Coulombic Interactions in Macromolecular Systems - American

14 Ion-Hopping Kinetics in Three-Arm Star Polyisobutylene-Based Model Ionomers 1

2,4

Masanori Hara1,3, Adi Eisenberg , Robson F. Storey 1

Downloaded by NANYANG TECHNOLOGICAL UNIV on June 15, 2016 | http://pubs.acs.org Publication Date: March 28, 1986 | doi: 10.1021/bk-1986-0302.ch014

2

and Joseph P. Kennedy

Department of Chemistry, McGill University, Montreal, Quebec, Canada H3A Institute of Polymer Science, University of Akron, Akron, OH 44325

2K6

2

Three-armed ionically terminated "stars" are used to determine the kinetics of ion-hopping or interchange. These species form a network in which the ionic aggregates provide the weak links. Stress relaxation techniques are utilized, and from the relaxation times the kinetic parameters are calculated. For the sodium sulfonate terminal groups, the first order rate constant is given by k = 7.11 x 10 exp (-94,000/RT) with ΔΗ in Joules/mol. 9

I t has l o n g been r e c o g n i z e d t h a t the f l o w of ionomers, because o f the p r e s e n c e o f i o n i c a g g r e g a t e s , i s r e l a t e d to the r a t e a t which i o n i c groups ( p a i r s ) remove themselves from one a g g r e g a t e and move to another (1-2). S i n c e ionomers have r e c e n t l y become of g r e a t i n d u s t r i a l and academic i n t e r e s t , and s i n c e the f l o w o f ionomers r e p r e s e n t s a fundamental problem i n s t u d y o f m e c h a n i c a l p r o p e r t i e s as w e l l as the p r o d u c t i o n o f these m a t e r i a l s , i t has been o f g r e a t i n t e r e s t to d e t e r m i n e the r a t e a t which t h i s i o n - h o p p i n g process occurs. Attempts to study t h i s fundamental p r o c e s s i n b u l k ionomers have not y i e l d e d much q u a n t i t a t i v e i n f o r m a t i o n , b e c a u s e , i n most ionomer systems, two d i f f e r e n t types o f a g g r e g a t e s a r e thought t o be p r e s e n t i . e . m u l t i p l e t s and c l u s t e r s ( 3 - 4 ) . M u l t i p l e t s a r e r e l a t i v e l y s m a l l i o n i c a g g r e g a t e s c o n s i s t i n g of a few i o n p a i r s , w h i l e c l u s t e r s can be q u i t e l a r g e and presumably a l s o c o n t a i n c o n s i d e r a b l e o r g a n i c c h a i n m a t e r i a l . Thus, i n o r d e r to o b t a i n i n f o r m a t i o n about i o n - h o p p i n g k i n e t i c s , i t i s n e c e s s a r y to work on a system i n which o n l y m u l t i p l e t s a r e p r e s e n t . Since conventional ionomers, i . e . , polymers c o n t a i n i n g a l a r g e c o n c e n t r a t i o n (4-12 mole %) of randomly p l a c e d i o n i c groups, g e n e r a l l y c o n t a i n b o t h c l u s t e r s and m u l t i p l e t s ( 5 - 6 ) , o t h e r systems have to be c o n s i d e r e d . D i f u n c t i o n a l l y t e r m i n a t e d polymer c h a i n s a r e , o b v i o u s l y , o f great i n t e r e s t i n t h i s context. T e y s s i e and coworkers (7-15) have i n v e s t i g a t e d the p r o p e r t i e s of h a l a t o - t e l e c h e l i c systems e x t e n s i v e l y , i n c l u d i n g a number o f r h e o l o g i c a l s t u d i e s . They found t h a t , f o r 3

Current address: Rutgers, The State University of New Jersey, Department of Mechanics and Materials Science, Piscataway, NJ 08854 Current address: Department of Polymer Science, University of Southern Mississippi, Hattiesburg, MS 39406-0076

4

0097-6156/ 86/ 0302-0176S06.00/ 0 © 1986 American Chemical Society

Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.


14.

HARAETAL.

