Gas Transport and Cooperative Main-Chain Motions in Glassy Polymers

5

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Gas Transport and Cooperative Main-Chain Motions in Glassy Polymers DANIEL RAUCHER and MICHAEL D. SEFCIK Monsanto Company, St. Louis, MO 63167

Carbon-13 rotating-frame relaxation rate measurements are used to elucidate the mechanism of gas transport in glassy polymers. The nmr relaxa tion measurements show that antiplasticization -plasticization of a glassy polymer by a low molecular weight additive effects the cooperative main-chain motions of the polymer. The correlation of the diffusion coefficients of gases with the main-chain motions in the polymer-additive blends shows that the diffusion of gases in polymers is controlled by the cooperative motions, thus provid ing experimental verification of the molecular theory of diffusion. Carbon-13 nmr relaxation measurements also show that the presence of a permanent gas alters the cooperative motions of the polymeric chains. These changes in main-chain molecular motions correlate with changes in the diffusion coefficient of the gas in the polymer, thus providing evidence that the diffusion coeffi cient is dependent on the gas-polymer-matrix composition. Experimental results show that sorbed gases interact with polymeric chains, inducing changes in the structural and dynamic properties of the polymer. These properties and the interchain forces controlling them determine many of the physical characteristics of the matrix, including the solubility and diffusion coefficients. These results are inconsistent with the assumptions and the physical interpretations implicit in the dual -mode sorption and transport model, and strongly suggest that the sorption and transport of gases in glassy polymers should be represented by concentration (composition) dependent solubility and diffusion coefficients. 0097-6156/83/0223-0089$06.50/0 © 1983 American Chemical Society


90

I N D U S T R I A L GAS

SEPARATIONS

The t r a n s p o r t o f g a s i n p o l y m e r s h a s b e e n s t u d i e d f o r o v e r 150 y e a r s ( 1 ) . Many o f t h e c o n c e p t s d e v e l o p e d i n 1866 b y Graham (2) are s t i l l accepted today. Graham p o s t u l a t e d that the mechanism o f t h e p e r m e a t i o n p r o c e s s i n v o l v e s t h e s o l u t i o n o f t h e gas i n t h e u p s t r e a m s u r f a c e o f t h e membrane, d i f f u s i o n t h r o u g h t h e membrane f o l l o w e d b y e v a p o r a t i o n f r o m t h e downstream membrane surface. T h i s i s t h e b a s i s f o r t h e " s o l u t i o n - d i f f u s i o n " model w h i c h i s u s e d e v e n t o d a y i n a n a l y z i n g g a s t r a n s p o r t phenomena i n p o l y m e r i c membranes. In 1879, v o n W r o b l e w s k i ( 3 ) showed t h a t s o r p t i o n and t r a n s p o r t o f g a s e s i n p o l y m e r s f o l l o w e d H e n r y ' s and F i c k ' s l a w s , respectively, C = σο ρ

(1)

J

(2)

= -Do ( d C / d x )

where C i s t h e e q u i l i b r i u m g a s c o n c e n t r a t i o n i n t h e p o l y m e r , J i s t h e p e r m e a t i o n f l u x , σο a n d Do a r e t h e s o l u b i l i t y a n d d i f f u s i o n coefficients, r e s p e c t i v e l y , and ρ i s t h e gas p r e s s u r e . By a s s u m i n g t h a t Do i s c o n s t a n t , v o n W r o b l e w s k i showed t h a t t h e steady s t a t e f l u x i s given by, J

= σο Do (p/£) = Po (p/£)

where ρ i simplicity, and 1 is coefficient

(3)

s the pressure d i f f e r e n c e a c r o s s t h e membrane ( f o r t h e downstream p r e s s u r e h a s b e e n assumed t o be z e r o ) , t h e t h i c k n e s s o f t h e membrane. The p e r m e a b i l i t y a t s t e a d y s t a t e , P o , i s d e f i n e d b y eq. ( 3 ) a s ,

Po = σο Do

(4)

Daynes ( 4 ) ( 1 9 2 0 ) showed t h a t a t r a n s i e n t p e r m e a t i o n e x p e r i m e n t , u n d e r t h e b o u n d a r y c o n d i t i o n s C ( x ; t ) = 0 f o r t 0 g i v e s two c h a r a c t e r i s t i c p a r a m e t e r s , t h e p e r m e a b i l i t y , P o , a n d t h e t i m e - l a g , Θο. Daynes showed t h a t t h e s o l u t i o n o f F i c k ' s law under these boundary c o n d i t i o n s y i e l d s , Do

2

= £ /6θο

(5)

T h u s , P o , Do, a n d σο [ b y e q . ( 4 ) ] c o u l d be m e a s u r e d i n a s i n g l e time-lag experiment. Eqs. ( l ) - ( 5 ) a r e s t i l l t h e b a s i c s o r p t i o n and t r a n s p o r t equations used today f o r "ideal" systems, penetrant-polymer s y s t e m s i n w h i c h b o t h σο a n d Do a r e p r e s s u r e a n d c o n c e n t r a t i o n independent. T h i s " i d e a l " b e h a v i o r i s o b s e r v e d i n s o r p t i o n and t r a n s p o r t o f p e r m a n e n t a n d i n e r t g a s e s i n p o l y m e r s w e l l above t h e i r Tg.

5.

RAUCHER

AND

SEFCIK

Transport

and

Main-Chain

91

Motions

B a r r e r ( 6 ) (1937) showed t h a t d i f f u s i o n i s an a c t i v a t e d p r o c e s s and t h a t t h e d i f f u s i o n c o e f f i c i e n t had an A r r h e n i u s f o r m ,


Do = Do e x p ( - E d / R T )

(6)

where Ed i s t h e a c t i v a t i o n e n e r g y o f d i f f u s i o n and Do i s t h e frequency or p r e - e x p o n e n t i a l f a c t o r . A number o f a t t e m p t s have b e e n made t o e x p l a i n t h e t e m p e r a t u r e dependence o f t h e d i f f u s i o n coefficient. The z o n e - t h e o r y o f B a r r e r ( 7 ) , t h e f r e e v o l u m e t h e o r i e s ( 8 , 9 ) , and t h e m o l e c u l a r t h e o r i e s o f d i f f u s i o n ( 1 0 , 1 1 , 12), a l t h o u g h d i f f e r i n g i n many a s p e c t s , have i n common t h e c o n n o t a t i o n t h a t the a c t i v a t i o n energy of d i f f u s i o n i s the energy needed f o r c h a i n s e p a r a t i o n , t h r o u g h c o o p e r a t i v e m o t i o n s , o f s u f f i c i e n t s i z e to allow the penetrant to execute a d i f f u s i o n a l jump. S e c t i o n IA summarizes t h e m o l e c u l a r model o f d i f f u s i o n o f P a c e and D a t y n e r ( 1 2 ) w h i c h p r o p o s e s t h a t t h e d i f f u s i o n o f g a s e s i n a p o l y m e r i c m a t r i x i s d e t e r m i n e d by t h e c o o p e r a t i v e m a i n - c h a i n motions of the polymer. I n S e c t i o n I B we r e p o r t c a r b o n - 1 3 nmr r e l a x a t i o n measurement w h i c h show t h a t t h e d i f f u s i o n o f g a s e s i n p o l y ( v i n y l c h l o r i d e ) (PVC) - t r i c r e s y l p h o s p h a t e (TCP) s y s t e m s i s c o n t r o l l e d by t h e c o o p e r a t i v e motions o f t h e polymer c h a i n s . The c o r r e l a t i o n of the phenomenological d i f f u s i o n c o e f f i c i e n t s w i t h the c o o p e r a t i v e m a i n - c h a i n m o t i o n s o f t h e p o l y m e r p r o v i d e s an experimental v e r i f i c a t i o n f o r the molecular d i f f u s i o n model. S e c t i o n I I A summarizes t h e p h y s i c a l a s s u m p t i o n s and the r e s u l t i n g mathematical d e s c r i p t i o n s of the "concentrationd e p e n d e n t " ( 5 ) and "dual-mode" ( 1 3 ) s o r p t i o n and t r a n s p o r t m o d e l s which describe the behavior of "non-ideal" penetrant-polymer systems, systems which exhibit nonlinear, pressure-dependent sorption and transport. In S e c t i o n I I B we elucidate the mechanism o f t h e " n o n - i d e a l " d i f f u s i o n i n g l a s s y p o l y m e r s b y c o r r e l a t i n g the phenomenological d i f f u s i o n c o e f f i c i e n t of C0 i n PVC w i t h t h e c o o p e r a t i v e m a i n - c h a i n m o t i o n s o f t h e p o l y m e r i n t h e presence of the penetrant. We report carbon-13 relaxation measurements w h i c h d e m o n s t r a t e t h a t C O 2 a l t e r s t h e c o o p e r a t i v e m a i n - c h a i n m o t i o n s o f PVC. T h e s e changes c o r r e l a t e w i t h changes i n the d i f f u s i o n c o e f f i c i e n t of C0 i n the polymer, thus prov i d i n g experimental evidence t h a t the d i f f u s i o n c o e f f i c i e n t i s c o n c e n t r a t i o n dependent. 2

2

I.

