Physical Processes - Isoconversional Kinetics of Thermally Stimulated Processes - page 100

Fig. 3.34 Theoretically
predicted dependencies of
E ʱ on ʱ for melt and glass
crystallization. (Adapted
from Vyazovkin [ 91 ] with
permission of Elsevier)

ER t
T

∂ ln( dd

1

α / )

=− .

(3.47)

α

−

∂

α

By virtue of the isoconversional principle (Eq. 1.12), the right-hand side of Eq. 3.47
is equal to that of Eq. 3.45. It means that E ʱ has the same form as E in Eq. 3.45. Since
an increase in ʱ is equivalent to a decrease in T for the melt crystallization and to in-
crease in T for the glass crystallization, the E ʱ versus ʱ dependencies take the forms
displayed in Fig. 3.34 . Having the opposite signs, the E ʱ values for the melt and glass
crystallization tend toward zero as crystallization progresses from ʱ = 0 to 1.
To conclude this section, two short comments need to be made about limitations
of the above derivations. First, the derivations have been performed for a spherical,
i.e., three-dimensional type of nucleus. Changing the assumption about the nucleus
shape to a two-dimensional type, such as a disk, would introduce important changes
into some of the equations [ 93 ]. For instance, Δ G * would be inversely proportional to
Δ G V , not to (Δ G V ) 2 as in the case of the three-dimensional type of nucleus (Eq. 3.36).
Consecutively, (Δ T ) 2 in Eq. 3.44 would change to Δ T . Second, the derivations have
been made under the assumption of homogeneous nucleation, i.e., when a nucle-
us is formed inside the melt phase. However, the nuclei can form on the substrate
(e.g., solid impurity, container wall, etc.), i.e., heterogeneously. The free energy bar-
rier of heterogeneous nucleation is substantially smaller than that of homogeneous
nucleation. For example, when a spherical nucleus is formed homogeneously, the free
energy barrier of its formation is proportional to the surface free energy of the entire
surface of the sphere. However, when the nucleus is formed at the substrate, it as-
sumes the shape of a spherical cup, which at the same radius has a smaller surface area
than the whole sphere. As a result, the free energy barrier to heterogeneous nucleation,
G * het , is always smaller than that for the homogeneous one, by some geometrical fac-
tor, f ( ʘ ) < 1:

* * ( .

∆ GGf

het

= ∆ Θ

(3.48)