p r a c t i c e it i s o f t e n c a l l e d by t h a t

TM.

name. 1.8

COLD FLOW OF POLYMERS

The m e c h a n i c a l t r a n s i t i o n u s u a l l y i s examined w i t h s m a l l stresses a n d s t r a i n s , so t h a t t h e p o l y m e r ' s s t r u c t u r e d o e s n o t change. Q u a l i t a t i v e l y new phenomena n o r m a l l y a r i s e w i t h l a r g e stresses: c o l d flow and f a i l u r e .

L a r g e stresses s u b s t a n t i a l l y a f f e c t t h e

r e l a x a t i o n t i m e , and t h e deformations t h a t arise a l t e r t h e polymer's s t r u c t u r e ( o r i e n t a t i o n , etc. 1.

Rubbery d e f o r m a t i o n a l s o c a n b e

o b s e r v e d i n r i g i d p o l y m e r s (amorphous a n d c r y s t a l l i n e ) , b u t o n l y f o r l o a d s t h a t e x c e e d a c e r t a i n l i m i t , t h e s o - c a l l e d lower l i m i t

o r s t r e n g t h o f c o l d f l o w , uB. B e l o w t h i s l i m i t , a r i g i d polymer d e f o r m s l i k e a n o r d i n a r y l o w - m o l e c u l a r - w e i g h t s o l i d ; above i t , p l a s t i c d e f o r m a t i o n d e v e l o p s , i n t h i s case c a l l e d c o l d f l o w . Cold f l o w a p p e a r s i n c r y s t a l l i n e p o l y m e r s below t h e m e l t i n g p o i n t and i n amorphous p o l y m e r s below t h e g l a s s t r a n s i t i o n t e m p e r a t u r e .

It

i s c h a r a c t e r i z e d by r e v e r s i b i l i t y i n l a r g e f o r c e d d e f o r m a t i o n s ; however, d e f o r m a t i o n r e v e r s i b i l i t y i s o b s e r v e d o n l y a f t e r h e a t i n g t o t e m p e r a t u r e s t h a t are n e a r t h e g l a s s t r a n s i t i o n t e m p e r a t u r e o r near t h e melting point.

The phenomenon o f c o l d f l o w i s e s s e n t i a l

f o r u n d e r s t a n d i n g t h e f r i c t i o n p r o p e r t i e s o f r i g i d polymer, b e c a u s e it e x p l a i n s t h e f o r m a t i o n o f t h e a c t u a l c o n t a c t a r e a o f a r i g i d

polymer u n d e r stress. Cold f l o w i n amorphous g l a s s y p o l y m e r s was f i r s t e l u c i d a t e d by A l e k s a n d r o v ( 5 6 ) a n d l a t e r i n v e s t i g a t e d i n d e t a i l by L a z u r k i n (57). These s c i e n t i s t s c o n t i n u e d t o conduct e x p e r i m e n t a l and t h e o r e t i c a l i n v e s t i g a t i o n o f c o l d f l o w (see a l s o 5 8 ) .

They p r o p o s e d t h e

r e l a x a t i o n c o n c e p t o f t h i s phenomenon, which i s now g e n e r a l l y accepted. L a z u r k i n , B a r t e n e v , a n d t h e i r co-workers (59,601 c a r r i e d o u t a thorough experimental i n v e s t i g a t i o n of c o l d flow i n r u b b e r l i k e p o l y m e r s a n d r u b b e r s a t low t e m p e r a t u r e s . The c o l d f l o w o f c r y s t a l l i n e p o l y m e r s w a s f i r s t d e s c r i b e d by C a r o t h e r s and H i l l , a n d w a s i n v e s t i g a t e d i n d e t a i l by M i k l o w i t a n d o t h e r s ( 6 1 , 6 2 1 .

20

C o n t r i b u t i o n s t o t h e s t u d y o f t h i s phenomenon h a v e a l s o b e e n made by K a r g i n a n d S o g o l o v (12,63) a n d o t h e r s (64). The l o w e r l i m i t o f c o l d f l o w i n c r e a s e s w i t h i n c r e a s i n g r a t e o f deformation or decreasing loading t i m e , r e l a x a t i o n phenomenon ( 6 5 ) .

indicating t h a t it i s a

Hence, a f i s a f u n c t i o n o f d e f o r m a t i o n

rate 0

f

= A - l o g &

where

E

i s t h e s t r a i n rate.

When t h e t e m p e r a t u r e rises t o a c e r t a i n v a l u e , u f becomes z e r o ( F i g . 1.8). For s l o w d e f o r m a t i o n s , t h i s t e m p e r a t u r e p r a c t i c a l l y c o i n c i d e s w i t h t h e g l a s s t e m p e r a t u r e ; i t h a s t o b e lower t h a n t h e mechanical t r a n s i t i o n temperature because t h a t temperature depends upon t h e r a t e o f polymer d e f o r m a t i o n s t r e t c h i n g .

