Biomedical Engineering Reference
In-Depth Information
Fig. 2.6
Scheme of the shadow effect of the substrate in heterogeneous nucleation. Reprinted with
permission from ref. [
8
]. Copyright (2004) Springer
and
1
2
.1
m/ at R
0
1;
f
00
.m; R
0
/
D
f
00
.m/
D
(2.23)
where
B
is the kinetic constant and
N
ı
denotes the density of substrates (or “seeds”).
The growth of nuclei is subject to collisions between growth units and the surfaces of
the nuclei, followed by incorporation of the growth units in the nuclei (cf. Fig.
2.6
).
In the case of homogeneous nucleation, growth units can be incorporated into the
nuclei from all directions. However, in the case of heterogeneous nucleation, the
presence of substrates will block the collision of growth units with the surfaces of
these nuclei.
f
00
(
m
,
R
0
) in the preexponential term describes the ratio between the
average effective collision in the presence of substrates and that of homogeneous
nucleation (no substrate).
Both
f
(
m
,
R
0
)and
f
00
(
m
,
R
0
) are functions of
m
and
R
0
.When
R
0
!
0or
m
D
1,
f
(
m
,
R
0
)
,f
00
(
m
,
R
0
)
D
1. This is equivalent to the case of homogeneous nucleation.
In the case where
m
!
1and
R
1, one has
f
(
m
,
R
0
)
,f
00
(
m
,
R
0
)
D
0. Normally,
heterogeneous nucleation occurs when
m
is within the range between 1 and
1, or
f
(
m
,
R
0
) is between 0 and 1, depending on the nature of the substrate surface and
supersaturation.
Notice that for homogeneous nucleation, one has
f
00
(
m
,
R
0
)
D
f
(
m
,
R
0
)
D
1, and
4
a
(
R
s
)
2
N
o
!
1. In this case, (
2.20
)isconvertedto:
D
B exp
:
16
cf
2
3kT S
m
T
2
J
(2.24)
This implies that (
2.20
) is applicable to describe both homogeneous and hetero-
geneous nucleation.
One of the most common ways to describe the kinetics of nucleation is to
measure the induction time
t
s
of nucleation at different supersaturations. By
definition, one has
1
t
s
V
;
J
D
(2.25)
