Selective Quenching of Large-Scale Molecular Motions by Cross

Chapter

3

Downloaded by UNIV OF SOUTHERN CALIFORNIA on June 15, 2013 | http://pubs.acs.org Publication Date: April 18, 1988 | doi: 10.1021/bk-1988-0367.ch003

Selective Quenching of Large-Scale Molecular Motions by Cross-Linking in the Strained State Ole Kramer Department of Chemistry, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark

It is shown without the need of a molecular theory of rubber elasticity or any assumption that chain entangling gives a large contribution to the equilibrium modulus. The contribution is equal to the rubber plateau modulus at fairly high degrees of cross-linking. High-vinyl polybutadiene was cross-linked in the strained state. This allows prevention of disentanglement by reptation and similar large scale molecular motions as well as experimental separation of the elastic contributions from cross-links and chain entangling. It is also argued that the molecular motions at the entrance to the rubber plateau must be local motions which involve chain segments smaller than the molecular weight between cross-links, only. The l a r g e s c a l e m o l e c u l a r motions which take p l a c e i n t h e rubber p l a t e a u and t e r m i n a l zones o f an u n c r o s s - l i n k e d l i n e a r polymer g i v e r i s e t o s t r e s s r e l a x a t i o n and t h e r e b y energy d i s s i p a t i o n . F o r narrow m o l e c u l a r weight d i s t r i b u t i o n e l a s t o m e r s n o n - c a t a s t r o p h i c r u p t u r e o f the m a t e r i a l i s caused by t h e d i s e n t a n g l e m e n t p r o c e s s e s which o c c u r i n t h e t e r m i n a l zone, e.g., by t h e r e p t a t i o n p r o c e s s . In p r a c t i c a l terms i t means t h a t t h e 'green s t r e n g t h ' o f t h e e l a s t o m e r i s poor. The green s t r e n g t h can be improved by b r o a d e n i n g t h e m o l e c u l a r weight d i s t r i b u t i o n o r a l t e r n a t i v e l y by p r e v e n t i n g t h e m o l e c u l a r motions r e s p o n s i b l e f o r r u p t u r e from o c c u r i n g . Thus, u n d e r s t a n d i n g and c o n t r o l l i n g t h e s e m o l e c u l a r motions i s o f both fundamental and practical interest. The s t r e s s r e l a x a t i o n p r o p e r t i e s o f a h i g h m o l e c u l a r weight p o l y b u t a d i e n e w i t h a narrow m o l e c u l a r weight d i s t r i b u t i o n a r e shown i n F i g u r e 1. The b e h a v i o r i s shown i n terms o f t h e apparent rubber e l a s t i c i t y s t r e s s r e l a x a t i o n modulus f o r t h r e e d i f f e r r e n t e x t e n s i o n r a t i o s and t h e experiment i s c a r r i e d on u n t i l r u p t u r e i n a l l t h r e e c a s e s . A v e r y wide rubber p l a t e a u e x t e n d i n g over n e a r l y 6 decades i n time i s o b s e r v e d f o r t h e s m a l l e s t e x t e n s i o n r a t i o . However, t h e p l a t e a u i s o b s e r v e d t o become narrower w i t h i n c r e a s i n g e x t e n s i o n

0097-6156/88/0367-0048$06.00/0 © 1988 American Chemical Society In Cross-Linked Polymers; Dickie, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1988.


3. KRAMER

Selective Quenching of Large-Scale Molecular Motions

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F i g u r e 1. S t r e s s r e l a x a t i o n c u r v e s f o r t h r e e d i f f e r e n t e x t e n s i o n r a t i o s . U n c r o s s - l i n k e d h i g h - v i n y l p o l y b u t a d i e n e w i t h a weight average m o l e c u l a r weight o f 2 m i l l i o n and a r e f e r e n c e temperature o f 283 K. G i s the apparent rubber e l a s t i c i t y modulus c a l c u l a t e d from c l a s s i c a l a f f i n e t h e o r y . ( S o l i d l i n e i s d a t a from Réf. 1).

In Cross-Linked Polymers; Dickie, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1988.

