12 Molecular Theories of the Interdomain
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Contribution to the Deformation of Multiple Domain Polymeric Systems R I C H A R D J. G A Y L O R D Department of Metallurgy and Mining Engineering and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801
A number of theories of the
contribution
of
interdomain
polymeric material to the stress-strain, modulus, and swelling
behavior
of
block
copolymers
polymers are examined. The mathematical
details of
and
semicrystalline
conceptual foundation and
each theory are
the
summarized.
critique is then made of each theory in terms of the
A
validity
of the theoretical model, the mathematical development of the theory, and the ability of the theory to explain experimental findings.
À
n u m b e r of p o l y m e r i c systems e x h i b i t d o m a i n f o r m a t i o n . T h i s results i n some p o l y m e r i c m a t e r i a l b e i n g c o n f i n e d i n regions b e t w e e n
domains.
the
T h e d e f o r m a t i o n p r o p e r t i e s of these systems d e p e n d o n
the
types of p o l y m e r chains l y i n g b e t w e e n the d o m a i n s , as w e l l as o n the shape a n d spatial arrangement
of the d o m a i n s .
S e v e r a l theories
have
b e e n p r o p o s e d to date f o r the c o n t r i b u t i o n of the i n t e r d o m a i n m a t e r i a l to different d e f o r m a t i o n properties i n s e m i c r y s t a l l i n e p o l y m e r s a n d b l o c k c o p o l y m e r s . W e w i l l present a n d a n a l y z e these theories h e r e i n . Semicrystalline Modulus.
Polymers Jackson
et a l . ( I )
calculate
the c o n t r i b u t i o n of
amor-
p h o u s m a t e r i a l to the shear m o d u l u s of a s e m i c r y s t a l l i n e p o l y m e r
by
a s s u m i n g that o n l y tie chains ( chains w h o s e ends are a t t a c h e d to different crystallites ) c o n t r i b u t e G a u s s i a n statistics.
to t h e m o d u l u s a n d that these chains
follow
T h e y assume that the chains d e f o r m affinely.
p r e d i c t e d m o d u l u s values
are
lower than
the
observed
values.
0-8412-0457-8/79/33-176-231$05.00/0 © 1979 American Chemical Society
Cooper and Estes; Multiphase Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
The The
232
MULTIPHASE POLYMERS
authors note three possible effects w h i c h w e r e n e g l e c t e d i n t h e i r treatment:
(a)
medium.
the
crystallites
act
as
rigid
inclusions
within
an
shape, a n d t h e i r d i s p e r s i o n i n the s u r r o u n d i n g m e d i u m ; ( b ) m o l e c u l e s are h i g h l y e x t e n d e d statistics;
elastic
T h i s effect d e p e n d s o n the v o l u m e f r a c t i o n of crystals, t h e i r
(c)
a n d therefore
the
tie
d o not f o l l o w G a u s s i a n
t h e crystals i n t r o d u c e interfaces w h i c h are
impenetrable
to the a m o r p h o u s chains a n d t h e r e b y l i m i t their configurations. N i e l s o n a n d S t o c k t o n (2) crystallites as r i g i d Jackson
fillers
a t t e m p t to a c c o u n t f o r the effect of the
b y m u l t i p l y i n g t h e shear m o d u l u s result
et a l . b y the " f i l l e r effect" c o r r e c t i o n t e r m w h i c h h a d
of
been
d e r i v e d b y G u t h a n d S m a l l w o o d f o r the Y o u n g s m o d u l u s . T h e p r e d i c t e d shear m o d u l u s values are s t i l l too l o w . T h e authors e x p l a i n this b y the fact t h a t : ( a ) at l o w c r y s t a l l i n i t i e s , a m o r p h o u s c h a i n entanglements
may
b e significant; ( b ) at h i g h c r y s t a l l i n i t i e s , the crystallites m a y i m p i n g e o n e a c h other a n d f o r m a c o n t i n u o u s c r y s t a l phase; a n d ( c ) at h i g h c r y s t a l linities, the tie chains m a y b e v e r y short a n d f o l l o w n o n - G a u s s i a n b e h a v i o r . T h e authors also p o i n t o u t that w h i l e e x p e r i m e n t a l l y the m o d u l u s of c r y s t a l l i n e p o l y m e r s decreases w i t h t e m p e r a t u r e , t h e o r y p r e d i c t s the opposite effect.
