Biomedical Engineering Reference
In-Depth Information
with
16
cf
2
3ŒkT
G
homo
D
(7.16)
.1
C
/
2
ln
"
2
#
1
mR
0
w
2
R
0
m
w
R
0
m
w
f.m;R
0
/
D
1
2
C
1
C
1
2
R
03
2
3
C
2
2
mR
02
R
0
m
C
3
1
(7.17)
w
and
h
i
1=2
1
C
R
0
2
2R
0
m
w
D
:
(7.18)
Here
R
0
is actually the dimensionless radius of curvature of the substrate with
reference to the radius of the critical nucleus
r
c
. Note that the factor
f
(
m
,
R
0
)varies
from 1 to 0. To acquire the details of (
7.17
,
7.18
), see Chap. 2 in this topic or Refs.
[
54
,
56
]. Obviously, this factor plays an important role in the determination of the
heterogeneous nucleation barrier
G
heter
. One can see from (
7.9
) that the influence
of foreign particles on the nucleation barrier can be fully characterized by this factor.
Figure
7.3
ashowsthat
f
(
m
,
R
0
) is a function of
R
0
for a given
m
.When
R
0
!
0,
f
(
m,R
0
)
1, implying that the foreign body “vanishes” completely as a nucleating
substrate. In practice, if foreign bodies are too small, e.g., clusters of several
molecules, nucleation on these substrates will not be stable. Then, they play no
role in lowering the nucleation barrier. On the other hand, if
R
'
D
1, the foreign
body can be treated as a flat substrate with respect to the critical nuclei. In this case,
f
(
m
,
R
0
)
D
f
(
m
) is solely a function of
m
,and(
7.17
) is then reduced to
f.m;R
0
/
D
f.m/
D
1
4
.2
3m
C
m
3
/
(7.19)
f
(
m
) as a function of
m
is given in Fig.
7.3
b.
7.3.2
Kinetics of Nucleation
The commonly accepted nucleation processes can be regarded as follows: on
the substrate surface, some molecular processes occur due to transient visiting
molecules that adsorb, form short-lived unions, break up, desorb, etc. An instanta-
neous census would show some distributions of subcritical nuclei (or clusters) with
1, 2, 3,
:::
molecules per cluster as,
Monomer
•
Dimer
•
.n
1/
mer
•
n
mer
•
.n
C
1/
mer
(7.20)
