Physical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 97

T=T m E=0 T=T g

1

2

T=T max

Melt:
E < 0

Glass:
E > 0

T -1

Fig. 3.31 Arrhenius plot for nucleation in temperature range T g - T m (1 melt nucleation, 2 glass
nucleation). (Adapted from Vyazovkin [ 91 ] with permission of Elsevier)

The nucleation rate constant is commonly expressed in the Arrhenius form:

*

− ∆ G

RT

(3.41)

wT w () exp = ,

0

where w 0 is the preexponential factor. However, the temperature dependence of the
nucleation constant is more complex than that of the regular rate constant (Eq. 1.2)
because Δ G * depends strongly on Δ T, whose magnitude changes with temperature.
This causes the Arrhenius plots of ln w ( T ) versus T −1 to be nonlinear (Fig. 3.31 ) [ 91 ].
Also, as supercooling decreases with decreasing temperature, both critical nucleus
size (Eq. 3.40) and energy barrier (Eq. 3.39) decrease so that the nucleation rate
constant increases. Therefore, it demonstrates a negative (or anti-Arrhenian) tem-
perature dependence. Figure 3.31 shows an Arrhenius plot for the nucleation rate.
The plot has a positive slope that corresponds to the negative temperature depen-
dence. When the melt crystallization data are fitted to the Arrhenius equation, the
fit yields a negative value of the effective activation energy. Also, the slope varies
strongly with the temperature, reaching infinity at T = T m (Eq. 3.39).
Just below the melting point, the nucleation rate quickly increases with decreas-
ing temperature (Fig. 3.32 ). However, the nucleation rate does not increase indefi-
nitely. It passes through a distinct maximum at a certain temperature, T max . Below
this temperature, the nucleation rate starts to decrease with decreasing temperature.
This happens because the molecular mobility decreases with temperature. The melt
becomes increasingly more viscous, creating an energy barrier, E D , to diffusion of
molecules across the phase boundary. Introduction of the respective energy term
into Eq. 3.41 gives rise to the Turnbull and Fisher equation [ 92 ]:

*

− ∆ G

RT

− E

RT

(3.42)

D

wT w () exp = exp ,

0