G a u s s i a n elasticity
H o w e v e r , if the d e g r e e of c r y s t a l -
l i n i t y decreases w i t h t e m p e r a t u r e t h e n one c a n p r e d i c t a negative t e m p e r a t u r e coefficient of the m o d u l u s , u s i n g G a u s s i a n statistics. K r i g b a u m et a l . (3)
a c c o u n t f o r the fact that tie m o l e c u l e s m a y b e i n
a h i g h l y e x t e n d e d state e v e n i n the absence of a n e x t e r n a l
macroscopic
strain, b y u s i n g inverse L a n g e v i n c h a i n statistics to c a l c u l a t e the Y o u n g ' s m o d u l u s . It is a s s u m e d that i n t h e u n d e f o r m e d state, the crystallites are randomly oriented
( i n the p r e v i o u s t w o theories, t h e a r r a n g e m e n t
of
crystallites is u n s p e c i f i e d , a l t h o u g h it is p r e s u m a b l y r a n d o m ). A n a d d i t i o n a l a s s u m p t i o n is that w h i l e the o v e r a l l d e f o r m a t i o n of the c r y s t a l l i n e p o l y m e r is affine, t h e crystallites themselves
semi-
d o not d e f o r m .
T h e r e f o r e , the d e f o r m a t i o n of t h e tie m o l e c u l e s is greater t h a n affine. C r y s t a l shear a n d r e o r i e n t a t i o n u n d e r d e f o r m a t i o n are b o t h
neglected.
T h e expression w h i c h they o b t a i n f o r the Y o u n g ' s m o d u l u s contains
the
d e g r e e of c r y s t a l l i n i t y a n d the t o t a l n u m b e r of segments i n the s e m i c r y s t a l l i n e c h a i n as v a r i a b l e s . It p r e d i c t s that the Y o u n g ' s m o d u l u s s h o u l d increase w i t h c r y s t a l l i n i t y a n d that, for constant c r y s t a l l i n i t y , the m o d u l u s is p r o p o r t i o n a l to t e m p e r a t u r e . using a relation between
T h e i r t h e o r y is f u r t h e r d e v e l o p e d
the degree of c r y s t a l l i n i t y a n d
by
temperature
w h i c h they h a d p r e v i o u s l y d e r i v e d f o r the c r y s t a l l i z a t i o n of f o l d e d - c h a i n crystallites i n a n i s o t r o p i c , u n d e f o r m e d s a m p l e (4).
T h e final expression
p r e d i c t s a decrease i n the Y o u n g s m o d u l u s w i t h i n c r e a s i n g t e m p e r a t u r e . L o h s e a n d G a y l o r d (5)
h a v e e x a m i n e d t h e role of c r y s t a l l i t e i m p e n -
e t r a b i l i t y o n the Y o u n g s m o d u l u s . T h e crystallites are a s s u m e d to f o r m p a i r s of p a r a l l e l l a m e l l a e . T w o m a c r o s c o p i c m o r p h o l o g i e s are c o n s i d e r e d :
Cooper and Estes; Multiphase Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
12.
GAYLORD
the
stacked
233
Deformation of Multiple Domain Polymers lamellae
structure,
i n w h i c h a l l the lamellae
pairs are
p a r a l l e l to e a c h other, a n d t h e s p h e r u l i t i c s t r u c t u r e , i n w h i c h t h e l a m e l l a e p a i r s a r e d i s t r i b u t e d i n a s p h e r i c a l l y s y m m e t r i c m a n n e r a l o n g the r a d i i of the s p h e r u l i t e .
