Transport Properties of Polymers - American Chemical Society

12

Transport Properties o f Polymers D. R. PAUL

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 17, 2017 | http://pubs.acs.org Publication Date: September 25, 1985 | doi: 10.1021/bk-1985-0285.ch012

Department of Chemical Engineering, University of Texas, Austin, TX 78712

Thermodynamics of Solubility Mathematics of Diffusion Factors Affecting Solubility and Transport Crystallinity, Fillers, and Morphology Temperature and Transitions Penetrant Size Penetrant Concentration-Plasticization Polymer Molecular Structure Relaxation-Controlled Transport Applications of Transport Concepts Barrier Materials Devolatilization Additive Migration Dyeing Membrane Separations Controlled-Release Technology

The transport of small molecules, such as gases, vapors, or liquids, in polymers has been of intense interest to engineers and scientists for many years. Early studies of gas transport in elastomers were motivated by concerns about gas loss or interchange from automobile inner tubes, rubber balloons, cable insulation, and foam rubber (1). The introduction of butyl rubber greatly reduced some of these problems because of its much lower permeability than natural rubber. Subsequent interest focused on the transport in plastics as they became more widely used for packaging applications that required certain barrier characteristics. This interest is illustrated by the more recent strong economic and safety incentives to replace glass with plastics for carbonated beverage bottles (2) and the subsequent search for "high barrier" polymers for these applications. Concerns about migration of additives, solvents, and 0097~6156/85/0285-Ό253$06.75/0 © 1985 American Chemical Society

Tess and Poehlein; Applied Polymer Science ACS Symposium Series; American Chemical Society: Washington, DC, 1985.

254

applied polymer science

residual monomers have introduced additional reasons for interest in this subject. The purpose of this chapter is to outline some of the basic issues and principles that pertain to the transport of small molecules in polymers and to introduce a few examples in which transport properties of polymers may be important or lead to new uses for polymers.


Thermodynamics of Solubility In order to be transported through bulk polymers, small molecules must first dissolve in them. Consequently, the practical indicators of transport rates such as permeability coefficients include a thermodynamic part to characterize solubility. The extent to which small molecules w i l l be sorbed into a polymer at equilibrium depends on the entropy and enthalpy of mixing and the activity of these molecules in the environment with which the polymer is equilibrated. These points are best illustrated by considering a polymer surrounded by a vapor of component 1 at a partial pressure of pp If the saturation vapor pressure of pure 1 is pj*, then the activity in the vapor phase is a^ = pi/p^*. Flory (3) has developed a thermodynamic model for mixing small molecules of molar volume with large polymer molecules of molar volume Ϊ^· This model combines an estimate for the entropy of mixing with a measure of the enthalpy of mixing expressed in terms of an interaction parameter and results in the following expression for phase equilibrium:

in a = In φ + (1 - —)Φ + Χι Φ 1

χ

2

(!)

2

V

2 =

where are volume fractions. For binary systems, + 2 1· Equation 1 is one form of what is now called the Flory-Huggins equation. When the polymer has a relatively large molecular weight, the ratio of molar volumes is small compared to unity and can be neglected. Figure 1 shows the extent of sorption of component 1 into the polymer as a function of its activity for various values of the Flory-Huggins interaction parameter as calculated from Equation 1. Note that the extent of sorption increases as becomes smaller, and for χ^ < 0.5 the polymer would dissolve in liquid 1. For small activities, the extent of sorption is proportional to p^; however, the concentration of sorbed vapor curves upward at higher activities. Ideally, χ^ is a measure of the heat of mixing of the sorbed molecules with the polymer; the less endothermic this process is, the greater the extent of sorption w i l l be. For nonpolar compo nents, the interaction parameter can be estimated by using solubility parameter theory (4)


12.

PAUL

Transport Properties of Polymers

X l

=— (6 6 ) r

2

2

255

(2)


RT where the s o l u b i l i t y parameter for each component, 6^, i s the square root of the cohesive energy density of that component. (See (5) for a t a b u l a t i o n of values.) For vapors at very low a c t i v i t i e s or f o r gases, the l i m i t i n g relationship C = S

(3)

P l

known as Henry's law i s useful for describing the sorption isotherm i n l i q u i d s or rubbery polymers. Here, C i s the concentration of gas or vapor i n e q u i l i b r i u m with the gas phase at p a r t i a l pressure p p and S i s a s o l u b i l i t y c o e f f i c i e n t . S increases as the gas becomes more c o n d e n s i b l e , t h a t i s , has a h i g h e r b o i l i n g p o i n t or c r i t i c a l temperature. The f o l l o w i n g e m p i r i c a l r e l a t i o n has proved useful (6) log S = log S° + m(e/k)

(4)

where e/k i s the Lennard-Jones p o t e n t i a l w e l l depth describing the c o h e s i v e f o r c e s between gas m o l e c u l e s and i s a p p r o x i m a t e l y p r o p o r t i o n a l to the c r i t i c a l temperature (7). The parameter m i s approximately 0.01 K~* (e/k has units of K) and S° ranges from 0.005 to about 0.02 cnr(STP)/cnr atm and depends on the polymer (6). Gas s o r p t i o n i n p o l y m e r s b e l o w t h e i r g l a s s t r a n s i t i o n temperature (T ) i s more complex as shown i n F i g u r e 2 f o r C O 2 i n p o l y c a r b o n a t e %T = 145 ° C ) . Here, the i s o t h e r m i s not s t r a i g h t such as that suggested by Henry's law (Equation 3) and i s curved i n the o p p o s i t e manner to t h a t p r e d i c t e d by the F l o r y - H u g g i n s theory (see F i g u r e 1). Isotherms such as those i n F i g u r e 2 are d e s c r i b e d by a model envisioning dual sorption mechanisms (8, 9): C' C