DIFFUSION AND

A.

Diffusion

COOPERATIVE MAIN-CHAIN MOTIONS

Theory

R e c e n t l y P a c e and D a t y n e r ( 1 2 ) a d v a n c e d a m o l e c u l a r t h e o r y of diffusion that correlates the d i f f u s i o n o f gases in a p o l y m e r i c m a t r i x w i t h the c o o p e r a t i v e motions of the polymer chains. The t h e o r y p r o p o s e s t h a t t h e d i f f u s a n t m o l e c u l e c a n move


92


SEPARATIONS

t h r o u g h t h e p o l y m e r m a t r i x i n two d i s t i n c t ways: (a) s l i d i n g along t h e a x i s o f i n t e r c h a i n channels o r bundles formed by adjacent polymer c h a i n s , o r (b) jumping a t r i g h t angles t o t h e polymer chains whenever adjacent chains are sufficiently separated. The f i r s t p r o c e s s h a s a s m a l l e r a c t i v a t i o n e n e r g y t h a n t h e s e c o n d a n d t h e r e f o r e o c c u r s much more r a p i d l y . This i s t r u e because o n l y l i m i t e d main-chain motions a r e necessary t o create i n t e r c h a i n channels o f molecular dimensions i n d i s o r d e r e d glasses. The p e n e t r a n t d i f f u s e s f r e e l y w i t h i n t h e s e i n t e r c h a i n c h a n n e l s , w h i c h may be bounded b y r e g i o n s o f d e n s e l o c a l p a c k i n g . The d i f f u s a n t c a n p r o c e e d t o t h e n e x t r e g i o n o f f a c i l e d i f f u s i o n only when two a d j a c e n t polymer chains undergo sufficient c o o p e r a t i v e motions t o cause an i n t e r c h a i n s e p a r a t i o n g r e a t e r than t h e molecular dimensions of the diffusant. Thus, p r o c e s s (b) i s t h e r a t e d e t e r m i n i n g s t e p i n d i f f u s i o n and t h e a c t i v a t i o n energy o f d i f f u s i o n i s t h e a c t i v a t i o n energy o f t h e c h a i n separation. I n o t h e r words, t h e phenomenological diffusion p r o c e s s i s t h e r e s u l t o f two d y n a m i c e v e n t s : the f i r s t , highfrequency motions ( > 1 0 Hz) o f t h e d i f f u s a n t w i t h i n i n t e r c h a i n channels; and t h e second, l o w - f r e q u e n c y , cooperative motions (10 -10 Hz) o f t h e polymer c h a i n s t o a l l o w t h e d i f f u s a n t t o enter another i n t e r c h a i n channel. The f o r m e r m o t i o n s have b e e n observed e x p e r i m e n t a l l y i n d i f f u s i o n experiments (_14, 1 5 ) , t h e l a t t e r a r e reported here. The activation energy f o r a symmetrical separation of polymer c h a i n centers by a d i s t a n c e o f ζ over a l e n g t h χ i s given by ( 1 2 ) , 1 0

4

8

(7)

where f ( z ) i s t h e a v e r a g e i n t e r c h a i n p o t e n t i a l p e r u n i t l e n g t h , β i s t h e a v e r a g e e f f e c t i v e s i n g l e c h a i n - b e n d i n g modulus p e r u n i t l e n g t h , and ρ i s t h e e q u i l i b r i u m c h a i n s e p a r a t i o n . Solution of eq. ( 7 ) f o r ζ = d, where d i s t h e minimum c h a i n s e p a r a t i o n w h i c h w i l l a l l o w t r a n s v e r s e passage o f a p e n e t r a n t , g i v e s t h e a c t i v a t i o n energy o f d i f f u s i o n , Ed = ΔΕ = 3.11 Γ

3

/

4

β

1 / 4

d 5/

4

(8)

where Γ and β a r e p a r a m e t e r s w h i c h c h a r a c t e r i z e t h e i n t e r c h a i n c o h e s i o n and c h a i n s t i f f n e s s , r e s p e c t i v e l y . The v a l u e o f Γ c a n be e s t i m a t e d f r o m p o l y m e r d e n s i t y a n d c o h e s i v e e n e r g y d e n s i t y , and β c a n be a p p r o x i m a t e d from t h e polymer c h a i n backbone g e o m e t r y and bond r o t a t i o n p o t e n t i a l s . The t h e o r y makes t h e u n i q u e p r e d i c t i o n t h a t t h e a c t i v a t i o n e n e r g y o f d i f f u s i o n depends n e a r l y l i n e a r l y on t h e p e n e t r a n t d i a m e t e r . T h i s p r e d i c t i o n has been r e c e n t l y c o n f i r m e d e x p e r i m e n t a l l y by Berens and Hopfenberg (16).

5.

RAUCHER A N D SEFCIK

Transport

and

Main-Chain

93

Motions

From t h i s m o l e c u l a r t h e o r y , we see t h a t t h e diffusion c o e f f i c i e n t depends on t h e f r e q u e n c y o f c o o p e r a t i v e m a i n - c h a i n m o t i o n s o f t h e p o l y m e r , \>, w h i c h c a u s e c h a i n s e p a r a t i o n s e q u a l t o or g r e a t e r t h a n t h e p e n e t r a n t d i a m e t e r . P a c e and D a t y n e r w e r e a b l e t o e s t i m a t e ν b y a d o p t i n g an A r r h e n i u s r a t e e x p r e s s i o n i n w h i c h t h e p r e - e x p o n e n t i a l f a c t o r , A, i s a f u n c t i o n o f b o t h ΔΕ and T. The d i f f u s i o n c o e f f i c i e n t i s g i v e n b y ,


Do = i L

2

ν

=

i L

2

A exp(-AE/RT)

(9)

2

where L i s t h e m e a n - s q u a r e jump d i s p l a c e m e n t . The c h a r a c t e r istic frequency of the d i f f u s i o n process e s t a b l i s h e s a time scale (τ = 1/v) f o r jumps o f t h e d i f f u s a n t m o l e c u l e i n t h e polymeric matrix. H e n c e , L i s t h e mean d i s p l a c e m e n t i n t h e a v e r a g e p o s i t i o n o f t h e gas m o l e c u l e f o l l o w i n g an a c t i v a t e d jump. (The a v e r a g e p o s i t i o n o f a p e n e t r a n t m o l e c u l e i s d e t e r m i n e d b y i t s random m o t i o n w i t h i n t h e r e g i o n o f f a c i l e d i f f u s i o n , p r o c e s s ( a ) , and a v e r a g e d o v e r t i m e τ.) The a v e r a g e jump d i s t a n c e , L, i s not p r e d i c t a b l e w i t h i n the l i m i t s of the theory. B.

D i f f u s i o n Mechanism i n Polymers

- Experimental Evidence

1.