I t is a s s u m e d , as i n t h e K r i g b a u m et a l . m o d e l , t h a t
the o v e r a l l d e f o r m a t i o n o f the m a t e r i a l is affine, w h i l e the l a m e l l a e a r e undeformable. model.
C r y s t a l shear a n d r e o r i e n t a t i o n are i n c o r p o r a t e d i n t o t h e
T o c o n s i d e r i m p e n e t r a b i l i t y effects, a l a m e l l a r surface, a l t h o u g h
n o t i n f i n i t e , is a p p r o x i m a t e d b y a n i n f i n i t e p l a n e . statistics
The
configurational
of a c h a i n confined b y infinite, parallel, impenetrable
(6; 7) a r e t h e n u s e d .
I t is p r e d i c t e d that c i l i a ( c h a i n s
w i t h one
walls end
a t t a c h e d to a c r y s t a l surface ), loops ( chains w i t h b o t h ends a t t a c h e d t o the s a m e c r y s t a l s u r f a c e ) ,
a n d u n a t t a c h e d chains a l l c o n t r i b u t e t o t h e
Y o u n g ' s m o d u l u s as a result o f d o m a i n i m p e n e t r a b i l i t y effects a n d t h a t t h e i r c o n t r i b u t i o n decreases w i t h d e c r e a s i n g c h a i n c o n t o u r l e n g t h . T i e m o l e c u l e s , w h e n t h e y are v e r y large, c o n t r i b u t e t o the Y o u n g ' s m o d u l u s i n t h e same m a n n e r as the o t h e r types o f chains, b u t w h e n the c o n t o u r l e n g t h b e c o m e s v e r y s m a l l , t h e m o d u l u s b e g i n s t o rise w i t h a f u r t h e r decrease i n c h a i n c o n t o u r l e n g t h .
T h e m o d u l u s is a l w a y s greater i n a
s t a c k e d l a m e l l a r s t r u c t u r e t h a n i n a s p h e r u l i t e f o r e a c h t y p e of a m o r p h o u s chain.
T h e authors also c a l c u l a t e the Y o u n g ' s m o d u l u s d e p e n d e n c e o n
temperature,
at constant
crystallinity, w i t h
the use of the Rotational
I s o m e r i c State scheme.
I t is p r e d i c t e d that the Y o u n g ' s m o d u l u s o f c i l i a ,
loops, a n d unattached
chains s h o u l d a l w a y s decrease w i t h
temperature.
increasing
T h e tie m o l e c u l e shows the same b e h a v i o r at l o w t e m p e r a -
tures b u t at h i g h temperatures f o l l o w s G a u s s i a n b e h a v i o r as its Y o u n g ' s m o d u l u s b e g i n s t o rise w i t h f u r t h e r t e m p e r a t u r e increases. t i o n o f a decrease i n m o d u l u s w i t h i n c r e a s i n g t e m p e r a t u r e c r y s t a l l i n i t y agrees w i t h e x p e r i m e n t .
The predicat constant
N o a t t e m p t is m a d e b y the authors
to relate the d e g r e e o f c r y s t a l l i n i t y to the t e m p e r a t u r e .
Block
Copolymers
Stress—Strain Relation for Uniaxial Extension.
Leonard ( 8 ) has
c a l c u l a t e d the stress-strain r e l a t i o n f o r a n i n t e r d o m a i n tie m o l e c u l e i n a s p h e r i c a l d o m a i n m o r p h o l o g y . H e first w r i t e s the t o t a l e n t r o p y o f d e f o r m a t i o n as the s u m o f the e n t r o p y o f d e f o r m a t i o n w h i c h o n e gets f r o m G a u s s i a n e l a s t i c i t y t h e o r y a n d the e n t r o p y o f d o m a i n f o r m a t i o n ( i n b o t h terms, the extension ratio refers t o the i n t e r d o m a i n s t r a i n a n d n o t to t h e macroscopic
strain).