= k

D

H

b p

p +

(5) 1 + b p

The f i r s t term i s s i m p l y Henry's law (kn may be equated to S i n E q u a t i o n 3), and the second term i s or the Langmuir form and c h a r a c t e r i z e d by c a p a c i t y , C'JJ, and a f f i n i t y , b, parameters. The l a t t e r mechanism i s b e l i e v e d t o r e s u l t from s o r p t i o n i n t o nonequilibrium regions of free volume e x i s t i n g i n the glassy state, and the s i z e of t h i s terra increases i n proportion to the value of T r e l a t i v e to the o b s e r v a t i o n temperature (6_). Vapors sorbed i n t o glassy polymers may e x h i b i t isotherms that have the dual sorption shape at low a c t i v i t i e s and change to the Flory-Huggins form at high a c t i v i t i e s (10). The s o l u b i l i t y of s o l i d s , f o r example, an a n t i o x i d a n t , i n a polymer are affected by the a d d i t i o n a l contribution of t h e i r free energy of fusion. This effect may be combined with Equation 1 to obtain the f o l l o w i n g p r e d i c t i o n of s o l u b i l i t y , (j)-^, at unit a c t i v i t y of the s o l i d (11):


256

APPLIED POLYMER SCIENCE


T

0

Figure 1.

0.2 0.4 0.6 0.8 Vapor Activity = a, = p,/p,*

1.0

Vapor sorption isotherms i n a polymer as computed from F l o r y - H u g g i n s theory by u s i n g i n t e r a c t i o n parameters shown.

Pressure (otm)

Figure 2.

S o r p t i o n isotherm f o r C0o i n g l a s s y p o l y c a r b o n a t e 35 °C.


at

12.

PAUL

257


AH

- £n

=

v

T

f

(1 - _ ) RT T

1

+ (1 - _ )

+

Z v

(6)

X l

2

where AHf and T are the heat of f u s i o n and m e l t i n g p o i n t of component 1, and i t has been assumed that 4>2 i s approximately 1. Downloaded by UNIV OF CALIFORNIA SAN DIEGO on January 17, 2017 | http://pubs.acs.org Publication Date: September 25, 1985 | doi: 10.1021/bk-1985-0285.ch012

m

Mathematics of Diffusion The r a t e of t r a n s p o r t i s d e s c r i b e d by F i c k ' s f i r s t law (12) which for most purposes can be w r i t t e n as N

. - D^£ 3x

X

(7)

where i s the f l u x of s p e c i e s 1, and D i s the d i f f u s i o n c o e f f i cient which depends on the nature of both components, temperature, and p o s s i b l y c o n c e n t r a t i o n of s p e c i e s 1, C. For d e s c r i p t i o n of unsteady-state problems, i t i s necessary to combine Equation 7 with a d i f f e r e n t i a l mass balance to obtain F i c k ' s second law: i£ =- ! ! i 3t 3x

=

i _ 3x

(

D

^ ) 3x

(

8

)

Many s o l u t i o n s to Equations 7 and 8 w i t h d i f f e r e n t boundary c o n d i t i o n s have been c o m p i l e d (13), and i n f o r m a t i o n about the d i f f u s i o n c o e f f i c i e n t s for polymer systems i s extensive (14). When D i s a c o n s t a n t , E q u a t i o n 8 can be s i m p l i f i e d by f a c t o r i n g t h i s c o e f f i c i e n t outside the d i f f e r e n t i a l operator. The k i n e t i c s of sorption of a penetrant i n t o a polymer f i l m of t h i c k n e s s £ s e r v e s to i l l u s t r a t e problems of n o n s t e a d y - s t a t e diffusion. At t < 0, C = 0 f o r a l l x; whereas, at t > 0, the surfaces at x = 0 and £ assume the value which after enough time i s the v a l u e a c h i e v e d f o r a l l x, t h a t i s , e q u i l i b r i u m . If M denotes the t o t a l amount of permeant t h a t has entered the f i l m at time t , and M^ denotes the amount at e q u i l i b r i u m = A £ C^, then the s o l u t i o n to Equation 8 for these boundary conditions i s t

M

t

Moo

, = 1 -

2

8 2_J n=0

2

-D(2n+l) 7T t/£

2

e 2

(2n+l) 7r

W

2

T h i s e q u a t i o n , t y p i c a l of s o l u t i o n s to F i c k ' s second l a w , i s d i f f i c u l t to manipulate, and c e r t a i n approximate r e s u l t s v a l i d near the beginning or end of t h i s process such as the f o l l o w i n g

*ty

12

M

t for 0 < — < 0.6


(10)

258


M _ = 1 - _ t

_ 2 /o2

8

7T

D

Dr t

M for 0.4 < — < 1 t

A

e

(11)

are h e l p f u l f o r q u i c k c a l c u l a t i o n s . Note t h a t M i n i t i a l l y i s proportional to vt but approaches asymptotically i n accordance with an exponential function. Another common s i t u a t i o n i s transient permeation i l l u s t r a t e d i n F i g u r e 3. I n i t i a l l y , C = 0 f o r a l l x but i s i n c r e a s e d to and h e l d at C at x = 0 t h e r e a f t e r . The response observed i s the amount of penetrant Q that has emerged from the downstream surface after time t. There i s a time l a g 0 associated with the transient buildup of a steady c o n c e n t r a t i o n p r o f i l e , a f t e r which a steady r a t e of permeation i s established as given by Equation 7.


t

Q

t

ADC

0

(12) when D i s constant.