C o o p e r a t i v e M a i n - C h a i n M o t i o n s i n PVC-TCP

I t i s now w e l l e s t a b l i s h e d t h a t TCP a c t s as an a n t i p l a s t i c i z e r o f PVC a t l o w c o n c e n t r a t i o n s and as a p l a s t i c i z e r a t h i g h concentrations. The observed v a r i a t i o n s i n tensile modulus, tensile strength, impact strength, ultimate elongation, and 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 o f PVC w i t h a d d i t i v e c o n c e n t r a t i o n have been a t t r i b u t e d to the antiplasticization-plasticization phenomenon ( 1 7 , 18, 1 9 ) . A n t i p l a s t i c i z a t i o n o f PVC a l s o r e s u l t s in a d e c r e a s e i n gas p e r m e a b i l i t y , and p l a s t i c i z a t i o n i n an i n c r e a s e i n gas p e r m e a b i l i t y (Γ7). B a s e d on c h a n g e s o f s e c o n d a r y - l o s s t r a n s i t i o n s o f a n t i p l a s t i c i z e d p o l y m e r s , Robeson (20) s p e c u l a t e d t h a t a d d i t i v e s a l t e r e d the c o o p e r a t i v e m o t i o n s o f t h e p o l y m e r c h a i n s , and t h e s e a l t e r a t i o n s were r e s p o n s i b l e f o r t h e o b s e r v e d c h a n g e s i n d i f f u s i v i t y . In p a r t i c u l a r , Robeson suggested t h a t the a d d i t i o n o f a n t i p l a s t i c i z e r s to polymers r e s t r i c t e d m o l e c u l a r f l e x i b i l i t y of chains t h e r e b y r e s t r i c t i n g d i f f u s i o n o f p e n e t r a n t s . K i n j o (18) attri buted the antiplasticization effect of several polar lowm o l e c u l a r w e i g h t a d d i t i v e s i n PVC t o c o h e s i o n o f t h e a d d i t i v e to t h e polymer c h a i n by d i p o l e - d i p o l e i n t e r a c t i o n s . He p r o p o s e d t h a t m o t i o n s o f t h e c o h e r i n g p a r t s o f t h e PVC w o u l d be e x t r e m e l y h i n d e r e d , r e d u c i n g t h e m e c h a n i c a l β-dispersion and increasing tensile strength. I t was s u g g e s t e d i n t h e l i t e r a t u r e ( 2 1 J t h a t high c o n c e n t r a t i o n s o f TCP i n PVC decrease the interchain

94



interactions i n t h e polymer, thus motions, r e s u l t i n g i n p l a s t i c i z a t i o n

SEPARATIONS

enhancing t h e main-chain of the polymeric matrix.

We have b e e n i n t e r e s t e d i n t h e n a t u r e o f c o o p e r a t i v e m o t i o n s in polymers f o r some t i m e a n d h a v e u s e d c a r b o n - 1 3 n u c l e a r magnetic resonance f o r examining main-chain motions i n s o l i d s (22-27). C a r b o n - 1 3 nmr w i t h c r o s s - p o l a r i z a t i o n a n d m a g i c - a n g l e sample s p i n n i n g i s a h i g h - r e s o l u t i o n , h i g h - s e n s i t i v i t y t e c h n i q u e for s o l i d s ( F i g u r e 1 ) . The c o o p e r a t i v e m a i n - c h a i n m o t i o n s i n g l a s s y p o l y m e r s c a n be e v a l u a t e d f r o m c a r b o n - 1 3 r o t a t i n g - f r a m e r e l a x a t i o n r a t e s , [ R i P ( C ) ] . D e t a i l s o f t h e nmr e x p e r i m e n t a r e reported elsewhere (28). The a v e r a g e r e l a x a t i o n r a t e o f t h e m e t h y l e n e - a n d m e t h i n e c a r b o n s i n PVC d e c r e a s e s w i t h t h e a d d i t i o n o f TCP up t o a b o u t 15 w e i g h t % ( T a b l e I ) . B a s e d o n s t a n d a r d r e l a x a t i o n r a t e t h e o r y (29), reduced r e l a x a t i o n r a t e s a r e i n d i c a t i v e o f a s h i f t i n t h e average c o o p e r a t i v e motions t o lower f r e q u e n c i e s . These r e s u l t s prove t h a t a n t i p l a s t i c i z a t i o n i s a s s o c i a t e d w i t h reduced cooper a t i v e motions o f t h e polymer c h a i n s . As t h e c o n c e n t r a t i o n o f TCP i n PVC i s i n c r e a s e d above 15 w e i g h t %, t h e a v e r a g e r e l a x a t i o n r a t e o f t h e methylene- and methine-carbons i n PVC i n c r e a s e s (Table I ) , showing t h a t p l a s t i c i z a t i o n l e a d s t o i n c r e a s e d cooper a t i v e motions i n t h e g l a s s . Our results show f o r the f i r s t time the effect of a n t i p l a s t i c i z a t i o n - p l a s t i c i z a t i o n on p o l y m e r p r o p e r t i e s a t t h e molecular l e v e l . These r e s u l t s p r o v i d e e x p e r i m e n t a l e v i d e n c e f o r the assumptions o f Robeson (20) and K i n j o (18) t h a t a t t r i b u t e d the c h a n g e s i n p o l y m e r p r o p e r t i e s w i t h a n t i p l a s t i c i z a t i o n p l a s t i c i z a t i o n t o changes i n t h e i n t e r c h a i n c o h e s i o n o f t h e polymer. 2.

T r a n s p o r t i n PVC-TCP

Hydrogen and c a r b o n monoxide p e r m e a b i l i t y d a t a f o r f i l m s o f PVC a n d PVC c o n t a i n i n g TCP a r e g i v e n i n T a b l e I . The p e r m e a b i l i t y c o e f f i c i e n t s , o f b o t h g a s e s , d e c r e a s e a s TCP c o n c e n t r a t i o n i s i n c r e a s e d t o a b o u t 15 w e i g h t %, a n d t h e n i n c r e a s e s h a r p l y w i t h additive concentration. S i m i l a r r e s u l t s have b e e n r e p o r t e d f o r C 0 a n d H 0 p e r m e a b i l i t i e s i n t h e PVC-TCP s y s t e m ( 1 7 ) . T a b l e I l i s t s a l s o t h e a p p a r e n t d i f f u s i o n c o e f f i c i e n t s , Da, c a l c u l a t e d f r o m t h e t i m e l a g , Θ. The dependence o f t h e a p p a r e n t d i f f u s i o n c o e f f i c i e n t s on a d d i t i v e c o n c e n t r a t i o n i s s i m i l a r t o the d e p e n d e n c e o f t h e p e r m e a b i l i t y c o e f f i c i e n t s on t h i s c o n c e n tration. We w i l l show l a t e r t h a t t h e d i f f u s i o n c o e f f i c i e n t s i n PVC a r e d e p e n d e n t o n t h e g a s c o n c e n t r a t i o n i n t h e p o l y m e r , a n d we emphasize here t h a t t h e v a l u e s quoted i n T a b l e I a r e t h e apparent d i f f u s i o n c o e f f i c i e n t s and a r e n o t t h e v a l u e s o f e i t h e r t h e r e a l or e f f e c t i v e d i f f u s i o n c o e f f i c i e n t s , D a n d D, r e s p e c t i v e l y . We r e c o g n i z e t h a t Da a c t u a l l y u n d e r e s t i m a t e s b o t h D a n d D, b u t we c a n s t i l l u s e t h e a p p a r e n t d i f f u s i o n c o e f f i c i e n t s t o show t h e 2

2


5.

R A U C H E R A N D SEFCIK

Transport

and Main-Chain

Motions

95

τ—>—ι—ι—ι—ι—ι—ι—ι—j—ι—ι—ι—ι—ι—ι—Γ 200 100 0 PPM F i g u r e 1. C r o s s - p o l a r i z a t i o n a n d m a g i c - a n g l e s p i n n i n g C-NMR s p e c t r a o f PVC-TCP s y s t e m s . The 15.08 MHz s p e c t r a w e r e o b t a i n e d w i t h 1 msec c o n t a c t t i m e a n d l800 Hz s p i n n i n g s p e e d .

a. b.

4..44 ± .23 4..73 ± .58 15. .6 ± 2 .20 22. .5 ± 1 .40

.0180 ± .0002 .0274 ± .0016 .1050 ± .0010 .3740 ± .0030

1..69 ± .01

2..16 ± .04

2..86 ± .02

3..61 ± .01

15.0

20.1

30.8

40.0

7

2

l 3

.2880 ± .0100

.0535 ± .0023

carbons.

