Physical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 76

where C is the temperature-independent preexponential factor, E is the activation
energy, and R is the gas constant. It follows from Eq. 3.14 that

dln
d

τ
1

(3.15)

ER = .

−

T

This means that for a noncooperative process, the plot of ln ˄ versus T −1 should be a
straight line whose slope is E/R .
The temperature dependence of a cooperative process, such ʱ -relaxation, fol-
lows the Vogel-Tammann-Fulcher (VTF) equation:

A
TT

(3.16)

τ= B exp ,

−

0

where B is the temperature-independent preexponential factor, A is a constant, and
T 0 is a reference temperature, typically well below T g . By using the same principle
as in Eq. 3.15, we can derive the activation energy from Eq. 3.16:

2

ER AT
TT

(3.17)

= .

( )

2

−

0

Equation 3.17 suggests that the activation energy and, thus, the slope of the plot of
ln ˄ versus T −1 increase with decreasing temperature. A similar result is obtained [ 32 ]
from the Williams-Landel-Ferry (WLF) equation:

2

CCT
CTT

(3.18)

12

E = 2 303 . R ,

2

( +− )

2 g

where C 1 = 17.44 and C 2 = 51.6 are universal constants. Both VTF and WLF equa-
tions are applied commonly to describe the temperature dependence of viscosity of
supercooled liquids. Although the equations can be used interchangeably, the WLF
equation is most frequently applied to polymers, whereas the VTF equation to low
molecular weight species.
The non-Arrhenius type of the temperature dependence (i.e., the VTF or WLF
dependence) breaks down in the vicinity of T g , below which the dependence takes
the Arrhenius form (Fig. 3.12a ). This is empirically found in a variety of liquids
[ 3 ]. An explanation of this phenomenon is provided by the theoretical results of
Di Marzio and Yang [ 33 ]. The change in the type of the temperature dependence
around T g has important implications for estimating experimental activation ener-
gies (Fig. 3.12b ). That is, above T g , the transition from the glass to liquid phase
should demonstrate large values of the activation energy that decrease with tem-
perature. However, when glass relaxes to supercooled liquid below T g , one should