In t h i s case, the time lag i s 1

I 9 = _ 6D

(13)

A complete expression for Q i s a v a i l a b l e but w i l l not be given here (13). For gases, we may d e f i n e the f o l l o w i n g p e r m e a b i l i t y coefficient /dQ \ DC P=JL(_L) (14) p A \dt /ss p t

£

t

0

0

Q

When Henry's law a p p l i e s , that i s , Equation 3, P = DS

(15)

Thus, from a transient permeation experiment we can get both D and S s e p a r a t e l y ; whereas, a s t e a d y - s t a t e experiment o n l y g i v e s t h e i r product. Factors Affecting S o l u b i l i t y and Transport C r y s t a l l i n i t y , F i l l e r s , and Morphology. The s o l u b i l i t y of low molecular weight compounds i s extremely s m a l l i n the c r y s t a l l i t e s of polymers i n comparison to that i n the amorphous regions of the same polymer (15). Thus, e q u i l i b r i u m s o r p t i o n i n s e m i c r y s t a l l i n e polymers i s l e s s than that for corresponding completely amorphous ones. For the same reasons c r y s t a l l i n e polymers are more chemical r e s i s t a n t than amorphous ones. As a good approximation for gases, the Henry's law s o l u b i l i t y c o e f f i c i e n t of a semicrystal l i n e polymer i s r e l a t e d to that for the same polymer i n the amorphous state, S , by the f o l l o w i n g : a

S = * S a

a


(16)

12.


PAUL

259

where i s the volume f r a c t i o n of the polymer t h a t e x i s t s i n the amorphous state. The l a t t e r i s r e l a t e d to the volume f r a c t i o n that i s c r y s t a l l i n e , or the c r y s t a l l i n i t y , by = 1 - 4> . S i m i l a r c o n s i d e r a t i o n s a p p l y to nonsorbing f i l l e r s f r e q u e n t l y used i n polymers, for example, g l a s s f i b e r s and minerals. C r y s t a l s or f i l l e r p a r t i c l e s that do not sorb the penetrant w i l l obviously be impermeable to them and, thus, reduce transport rates i n the composite. Figure 4 i l l u s t r a t e s t h i s effect for matrices i n which such p a r t i c l e s are arranged i n an ordered and i n a random manner. The rate of transport i n such systems w i l l be slower than when such p a r t i c l e s are absent because of the reduced area f o r transport and the r e s u l t i n g more tortuous path for permeation. The 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 f o r such cases w i l l be a f a c t o r K smaller than i n the pure amorphous material D . The permeability c o e f f i c i e n t i n a c r y s t a l l i n e polymer should then be given by a


a

c

a

P

KS

D

17

c = *a a a

< >

Various theories and e m p i r i c a l expressions are a v a i l a b l e (14, 16) for e s t i m a t i n g K f o r s p e c i a l s i t u a t i o n s . O b v i o u s l y , t h i s f a c t o r w i l l depend on the volume f r a c t i o n of impermeable p a r t i c l e s , t h e i r shape, and t h e i r arrangement i n space, t h a t i s , morphology. C r y s t a l l i n e polymers are much better b a r r i e r s to permeation than are e q u i v a l e n t amorphous polymers by v i r t u e of the o b s t r u c t i o n to transport caused by t h e i r c r y s t a l l i t e s . Often t h e i r resistance to permeation can be further improved by stretching or drawing so that the c r y s t a l s are c o n v e r t e d from a random arrangement to a more ordered array such as that i l l u s t r a t e d i n Figure 4. Temperature and T r a n s i t i o n s . D i f f u s i o n i n s o l i d s and l i q u i d s i s extremely affected by temperature. Like reaction rates, diffusion may be thought of as an a c t i v a t e d process f o l l o w i n g an expression of the Arrhenius form D = D ~E/RT Q e

( 1 8

)

where the a c t i v a t i o n energy E depends on the polymer and s i z e of the penetrant and i s of the order of 10 k c a l / m o l . Because of these c o n s i d e r a t i o n s i t i s c o n v e n i e n t to p l o t e x p e r i m e n t a l data i n the form of log D versus 1/T as i l l u s t r a t e d i n Figure 5 except here the 1/T s c a l e i s r e v e r s e d so t h a t temperature i n c r e a s e s a l o n g the abscissa. Far from any t r a n s i t i o n s , these p l o t s are l i n e a r and thus show agreement with Equation 18. The l e f t p a r t of F i g u r e 5 i s f o r d i f f u s i o n of c y c l o p r o p a n e i n high-density polyethylene (17), a h i g h l y c r y s t a l l i n e polymer that m e l t s at 135 °C. Upon c o o l i n g below T , D decreases i n a dramatic manner because of the development of c r y s t a l s that impede d i f f u s i o n . Further below T , another Arrhenius form i s established. Note that the l i n e s above and below T are not p a r a l l e l which shows that the effect of c r y s t a l l i n i t y i s more than a simple geometric obstruction to d i f f u s i o n i n t h i s case. The r i g h t p a r t of F i g u r e 5 i s f o r d i f f u s i o n of C O 2 i n an a e r y l o n i t r i l e / m e t h y 1 a c r y l a t e (AN/MA) copolymer (18) having a g l a s s t r a n s i t i o n near 65 °C. The diffusion c o e f f i c i e n t shows a change i n s l o p e on the A r r h e n i u s p l o t i n t h i s v i c i n i t y as i s often observed on t r a v e r s i n g the T . Such breaks i n m

ffl

ffl


260



Figure 4.

Diffusion through media containing impermeable p a r t i c l e s (ordered on the l e f t and random on the r i g h t ) .



12. PAUL

261


50

Figure 5 .

C

60

70

80

E f f e c t of temperature and t r a n s i t i o n s on penetrant d i f f u s i o n c o e f f i c i e n t s . Left: cyclopropane i n h i g h d e n s i t y p o l y e t h y l e n e to i l l u s t r a t e e f f e c t of m e l t i n g point. Right: CO2 i n AN/MA copolymer to show effect of g l a s s t r a n s i t i o n . Note: reversed 1/T scale i s used.