430

250

155

.0293 ± .0004

140

170

235

(C)> χ se

120

and Da i n c r e a s e ( F i g . 2 ) . The d i l u t i o n o f t h e c h a i n s b y t h e l o w m o l e c u l a r w e i g h t a d d i t i v e decreases the interchain potential, thereby increasing the f r e q u e n c y o f m a i n - c h a i n m o t i o n s , and i n c r e a s i n g t h e d i f f u s i o n coefficients. In c o n c l u s i o n , the average r o t a t i n g - f r a m e r e l a x a t i o n r a t e of the m e t h y l e n e - and m e t h i n e - c a r b o n s c o r r e l a t e w i t h the apparent diffusion coefficients for H and CO i n PVC when t h e m a i n - c h a i n m o l e c u l a r m o t i o n s o f t h e p o l y m e r a r e a l t e r e d b y an a d d i t i v e . (Fig. 2). These r e s u l t s p r o v i d e e x p e r i m e n t a l e v i d e n c e that m a i n - c h a i n c o o p e r a t i v e motions c o n t r o l the d i f f u s i o n o f gases t h r o u g h p o l y m e r s . I n S e c t i o n I I B we w i l l show t h a t p e r t u r b a t i o n of p o l y m e r i c c o o p e r a t i v e motions i s not r e s t r i c t e d t o c l a s s i c a l plasticizing additives. 5

8

2

4

5

x

2

II.

SORPTION AND

TRANSPORT I N GLASSY POLYMERS

N o n l i n e a r , p r e s s u r e - d e p e n d e n t s o l u b i l i t y and p e r m e a b i l i t y i n p o l y m e r s have b e e n o b s e r v e d f o r o v e r 40 y e a r s . M e y e r , Gee and t h e i r c o - w o r k e r s ( 5 ) r e p o r t e d p r e s s u r e - d e p e n d e n t s o l u b i l i t y and diffusion coefficients i n rubber-vapor systems. Crank, Park, Long, Barrer, and their co-workers (5) observed p r e s s u r e d e p e n d e n t s o r p t i o n and t r a n s p o r t i n g l a s s y p o l y m e r - v a p o r s y s t e m s . S o r p t i o n and t r a n s p o r t m e a s u r e m e n t s o f g a s e s i n g l a s s y p o l y m e r s show t h a t t h e s e p e n e t r a n t - p o l y m e r s y s t e m s do n o t o b e y t h e " i d e a l s o r p t i o n and t r a n s p o r t e q s . ( l ) - ( 5 ) . The o b s e r v a b l e v a r i a b l e s , 1 1

98

INDUSTRIAL

GAS

SEPARATIONS

5


ΙΟ ΓΤ

F i g u r e 2. The dependence o f t h e m a i n - c h a i n r o t a t i n g - f r a m e relaxa t i o n r a t e , and apparent d i f f u s i o n c o e f f i c i e n t s o f a n d CO, o n t h e c o n c e n t r a t i o n o f TCP i n PVC. The r e l a x a t i o n r a t e s w e r e m e a s u r e d a t 3k k H z a n d 26 °C. The d i f f u s i o n c o e f f i c i e n t s w e r e m e a s u r e d a t 270 cm-Hg a n d 27 °C.

5.

RAUCHER

AND

SEFCIK

Transport

and Main-Chain

99

Motions


s o l u b i l i t y c o e f f i c i e n t , p e r m e a b i l i t y , and d i f f u s i o n t i m e - l a g a r e p r e s s u r e dependent (13). A number o f attempts have b e e n made t o e x p l a i n t h e n o n l i n e a r , p r e s s u r e - d e p e n d e n t s o r p t i o n and t r a n s p o r t i n p o l y m e r s . T h e s e e x p l a n a t i o n s may be c l a s s i f i e d a s " c o n c e n t r a t i o n - d e p e n d e n t " (5) a n d " d u a l - m o d e " ( 1 3 ) s o r p t i o n and t r a n s p o r t m o d e l s . These models differ i n their physical a s s u m p t i o n s and i n t h e i r m a t h e m a t i c a l d e s c r i p t i o n s o f t h e s o r p t i o n and t r a n s p o r t i n penetrant-polymer systems. A.

S o r p t i o n and T r a n s p o r t

Models

1.

C o n c e n t r a t i o n - D e p e n d e n t S o r p t i o n and T r a n s p o r t

Model

P r e s s u r e - d e p e n d e n t s o r p t i o n and t r a n s p o r t p r o p e r t i e s i n p o l y m e r s c a n be a t t r i b u t e d t o t h e p r e s e n c e o f t h e p e n e t r a n t i n t h e p o l y m e r . C r a n k ( 3 2 ) s u g g e s t e d i n 1953 t h a t t h e " n o n - i d e a l " b e h a v i o r o f p e n e t r a n t - p o l y m e r systems c o u l d a r i s e from s t r u c t u r a l and d y n a m i c c h a n g e s o f t h e p o l y m e r i n r e s p o n s e t o t h e p e n e t r a n t . As t h e p r o p e r t i e s o f t h e p o l y m e r a r e d e p e n d e n t o n t h e n a t u r e a n d concentration of the penetrant, t h e s o l u b i l i t y and d i f f u s i o n c o e f f i c i e n t a r e a l s o c o n c e n t r a t i o n - d e p e n d e n t . The c o n c e n t r a t i o n d e p e n d e n t s o r p t i o n and t r a n s p o r t m o d e l s u g g e s t s t h a t " n o n - i d e a l " p e n e t r a n t - p o l y m e r s y s t e m s s t i l l obey H e n r y ' s and F i c k ' s l a w s , a n d d i f f e r from t h e " i d e a l " systems o n l y by t h e f a c t t h a t σ and D a r e c o n c e n t r a t i o n dependent,

C = σ ρ = σο [1 + g ( C ) ]

ρ

(10)

J = -D ( d C / d x ) = -Do [1 + f ( C ) ] ( d C / d x )

(11)

where g ( C ) a n d f ( C ) a r e f u n c t i o n s d e s c r i b i n g t h e c o n c e n t r a t i o n d e p e n d e n c e o f σ a n d D, r e s p e c t i v e l y . S o l u t i o n o f eq. (11) under the boundary c o n d i t i o n s o f t h e t r a n s i e n t permeation experiment yields, Ρ = D σ = Do [1 + f ' ( C ) ] σο [1 + = Po [1 + f ' ( C ) ] 2

g(C)]

[1 + g ( C ) ]

θ = (£ /6Do) [1 + F ( C ) ]

= θο [1 + F ( C ) ]

(12) (13)

where D, t h e e f f e c t i v e d i f f u s i o n c o e f f i c i e n t , i s d e f i n e d by D = ( 1 / C ) J DdC, and t h e f u n c t i o n s f ' ( C ) and F ( C ) d e s c r i b e t h e c o n c e n t r a t i o n d e p e n d e n c e o f D a n d Θ, r e s p e c t i v e l y . C r a n k , _ P a r k , L o n g , B a r r e r a n d t h e i r c o - w o r k e r s ( 5 ) have shown t h a t D a n d D c a n be r e p r e s e n t e d a s e x p o n e n t i a l f u n c t i o n s o f penetrant concentration. Aitken and Barrer (33) used successfully a linear expression t o describe the concentration

100


SEPARATIONS

d e p e n d e n c e o f D. F r i s c h ( 3 4 ) d e v e l o p e d e x p l i c i t e x p r e s s i o n s Θ f o r c a s e s where D i s c o n c e n t r a t i o n - d e p e n d e n t .


2.