This
s u m is t h e n
S m a l l w o o d " f i l l e r effect" c o r r e c t i o n t e r m .
multiplied b y the Guth
and
T h e stress is c a l c u l a t e d b y
d i f f e r e n t i a t i n g the e n t r o p y e x pre ssi on w i t h respect t o the l e n g t h o f t h e d e f o r m e d i n t e r d o m a i n r e g i o n . T h e r a t i o o f the l e n g t h o f the u n d e f o r m e d i n t e r d o m a i n r e g i o n to the i n i t i a l o v e r a l l s a m p l e l e n g t h is set e q u a l t o t h e
Cooper and Estes; Multiphase Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
234
MULTIPHASE
POLYMERS
v o l u m e f r a c t i o n o f i n t e r d o m a i n m a t e r i a l , r a i s e d to the o n e - t h i r d p o w e r . L e o n a r d s c a l c u l a t i o n contains a great m a n y flaws: i t is incorrect to use t h e e n t r o p y of d o m a i n f o r m a t i o n i n t h e stress c a l c u l a t i o n because d o m a i n f o r m a t i o n occurs p r i o r to d e f o r m a t i o n a n d does n o t c h a n g e thereafter; the stress c a l c u l a t i o n s h o u l d b e p e r f o r m e d b y d i f f e r e n t i a t i n g w i t h respect to t h e m a c r o s c o p i c s a m p l e d i m e n s i o n rather t h a n t h e i n t e r d o m a i n d i m e n s i o n ; t h e expression r e l a t i n g the i n t e r d o m a i n a n d m a c r o s c o p i c
extension
ratios is i n c o r r e c t since i t fails to p r e d i c t that the f o r m e r q u a n t i t y b e c o m e s u n i t y as t h e latter q u a n t i t y goes to one; a n d , the r e l a t i o n g i v e n f o r t h e ratio of the i n i t i a l o v e r a l l s a m p l e l e n g t h to t h e i n i t i a l i n t e r d o m a i n l e n g t h fails to change w h e n t h e n u m b e r a n d size of the d o m a i n s are v a r i e d w h i l e the total v o l u m e f r a c t i o n of i n t e r d o m a i n m a t e r i a l is k e p t constant. Meier
(9)
has m o d e l e d the s p h e r i c a l d o m a i n m o r p h o l o g y b y a
s i m p l e c u b i c lattice i n w h i c h d o m a i n s are a r r a n g e d o n the lattice sites. T h e tie m o l e c u l e s r u n b e t w e e n nearest-neighbor d o m a i n s a n d are a s s u m e d to b e c o n f i n e d b y pairs of infinite, p a r a l l e l w a l l s .
T h e extension
f o r t h e i n t e r d o m a i n r e g i o n is set e q u a l to the m a c r o s c o p i c
ratio
extension
ratio d i v i d e d b y t h e v o l u m e f r a c t i o n of the i n t e r d o m a i n m a t e r i a l .
The
r a t i o of the i n i t i a l i n t e r d o m a i n d i m e n s i o n to t h e d o m a i n d i m e n s i o n is set e q u a l to the ratio of the v o l u m e fractions of the i n t e r d o m a i n a n d d o m a i n material.
