Biomedical Engineering Reference
In-Depth Information
The free energy change to form a cluster of
n
D
1, 2, 3, ::: molecules can be
found by thermodynamic considerations, since it is defined as
G
D
G
fin
G
ini
(2.3)
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 the cluster
formation, respectively).
G
D
n
C
ˆ
n
;
(2.4)
where ˚
n
is the total surface energy of the
n
-sized cluster.
G
reaches its maximum
G
*
at
r
D
r
c
,or
n
D
n
*
. The cluster of
n
*
molecules
is the critical nucleus,
r
c
is the radius of curvature of the critical nuclei, and
G
*
is
the nucleation barrier. One of the major problems in the nucleation theory is to find
G
*
, which, physically, is the energy barrier of nucleation.
The occurrence of foreign bodies in the nucleation system normally reduces the
interfacial (or surface) free energy, which therefore will also lower the nucleation
barrier. Under a given condition, if the probability of creating a nucleus is
homogeneous throughout the system, the nucleation is considered to be homo-
geneous nucleation. Otherwise, it is considered to be heterogeneous nucleation.
In heterogeneous nucleation on solid or liquid surfaces, microclusters, dusts, and
macromolecules, the property of these foreign bodies is an additional factor upon
which this barrier and rate depend. Let G
homo
, be the homogeneous nucleation
barrier, and G
hetero
be the heterogeneous nucleation barrier (the nucleation barrier
in the presence of a foreign body.) We can then define here a factor describing the
lowering of the nucleation barrier due to the foreign body:
G
heter
f
D
G
homo
:
(2.5)
In the following, G
heter
and
f
are derived [
49
].
AsshowninFig.
2.4
b, we assume that nucleation occurs at a foreign body with
aradiusof
R
s
. The mother phase is represented by subscript
f
, the cluster of the
crystalline phase by
c
, and the foreign body by
s
. If we denote the volume by
V
and the surface area of the foreign body by
S
, then the free energy of formation
of a cluster of radius
r
on a nucleating particle of radius
R
s
is given, according to
(
2.4
), by
G
D
V
c
=
C
cf
S
cf
C
.
sf
sc
/S
sc
;
(2.6)
where
ij
is the surface free energy between phases
i
and
j
and is the volume per
structural unit. Assume that the concept of contact angle can still be applied in this
case. We have then
sf
sc
cf
m
D
cos .
1
6
m
6
1/:
(2.7)
