Industrial Gas Separations - ACS Publications - American Chemical

3 Implications of the Dual-Mode Sorption and Transport Models for Mixed Gas Permeation

Downloaded by UNIV OF LEEDS on July 6, 2015 | http://pubs.acs.org Publication Date: June 16, 1983 | doi: 10.1021/bk-1983-0223.ch003

R. T. CHERN, W. J. KOROS, E. S. SANDERS, S. H. CHEN, and H. B. HOPFENBERG North Carolina State University, Department of Chemical Engineering, Raleigh, NC 27650

The concept of unrelaxed volume in glassy polymers is used to interpret sorption and transport data for pure and mixed penetrants. A review of recent sorption and permeation data for mixed penetrants indicates that competition for sorption sites associated with unrelaxed gaps between chain segments is a general feature of gas/glassy polymer systems. This observation provides convincing support for the use of the Langmuir isotherm to describe deviations from simple Henry's law sorption behavior. The observed flux depressions of a component in a mixture, relative to its pure component value at an equivalent partial pressure driving force, derives from the above sorption competition mechanism which influences the effective concentration gradient driving diffusion across the membrane. The competition among penetrants for excess volume described above should be an important consideration for modeling essentially all permselective gas separation membranes. Significant plasticizing effects may mitigate flux reductions caused by the above competitive effects at high pressures in plasticization-prone polymers, but would also lead to selectivity losses which are highly undesirable. The permeation behavior of stiff-chain, plasticization-resistant polymers which are likely to comprise the next generation of gas separation polymers w i l l be appropriately treated by the model discussed here. 0097-6156/83/0223-0047$07.75/0 © 1983 American Chemical Society

In Industrial Gas Separations; Whyte, T., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.


48

INDUSTRIAL

GAS

SEPARATIONS

H i g h membrane p e r m s e l e c t i v i t y i s g e n e r a l l y a s s o c i a t e d w i t h a r i g i d c h a i n backbone of t h e c o n s t i t u e n t p o l y m e r , p r o v i d i n g s i e v i n g on a m o l e c u l a r s c a l e ( 1 , 2 ) . At h i g h u p s t r e a m d r i v i n g p r e s s u r e ( i . e . , at high sorbed penetrant concentration), p l a s t i c i z a t i o n of t h e p o l y m e r may r e d u c e t h e p e r m s e l e c t i v i t y of t h e membrane. A l t h o u g h p l a s t i c i z a t i o n w i l l o c c u r a t s u f f i c i e n t l y high sorbed c o n c e n t r a t i o n s f o r e s s e n t i a l l y a l l g l a s s y m a t e r i a l s , s e l e c t i o n of a membrane m a t e r i a l w i t h a v e r y h i g h 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 and e x t r a o r d i n a r y i n h e r e n t backbone r i g i d i t y should m i n i m i z e e f f e c t s a s s o c i a t e d w i t h plasticization. O f t e n t h e r a t i o of p u r e component p e r m e a b i l i t i e s a t an a r b i t r a r y p r e s s u r e f o r t h e components of i n t e r e s t i s u s e d as an i n d i c a t i o n of t h e p o t e n t i a l s e l e c t i v i t y of a c a n d i d a t e membrane. W h i l e t h i s a p p r o a c h o f f e r s a u s e f u l approximation, t h e a c t u a l s e l e c t i v i t y and p r o d u c t i v i t y ( f l u x ) o b s e r v e d i n t h e m i x e d gas c a s e may be s u r p r i s i n g l y d i f f e r e n t t h a n p r e d i c t e d on t h e b a s i s of t h e p u r e component d a t a . As shown i n F i g u r e 1, p r i o r t o t h e o n s e t of p l a s t i c i z i n g , t h e p r e s e n c e of a s e c o n d component Β can d e p r e s s t h e o b s e r v e d p e r m e a b i l i t y of a component A r e l a t i v e t o i t s p u r e component v a l u e a t a g i v e n u p s t r e a m d r i v i n g p r e s s u r e of component A. I n t h e words o f Pye e t a l . ( 3 ) , t h e p e r m e a b i l i t y of a membrane t o a component A may be r e d u c e d due t o the s o r p t i o n of a s e c o n d component Β i n t h e p o l y m e r w h i c h ". · · e f f e c t i v e l y r e d u c e s t h e m i c r o v o i d content o f t h e f i l m and t h e a v a i l a b l e d i f f u s i o n p a t h s f o r t h e non r e a c t i v e g a s e s " . The p r e s e n t p a p e r w i l l o f f e r a d d i t i o n a l e x p e r i m e n t a l and t h e o r e t i c a l i n s i g h t i n t o t h i s i n t e r e s t i n g e f f e c t t h a t i s c h a r a c t e r i s t i c of g l a s s y p o l y m e r s . U n r e l a x e d Volume - R e l a t i o n To P r o p e r t i e s Of G l a s s y P o l y m e r s

S o r p t i o n And

Transport

The c o n c e p t of " u n r e l a x e d v o l u m e " , Vg-V^, i l l u s t r a t e d i n F i g u r e 2 may be u s e d t o i n t e r p r e t s o r p t i o n and t r a n s p o r t d a t a i n g l a s s y p o l y m e r s e x p o s e d t o p u r e and m i x e d p e n e t r a n t s . The e x t r a o r d i n a r i l y long r e l a x a t i o n times f o r segmental motion i n t h e g l a s s y s t a t e l e a d t o t r a p p i n g of n o n e q u i l i b r i u m c h a i n c o n f o r m a t i o n s i n quenched g l a s s e s , t h e r e b y p e r m i t t i n g m i n i s c u l e g a p s t o e x i s t between c h a i n s e g m e n t s . These gaps can be r e d i s t r i b u t e d by p e n e t r a n t i n s o - c a l l e d " c o n d i t i o n i n g " t r e a t m e n t s d u r i n g t h e i n i t i a l e x p o s u r e of t h e p o l y m e r t o h i g h p r e s s u r e s of t h e p e n e t r a n t ( 4 ) . F o l l o w i n g s u c h i n i t i a l e x p o s u r e t o p e n e t r a n t , s e t t l e d i s o t h e r m s c h a r a c t e r i z e d by c o n c a v i t y t o t h e p r e s s u r e a x i s a t l o w p r e s s u r e s and a t e n d e n c y t o a p p r o a c h l i n e a r h i g h p r e s s u r e l i m i t s a r e o b s e r v e d as shown i n F i g u r e 3. Reference to penetrant-induced conditioning e f f e c t s have been made c o n t i n u o u s l y s i n c e t h e v e r y e a r l i e s t i n v e s t i g a t i o n s of p e n e t r a n t / g l a s s y p o l y m e r s o r p t i o n



3.

CHERN

ET

AL.

Dual-Mode

Sorption

simple

and Transport

Models

49

dual mode^ iplasticizing

1 by A or Β /permeability \ of A

increasing

upstream

partial pressure of A

Figure 1. P e r m e a b i l i t y o f a glassy polymer t o penetrant A i n t h e p r e s e n c e o f v a r y i n g p a r t i a l p r e s s u r e s o f p e n e t r a n t B.



50

INDUSTRIAL

GAS

SEPARATIONS

(gm)

\y

^equilibrium volume densitied

glass

-)

/

Temperature -— Figure 2. Schematic r e p r e s e n t a t i o n o f the unrelaxed volume, Vg - V j , i n a g l a s s y p o l y m e r . Note t h a t t h e u n r e l a x e d volume disappears at the glass t r a n s i t i o n temperature, Τ g'


CHERN E T A L .

Dual-Mode

Sorption

and Transport

Models


3.


51

52

INDUSTRIAL GAS SEPARATIONS

behavior (2>4.».5»6)· " o v e r s w e l l s " the polymer with a h i g h l y s o r b i n g penetrant such as a high a c t i v i t y vapor and then removes the p e n e t r a n t , the excess volume which i s introduced w i l l tend to r e l a x q u i c k l y at f i r s t , followed by a very slow, l o n g term approach toward e q u i l i b r i u m ( 6 ) . At extremely high gas c o n d i t i o n i n g p r e s s u r e s , s u b s t a n t i a l s w e l l i n g of the polymer sample can occur with a r e s u l t a n t i n c r e a s e i n the value of Vg-V^ ( 5 ) . For the case of c o n d i t i o n i n g with gases such as (X>2 at pressures l e s s than 30 atm, however, c o n s o l i d a t i o n i n the absence of penetrant f o l l o w i n g the c o n d i t i o n i n g treatment i s t y p i c a l l y unmeasurable s i n c e penetrant s o r p t i o n uptake i s not very extensive (< 4-5% by weight) (4*^5)· As a r e s u l t , i n such cases r e d i s t r i b u t i o n of the o r i g i n a l l y present intersegmental gaps may be the primary process o c c u r r i n g during the f i r s t exposure of the polymer to h i g h pressures of penetrant as shown i n Figure 3 ( 4 ) . An i n t e r p r e t a t i o n of the observed c o n d i t i o n i n g behavior t h a t occurs during the primary penetrant exposure i n the absence of l a r g e s w e l l i n g e f f e c t s may be couched i n terms of coalesence of packets of the o r i g i n a l intersegmental gaps* R e d i s t r i b u t i o n of chain conformations, c o n s i s t e n t with optimal accomodation of the penetrant i n the unrelaxed volume between c h a i n segments may permit t h i s process during the c o n d i t i o n i n g treatment. This rearrangement would tend to produce a more or l e s s d e n s i f i e d matrix with a small volume f r a c t i o n of e s s e n t i a l l y uniformly d i s t r i b u t e d molecular s c a l e gaps or " h o l e s " throughout the m a t r i x . In such a s i t u a t i o n , one can a p p r e c i a t e the meaning of two s l i g h t l y d i f f e r e n t molecular environments i n the g l a s s i n which s o r p t i o n of gas may o c c u r . Consider f i r s t the l i m i t i n g case i n which a h i g h l y annealed, t r u l y e q u i l i b r i u m d e n s i f i e d g l a s s c h a r a c t e r i z e d by "V^" i n F i g u r e 2 i s exposed to a given pressure of a penetrant. In t h i s c a s e , a l l gaps are m i s s i n g , but there w i l l c l e a r l y s t i l l be a c e r t a i n c h a r a c t e r i s t i c s o r p t i o n c o n c e n t r a t i o n ( C ) t y p i c a l of true molecular d i s s o l u t i o n i n the d e n s i f i e d g l a s s , s i m i l a r to that observed i n low molecular weight l i q u i d s or rubbers (above T g ) . Next, consider a corresponding c o n d i t i o n e d n o n e q u i l i b r i u m g l a s s ( i l l u s t r a t e d by "Vg" i n Figure 2) c o n t a i n i n g unrelaxed volume i n which the surrounding matrix has been more or l e s s d e n s i f i e d by the coalesence of gaps to form molecular s c a l e berths f o r penetrant. A local equilibrium requirement leads to an average l o c a l c o n c e n t r a t i o n of penetrant held i n the uniformly d i s t r i b u t e d molecular s c a l e gaps ( C ) i n e q u i l i b r i u m with the " d i s s o l v e d " c o n c e n t r a t i o n (Cn) at any given e x t e r n a l penetrant pressure or a c t i v i t y . T h i s simple p h y s i c a l model can be d e s c r i b e d a n a l y t i c a l l y up to reasonably high pressues ( g e n e r a l l y f o r pressures l e s s than or equal to the maximum c o n d i t i o n i n g pressure employed (4) ) i n terms of the sum of Henry's law f o r C and a Langmuir isotherm for C .