(Y

E

E

2 Y

d

T.*C

F i g . 1 . 8 T e m p e r a t u r e dependence o f t h e t e n s i l e c o l d f l o w s t r e s s a f (0) a n d s t r e n g t h f o r b r i t t l e f r a c t u r e , ag ( 0 1 , o f b u t a d i e n e s t y r e n e r u b b e r SBR-30. The r e l a x a t i o n t i m e

T,

which c h a r a c t e r i z e s t h e segment

r e a r r a n g e m e n t r a t e , d e p e n d s upon stress a n d t e m p e r a t u r e a c c o r d i n g to t h e equation T

=

Toexp [ ( U - a o ) / k T l 0

,

(1.14)

21

where Uo i s t h e a c t i v a t i o n e n e r g y f o r stress u = 0 , d e t e r m i n e d by p o t e n t i a l b a r r i e r s t h a t s e g m e n t s overcome i n t r a n s f e r from o n e s u p e r p o s i t i o n s i t e t o a n o t h e r , and g i s a c o n s t a n t t h a t depends upon t h e s i z e o f a k i n e t i c u n i t , i . e . , a c h a i n segment. From t h i s e q u a t i o n , it f o l l o w s t h a t f o r low t e m p e r a t u r e s a n d s m a l l stresses,

T

i s longer than t h e t e s t i n g t i m e :

deformations of chains appear "frozen".

t h e r e f o r e , rubbery

The r e l a x a t i o n t i m e

T,

c o m p a r a b l e t o t h e o b s e r v a t i o n t i m e ( o r t o t h e i n v e r s e magnitude o f t h e d e f o r m a t i o n r a t e ) a t which r u b b e r y d e f o r m a t i o n " a p p e a r s " , may b e a t t a i n e d e i t h e r by a r i s e i n t e m p e r a t u r e t o t h e m e c h a n i c a l t r a n s i t i o n t e m p e r a t u r e TM o r by a n i n c r e a s e i n stress a c c o r d i n g t o Eq. (1.14). Hence, i t f o l l o w s t h a t u f depends upon t h e o b s e r v a t i o n t i m e o r upon t h e d e f o r m a t i o n r a t e . From Eq. ( 1 . 1 4 ) i t a l s o f o l l o w s t h a t u f i s t h e b o u n d a r y c o n d i t i o n o f t h e stress s t a t e where r u b b e r y d e f o r m a t i o n can d e v e l o p s l o w l y a t s m a l l e r

stresses.

T h i s i s d e f i n e d a s t h e c r e e p o f polymers.

E i n b i n d e r a n d co-workers

(66,67) have shown t h a t t h e r h e o l o g i c a l

a n d m e c h a n i c a l p r o p e r t i e s o f p o l y m e r s e s s e n t i a l l y depend upon t h e h y d r o s t a t i c component o f t h e p r e s s u r e : t h e r e f o r e ,

the characteristics

o f p o l y m e r i c m a t e r i a l s even u n d e r s i m p l e s t r e t c h i n g w i l l d i f f e r from t h o s e under s i m p l e c o m p r e s s i o n .

Rheological p r o p e r t i e s d i f f e r

c o n s i d e r a b l y under h i g h compression. 1.9

PROLONGED STRENGTH (FATIGUE) OF POLYMERS

Polymer wear u n d e r f r i c t i o n d e p e n d s as much upon t h e f r i c t i o n f o r c e a s upon t h e m a t e r i a l ' s s t r e n g t h p r o p e r t i e s - - i n

particular,

i t depends upon p r o p e r t i e s r e l a t e d t o p r o l o n g e d s t r e s s e d c o n d i t i o n s . C y c l i c d e f o r m a t i o n on f r i c t i o n - s u r f a c e a s p e r i t i e s l e a d s t o t y p i c a l fatigue effects. Therefore, f a t i g u e p r o p e r t i e s can never b e ignored i n t h e i n v e s t i g a t i o n of f r i c t i o n p r o p e r t i e s i n s o l i d s . The p r e s e n t s t a t e o f knowledge o f polymer s t r e n g t h h a s been t r e a t e d e l s e w h e r e (68). I n t h i s s e c t i o n , p r o b l e m s o f s t r e n g t h w i l l be pursued o n l y to t h e degree t h a t i s n eces s ar y f o r f u r t h e r

e x a m i n a t i o n o f polymer f r i c t i o n a n d wear p r o p e r t i e s . T h e r e a r e two f u n d a m e n t a l f a i l u r e mechanisms f o r s o l i d s and polymers: athermal and thermal-fluct u atin g ( 5 2 ) . Griffith s t u d i e d and d e s c r i b e d a t h e r m a l f a i l u r e i n s o l i d s u n d e r t h e o p e r a t i o n of e x t e r n a l f o r c e , b u t d i d n o t c a l c u l a t e t h e i n t e r m i t t e n t b r e a k i n g o f i n t e r a t o m i c bonds d u r i n g t h e t h e r m a l motion p r o c e s s ( s t r i c t l y s p e a k i n g , h e t r e a t e d t h e problem a t a b s o l u t e z e r o ) .

22

For r e a l s o l i d s c o n t a i n i n g d e f e c t s , a c c o r d i n g t o G r i f f i t h , f a i l u r e o c c u r s when t h e stress o f t h e t i p s o f t h e w o r s t c r a c k s a t t a i n c r i t i c a l v a l u e t o t o stress c o n c e n t r a t i o n . The t h e r m a l - f l u c t u a t i n g mechanism p r e s u p p o s e s t h e f a c t t h a t random bond b r e a k i n g o c c u r s among atoms,

i.e.,

by t h e i n c r e a s e i n

k i n e t i c e n e r g y o f atoms t o a c r i t i c a l magnitude e n e r g y f o r bond b r e a k i n g ) i n t h e r m a l motion.