49

50

CROSS-LINKED POLYMERS


r a t i o . I f i t i s t r u e t h a t r e p t a t i o n (2) i s r e s p o n s i b l e f o r t h e d e c r e a s e o f modulus i n the t e r m i n a l zone, the o b s e r v e d b e h a v i o r i n d i c a t e s e i t h e r t h a t the r e p t a t i o n p r o c e s s i s s t r a i n dependent o r t h a t an a d d i t i o n a l r e l a x a t i o n p r o c e s s becomes important a t l a r g e s t r a i n s . The l a t t e r p o s s i b i l i t y has l e d t o the p r o p o s a l o f a new type of m o l e c u l a r motion by D o i and Edwards ( 3 ) . R e p t a t i o n . G r a d u a l d i s e n t a n g l e m e n t by the r e p t a t i o n p r o c e s s i s i l l u s t r a t e d i n F i g u r e 2. The r e p t a t i o n mechanism was proposed by de Gennes i n 1971 ( 2 ) . The main i d e a i s the f o l l o w i n g : L a t e r a l d i f f u s i o n of a l i n e a r c h a i n meets h i g h r e s i s t a n c e s i n c e the o t h e r c h a i n s cannot be i n t e r s e c t e d . The l e n g t h w i s e d i f f u s i o n a l o n g the o v e r a l l c o n t o u r o f the c h a i n meets l i t t l e r e s i s t a n c e which means t h a t i t i s a s u r p r i s i n g l y f a s t p r o c e s s u n l e s s the m o l e c u l a r weight becomes e x c e e d i n g l y h i g h . In the p r e s e n t c o n t e x t , i t i s important t h a t r e p t a t i o n can be p r e v e n t e d by a s i n g l e c r o s s - l i n k anywhere a l o n g the chain. Contour Length R e l a x a t i o n . D o i and Edwards have proposed an a d d i t i o n a l , f a s t e r r e l a x a t i o n mechanism f o r which t h e y use the term 'contour l e n g t h r e l a x a t i o n ' ( 5 ) . As shown i n F i g u r e 3 , c o n t o u r l e n g t h r e l a x a t i o n i s a p r o c e s s by which a deformed l i n e a r (or s t a r shaped) c h a i n s h o u l d r e t r a c t towards the c e n t e r o f mass o f the c h a i n . S i n c e the o v e r a l l c o n t o u r l e n g t h i n c r e a s e s upon d e f o r m a t i o n , the p r o p o s a l by D o i and Edwards i s t h a t the deformed c h a i n would want t o resume the same c h a i n d e n s i t y a l o n g the o v e r a l l c o n t o u r o f the c h a i n as t h a t of the undeformed c h a i n ( 5 ) . The o v e r a l l c o n t o u r l e n g t h i s u s u a l l y taken as the ' p r i m i t i v e p a t h ' . The p r i m i t i v e p a t h i s d e f i n e d as the s h o r t e s t p a t h c o n n e c t i n g the two ends o f the c h a i n w i t h the same t o p o l o g y as the c h a i n i t s e l f r e l a t i v e t o the s u r r o u n d i n g o b s t a c l e s (7). The p r i m i t i v e p a t h i s t h e r e f o r e much s m a l l e r than the c o n t o u r l e n g t h a l o n g the backbone o f the c h a i n . Contour l e n g t h r e l a x a t i o n i s a p r o c e s s which s h o u l d p l a y no r o l e at s m a l l d e f o r m a t i o n s . However, i t s h o u l d c o n t r i b u t e s u b s t a n t i a l l y t o the n o n - l i n e a r p r o p e r t i e s a t l a r g e d e f o r m a t i o n s . The o v e r a l l c o n t o u r l e n g t h o f the deformed c h a i n s h o u l d c o n t r a c t t o the e q u i l i b r i u m length before r e p t a t i o n begins, thereby r e l a x i n g a s u b s t a n t i a l f r a c t i o n o f the s t r e s s ( 5 ) . A c c o r d i n g t o the D o i and Edwards model, the r e l a x a t i o n time f o r r e p t a t i o n i s much l o n g e r than the r e l a x a t i o n time f o r the c o n t o u r l e n g t h r e l a x a t i o n p r o c e s s , e s p e c i a l l y i n the case o f v e r y h i g h m o l e c u l a r weight polymers. The r a t i o i s 6 N , where Ν i s the number o f s t e p s o f the p r i m i t i v e p a t h (one s t e p c o r r e s p o n d s a p p r o x i m a t e l y t o the d i s t a n c e between two c o n s e c u t i v e entanglement p o i n t s ) ( 8 ) . Contour l e n g t h r e l a x a t i o n i s a more d i f f i c u l t p r o c e s s t o quench by c r o s s - l i n k i n g than the r e p t a t i o n p r o c e s s . R e f e r r i n g t o F i g u r e 3 , i t can be seen t h a t a s i n g l e c r o s s - l i n k e x a c t l y a t the m i d p o i n t o f the c h a i n would have no e f f e c t whatsoever. A c r o s s - l i n k near each o f the two c h a i n ends would p r e v e n t the c o n t o u r l e n g t h r e l a x a t i o n p r o c e s s c o m p l e t e l y . U s i n g random c r o s s - l i n k i n g , f a i r l y h i g h degrees o f c r o s s - l i n k i n g are r e q u i r e d t o p r e v e n t the c o n t o u r length r e l a x a t i o n process a l t o g e t h e r . E f f e c t of C r o s s - l i n k i n g The e f f e c t o f c r o s s - l i n k i n g i n the u n s t r a i n e d s t a t e i s shown i n F i g u r e 4 f o r a h i g h - v i n y l p o l y b u t a d i e n e w i t h a m o l e c u l a r weight o f