262



the curve are not always observed e s p e c i a l l y for s m a l l gas molecules (19). D i s a c t u a l l y h i g h e r below T than what one would e s t i m a t e from extrapolating data from above T^. The AN/MA copolymer mentioned E a r l i e r i s of i n t e r e s t technol o g i c a l l y because of the very good b a r r i e r properties exhibited by polymers containing acrylonitrile—compare the D for t h i s copolymer w i t h those f o r p o l y e t h y l e n e on the l e f t . However, because pure p o l y a c r y l o n i t r i l e i s e s s e n t i a l l y impossible to melt process, i t i s necessary to copolymerize with other monomers to obtain processable materials with the r e s u l t being a s a c r i f i c e of b a r r i e r properties. Penetrant S i z e . One view of d i f f u s i o n i n polymers e n v i s i o n s penetrant molecules hopping i n random steps f o l l o w i n g the opening of a h o l e or t u n n e l of s u f f i c i e n t s i z e f o r the penetrant m o l e c u l e to pass, and the energy a s s o c i a t e d w i t h such a passage i s the a c t i v a t i o n energy i n E q u a t i o n 18. Somewhat different models have a l s o been proposed (14); however, they would a l l agree that the s i z e of the penetrant m o l e c u l e s h o u l d g r e a t l y a f f e c t the v a l u e of D. This fact i s n i c e l y demonstrated by the extensive data i n Figure 6 for various s m a l l molecules i n p o l y ( v i n y l c h l o r i d e ) (20). Here the s i z e of the various penetrant molecules i s expressed by the van der Waals volume, b, obtained from pressure-volume-temperature behavior i n the gaseous s t a t e . Note t h a t D v a r i e s i n t h i s p l o t by n e a r l y 10 orders of magnitude. S p e c i a l techniques are r e q u i r e d to study systems with d i f f u s i o n c o e f f i c i e n t s l e s s than about 1 0 cnr/s. - 1

Penetrant C o n c e n t r a t i o n - P l a s t i c i z a t i o n . E a r l i e r , i n seeking s o l u t i o n s to F i c k ' s laws, the d i f f u s i o n c o e f f i c i e n t was assumed to be a constant independent of penetrant c o n c e n t r a t i o n . This assumption i s good f o r gases and other m o l e c u l e s of very l i m i t e d solubility. However, f o r vapors or l i q u i d s t h a t may sorb i n s i g n i f i c a n t amounts (see F i g u r e 1) t h i s assumption u s u a l l y i s not the case (14). A d d i t i o n of s m a l l m o l e c u l e s i n more than d i l u t e amounts changes the environment of the polymer segments and causes t h e i r m o t i o n s t o be more r a p i d . This condition i s c a l l e d p l a s t i c i z a t i o n and may be thought of as r e s u l t i n g from an increase i n free volume. As a r e s u l t of t h i s , d i f f u s i o n of the penetrant m o l e c u l e i s more r a p i d or D i s i n c r e a s e d . Thus, i n g e n e r a l , D depends on concentration, and sometimes i n a very strong manner, for example, e x p o n e n t i a l l y : D = D

c = 0

e

^

C

(19)

A discussion of various models for concentration dependent diffusion i s a v a i l a b l e (21), and compilations of s o l u t i o n s to F i c k ' s laws for some of these cases e x i s t (13, 14). Polymer M o l e c u l a r S t r u c t u r e . We may expect polymer m o l e c u l a r s t r u c t u r e to have an enormous e f f e c t on the r a t e of t r a n s p o r t of s m a l l m o l e c u l e s through them. T h i s f a c t may be most e a s i l y appreciated by examining gas transport because s o l u b i l i t y of these m o l e c u l e s i s q u i t e low and, thus, not a f f e c t e d by p l a s t i c i z a t i o n effects as mentioned e a r l i e r . Table I gives the oxygen permeability i n a wide range of polymers at 25 °C as c o l l e c t e d from a v a r i e t y of published (22) and unpublished sources. Among the o l e f i n s l i s t e d ,



12. PAUL

263


-4

-

\r

H

PVC 30*C

2

-HjO

~

-8

E

a -10

Kr-J \

\ -12

\r-H C-CHCl \ |HCH)CO CjH^ON-? \ l n-C4Hio 2

32

r

V

s

n-CsMrOH-T £ 4

r 1

{

-16 0.05

0.10

VAN DER WAALS

Figure 6.

0.15

0.20

b (liter / m o l e )

E f f e c t of p e n e t r a n t s i z e , expresssed as van der Waals volume, on d i f f u s i o n c o e f f i c i e n t i n p o l y ( v i n y l c h l o r i d e ) at 30 °C. (Reproduced w i t h p e r m i s s i o n from Reference 20.)


264



Table I .

Oxygen Permeability i n Polymers of Various Molecular Structures at 25 °C

Polymertype Olefins

2

Polymer

(cm^STP) cm/cm scm Hg) 1.4

polypropylene low-density polyethylene (p=0.92g/cm )

3.0

high-density polyethylene (p=0.96g/cm ) ethylene-propylene elastomer

0.58 14

natural fCH CH=CH CH 4

25

3

3

Rubber

2

2

1.3

butyl fCH -C(CH ) 4 2

3

2

silicone f0-Si(CH ) 4 3

Halogenated

poly(vinyl

fluoride)

poly(vinylidene

Highly polar

2

fluoride)

650 0.18 0.028

p o l y ( v i n y l chloride)

0.055

poly(vinylidene chloride)

0.0035

polyacrylonitrile

0.00025

p o l y ( v i n y l a l c o h o l ) , dry

-lO"