(the activation

For t h i s b a s i c reason,

bond b r e a k i n g i s f l u c t u a t i n g , b u t e x t e r n a l f o r c e r e d u c e s t h e p o t e n t i a l b a r r i e r , t h a t i s , i n c r e a s e s t h e p r o b a b i l i t y o f bond breaking. The c o n c e p t i o n o f t h e f a i l u r e p r o c e s s i n s o l i d s and p o l y m e r s a s due t o t h e r m a l f l u c t u a t i o n s h a s been i n t r o d u c e d e l s e w h e r e ( 5 6 , 6 9 , 7 0 ) . Zhurkov and h i s co-workers have c o r r o b o r a t e d t h i s c o n c e p t i o n experimentally i n recent years.

From i t h a s d e v e l o p e d t h e g e n e r a l

form o f t h e t i m e - t e m p e r a t u r e dependence o f s t r e n g t h .

In the

a t h e r m a l f a i l u r e mechanism i n s o l i d s , t i m e dependence o f s t r e n g t h

i s p r a c t i c a l l y absent. The f a i l u r e p r o c e s s must n o t b e examined w i t h o u t t a k i n g i n t o consideration the relaxation effects. of t h e s e e f f e c t s i s m e c h a n i c a l l o s s e s .

One o f t h e b a s i c m a n i f e s t a t i o n s According t o B a r t e n e v ' s

c l a s s i f i c a t i o n (52,681 i t i s p o s s i b l e t o d i s t i n g u i s h c o n d i t i o n a l l y two t y p e s o f losses:

" s u r f a c e " a n d "volume".

The f i r s t t y p e o f

loss, c h a r a c t e r i s t i c o f a n y s o l i d ( t h e o r e t i c a l l y i n c l u d i n g even b r i t t l e o n e s ) , i s r e l a t e d to t h e n e c e s s a r y energy s h i f t a f t e r t h e t r a n s f e r o f a b r e a k i n g bond across a p o t e n t i a l b a r r i e r and t o t h e i m p o s s i b i l i t y of f u l l y q u a s i s t a t i c processes.

In inelastic, plastic,

a n d o t h e r m a t e r i a l s , l o s s e s d e v e l o p t h a t are d e t e r m i n e d by l o c a l i n e l a s t i c d e f o r m a t i o n s , which o c c u r even i n e x t e r n a l l y i n d u c e d b r i t t l e failure. Relaxation p r o c e s s e s determine t h e c h a r a c t e r o f f a i l u r e and

a l t e r i t s mechanism ( 7 1 ) .

Because t e m p e r a t u r e a f f e c t s t h e r e l a x a t i o n

p r o p e r t i e s of p o l y m e r s , i t i s p o s s i b l e t o d i s t i n g u i s h t h r e e t e m p e r a t u r e zones ( F i g . 1 . 9 ) c o r r e s p o n d i n g t o t h e d i f f e r e n t f a i l u r e mechanisms. The g e n e r e 1 t h e r m o - f l u c t u a t i n g

mechanism i s o p e r a t i v e f o r a l l t h r e e

z o n e s , b u t u n d e r d i f f e r e n t c o n d i t i o n s ( 6 9 ,7 2 , 7 3 )

.

The l o w e r b r i t t l e t e m p e r a t u r e TB of a r e a l polymer i s d e t e r m i n e d by t h e development o f one o r s e v e r a l c r a c k s i n i t i a t e d by t h e m o s t catastropic defect.

Micro-deformations i n t h e e l a s t i c range and

m e c h a n i c a l losses a r e c o m p a r a t i v e l y s m a l l ,

local f o r c e d - e l a s t i c

d e f o r m a t i o n s c a n b e r e a l i z e d i n t h e r i g i d s t a t e below t h e g l a s s

23 melting

delor mat ion

Fig. 1 . 9 T e n s i l e s t r e n g t h o f amorphous p o l y m e r s o v e r a w i d e t e m p e r a t u r e uB i s t h e b r i t t l e f a i l u r e stress: u f i s t h e c o l d f l o w stress: range. u e l i s t h e s t r e n g t h i n t h e r u b b e r y s t a t e : up i s t h e p l a s t i c i t y f l o w stress. TB, T Tp! a n d Tf are t h e b r i t t l e , g l a s s t r a n s i t i o n , p l a s t i c i t y , a n i f fluid-flow temperatures. temperature T

9'

and local cold-flow deformations occur i n p r o p o r t i o n

t o t h e r i s e i n t e m p e r a t u r e a t p o i n t s o f stress c o n c e n t r a t i o n ( o n microcrack t i p s ) .

A t t h o s e t i p s , t h e r e a p p e a r areas i n which

f i b r o u s m a t e r i a l form a n d a r e e l o n g a t e d d u r i n g c o l d f l o w . M e c h a n i c a l losses b a s i c a l l y a r e d e t e r m i n e d by l o s s e s i n l o c a l cold-flow micro-deformations.

A s t h i s process develops, with rise

i n temperature, so-called "crazing" appears.