KRAMER


J A F i g u r e 2. Four s u c c e s s i v e s i t u a t i o n s o f a r e p t a t i n g c h a i n . The s o l i d l i n e i n d i c a t e s the o v e r a l l c o n t o u r o f the c h a i n . T e r m i n a l s t r e s s r e l a x a t i o n s h o u l d o c c u r when the c h a i n g r a d u a l l y d i s e n g a g e s from i t s o r i g i n a l s u r r o u n d i n g s , (a) The i n i t i a l c o n f o r m a t i o n of the p r i m i t i v e c h a i n and the o r i g i n a l tube, (b) and (c) As the c h a i n moves r i g h t o r l e f t , some p a r t s o f the c h a i n l e a v e the o r i g i n a l t u b e . The p a r t s o f the o r i g i n a l tube which have become empty o f the c h a i n d i s a p p e a r ( d o t t e d l i n e ) . (d) The conformation a t a l a t e r time t . R e p t a t i o n may be p r e v e n t e d by a s i n g l e c r o s s - l i n k anywhere a l o n g the c h a i n . (Reproduced w i t h p e r m i s s i o n from Ref. 4. C o p y r i g h t 1986 Oxford University Press.)


51

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F i g u r e 3 . Contour l e n g t h r e l a x a t i o n a f t e r a l a r g e s t e p s t r a i n , (a) B e f o r e d e f o r m a t i o n , t h e c o n f o r m a t i o n i s i n e q u i l i b r i u m , (b) Immediately a f t e r d e f o r m a t i o n , t h e c h a i n and t h e tube a r e deformed a f f i n e l y . (c) A f t e r time Τ , t h e c h a i n c o n t r a c t s i n s i d e t h e tube and r e c o v e r s t h e e q u i l i b r i u m " c o n t o u r l e n g t h , (d) A f t e r time Τ , the c h a i n escapes from t h e deformed tube by r e p t a t i o n . The c o n t o u r l e n g t h r e l a x a t i o n p r o c e s s i s more d i f f i c u l t t o quench by c r o s s l i n k i n g than t h e r e p t a t i o n p r o c e s s . (Reproduced w i t h p e r m i s s i o n from R e f . 6 . C o p y r i g h t 1 9 8 0 John W i l e y & Sons.)

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Figure 4. Stress relaxation curves for high-vinyl polybutadiene with a weight average molecular weight of 291,000 and a reference temperature of 263 K. The cross-linked sample has about 60 cross-links per chain. (Reproduced with permission from ref. 9. Copyright 1985 O. Kramer.)



3.