111

Three-Arm Star Polyisobutylene-Based Ionomers

p o l y b u t a d i e n e h a l a t o - t e l e c h e l i c systems w i t h d i v a l e n t c a t i o n s (Ba, Ca, Zn, and Mg), o n l y one r e l a x a t i o n mechanism c h a r a c t e r i s t i c o f i o n i c a g g r e g a t e s worked w i t h o u t c h a i n f l o w due t o t h e l a y e r e d structures of multiplets. The a c t i v a t i o n e n e r g i e s o b t a i n e d a r e from 62.8 KJoul/mole f o r Ba s a l t t o 128 KJoul/mole f o r Mg s a l t . However, i n general, i t i s probable that i o n i c aggregation i n the h a l a t o t e l e c h e l i c systems w i t h monovalent c a t i o n s c o n s i s t s m a i n l y o f i o n q u a r t e t s ( i n v o l v i n g o n l y two c h a i n e n d s ) , as shown s c h e m a t i c a l l y i n F i g u r e 1. I n t h i s c a s e , i s o l a t i o n o f t h e i o n - h o p p i n g mechanism from a d d i t i o n a l r e l a x a t i o n mechanisms, e.g., normal c h a i n f l o w , c a n not be a s s u r e d . I t i s t h e r e f o r e d e s i r a b l e t o s t u d y a system i n which t h e polymer i s p r e s e n t as a c o v a l e n t l y c r o s s l i n k e d network, the c h a i n s o f which c o n t a i n "weak o r t h e r m o l a b i l e l i n k a g e s i n t h e form o f i o n q u a r t e t s . A t h r e e - a r m s t a r polymer c a r r y i n g t e r m i n a l i o n i c groups, a l s o shown i n F i g u r e 1, c a n y i e l d such a system p r o v i d e d t h e i o n i c groups a r e p r e s e n t i n s m a l l enough c o n c e n t r a t i o n so t h a t c l u s t e r i n g does n o t o c c u r . Such a system would form a n e t work which s a t i s f i e s a l l t h e r e q u i r e m e n t s f o r t h e measurement o f ion-hopping k i n e t i c s . Three-arm s t a r ionomers o f t h i s type have r e c e n t l y become a v a i l a b l e (16-17). They c o n s i s t o f p o l y i s o b u t y l e n e (PIB) c h a i n segments emanating from a c e n t r a l p h e n y l r i n g and c a r r y i n g m e t a l s u l f o n a t e groups a t t h e t h r e e c h a i n ends. M o l e c u l a r w e i g h t s o f t h e s e polymers range from 6,000 t o 20,000. The c h e m i s t r y o f these systems has been d e s c r i b e d e x t e n s i v e l y . I t has been shown t h a t t h e m a t e r i a l s behave l i k e c o n v e n t i o n a l c o v a l e n t l y c r o s s l i n k e d r u b b e r s , and i n v e s t i g a t i o n s o f t h e s e m a t e r i a l s a r e c o n t i n u i n g (18-23). A w e l l d e f i n e d way o f o b t a i n i n g k i n e t i c d a t a f o r c h e m i c a l r e a c t i o n s which l e a d t o c h a i n s c i s s i o n i n b u l k network polymers i s by s t r e s s r e l a x a t i o n . The f o r m a l i s m o f t h i s method h a s been d e v e l o p e d by T o b o l s k y and o t h e r s (24-26), and c a n be summarized as follows. From t h e k i n e t i c t h e o r y o f r u b b e r e l a s t i c i t y , t h e s t r e s s , f ( t ) , a t time t i s r e l a t e d t o t h e i n i t i a l s t r e s s , f ( 0 ) , by where N ( t ) i s 11

f(t) fCO)

NCO N(0)

=

m u

;

the number o f network c h a i n s p e r u n i t volume s u p p o r t i n g the s t r e s s a t time L . When t h e network c h a i n s a r e u n d e r g o i n g an i n t e r c h a n g e r e a c t i o n , f i r s t - o r d e r r e a c t i o n r a t e law c a n be a p p l i e d , i . e . -

f

^

=

k

N(t)

(2)

where k denotes t h e r a t e c o n s t a n t o f t h e i n t e r c h a n g e r e a c t i o n . I n t e g r a t i o n o f E q u a t i o n 2 w i t h t h e boundary c o n d i t i o n N(t)=N(0) a t t=0 g i v e s =

exp(- k t ) = exp(-t/T)

