Chemistry Reference
In-Depth Information
time,
, can be described using the Avrami
equation,
t
In Equation (3),
T
d
is the equilibrium
dissolution temperature of a chain with
infinite length (this parameter depends on
solvent and polymer type), T
S
is a super-
cooling temperature (to account for super-
cooling during Crystaf analysis), and
t
n
X
ð
t
Þ¼
1
exp
ð
k
Þ
(1)
where
are the Avrami constants.
This equation can be used to describe
crystallization from polymer melts and
from polymer solutions.
[12-14]
The fractio-
nation in Crystaf is not isothermal; how-
ever, for a group of homopolymers with
similar chain microstructures, the range of
crystallization temperatures during Crystaf
analysis is very narrow. Therefore, we first
make the assumption that both Avrami
parameters can be considered constant for
each homopolymer at each condition.
The Avrami exponent, n, is known to be
constant over a range of temperatures, so it
should be a constant over a range of cooling
rates as well. However, the Avrami para-
meter
n
and
k
is
a constant that is inversely proportional
to the enthalpy of fusion. To reduce the
number of parameters in the model, Equa-
tion (3) is rearranged as follows,
a
B
r
T
d
ðrÞ¼A
(4)
where
A ¼ T
d
T
S
and
B ¼ T
d
a
(5)
Since both
T
d
and
T
S
are essentially
constant for a given polymer/solvent com-
bination, we expect the value of the
parameter A to remain constant. Similarly,
because
depends greatly on the crystalliz-
ation temperature and, therefore, on cool-
ing rate. The parameter
k
is a constant that is inversely
proportional to the enthalpy of fusion, the
parameter B should also remain constant
for all cooling rates.
We also have to consider the difference
between the temperature measured in
the Crystaf oven and the temperature inside
the crystallization vessel (temperature lag,
T
l
). For our Crystaf instrument, the emp-
irical relation between the temperature
lag and cooling rate was reported in our
previous publication:
[10]
a
used in the model
for homopolymers should be considered
an effective or apparent parameter (i.e., an
average value measured over a range of
temperatures) at each cooling rate. We will
show that, despite of this simplification,
we can still use the model to describe the
Crystaf profiles very well for the polyethy-
lene samples studied in this investigation.
To use Equation (1), we must establish
a relationship between the crystallization
temperature,
k
T
C
, and the crystallization
T
l
¼
5
:
02
CR
0
:
05
(6)
time,
t. Generally, a slow, constant cooling
This empirical equation was established
for a cooling rate range of 0.02-1
rate,
, is used during Crystaf analysis.
Therefore, the relationship between crys-
tallization temperature and time can be
simply written as:
CR
C/min,
which covers the experimental conditions
studied in the present investigation.
Considering the temperature lag in the
system, the initial condition for Equation
(2) is T
C
(0)
¼
T
d
(r)-T
l
. We must integrate
Equation (2) with this initial condition to
obtain the relationship between crystalliz-
ation temperature and time:
8
d
T
C
d
t
¼CR
(2)
t
¼
0), the
crystallization temperature should be equal
to the dissolution temperature,
At the onset of crystallization (
T
d
. For the
case of homopolymers, T
d
is a function of
kinetic chain length, r. The modified Gibbs-
Thomson equation introduced by Beigzadeh
et al.
[7]
was used for this purpose:
t ¼
ð
T
d
ð
r
Þ
T
l
Þ
T
C
CR
(7)
We obtain our final equation relating
the degree of crystallinity as a function of
chain length, cooling rate, and crystallization
h
i
r
a
r
T
d
ðrÞ¼T
d
T
S
(3)