KRAMER


2 9 1 0 0 0 which e x h i b i t s a much narrower p l a t e a u zone than t h a t shown i n F i g u r e 1 . I t can be seen t h a t the t e r m i n a l zone has d i s a p p e a r e d i n the case o f the c r o s s - l i n k e d sample. I t means t h a t the m o l e c u l a r motions r e s p o n s i b l e f o r t e r m i n a l r e l a x a t i o n have been quenched by the c r o s s - l i n k s . A l t h o u g h the two c u r v e s w i l l converge i n the r u b b e r - t o - g l a s s t r a n s i t i o n r e g i o n , they seem t o be s u p e r p o s a b l e by a v e r t i c a l s h i f t at the e n t r a n c e t o the rubber p l a t e a u r e g i o n . T h i s means t h a t the r a t e s of r e l a x a t i o n i n t h i s r e g i o n are i n s e n s i t i v e t o the p r e s e n c e of c h e m i c a l c r o s s - l i n k s . U n l i k e the p r o c e s s e s r e s p o n s i b l e f o r t e r m i n a l r e l a x a t i o n which c o u l d be p r e v e n t e d by c r o s s - l i n k i n g , the m o l e c u l a r motions r e s p o n s i b l e f o r r e l a x a t i o n a t the e n t r a n c e t o the p l a t e a u zone must be r a t h e r l o c a l p r o c e s s e s . They must i n v o l v e c h a i n segments s m a l l e r than the m o l e c u l a r weight between cross-links. The degree of c r o s s - l i n k i n g f o r the c r o s s - l i n k e d sample i n F i g u r e 4 i s f a i r l y h i g h , o f the o r d e r o f 6 0 c r o s s - l i n k e d p o i n t s p e r c h a i n . I t means t h a t the e q u i l i b r i u m modulus i s h i g h e r than the p l a t e a u modulus of the u n c r o s s - l i n k e d polymer and t h a t the e q u i l i b r i u m modulus i s reached q u i c k l y . E l a s t i c e q u i l i b r i u m i s reached i n l e s s than two minutes a t 3 2 3 Κ which i s 6 8 Κ above the 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 . At low degrees o f c r o s s - l i n k i n g , the e q u i l i b r i u m modulus would be lower than the p l a t e a u modulus and the time t o r e a c h e l a s t i c e q u i l i b r i u m much l o n g e r . The C o n t r i b u t i o n o f C h a i n E n t a n g l i n g t o the E q u i l i b r i u m Modulus. An important q u e s t i o n i s whether c h a i n e n t a n g l i n g c o n t r i b u t e s t o the e q u i l i b r i u m modulus of c r o s s - l i n k e d e l a s t o m e r s . I t i s a problem which i s o f both p r a c t i c a l and t h e o r e t i c a l i n t e r e s t and which has become the s o u r c e o f much c o n t r o v e r s y . In h i s new t h e o r y ( 1 0 , 1 1 ) , F l o r y s p e c i f i c a l l y made the assumption t h a t c h a i n e n t a n g l i n g makes no c o n t r i b u t i o n t o the modulus a t e l a s t i c e q u i l i b r i u m . T h i s assumption seems t o be c o n f i r m e d by a number of experiments which p r i m a r i l y are based on e n d - l i n k e d networks ( 1 2 , 1 3 ) . Other workers b e l i e v e t h a t the c r o s s - l i n k s t r a p the e n t a n g l e d s t r u c t u r e t h e r e b y p r e v e n t i n g d i s e n t a n g l e m e n t . U s i n g the L a n g l e y method ( 1 4 ) , G r a e s s l e y and cowor k e r s ( 1 5 , 1 6 ) found t h a t the c o n t r i b u t i o n from c h a i n e n t a n g l i n g de pends on the degree of c r o s s - l i n k i n g and t h a t the c o n t r i b u t i o n appro aches the rubber p l a t e a u modulus i n h i g h l y c r o s s - l i n k e d networks where the s t r u c t u r e i s t r a p p e d almost c o m p l e t e l y . F e r r y and coworkers found the same r e s u l t , u s i n g the two-network method ( 1 7 , 1 8 ) . Comparison o f F i g u r e s 1 and 4 shows t h a t an i n c r e a s e i n the m o l e c u l a r weight from 2 9 1 0 0 0 t o 2 m i l l i o n d e l a y s t e r m i n a l r e l a x a t i o n by more than two o r d e r s of magnitude. T h i s i s i n agreement w i t h the p r e d i c t i o n f o r r e p t a t i o n which g i v e s a dependence on m o l e c u l a r weight t o the 3 t o 3 . 4 power ( 1 9 ) . The p l a t e a u modulus i n F i g u r e 1 i s f a i r l y constant u n t i l disentanglement sets i n . I t i n d i c a t e s that chain e n t a n g l i n g would c o n t r i b u t e s t r o n g l y t o the e q u i l i b r i u m modulus i f disentanglement i s prevented. The importance of t h i s o b s e r v a t i o n may be r e i n f o r c e d by c o n s i d e r i n g a h y p o t h e t i c a l experiment, namely s t r e s s r e l a x a t i o n o f a l i n e a r polymer w i t h a m o l e c u l a r weight o f 2 b i l l i o n , i . e . , 1 0 0 0 times h i g h e r t h a n t h a t o f the polymer i n F i g u r e 1. A l t h o u g h such a polymer may be d i f f i c u l t t o make and handle e x p e r i m e n t a l l y , i t s s t r e s s r e l a x a t i o n p r o p e r t i e s can be p r e d i c t e d r e l i a b l y by s c a l i n g ( 2 0 ) . R e l a t i v e t o F i g u r e 1 , such an i n c r e a s e i n m o l e c u l a r weight would move