o

n

e

D

H

n

H


3.

Dual-Mode

CHERN E T A L .

C = C + C D

C = k P D

+

Sorption

and Transport

53

Models (1)

H

C

b

H P 1 + bp

(2)

where kn i s the Henry's law constant that c h a r a c t e r i z e s s o r p t i o n of penetrant i n the d e n s i f i e d regions which comprise most of the m a t r i x . The parameter b c h a r a c t e r i z e s the a f f i n i t y of the penetrant f o r the i n t e r c h a i n gap regions i n the polymer. The parameter C i s the Langmuir s o r p t i o n c a p a c i t y of the g l a s s y matrix and can be i n t e r p r e t e d d i r e c t l y i n terms of F i g u r e 2 i n which the unrelaxed volume, V ~V^, corresponds to the summation of a l l of the molecular s c a l e gaps i n the g l a s s . As shown i n Figure 4, the Langmuir c a p a c i t y of g l a s s y polymers tends to approach zero i n the same way that Vg ~V^, approaches zero at the g l a s s t r a n s i t i o n temperature (see F i g u r e 2 ) . This q u a l i t a t i v e o b s e r v a t i o n can be extended to a q u a n t i t a t i v e statement i n cases f o r which the e f f e c t i v e molecular volume of the penetrant i n the sorbed s t a t e can be e s t i m a t e d . As a f i r s t approximation, one may assume that the e f f e c t i v e molecular volume of a sorbed C(>2 molecule i s 80 i n the range of temperature from 25°C to 8 5 ° C . This molecular volume corresponds to an e f f e c t i v e molar volume of 49 cc/mole of CO2 molecules and i s s i m i l a r to the p a r t i a l molar volume of CO2 i n v a r i o u s s o l v e n t s , i n s e v e r a l z e o l i t e environments, and even as a pure s u b c r i t i c a l l i q u i d (See Table 1) ( 4 » 8 ) . The i m p l i c a t i o n here i s not that more than one CO2 molecule e x i s t s i n each molecular s c a l e gap, but rather that the e f f e c t i v e volume occupied by a CO2 molecule i s roughly the same i n the polymer sorbed s t a t e , i n a saturated z e o l i t e sorbed s t a t e and even i n a d i s s o l v e d or l i q u i d - l i k e s t a t e s i n c e a l l of these volume estimates tend to be s i m i l a r f o r m a t e r i a l s that are not too much above t h e i r c r i t i c a l temperatures. With the above approximation, the p r e d i c t i v e expression given below f o r can be compared to independently measured values f o r t h i s parameter from s o r p t i o n measurements.


H

Ci,

P*

(3)

where ρ i s the equivalent d e n s i t y of CO2 ( ~ l / 4 9 m o l e / c c ) d i s c u s s e d above. The comparison of measured (4,9-11) and p r e d i c t e d C ^ ' s c a l c u l a t e d from Eq ( 3 ) u s i n g reported d i l a t o m e t r i c parameters (12-15) f o r the v a r i o u s g l a s s y polymers i s shown i n Figure 5. The c o r r e l a t i o n i s c l e a r l y i m p r e s s i v e . A p p l i c a t i o n of Eq ( 3 ) to h i g h l y s u p e r c r i t i c a l gases i s somewhat ambiguous s i n c e the e f f e c t i v e molecular volume of sorbed gases under these c o n d i t i o n s i s not e a s i l y e s t i m a t e d . A



20

-

40

40

60

60

80

120 20

40

80

100

120

2

180

60

20

80

40

100

T e m p e r a t u r e (°C)

140

(d ϊ Polycarbonate / C 0

100

60

20

80

40

100

60

120

80

140

100

F i g u r e U. L a n g m u i r s o r p t i o n c a p a c i t y , C^, a s a f u n c t i o n o f temp e r a t u r e f o r s e v e r a l p o l y m e r / p e n e t r a n t s y s t e m s . N o t e t h a t c£ d i s a p p e a r s n e a r Τ . ( R e p r o d u c e d w i t h p e r m i s s i o n f r o m R e f . 7. C o p y r i g h t J o h n W i l e y & S o n s , 198l.)

20

PET/CO,

(α )


120

CHERN ET A L .


T a b l e 1:

Dual-Mode

Sorption

and Transport

Models

E f f e c t i v e M o l a r o r P a r t i a l M o l a r Volume o f CO2 i n V a r i o u s E n v i r o n m e n t s a t 25°C.

Environment

Molar or P a r t i a l Molar Volume ( c c / m o l e )

Ref

Carbon Tetrachloride

48.2

8

Chlorobenzene

44.6

8

Benzene

47.9

8

Acetone

44.7

8

Methyl Acetate

44.5

8

4A o r 5A Z e o l i t e at s a t u r a t i o n of capacity

52.4

4



56

INDUSTRIAL

GAS

SEPARATIONS

f

ce (STP) Λ [ce polymery

F i g u r e 5. Q u a n t i t a t i v e c o m p a r i s o n o f e x p e r i m e n t a l l y m e a s u r e d values o f f o r CO^ i n v a r i o u s p o l y m e r s w i t h t h e p r e d i c t i o n s o f E q u a t i o n 3. Π · p o l y ( e t h y l e n e t e r e p h t h a l a t e ) a t 25, 35, *+5, 55, 65, 75, a n d 85 ° C j , £ · , p o l y ( b e n z y l m e t h a c r y l a t e ) a t 30 ° C ] , ["φ , p o l y ( p h e n y l m e t h a c r y l a t e ) a t 35, 50, a n d 75 ° C j , C 0 . p o l y C a c r y l o n i t r i l e ) a t 35, 55, a n d 65 °çO>[p, Φ p o l y c a r b o n a t e a t 35, 55, a n d 75 °C: t h e Ο s y m b o l s r e f e r t o p r e d i c t i o n s u s i n g d i l a t o m e t r i c c o e f f i c i e n t s f r o m R e f . lh; t h e 0 s y m b o l s r e f e r t o p r e d i c t i o n s u s i n g c o e f f i c i e n t s f r o m R e f . 15j. Π Θ, p o l y ( e t h y l m e t h a c r y l a t e ) a t 30, Uo, a n d 55 ° θ Ί , Γ θ , p o l y ( m e t h y l methacrylate) a t 35, 55, 80 a n d 100 °CQ.


3.

C H E R N ET A L .

Dual-Mode

Sorption

and

Transport

57

Models


s i m i l a r problem e x i s t s i n a p r i o r i e s t i m a t e s of p a r t i a l m o l a r v o l u m e s of s u p e r c r i t i c a l components e v e n i n l o w m o l e c u l a r weight l i q u i d s (16). The p r i n c i p l e upon w h i c h Eq ( 3 ) i s b a s e d r e m a i n s v a l i d , h o w e v e r , and w h i l e t h e t o t a l amount of u n r e l a x e d v o l u m e may be a v a i l a b l e f o r a p e n e t r a n t , t h e m a g n i t u d e of d e p e n d s s t r o n g l y on how c o n d e n s i b l e t h e p e n e t r a n t i s , s i n c e t h i s f a c t o r d e t e r m i n e s 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 component c a n u t i l i z e t h e a v a i l a b l e v o l u m e . Transport. A companion t r a n s p o r t model t h a t a l s o a c k n o w l e d g e s t h e f a c t t h a t p e n e t r a n t may e x e c u t e d i f f u s i v e jumps i n t o and o u t of t h e two s o r p t i o n e n v i r o n m e n t s e x p r e s s e s t h e l o c a l f l u x , N, a t any p o i n t i n t h e p o l y m e r i n t e r m s of a two p a r t c o n t r i b u t i o n ( 1 7 - 2 0 ) : (4)

Ν 3x

3x

where Dn and D r e f e r t o t h e m o b i l i t y of t h e d i s s o l v e d and Langmuir s o r b e d components, r e s p e c t i v e l y . I t i s t y p i c a l l y f o u n d t h a t Dp i s c o n s i d e r a b l y l a r g e r t h a n D^ e x c e p t f o r non c o n d e n s i b l e g a s e s s u c h as h e l i u m ( 1 4 ) . The above e x p r e s s i o n c a n be w r i t t e n i n t e r m s of F i c k ' s l a w w i t h an e f f e c t i v e diffusion coefficient, D ( C ) , t h a t i s d e p e n d e n t on l o c a l concentration: H

e f f

Ν = - D

e f f

(C)

(5)

™ 3x

The d u a l m o b i l i t y m o d e l e x p r e s s e s 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 y o f D f f ( C ) i n t e r m s of t h e l o c a l c o n c e n t r a t i o n o f d i s s o l v e d p e n e t r a n t , C > as shown i n Eq ( 6 ) : e

D

FK 1 + D

e f f

(O

=

(1 +

aC V D

(6)

Dn 1 +

(1 +

aC ) D

where F = D H / D , Κ Ξ C ^ b / k and α = b / k . T h i s model e x p l a i n s c o n c e n t r a t i o n d e p e n d e n c y of t h e l o c a l d i f f u s i o n c o e f f i c i e n t s u c h as t h a t shown f o r C 0 i n poly(ethylene terephthalate) (PET) i n F i g u r e 6 ( 2 1 ) i n t e r m s of a p r o g r e s s i v e i n c r e a s e i n the f r a c t i o n of the l o c a l c o n c e n t r a t i o n present i n the h i g h e r m o b i l i t y H e n r y ' s l a w e n v i r o n m e n t as t h e l o c a l L a n g m u i r c a p a c i t y s a t u r a t e s at i n c r e a s i n g l y higher pressures. The p o i n t s i n F i g u r e 6 were e v a l u a t e d f r o m t h e p h e n o m e n o l o g i c a l p e r m e a b i l i t y and s o r p t i o n c o n c e n t r a t i o n d a t a d

D

D

2


I N D U S T R I A L GAS

SEPARATIONS


58


3.