After the "crazing"

come f a i l u r e c r a c k s , i n w h i c h t h e m o s t e l o n g a t e d f i b e r s a r e t o r n (74). With f u r t h e r r i s e i n t e m p e r a t u r e , t h e polymer p a s s e s t o t h e r u b b e r y s t a t e . The f a i l u r e mechanism i s d e t e r m i n e d by t h e amount s t r a i n and i t s d i s t r i b u t i o n throughout t h e material. These s t r a i n s form a s a r e s u l t o f m i c r o - s t r a t i f i c a t i o n i n p l a c e s of stress conc e n t r a t i o n , which a r e r e l a t e d t o m i c r o - v i s c o u s d e f o r m a t i o n s . These d e f o r m a t i o n s d e t e r m i n e t h e major p a r t o f m e c h a n i c a l l o s s e s d u r i n g f a i l u r e of vulcanized rubbers. T h e o r i e s of polymer f a i l u r e i n a l l t h r e e b a s i c z o n e s have n o t b e e n worked o u t t o t h e same d e g r e e . F l u c t u a t i o n t h e o r y h a s been a p p l i e d t o t h e b r i t t l e - f a i l u r e zone (75,761. A c c o r d i n g t o t h i s t h e o r y , t h e b r e a k i n g o f polymer c h e m i c a l a n d i n t e r m o l e c u l a r bonds o c c u r s as a r e s u l t of thermal f l u c t u a t i o n s , and t h e d u r a b i l i t y t of a material under s t r e t c h i n g (stress u = c o n s t ) i s expressed by t h e f o l l o w i n g a p p r o x i m a t e f o r m u l a

24

(1.15)

where L i s t h e l i n e a r dimension o f a t r a n s v e r s e c r o s s - s e c t i o n o f

41 i s t h e s c a l e f a c t o r , k i s t h e Boltzmann c o n s t a n t , i s t h e c r i t i c a l rate of c r a c k C growth, w i s t h e f l u c t u a t i o n c a p a c i t y , Z i s t h e c o e f f i c i e n t o f stress c o n c e n t r a t i o n a t a m i c r o f i s s u r e p e a k , u i s t h e t e n s i l e stress, u i s the c r i t i c a l s t r e s s , uo i s t h e s a f e s t r e s s , U i s C 0 t h e " z e r o " a c t i v a t i o n e n e r g y d u r i n g f a i l u r e , and q i s t h e t h e sample,

T is the absolute temperature, v

temperature c o e f f i c i e n t o f t h e a c t i v a t i o n energy.

I n c o n c i s e form

w e o b t a i n , k e e p i n g i n mind t h a t t h i s f o r m u l a i s c o r r e c t f o r t e n s i l e stresses from u 0 t o uc,

where A

0

depends weakly upon t e m p e r a t u r e .

The c u r v e i n F i g .

1.10 f i t s Eq.

tends t o i n f i n i t y ; f o r u

+

(1.16).

For u

-+

u

0

the durability

uc i t i s d e t e r m i n e d by t h e c r i t i c a l r a t e

As can b e s e e n , t h e e q u a t i o n i n t h e s i g n i f i c a n t zone o f d u r a b i l i t y change a p p r o x i m a t e s a l i n e a r r e l a t i o n s h i p

of c r a c k growth.

between t h e d u r a b i l i t y l o g a r i t h m l o g t a n d t h e stress u ( Z h u r k o v ' s equation) (1.17) where y = wZ+,

A

=

Ao/(u-uo).

For s m a l l stresses, a c c o r d i n g t o t h i s t h e o r y , t h e zone o f a p p l i c a b i l i t y o f t h e d u r a b i l i t y Eq.

(1.17)

is limited t o safe

stresses uo ( F i g . 1.10). I f f a i l u r e i s d e t e r m i n e d by c r a c k g r o w t h , t h e n t h e s a f e stress i s p r o p o r t i o n a l t o t h e s u r f a c e f r e e e n e r g y c1 o f a r i g i d polymer i n a g i v e n e n v i r o n m e n t ( 6 8 , 7 7 ) : u :: a/X0, where X i s t h e i n t e r a t o m i c d i s t a n c e . 0

The zone o f b r i t t l e ( l o w - t e m p e r a t u r e )

polymer f a i l u r e h a s b e e n

s t u d i e d by Peschansksaya and S t e p a n o v ( 7 8 ) a n d o t h e r s . Uo, a c c o r d i n g t o Zhurkov a n d i s a p p r o x i m a t e l y e q u a l t o t h e c h e m i c a l bond s t r e n g t h o f a polymer. Sample i m p e r f e c t i o n s , t h e s c a l e f a c t o r , The " z e r o " e n e r g y b a r r i e r ,

a s s o c i a t e s (70,79-82)

,

25

F i g . 1 . 1 0 T i m e dependence o f d u r a b i l i t y f o r r i g i d polymer. according t o Eq. (1.17).

Plot

polymer o r i e n t a t i o n , and p l a s t i c i z a t i o n do n o t s i g n i f i c a n t l y change

A l l t h e s e f a c t o r s show up more s t r o n g l y i n v a l u e y, t h e v a l u e Uo. t h e s t r u c t u r e - s e n s i t i v e coe f f i c i e n t I n t h e rubbery s t a t e , t h e d u r a b i l i t y of polymers i s c h a r a c t e r i z e d

.

by a power l a w ( 8 3 )

t = B u - ~= Bou-b where B

0

e x p (U/kT)

(1.18)

and b are c o n s t a n t s t h a t have no d i r e c t p h y s i c a l meaninq

b u t i n a complex way t a k e i n t o a c c o u n t t h e way i n which b a s i c f l u c t u a t i o n s add up d u r i n g f a i l u r e . An a n a l o g o u s r e l a t i o n i s o b s e r v e d f o r dynamic f a t i g u e i n v u l c a n i z e d r u b b e r s ( F i g . 1.11).