Dual-Mode S o r p t i o n a n d T r a n s p o r t

for

Model

B a r r e r , B a r r i e , a n d S l a t e r ( 3 5 ) s u g g e s t e d i n 1958 t h a t nonlinear s o r p t i o n isotherms o f gases and v a p o r s i n g l a s s y p o l y m e r s c o u l d be e x p l a i n e d b y i n v o k i n g two d i s t i n c t mechanisms of s o r p t i o n . They d e s c r i b e d t h i s dual-mode s o r p t i o n a s a r i s i n g f r o m o r d i n a r y d i s s o l u t i o n ( H e n r y ' s l a w mode) p l u s a b s o r p t i o n i n pre-existing "holes" ( L a n g m u i r ' s mode). M i c h a e l s , V i e t h , and Barrie (36) suggested i n 1963 t h a t o n l y t h e d i s s o l v e d g a s (Henry's p o p u l a t i o n ) was i n v o l v e d i n t h e t r a n s p o r t process. V i e t h and S l a d e k (37) extended t h i s i d e a and d e v e l o p e d t h e t o t a l - i m m o b i l i z a t i o n model o f g a s t r a n s p o r t . The t r a n s p o r t model, assuming complete i m m o b i l i z a t i o n o f t h e absorbed gas ( L a n g m u i r ' s p o p u l a t i o n ) , was m o d i f i e d b y P e t r o p o u l o s (38) and l a t e r by P a u l and Koros ( 3 9 ) . They s u g g e s t e d t h a t t h e L a n g m u i r p o p u l a t i o n was n o t i m m o b i l i z e d b u t h a d a f i n i t e m o b i l i t y . The dual-mode s o r p t i o n a n d t r a n s p o r t m o d e l d e s c r i b e s t h e s o r p t i o n o f a gas i n a g l a s s y polymer b y a c o m b i n a t i o n o f Henry's and L a n g m u i r ' s i s o t h e r m s , e q . ( 1 4 ) , and t h e g a s t r a n s p o r t b y F i c k ' s law, eq. ( 1 5 ) ,

C = C

D

+ C

H

C'bp = k p + — 2 1 + bp D

dC

J = -Dp

— dx

(14)

dC -

DJJ

—

(15)

dx

where and a r e t h e c o n c e n t r a t i o n s o f Henry's and Langmuir's p o p u l a t i o n s , r e s p e c t i v e l y , D^ a n d are the respective diffusion c o e f f i c i e n t s o f t h e two p o p u l a t i o n s , and b a r e t h e c a p a c i t y and a f f i n i t y p a r a m e t e r s o f t h e L a n g m u i r s o r p t i o n , r e s p e c t i v e l y , and k_ i s H e n r y ' s l a w s o l u b i l i t y c o e f f i c i e n t . S o l u t i o n o f eq. (15) T o r t h e b o u n d a r y c o n d i t i o n s o f t h e t r a n s i e n t p e r m e a t i o n experiment y i e l d s (39),

p=

VD

( 1

θ =

+

[1 + f ( K , F, b p ) ]

( 1 6 )

(17)

D where Κ = C^b/k^, F = cumbersome t o r e p r o d u c e h e r e .

and t h e f u n c t i o n s y m b o l i z e d

i s too

5.

RAUCHER A N D SEFCIK

Transport

and Main-Chain

101

Motions

In summary, t h e dual-mode s o r p t i o n a n d t r a n s p o r t model assumes (13) : (a) Two modes o f s o r p t i o n - n o r m a l H e n r y ' s mode d i s s o l u t i o n , a n d L a n g m u i r s mode a b s o r p t i o n i n t o u n r e l a x e d - v o l u m e f r o z e n i n the g l a s s y s t a t e . (b) The two g a s p o p u l a t i o n s a r e i n a d y n a m i c e q u i l i b r i u m w i t h each other. ( c ) The two g a s p o p u l a t i o n s have constant but different diffusion coefficients. Langmuir's p o p u l a t i o n g e n e r a l l y has considerably less diffusional mobility than Henry's population. (d) G a s e s , p a r t i c u l a r l y p e r m a n e n t g a s e s , do n o t i n t e r a c t w i t h the polymer m a t r i x , thus Henry's law s o l u b i l i t y c o e f f i c i e n t , kp, and t h e d i f f u s i o n coefficients of both Henry's p o p u l a t i o n , D^, a n d L a n g m u i r ' s p o p u l a t i o n , DJJ, a r e p r e s s u r e and concentration-independent.


1

I n t h e dual-mode s o r p t i o n a n d t r a n s p o r t m o d e l t h e p r e s s u r e d e p e n d e n c e o f σ (= C / p ) , Ρ a n d θ i n g a s - g l a s s y p o l y m e r s y s t e m s a r i s e s from t h e p r e s s u r e - d e p e n d e n t d i s t r i b u t i o n o f t h e s o r b e d gas m o l e c u l e s between Langmuir s i t e s and Henry's l a w d i s s o l u t i o n . A l t h o u g h k p , Dp a n d D^ a r e assumed t o be c o n s t a n t , t h e a v e r a g e o r e f f e c t i v e s o l u b i l i t y and d i f f u s i o n c o e f f i c i e n t s o f t h e e n t i r e e n s e m b l e o f g a s m o l e c u l e s change w i t h p r e s s u r e a s t h e r a t i o o f H e n r y ' s t o L a n g m u i r ' s p o p u l a t i o n , C^/C^, c h a n g e s continuously w i t h p r e s s u r e [eq. ( 1 4 ) ] . B.

D i f f u s i o n Mechanism i n G l a s s y

1. C o o p e r a t i v e

Polymers - Experimental

Evidence

Main-Chain Motions i n PVC-CO2

In S e c t i o n I B we showed that carbon-13 rotating-frame relaxation measurements c a n be u s e d t o measure cooperative main-chain motions i n polymers (28). We r e p o r t h e r e t h e e f f e c t o f C 0 o n t h e m a i n - c h a i n m o t i o n s o f PVC. Carbon-13 r o t a t i n g - f r a m e r e l a x a t i o n r a t e s , w e r e d e t e r m i n e d on s t a t i c s a m p l e s u s i n g c r o s s - p o l a r i z a t i o n t e c h n i q u e s (40). W i t h o u t m a g i c - a n g l e sample s p i n n i n g , t h e m e t h y l e n e - and methine-carbon resonances a r e u n r e s o l v e d , so t h e r e l a x a t i o n r a t e s reported here a r e t h e average relaxation rates f o r the combination l i n e . The < R i p ( C ) > , a t 37 k H z , f o r PVC i n v a c u o a n d i n t h e p r e s e n c e o f C 0 a r e g i v e n i n T a b l e I I . Two m a j o r e f f e c t s a r e o b s e r v e d ; t h e p r e s e n c e o f r e l a t i v e l y s m a l l amounts o f C 0 i n c r e a s e s t h e o f PVC, a n d e x p o s u r e o f PVC t o C 0 c a u s e s a long-term i n c r e a s e i n . The f i r s t phenomena w i l l be d i s c u s s e d i n terms o f i t s e f f e c t on t r a n s p o r t p r o p e r t i e s , and t h e second i n terms o f h i s t o r y - d e p e n d e n t properties. The < R p ( C ) > f o r " c o n d i t i o n e d " PVC i n v a c u o i s 154 s e c " . The p r e s e n c e o f C 0 l e a d s t o n o t i c e a b l e i n c r e a s e s i n . At 100 t o r r t h e < R p ( C ) > o f PVC i s i n c r e a s e d t o 158 s e c . 2

2

2

2

1

x

2

- 1

x

102

INDUSTRIAL

GAS S E P A R A T I O N S

TABLE I I


R e l a x a t i o n Rates

Sequence o f Experiment

i n PVC-C0

Sample Description

2

Relaxation Rate s e c " 1

1

1

i n vacuo

138 ± 3

3

degassing* a f t e r f i r s t e x p o s u r e t o 800 t o r r C 0 - i n vacuo

151 ± 4

degassing a f t e r repeated exposure t o C 0 - i n vacuo

154 ± 3

5

2

7

2

C

5

100 t o r r C 0

6

200 t o r r C 0

2

163 ± 8

4

400 t o r r C 0

2

166 ± 3

2

800 t o r r C 0

2

183 ± 4

2

158 ± 5

a)

Least-squares f i to f r e l a x a t i o n data c o l l e c t e d f o r times b e t w e e n 0.05 a n d 1 msec; e r r o r l i m i t s a r e one s t a n d a r d d e v i a t i o n ; Ηχ(0) = 37 kHz. A v e r a g e f o r m e t h y l e n e a n d methine carbons.

b)

A l l s a m p l e s were d e g a s s e d f o r 24 h o u r s

c)

M e a s u r e d a f t e r 12 h o u r s e q u i l i b r a t i o n a t s t a t e d p r e s s u r e . The c o n c e n t r a t i o n o f C 0 r a n g e s f r o m a b o u t 1 C 0 m o l e c u l e p e r 1200 r e p e a t u n i t s a t 100 mmHg t o a b o u t 1 C 0 p e r 200 r e p e a t u n i t s a t 800 mmHg (19). 2

3

at 10~ torr.

2

2

3

5.


