Physical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 98

1

2

3

T

Fig. 3.32 Temperature dependence of nucleation rate (1 exp(− ΔG * / RT ), 2 exp(− E D / RT ), 3 product
of 1 and 2). (Adapted from Vyazovkin [ 91 ] with permission of Elsevier)

where E D is the activation energy of the diffusion process. Unlike the Δ G * term,
the E D term represents a typical Arrhenius temperature dependence (Fig. 3.32 ). The
product of these two terms (Eq. 3.42) yields a temperature dependence that dem-
onstrates a maximum in the nucleation rate. Below T max , the process becomes con-
trolled by diffusion that results in a dramatic decrease of the nucleation rate.
If for a particular compound the maximum nucleation rate is not very large, the
respective melt can be readily turned into the glass phase on cooling. Glasses can
crystallize on heating. Once the temperature rises above the glass transition tem-
perature, the glass relaxes turning into the metastable supercooled liquid. As tem-
perature continues to rise, the molecular mobility increases, promoting nucleation
and crystallization of the supercooled liquid. The glass crystallization on heating is
frequently called “cold crystallization.” Cold crystallization normally occurs below
T max . In this temperature range, the nucleation rate increases with increasing tem-
perature because the rate is limited by diffusion. The corresponding Arrhenius plot
(Fig. 3.31 ) has the regular negative slope that represents a positive (or Arrhenian)
temperature dependence. Fitting glass crystallization data to the Arrhenius equation
yields a positive value of the effective activation energy. Note that the slope of the
Arrhenius plot decreases with increasing temperature.
To better understand the temperature dependence of the effective activation en-
ergy for the process of nucleation in the melt and glass crystallization, we can use
Eq. 3.42 to derive a theoretical expression for E versus T . The effective activation
energy is generally defined as the logarithmic derivative of the rate constant with
respect to the reciprocal temperature:

ER wT
T

dln
d

() .
1

(3.43)

=−

−