Biomedical Engineering Reference
In-Depth Information
f
D
G
heter
G
homo
(7.9)
As shown in Fig.
7.2
b, we assume that nucleation occurs on a rough substrate
or foreign body with a radius of curvature
R
s
. The fluid phase is denoted by the
subscript
f
, 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 foreign particle of radius
R
s
is given, according
to (
7.8
), by
G
D
V
c
C
cf
S
cf
C
.
sf
sc
/S
sc
(7.10)
where
”
ij
is the surface free energy between phases
i
and
j
,and
is the volume per
structural unit. We have then
m
D
.
sf
sc
/
cf
cos
.
1
6
m
6
1/
(7.11)
G
heter
, we can substitute (
7.11
)andthe
expressions of
V
c
,
S
cf
,and
S
sc
into (
7.10
) and require that
To evaluate the critical free energy
@G
@r
D
0
(7.12)
Solving (
7.12
) should, in principle, give the value of the critical radius. Never-
theless, it will involve a complicated and tedious treatment. On the other hand, a
much simpler approach based on the thermodynamics principles can be adopted: it
is known that a critical nucleus is a stable nucleus with the maximum curvature for a
given thermodynamic condition. Under such an experimental condition, the size of
the critical nucleus is the same for homogeneous and heterogeneous nucleation due
to the Gibbs-Thomson effect [
52
-
55
]. We have then [
54
,
56
] the radius of critical
nuclei,
r
c
D
2
cf
(7.13)
Referring to Fig.
7.2
c and taking
s
s
s
R
0
D
R
r
c
D
R
cf
D
R
kT
.1
C
/
cf
ln
;
(7.14)
the free energy of formation of a critical nucleus is given according to (
7.9
)by
G
heter
D
G
homo
f.m;R
0
/
(7.15)