53


54


the t e r m i n a l zone t o the r i g h t by about 10 o r d e r s o f magnitude, making the rubber p l a t e a u n e a r l y e n t i r e l y f l a t f o r about 15 o r d e r s o f magnitude i n t i m e . I t would be p o s s i b l e t o move the t e r m i n a l zone t o s h o r t e r times by an i n c r e a s e i n t e m p e r a t u r e . However, i t would be i m p o s s i b l e t o p e r f o r m s t r e s s r e l a x a t i o n experiments a t h i g h enough temperatures and a t l o n g enough times t o r e a c h the t e r m i n a l zone. At l e a s t w i t h o u t c a u s i n g t h e r m a l d e g r a d a t i o n o f the polymer. Thus, the e l a s t i c e f f e c t o f c h a i n e n t a n g l i n g would not r e l a x below the rubber p l a t e a u v a l u e i n the e x p e r i m e n t a l l y a c c e s s i b l e time range s i n c e no d i s e n t a n g l e m e n t would o c c u r . The e x p e r i m e n t a l s t r e s s r e l a x a t i o n r e s u l t s p r e s e n t e d i n F i g u r e s 1 and 4 combined w i t h the c o n c l u s i o n s from the h y p o t h e t i c a l case c l e a r l y i n d i c a t e the f o l l o w i n g : There i s no reason t o b e l i e v e t h a t the e l a s t i c c o n t r i b u t i o n from c h a i n e n t a n g l i n g r e l a x e s t o zero a t e l a s t i c e q u i l i b r i u m u n l e s s the h i g h l y e n t a n g l e d s t r u c t u r e i s a l l o w e d t o d i s e n t a n g l e . The s i m p l e s t method t o p r e v e n t d i s e n t a n g l e m e n t i s t o permanently t r a p the e n t a n g l e d s t r u c t u r e by the i n t r o d u c t i o n of c h e m i c a l c r o s s - l i n k s i n the form o f c o v a l e n t bonds. I f we a c c e p t an e l a s t i c c o n t r i b u t i o n from c h a i n e n t a n g l i n g i n c r o s s - l i n k e d networks, the problem i s t o f i n d the r e l a t i v e magnitudes o f the c o n t r i b u t i o n s from c h a i n e n t a n g l i n g and from c r o s s - l i n k s . S i n c e the two e f f e c t s work i n p a r a l l e l i n o r d i n a r y networks, i t i s n e c e s s a r y t o know the c o n c e n t r a t i o n o f e f f e c t i v e c r o s s - l i n k s and t o have a m o l e c u l a r t h e o r y which c o r r e c t l y r e l a t e s the modulus t o the c o n c e n t r a t i o n o f c r o s s - l i n k s . The c o n t r i b u t i o n from c h a i n e n t a n g l i n g i s then found as the d i f f e r e n c e between the o b s e r v e d and the c a l c u l a t e d modulus. T h i s seems t o be an almost h o p e l e s s t a s k u n l e s s the network s t r u c t u r e i s v e r y simple and the c o n t r i b u t i o n from c h a i n entangling i s large. The c h a l l e n g e i s t h e r e f o r e t o d e v e l o p an experiment which a l l o w s an e x p e r i m e n t a l s e p a r a t i o n o f the c o n t r i b u t i o n s from c h a i n e n t a n g l i n g and c r o s s - l i n k s . The Two-Network method d e v e l o p e d by F e r r y and coworkers (17,18) i s such a method. C r o s s - l i n k i n g o f a l i n e a r polymer i n the s t r a i n e d s t a t e c r e a t e s a composite network i n which the o r i g i n a l network from c h a i n e n t a n g l i n g and the network c r e a t e d by c r o s s - l i n k i n g i n the s t r a i n e d s t a t e have d i f f e r e n t r e f e r e n c e s t a t e s . We have s i m p l i f i e d the Two-Network method by u s i n g such c o n d i t i o n s t h a t no m o l e c u l a r t h e o r y i s needed (1,21). S i m p l i f i e d Two-Network Method. As s t a t e d above, i n t r o d u c t i o n o f c r o s s - l i n k s i n the s t r a i n e d s t a t e means t h a t the o r i g i n a l h i g h l y e n t a n g l e d network and the c r o s s - l i n k network have d i f f e r e n t r e f e r e n c e s t a t e s (17,21). The r e f e r e n c e s t a t e f o r the c r o s s - l i n k network i s the s t a t e i n which the c r o s s - l i n k s are i n t r o d u c e d . R e l a t i v e t o t h a t s t a t e we s h o u l d expect no c o n t r i b u t i o n from the c r o s s - l i n k s t o the e x t e r n a l s t r e t c h i n g f o r c e . By c o n d u c t i n g a l l f o r c e measurements r e l a t i v e t o t h a t s t a t e , a l l f o r c e s and changes i n f o r c e s must come from the o t h e r network, the o r i g i n a l entanglement network. The s i m p l i f i e d two-network experiment i s performed i n the f o l l o w i n g manner: A t h i n s t r i p o f the u n c r o s s - l i n k e d polymer i s s t r e t c h e d by about 60% and m a i n t a i n e d w i t h c o n s t a n t l e n g t h t h r o u g h o u t the remainder o f the experiment. The f o r c e i s m o n i t o r e d a t a l l t i m e s . A f t e r a p r e d e t e r m i n e d r e l a x a t i o n p e r i o d , the temperature i s d e c r e a s e d t o below the g l a s s t r a n s i t i o n temperature t o quench a l l o v e r a l l c o n f o r m a t i o n a l changes. The sample i s c r o s s - l i n k e d i n the g l a s s y



