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