Chemistry Reference
In-Depth Information
If a compliance were being measured at a series of temperatures
T
, the data
could be reduced to a reference temperature
T
by
Þ 5
ρð
T
Þ
T
ρð
T;
t
J
ð
T
1
;
t
T
1
J
(4-71)
T
1
Þ
a
T
where
(
T
1
) is the material density at temperature
T
1
.
It is common practice now to use the glass transition temperature measured by
a very slow rate method as the reference temperature for master curve construc-
tion. Then the shift factor for most amorphous polymers is given fairly well by
ρ
C
1
ð
T
2
T
g
Þ
log
10
a
T
52
(4-72)
C
2
1
T
2
T
g
where the temperatures are in Kelvin.
Equation (4-72)
is known as the WLF equa-
tion, after the initials of the researchers who proposed it
[7]
. The constants
C
1
and
C
2
depend on the material. “Universal” values are
C
1
5
51
.
6
C.
17
.
4 and
C
2
5
100
C. If a different reference
temperature is chosen, an equation with the same form as
Eq. (4-72)
can be used,
but the constants on the right-hand side must be reevaluated.
Accumulation of long-term data for design with plastics can be very inconve-
nient and expensive. The equivalence of time and temperature allows information
about mechanical behavior at one temperature to be extended to longer times by
using data from shorter time studies at higher temperature. It should be used with
caution, however, because the increase of temperature may promote changes in
the material, such as crystallization or relaxation of fabrication stresses that affect
mechanical behavior in an irreversible and unexpected manner. Note also that the
master curve in the previous figure is a semilog representation. Data such as that
in the left-hand panel is usually readily shifted into a common relation but it is
not always easy to recover accurate stress level values from the master curve
when the time scale is so compressed.
The following simple calculation illustrates the very significant temperature
and time dependence of viscoelastic properties of polymers. It serves as a conve-
nient, but less accurate, substitute for the accumulation of the large amount of
data needed for generation of master curves.
The expression given holds between
T
g
and
T
g
1
EXAMPLE 4-3
Suppose that a value is needed for the compliance (or modulus) of a plastic article for 10
years' service at 25
C. What measurement time at 80
C will produce an equivalent figure?
We rely here on the use of a shift factor, a
T
, and
Eq. (4-69)
. Assume that the temperature
dependence of the shift factor can be approximated by an Arrhenius expression of the form
exp
ΔH
R
1
T
2
1
T
0
a
T
5