S t a t i c and dynamic f a t i g u e ( i n c y c l i c deforma-

t i o n s ) a r e c h a r a c t e r i z e d by t h e same v a l u e o f c o n s t a n t b i n t h e d u r a b i l i t y f o r m u l a , b u t by d i f f e r e n t v a l u e s o f B a n d B ' , so t h a t it

i s p o s s i b l e t o w r i t e f o r t h e dynamic d u r a b i l i t y (1.19) where B '

i s a c o n s t a n t t h e depends e x p o n e n t i a l l y upon t e m p e r a t u r e ,

and b i s a c o n s t a n t t h a t d o e s n o t depend upon t e m p e r a t u r e and t h e normal stress o p e r a t i o n of a v u l c a n i z e d r u b b e r - - n o r , upon t h e d e f o r m a t i o n f r e q u e n c y v .

and c o n s e q u e n t l y ,

26

log t (scc)

F i g . 1.11 R e l a t i o n s among maximum stresses i n a c y c l e a n d d u r a b i l i t y f o r v u l c a n i z e d SBR-30 f o r d i f f e r e n t modes o f d e f o r m a t i o n s t r e t c h i n g : (1) C y c l i c s t r e t c h i n g w i t h c o n s t a n t maximum d e f o r m a t i o n : (2) The same, w i t h a c o n s t a n t maximum l o a d : ( 3 ) The same, w i t h c o n s t a n t maximum stress: (4) S t a t i c deformation under a given load. I f , a l l o w i n g f o r t h e number o f c y c l e s u n t i l f a i l u r e ( n = v t ' ), then it i s e a s i l y seen t h a t t h e

w e assume m = b and C = v B ' , f o r m u l a num = C a n d E q .

( 1 . 1 9 ) e x p r e s s t h e same dynamic f a t i g u e

l a w f o r a vulcanized rubber.

Because t h e c o n s t a n t C d e p e n d s weakly

upon f r e q u e n c y , t h e c o n s t a n t B ' = C/v t o frequency.

is inversely proportional

D u r i n g t h e t r a n s i t i o n t o t h e s t a t i c mode, when

v * 0 , t h i s c o r r e l a t i o n l o s e s meaning, b e c a u s e t h e c o n s t a n t B o u g h t n o t t o a p p r o a c h i n f i n i t y , b u t r a t h e r some c o n s t a n t Bo. Because b > > 1, l a r g e c h a n g e s i n d u r a b i l i t y c o r r e l a t e d w i t h

s m a l l changes i n s t r e n g t h .

C o n s e q u e n t l y , d u r a b i l i t y i s a more

s e n s i t i v e c h a r a c t e r i s t i c o f f a t i g u e t h a n s t r e n g t h . Hence, i n e n g i n e e r i n g t h e number o f c y c l e s u n t i l f a i l u r e i s u s e d a s a c h a r a c t e r i s t i c o f dynamic f a t i g u e . I n p a r t i c u l a r , the k i n e t i c n a t u r e of f a i l u r e appears i n t h e r e l a t i o n o f polymer d u r a b i l i t y t o t i m e o f e x p e r i m e n t a t i o n ( s t a t i c l o a d s , s t r e t c h i n g w i t h d i f f e r e n t r a t e s , r e p e a t e d and m u l t i p l e d e f o r m a t i o n s , etc. 1.

T h i s allows t h e r e c a l c u l a t i o n o f d u r a b i l i t y a f t e r e a c h stress o p e r a t i o n . Methods f o r s u c h r e c a l c u l a t i o n a r e d e s c r i b e d e l s e w h e r e (68).