Transport

and

Main-Chain

103

Motions

H i g h e r p r e s s u r e s o f C O 2 cause c o r r e s p o n d i n g l y l a r g e r i n c r e a s e s i n . A t 800 t o r r a n i n c r e a s e o f 1 9 % t o 183 s e c " is observed. Based on standard relaxation rate theory (29), increased r e l a x a t i o n rates are i n d i c a t i v e of a s h i f t i n the average cooperative main-chain motions t o h i g h e r frequencies. C o n v e r s e l y , t h i s means t h a t e v e n s m a l l amounts o f C 0 increase the c o o p e r a t i v e m o t i o n s o f t h e p o l y m e r c h a i n s . Gravimetric measurements by Berens ( 4 1 ) show t h a t C0 s o r p t i o n b y PVC r a n g e s f r o m a b o u t 0.5 mg C 0 / g r a m o f PVC a t 100 t o r r t o a b o u t 3.5 mg C 0 / g r a m o f PVC a t 800 t o r r o f C 0 . The r e a s o n t h a t s u c h s m a l l amounts o f C 0 a r e so e f f e c t i v e i n a l t e r i n g t h e c o o p e r a t i v e m o t i o n s o f PVC r e s u l t s f r o m t h e f a c t t h a t gases have v e r y h i g h m o l e c u l a r m o b i l i t i e s i n g l a s s y polymers (correlation time between 10~ -10~ sec) and can sample extended areas of the polymer on the time scale of the c o o p e r a t i v e motions of the polymer ( 1 0 ~ - 1 0 ~ s e c ) . 1

x

2

2

2

2

2


2

1 0

1 2

5

8

The e x p e r i m e n t a l d a t a i n T a b l e I I shows t h a t < R p ( C ) > o f PVC becomes l o n g e r a f t e r s u c c e s s i v e e x p o s u r e s o f t h e s a m p l e t o C 0 . T h i s i n d i c a t e s t h a t e x p o s u r e t o c a r b o n d i o x i d e c a u s e s a change i n the m o l e c u l a r p a c k i n g o f PVC which allows more m a i n - c h a i n motions. There i s now abundant evidence that exposure of p o l y m e r s t o gases r e s u l t s i n changes w h i c h p e r s i s t l o n g a f t e r t h e gases are removed. These changes are r e f l e c t e d i n altered p h y s i c a l p r o p e r t i e s o f t h e p o l y m e r ( 4 2 - 4 6 ) . The " c o n d i t i o n i n g " o f a p o l y m e r b y e x p o s u r e t o a gas c a n be t h o u g h t o f i n t h e same t e r m s as a n n e a l i n g . I n a n n e a l i n g , the i n c r e a s e d thermal energy a l l o w s the polymer s u f f i c i e n t segmental m o b i l i t y t o e l i m i n a t e e n e r g e t i c a l l y u n f a v o r a b l e c o n f o r m a t i o n s w h i c h were f r o z e n i n t o t h e s o l i d on r a p i d c o o l i n g f r o m t h e m e l t ( 4 7 ) . E x p o s u r e t o a gas can have t h e o p p o s i t e e f f e c t . S i n c e t h e p o l y m e r must s w e l l t o accommodate the sorbed gas m o l e c u l e s , the polymer becomes stressed and ultimately r e a c h e s a new equilibrium condition d e t e r m i n e d b y t h e p r e s e n c e o f t h e p e n e t r a n t . The c o n c e n t r a t i o n and t h e n a t u r e o f t h e p e n e t r a n t d e t e r m i n e t h e r a t e o f r e l a x a t i o n j u s t as t e m p e r a t u r e d e t e r m i n e s t h e r a t e o f r e l a x a t i o n i n t h e annealing experiment. x

2

S i n c e t h e p e n e t r a n t enhances cooperative motions of the p o l y m e r , t h e r a t e o f s t r u c t u r a l change i n t h e p o l y m e r , m e a s u r e d by various relaxation experiments, w i l l be greater during s o r p t i o n , o r i n t h e p r e s e n c e o f t h e p e n e t r a n t , t h a n i t w i l l be d u r i n g d e s o r p t i o n , o r i n the absence o f the p e n e t r a n t ( 3 2 ) . Because o f t h i s , " c o n d i t i o n i n g " o f a polymer r e s u l t s i n l o n g - t e r m o r " p e r m a n e n t " changes i n t h e p o l y m e r . The r e s u l t s o f s o r p t i o n and t r a n s p o r t e x p e r i m e n t s t h e r e f o r e depend on t h e s t a t i c and dynamic s t a t e o f t h e p o l y m e r , w h i c h a r e d e t e r m i n e d by t h e t h e r m a l h i s t o r y o f t h e sample and b y t h e p r e s s u r e and d u r a t i o n of previous exposures t o sorbents. Wonders and P a u l ( 4 2 ) h a v e described these phenomenological e f f e c t s in detail f o r the C 0 - p o l y c a r b o n a t e system. 2

104

2.

INDUSTRIAL

Diffusion

and C o o p e r a t i v e

GAS

SEPARATIONS

Main-Chain Motions i n PVC-CO2

In S e c t i o n I B we p r e s e n t e d experimental evidence that diffusion coefficients correlate w i t h PVC m a i n - c h a i n p o l y m e r m o t i o n s . T h i s r e l a t i o n s h i p has a l s o been j u s t i f i e d t h e o r e t i c a l l y (12). I n t h e p r e v i o u s s e c t i o n we d e m o n s t r a t e d t h a t t h e p r e s e n c e of C O 2 e f f e c t s t h e cooperative main-chain motions o f t h e polymer. The i n c r e a s e i n < R p ( C ) > w i t h i n c r e a s i n g g a s c o n c e n t r a t i o n means t h a t t h e r e a l d i f f u s i o n c o e f f i c i e n t [D i n e q . ( 1 1 ) ] must a l s o increase with concentration. The nmr r e s u l t s r e f l e c t t h e r e a l d i f f u s i o n c o e f f i c i e n t s , s i n c e t h e gas c o n c e n t r a t i o n i s u n i f o r m t h r o u g h o u t t h e p o l y m e r sample u n d e r t h e s t a t i c g a s p r e s s u r e s a n d e q u i l i b r i u m c o n d i t i o n s o f t h e nmr m e a s u r e m e n t s . Unfortunately, the r e a l d i f f u s i o n c o e f f i c i e n t , t h e d i f f u s i o n c o e f f i c i e n t i n t h e a b s e n c e o f a c o n c e n t r a t i o n g r a d i e n t , c a n n o t be d e t e r m i n e d f r o m c l a s s i c a l s o r p t i o n and t r a n s p o r t d a t a w i t h o u t t h e a i d o f a t r a n s p o r t model. W i t h o u t p r e j u s t i c e t o a n y p a r t i c u l a r m o d e l , we c a n o n l y u s e t h e r e l a t i v e change i n t h e r e a l d i f f u s i o n c o e f f i c i e n t t o i n d i c a t e t h e r e l a t i v e change i n t h e a p p a r e n t d i f f u s i o n c o e f f i cient. Apparent d i f f u s i o n c o e f f i c i e n t s o f C 0 i n PVC have b e e n reported by T o i ( 4 8 ) . S e m i l o g p l o t s o f both the apparent d i f f u s i o n c o e f f i c i e n t , Da, and t h e r e l a x a t i o n r a t e , < R i P ( C ) > , a s a f u n c t i o n o f C O 2 p r e s s u r e i n t h e r a n g e o f 0-800 t o r r a r e shown i n F i g u r e 3. Of c o u r s e , t h e r e i s no r e a s o n t o e x p e c t t h e m a g n i t u d e o f t h e change i n t h e r o t a t i n g - f rame r e l a x a t i o n r a t e t o be e q u a l t o t h e m a g n i t u d e o f t h e change i n D o r Da. Both t h e relaxation r a t e and t h e d i f f u s i o n coefficient depend on t h e frequency o f t h e main-chain cooperative motions b u t i n d i f f e r e n t ways ( 2 8 , 2 9 , 4 9 ) . N e v e r t h e l e s s , t h e s i m i l a r i t y i n t h e d e p e n dence o f b o t h t h e r e l a x a t i o n r a t e and t h e a p p a r e n t d i f f u s i o n c o e f f i c i e n t on C O 2 p r e s s u r e s u p p o r t s t h e o b s e r v a t i o n t h a t mainc h a i n molecular motions p l a y a major r o l e i n determining t h e d i f f u s i o n c o e f f i c i e n t o f a gas t h r o u g h a polymer m a t r i x ( 2 8 ) . The e x p e r i m e n t a l e v i d e n c e p r e s e n t e d h e r e and i n t h e l i t e r a t u r e ( 1 5 ) show t h a t t h e r e a l d i f f u s i o n c o e f f i c i e n t depends o n c o n c e n t r a t i o n . These r e s u l t s a r e i n c o m p a t i b l e w i t h t h e n o t i o n o f concentration-independent diffusion coefficients f o r the diss o l v e d and L a n g m u i r s o r b e d m o l e c u l e s [D^ a n d i n equation (15)] as p r o p o s e d b y t h e dual-mode s o r p t i o n a n d t r a n s p o r t model ( 1 3 ) .


x

2

C.

