Chemistry Reference
In-Depth Information
Equation 5.1 describes the nucleation rate in terms of the activation energy for molecular
transport from the amorphous (undercooled) melt phase to the growing nucleus (
Δ
G
a
)
G
∗
).
A
is the pre-exponential factor,
T
is
the temperature, and
k
B
is the Boltzmann constant.
and the thermodynamic barrier for nucleation (
Δ
G
∗
depends on the balance between
Δ
Δ
the energy required to form a new interface (
G
s
) and the bulk free energy difference
between the crystalline and amorphous phases (
Δ
G
v
):
Δ
G
*
Δ
G
s
Δ
G
v
:
(5.2)
Δ
G
s
is a positive (unfavorable to nucleation) quantity and is proportional to
r
2
, the radius
of the nucleus, and
Δ
G
v
is a negative (favorable to nucleation) quantity proportional
to
r
3
. For a spherical nucleus, Equation 5.2 can be expanded to
4
3
π
r
2
r
3
Δ
G
*
4
π
γ
Δ
G
v
;
(5.3)
where
is the interfacial tension between the nucleus and the supercooled liquid. From
Equation 5.3, it becomes apparent that there is a critical radius that must be exceeded for
the nucleus to be thermodynamically favorable and thus to persist.
The difference in free energy between the liquid and the crystal can be estimated
from enthalpy of fusion (
γ
Δ
H
fus
) and melting temperature (
T
m
) of the crystal [3]:
G
v
Δ
H
fus
T
m
T
T
T
2
m
Δ
:
(5.4)
It can be seen from Equations 5.1
5.4 that several variables are important in determining
the rate of nucleation from amorphous systems, including the heat of fusion of the
crystalline phase and the crystal
-
amorphous interfacial energy. Temperature clearly also
plays an extremely important role in determining the rate of nucleation, whereby an
increase in undercooling (
T
m
T
) increases the driving force for nucleation. In practice, it
is observed that the nucleation rate increases with the degree of undercooling, reaches a
maximum, and then decreases. This behavior arises because of the decreased molecular
mobility that occurs as the temperature decreases and the viscosity increases, as shown in
Figure 5.2. This is accounted for in Equation 5.1 by the kinetic term
-
G
a
that depends on
the molecular mobility of the crystallizing species. Thus, the observed nucleation rate
will be a complex interplay of thermodynamic, kinetic, and structural factors. In addition,
since homogeneous nucleation is rarely encountered in practice, the presence of
impurities, foreign particles, and surfaces can also impact the observed nucleation
behavior [2].
Δ
5.2.2 Crystal Growth
Once nucleation has occurred, stable nuclei undergo crystal growth, which involves the
successive transfer of molecules from the undercooled liquid phase to the surface of
the growing nuclei. Crystal growth from the melt state is inherently very complex, and