3.

KRAMER


s t a t e w i t h h i g h energy ( 1 0 MeV) e l e c t r o n s , t h e r e b y p e r m a n e n t l y t r a p p i n g the r e m a i n i n g e n t a n g l e d s t r u c t u r e . The t e m p e r a t u r e i s t h e n r a i s e d t o a temperature o f 8K above t h e g l a s s t r a n s i t i o n temperature i n o r d e r t o a l l o w the t r a p p e d f r e e r a d i c a l s t o r e a c t . In the next s t e p , the t e m p e r a t u r e i s r a i s e d t o the s t r e s s r e l a x a t i o n temperature, a l l o w i n g a d i r e c t comparison o f t h e f o r c e s b e f o r e , f , and a f t e r c r o s s - l i n k i n g , f . The sample i s then a l l o w e d t o r e a c h e l a s t i c e q u i l i b r i u m . T h i s i s a c c o m p l i s h e d by f i r s t h e a t i n g t h e sample t o a h i g h e r temperature ( 3 2 3 K) and s u b s e q u e n t l y c o o l i n g i t back t o the s t r e s s r e l a x a t i o n temperature t o measure t h e e q u i l i b r i u m f o r c e , f . The t h r e e f o r c e s f , f and f may now be d i r e c t l y compared e c e a t t h e same e x t e n s i o n and t e m p e r a t u r e . F i g u r e 5 shows the measured f o r c e s f o r a t y p i c a l experiment f o r a h i g h - v i n y l p o l y b u t a d i e n e w i t h a m o l e c u l a r weight o f 2 m i l l i o n . The s t r e s s r e l a x a t i o n t e m p e r a t u r e i s 2 8 3 Κ i n t h i s c a s e , i . e . , 2 8 Κ above t h e g l a s s t r a n s i t i o n temperature i n o r d e r t o get c l o s e t o the t e r m i n a l zone. The r e s u l t s f o r t h r e e d i f f e r e n t d e g r e e s o f c r o s s - l i n k i n g a r e shown i n T a b l e 1 . The q u a n t i t y 1 0 0 ( f - f ) / f i s a measure o f t h e p e r c e n t a g e change i n the s t r e t c h i n g f o r c e caused by c r o s s - l i n k i n g and the subsequent r e l a x a t i o n d u r i n g h e a t i n g t o the s t r e s s r e l a x a t i o n t e m p e r a t u r e . The q u a n t i t y 1 0 0 ( f - f ) / f i s a measure o f t h e r e l a t i v e d i f f e r e n c e between the s t r e s s r e l a x a t i o n f o r c e and the e q u i l i b r i u m f o r c e . T a b l e 1 shows v e r y s m a l l v a l u e s o f the q u a n t i t y 100(f-f )/f f o r a l l t h r e e degrees o f c r o s s - l i n k i n g . T h i s p r o v e s t h a t the c r o s s l i n k s do n o t c o n t r i b u t e t o the s t r e t c h i n g f o r c e i n t h i s experiment where t h e sample i s h e l d a t c o n s t a n t l e n g t h . They s e r v e the purpose of t r a p p i n g the e n t a n g l e d s t r u c t u r e , o n l y . The s m a l l v a l u e s o f t h e q u a n t i t y 1 0 0 ( f - f ) / f show t h a t t h e e q u i l i b r i u m f o r c e i s o n l y s l i g h t l y s m a l l e r than the s t r e s s r e l a x a t i o n f o r c e j u s t b e f o r e c r o s s - l i n k s a r e introduced i n a l l three cases. T h i s demonstrates t h a t the trapped entangled s t r u c t u r e gives r i s e t o a large e q u i l i b r i u m force, since the c r o s s - l i n k s c o n t r i b u t e n o t h i n g t o the measured f o r c e s i r r e s p e c t i v e o f degree o f c r o s s - l i n k i n g . c

T a b l e 1 . R e s u l t s from c r o s s - l i n k i n g i n the w i t h e x t e n s i o n r a t i o s o f about 1 . 6 .

