Physical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 88

The equation is not very accurate [ 66 ] because it was derived assuming that the tem-
perature dependence of the relaxation time obeys the WLF equation [ 32 ]. However,
below T g , the relaxation processes tend to follow [ 33 , 67 ] the Arrhenius equation
that predicts much weaker temperature dependence. In addition, the rate of aging
can differ substantially for different materials. For instance, the aging data [ 68 ] for
polycarbonate and PS suggest that at similar T g − T, the former reaches equilibrium
almost ten times faster than the latter. Nonetheless, Eq. 3.26 can provide a reason-
able estimate for the magnitude of the aging time, i.e., hours, days, and months.
To perform the isoconversional calculations, experimentally measured curves
of the recovered enthalpy versus aging time need to be converted to the curves of
the conversion versus time. For any given aging time, the extent of aging, i.e., the
extent of conversion from the glass to supercooled liquid phase, is determined as:

∆
∆

Ht
H

a () ,

(3.27)

α=

∞

where Δ H ( t ) is the enthalpy measured at the aging time, t , and Δ H ∞ in the equilib-
rium (plateau) value. Since aging runs are conducted isothermally, the activation
energy can be evaluated straightforwardly by Eq. 3.28

ER t
T

dln
d

α

(3.28)

=

1 ,

α

−

i.e., as the slope of a plot of the natural logarithm of the time, t a , to reach a given
extent of aging, ʱ , against the reciprocal aging temperature. By repeating this
procedure for a series of the conversions, one obtains a dependence of E ʱ on ʱ .
The use of Eq. 3.28 requires determining the time to reach a given extent of con-
version at different aging temperatures. Unfortunately, it is impossible to measure
reliably small extents of conversion. The aging rate is the fastest in the initial moments
so that the smallest values of conversion are experimentally detected after only a few
minutes of aging. For instance, 4-min aging of Mt resulted in ʱ being about 0.27
(Fig. 3.23 ). The values of t ʱ can be found by interpolating the discrete experimental
dependence of ʱ versus t by the Kohlrausch-Williams-Watts (KWW) equation [ 3 , 4 ]

γ

=− −

t

α 1exp . (3.29)

τ

ef

The equation has two fit parameters: ˄ ef , which is the effective relaxation time, and
ʳ, which is the stretch exponent. The KWW equation is generally found to describe
accurately the relaxation kinetics of glasses, although it is commonly found that
[ 69 ] the parameter ʳ varies systematically with temperature. Once the values of ˄ ef
and ʳ are estimated, Eq. 3.29 can be used to find t ʱ for any ʱ .
The isoconversional plots of ln t ʱ versus T −1 for aging of Mt glass are seen in
Fig. 3.24 . The most remarkable feature of these plots is that their slopes increase