6



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c r y s t a l l i n i t y ranges from none for the ethylene-propylene copolymer to very high for polypropylene and high-density polyethylene. This range i s a major factor i n v a r i a t i o n of oxygen permeation shown by these materials. More than a 400-fold spread e x i s t s i n permeability among the three rubbers shown i n Table I . Butyl rubber, b a s i c a l l y a polymer composed of isobutylene with s m a l l amounts of the comonomer isoprene to g i v e unsaturation for v u l c a n i z a t i o n , i s widely known for i t s low r a t e of gas t r a n s p o r t b e l i e v e d to occur because of the s l u g g i s h segmental motions of i t s c h a i n . These s l u g g i s h motions r e s u l t from the s t e r i c hindrance caused by the two pendent methyl groups on every other chain carbon atom. Although s i l i c o n e rubber has a s i m i l a r s u b s t i t u t i o n p a t t e r n , the ether oxygen c o n n e c t i n g these u n i t s g i v e s r i s e to f a c i l e segmental m o b i l i t y . S i l i c o n e rubber i s one of the most permeable polymers known. H a l o g e n a t i o n reduces p e r m e a b i l i t y as T a b l e I shows. S i n g l e halogen s u b s t i t u t i o n , p o l y ( v i n y l f l u o r i d e ) and p o l y ( v i n y l c h l o r i d e ) r e s u l t s i n polymers l e s s permeable than any p o l y o l e f i n , even though p o l y o l e f i n s may be c o n s i d e r a b l y more c r y s t a l l i n e . Double s u b s t i t u t i o n as i n p o l y ( v i n y l i d e n e f l u o r i d e ) and p o l y ( v i n y l i d e n e c h l o r i d e ) produces even more dramatic reductions i n permeation rates p a r t l y because these polymers can be rather c r y s t a l l i n e , but mainly because of the s i m i l a r e f f e c t of double s u b s t i t u t i o n on the same backbone carbon as seen f o r b u t y l rubber. The two h i g h l y p o l a r polymers l i s t e d i n T a b l e I have e x t r a o r d i n a r i l y low oxygen p e r m e a b i l i t i e s because of the s t r o n g d i p o l a r f o r c e s i n p o l y a c r y l o n i t r i l e and the s t r o n g hydrogen bonding i n p o l y ( v i n y l a l c o h o l ) . The very low permeability for p o l y ( v i n y l a l c o h o l ) only e x i s t s when the polymer i s dry. P o l y ( v i n y l a l c o h o l ) i s , of course, q u i t e h y g r o s c o p i c ; sorbed moisture breaks up the s t r o n g bonding between c h a i n s , or p l a s t i c i z e s the polymer, and causes l a r g e increases i n oxygen permeation. T a b l e I I shows the p r o p e r t i e s of four s t r u c t u r a l l y s i m i l a r polymers (some are i s o m e r i c ) and the 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 f o r a s i m p l e gas, argon, i n them (23). By comparison, the Henry's law s o l u b i l i t y c o e f f i c i e n t , S, varies r e l a t i v e l y l i t t l e among these polymers when c o n t r a s t e d to the n e a r l y 2 0 0 - f o l d v a r i a t i o n i n d i f f u s i o n c o e f f i c i e n t s that e x i s t s . For these polymers, the transport parameters change i n the d i r e c t i o n one would expect on the basis of the cohesive energy densities (CED) of these m a t e r i a l s , t h a t i s , t r a n s p o r t s l o w s down as the s t r e n g t h of the cohesive force f i e l d increases. However, t h i s trend i s o n l y q u a l i t a t i v e and i s by no means g e n e r a l when polymers of d i v e r s e structures are considered. In fact, no general way i s yet known to p r e d i c t permeation r a t e s through polymers from knowledge of t h e i r molecular structures. Some free volume c o r r e l a t i o n s (24) may be used for very rough approximations; however, such p r e d i c t i o n remains one of the unsolved problems i n applied polymer science. Although m o l e c u l a r m o b i l i t y and c o h e s i v e f o r c e s are c o n s i d e r e d to be c o n t r i b u t i n g p a r a m e t e r s , as t h e y a r e i n d e t e r m i n i n g o t h e r properties, i t i s i n s t r u c t i v e to contemplate the f o l l o w i n g example (9). Poly(phenylene oxide) has r i g i d aromatic u n i t s i n i t s backbone and e x h i b i t s a T of 220 °C, but i t i s g r e a t e r than an order of magnitude more permeable than b u t y l rubber which has a T of -76 °C. g

8



p o l y ( v i n y l methyl ether)

poly(methyl acrylate) -23

3

28

p o l y ( v i n y l acetate)

81.3

102.5

109.2

127.5

20

p o l y ( v i n y l methyl ketone)

Polymer

3

8 (°C)

T

3

2.24

0.50

0.20

0.024

P x 10 cm (STP) cm cm sec cm Hg

10

60.6

5.7

1.6

0.32

2

8

D x 10 (cm /sec)

S o l u b i l i t y and Transport of Argon at 30 °C i n Four S t r u c t u r a l l y Similar Polymers

CED (cal/cra )

Table I I .