7

Sample Dose/kGy

9

strained

state

8

50

100

200

f-f 100,

%

3.0

2.9

3.6

100,

%

8.9

7.9

5.4

f f-f f

SOURCE: Data arc from réf. 1.


55

56



1.0

I

I

I

Sample 8

'

ν

1,2-POLYBUTADIENE M = 2-10 Τ * 283 Κ 6

0.8 Uncross-linked

ο ο

Q

0.6

ο

-

Cross-linked

0.4

h 0

ι 1

I 2 log

I 3

κ.

t/s

Figure 5 . S t r e t c h i n g f o r c e a t c o n s t a n t l e n g t h and a temperature o f 2 8 3 Κ f o r the same polymer as i n F i g u r e 1. The sample i s u n c r o s s - l i n k e d f o r the f i r s t 1 0 0 0 s and t h e n r a p i d l y c o o l e d t o the g l a s s y s t a t e and c r o s s - l i n k e d w i t h h i g h energy e l e c t r o n s . The force f i s o b t a i n e d by subsequent h e a t i n g t o the s t r e s s r e l a x a t i o n temperature o f 2 8 3 K. The e q u i l i b r i u m f o r c e f at 2 8 3 Κ i s o b t a i n e d a f t e r h e a t i n g t o 3 2 3 Κ f o l l o w e d by c o o l i n g back t o 2 8 3 K. (Reproduced w i t h p e r m i s s i o n from Ref. 1 . C o p y r i g h t 1 9 8 8 E l s e v i e r Applied

Science P u b l i s h e r s . )



3.

KRAMER


Thus, the s i m p l i f i e d Two-Network experiment shows by a d i r e c t comparison o f f o r c e s a t c o n s t a n t l e n g t h t h a t the t r a p p e d e n t a n g l e d s t r u c t u r e o f a w e l l c r o s s - l i n k e d e l a s t o m e r c o n t r i b u t e s t o the e q u i l i b r i u m modulus by an amount t h a t i s a p p r o x i m a t e l y e q u a l t o the rubber p l a t e a u modulus. The modulus c o n t r i b u t i o n from the t r a p p e d e n t a n g l e d s t r u c t u r e w i l l be l e s s f o r lower m o l e c u l a r weights and e s p e c i a l l y a t low degrees o f c r o s s - l i n k i n g (14). The s m a l l v a l u e s o f the q u a n t i t y 1 0 0 ( f - f ) / f p r o v e t h a t c h a i n s c i s s i o n i s absent o r s m a l l f o r t h i s system. T h i s i s important s i n c e the i n t e r p r e t a t i o n o f the e x p e r i m e n t a l r e s u l t s would be d i f f i c u l t or even i m p o s s i b l e i n the case o f s u b s t a n t i a l c h a i n s c i s s i o n . I t s h o u l d a l s o be s t r e s s e d t h a t the f u n c t i o n a l i t y o f the c r o s s - l i n k s need not be known, and t h a t a d i s t r i b u t i o n o f c r o s s - l i n k f u n c t i o n a l i t i e s as w e l l as inhomogeneous c r o s s - l i n k i n g i s unimportant. The r e a s o n b e i n g t h a t the c r o s s - l i n k s m e r e l y s e r v e the purpose o f t r a p p i n g the entangled s t r u c t u r e . F i n a l l y , i t s h o u l d be p o i n t e d out t h a t no m o l e c u l a r t h e o r y of rubber e l a s t i c i t y i s r e q u i r e d and t h a t no assumptions were made i n o r d e r t o r e a c h above c o n c l u s i o n s . The S i m p l i f i e d Two-Network Method and Contour Length R e l a x a t i o n . An attempt was made t o s t u d y the c o n t o u r l e n g t h r e l a x a t i o n p r o c e s s by the s i m p l i f i e d Two-Network method. I t i s p o s s i b l e t o make samples which have been c r o s s - l i n k e d i n the s t r a i n e d s t a t e b e f o r e the contour length r e l a x a t i o n process takes p l a c e (Batsberg, W.; G r a e s s l e y , W.W.; Kramer, 0., u n p u b l i s h e d d a t a ) . However, a l l samples which were t e s t e d a t l a r g e e x t e n s i o n r a t i o s broke b e f o r e e l a s t i c e q u i l i b r i u m was reached. The f i r s t d i r e c t e v i d e n c e o f the c o n t o u r l e n g t h r e l a x a t i o n p r o c e s s has been o b t a i n e d on h i g h m o l e c u l a r weight p o l y ( e t h y l e t h y l e n e ) , u s i n g s m a l l a n g l e n e u t r o n s c a t t e r i n g on h i g h l y s t r a i n e d samples (22). Conclusions C r o s s - l i n k i n g i n the s t r a i n e d s t a t e can be used t o quench the l a r g e s c a l e m o l e c u l a r motions s e l e c t i v e l y . I t i s found t h a t c h a i n e n t a n g l i n g c o n t r i b u t e s s t r o n g l y t o the e q u i l i b r i u m modulus i f c r o s s - l i n k s are i n t r o d u c e d b e f o r e d i s e n t a n g l e m e n t t a k e s p l a c e . The c o n t r i b u t i o n from c h a i n e n t a n g l i n g a t e l a s t i c e q u i l i b r i u m i s a p p r o x i m a t e l y e q u a l t o the rubber p l a t e a u modulus f o r e l a s t o m e r s o f v e r y h i g h m o l e c u l a r weight and f a i r l y h i g h degrees o f c r o s s - l i n k i n g . T h i s r e s u l t i s o b t a i n e d w i t h o u t the need o f a m o l e c u l a r t h e o r y o f rubber e l a s t i c i t y and w i t h o u t making any assumptions. C r o s s - l i n k i n g i n the s t r a i n e d s t a t e c o u l d not be used t o study the c o n t o u r l e n g t h relaxation p r o c e s s due t o sample r u p t u r e a t l a r g e e x t e n s i o n s . The m o l e c u l a r motions r e s p o n s i b l e f o r s t r e s s r e l a x a t i o n a t the e n t r a n c e t o the rubber p l a t e a u zone are argued t o be l o c a l p r o c e s s e s which i n v o l v e c h a i n segments s m a l l e r than the m o l e c u l a r weight between cross-links, only. Acknowledgment Support from the Danish C o u n c i l f o r N a t u r a l S c i e n c e and the Danish C o u n c i l f o r S c i e n c e and I n d u s t r i a l R e s e a r c h under the FTU program a r e g r a t e f u l l y acknowledged.