C H E R N ET A L .

Dual-Mode

Sorption

and

Transport

59

Models

u s i n g a method t h a t does not depend on t h e d u a l mode m o d e l i n any way ( 2 2 ) . I n t e r e s t i n g l y , h o w e v e r , t h e l i n e t h r o u g h t h e d a t a p o i n t s corresponds t o the p r e d i c t i o n s of D f f ( C ) u s i n g Eq ( 6 ) a l o n g w i t h t h e i n d e p e n d e n t l y d e t e r m i n e d d u a l mode parameters f o r t h i s system (21,23). I t i s also important to n o t e t h a t the form of d a t a i n t h i s p l o t which e x h i b i t s a tendency t o asymptote a t h i g h p r e s s u r e s i s not t y p i c a l of plasticization. F i n a l l y , i f one c o n s i d e r s t h e l o w c o n c e n t r a t i o n r e g i o n o f F i g u r e 6, i t i s c l e a 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 i s s u r p r i s i n g l y c o n c e n t r a t i o n dependent. For example, D f f ( C ) i n c r e a s e s by more t h a n 35% as t h e l o c a l s o r b e d c o n c e n t r a t i o n r i s e s f r o m 0.142 c c ( S T P ) / c c p o l y m e r ( o r 0.00020 wt. f r a c t i o n ) a t 50 mm Hg t o 1.7 c c ( S T P ) / c c p o l y m e r ( o r 0.0025 wt. f r a c t i o n ) a t 760 mm Hg. The above c o n c e n t r a t i o n s a r e e x t r e m e l y d i l u t e , c o r r e s p o n d i n g t o l e s s t h a n 1 CO /1200 PET r e p e a t u n i t s and 1 C0 /98 repeat u n i t s r e s p e c t i v e l y ( 4 ) . e


e

2

2

An e v e n more d r a m a t i c c a s e c o r r e s p o n d s t o C 0 i n PVC i n w h i c h D f ( C ) r i s e s by 86% as t h e l o c a l c o n c e n t r a t i o n i n c r e a s e s f r o m 1 C 0 / 4 4 0 0 r e p e a t u n i t s a t 100 mm Hg t o 1 C 0 / 1 3 0 0 r e p e a t u n i t s a t 500 mm Hg a t 40°C ( 2 4 ) . The above v a l u e s o f C 0 s o l u b i l i t y a r e b a s e d on measurements by T o i ( 2 4 ) and T i k h o m o r o v e t a l . ( 2 5 ) w h i c h a r e i n good a g r e e m e n t . At such e x t r a o r d i n a r i l y low penetrant c o n c e n t r a t i o n s , p l a s t i c i z a t i o n of t h e o v e r a l l m a t r i x i s c e r t a i n l y n o t anticipated. Motions i n v o l v i n g r e l a t i v e l y few r e p e a t u n i t s are b e l i e v e d t o g i v e r i s e t o most s h o r t t e r m g l a s s y s t a t e properties. I n r u b b e r y p o l y m e r s , on t h e o t h e r h a n d , l o n g e r c h a i n c o n c e r t e d motions occur over r e l a t i v e l y s h o r t time s c a l e s , and one e x p e c t s p l a s t i c i z a t i o n t o be e a s i e r t o i n d u c e i n t h e s e materials. I n t e r e s t i n g l y , no known t r a n s p o r t s t u d i e s i n r u b b e r s have i n d i c a t e d p l a s t i c i z a t i o n a t the low s o r p t i o n l e v e l s noted a b o v e f o r PVC and PET. The d i s c u s s i o n d i r e c t l y f o l l o w i n g Eq ( 6 ) p r o v i d e s a s i m p l e , p h y s i c a l l y reasonable e x p l a n a t i o n f o r the p r e c e d i n g o b s e r v a t i o n s of marked c o n c e n t r a t i o n dependence o f D f f ^ ^ i y concentrations. C l e a r l y , a t some p o i n t , t h e a s s u m p t i o n o f c o n c e n t r a t i o n i n d e p e n d e n c e of D and D^ i n Eq ( 6 ) w i l l f a i l ; h o w e v e r , f o r o u r work w i t h " c o n d i t i o n e d " p o l y m e r s a t C 0 p r e s s u r e s b e l o w 300 p s i , s u c h e f f e c t s a p p e a r t o be n e g l i g i b l e . Due t o t h e c o n c a v e shape of t h e s o r p t i o n i s o t h e r m , e v e n a t a C 0 p r e s s u r e of 10 atm, t h e r e w i l l s t i l l be l e s s t h a n one C 0 m o l e c u l e p e r t w e n t y PET r e p e a t u n i t s a t 35°C. S t e r n ( 2 6 ) has d e s c r i b e d a g e n e r a l i z e d f o r m of t h e d u a l mode t r a n s p o r t m o d e l t h a t p e r m i t s h a n d l i n g s i t u a t i o n s i n w h i c h n o n - c o n s t a n c y o f Dp and DL| m a n i f e s t t h e m s e l v e s . I t i s r e a s o n a b l e t o assume t h a t t h e n e x t g e n e r a t i o n of gas s e p a r a t i o n membrane p o l y m e r s w i l l be e v e n more r e s i s t a n t t o p l a s t i c i z a t i o n t h a n p o l y s u l f o n e , and c e l l u l o s e a c e t a t e , so t h e a s s u m p t i o n of c o n s t a n c y of t h e s e t r a n s p o r t p a r a m e t e r s w i l l be e v e n more f i r m l y j u s t i f i e d . 2

e f

2

2

2

C

a t

r

e

a

t

i

v

e

l

l

o

w

e

n

2

2

2


60

INDUSTRIAL

GAS

SEPARATIONS

A l t h o u g h n o t n e c e s s a r y i n terms o f p h e n o m e n o l o g i c a l applications, i t i sinteresting to consider possible molecular m e a n i n g s o f t h e c o e f f i c i e n t s , Dn a n d DJJ. I f two p e n e t r a n t s e x i s t i n a p o l y m e r i n t h e two r e s p e c t i v e modes d e s i g n a t e d by "D" and "H" t o i n d i c a t e t h e " d i s s o l v e d " ( H e n r y ' s l a w ) a n d t h e " h o l e " (Langmuir) environments, then the molecules can execute d i f f u s i v e movements w i t h i n t h e i r r e s p e c t i v e modes o r t h e y may e x e c u t e intermode jumps.


"D"

• "D"

" d i s s o l v e d " t o " d i s s o l v e d " jump

"H"

" d i s s o l v e d " t o " h o l e " jump

"H"

' h o l e " t o " h o l e " jump

"D"

' h o l e " t o " d i s s o l v e d " jump

C l e a r l y , t h e t r u e c h a r a c t e r ( a c t i v a t i o n e n e r g y , e n t r o p y and jump l e n g t h ) o f t h e p h e n o m e n o l o g i c a l l y o b s e r v e d Dn w i l l be a w e i g h t e d a v e r a g e o f t h e r e l a t i v e f r e q u e n c y o f "D"+"D" and "D"+"H" j u m p s , and l i k e w i s e f o r D i n terms o f "H"->-"H" and "H"+"D" j u m p s . G i v e n t h e r e l a t i v e l y d i l u t e o v e r a l l volume f r a c t i o n a s s o c i a t e d w i t h the n o n e q u i l i b r i u m gaps w h i c h c o m p r i s e t h e "H" e n v i r o n m e n t (< 4 t o 5% on a volume b a s i s ) , one may t o a f i r s t a p p r o x i m a t i o n assume t h a t most d i f f u s i v e jumps o f a p e n e t r a n t f r o m a "D" e n v i r o n m e n t r e s u l t i n movement t o a n o t h e r "D" e n v i r o n m e n t and most d i f f u s i v e jumps f r o m "H" e n v i r o n m e n t s r e s u l t i n movement t o a "D" e n v i r o n m e n t . The o b s e r v e d a c t i v a t i o n e n e r g i e s , e n t r o p i e s and jump l e n g t h s , t h e r e f o r e , have f a i r l y w e l l - d e f i n e d meanings on a m o l e c u l a r s c a l e . One c a n e a s i l y show t h a t t h e a p p r o p r i a t e e q u a t i o n d e r i v e d f r o m t h e d u a l mode s o r p t i o n and t r a n s p o r t models f o r t h e s t e a d y s t a t e p e r m e a b i l i t y o f a p u r e component i n a g l a s s y p o l y m e r i s g i v e n by Eq ( 7 ) ( 1 8 ) when t h e downstream r e c e i v i n g p r e s s u r e i s e f f e c t i v e l y z e r o and t h e u p s t r e a m d r i v i n g p r e s s u r e i s p. H

Ρ = k Dn[l + D

L

F

K

1 + bp

1

(7)

J

The f i r s t t e r m i n Eq ( 7 ) d e s c r i b e s t r a n s p o r t r e l a t e d t o t h e Henry's l a w environment, w h i l e t h e second term i s r e l a t e d t o t h e Langmuir environment. The t e n d e n c y f o r t h e p e r m e a b i l i t y t o a s y m p t o t i c a l l y approach t h e l i m i t i n g v a l u e of k Dn a t h i g h pressures d e r i v e s from the f a c t that a f t e r s a t u r a t i o n of the upstream Langmuir c a p a c i t y a t h i g h p r e s s u r e s , a d d i t i o n a l pressure increases r e s u l t i n a d d i t i o n a l f l u x contributions only f r o m t h e t e r m r e l a t e d t o Henry's l a w w h i c h c o n t i n u e s t o i n c r e a s e as u p s t r e a m p r e s s u r e i n c r e a s e s . The r e m a r k a b l e e f f i c a c y o f t h e d u a l mode s o r p t i o n and n



3.