C o n c e n t r a t i o n - D e p e n d e n t S o r p t i o n and T r a n s p o r t i n Glassy Polymers M

M

The b a s i c d i f f e r e n c e b e t w e e n c o n c e n t r a t i o n - d e p e n d e n t and "dual-mode" m o d e l s i s i n t h e i r a s s u m p t i o n a b o u t p e n e t r a n t - p o l y m e r interactions. Concentration-dependent s o r p t i o n and t r a n s p o r t models a r e based on t h e a s s u m p t i o n t h a t t h e c o n c e n t r a t i o n dependence o f t h e s o l u b i l i t y and d i f f u s i o n c o e f f i c i e n t s arises


5.


F i g u r e 3. The c o e f f i c i e n t on m e a s u r e d a t 37 m e a s u r e d a t kO

Transport

and Main-Chain

Motions

105

dependence o f r e l a x a t i o n r a t e and a p p a r e n t d i f f u s i o n C0 p r e s s u r e i n PVC. The r e l a x a t i o n r a t e s w e r e kHz a n d 26 °C. The d i f f u s i o n c o e f f i c i e n t s , ° C , a r e f r o m Réf. U8. 2


106


SEPARATIONS

from p e n e t r a n t - p o l y m e r i n t e r a c t i o n s . On t h e o t h e r h a n d , t h e dual-mode s o r p t i o n and t r a n s p o r t model assumes t h a t t h e r e a r e no gas-polymer i n t e r a c t i o n s , thus assuming concentration-independent s o l u b i l i t y and d i f f u s i o n c o e f f i c i e n t s - k^, D^ a n d D^, r e s p e c tively. Experimental r e s u l t s presented i n t h i s w o r k and i n t h e l i t e r a t u r e a r e i n c o n s i s t e n t w i t h t h e assumptions and t h e p h y s i c a l i n t e r p r e t a t i o n s i m p l i c i t i n t h e dual-mode s o r p t i o n a n d t r a n s p o r t model, and s t r o n g l y s u g g e s t t h a t t h e s o r p t i o n and t r a n s p o r t i n g a s - g l a s s y p o l y m e r s y s t e m s s h o u l d be p r e s e n t e d b y a c o n c e n t r a tion-dependent model: (a) E v i d e n c e f o r g a s - g l a s s y p o l y m e r i n t e r a c t i o n s i s a b u n d a n t . The r e s u l t s presented i n T a b l e I I show t h a t e v e n s m a l l amounts o f g a s a f f e c t t h e c o o p e r a t i v e m a i n - c h a i n m o l e c u l a r motions o f g l a s s y polymers. Evidence that t h e presence o f g a s e s i n p o l y m e r c a u s e s t r u c t u r a l a n d d y n a m i c changes c a n be s e e n i n t h e d e p r e s s i o n o f t h e Tg ( 4 2 , 4 3 , 4 4 ) , a n d i n t h e increased viscoelatic relaxation rates ( 4 3 , 44) o f polymer-gas systems. Further, " c o n d i t i o n i n g " o f polymers, t h e g a s - i n d u c e d s t r u c t u r a l changes i n p o l y m e r s b y e x p o s u r e t o p e n e t r a n t , was shown b y Wonders a n d P a u l ( 4 2 ) , a s w e l l a s by o u r work (40) ( T a b l e I I ) . (b) The e f f e c t o f g a s on t h e c o o p e r a t i v e m a i n - c h a i n m o t i o n s o f g l a s s y p o l y m e r s ( T a b l e I I ) shows t h a t t h e m o l e c u l a r level d i f f u s i o n p r o c e s s i s c o n c e n t r a t i o n d e p e n d e n t , a s must be t h e phenomenological d i f f u s i o n c o e f f i c i e n t . ( c ) T h e r e i s o n l y one p o p u l a t i o n o f s o r b e d gas i n a g l a s s y polymer a t any g i v e n p r e s s u r e . A l l spectroscopic analyses ( 1 5 , 2 2 ) , i n c l u d i n g t h e work o f A s s i n k ( 1 4 ) , a r e c o n s i s t e n t w i t h a l l o f t h e sorbed gas m o l e c u l e s b e i n g i n a s i n g l e state. Nuclear magnetic spin-spin and spin-lattice r e l a x a t i o n measurements o f s o r b e d g a s e s c o n s i s t e n t l y show s i n g l e e x p o n e n t i a l decays i n d i c a t i n g t h a t a l l o f t h e gas m o l e c u l e s a r e r e l a x e d b y t h e same mechanism. T h e s e s t u d i e s a l s o show t h a t t h e r e l a x a t i o n t i m e s i n c r e a s e w i t h i n c r e a s i n g e q u i l i b r i u m gas p r e s s u r e . T h e r e a r e a t l e a s t two p o s s i b l e explanations f o r these results: 1) t h e r e i s a single p o p u l a t i o n o f gas m o l e c u l e s which i n t e r a c t w i t h t h e polymer m a t r i x ; o r 2 ) t h e r e a r e two p o p u l a t i o n s w i t h d i f f e r e n t m o b i l i t i e s i n r a p i d e x c h a n g e a n d whose r e l a t i v e p o p u l a t i o n s vary with pressure. The r e s u l t s p r e s e n t e d i n Section IIB are c o n s i s t e n t only w i t h t h e f i r s t e x p l a n a t i o n (40). The work o f A s s i n k (14) has been f r e q u e n t l y c i t e d as p r o o f f o r t h e e x i s t e n c e o f two d i s t i n c t s o r p t i o n modes i n g l a s s y polymers (50). This i s n o t t h e case. Measuring the s p i n - s p i n r e l a x a t i o n t i m e ( T ) o f ammonia i n p o l y s t y r e n e , Assink showed t h a t t h e ammonia i s r e l a x e d b y a s i n g l e e x p o n e n t i a l p r o c e s s and t h a t T increases with increasing concentration o f sorbed ammonia. A s s i n k eliminated the 2