3

7.38

3.70

8.80

12.6

3

S x 10* cm rSTP} cm cm Hg


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R e l a x a t i o n - C o n t r o l l e d T r a n s p o r t . In the discussions thus far, i t has been assumed that l o c a l l y the mechanical state of the polymerpenetrant system a d j u s t s r a p i d l y to the changes imposed and t h a t s i m p l e d i f f u s i o n i s the r a t e - c o n t r o l l i n g mechanism of penetrant transport. However, for glassy polymers these conditions may not be met when c e r t a i n vapors or l i q u i d s are sorbed (25). For example, sorption of methanol i n t o poly(methyl methacrylate) sheets sets up c o n d i t i o n s of c o n s t r a i n e d s w e l l i n g and generates internal s t r e s s e s (26) t h a t change the mode of t r a n s p o r t to what has been c a l l e d Case I I b e h a v i o r or "anomalous" d i f f u s i o n . In t h i s s i t u a t i o n , time-dependent s t r u c t u r a l rearrangements or r e l a x a t i o n s become the dominant process and g i v e r i s e to deviations from Fickean b e h a v i o r . Case I I t r a n s p o r t i s c h a r a c t e r i z e d by a sharp f r o n t moving i n t o the polymer at constant v e l o c i t y . Behind the f r o n t e x i s t s s w o l l e n polymer e s s e n t i a l l y at e q u i l i b r i u m penetrant c o n c e n t r a t i o n , and i n advance of the f r o n t i s unpenetrated g l a s s y polymer. As a r e s u l t , mass increase from sorption varies l i n e a r l y i n time i n c o n t r a s t to the s q u a r e - r o o t dependence expected from F i c k ' s laws (Equation 10). S o r p t i o n may be accompanied by c r a z i n g or even catastrophic cracking i n some cases. Applications of Transport Concepts The sorption and transport c h a r a c t e r i s t i c s of polymers are important i s s u e s i n s e l e c t i n g the proper polymer f o r c e r t a i n a p p l i c a t i o n s . T h i s b e h a v i o r may be t a i l o r e d to the a p p l i c a t i o n by p r o c e s s i n g technique, chemical modification, or p h y s i c a l design. The f o l l o w i n g are s e l e c t e d i l l u s t r a t i o n s chosen to suggest the wide range of a p p l i c a t i o n s for which such c h a r a c t e r i s t i c s may be c r i t i c a l or form the basis for a useful product. B a r r i e r M a t e r i a l s . Polymers are often used as b a r r i e r s to keep s m a l l m o l e c u l e s i n or to keep them out. One common example i s rubber tubes for t i r e s or more recently the inner l i n e r of tubeless t i r e s . The purpose of such a m a t e r i a l i s to c o n t a i n a i r under p r e s s u r e to m a i n t a i n t i r e i n f l a t i o n . From the data i n T a b l e I , i t i s c l e a r t h a t b u t y l rubber i s a much b e t t e r m a t e r i a l f o r t h i s purpose than n a t u r a l rubber. Because of t h i s , b u t y l rubber has e n t i r e l y displaced natural rubber from t h i s market. Aside from i t s p r o h i b i t i v e p r i c e , s i l i c o n e rubber would be t o t a l l y unsatisfactory for t h i s use because of i t s high gas permeability. Foodstuffs are packaged i n a v a r i e t y of polymeric f i l m products. One purpose i s to keep oxygen out i n order to p r e v e n t decay. From Table I , i t i s c l e a r that p o l y ( v i n y l i d e n e chloride)-based polymers would be a good c h o i c e of m a t e r i a l s because of t h e i r low oxygen permeability. For t h i s reason, p o l y ( v i n y l i d e n e c h l o r i d e ) forms the basis of one of the very commonly used foodwraps. P o l y o l e f i n f i l m s are a l s o used f o r food packaging even though they do not have e x c e p t i o n a l l y good resistance to gas permeation; however, because of t h e i r nonpolar c h a r a c t e r they are good b a r r i e r s to water vapor transport. T h i s c h a r a c t e r i s t i c i s an important one f o r a v o i d i n g dehydration of foods, e s p e c i a l l y vegetables and f r u i t s . Recently, there has been a trend to package carbonated beverages i n p l a s t i c b o t t l e s because of t h e i r l i g h t e r weight and i n c r e a s e d safety from breakage. In t h i s a p p l i c a t i o n , the polymer chosen must



268


be a good b a r r i e r i n order to a v o i d l o s s of c a r b o n a t i o n by C O 2 permeation and to prevent atmospheric oxygen from e n t e r i n g the b o t t l e and s p o i l i n g the contents. Although dry p o l y ( v i n y l a l c o h o l ) i s an e x c e l l e n t b a r r i e r t o gas t r a n s p o r t , i t c o u l d n o t be contemplated for t h i s use because i t would s w e l l or d i s s o l v e i n the water that i s to be contained. A c r y l o n i t r i l e - b a s e d polymers were o r i g i n a l l y c o n s i d e r e d f o r t h i s use because of t h e i r e x c e l l e n t b a r r i e r p r o p e r t i e s , but they were abandoned because of concerns about the p o t e n t i a l h e a l t h hazards a s s o c i a t e d w i t h l e a c h i n g of r e s i d u a l monomer from p o l y m e r i z a t i o n i n t o the b o t t l e c o n t e n t s . Today, b o t t l e s for carbonated beverages are based on poly(ethylene terephthalate) which appears to be the best choice i n terms of a l l of the c h a r a c t e r i s t i c s needed for t h i s a p p l i c a t i o n . There i s some l o s s of CO2 from the b o t t l e during prolonged storage, and Figure 7 shows a c a l c u l a t i o n for a t y p i c a l 2-L b o t t l e . The c a l c u l a t i o n shows the CO2 pressure l o s s broken down i n t o that which occurs by sorption of gas i n t o the b o t t l e w a l l and by e v e n t u a l permeation from the e x t e r i o r of the b o t t l e s u r f a c e . The former accounts f o r a s i g n i f i c a n t f r a c t i o n of the l o s s ; hence, one must be concerned with s o l u b i l i t y as w e l l as transport behavior. Devolatilization. F o l l o w i n g p o l y m e r i z a t i o n of many p o l y m e r s , removal of r e s i d u a l monomer or s o l v e n t s i s necessary. Ultimately t h i s removal becomes c o n t r o l l e d by d i f f u s i o n from the polymer no matter what type of process i s employed (27). Many processes use a s t e a m - s t r i p p i n g o p e r a t i o n f o r t h i s purpose, and o t h e r s employ d e v o l a t i l i z a t i o n i n a vented extruder. Removal of r e s i d u a l v i n y l c h l o r i d e monomer from p o l y ( v i n y l c h l o r i d e ) was an i s s u e of much concern f o l l o w i n g the discovery that t h i s monomer i s a carcinogen. A d d i t i v e Migration. A v a r i e t y of a d d i t i v e s such as antioxidants and UV s t a b i l i z e r s are used i n polymers to f a c i l i t a t e t h e i r processing or to p r o l o n g t h e i r u s e f u l l i f e . These low m o l e c u l a r weight a d d i t i v e s may be l o s t by migration during processing, and thus t h e i r b e n e f i t i s l o s t , or they may migrate i n t o f o o d s t u f f s when these polymers are used as packaging m a t e r i a l s . For s e v e r a l y e a r s , the National Bureau of Standards has conducted an extensive program to develop r a t i o n a l models to e v a l u a t e these p o s s i b i l i t i e s by u s i n g p r i n c i p l e s of s o l u b i l i t y and d i f f u s i o n f o r making r e g u l a t o r y d e c i s i o n s (28). Dyeing. Most f i b e r s are colored by a dyeing operation during which a dye molecule diffuses into the f i b e r structure (29). N a t u r a l l y , the time r e q u i r e d f o r t h i s process to occur i s an important i s s u e and i s one of the f a c t o r s c o n s i d e r e d i n d e v e l o p i n g f i b e r s . To a t t r a c t enough dye i n t o the f i b e r to d e v e l o p the depth of shades d e s i r e d and to h o l d i t i n the f i b e r d u r i n g l a u n d e r i n g , many synthetic fibers have i o n i c dye s i t e s to a t t r a c t oppositely charged dye m o l e c u l e s . Other mechanisms such as r e a c t i v e and d i s p e r s e dyeing are a l s o used. To accept a c a t i o n i c or basic dye, the f i b e r must have s t r o n g a c i d groups i n i t s s t r u c t u r e , whereas a n i o n i c or acid dyes require basic groups. Condensation polymers such as nylon have unreacted end groups t h a t can s e r v e t h i s purpose. Addition polymers such as the a c r y l i c s may have s u l f o n i c acid groups at the


