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Brandt, W.W. J. Phys. Chem. 1985, 63, 1080. 14. Barrer, R.M. J. Phys. Chem. 1957, 61, 178. 15. Brandt, W.W.; Anysas, B.A. J. Appl. Polym. Sci. 1963, 7...
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Chapter 2

Fundamentals of Gas Diffusion in Rubbery and Glassy Polymers

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S. A. Stern and S. Trohalaki Department of Chemical Engineering and Materials Science, Syracuse University, Syracuse, NY 13244-1190

This paper reviews some of the more important models and mechanisms of gas diffusion in rubbery and glassy poly­ mers in light of recent experimental data. Diffusion (transport) of gases in polymers is an important, and in some cases, controlling factor in a number of important applica­ tions, such as protective coatings, membrane separation processes, and packaging for foods and beverages. Therefore, a better under­ standing of the mechanisms of gas diffusion in polymers is highly desirable in order to achieve significant improvements in these applications and to develop new ones. From a formal (macroscopic) viewpoint, the diffusion process can be described in many cases of practical interest by Fick's two laws (1-5). These laws are represented by the following equations for the isothermal diffusion of a substance in or through a v-dimensional, hyperspherical polymer body of sufficiently large area [v=l for a slab or membrane (film), v=2 for a hollow cylinder, and V=3 for a spherical shell] (2) : ( V

J = -ω r v

X )

D

3

^ ^ dr

(1)

and 3c 3t

=

1 3 ( v-l 3C rV 3r~ ( D r

R

ν

average and

to

T

12

< >

parameters system.

which

depend o n l y

F o r low p e n e t r a n t

on

con­

v ^ i s g i v e n by

i s the

f

(T,v) = v

volume

fractional f

f

(Τ,0)+γ(Τ)ν ,

fraction

free

γ (Τ) [= ( 3 v / 3 v )

volume

Combining

of of

Equations

12

F i g u r e 2, F u j i t a

(13)

the the

] i s a measure

i n i n c r e a s i n g the f r e e

ν -4 0, c f .

D,

r

are c h a r a c t e r i s t i c

v

where

coefficient,

by

the nature of the penetrant/polymer

ness

a

density.

a) The Model o f

T,

in

for diffusion

form o f d i s c o n t i n u o u s v o i d s .

i n t h e m o l e c u l a r models, but

distribution

and (18)

i n v e s t i g a t o r s argue t h a t t h e t o t a l f r e e

redistributed

second

here

theory

penetrant, pure

of

polymer

the

v (T,0) f

at

is

the

temperature

penetrants

effective­

volume. and

13

and

noting

and K i s h i m o t o

that

(20^) f o u n d

D^

-> D(v=0)

that

In Barrier Polymers and Structures; Koros, W.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

as

S T E R N & TROHALAKI

Gas Diffusion in Rubbery and Glassy Polymers

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2.

In Barrier Polymers and Structures; Koros, W.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

31

32

BARRIER POLYMERS AND STRUCTURES

ν in I

[v^(T,0)]

(T, 0)

^

B

+

( 1 4 )

B Y(T)v

d

d

and

I n f ^ ) RT

= In A

J

- - ^ J — ν (T,0)

d

(15) -1

A,. d

Β . and γ d

1/v

together

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because

of

small,

can be with

the

a

The

plot

fractional

applicable 100

free

incurred.

l/v (T,0),

plot

A plot

is

f

but,

usually

very

o f E q u a t i o n 14 i s

volume

of

pure

polymer

as

a

function

of

can be f o u n d from

(T,0) = v^ (T ,0)+Ot(T-T ) fs s s

t o polymers

only

fractional

free

The

versus

latter

versus

(16)

i n the temperature

range Tg < Τ < Tg

+

volume, v ^ (T ,0) o f p u r e polymer i s a fs s r e f e r e n c e s t a t e a t t e m p e r a t u r e Τ , and a , a parameter c h a r a c t e r i s t i c s of

K.

the

[ln(D/D(0)3 Τ

2.

temperature, v ^ ( T , 0 ) ,

f

are

of

ln[D(0)/RT]

from

errors

i n Figure

ν

of

intercept

substantial

illustrated

o b t a i n e d from a p l o t

t h e polymer,

viscosity fied

with

Τ

^ e

0.048 C Ferry

so

,

This

1

but

fails

acetate)

model

trant

α

assume

to

suggested

Tg can be

values of

identi­

0.025

coefficients

valid that

the

for

nature

the

of

thermal

Landel,

and

and

with

expansion

of

a

strong

of

the the

difference

water

rather

h i s t h e o r y as

critical

between (20) .

than

water

to

a

size

in

such

poly(vinyl

be due

to the

failure

of

f o r small

independent for

with

depend­

(20,22,24-27),

T h i s may

inappropriate

i s largely hole

systems

concentration

i n amorphous polymers,

m o l e c u l e s , whose d i f f u s i o n because

Williams,

penetrant/polymer

exhibit

vapors

describe

viewed

by

i s i n r e a s o n a b l e agreement

and i n p o l y (methyl a c r y l a t e )

Fujita

tration,

Alternatively, and

Tg.

is

organic

hydrogen-bonding model.

the

coefficients as

as

value of α

between

Fujita s

such

(Tg,0)

fs

above and below

diffusion

(21,22).

v_

respectively,

difference

ence,

that

s

(2_3) .

polymer

can be e v a l u a t e d from t h e dependence o f s t e a d y - f l o w

on t e m p e r a t u r e

of

the

pene­

concen­

penetrants i s

In Barrier Polymers and Structures; Koros, W.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

2.

