Chapter
10
Deformation Kinetics of Cross-Linked Polymers
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T. S. Chow Xerox Corporation, Webster Research Center, Webster, NY 14580
A unified approach to the glass transition, viscoelastic response and yield behavior of crosslinking systems is presented by extending our statistical mechanical theory of physical aging. We have (1) explained the transition of a WLF dependence to an Arrhenius temperature dependence of the relaxation time in the vicinity of Tg, (2) derived the empirical Nielson equation for Tg, and (3) determined the Chasset and Thirion exponent (m) as a function of cross-link density instead of as a constant reported by others. In addition, the effect of crosslinks on yield stress is analyzed and compared with other kinetic effects -- physical aging and strain rate.
The time and temperature dependent properties of c r o s s l i n k e d polymers including epoxy resins ( 1 - 3 ) and rubber networks ( 4 - 7 ) have been studied i n the past. Crosslinking has a strong e f f e c t on the g l a s s t r a n s i t i o n temperature (Tg), on v i s c o e l a s t i c response, and on p l a s t i c deformation. Although experimental observations and e m p i r i c a l expressions have been made and proposed, respectively, progress has been slow i n understanding the nonequilibrium mechanisms responsible f o r the time dependent behavior. The purpose o f t h i s paper i s to e s t a b l i s h the fundamental l i n k s between the glass t r a n s i t i o n , v i s c o e l a s t i c r e l a x a t i o n , and y i e l d s t r e s s by i n v e s t i g a t i n g the r e l a x a t i o n processes i n polymers. The r e l a t i o n s h i p between temperature and relaxation time scale i s represented by a s h i f t factor ( a ) . At temperature Τ
0097-6156/88/0367-0124$06.00/0 © 1988 American Chemical Society In Cross-Linked Polymers; Dickie, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1988.
Downloaded by MICHIGAN STATE UNIV on February 18, 2015 | http://pubs.acs.org Publication Date: April 18, 1988 | doi: 10.1021/bk-1988-0367.ch010
10. C H O W
Deformation Kinetics of Cross-Linked Polymers
125
> Tg, the stress relaxation data can be described by the WLF temperature dependence (8). However, an Arrhenius type of dependence i s usually observed f o r v i s c o e l a s t i c response i n the glassy region. The phenomenon i s generally true f o r amorphous polymers as well as c r o s s l i n k i n g systems. A typical transition from a WLF dependence to an Arrhenius temperature dependence f o r an epoxy r e s i n (]_) i s shown i n Figure 1. The change i n the relaxation mechanism f o r deformation near Tg w i l l be explained by our physical aging theory (9-12) recently developed f o r amorphous polymers. The same basic approach w i l l be extended here to discuss the e f f e c t of c r o s s l i n k i n g as a chemical aging process. The relaxation time w i l l then be used to determine the above mentioned physical properties of crosslinked polymers. RELAXATION TIME On the basis of the idea of continuous conversion of the number of holes (free volumes) and the number of phonons i n a polymer l a t t i c e , we have introduced (2) the physical picture of quantized hole energy states with i = 1, 2, . . . L. The problem i s to determine the d i s t r i b u t i o n of the ensemble characterized by a set of hole numbers { n ^ with Σ ^ ^ = η. The r a t i o of ni/N = f i i s the i t h c o n t r i b u t i o n to the free volume f r a c t i o n ( f = E i f j . ) . Minimizing the excess Gibbs free energy due to hole introduction with respect to n^, the equilibrium d i s t r i b u t i o n of the free volume f r a c t i o n i s obtained (J)
ftT) = f f
exp
R VΤ
7]
Τ r
where ε" = Σι e i f j / f i s the mean hole energy, R i s the gas constant and the subscript r refers to the condition at Τ = T which i s a f i x e d quantity near Tg. The nonequilibrium glassy state, ô(t) = f ( t ) - f , i s determined by solving the k i n e t i c equations which describe the l o c a l motion of h o l e s i n response to m o l e c u l a r f l u c t u a t i o n s during v i t r i f i c a t i o n and physical aging. The s o l u t i o n i s (11) r
where t i s the physical aging time and q i s the cooling (< 0) rate. The relaxation function (J>(t) has been derived (10) as the p r o b a b i l i t y of the holes having not reached t h e i r equilibrium states f o r a quenched and annealed glass, and has the form e
In Cross-Linked Polymers; Dickie, R., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1988.
126
CROSS-LINKED POLYMERS
(3)
p
4>(t) = exp[-(t/x) ] where
β
defines
relaxation temperature, applied
the
time
χ
shape
in
stresses
relaxation
Downloaded by MICHIGAN STATE UNIV on February 18, 2015 | http://pubs.acs.org Publication Date: April 18, 1988 | doi: 10.1021/bk-1988-0367.ch010
time
= x
r
the
hole
3
is
instead
of
an
χ i s related
a
The
function
of
(δ), c r o s s l i n k d e n s i t y The
constant
macroscopic
t o t h a t o f the in
spectrum.
as
experimental
function.
a which r e s u l t s
r
as
energy
treated
nonequilibrium glassy state
Kohlrausch-Williams-Watts x
of
Equation
local
and
in
the
(global)
state
(λ) by
χ =
(11)
(4)
The has
above
to
When
be
δ
=
0,
equation acquire
equation
modified Equation
which deep
that
include 4
can
known
be
into
mechanism shown i n F i g u r e
1, we
Tg which i s approximated
the
be
to
insight
of
a
is
suggests
to
written valid
α
= εξ/ΗΤ 2
Γ
Γ
The
>
(H):
ε = 2.51
poly(vinyl been chosen state from
for the
shown
shear
(12)
calculated
slope:
E
in
cooling we
r
volume
d l o g a/âT
a
T
measurement
F o l l o w i n g E q u a t i o n 5,
2 r
the
the
WLF
order
to
deformation
4 i n the
vicinity
(5)
Equations
f
r
2
(J4_) w i t h rate
i n the
has
x
These
relaxation Figure
a
5
= 25 min
r
parameters
and
are
and
is
i n p u t parameters
= 0.0336 and
= 308K.
3
2,
by u s i n g the
the
(297K) =
glassy state
for have
equation
experimental 1.
We
effect
on
(T
-
little
o b t a i n the a c t i v a t i o n
= - RT
of In
2
of
curve
circles
the
form Tg.
in
( 13)
contribution.
Γ
prediction
(PVAc) w i t h
creep
that
equation
φΐ
Γ
the s o l i d
The
>
change
α (Τ - Τ ) + δ
t o d e s c r i b e the PVAc.
the
Τ
look at Equation
k c a l / m o l , β = 0.48,
acetate)
in
for
the
Γ
shown i n F i g u r e 2 as
Doolittle
by
lna= -
where
the
nonequilibrium
< Tg
of
data have the 10K).
energy
(6)
a In a(T, δ)/3Τ = (1 - μ) έ7βΤ
where
μ = _
-Ι α
(7)
άδ(Τ,δ)/6Τ
r
It
reaches
approaches ε/β?
Γ
=
a
constant
zero
155.6
to
for Τ 30.5
value
> T . g
kcal/mol
of The =
0.8
for
change
of
(1
-
Τ E
μ)?/βΓ
Γ
a