27

RE FE REHCES

1 V.V. K o r s h a k , Khimiya V y s o k o m o l e k u l y a r n y k h S o y e d i n e n i i ( C h e m i s t r y o f H i g h - M o l e c u l a r Compounds) , AN SSSR, MOSCOW, 1950. 2 I . P . L o s e v a n d Y e . B . T r o s t y a n s k a y a , Khimiya S i n t y e t i o h e s k i k h P o l i m e r o v ( C h e m i s t r y o f S y n t h e t i c P o l y m e r s ) , K h i m i y a , MOSCOW, 1964. 3 F.W. B i l l m e y e r , J r . , T e x t b o o k o f P o l y m e r S c i e n c e , 2nd e d n . , I n t e r s c i e n c e , N e w York, 1 9 7 1 4 B. C o l d i n g , P o l y m e r s a n d R e s i n s , T h e i r C h e m i s t r y a n d C h e m i c a l E n g i n e e r i n g , Van N o s t r a n d - R e i n h o l d , N e w York, 1 9 5 9 . 5 S.Ye. B r e s l e r a n d B.L. Y e r u s a l i m s k y , Khimiya i F i z i k a Makromolekul ( M a c r o m o l e c u l e C h e m i s t r y a n d P h y s i c s ) , Nauka, MOSCOW, 1965. 6 S. Y a . F r e n k e l , Vvedeniye v S t a t i s t i c h e s k u y u T e o r i y u P o l i m e r i z a t s i i ( I n t r o d u c t i o n t o t h e S t a t i s t i c a l T h e o r y of P o l y m e r i z a t i o n ) , Nauka, MOSCOW, 1965. 7 B.A. D o g a d k i n , K h i m i y a i F i z i k a Kauchukov ( C h e m i s t r y a n d P h y s i c s o f N a t u r a l R u b b e r s ) , G o s k h i m i z d a t , 1947. 8 M.V. V o l k e n s h t e i n , K o n f i g u r a t s i o n n a y a S t a t i s t i k a P o l i m e r n y k h T s e p e i ( C o n f i g u r a t i o n S t a t i s t i c s o f P o l y m e r C h a i n s ) , AN SSSR, MOSCOW, 1959. 9 T.M. B i r s h t e i n a n d O . B . P t s i t s y n , K o n f o r m a t s i i Makromolekul ( M a c r o m o l e c u l e C o n f o r m a t i o n s ) , Nauka, Moscow, 1 9 6 4 . 1 0 T . I . S o g o l o v a a n d G.L. S l o n i m s k y , Zh. V s e s . Khim. Ova., 6 ( 1 9 6 1 ) 3 8 9 . 11 A.P. A l e k s a n d r o v a n d Yu.S. L a z u r k i n , Zh. Tekh. F i z . , 9 ( 1 9 3 9 ) 1 2 4 9 . 1 2 V.A. K a r g i n a n d T . I . S o g o l o v a , Zh. Tekh. F i z . , 2 3 ( 1 9 4 9 ) 5 3 8 . 1 3 V.A. K a r g i n a n d G.L. S l o n i m s k i i , K r a t k i y e O c h e r k i PO F i z k o - K h i m i i P o l i m e r o v ( S h o r t E s s a y s o n P o l y m e r P h y s i c a l C h e m i s t r y ) I Khimiya, 1967. 1 4 M.V. V o l k e n s h t e i n , Dokl. Akad, Nauk SSSR, 7 8 ( 1 9 5 1 ) 8 7 9 ; 1 2 5 ( 1 9 5 9 )523. 1 5 L.R.G. T r e l o a r , T h e P h y s i c s o f R u b b e r E l a s t i c i t y , 2 n d e d n . , C l a r e n d o n P r e s s , O x f o r d , 1967. 1 6 V.A. K a r g i n , A . I . K i t a i g o r o d s k y a n d G.L. S l o n i m s k y , K o l l o i d n . Zh. , 1 9 (1957) 131. 1 7 C. T e n f o r d , P h y s i c a l C h e m i s t r y o f Macromolecules, Wiley, N e w York, 1961. 1 8 Y a . 1 . F r e n k e l , 2. P h y s . , 3 5 ( 1 9 2 6 ) 6 6 4 . 19 Y a . 1 . Frenkel, Kineticheskaya Teoriya Zhidkosti (Liquid K i n e t i c T h e o r y ) , AN SSSR, 1 9 4 5 ; V v e d e n i y e v T e o r i y u M e t a l l o v ( I n t r o d u c t i o n t o Metal T h e o r y ) , G o s t e k h i z d a t , 1 9 5 0 . 20 H. E y r i n g , J. Chem. P h y s . , 4 ( 1 9 3 6 ) 2 8 3 . 2 1 G.M. B a r t e n e v , Zh. F i z . Khim., 2 9 ( 1 9 5 5 ) 2 0 0 7 . 22 P.A. R e b i n d e r , i n F i z i k o - K h i m i c h e s k a y a Mekhanika D i s p e r s n y k h S t r u k t u r ( P h y s i c o - C h e m i c a l M e c h a n i c s of D i s p e r s e d S t r u c t u r e s ) , Nauka, M o s c o w , 1 9 6 6 , p. 3. 2 3 P.A. R e b i n d e r a n d I . N . Vlodavets, i n P r o b l y e m y F i z i k o - K h i m i c h e s k o i Mekhaniki Voloknistykh i P o r i s t y k h Dispersynykh S t r u k t u r i Materialov ( P r o b l e m s o f P h y s i c o - C h e m i c a l M e c h a n i c s o f F i b r o u s a n d P o r o u s D i s p e r s e d S t r u c t u r e s a n d Materials) , Z i n a t n y e , R i g a , 1967, p . 3. 2 4 V.A. F e d o t o v a , Kh.Kh. K h a d z h a y e v a n d P.A. R e b i n d e r , Dokl. Akad. Nauk SSSR, 1 7 0 ( 1 9 6 6 ) 1 1 3 3 . 25 L.A. Adburagimova, P.A. R e b i n d e r a n d N . N . S e r b - S e r b i n a , K o l l o i d n . Zh. , 1 7 ( 1 9 5 5 ) 184.

