Biomedical Engineering Reference
In-Depth Information
During nucleation, if the probability of creating critical nuclei is uniform
throughout the system, nucleation is defined as
homogeneous nucleation
.Otherwise,
it is defined as
heterogeneous nucleation
. Notice that what has been mentioned
above is normally referred to as 3D nucleation. During crystal growth, the so-called
2D nucleation will take place at the growing crystal surfaces in order to create
new growing layers. When the growth of a crystal surface occurs under its critical
roughening temperature, there will be a non-zero free energy in correspondence to
the interfacial free energy in 3D. The so-called step free energy is associated with
the creation of a step of unit length at the surface. Such a crystal face has atomically
a smooth surface. Due to the step free energy, the creation of a new layer on the
existing layer of the crystal surface should overcome a free energy barrier, similar
to 3D nucleation barrier. If the crystals are free of screw or mixed dislocations, they
grow by the mechanism of 2D nucleation [
16
]andthegrowthrate
R
g
is largely
determined by the 2D nucleation rate. Although they are not exactly the same, both
3D and 2D nucleation share many common features in almost all aspects [
50
,
51
]
Therefore, the analysis on 2D nucleation can be applicable to 3D nucleation, and
vice versa.
As one of the most important factors, the nucleation rate
J
is determined by the
height of the free energy barrier, the so-called nucleation barrier. The free energy
change associated with the formation of a cluster of molecules can be found from
thermodynamic considerations, since it is defined as
G
D
G
fin
G
ini
(7.7)
for a system at constant pressure and temperature (
G
ini
and
G
fin
denotes the Gibbs
free energies of the system in the initial and final states before and after cluster
formation, respectively). If
M
is the number of solute molecules in the system, one
has then
G
D
n
C
ˆ
n
(7.8)
where
˚
n
is the total surface energy of the
n
-sized cluster (except for the nucleation
of bubbles when
˚
n
contains also pressure-volume terms).The function
G
reaches
n
*
(
r
and
r
c
are the radius and the critical radius
of the cluster, respectively). A cluster of
n
*
molecules is a critical nucleus,
r
c
is the
radius of curvature of that critical nucleus, and
G
*
its maximum
at
r
D
r
c
or
n
D
G
*
is the nucleation barrier.
The occurrence of a foreign body in the system normally reduces the interfacial
(or surface) free energy between the substrate and the nucleating phase; it will then
lower the nucleation barrier according to (
7.8
). Given
G
homo
, the homogeneous
G
heter
, the heterogeneous nucleation barrier, we can define
an interfacial correlation factor
f
, describing the reduction of the nucleation barrier
due to the occurrence of foreign body as
nucleation barrier, and