CHERN ET A L .

Dual-Mode

Sorption

and Transport

Models

61

t r a n s p o r t m o d e l f o r d e s c r i p t i o n o f p u r e component d a t a h a s been i l l u s t r a t e d by p l o t s o f t h e l i n e a r i z e d f o r m s o f Eq ( 2 ) and Eq ( 7 ) f o r a w i d e number o f p o l y m e r / p e n e t r a n t s y s t e m s (4>5,9.» 10,22,24). These l i n e a r i z e d p l o t s a r e s t r i n g e n t t e s t s o f t h e a b i l i t y of t h e proposed f u n c t i o n a l forms t o d e s c r i b e t h e p h e n o m e n o l o g i c a l d a t a . A s s i n k (27) has a l s o i n v e s t i g a t e d t h e d u a l mode m o d e l u s i n g a p u l s e d NMR t e c h n i q u e and c o n c l u d e d that: "We have been a b l e t o d e m o n s t r a t e the b a s i c v a l i d i t y of the a s s u m p t i o n s on w h i c h t h e d u a l mode m o d e l i s b a s e d and we have shown t h e u s e f u l n e s s o f NMR r e l a x a t i o n techniques i n the study of t h i s model." Whereas t h e d u a l s o r p t i o n and t r a n s p o r t m o d e l d e s c r i b e d above u n i f i e s i n d e p e n d e n t d i l a t o m e t r i c , 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 c h a r a c t e r i z i n g t h e g l a s s y s t a t e , an a l t e r n a t e m o d e l o f f e r e d r e c e n t l y by R a u c h e r and S e f c i k p r o v i d e s a n e m p i r i c a l and f u n d a m e n t a l l y c o n t r a d i c t o r y f i t o f s o r p t i o n , d i f f u s i o n and s i n g l e component p e r m e a t i o n d a t a i n t e r m s o f p a r a m e t e r s w i t h a m b i g u o u s p h y s i c a l m e a n i n g s ( 2 8 ) . The d e t a i l e d e x p o s i t i o n o f t h e d u a l mode model and t h e d e m o n s t r a t i o n o f t h e p h y s i c a l s i g n i f i c a n c e and c o n s i s t e n c y o f t h e v a r i o u s e q u i l i b r i u m and t r a n s p o r t parameters i n t h e model i n t h e p r e s e n t paper p r o v i d e a b a c k d r o p f o r s e v e r a l b r i e f comments p r e s e n t e d i n t h e A p p e n d i x r e g a r d i n g t h e m o d e l o f R a u c h e r and S e f c i k . M i x e d Component S o r p t i o n

and T r a n s p o r t

A r g u m e n t s s i m i l a r t o t h o s e p r e s e n t e d above f o r p u r e c o m p o n e n t s have b e e n e x t e n d e d t o g e n e r a l i z e t h e e x p r e s s i o n s g i v e n i n E q u a t i o n s 2 and 7 t o a c c o u n t f o r t h e c a s e o f m i x e d penetrants (29,30). The a p p r o p r i a t e e x p r e s s i o n s a r e g i v e n below: C

A

=

k

DAPA

+

C

1

k

D

DA DA

1

b

HA APA b

(8) b

+ APA + BPB F

+ 1

b

_

K

A A

(9) b

+ APA + BPB

I n t h e above e x p r e s s i o n s , "A" r e f e r s t o t h e component o f p r i m a r y i n t e r e s t w h i l e "B" r e f e r s t o a s e c o n d " c o m p e t i n g " component. The p e r m e a b i l i t y o f a p o l y m e r t o a p e n e t r a n t depends on t h e m u l t i p l i c a t i v e c o n t r i b u t i o n o f a s o l u b i l i t y and a m o b i l i t y t e r m . T h e s e two f a c t o r s may be f u n c t i o n s o f l o c a l p e n e t r a n t c o n c e n t r a t i o n i n t h e g e n e r a l c a s e as i n d i c a t e d by t h e d u a l mode model. Robeson ( 3 1 ) has p r e s e n t e d d a t a f o r C 0 p e r m e a t i o n i n 2



62


SEPARATIONS

p o l y c a r b o n a t e i n w h i c h b o t h s o l u b i l i t y and d i f f u s i v i t y a r e r e d u c e d due t o a n t i p l a s t i c i z a t i o n c a u s e d by t h e p r e s e n c e o f t h e s t r o n g l y i n t e r a c t i n g 4-4" d i c h l o r o d i p h e n y l s u l f o n e . On t h e o t h e r hand, s o r p t i o n of a l e s s s t r o n g l y i n t e r a c t i n g p e n e t r a n t , s u c h as a h y d r o c a r b o n , may a f f e c t p r i m a r i l y o n l y t h e s o l u b i l i t y f a c t o r w i t h o u t s i g n i f i c a n t l y changing the i n h e r e n t m o b i l i t y of t h e p e n e t r a n t i n e i t h e r of t h e two modes. F l u x r e d u c t i o n i n t h i s l a t t e r c o n t e x t occurs s i m p l y because the c o n c e n t r a t i o n d r i v i n g f o r c e of p e n e t r a n t A i s reduced. This r e s u l t s from the e x c l u s i o n o f A by component Β f r o m L a n g m u i r s o r p t i o n s i t e s w h i c h were p r e v i o u s l y a v a i l a b l e t o p e n e t r a n t A i n t h e a b s e n c e of p e n e t r a n t B. C o n s i s t e n t w i t h the p r e c e d i n g d i s c u s s i o n c o n c e r n i n g s o r p t i o n and f l u x r e d u c t i o n s by r e l a t i v e l y non i n t e r a c t i n g p e n e t r a n t s , t h e d a t a shown i n F i g u r e 7 c l e a r l y i l l u s t r a t e t h e p r o g r e s s i v e e x c l u s i o n of C 0 from Langmuir s o r p t i o n s i t e s i n p o l y ( m e t h y l m e t h a c r y l a t e ) (PMMA) as e t h y l e n e p a r t i a l p r e s s u r e ( p ) i s i n c r e a s e d i n t h e p r e s e n c e o f an e s s e n t i a l l y c o n s t a n t C 0 p a r t i a l p r e s s u r e o f p - 1.53±0.04 atm ( 3 2 ) . The t e n d e n c y o f t h e C 0 s o r p t i o n shown i n F i g u r e 7 t o d e c r e a s e m o n o t o n i c a l l y w i t h e t h y l e n e pressure provides i m p r e s s i v e support f o r the " c o m p e t i t i o n " c o n c e p t on w h i c h Eq ( 8 ) and Eq ( 9 ) a r e b a s e d . P e r m e a t i o n d a t a a r e not a v a i l a b l e f o r t h i s system t o determine i f c h a n g e s i n t h e v a l u e o f Dp and D^ o c c u r i n t h e m i x e d p e n e t r a n t s i t u a t i o n ; h o w e v e r , f o r t h e c a s e o f t h e r e l a t i v e l y non i n t e r a c t i n g e t h y l e n e a t t h e p r e s s u r e s s t u d i e d , any s u c h e f f e c t s a r e e x p e c t e d t o be m i n o r . The d a t a shown i n F i g u r e 8 i l l u s t r a t e t h e r e d u c t i o n i n p e r m e a b i l i t y o f p o l y c a r b o n a t e t o C 0 c a u s e d by c o m p e t i t i o n b e t w e e n i s o p e n t a n e and C 0 f o r L a n g m u i r s o r p t i o n s i t e s ( 3 3 ) . The f l u x d e p r e s s i o n shown i n F i g u r e 8 was f o u n d t o be r e v e r s i b l e , w i t h the p e r m e a b i l i t y r e t u r n i n g t o the pure C 0 l e v e l a f t e r s u f f i c i e n t e v a c u a t i o n of the i s o p e n t a n e - c o n t a m i n a t e d membrane. E v e n a t t h e l o w i s o p e n t a n e p a r t i a l p r e s s u r e c o n s i d e r e d (117 mm Hg) t h e t e n d e n c y i s c l e a r f o r t h e C 0 p e r m e a b i l i t y t o be d e p r e s s e d f r o m i t s p u r e component l e v e l toward the the l i m i t i n g v a l u e ( P = DA DA^ c o r r e s p o n d i n g t o complete e x c l u s i o n of C 0 from the Langmuir environment. F u r t h e r i n c r e a s e s i n the isopentane p a r t i a l pressure i n the feed should e v e n t u a l l y complete the d e p r e s s i o n of C 0 p e r m e a b i l i t y t o i t s l i m i t i n g v a l u e o f 4.57 B a r r e r s u n l e s s p l a s t i c i z i n g e f f e c t s s e t i n at the h i g h e r isopentane l e v e l s . T h i s s u g g e s t s , f o r example, t h a t the e f f e c t i v e C 0 p e r m e a b i l i t y a t a C 0 p a r t i a l p r e s s u r e of 2 atm c o u l d be r e d u c e d by as much as 36% due t o s m a l l amounts o f such hydrocarbons i n the f e e d stream. 2

B

2

A

2

2

2

2

2

k

D

A

2

2

2

2

The l i n e s drawn t h r o u g h t h e d a t a i n F i g u r e 8 were c a l c u l a t e d f r o m Eq ( 7 ) and Eq ( 9 ) f o r t h e p u r e and m i x e d p e n e t r a n t f e e d s i t u a t i o n s , r e s p e c t i v e l y , u s i n g t h e same C 0 m o d e l p a r a m e t e r s i n b o t h c a s e s . The a f f i n i t y c o n s t a n t of


2

CHERN ET A L .