28 26 G.M. B a r t e n e v a n d N.V. E r m i l o v a , i n F i z i k o - K h i m i c h e s k a y a Mekhanika Dispersynkh S t r u k t u r (Physico-Chemical Mechanics o f D i s p e r s e d S t r u c t u r e s ) , Nauka, MOSCOW, 1 9 6 6 , pp. 371, 378. 27 G.M. B a r t e n e v a n d Yu.V. Z e l e n e v , Mekh. P o l i m . , ( n o . 1)( 1 9 6 9 1 3 0 . 28 G . V . V i n o g r a d o v , A . Y a . M a l k i n , N.V. P r o z o r o v s k a y a a n d V.A. K a r g i n , Dokl. Akad. Nauk SSSR, 1 5 0 ( 1 9 6 3 ) 5 7 4 ; 1 5 5 ( 1 9 6 4 ) 4 0 6 . 29 G.V. V i n o g r a d o v a n d A . Y a . M a l k i n , J. Polym. S c i . , A 2 ( 1 9 6 4 ) 2 3 5 7 . 30 G.V. V i n o g r a d o v , B . A . Dogadkin, N.V. P r o z o r o v s k a y a a n d A.P. Neverov, K o l l o i d n . Zh. , 26 ( 1 9 6 4 ) 567. 3 1 A . I . Leonov a n d G.V. V i n o g r a d o v , Dokl. Akad. Nauk SSSR, 1 5 5 ( 1 9 6 4 ) 406. 32 E. M u s t a f a e v , A . Y a . M a l k i n , Ye.P. P l o t n i k o v a a n d G.V. V i n o g r a d o v , Vysokomol. S o e d i n . , 6 ( 1 9 6 4 ) 1515. 3 3 A.V. T o b o l s k y , S t r u c t u r e a n d P r o p e r t i e s of P o l y m e r s , W i l e y , N e w York, 1960. 34 B.A. Dogadkin a n d T.N. T a r a s o v a , i n V u l k a n i z a t s i y a R e z i n ( T h e V u l c a n i z a t i o n of R u b b e r s ) , G o s k h i m i z d a t , 1 9 5 4 , p. 1 1 3 . 35 V.A. K a r g i n a n d T . I . S o g o l o v a , Dokl. Akad, Nauk SSSR, 1 0 8 ( 1 9 5 6 ) 6 6 2 : P r o b l y e m y F i z i c h e s k o i K h i m i i , V o l . 1, G o s k h i m i z d a t , 1958, p. 1 8 . 36 N . I . S h i s h k i n a n d M.F. M i l a g i n , F i z . T v e r d . T e l a , 4 ( 1 9 6 2 ) 2 6 8 1 . 37 P . P . Kobeko, Amorfnyye V e s h c h e s t v a (Amorphous S u b s t a n c e s ) , AN SSSR, MOSCOW, 1952. 38 J . D . F e r r y , V i s c o e l a s t i c P r o p e r t i e s o f P o l y m e r s , 2nd e d n . , W i l e y , N e w York, 1970. 39 G.L. S l o n i m s k y a n d L.B. R o g o v i n a , Vysokomol S o e d i n . , 6 ( 1 9 6 4 ) 3 1 5 , 620. 40 G.L. S l o n i m s k y , Zh. Tekh. F i z . , 9 ( 1 9 3 9 1 1 7 9 1 ; Dokf. Akad. Nauk SSSR, 1 4 0 ( 1 9 6 1 ) 343. 4 1 A.P. B r o n s k y , P r i k l . M a t . Mekh., 5 ( 1 9 4 1 ) 1 3 2 . 42 G.M. B a r t e n e v a n d Yu.V. Z y e l e n e v , Vysokomol. S o e d i n . , 1 4 A ( n o . 5 ) ( 1 9 7 2 ) 912. Polym. S c i . , C , P o l y m e r S y m p o s i a " T r a n s i t i o n s a n d 4 3 R.F. B o y e r , R e l a x a t i o n s i n P o l y m e r s " , N o . 4 , 1966. 4 4 G.M. B a r t e n e v a n d Yu.V. Z e l e n e v , J. Mater. S c i . E n g . , 2 ( 1 9 6 7 ) 1 3 7 . 45 G.M. B a r t e n e v a n d Yu.V. Z e l e n e v , Vysokomol. S o e d i n . , 4 ( 1 9 6 2 ) 6 6 . 46 P.P. Kobeko, Y e . V . K u v s h i n s k y a n d G . I . G u r e v i c h , I z v . Akad. Nauk SSSR, S e r . F i z . , ( N o . 3 ) ( 1 9 3 7 ) 329. 47 H. Leadermann, J. T e x t . R e s . , 1 1 ( 1 9 4 1 ) 1 7 1 . 48 G. F u l c h e r , J. Am. C e r a m . SOC., 8 ( 1 9 2 5 ) 3 3 9 . 49 E . I . F r e n k i n a n d Yu.G. Yanovsky, i n U s p e k h i R e o l o g i i P o l i m e r o v (Advances i n P o l y m e r R h e o l o g y ) , Khimiya, MOSCOW, 1 9 7 0 , p . 269. 50 G. Tamman, S t e k l o o b r a z n o y e S o s t o y a n i y e ( T h e G l a s s y S t a t e ) , ONTI, 1935. 5 1 N . I . S h i s h n k i n , Zh. Tekh. F i z . , 25 (19551188; 26 ( 1 9 5 6 ) 1 4 6 1 ; F i z . T v e r d . T e l a . , 2 ( 1 9 6 0 ) 350, 358. 52 G.M. B a r t e n e v , S t r o y e n i y e i M e k h a n i c h e s k i y e S v o i s t v a N e o r g a n i c h e s k i k h S t e k o l (The S t r u c t u r e and Mechanical P r o p e r t i e s o f I n o r g a n i c Glasses), Groningen, 1970. 5 3 M.V. V o l k e n s h t e i n a n d O . B . P t i t s y n , Zh. Tekh. F i z . , 2 6 ( 1 9 5 6 ) 2 2 0 4 . 54 S.N. Zhurkov a n d B . Y a . L e v i n , Khimiya i F i z i k o - K h i m i y a Vysokomolekulyarnykh S o y e d i n e n i i (Chemistry a n d P h y s i c a l C h e m i s t r y of High P o l y m e r s ) , AN SSSR, MOSCOW, 1 9 5 2 , pp. 280-289. 55 G.M. B a r t e n e v , Dokl. Akad. Nauk SSSR, 1 1 0 ( 1 9 5 6 ) 8 0 5 ; i n S t e k l o o b r a z n o y e S o s t o y a n i y e ( T h e G l a s s y S t a t e ) , AN SSSR, MOSCOW, p. 147. 56 A.P. A l e k s a n d r o v , T r u d y I a n d I1 i K o n f e r e n t s i i PO Vysokomelekulyarnym S o y e d i n e n i y a m ( T r a n s a c t i o n s of t h e F i r s t a n d S e c o n d C o n f e r e n c e s o n High P o l y m e r s ) , AN SSSR, MOSCOW, 1 9 4 5 , p. 49. 57 Yu.S. L a z u r k i n a n d R.L. F o g e l s o n , Zh. Tekh. F i z . , 2 1 ( 1 9 5 1 ) 2 6 7 .