57


58


Literature Cited 1. Batsberg, W.; Hvidt, S.; Kramer, O.; Fetters, L.J. In Biological and Synthetic Polymer Networks; Kramer, O., Ed.; Elsevier Applied Science Publishers: London, 1988; p xx 2. de Gennes, P.G. J. Chem. Phys. 1971, 55, 572. 3. Doi, M.; Edwards, S.F. J. Chem. Soc. Faraday Trans. 2 1978, 74, 1789, 1802 and 1818. 4. Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986, p 194. 5. Ibid., p 247. 6. Doi, M. J. Polym. Sci., Polym. Phys. Ed. 1980, 18, 1891. 7. Ref. 4, p 192. 8. Graessley, W.W. Adv. Polym. Sci. 1982, 47, 67. 9. Kramer, O. Brit. Polym. J. 1985, 17, 129. 10. Flory, P.J. Proc. Roy. Soc. London (A) 1976, 351, 351. 11. Flory, P.J. J. Chem. Phys. 1977, 66, 5720. 12. Mark, J.E. Makromol. Chem. 1979, Suppl. 2, 87. 13. Llorente, M.A.; Mark, J.E. Macromolecules 1980, 13, 681. 14. Langley, N.R. Macromolecules 1968, 1, 348. 15. Dossin, L.M.; Graessley, W.W. Macromolecules 1979, 12, 123. 16. Pearson, D.S.; Graessley, W.W. Macromolecules 1980, 13, 1001. 17. Kramer, O.; Carpenter, R.L.; Ty, V.; Ferry, J.D. Macromolecules 1974, 7, 79. 18. Ferry, J.D. Polymer 1979, 20, 1343. 19. Doi, M. J. Polym. Sci., Polym. Lett. Ed. 1981, 19, 265. 20. de Gennes, P.G. Scaling Concepts in Polymer Science; Cornell Univ. Press: Ithaka, 1979. 21. Batsberg, W.; Kramer, O. J. Chem. Phys. 1981, 74, 6507. 22. Mortensen, K.; Batsberg, W.; Kramer, O.; Fetters, L.J. In Biological and Synthetic Polymer Networks; Kramer, Ο., Ed.; Elsevier Applied Science Publishers: London, 1988; p. xx. RECEIVED October 7, 1987


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