Dual-Mode

Sorption

and

Transport

Models

63


3.

F i g u r e 8. P e r m e a b i l i t y o f L e x a n p o l y c a r b o n a t e t o c a r b o n d i o x i d e a t 35 °C a s a f u n c t i o n o f C 0 p a r t i a l p r e s s u r e . Q o . i n t h e p r e s e n c e o f 117 mm. Hg o f i s o p e n t a n e i n t h e f e e d ; # , p u r e C 0 Γ] 2

2


64


SEPARATIONS

1

i s o p e n t a n e was e s t i m a t e d f o r u s e i n Eq ( 9 ) t o be 13.8 a t m " (33). The e x c e l l e n t f i t o f t h e d a t a s u g g e s t s t h a t Dn a n d D f o r C 0 i n t h e m i x e d f e e d c a s e a r e n o t a f f e c t e d m e a s u r a b l y by t h e p r e s e n c e o f t h e r e l a t i v e l y n o n - i n t e r a c t i n g i s o p e n t a n e . The s o r p t i o n and t r a n s p o r t parameters f o r C 0 i n t h e p o l y c a r b o n a t e s a m p l e u s e d i n t h e above s t u d y a r e r e p o r t e d i n T a b l e 2. R

2

2


Conclusions The c o n f l u e n c e o f c u r r e n t v i e w s o f t h e g l a s s y s t a t e w i t h r e c e n t d a t a f o r s o r p t i o n and t r a n s p o r t o f m i x e d p e n e t r a n t s i n g l a s s y polymers p r o v i d e s a complementary framework u s e f u l i n both areas. S p e c i f i c a l l y , t h e concept o f " u n r e l a x e d volume" i n v o k e d t o e x p l a i n c o n c a v i t y i n t h e s o r p t i o n i s o t h e r m may e v e n t u a l l y be v a l u a b l e i n r a t i o n a l i z i n g d i f f e r e n t c r e e p , i m p a c t s t r e n g t h and p h y s i c a l a g i n g b e h a v i o r o f g l a s s y p o l y m e r s . The d u a l mode s o r p t i o n and t r a n s p o r t model p r o v i d e s a v a l u a b l e b a s i s f o r b o t h q u a l i t a t i v e and q u a n t i t a t i v e a n a l y s i s o f g a s s e p a r a t o r o p e r a t i o n i n t h e p r e p l a s t i c i z i n g regime. Although p l a s t i c i z i n g e f f e c t s a r e l i k e l y t o be o f c o n s i d e r a b l e i m p o r t a n c e a t h i g h p r e s s u r e s i n gas s e p a r a t o r s w i t h t h e p r e s e n t g e n e r a t i o n o f membrane m a t e r i a l s , one m i g h t a n t i c i p a t e t h a t t h e n e x t g e n e r a t i o n o f m a t e r i a l s b a s e d on v e r y h i g h T , h i g h l y r i g i d backbone polymers w i l l m i n i m i z e s e n s i t i v i t y t o such d e l e t e r i o u s e f f e c t s . I f s u c h s t i f f - c h a i n m a t e r i a l s c a n be f o r m u l a t e d i n t o u s e f u l membranes, phenomena a s s o c i a t e d w i t h d u a l mode s o r p t i o n and t r a n s p o r t w i l l be o f p r i m a r y i m p o r t a n c e o v e r t h e f u l l r a n g e of o p e r a t i n g p r e s s u r e s , s i n c e the t r a n s i t i o n t o p l a s t i c i z i n g b e h a v i o r shown i n F i g u r e 1 w o u l d p r e s u m a b l y r e q u i r e v e r y h i g h sorbed concentration l e v e l s . Such p l a s t i c i z a t i o n - r e s i s t a n t membranes m i g h t e v e n m a i n t a i n r e a s o n a b l e s e l e c t i v i t y i n l i q u i d / l i q u i d permeation s e p a r a t i o n s such as p e r v a p o r a t i o n . g

Appendix: Comments C o n c e r n i n g The " M a t r i x " M o d e l F o r S o r p t i o n and D i f f f u s i o n i n G l a s s y P o l y m e r s I n t h e p r e c e d i n g d i s c u s s i o n we have p r e s e n t e d a model w i t h p h y s i c a l l y interprétable p a r a m e t e r s t o e x p l a i n s o r p t i o n d a t a f o r p e n e t r a n t s i n g l a s s y p o l y m e r s . The b a s i s o f t h e m o d e l , t h e concept of " u n r e l a x e d volume" i s a l s o u s e f u l i n u n d e r s t a n d i n g many o t h e r p r o p e r t i e s o f g l a s s y p o l y m e r s s u c h as i m p a c t s t r e n g t h , p h y s i c a l a g i n g and c r e e p , w h i c h a r e r e l a t e d t o t h e non e q u i l i b r i u m n a t u r e of these m a t e r i a l s . Moreover, t h i s model y i e l d s a physically consistent expression f o r the l o c a l 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 , D f £ ( C ) , w h i c h shows c o n c e n t r a t i o n dependence a t s o r b e d c o n c e n t r a t i o n s w e l l b e l o w t h e p o i n t a t w h i c h s u b s t a n t i a l i n t e r a c t i o n o f one p e n e t r a n t w i t h t h e m a t r i x i s g e n e r a l l y expected t o s i g n i f i c a n t l y f a c i l i t a t e the movement o f a n o t h e r p e n e t r a n t . e


3.

CHERN ET A L .

Dual-Mode

Sorption

and

Transport

Models

Table 2 S o r p t i o n and t r a n s p o r t p a r a m e t e r s o f C 0 i n L e x a n p o l y c a r b o n a t e a t 35°C


2

k

D

b

cc(STP) cc-atm

atm"

1

cc(STP) cc

0.6751

0.2563

17.61

polymer η

cm^/sec

5.1484xl0"

8

2 cm^/sec

5.8266xl0"

9


65

66


SEPARATIONS

The d u a l mode model assumes t h a t t h e h e t e r o g e n e i t y t h a t c h a r a c t e r i z e s t h e g l a s s y p o l y m e r s t a t e c a n be t r e a t e d i n a p h y s i c a l l y r e a s o n a b l e f a s h i o n u s i n g t h e c o n c e p t o f two b a s i c a l l y d i f f e r e n t e n v i r o n m e n t s i n t h e p o l y m e r . One e n v i r o n m e n t corresponds t o t h a t i n which a penetrant molecule i s sorbed b e t w e e n c h a i n segments t h a t have a c h i e v e d e s s e n t i a l l y t h e i r e q u i l i b r i u m c o n f o r m a t i o n s and i n t e r c h a i n d i s t a n c e s ( p e r h a p s w i t h the a i d of the invading penetrant during the " c o n d i t i o n i n g " step shown i n F i g u r e 3 ) . A s e c o n d e n v i r o n m e n t c o r r e s p o n d s t o t h a t i n which a penetrant molecule f i n d s i t s e l f sorbed i n a r e g i o n of l o c a l i z e d l o w e r d e n s i t y ( p e r h a p s due t o l o c a l c h a i n k i n k s o r o t h e r i m p e d i m e n t s t o r a p i d r e l a x a t i o n ) . The m o d e l r e q u i r e s , h o w e v e r , t h a t t h e l o c a l c o n c e n t r a t i o n s o f Cn and C i n Eq ( 1 ) a d j u s t themselves c o n s i s t e n t w i t h the requirement that there i s o n l y one c h e m i c a l p o t e n t i a l a p p l i c a b l e t o t h e p e n e t r a n t , and a t s o r p t i o n e q u i l i b r i u m t h i s chemical p o t e n t i a l i s a l s o i d e n t i c a l t o t h a t o f t h e p e n e t r a n t i n t h e e x t e r n a l gas p h a s e . Any s i g n i f i c a n t energetic heterogeneity associated with sorption i n t h e n o n e q u i l i b r i u m gaps r e g i o n s s h o u l d m a n i f e s t i t s e l f as a f a i l u r e i n t h e Langmuir form t o f i t t h e d e v i a t i o n from t h e s i m p l e H e n r y ' s l a w m o d e l , s i n c e t h e L a n g m u i r f o r m assumes a u n i f o r m s i t e a f f i n i t y . Repeated t e s t s of t h e l i n e a r i z e d form of t h e L a n g m u i r c o n t r i b u t i o n t o s o r p t i o n have shown e x c e l l e n t c o n f o r m i t y t o t h e m o d e l . Any s u c h e n e r g e t i c h e t e r o g e n i t y i s , t h e r e f o r e , of n e g l i g i b l e importance w i t h respect t o s o r p t i o n modeling. I n t h e i r r e c e n t p a p e r , R a u c h e r and S e f c i k i n t e r p r e t D r . R. A s s i n k ' s p u l s e d NMR s t u d y o f ammmonia s o r p t i o n i n p o l y styrene (27)and i n d i c a t e t h a t : " S p e c t r o s c o p i c a n a l y s e s of gas molecules w i t h i n polymer matrices a r e c o n s i s t e n t w i t h a l l of t h e gas m o l e c u l e s b e i n g i n a s i n g l e s t a t e " ( 2 8 ) . One i s g i v e n t h e i n d i r e c t and e r r o n e o u s i m p r e s s i o n f r o m R a u c h e r ' s and S e f c i k ' s r e f e r e n c e t o A s s i n k ' s work t h a t t h e A s s i n k s t u d y d i d n o t s u p p o r t t h e c o n c e p t s o f t h e d u a l mode m o d e l . On t h e c o n t r a r y , t h e c o n c l u s i o n s of Dr. A s s i n k w i t h regard t o t h i s i s s u e completely s u p p o r t t h e d u a l mode model p r e m i s e s as shown b e l o w i n a d i r e c t quote of t h e complete c o n c l u s i o n s from t h e A s s i n k paper ( 2 7 ) : "We have been a b l e t o c r i t i c a l l y examine t h e d u a l mode model by p u l s e d NMR r e l a x a t i o n t e c h n i q u e s . The p r e s s u r e dependence o f t h e c o n c e n t r a t i o n o f s o r b e d g a s was c o n s i s t e n t w i t h t h e d u a l mode model w h i l e t h e r e l a x a t i o n data addressed i t s e l f t o t h e v a l i d i t y o f t h e a s s u m p t i o n s made by t h e m o d e l . The a s s u m p t i o n o f r a p i d i n t e r c h a n g e was f o u n d t o be v a l i d f o r t h i s s y s t e m w h i l e t h e a s s u m p t i o n o f an i m m o b i l e a d s o r b e d phase c o u l d introduce a small error i n the a n a l y s i s . I t s h o u l d be p o s s i b l e t o r e d u c e t h i s e r r o r by more e x a c t measurements o f t h e c o n c e n t r a t i o n o f s o r b e d gas a s c l a s s i c a l p r e s s u r e e x p e r i m e n t s c o u l d


H



3.

