Chemistry Reference
In-Depth Information
where the activation energy,
, may be taken as 0.12 MJ/mol, which is a typical value for
relaxations in semicrystalline polymers and in glassy polymers at temperatures below
ΔH
T
g
.
(The shift factor could also have been calculated from the WLF relation if the temperatures
had been around
T
g
of the polymer.) In the present case:
"
#
1
353
2
a
T
5exp
0
:
12
3
10
6
8:31
1
298
50:53310
23
The measurement time required at 80
Cis[0.53310
20
] [10 years] [365 days/year]
[24 hours/day] 545.3 hours, to approximate 10 years' service at 25
C.
4.8
Dynamic Mechanical Behavior at Thermal Transitions
The storage modulus
G
0
(
) behaves like a modulus measured in a static test and
decreases in the glass transition region (cf.
Fig. 4.8
). The loss modulus
G
ω
v
ω
(
) and tan
δ
go through a maximum under the same conditions, however.
Figure 4.20
shows
some typical experimental data.
T
g
can be identified as the peak in the tan
or the
loss modulus trace. These maxima do not coincide exactly. The maximum in tan
δ
is
at a higher temperature than that in
Gv
δ
is the ratio of
G
0
(
(
ω
), because tan
δ
ω
)and
Gv
(
)(
Eq. 4-49
) and both these moduli are changing in the transition region. At low
frequencies (about 1 Hz) the peak in tan
ω
is about 5
C warmer than
T
g
from static
measurements or the maximum in the loss modulus
δ
temperature curve.
or the loss modulus at the glass-to-
rubber transition is explained as follows. At temperatures below
T
g
the polymer
behaves elastically, and there is little or no flow to convert the applied energy
into internal work in the material. Now
h
, the energy dissipated as heat per unit
volume of material per unit time because of flow in shear deformation, is
The development of a maximum in tan
δ
2
h
5τ
d
γ=
dt
5ηð
d
γ=
dt
Þ
(4-73)
[To check this equation by dimensional analysis in terms of the fundamental
units mass (
m
), length (
l
), and time (
t
):
ml
2
1
t
2
2
t
2
1
ml
2
1
t
2
1
mlt
2
2
τ 5
;
d
γ=
dt
5
;
η5
;
force
5
;
ml
2
t
2
2
ml
2
1
t
2
3
work
5
;
work
=
volume
=
time
5
5
Eq
: ð
4-73
Þ:
Thus the work dissipated is proportional to the viscosity of the material at
fixed straining rate
d
/dt
are
vanishingly small and
h
is negligible. As the structure is loosened in the transition
region,
γ
/dt
. At low temperatures,
η
is very high but
γ
and
d
γ
η
decreases but
d
γ
/dt
becomes much more significant so that
h
(and the
loss modulus and tan
) increases. The effective straining rate of polymer seg-
ments continues to increase somewhat with temperature above
T
g
but
δ
η
, which
measures the resistance to flow, decreases at the same time. The net result is a
diminution of damping and a fall-off of the magnitudes of the storage modulus
and tan
δ
.