2

RAUCHER

AND

Transport

SEFCIK

and Main-Chain

Motions

107

obvious interpretation of gas-polymer interactions by a s s u m i n g t h a t a t t h e e x p e r i m e n t a l p r e s s u r e s o f up t o 6 atm. " t h e c o n c e n t r a t i o n o f d i s s o l v e d gas was n o t s u f f i c i e n t t o appreciably plasticize the polymer". He proceeded t o a n a l y z e h i s e x p e r i m e n t a l r e s u l t s i n t e r m s o f t h e dual-mode sorption t h e o r y b y a t t r i b u t i n g d i f f e r e n t m o b i l i t i e s and t h e r e f o r e d i f f e r e n t T ' s t o H e n r y ' s and L a n g m u i r ' s p o p u l a tions. A s s i n k p l o t t e d T as a f u n c t i o n o f t h e mole f r a c t i o n o f t h e gas i n t h e L a n g m u i r s i t e s , as d e t e r m i n e d from t h e dual-mode m o d e l , s o t h e s l o p e o f t h e l i n e s h o u l d g i v e 1/T f o r t h e L a n g m u i r f r a c t i o n o f t h e ammonia and t h e i n t e r c e p t w o u l d be t h e 1/T f o r t h e d i s s o l v e d o r H e n r y ' s f r a c t i o n o f t h e g a s . A l e a s t - s q u a r e s a n a l y s i s o f t h e d a t a gave a s l o p e o f 235 s e c and a n i n t e r c e p t o f -5.2 s e c . A n e g a t i v e relaxation rate i s physically impossible. Rather than abandoning t h e attempt t o r e p r e s e n t t h e data by d u a l s o r p t i o n - m o b i l i t y model, A s s i n k a r b i t r a r i l y assigned a v a l v e o f 12 s e c f o r 1/T o f t h e d i s s o l v e d m o l e c u l e s m a k i n g t h e mobility o f the Langmuir molecules l/20th that of the dissolved molecules. The r e s u l t s of Zupancic, e t . a l . ( 1 5 ) , a r e a l s o relèvent here. Zupancic i n v e s t i g a t e d d i f f u s i o n o f butane i n l i n e a r polyethylene using pulsed magnetic field gradient experiments to measure directly the real diffusion c o e f f i c i e n t a t 23°C. They showed t h a t t h e r e a l d i f f u s i o n c o e f f i c i e n t i n c r e a s e d w i t h e q u i l i b r i u m b u t a n e p r e s s u r e as did the s p i n - s p i n r e l a x a t i o n time. As t h e p o l y e t h y l e n e was above i t s T g , t h e s e r e s u l t s c a n n o t be a t t r i b u t e d t o d u a l mode behavior (13). A single population of butane i n t e r a c t i n g w i t h the p o l y e t h y l e n e accounts f o r both the change i n t h e d i f f u s i o n c o e f f i c i e n t and t h e change i n spin-spin relaxation. The o n l y e x c e p t i o n t o a s i n g l e p o p u l a t i o n o f s o r b e n t i n a g l a s s y p o l y m e r was o b s e r v e d i n t h e w a t e r - c e l l u l o s e a c e t a t e s y s t e m (5_1, 5 2 ) . I n t h i s s y s t e m two r e s o n a n c e f r e q u e n c i e s and two r e l a x a t i o n r a t e s f o r w a t e r were o b s e r v e d . However, t h e two d y n a m i c s t a t e s o f t h e w a t e r i n t h i s s y s t e m a r e due to s p e c i f i c hydrogen bonding i n t e r a c t i o n s r a t h e r than sorpt i o n i n Langmuir type h o l e s . T h e r e i s no m o d e l - i n d e p e n d e n t o r p h y s i c a l e v i d e n c e f o r t h e e x i s t e n c e o f Langmuir o r o t h e r s o r p t i o n s i t e s i n g l a s s y polymers which could account for multiple-site sorption. The i n t e r p r e t a t i o n o f f r e e volume as m o l e c u l a r - s i z e v o i d s responsible f o r Langmuir sorption are only partially successful. W h i l e f r e e volume has been equated w i t h t h e L a n g m u i r c a p a c i t y , C^, i n C 0 - g l a s s y p o l y m e r s y s t e m s ( 1 3 , 46, 5 0 , 5 3 ) t h e c o r r e l a t i o n f a i l s t o h o l d f o r o t h e r g a s e s (54, 5 5 ) . Koros e t . a l . (50) e x p l a i n t h e e q u i v a l e n c e o f the "unrelaxed volume" t o t h e Langmuir capacity by 2

2


2

2

- 1

- 1

- 1

2

2

108

INDUSTRIAL

GAS

SEPARATIONS

suggesting t h a t t h e t o t a l "unrelaxed volume" i s u n i f o r m l y d i s t r i b u t e d a s m o l e c u l a r s c a l e gaps a n d t h a t t h e s e gaps a r e Langmuir s o r p t i o n s i t e s f o r s i n g l e 0 0 m o l e c u l e s . This p h y s i c a l model i m p l i e s t h a t t h e average Langmuir s i t e has t h e same m o l e c u l a r v o l u m e a s t h e e f f e c t i v e m o l e c u l a r v o l u m e o f C O 2 m o l e c u l e s when s o r b e d o n z e o l i t e s o r i n l i q u i d s ( 8 0 Â ) ( 5 0 , 5 4 ) . Methane i s s l i g h t l y l a r g e r t h a n C 0 s o i t i s r e a s o n a b l e t h a t n o t a l l s i t e s a c c e s s i b l e t o C O 2 would be accessible t o CH . Indeed, t h e Langmuir saturation c a p a c i t i e s o f p o l y c a r b o n a t e (54) and p o l y s u l f o n e ( 5 5 ) f o r C H a r e a b o u t 1/2 t h e o f C 0 c a p a c i t i e s . However, a r g o n and nitrogen a r e both s l i g h t l y s m a l l e r than C O 2 , so i t s h o u l d be p o s i b l e t o s o r b one o f t h e s e m o l e c u l e s i n t o e a c h Langmuir s i t e under s a t u r a t i o n c o n d i t i o n s , b u t t h e CJl s f o r Ar and N i n p o l y c a r b o n a t e (54) and p o l y s u l f o n e ( f t ) a r e a g a i n l e s s t h a n 1/2 t h e f o r C O 2 i n t h e same p o l y m e r s . To a c c e p t t h e p h y s i c a l r a t i o n a l o f t h e dual-mode m o d e l w o u l d i m p l y t h a t C O 2 i s u n i q u e among a l l p e n e t r a n t s i n p r o b i n g t h e "unrelaxed volume" o f polymers. The d i s c r e p a n c y i n the values o f f o r d i f f e r e n t gases cannot be a t t r i b u t e d t o t h e condensibility of the penetrants, since condensibility determines the r e l a t i v e e f f i c i e n c y w i t h which the penetrant can u t i l i z e t h e a v a i l a b l e volume a n d s o w o u l d e f f e c t o n l y the value o f t h e Langmuir a f f i n i t y parameter, b ( 1 3 ) . Dilatometric measurements (56) o f the swelling of " c o n d i t i o n e d " p o l y c a r b o n a t e b y C O 2 show t h a t t h e r e l a t i v e swelling o f t h e polymer drops d r a m a t i c a l l y as t h e C 0 pressure i s i n c r e a s e d . T h e s e r e s u l t s c a n n o t be r e c o n c i l e d with the dual-mode sorption model, which assumes preferential sorption into pre-existing sorption sites at low p r e s s u r e s ( s o r p t i o n t h a t s h o u l d have l i t t l e e f f e c t o n the polymer d i m e n s i o n s ) and m a i n l y Henry's t y p e d i s s o l u t i o n at high pressures ( d i s s o l u t i o n that should r e s u l t i n t h e swelling o f the polymer). 2

3

2

4


4

2

f

2

(e)

2

The dual-mode s o r p t i o n a n d t r a n s p o r t m o d e l p r o v i d e s an adequate mathematical description o f the phenomenological s o r p t i o n a n d t r a n s p o r t i n g l a s s y p o l y m e r s ( 1 3 ) . However, t h e results presented here invalidate the physical assumptions i m p l i c i t t o t h e dual-mode m o d e l r e g a r d i n g no p e n e t r a n t - p o l y m e r i n t e r a c t i o n s a n d t h e e x i s t e n c e o f two d i s t i n c t p o p u l a t i o n s o f sorbed gas m o l e c u l e s . In the i n i t i a l formulation of the dual-sorption-mobility model, the authors cautioned against a t t a c h i n g p h y s i c a l s i g n i f i c a n c e t o t h e mathematical parameters ( 3 9 , 5 7 ) . S i n c e t h a t t i m e no p h y s i c a l e v i d e n c e h a s a p p e a r e d t o support t h e e x i s t e n c e o f Langmuir s o r p t i o n s i t e s o r concentrat i o n - i n d e p e n d e n t d i f f u s i o n and s o l u b i l i t y c o e f f i c i e n t s i n g l a s s y polymers. Nevertheless, the mathematical parameters d e s c r i b i n g t h e s e two s t a t e s have t a k e n o n a p h y s i c a l s i g n i f i c a n c e o u t o f proportion with r e a l i t y .

5.


Transport

and Main-Chain

Motions

109

I n t h e f o l l o w i n g c h a p t e r we p r e s e n t t h e m a t r i x m o d e l o f g a s s o r p t i o n and d i f f u s i o n i n g l a s s y p o l y m e r s w h i c h i s based on t h e o b s e r v a t i o n t h a t gas m o l e c u l e s i n t e r a c t w i t h t h e p o l y m e r , t h e r e b y a l t e r i n g t h e s o l u b i l i t y and d i f f u s i o n c o e f f i c i e n t s o f t h e polymer matrix.


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Gas Transport and Cooperative Main-Chain Motions in Glassy Polymers

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