(3)

where k=l/x and τ denotes t h e r e l a x a t i o n t i m e . Combining E q u a t i o n 1 and E q u a t i o n 3 y i e l d s t h e law f o r M a x w e l l i a n d e c a y . F i n a l l y , the r a t e constant k f o r the interchange r e a c t i o n can be e x p r e s s e d by t h e f o l l o w i n g e q u a t i o n


178

COULOMBIC INTERACTIONS IN MACROMOLECULAR SYSTEMS

k = A exp(-AE /RT)

(4)

a

where A E i s t h e a c t i v a t i o n energy o f t h e r e a c t i o n and R i s t h e gas constant. The assumptions made i n a p p l y i n g t h e s e e q u a t i o n s t o i o n - h o p p i n g r e a c t i o n s a r e t h a t a l l t h e network c h a i n s a r e o f u n i f o r m l e n g t h and t h a t t h e i o n - h o p p i n g mechanism i s r e f l e c t e d o n l y i n t h e " u l t i m a t e e q u i v a l e n t " Maxwell element.


a

Experimental Three-arm s t a r PIB ionomers were p r e p a r e d by t e c h n i q u e s d e s c r i b e d e x t e n s i v e l y b e f o r e (16-17). Some r e l e v a n t parameters f o r t h e sample a r e shown i n T a b l e I . The 3-arm PIB ionomers n e u t r a l i z e d by NaOH were d r i e d a t room temperature under vacuum f o r 1 month, w h i l e ano t h e r s e r i e s o f Na s a l t s were d r i e d a t 140°C f o r a week, Samples were t h e n c o m p r e s s i o n molded a t 150°C under an a p p l i e d l o a d o f 4000 l b s / in2. T y p i c a l d i m e n s i o n s o f t h e r e c t a n g u l a r specimens f o r s t r e s s r e l a x a t i o n s t u d i e s were 3.0 χ 6.0 χ 40 (mm). The e x p e r i m e n t s were performed on a s t r e s s r e l a x o m e t e r (27) under a n i t r o g e n atmosphere. The temperature i n s i d e t h e sample chamber was c o n s t a n t w i t h i n ± 0.2°C i n t h e range o f 60 t o 150°C. The s t r e t c h i n g mode was u t i l i z e d and t h e e l o n g a t i o n was c a . 10% o f total length. P r e v i o u s s t u d i e s o f ionomers have shown t h a t dynamic m e c h a n i c a l t e s t s a r e most s e n s i t i v e to t h e p r e s e n c e o f l a r g e c l u s t e r s . For this r e a s o n , dynamic m e c h a n i c a l t e s t was performed u s i n g a c o m p u t e r i z e d t o r s i o n pendulum ( 2 8 ) . The f r e q u e n c i e s v a r i e d from c a . 3 Hz f o r t h e g l a s s y r e g i o n t o c a . 0.1 Hz f o r the low modulus r e g i o n . The h e a t i n g r a t e was u s u a l l y 0.6°C/min w i t h a temperature c o n t r o l o f ± 1 ° C . Results F i g u r e 2 shows t h e shear s t o r a g e modulus, G', t h e shear l o s s modulus, G", and tan 6 f o r the sample d r i e d a t room t e m p e r a t u r e . The p r i m a r y t r a n s i t i o n i s o b s e r v e d around -55°C and a w e l l - d e f i n e d r u b b e r y p l a t e a u i s o b s e r v e d from -20 t o 140°C. The r e s u l t s o f t h e s t r e s s r e l a x a t i o n e x p e r i m e n t s a t v a r i o u s t e m p e r a t u r e s , p l o t t e d as l o g E ( t ) v s . t , a r e shown i n F i g u r e 3. I t i s s e e n t h a t t h e t e r m i n a l p o r t i o n s o f a l l t h e c u r v e s a r e i n d e e d l i n e a r ; t h u s , t h e maximum r e l a x a t i o n time a t each temperature, Tm, c a n be o b t a i n e d from t h e s l o p e o f t h e curves. The maximum r e l a x a t i o n times a r e p l o t t e d as l o g T v s . 1/T i n F i g u r e 4. L i n e a r i t y i s o b s e r v e d w i t h i n e x p e r i m e n t a l e r r o r o v e r t h r e e o r d e r s o f magnitude o f t i m e . The a c t i v a t i o n energy e s t i m a t e d from t h e s l o p e o f t h e c u r v e i s 94 K J o u l / m o l e . F i g u r e 4 a l s o shows the d a t a f o r samples d r i e d a t h i g h t e m p e r a t u r e . I t i s seen t h a t t h e s l o p e s o f t h e two s e t s o f d a t a a r e i d e n t i c a l . The p o s i t i o n o f t h e p l o t f o r t h e sample d r i e d a t h i g h temperature i s , however, much h i g h e r t h a n f o r t h a t d r i e d a t a lower t e m p e r a t u r e . These p l o t s a r e v e r y r e m i n i s c e n t o f c l a s s i c a l r h e o l o g i c a l d a t a from which k i n e t i c parameters o f t h e i o n i n t e r c h a n g e p r o c e s s c a n be c a l c u l a t e d (29-33). The k i n e t i c d a t a o b t a i n e d f o r t h e i o n - h o p p i n g r e a c t i o n o f 3-arm s t a r PIB ionomer i s k = 7.11 χ 1 0 exp(-94,100/RT) f o r t h e sample d r i e d a t room temperature and k = 6.90 χ 1 0 exp(-94,100/Rt) f o r t h e sample d r i e d a t high temperatures. r