U s i n g this t h r e e - c h a i n m o d e l , M e i e r calculates t h e stress-strain
r e l a t i o n b y d i f f e r e n t i a t i n g his e n t r o p y
expression
w i t h respect to t h e
i n t e r d o m a i n extension ratio. T h e M e i e r c a l c u l a t i o n has some d i f f i c u l t i e s : the i n t e r d o m a i n d e f o r m a t i o n fails to v a n i s h i n the absence of a n a p p l i e d m a c r o s c o p i c d e f o r m a t i o n ; the r e l a t i o n b e t w e e n the ratio of t h e d o m a i n d i m e n s i o n to the i n i t i a l i n t e r d o m a i n d i m e n s i o n a n d the ratio of v o l u m e fractions is incorrect; a n d the d i f f e r e n t i a t i o n s h o u l d b e c a r r i e d o u t w i t h respect to the m a c r o s c o p i c extension ratio. Gaylord and Lohse
(10)
have c a l c u l a t e d t h e stress-strain
relation
f o r c i l i a a n d tie m o l e c u l e s i n a s p h e r i c a l d o m a i n m o r p h o l o g y u s i n g t h e same t y p e of t h r e e - c h a i n m o d e l as M e i e r . It is a s s u m e d that the o v e r a l l s a m p l e d e f o r m a t i o n is affine w h i l e the d o m a i n s are u n d e f o r m a b l e .
It is
p r e d i c t e d that t h e stress increases r a p i d l y w i t h i n c r e a s i n g s t r a i n f o r b o t h types of chains.
T h e rate of stress rise is greatly a c c e l e r a t e d as the r a t i o
of t h e d o m a i n thickness to t h e i n i t i a l i n t e r d o m a i n separation increases. T h e results i n d i c a t e that i t is n o t correct to use the stress-strain
equation
o b t a i n e d b y G a u s s i a n elasticity theory, e v e n i f i t is m u l t i p l i e d b y a " f i l l e r effect" c o r r e c t i o n
term.
N o connection
d i m e n s i o n s a n d t h e v o l u m e fractions
is m a d e
between
the
initial
of the d o m a i n a n d i n t e r d o m a i n
m a t e r i a l i n this theory. Partial Molar E l a s t i c Free Energy of Swelling. L e o n a r d ( 8 ) c a l c u l a t e d t h e p a r t i a l m o l a r elastic free e n e r g y of s w e l l i n g f o r a n i n t e r d o m a i n tie m o l e c u l e i n a s p h e r i c a l d o m a i n m o r p h o l o g y . H e i n c l u d e d t h e e n t r o p y
Cooper and Estes; Multiphase Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
12.
GAYLORD
235
Deformation of Multiple Domain Polymers
of d o m a i n f o r m a t i o n a n d the e n t r o p y t e r m f r o m G a u s s i a n elasticity t h e o r y . T h e c u b e of the extension r a t i o f o r i s o t o p i c s w e l l i n g is t a k e n e q u a l t o the i n v e r s e of the v o l u m e f r a c t i o n of p o l y m e r i n the i n t e r d o m a i n r e g i o n . L e o n a r d stated t h a t h i s expression reduces to t h e F l o r y - R e h n e r e q u a t i o n w h e n the d o m a i n size goes to z e r o . f o r m a t i o n b y L e o n a r d is i n c o r r e c t .
T h e use of the e n t r o p y of d o m a i n
A d d i t i o n a l l y , L e o n a r d ' s final e q u a t i o n
f o r the p a r t i a l m o l a r elastic free e n e r g y of s w e l l i n g is i n c o r r e c t l y w r i t t e n . T h e a c t u a l expression n e v e r r e d u c e s to t h e F l o r y - R e h n e r e q u a t i o n ,
but
is p o s i t i v e o v e r t h e entire r a n g e of i n t e r d o m a i n p o l y m e r v o l u m e f r a c t i o n a n d goes to zero as the v o l u m e f r a c t i o n b e c o m e s zero. M e i e r (11 ) considers the s w e l l i n g of a tie m o l e c u l e i n b l o c k c o p o l y mers w i t h l a m e l l a r , c y l i n d r i c a l , a n d s p h e r i c a l d o m a i n m o r p h o l o g i e s .