CHERN

ET A L .

Dual-Mode

Sorption

and

Transport

67

Models

provide. We have been a b l e t o d e m o n s t r a t e t h e b a s i c v a l i d i t y o f t h e a s s u m p t i o n on w h i c h t h e d u a l mode m o d e l i s b a s e d and we have shown t h e u s e f u l n e s s of NMR r e l a x a t i o n t e c h n i q u e s i n t h e s t u d y of t h i s model." The o n l y " s m a l l e r r o r " , s u g g e s t e d i n A s s i n k ' s s t a t e m e n t c o n c e r n i n g t h e d u a l mode m o d e l ' s a s s u m p t i o n s , d e a l s w i t h t h e e a r l i e r a p p r o x i m a t i o n by V i e t h and S l a d e k ( 3 4 ) t h a t % was e q u a l to zero. The work of A s s i n k was p e r f o r m e d p r i o r t o t h e f o r m u l a t i o n of the d u a l m o b i l i t y model which e l i m i n a t e s t h i s a p p r o x i m a t i o n and a c c o m o d a t e s v a l u e s of DJJ > 0 ( 1 7 , 1 8 ) . I n t h e p r e s e n t symposium, D r . R a u c h e r a r g u e d t h a t t h e o b s e r v e d c u r v a t u r e i n gas s o r p t i o n i s o t h e r m s does n o t a r i s e f r o m a s i t e s a t u r a t i o n mechanism s u c h as we have d e s c r i b e d i n t h e p r e c e d i n g d i s c u s s i o n . The f o r m o f t h e s o r p t i o n i s o t h e r m and l o c a l c o n c e n t r a t i o n d e p e n d e n t d i f f u s i o n c o e f f i c i e n t p r o p o s e d by R a u c h e r and S e f c i k a r e g i v e n b e l o w : C =

σ

οΡ

1 +

D

e f f

(A-l) aC

( C ) = DQ[1

(A-2)

+ SC]

where D was i d e n t i f i e d as t h e z e r o c o n c e n t r a t i o n l i m i t of t h e d i f f u s i o n c o e f f i c i e n t and 3 was i d e n t i f i e d as a p l a s t i c i z a t i o n - r e l a t e d parameter which i n d i c a t e s the r e l a t i v e s e n s i t i v i t y of the l o c a l d i f f u s i o n c o e f f i c i e n t , D f f ( C ) t o the p r e s e n c e of o t h e r p e n e t r a n t s i n t h e v i c i n i t y . W h i l e no p h y s i c a l i n t e r p e t a t i o n was g i v e n t o t h e s o r p t i o n p a r a m e t e r s , t h e o p a r a m e t e r i s c l e a r l y e q u a l t o t h e l i m i t i n g s l o p e of t h e concentration versus pressure isotherm. D u r i n g a q u e s t i o n and a n s w e r p e r i o d f o l l o w i n g h i s p r e s e n t a t i o n , Dr. R a u c h e r e x p l i c i t l y r u l e d o u t an i n t e r p r e t a t i o n of t h e p a r a m e t e r α i n t e r m s o f a s i t e s a t u r a t i o n mechanism s i m i l a r t o t h e L a n g m u i r m o d e l . Drs. R a u c h e r and S e f c i k have s t a t e d : "The apparent c o n c e n t r a t i o n dependence of t h e s o r p t i o n c a p a c i t y i n t h e m a t r i x m o d e l r e s u l t s f r o m a d e c r e a s e i n a H e n r y ' s l a w - l i k e c o n s t a n t as the polymer a d j u s t s to the s o r b a t e " (28). The e x t e n t of t h e i n t e r p r e t a t i o n o f α o f f e r e d by t h e s e a u t h o r s i s t h a t : "a i s a p a r a m e t e r i n d i c a t i n g t h e m a g n i t u d e o f t h e change i n s o l u b i l i t y a r i s i n g f r o m changes i n the gas-polymer m a t r i x " . Specifically, what i s m i s s i n g f r o m t h e i r d i s c u s s i o n i s a s t a t e m e n t as t o what t h e n a t u r e of the h y p o t h e t i c a l p o l y m e r / p e n e t r a n t interactions are t h a t cause the c o n c a v i t y i n the i s o t h e r m . In other words, what change i s i n d u c e d i n t h e n a t u r e o f t h e p o l y m e r i n t h e p r e s e n c e of t h e p e n e t r a n t ? The "model" c o m p r i s e d o f Eq ( A - l ) and ( A - 2 ) has a s t r i c t l y e m p i r i c a l b a s i s . I t seems i n c u m b e n t u p o n t h e s e a u t h o r s t o e x p l a i n how t h e change i n t h e p o l y m e r m a t r i x makes i t more d i f f i c u l t t o i n s e r t a n o t h e r p e n e t r a n t Q

e

Q


68

INDUSTRIAL

GAS

SEPARATIONS

[Eq ( A - l ) ] , b u t a t t h e same t i m e makes i t e a s i e r f o r t h e p e n e t r a n t t o move t h r o u g h t h e m a t r i x [Eq (Α-2)]· The d u a l mode model o f f e r s 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 t h a t c o n c a v i t y i n t h e i s o t h e r m a r i s e s f r o m a s i t e s a t u r a t i o n mechanism r e l a t e d t o f i l l i n g o f u n r e l a x e d g a p s — a phenomenon w h i c h i s n o t o n l y e a s i l y v i s u a l i z e d a n d u n d e r s t o o d , b u t a l s o h a s been t e s t e d e x p l i c i t l y w i t h i n d e p e n d e n t l y o b t a i n e d d i l a t o m e t r i c data (See F i g u r e 5 ) . In an attempt t o j u s t i f y t h e assumption of p l a s t i c i z a t i o n p u t f o r t h i n t h e i r i n t e r p r e t a t i o n o f 3 i n Eq ( A - 2 ) , R a u c h e r a n d S e f c i k compare t r a n s p o r t d a t a and C NMR d a t a f o r t h e C 0 2 / V C s y s t e m . T h i s c o m p a r i s o n h a s s e v e r a l q u e s t i o n a b l e a s p e c t s . To r e l a t e l o c a l m o l e c u l a r c h a i n motions t o the d i f f u s i o n c o e f f i c i e n t o f a p e n e t r a n t , one s h o u l d u s e t h e s o - c a l l e d l o c a l effective coefficient, D ( C ) , s u c h as shown i n F i g u r e 5 r a t h e r t h a n a n a v e r a g e o r " 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 as was e m p l o y e d by t h e s e a u t h o r s . f f ^ ^ describes the e f f e c t s of the l o c a l sorbed c o n c e n t r a t i o n on t h e a b i l i t y of t h e average p e n e t r a n t t o respond t o a c o n c e n t r a t i o n o r chemical p o t e n t i a l gradient i nthat region. R a u c h e r and S e f c i k , on t h e o t h e r h a n d , b a s e t h e i r c o m p a r i s o n s b e t w e e n NMR d a t a and t r a n s p o r t d a t a o n t h e s o - c a l l e d "apparent" d i f f u s i o n c o e f f i c i e n t d e f i n e d by: 1 3

p


e f f

D

C

e

D

2

- * /6Θ

a

(A-3)

where Θ i s t h e o b s e r v e d t i m e l a g and I i s t h e membrane thickness. D d o e s n o t have a s i m p l e m e a n i n g e q u i v a l e n t t o a t r u e m o l e c u l a r m o b i l i t y u n l e s s both t h e time l a g and p e r m e a b i l i t y a r e independent of upstream p r e s s u r e . Since t h i s s i t u a t i o n i s n o t t y p i c a l l y o b s e r v e d i n g l a s s y p o l y m e r s , one must use D f f ( C ) f o r comparison w i t h complementary t e c h n i q u e s t h a t probe molecular motion. The s e r i o u s n e s s o f t h i s o v e r s i g h t i s a p p a r e n t i n S e f c i k and S c h a e f e r ' s a n a l y s i s o f T o i ' s t r a n s p o r t d a t a ( 2 4 ) i n terms o f t h e i r NMR r e s u l t s ( 2 8 ) . The v a l u e o f t h e s o - c a l l e d " 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 c a l c u l a t e d from T o i ' s time l a g data i n c r e a s e s by ~ 2 5 % f o r a n u p s t r e a m p r e s s u r e r a n g e between 100 mm Hg and 500 mm Hg. On t h e o t h e r h a n d , t h e v a l u e o f D (C) c a l c u l a t e d f r o m T o i ' s d a t a changes by 8 6 % o v e r t h e c o n c e n t r a t i o n r a n g e f r o m 100 t o 500 mm Hg. The d i f f e r e n c e i n t h e two above c o e f f i c i e n t s a r i s e s from t h e f a c t t h a t D i s an average of v a l u e s c o r r e s p o n d i n g t o a range of c o n c e n t r a t i o n s from t h e u p s t r e a m v a l u e t o t h e e s s e n t i a l l y z e r o c o n c e n t r a t i o n downstream v a l u e i n a t i m e l a g measurement. ff^ ^» band, has a w e l l - d e f i n e d p o i n t v a l u e a t e a c h s p e c i f i e d c o n c e n t r a t i o n and i s t y p i c a l l y e v a l u a t e d ( i n d e p e n d e n t o f any s p e c i f i c model o t h e r t h a n F i c k ' s l a w ) by d i f f e r e n t a t i o n o f s o l u b i l i t y and permeability data (22). A s e c o n d i s s u e c l o u d i n g t h e NMR i n t e r p r e t a t i o n o f T o i ' s a

e

e f f

a

D

C

o

n

t

n

e

o

t

n

e

r

e


3.

