Physical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 80

ing the glass transition region, the molecular motion intensifies, and the free volume
increases, initiating the ʱ -process. The latter involves considerable cooperativity
between the molecules and, thus, a large energy barrier as reflected in the large
value of E at the initial stages of the transition (Figs. 3.12b and 3.15 ). As tempera-
ture continues to rise, the free volume continues to increase. The molecular packing
becomes increasingly looser, allowing the molecules to move less dependently, i.e.,
in a less cooperative fashion. This relieves the energetic constrains, and the activa-
tion energy decreases.
A decrease in E is consistent with the predictions of the VTF and WLF equations
(Eqs. 3.17 and 3.18). A similar trend is predicted by the Adam-Gibbs equation[ 45 ]

* ∆

z
kT

µ

τ = A exp , (3.21)

B

where k B is the Boltzmann constant, Δ ʼ is the activation energy per particle, and z *
is the number of particles that cooperatively rearrange. In Eq. 3.21, z * is inversely
proportional to the configurational entropy that increases with T so that both z * and
the effective activation energy (i.e., z * Δ ʼ ) decrease with T .
Note that even before the first applications [ 30 , 46 ] of isoconversional methods
to the glass transition kinetics, the trend for the activation energy to decrease with
increasing temperature was observed in other studies [ 47 - 49 ]. In them, the activa-
tion energy was determined from the shift in the value of T g with the heating rate in
accord with the equation proposed by Moynihan et al.: [ 42 , 43 ]

dln
d

| β

1

| ,

(3.22)

ER
T

=−

−

g

where ʲ can be the rate of heating or cooling. However, the value of T g can be
defined in the order of its increase as the onset temperature, the temperature of the
midpoint step in the heat flow, and the endset temperature. For the glass transition
of sorbitol, Angell et al. [ 47 ] have found that Eq. 3.22 gives rise to a significantly
larger E when T g is determined as the onset temperature than when T g is estimated
as the temperature of the heat capacity peak. A similar effect was reported by Lacey
et al. [ 48 ] for PS oligomer and side-chain polysiloxane and by Hancock et al. [ 49 ]
for some pharmaceutical glasses, including poly(vinylpyrrolidone) (PVP), IM, and
several sugars. The observed temperature dependence of the activation energy sug-
gests that the plot of ln ʲ versus T g −1 should be nonlinear. The nonlinearity can be
quite obvious when T g is measured in a wide range of the heating rates as illustrated
in Fig. 3.16 for the glass transition in PS [ 30 ]. From this plot, we can see again that
the activation energy of the glass transition decreases with increasing temperature.
Our numerous applications of the isoconversional method to the glass transition
in a variety of systems indicate that the obtained activation energies are in reason-
able agreement with the activation energies obtained by other techniques, such as