The
statistics u s e d f o r the l a m e l l a r d o m a i n m o r p h o l o g y is that of a c h a i n confined between
a p a i r of i n f i n i t e , p a r a l l e l i m p e n e t r a b l e
c y l i n d r i c a l a n d s p h e r i c a l d o m a i n structures confined between respectively.
infinite, concentric
are
modeled
walls. by
a
cylinders and concentric
The chain
spheres,
T h e i n v e r s e of the v o l u m e f r a c t i o n of p o l y m e r i n the i n t e r -
d o m a i n r e g i o n is t a k e n e q u a l to the i s o t r o p i c s w e l l i n g r a t i o , r a i s e d to t h e first,
s e c o n d , a n d t h i r d p o w e r i n the l a m e l l a r , c y l i n d r i c a l , a n d s p h e r i c a l
domain morphologies, respectively.
T h e results i n d i c a t e t h a t the b e h a v i o r
of the p a r t i a l m o l a r elastic f r e e e n e r g y of s w e l l i n g as a f u n c t i o n of interd o m a i n p o l y m e r v o l u m e f r a c t i o n is q u i t e different i n t h e different m o r phologies.
A n objection
spherical domain models. constructs
can be
raised about
Meier's cylindrical a n d
M e i e r takes one p a r t i c u l a r d o m a i n a n d t h e n
a c o n f i n i n g s h e l l a r o u n d that d o m a i n , w h i c h passes t h r o u g h
nearest-neighbor
domains.
T h e tie c h a i n is t h e n c o n f i n e d b e t w e e n
d o m a i n a n d the s u r r o u n d i n g s h e l l . H o w e v e r , a tie m o l e c u l e is to t w o different d o m a i n s , a r o u n d e a c h of w h i c h one
the
attached
c a n construct
a
c o n f i n i n g s h e l l . A n i n t e r d o m a i n tie m o l e c u l e s h o u l d t h e r e f o r e b e c o n f i n e d to the v o l u m e d e f i n e d b y the i n t e r s e c t i o n of these t w o shells i f M e i e r s a p p r o a c h is to b e
consistent.
G a y l o r d a n d L o h s e (10)
h a v e e x a m i n e d the s w e l l i n g b e h a v i o r of tie
m o l e c u l e s , loops, c i l i a , a n d u n a t t a c h e d phologies.
chains i n different d o m a i n m o r -
E a c h c h a i n is c o n f i n e d b e t w e e n
a p a i r of i n f i n i t e , p a r a l l e l
i m p e n e t r a b l e w a l l s a l t h o u g h the d o m a i n s are n o t a s s u m e d to b e i n f i n i t e . T h e r e are one, t w o , a n d three o r t h o g o n a l p a i r s of p a r a l l e l w a l l s i n t h e l a m e l l a r , c y l i n d r i c a l , a n d s p h e r i c a l d o m a i n strucutres, r e s p e c t i v e l y .
The
r e l a t i o n b e t w e e n the s w e l l i n g extension r a t i o a n d the i n t e r d o m a i n p o l y m e r f r a c t i o n f o r the different d o m a i n m o r p h o l o g i e s is the s a m e as that u s e d by Meier.
T h e results i n d i c a t e t h a t c i l i a , l o o p s , a n d u n a t t a c h e d
a l l f a v o r s w e l l i n g o v e r t h e entire range of d i l u t i o n . d o m a i n separation
chains
If t h e i n i t i a l i n t e r -
is not too l a r g e r e l a t i v e to the i n t e r d o m a i n
chain
c o n t o u r l e n g t h , a tie m o l e c u l e also w i l l f a v o r s w e l l i n g at l o w degrees of
Cooper and Estes; Multiphase Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
MULTIPHASE POLYMERS
236
s w e l l i n g b u t w i l l o p p o s e s w e l l i n g a t h i g h levels o f d i l u t i o n .
W h e n the
tie c h a i n is short o r t h e i n i t i a l i n t e r d o m a i n s e p a r a t i o n is large, t h e curves are s i m i l a r t o those o b t a i n e d b y M e i e r , a n d s w e l l i n g is a l w a y s o p p o s e d . Young's Modulus.