CHERN

ET

Dual-Mode

AL.

Sorption

and

Transport

69

Models

40°C t r a n s p o r t d a t a l i e s i n t h e f a c t t h a t t h e c i t e d NMR data were c o l l e c t e d a t 26°C r a t h e r t h a n 40°C. I f one n e g l e c t s t h i s f a c t , S e f c i k and S c h a e f e r s u g g e s t t h a t t h e 5.7% i n c r e a s e i n t h e average r o t a t i n g frame r e l a x a t i o n r a t e , over the p r e s s u r e r a n g e (100 mm Hg t o 500 mm Hg) c o r r e s p o n d i n g t o T o i ' s measurements v a l i d a t e t h e i r c l a i m s c o n c e r n i n g p l a s t i c i z a t i o n . T h i s s u g g e s t i o n n e g l e c t s t h e f a c t t h a t t h e 25% i n c r e a s e i n D d o e s n o t m a t c h t h e 5.7% i n c r e a s e i n < R i ( C ) > . M o r e o v e r , r e c a l l t h a t D i s a c t u a l l y n o t e v e n t h e c o r r e c t c o e f f i c i e n t t o compare w i t h < R ( C ) > . The o b s e r v e d 86% i n c r e a s e i n D ( C ) i s i n poor a g r e e m e n t w i t h t h e 5.7% i n c r e a s e i n t h e NMR p a r a m e t e r o v e r t h e r a n g e o f p r e s s u r e s where b o t h t y p e s of d a t a a r e a v a i l a b l e . The c o n s i d e r a b l e d i s c r e p a n c y b e t w e e n t h e changes i n t h e two p a r a m e t e r s D f f ( ^ and < R i ( C ) > i s not i m p r e s s i v e p r o o f f o r p l a s t i c i a t i o n a t such low c o n c e n t r a t i o n s . This observation i s e s p e c i a l l y t r u e s i n c e p a r t o f t h e o b s e r v e d 5.7% i n c r e a s e i n < R l p ( C ) > may be a c c o u n t e d f o r by s p i n - s p i n e f f e c t s . The v a l u e o f < R ( C ) > r e q u i r e s c a u t i o u s i n t e r p r e t a t i o n i n t e r m s of b u l k p o l y m e r p r o p e r t i e s s u c h as 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 ; e s p e c i a l l y i n l i g h t of t h e p r e c e d i n g d i s c u s s i o n c o n c e r n i n g T o i ' s data. I t i s i m p o r t a n t t o remember t h a t e v e n a t 500 mm Hg, t h e r e i s o n l y one 0 0 m o l e c u l e f o r e v e r y 1300 PVC r e p e a t u n i t s i n t h e a b o v e s i t u a t i o n and y e t an i n c r e a s e o f 86% o c c u r s i n t h e v a l u e o f D f j f ( C ) b e t w e e n 100 mm Hg and 500 mm Hg. These c o n d i t i o n s a r e e x t r a o r d i n a r i l y d i l u t e , and t h e o b s e r v e d c o n c e n t r a t i o n d e p e n d e n c e of D f f ( C ) i s p e r f e c t l y c o n s i s t e n t w i t h a s i t e s a t u r a t i o n t y p e b e h a v i o r i n h e r e n t i n t h e d u a l mode m o d e l . We h a v e a l w a y s a c k n o w l e d g e d t h a t c o n c e n t r a t i o n d e p e n d e n c e of Dp and Dfl may become i m p o r t a n t f a c t o r s a t e l e v a t e d p r e s s u r e s f o r p l a s t i c i z a t i o n prone polymers. Such e f f e c t s , i f p r e s e n t have b e e n o f s e c o n d o r d e r i m p o r t a n c e i n o u r work w h i c h i s g e n e r a l l y p e r f o r m e d on " c o n d i t i o n e d " p o l y m e r s a t CO2 p r e s s u r e s no g r e a t e r t h a n 300 p s i . E v e n t u a l 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 and i s , h o w e v e r , n o t r e l a t e d t o t h e v i e w s of R a u c h e r and S e f c i k s u m m a r i z e d by Eq ( A - l ) and Eq ( A - 2 ) . These e q u a t i o n s r e q u i r e t h a t the s e e m i n g l y m u t u a l l y e x c l u s i v e p r o c e s s e s of a n t i p l a s t i c i z a t i o n (Eq A - l ) and p l a s t i c i z a t i o n (Eq A-2) occur simultaneously. p

a

p

a

l p

e f f


c

e

p

l p

2

e

e

n

F i n a l l y , i t was s u g g e s t e d by R a u c h e r and S e f c i k t h a t t h e e q u a t i o n s d e r i v e d f r o m t h e i r a n a l y s i s were e s s e n t i a l l y as e f f e c t i v e as t h e d u a l mode model f o r d e s c r i b i n g e x i s t i n g d a t a . T h i s s t a t e m e n t r e q u i r e s some s t r o n g q u a l i f i c a t i o n , s i n c e no l i n e a r e q u a t i o n can d e s c r i b e t h e i n f l e c t e d f o r m of t h e D f f ( C ) p l o t shown i n F i g u r e 6 w h i c h was e v a l u a t e d by g r a p h i c a l d i f f e r e n t i a t i o n of t h e p e r m e a b i l i t y and s o l u b i l i t y d a t a w i t h o u t r e f e r e n c e t o any p a r t i c u l a r m o d e l o t h e r t h a n F i c k ' s l a w ( 2 1 , 2 2 ) . The c u r v e t h r o u g h t h e d a t a , n e v e r t h e l e s s , c o r r e s p o n d s t o t h e d u a l mode model and p r o v i d e s a v e r y good d e s c r i p t i o n of t h e data. The f o r m of t h e s e d a t a i s n o t a t a l l c o n s i s t e n t w i t h e


70


SEPARATIONS

p l a s t i c i z a t i o n . T h i s same argument w i l l a p p l y f o r e s s e n t i a l l y a l l o f t h e c a s e s i n w h i c h t h e d u a l mode m o d e l h a s b e e n shown t o f i t the form of the l o c a l d i f f u s i v i t y . The d a t a p o i n t s c a n a t b e s t be made t o " s n a k e " a r o u n d t h e l i n e a r f i t o f t h e e x p r e s s i o n i n Eq (A-2) a n d t h e l i n e a r m o d e l becomes s e r i o u s l y i n e r r o r a s D f f b e g i n s t o a s y m p t o t e . L i m i t e d d a t a f o r two g a s / p o l y m e r s y s t e m s were shown by D r . R a u c h e r w h i c h i n d i c a t e d t h a t t h e d e s c r i p t i o n s o f p e r m e a b i l i t i e s a n d t i m e l a g s u s i n g t h e d u a l mode and m a t r i x m o d e l s i s n o t v e r y d i f f e r e n t f o r t h e s y s t e m s c h e c k e d . I f t h i s i s t r u e , i t i s s u r p r i s i n g s i n c e t h e d i f f e r e n t forms of D £ f ( C ) w o u l d be e x p e c t e d t o g i v e r i s e t o somewhat d i f f e r e n t t i m e l a g r e s u l t s when i n t e g r a t e d a c c o r d i n g t o F r i s c h ' s method (35). A s i m i l a r conclusion pertains to the permeablity, since e v e n i f t h e s o l u b i l i t y d a t a were p e r f e c t l y d e s c r i b e d , t h e m i s f i t of f f ( C ) s u g g e s t s t h a t t h e r e s h o u l d be a m i s f i t i n p e r m e a b i l i t y as w e l l . I t i s p o s s i b l e that a m i s f i t of D f f ( C ) v e r s u s C and o f C v e r s u s ρ c o u l d o f f s e t e a c h o t h e r . I n one o f t h e i r c o m p a r i s o n s b e t w e e n t h e m a t r i x model and t h e d u a l mode m o d e l , a somewhat m i s l e a d i n g p r e s e n t a t i o n o f d a t a i s u n i n t e n t i o n a l l y o f f e r e d i n F i g u r e 4 o f t h e R a u c h e r and S e f c i k paper: " M a t r i x M o d e l o f Gas 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 " ( 2 8 ) . What s h o u l d be compared i n t h e l e f t hand s i d e o f t h i s p l o t i s t h e d u a l mode m o d e l w i t h r e f i t t e d p a r a m e t e r s o v e r t h e same p r e s s u r e r a n g e as t h e " M a t r i x m o d e l " p a r a m e t e r s were r e f i t t e d o v e r . C l e a r l y , a d e t a i l e d s t a t i s t i c a l comparison o f t h e p e r m e a b i l i t y and t i m e l a g p r e d i c t i o n s a r i s i n g f r o m Eq ( A - l ) and Eq (A-2) must be made w i t h t h e l a r g e body o f e x p e r i m e n t a l d a t a a v a i l a b l e f o r g l a s s y systems b e f o r e a c o n c l u s i o n c a n be r e a c h e d r e g a r d i n g t h e e f f i c a c y o f t h e " m a t r i x " model f o r phenomenologically d e s c r i b i n g such d a t a i n g e n e r a l . On a more b a s i c l e v e l , s i n c e t h e m a t r i x m o d e l i m p l i c i t l y r e q u i r e s a somewhat i n c o n s i s t e n t i n t e r p r e t a t i o n f o r t h e v a r i o u s m o d e l p a r a m e t e r s i n Eq ( A - l ) and Eq ( A - 2 ) , i t becomes p r i m a r i l y a n e m p i r i c a l means o f r e p r o d u c i n g t h e o b s e r v e d p u r e component d a t a w i t h no f u n d a m e n t a l b a s i s f o r g e n e r a l i z a t i o n t o m i x t u r e s . One c o u l d , o f c o u r s e e n v i s i o n s e v e r a l e x t e n s i o n s b a s e d on a d d i t i o n a l α t e r m s i n t h e d e n o m i n a t o r o f Eq ( A - l ) and a d d i t i o n a l 3 t e r m s i n Eq ( A - 2 ) . Such an a p p r o a c h t o m i x t u r e permeation a n a l y s e s w o u l d be c o m p l e t e l y e m p i r i c a l a n d m i m i c t h e g e n e r a l i z a t i o n o f Eq ( 2 ) and Eq ( 7 ) ; h o w e v e r , w i t h o u t any physical justification. The g e n e r a l i z a t i o n s o f Eq ( 2 ) and Eq ( 7 ) were n a t u r a l o u t g r o w t h s o f t h e f u n d a m e n t a l p h y s i c a l b a s i s o f the Langmuir i s o t h e r m . The f a c t t h a t t h e m i x t u r e d a t a a r e s o c o n s i s t e n t w i t h Eq ( 7 ) and Eq ( 9 ) p r o v i d e s s t r o n g s u p p o r t f o r t h e p h y s i c a l b a s i s o f t h e d u a l mode m o d e l . An i m p o r t a n t v a l u e o f a p e r m e a t i o n model i s n o t s i m p l y i t s a b i l i t y t o c o r r e l a t e experimental d a t a , but r a t h e r t o provide a framework f o r understanding the p r i n c i p a l f a c t o r s c o n t r o l l i n g membrane p e r f o r m a n c e . The d u a l mode model i s d e r i v e d f r o m e