12. PAUL

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chain ends r e s u l t i n g from redox i n i t i a t i o n . A l t e r n a t i v e l y , i o n i c monomers may be copolymerized i n t o the polymer for t h i s purpose. Membrane S e p a r a t i o n s . Separation processes u s i n g p o l y m e r i c membranes (30) have become important techniques because of t h e i r s i m p l i c i t y and low consumption of energy i n comparison to a l t e r n a t i v e s such as d i s t i l l a t i o n . Membranes for u l t r a f i l t r a t i o n are porous, and no d i f f u s i v e t r a n s p o r t a c t u a l l y o c c u r s through the polymer i t s e l f . However, f o r s e p a r a t i o n at the m o l e c u l a r l e v e l , d i f f u s i o n through the polymer p r o v i d e s a p o s s i b l e mechanism f o r s e l e c t i v e passage of the desired s m a l l molecule. Reverse osmosis f o r d e s a l i n a t i o n of water can occur by t h i s mechanism, and l a r g e commercial processes using t h i s technique are now i n operation. Reverse osmosis i s s i m p l y the a p p l i c a t i o n of pressure on a s o l u t i o n i n excess of the osmotic pressure to create a d r i v i n g force t h a t r e v e r s e s the d i r e c t i o n of osmotic t r a n s f e r of the s o l v e n t , u s u a l l y water. The transport behavior can be analyzed e l e g a n t l y by using general theories of i r r e v e r s i b l e thermodynamics; however, a s i m p l i f i e d s o l u t i o n - d i f f u s i o n model accounts q u i t e w e l l f o r the a c t u a l d e t a i l s and mechanism i n most reverse osmosis systems. Most successful membranes for t h i s purpose sorb approximately 5 to 15% water at e q u i l i b r i u m . A thermodynamic a n a l y s i s shows t h a t the a p p l i c a t i o n of a pressure d i f f e r e n c e , Ap, to the water on the two sides of the membrane induces a d i f f e r e n t i a l concentration of water w i t h i n the membrane at i t s two faces i n accordance w i t h the f o l l o w i n g (31): C

V

A

w* w P

AC , JZ-JZ

(20)

W

RT where V i s the molar volume of water, and C * i s the e q u i l i b r i u m c o n c e n t r a t i o n of water i n the membrane. The g r a d i e n t of water i n the membrane leads to a d i f f u s i v e f l u x of water, J , i n accordance with F i c k ' s law: w

w

w

J

D

21

w = w— I

< >

Because for s a l t water, the osmotic pressure difference, Air, acts i n opposition to the applied h y d r a u l i c pressure, the water f l u x becomes D

J

V

W

=

C

w w w* W

W

(Ap - ATT)

(22)

RT£ where D i s the diffusion c o e f f i c i e n t for water i n the polymer. The s o l u t e or s a l t f l u x i s given by the f o l l o w i n g r e l a t i o n : w

J

D

K

s = s s «£, -

C

sb/*

(23)

provided the pressure d i f f e r e n t i a l does not contribute s i g n i f i c a n t l y to the d r i v i n g f o r c e f o r i t s t r a n s p o r t , as i s the u s u a l case. The



270

terms are defined as f o l l o w s : K i s the p a r t i t i o n c o e f f i c i e n t for s a l t between polymer and l i q u i d phases, D i s the s a l t d i f f u s i o n c o e f f i c i e n t i n polymer, C i s the s a l t c o n c e n t r a t i o n i n l i q u i d phase w i t h s u b s c r i p t s o and £ d e n o t i n g upstream and downstream, r e s p e c t i v e l y . The s e l e c t i v i t y c h a r a c t e r i s t i c s of a reverse osmosis membrane system i n terms of a r e j e c t i o n c o e f f i c i e n t can be defined as f o l l o w s : g

g


g

(24)