m

9

8


HARAETAL.


179


14.

F i g u r e 2.

1

V a r i a t i o n o f t h e s h e a r s t o r a g e modulus, G , t h e shear modulus, G", and t a n 6 f o r t h e sample d r i e d a t room temperature.


loss



180

F i g u r e 4.

Maximum r e l a x a t i o n time (Tm) v s . 1000/T f o r samples at room temperature (0) and a t 140°C ( Φ ) .


dried

14.

HARAETAL.


Table I . Relevant

parameters f o r 3-arm PIB s t a r

Parameter


M o l e c u l a r Weight Functionality Glass T r a n s i t i o n Temperature

181

ionomer.

Symbol

Value

Method

(Fn) (Tg)

6,900 3.2 -56°C -55°C

VPO titration DSC T.P.

T a b l e I I . Temperature dependence o f Em and Mc f o r samples d r i e d a t room t e m p e r a t u r e .

Temperature

e x p

Em

( a )

M c

Mc

e x p

/Mc

C a l

(

b

)

(N/m2) (°C) 61.4 1.05 χ 1 0 1.45 6,670 1.21 82.7 1.34 5,560 c 104.5 9.0 χ 1 0 1.91 8,780 2.48 124.5 7.3 11,400 2.52 138.2 7.4 11,600 t" J· M c P was o b t a i n e d from t h e e q u a t i o n Em = 3pRT/Mc. (b). cal c a l c u l a t e d t o be 4,600 (= 6,900 χ 2/3), assuming t h a t a l l i o n s a r e p r e s e n t as q u a r t e t s . 6

5

a

e x

M c

w

Table I I I .

a

s

Temperature dependence o f Em and Mc f o r samples a t 140°C.

(N/m2) 4.95 χ 1 0 4.2 4.4 4.3 4.5 3.5 3.5 2.2

(°C) 93.3 102.7 116.1 120.4 132.3 133.0 134.4 136.1

Mc

Em

Temperature

D

e x p

15,500 18,700 18,500 19,200 18,800 24,300 24,400 39,000

Mc

e x p

/Mc

dried

C a l

3.37 4.07 4.02 4.17 4.09 5.28 5.30 8.48

Discussion In t h e 3-arm s t a r PIB ionomers, most o f t h e i o n i c a g g r e g a t e s s h o u l d be m u l t i p l e t s due t o t h e low i o n c o n t e n t (2.4 mole %) ( 3 - 4 ) . F i g u r e 2 shows o n l y t h e s h o u l d e r f o r secondary t r a n s i t i o n i n t h e t a n δ ?.urve (around 3 0 ° C ) ; t h u s , o n l y v e r y s m a l l amounts o f i o n s e x i s t as c l u s t e r s ( 3 5 ) , i f they a r e p r e s e n t a t a l l . Most o f t h e i o n s a r e thus e x p e c t e d t o e x i s t a s s m a l l m u l t i p l e t s , i . e . q u a r t e t s . Moreover, i t i s d i f f i c u l t f o r t h e m o l e c u l e s t o f l o w due t o network c r o s s l i n k s r e s u l t i n g from t h e p r e s e n c e o f t h e t r i f u n c t i o n a l l i n k a g e s . As i s seen i n t h e G c u r v e i n F i g u r e 2, t h e w e l l - d e f i n e d r u b b e r y p l a t e a u ?