e

n

e

e


C H E R N ET A L .

3.

Dual-Mode

Sorption

and

Transport

71

Models


c o n s i s t e n t , p h y s i c a l d e s c r i p t i o n s o f t h e g l a s s y s t a t e and y i e l d s p a r a m e t e r s w h i c h make unambiguous m o l e c u l a r s c a l e s t a t e m e n t s r e g a r d i n g t h e g e n e r a l n a t u r e o f g l a s s y , amorphous m a t e r i a l s . I n summary, t h e two e x p r e s s i o n s o f f e r e d i n Eq ( A - l ) and Eq ( A - 2 ) p r o v i d e e m p i r i c a l f o r m s f o r c o r r e l a t i n g t h e g e n e r a l t r e n d s i n p u r e component s o r p t i o n and t r a n s p o r t d a t a . These e x p r e s s i o n s a r e not o f f e r e d i n terms of i n t e r n a l l y c o n s i s t e n t p h y s i c a l a r g u m e n t s and a p p e a r t o o f f e r no_ f u n d a m e n t a l b a s i s f o r understanding general glassy state behavior. F u r t h e r m o r e , no f u n d a m e n t a l approach i s a p p a r e n t f o r the t r e a t m e n t of the c r i t i c a l l y i m p o r t a n t mixed p e n e t r a n t problem u s i n g the model, s i n c e t h e v a r i o u s parameters i n t h e model l a c k t h e w e l l - d e f i n e d s i g n i f i c a n c e p r o v i d e d by t h e d u a l mode m o d e l . Nomenclature b^ D DJI F k Κ

1

A f f i n i t y c o n s t a n t o f component i f o r t h e p o l y m e r (atm"" ) D i f f u s i o n c o e f f i c i e n t of the Henry's l a w s p e c i e s (cm^/sec) D i f f u s i o n c o e f f i c i e n t of the Langmuir s p e c i e s (cm^/sec) R a t i o of L a n g m u i r and H e n r y ' s l a w d i f f u s i o n c o e f f i c i e n t s H e n r y ' s l a w c o n s t a n t (cm-* gas ( S T P ) / c m polymer-atm) C ^ b / k where i s t h e c a p a c i t y o f L a n g m u i r mode (cm-* g s (STP)/cm polymer) P£ U p s t r e a m d r i v i n g p r e s s u r e f o r p e n e t r a n t i ( a t m ) ? P e r m e a b i l i t y of i ( c m gas(STP)-cm/cm -sec-cmHg) Vg S p e c i f i c volume of t h e g l a s s y p o l y m e r ( c m / g ) Vj£ S p e c i f i c volume of t h e d e n s i f i e d g l a s s y p o l y m e r ( c m / g ) D

3

D

n

a

3

3

2

±

3

3

Acknowledgments The a u t h o r s g r a t e f u l l y a c k n o w l e d g e s u p p o r t o f t h i s work u n d e r NSF G r a n t No. CPE-79-18200 and ARO C o n t r a c t No. DAAG29-81-K-0039. A l s o t h e a s s i s t a n c e o f M r s . L e s l i e E d g e r t o n i n t y p i n g the manuscript i s acknowledged.

Literature Cited 1. 2. 3. 4. 5. 6.

Stannett, V. T.; Koros, W. J.; Paul, D. R.; Lonsdale, H. K. and Baker, R. W.; Adv. in Polymer S c i . , 1979, 32, 69. Barrer, R. M. and Barrie, J . Α.; J . Polym. S c i . , 1957, 23, 331. Pye, D. G.; Hoehn, Η. H. and Panar, M.; J . Appl. Polym. S c i . , 1976, 20, 287. Koros, W. J . and Paul, D. R.; J . Polym. Sci.-Polym. Phys. Ed., 1978, 16, 1947. Wonders, A. G. and Paul, D. R.; J. Membr. S c i . , 1979, 5, 63. Fechter, J . M.; Hopfenberg, Η. B. and Koros, W. J.; Polym. Engr. and S c i . , 1981, 21, 925.


ι** 7. 8. 9. 10. 11.


12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

INDUSTRIAL GAS SEPARATIONS Koros, W. J . and Paul, D. R.; J . Polym. S c i . , Polym. Phys. Ed., 1981, 19, 1655. Horiuti, J.; Sci. Papers of the Inst. of Phy. and Chem. Res. (Tokyo); 1931, 17, 126. Chan, A. H. and Paul, D. R., Polym. Engr. and S c i . , 1980, 20, 87. Huvard, G. S.; Stannett, V. T.; Koros, W. J. and Hopfenberg, Η. Β., J . Membr. S c i . , 1980, 6, 185. Chen, S. Η., M.S. Thesis, North Carolina State University, 1982. Lewis, O. G.; "Physical Constants of Linear Homopolymers," Springer Verlag, New York (1968). Kolb, H. J . and Izard, E.; J . Appl. Phys., 1949, 20, 564. Matuska, S. and Ishida, Y.; J . Polym. S c i . , Pt. C., 1960, 14, 247. Mercier, J . P.; Aklonis, J . J.; L i t t , M.; and Tobolsky, A. V.; J . Appl. Polym. S c i . , 1965, 9, 447. Sandler, Stanley I . ; "Chemical and Engineering Thermodynamics," J . Wiley and Sons, New York, 1977, pg. 435. Petropoulos, J . H.; J . Polym. S c i . , Pt. 2-A, 1970, 8, 1797. Paul, D. R. and Koros, W. J.; J . Polym. Sci.-Polym. Phys. Ed., 1976, 14, 675. Koros, W. J.; Chan, A. H. and Paul, D. R.; J. Membr. S c i . , 1977, 2 165. Barrie, J . Α.; Williams, M. L. and Munday, K.; Polym. Engr. and S c i . , 1980, 20, 20. Koros, W. J., Ph.D. Dissertation; "Sorption and Transport of Gases in Glassy Polymers," The University of Texas (Austin), 1977. Koros, W. J.; Paul, D. R. and Rocha, Α. Α.; J . Polym. S c i . , Polym. Phys. Ed., 1976, 14, 687. Koros, W. J. and Paul, D. R.; J . Poly. S c i . , Polym. Phys. Ed., 1978, 16, 2171. Toi, Κ.; Polym. Engr. and S c i . , 1980, 20, 30. Tikhomorov, B. P.; Hopfenberg, Η. Β.; Stannett, V. Τ. and Williams, J . L . ; Macromol. Chem., 1968, 118, 177. Stern, S. A. and Saxena, V.; J. Membr. S c i . , 1980, 7, 47. Assink, R. Α.; J . Polym. Sci.-Polym. Phys. Ed., 1975, 13, 1665. Raucher, D. and Sefcik, M.; Paper presented in this symposium. Koros, W. J.; J . Polym. Sci.-Polym. Phys. Ed., 1980, 18, 981. Koros, W. J.; Chern, R. T.; Stannett, V. T. and Hopfenberg, H. B.; J . Polym. Sci.-Polym. Phys. Ed., 1981, 19, 1513. Robeson, L. M.; Polym. Engr. and S c i . , 1969, 9, 277.


3. CHERN ET AL. 32.

33.

Dual-Mode Sorption and Transport Models

Sanders, E. S.; Koros, W. J.; Hopfenberg, H. B. and Stannett, V. T.; "Mixed Gas Sorption in Glassy Polymers: Equipment Design and Considerations and Preliminary Results," Submitted for Publication, J . Polym. S c i . - Phys. Ed. Chern, R. T.; Koros, W. J.; Hopfenberg, H. B. and Stannett, V. T.; "The Effects of Low Partial Pressures of Isopentane on the Permeability of Polycarbonate to CO ," accepted by J. Polym. S c i . - Phys. Ed. Vieth, W. R. and Sladek, K. J.; J . Coll. S c i . , 1965, 20, 1014. Frisch, H. L . ; J . Phys. Chern., 1957, 61, 93. Smith, G. N . ; M.S. Thesis, North Carolina State University, 1980. 2

34.


35. 36.

73

RECEIVED December

27, 1982


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