STERN & TROHALAKI

smaller

than

that

discrepancy proposed

2

However, shown

that

need

without

t o adequately

molecules

Stern,

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even

Klempner,

not c o i n c i d e

this

describe

i n r u b b e r y polymers,

and Kwei

correction,

This

(29), who

with

that f o r

Fujita's

model

the absorption k i n e t i c s of

e.g., o f CH , C H , 4

2

4

C H , and 3

8

i n p o l y e t h y l e n e (30) . Frisch,

volume model 3)

f o r viscous flow of t h e mixture. by F r i s c h ,

a r e f e r e n c e volume

been

small C0

required

was c o r r e c t e d

viscosity. has

Gas Diffusion in Rubbery and Glassy Polymers 33

and b i n a r y

membranes. the

gas m i x t u r e s

The e x t e n d e d

dependence

temperature the

of

polymer.

have

extended

of l i g h t

(_34_,35j

gases

(see

Fujita's

(31-33)

permeability

coefficients

of light

on c o m p o s i t i o n The v a l i d i t y

free-

(see

F i g u r e 4) t h r o u g h

model was f o u n d t o d e s c r i b e

f o r a variety

dependence

same

and coworkers

t o t h e permeation

gases

Figure polymer

satisfactorily

on

pressure

and

i n p o l y e t h y l e n e , as w e l l

f o r several

binary

o f t h e extended

mixtures

model

in

the

i s limited to

t o t a l p e n e t r a n t c o n c e n t r a t i o n s o f up t o 20-25 mol-%. The

effets

of c r y s t a l l i n i t y

have been t r e a t e d and F r i s c h

b)

and o f i n e r t

fillers

i n t h e context o f free-volume

on d i f f u s i o n

t h e o r y by K r e i t u s s

(36,32) ·

O t h e r Free-Volume Models

Other (_5) ,

free-volume by

Kumins

Free-volume polymers

models

have

and Kwei

models which

been

(38) ,

d i s c u s s e d by F r i s c h

and b y Rogers

are applicable

t o both

and S t e r n

and Machin rubbery

aredescribed i n a following section of this

(3_9) .

and g l a s s y

review.

DIFFUSION MODELS FOR GLASSY POLYMERS 1.

Effect

The

mechanisms

above

of

tration

the glass-transition

(1,3-8) .

i n the significant

the d i f f u s i o n

solubility

are very d i f f e r e n t

i . e . , when t h e polymers

respectively

reflected

State

o f gas d i f f u s i o n

and below

polymers, state,

o f G l a s s y Polymer

The d i f f e r e n c e

coefficient,

coefficients,

i n polymers

temperature,

are i n their

differences as w e l l

a t temperatures T^,

of the

"rubbery" o r " g l a s s y " i n t h e s e mechanisms i s

observed

i n t h e dependence

as o f t h e p e r m e a b i l i t y and

on t h e p e n e t r a n t

gas p r e s s u r e

o r concen-

and on t h e t e m p e r a t u r e .

In Barrier Polymers and Structures; Koros, W.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

34

BARRIER POLYMERS AND STRUCTURES

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1

1 -o

1

° .^

j

1

I

I 35*C

1

—Ο

ο—ο—



20°C

—o

. Λ.

°o

ο

X



°

2

o-

•Experimental Theoretical oo 00

oo

ο υ ο

ο

Ο

0.8 0-6

Figure

0

3a.

I 2

1 4

1 I ι 6 θ 10 MEAN PRESSURE^ (atm)

1 12

14

Comparison o f e x p e r i m e n t a l p e r m e a b i l i t y c o e f f i c i e n t s w i t h v a l u e s p r e d i c t e d by S t e r n , F r i s c h , and c o w o r k e r s ' e x t e n s i o n o f F u j i t a ' s f r e e - v o l u m e model f o r Ar i n polyethylene. ( S . A . S t e r n , S. R. Sampat, and S. S. K u l k a r n i , J . Polym. S c i . : P a r t B: Polym. Phys., 24, 2149, 1986, c o p y r i g h t 1986 John W i l e y & Sons, I n c . R e p r i n t e d by p e r m i s s i o n o f John W i l e y & Sons, Inc.) c

In Barrier Polymers and Structures; Koros, W.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

STERN & TROHALAKI

Gas Diffusion in Rubbery and Glassy Polymers 35

20r

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50*0

10 θ a

35 C

- Experimental -Theoretical

20