182

extends over a wide temperature range, which i s r e m i n i s c e n t o f a t y p i c a l c o v a l e n t l y c r o s s - l i n k e d rubber (36-37). In the present system, t h e modulus c o r r e s p o n d i n g t o t h e u l t i m a t e Maxweel element i s as l a r g e as K)6(N/m2), which i s c o n s i s t e n t w i t h t h e e x i s t e n c e o f c r o s s l i n k s r a t h e r t h a n s i m p l y t h e entanglement o f m o l e c u l e s . T a b l e I I shows t h e v a l u e s o f Em ( t h e modulus c o r r e s p o n d i n g t o t h e u l t i m a t e Maxwell e l e m e n t ) , and Mc ( t h e m o l e c u l a r weight o f t h e network c h a i n ) f o r t h e samples d r i e d a t a low t e m p e r a t u r e . It i s c l e a r from t h e t a b l e t h a t M c P / M c i s l a r g e r than 1, which means t h a t some o f t h e 3-arm PIB ionomers f u n c t i o n as d i f u n c t i o n a l p o l y mers, i . e . some o f t h e i o n s a r e p r e s e n t as f r e e p a i r s . I t i s also seen t h a t t h e h i g h e r t h e temperature, t h e more o f t h e polymer c h a i n s a c t as d i f u n c t i o n a l u n i t s . T h i s r e s u l t confirms that i o n i c a g g r e g a t e s a r e d i s r u p t e d by i n c r e a s i n g t h e temperature, a phenomenon fundamental t o t h e use o f t h e s e m a t e r i a l s as t h e r m o p l a s t i c e l a s t o mers. Moreover, t h i s r e s u l t p a r a l l e l s t h e f i n d i n g o f Neppel e t a l . (5-6) who showed t h a t c l u s t e r s decompose p r o g r e s s i v e l y t o m u l t i p l e t s as the temperature i s i n c r e a s e d .


e x

c a l

I t was o b s e r v e d t h a t d r y i n g t h e sample a t h i g h temperature i n c r e a s e d t h e m and d e c r e a s e d Em, w h i l e A E remained almost t h e same. The d e c r e a s e i n Em s u g g e s t s t h a t a n n e a l i n g a t h i g h temperature may have l e d t o some c h e m i c a l d e g r a d a t i o n o f t h e sample which i s f u r t h e r c o n f i r m e d by T a b l e I I I when one c o n s i d e r s t h e v a l u e s o f M c l / M c P . F o r t h i s r e a s o n , t h e samples d r i e d a t room temperature a r e c o n s i d e r e d most r e l i a b l e . In summary, i t i s shown t h a t t h e 3-arm s t a r PIB ionomer i s u s e f u l i n the study o f ion-hopping k i n e t i c s . The polymer c h a i n s a r e c r o s s l i n k e d by c o v a l e n t t r i f u n c t i o n a l l i n k a g e s and i o n i c a g g r e g a t e e x i s t m o s t l y as m u l t i p l e t s i n t h e m i d d l e o f t h e c h a i n s . The k i n e t i c c o n s t a n t s f o r i o n - h o p p i n g i n t h i s system a r e g i v e n by k = 7.11 χ 10^ exp(-94,100/RT). a