Gaylord a n d Lohse
(10)
have
examined the
Y o u n g s m o d u l u s of t h e v a r i o u s types o f i n t e r d o m a i n c h a i n s i n t h e different d o m a i n m o r p h o l o g i e s , u s i n g t h e m o d e l d e s c r i b e d i n the p r e v i o u s section. T h e results i n d i c a t e that i n a n y g i v e n m o r p h o l o g y , t h e Y o u n g s m o d u l u s behavior of the loop, cilium, a n d unattached
c h a i n a l l arise f r o m t h e
i m p e n e t r a b i l i t y o f t h e d o m a i n s a n d decreases w i t h
decreasing
chain
l e n g t h . T h e t i e m o l e c u l e s h o w s this same b e h a v i o r a t l o n g c h a i n c o n t o u r lengths
b u t a t sufficiently s m a l l c h a i n l e n g t h b e h a v e s i n a G a u s s i a n
elastic m a n n e r .
I t is p r e d i c t e d that the Young's m o d u l u s is greatest f o r a
l a m e l l a r d o m a i n structure s t r e t c h e d n o r m a l to t h e l a m e l l a r p l a n e . T h e c y l i n d r i c a l d o m a i n structure s t r e t c h e d n o r m a l to t h e c y l i n d r i c a l axis has a l o w e r m o d u l u s , a n d t h e m o d u l u s o f t h e s p h e r i c a l d o m a i n s t r u c t u r e is even lower.
T h e m o d u l u s is l o w e s t f o r c y l i n d e r s s t r e t c h e d
along the
c y l i n d r i c a l axis a n d l a m e l l a e s t r e t c h e d a l o n g t h e l a m e l l a r p l a n e .
I t also
is p r e d i c t e d that t h e m o d u l u s increases r a p i d l y w i t h i n c r e a s i n g v o l u m e f r a c t i o n o f d o m a i n m a t e r i a l a n d that, a t l o w temperatures, t h e m o d u l u s decreases w i t h i n c r e a s i n g t e m p e r a t u r e .
T h e s e last t w o p r e d i c t i o n s a r e i n
agreement w i t h experiment. Acknowledgment The
author
is a p p r e c i a t i v e
of m a n y
e n l i g h t e n i n g discussions o n
v a r i o u s aspects o f this w o r k w i t h D a v i d J . L o h s e .
This w o r k was sup-
ported, i n part, b y the U . S . E n e r g y Research a n d D e v e l o p m e n t A d m i n i s t r a t i o n u n d e r contract E R D A - E Y - 7 6 - C - 0 2 - 1 1 9 8 . Literature
Cited
1. Jackson, J. B., Flory, P. J., Chaing, R., Richardson, M . J., Polymer (1963) 4, 237. 2. Nielson, L . E . , Stockton, F. D . , J. Polym. Sci., Part A (1963) 1, 1995. 3. Krigbaum, W . R., Roe, R. J., Smith, K. J., Jr., Polymer (1964) 5, 533. 4. Roe, R. J., Smith, K. J., Jr., Krigbaum, W . R., J. Chem. Phys. (1961) 35, 1306. 5. Lohse, D . J., Gaylord, R. J., Polym. Eng. Sci. (1978) 18, 512. 6. Gaylord, R. J., Lohse, D . J., J. Chem. Phys. (1976) 65, 2779. 7. Lohse, D . J., Gaylord, R. J., J. Chem. Phys. (1977) 66, 3843. 8. Leonard, W . J., Jr., J. Polym. Sci., Polym. Symp. (1976) 54, 237. 9. Meier, D . J., Polym. Prepr., Am. Chem. Soc., Div. Polym. Chem. (1973) 14(1), 280. 10. Gaylord, R. J., Lohse, D . J., Polym. Eng. Sci. (1978) 18, 359. 11. Meier, D . J., J. Appl. Polym. Symp. (1974) 24, 67. RECEIVED April 14, 1978.
Cooper and Estes; Multiphase Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
