Barrier Polymers and Structures - ACS Publications - American

Chapter 2

Fundamentals of Gas Diffusion in Rubbery and Glassy Polymers

Downloaded by UNIV OF SYDNEY on September 1, 2014 | http://pubs.acs.org Publication Date: May 9, 1990 | doi: 10.1021/bk-1990-0423.ch002

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


2.


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


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


2.

STERN & TROHALAKI

smaller

than

that

discrepancy proposed

2

However, shown

that

need

without

t o adequately

molecules

Stern,


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 .


34

BARRIER POLYMERS AND STRUCTURES


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


STERN & TROHALAKI

Gas Diffusion in Rubbery and Glassy Polymers 35

20r


50*0

10 θ a

35 C

- Experimental -Theoretical

20

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