J.

29 58 G . M . B a r t e n e v , Usp. Khim. ( A d v a n c e s i n C h e m i s t r y ) , 2 4 ( 1 9 5 5 ) 8 1 5 . 59 Yu.S. L a z u r k i n , G . M . B a r t e n e v e t a l . , Vysokomol. S o e d i n . , 6 ( 1 9 6 4 ) 504. 60 G.M. B a r t e n e v a n d f1.V. Voyevodskaya, Kauch. R e z i n a N . 12 ( 1 9 6 4 ) 1 4 . 6 1 W.H. C a r o t h e r s a n d J . W . H i l l , J. Am. Chem. S O C . , 5 4 ( 1 9 3 2 ) 1 5 7 9 . 62 J. M i k l o w i t z , J. C o l l . S c i . , 2 ( 1 9 4 7 ) 1 9 3 . 6 3 V.A. K a r g i n a n d T . I . S o g o l o v a , Zh. F i z . Khim., 2 7 ( 1 9 5 3 ) 1 0 3 9 , 1 2 0 8 , 1213, 1 3 2 5 . 64 G . I . B a r e n b l a t t , P r i k l . M a t . Melch. , 2 8 ( 1 9 6 4 ) 1 0 4 8 . 6 5 G.M. B a r t e n e v , Yu.V. Z e l e n e v a n d I . S . L y a k h o v i c h , Mekh. P o l i m . , ( N o . 5 ) ( 1 9 7 1 ) 885. 66 S.B. E i n b i n d e r e t a l . , Dokl. Akad. Nauk SSSR, 1 5 9 ( N o . 6 ) ( 1 9 6 4 ) 1 2 4 4 . 67 S.B. E i n b i n d e r e t a l . , Mek. P o l i m . , ( N o . 1 ( 1 9 6 5 ) 6 5 . 68 G . M . B a r t e n e v a n d Yu.S. Zuyev, P r o c h n o s t ' i R a z r u s h e n i y e V y s o k o e l a s t i c h e s k i k h M a t e r i a l o v , Khimiya, MOSCOW, 1964; or S t r e n g t h a n d F a i l u r e o f V i s c o - e l a s t i c Materials, O x f o r d , 1968. 69 G . M . B a r t e n e v , Mek. P o l i m . , ( N o . 5 ) ( 1 9 6 6 ) 7 0 0 . 70 S . N . Zhurkov, N e o g a n i c h e s k i y e M a t e r i a l y ( I n o r g a n i c Materials) , 3 ( N o . 10) (1967) 1767. 7 1 G.M. B a r t e n e v , i n T r u d y M e z h d u n a r o d n o i K o n f e r e n t s i i PO Kauchuku i Rezine ( T r a n s a c t i o n s o f t h e I n t e r n a t i o n a l Conference on N a t u r a l a n d V u l c a n i z e d R u b b e r ) , Khimiya, Moscow, 1971, p. 13. 72 G.M. B a r t e n e v , Vysokomol. S o e d i n . , l l A ( 1 9 6 9 1 2 3 4 1 . 7 3 S.B. R a t n e r a n d Yu.1. B r o k h i n , Dokl. Akad. Nauk SSSR, 1 8 8 ( 1 9 6 9 ) 8 0 7 . 7 4 M . I . Bessonov and Ye.V. Kuvshinsky, F i z . Tverd. T e l a . , 1(1959)1441; Vysokomol. S o e d i n . , 2 ( 1 9 6 0 ) 397; F i z . T v e r d . T e l a . , 3 ( 1 9 6 1 ) 1 3 1 4 ; F i z . T v e r d . T e l a . , 3 ( 1 9 6 1 ) 607. 75 G . M . B a r t e n e v , I z v . Akad. Nauk SSSR, O t d . Tekh. Nauk, ( N o . 9 ) ( 1 9 5 5 ) 53. 76 G.M. B a r t e n e v a n d V . N . K o n d r a t y e v , F i z . -Khim. Mekh. Mater. , 4 ( 1 9 6 8 ) 201. 77 G.M. B a r t e n e v a n d I . V . Razumoyskaya, Dokl. Akad. Nauk SSSR, 150 (1963) 784). 78 N . I . P e s c h a n s k a y a a n d V.A. S t e p a n o v , F i z . T v e r d . T e l a . , 4 ( 1 9 6 2 ) 2 ? 2 6 . 79 S.N. Zhurkov a n d B . N . N a r x u l l a y e v , Zh. Tekh. F i z . , 2 3 ( 1 9 5 3 ) 1 6 7 7 . 80 V.R. R e g e l , A . I . S l u t s k e r a n d E . Y e . Tomashevsky, Usp. F i z . Nauk, 1 0 6 ( 1 9 7 2 ) 193. 8 1 S.N. Zhurkov a n d S.A. Abasov, Zh. V s e s . Khim. Ova., ( N o . 3 ) ( 1 9 6 1 ) 441. 82 S.N. Zhurkov a n d S.A. Abasov, F i z . T v e r d . T e l a . , (196212184. 8 3 G.M. B a r t e n e v a n d L.S. B r y u k h a n o v a , Zh. Tekh. F i z . , 2 8 ( 1 9 5 8 ) 2 8 7 .