c a

e x

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Sakamoto, K.; MacKnight, W.J.; and Porter, R.S. J . Polym. Sci. 1970, Part A-2 8, 277. Eisenberg, A. and Navratil, M. Macromolecules 1973, 6, 604. Eisenberg, A. Macromolecules 1970, 3, 147. Eisenberg, Α.; and King, M. "Ion-Containing Polymers"; Academic Press: New York 1977. Neppel, Α.; Butler, I.S.; and Eisenberg, A. Can. J . Chem. 1979, 57, 2518 Neppel, A.; Butler, I.S.; and Eisenberg, A. Macromolecules 1979, 12, 948. Broze, G.; Jerome, R.; and Teyssie, P. Macromolecules 1981, 14, Broze, G.; Jerome, R.; and Teyssie, P. Macromolecules 1982, 15, 920. Broze, G.; Jerome, R.; Teyssie, P.; and Marco, C. Polym. Bull. 1981, 4, 241. Broze, G.; Jerome, R.; Teyssie, P.; and Gallot, B. J . Polym. Sci., Polym. Lett. Ed. 1981, 19, 415. Broze, G.; Jerome, R.; and Teyssie, P. Macromolecules 1982, 15, 1300. Broze, G.; Jerome, R.; and Teyssie, P. J . Polym. Sci., Polvm. Lett. Ed. 1983, 21, 237. Eisenberg and Bailey; Coulombic Interactions in Macromolecular Systems ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

14. HARA ET AL. Three-Arm Star Polyisobutylene-Based Ionomers


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Broze, G.; Jerome, R.; Teyssie, P.; and Marco, C. Macromolecules 1983, 16, 996. 14. Broze, G.; Jerome, R.; and Teyssie, P. J . Polym. Sci. Polym. Ed. 1983, 21, 2205. 15. Broze, G.; Jerome, R.; Teyssie, P.; and Marco, C. Macromolecules 1983, 16, 1771. 16. Kennedy, J.P.; Ross, L.R.; Lackey, J . E . ; and Nuyken, O. Polym. Bull. 1981, 4, 67. 17. Kennedy, J.P.; and Storey, F. Am. Chem. Soc. Div. Org. Coat. Appl. Polym. Sci. 1982, 46, 182. 18. Kennedy, J.P.; Storey, R.; Mahajer, Y.; and Wilkes, G.L. Proc. IUPAC, Macro 82 1982 905. 19. Mohajer, Y.; Tyagi, D.; Wilkes, G.L.; Storey, R.; and Kennedy, J.P. ibid 1982, 906. 20. Mohajer, Y.; Tyagi, D.; Wilkes, G.L.; Storey, R.F.; and Kennedy, J.P. Polym. Bull. 1982, 8, 47. 21. Bagrodia, S.; Wilkes, G.L.; Mohajer, Y.; Storey, R.F.; and Kennedy, J.P. Polym. Bull. 1982, 8 281. 22. Bagrodia, S.; Mohajer, Y.; Wilkes, G.L.; Storey, R.F.; and Kennedy, J.P. Polym. Bull. 1983, 9, 174. 23. Tobolsky, A.V.; "Properties and Structures of Polymers"; John Wiley & Sons Inc.: New York, 1960. 24. Tobolsky, A.V.; J . Polym. Sci. 1964, 2 637. 25. Yu, H.; J . Polym. Sci. Polym. Lett. 1964, 2, 631. 26. Mahajer, Y.; Bagrodia, S.; Wilkes, G.L.; Storey, R.; and Kennedy, J.P. Polym. Bull. 27. Eisenberg, Α.; and Sasada, T. J . Polym. Sci. 1968, C-16, 3473. 28. Cayrol, Β. Ph.D. Thesis, McGill University, Montreal, 1972. 29. Stern M.D.; and Tobolsky, A.V. J . Chem. Phys. 1946, 14, 93. 30. Tobolsky, A.V.; Beevers, R.B.; and Owen, G.D. J . Colloid Sci. 1963, 18, 359. 31. Eisenberg, Α.; and Teter, L.A. J . Phys. Chem. 1967, 71, 2332. 32. Eisenberg, Α.; Saito, S.; and Teter, L.A. J. Polym. Sci. 1966, C-14, 323. 33. Eisenberg, Α.; and Saito, S. J . Macromol. Sci. 1968, A2, 799. 34. Rigdahl, M.; and Eisenberg, A. J . Polym. Sci. Polym. Phys. Ed. 1981, 19, 1641. 35. For example, in the case of sulfonated styrene ionomers (34), the peak height of primary transition is much higher than that of secondary transition below the critical concentration, while the hight of secondary transition peak exceeds that of primary transition peak above the critical concentration. 36. Tobolsky, A.V.; Lyons, P.F.; and Hata, N. Macromolecules 1968, 1, 515. 37. Agarwal, P.K.; Makowski, H.S.; and Lundberg, R.D. Macromolecules 1980, 13, 1679. RECEIVED January 24, 1986


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