The s a l t content on the downstream s i d e of the membrane i s d e t e r mined by the r e l a t i v e s a l t and water f l u x e s through the membrane. A combination of r e s u l t s from the s o l u t i o n - d i f f u s i o n model o u t l i n e d e a r l i e r with the d e f i n i t i o n of the r e j e c t i o n c o e f f i c i e n t gives D K RT tl ) "I D V C *(Ap-A7r) S

R = (1 +

w

w

S

(25)

w

Note t h a t the r e j e c t i o n i s independent of the membrane t h i c k n e s s , but the f l u x of water or p r o d u c t i v i t y i n c r e a s e s as the membrane becomes t h i n n e r . Reverse osmosis was not commercially p r a c t i c a l u n t i l techniques for increasing p r o d u c t i v i t y were developed. The p r i n c i p a l discovery (32) i n v o l v e d a c a s t i n g procedure t h a t r e s u l t s i n asymmetric membranes having a t h i n dense l a y e r of polymer, approximately 0.2u t h i c k , supported on a porous s u b l a y e r as i l l u s t r a t e d i n F i g u r e 8. These membranes are c a l l e d Loeb membranes (33). Current commercial membranes of t h i s type are made of c e l l u l o s e a c e t a t e , aromatic polyamides, and c e r t a i n composites that achieve water f l u x e s of the order of 1.0 nr/m day with NaCl rejections of 99% or more (27). As seen i n Equation 25, r e j e c t i o n increases with applied pressure. A f u r t h e r advance i n membrane t e c h n o l o g y was made by l e a r n i n g how to s p i n h o l l o w f i b e r s w i t h w a l l s of the Loeb-type s t r u c t u r e . These f i b e r s can be assembled i n t o modules resembling shell-and-tube heat exchangers such as i l l u s t r a t e d i n F i g u r e 9. T h i s process r e s u l t s i n a l a r g e s u r f a c e area per u n i t of volume and g r e a t l y reduces the c a p i t a l cost of such u n i t s i n comparison to other module configurations (30). S i m i l a r devices are now a v a i l a b l e commercially for gas separations (34) and are economical i n comparison to other processes because of t h e i r low energy consumption. The i n t r i n s i c separation factor for a polymer membrane for two gases i s given by the r a t i o of the permeability c o e f f i c i e n t s of the two gases: 2

?i

«12 = — (26) 2 Polymers with fixed i o n i c charges may be used for ion exchange. When formed into membranes, these materials e s s e n t i a l l y prevent the passage of i o n s of the same charge as those f i x e d to the polymer P



12.

111


PAUL

0

50

100

150

200

t (days)

Figure 7.

C a l c u l a t e d CO^ p r e s s u r e l o s s from p l a s t i c carbonated beverage b o t t l e . T o t a l l o s s i s comprised of C 0 t h a t has l e f t the exterior surface of b o t t l e w a l l plus that s t i l l contained i n the w a l l . 2

Thin Dense Skin Porous Support

Figure 8.

Schematic diagram of Loeb-type membrane.


272



backbone and a l l o w the passage of i o n s of o p p o s i t e s i g n (30). A l t e r n a t i o n of anionic and c a t i o n i c membranes as shown i n Figure 10 permits the use of an e l e c t r i c a l p o t e n t i a l to d r i v e the process known as e l e c t r o d i a l y s i s which can be used to desalinate water or to produce a concentrated e l e c t r o l y t e mixture from a more d i l u t e one. E l e c t r o c h e m i c a l membrane processes are p r e s e n t l y being used i n processes for producing c h l o r i n e and caustic soda (30). C o n t r o l l e d - R e l e a s e Technology. C o n v e n t i o n a l l y , chemicals such as drugs, pesticides, f e r t i l i z e r s , and herbicides are administered i n a periodic fashion and thus cause temporal concentration v a r i a t i o n s . The c o n c e n t r a t i o n extremes may range from dangerously h i g h to i n e f f e c t i v e l y low l e v e l s and a c h i e v e the optimum l e v e l f o r o n l y short periods of time. This c y c l i c a p p l i c a t i o n i n e f f i c i e n t l y uses these chemicals and r e s u l t s i n added expense and p o s s i b l y c e r t a i n side effects. During the l a s t two decades, the concept of c o n t r o l l e d d e l i v e r y or r e l e a s e has emerged as a means of s o l v i n g many of these problems. Often t h i s technology i n v o l v e s the use of polymers (35). Numerous concepts using polymers as the r a t e - c o n t r o l l i n g element have been d e v e l o p e d and c o m m e r c i a l i z e d (27, 30, 35). F i g u r e 11 i l l u s t r a t e s one approach that w i l l d e l i v e r a solute such as a drug at a constant r a t e f o r very l o n g times depending on the d e s i g n . T h i s instrument i s a r e s e r v o i r - t y p e d e v i c e i n which the s o l u t e i n the form of a suspension i s enclosed i n a membrane that c o n t r o l s i t s r e l e a s e . Because the r e s e r v o i r c o n t a i n s s o l i d drug p a r t i c l e s at u n i t a c t i v i t y , the d r i v i n g f o r c e f o r t r a n s p o r t remains f i x e d , and the f l u x through the membrane i s constant u n t i l a l l of the drug p a r t i c l e s have been d i s s o l v e d . By s e l e c t i o n of the polymer and the geometry of the membrane, the designer can engineer a product that meets the many different requirements of the a p p l i c a t i o n .


12.

273


PAUL


o o o

Anode

Cathode

4=1 Stripping Stream

Figure 10.

Figure 11.

Feed Solution

E l e c t r o d i a l y s i s . Cation and anion designations refer to membranes t h a t w i l l exchange c a t i o n s or a n i o n s , respectively.

Reservoir-type membrane device for c o n t r o l l e d release.


274


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Transport Properties of Polymers - American Chemical Society

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