Biomedical Engineering Reference
In-Depth Information
At the steady state,
J
0
, that is the formation rate of critical nuclei per unit volume
per unit time around a foreign particle is equal to the steady-state growth of clusters
on the surface of the particle. The nucleation rate can then be given in terms of
J
0
D
J
n
D
J
n
D
constant
D
criticalsizednucleiformed
=
unitvolume
time
:::
D
k
n
Z
n
k
nC1
Z
nC1
D
constant
:
(7.31)
Regarding the effect of the substrate on both the nucleation barrier and the
transport process, and the fact that the average nucleation rate in the fluid phase
depends on the density and size of the foreign particles present in the system, the
nucleation rate is given by [
49
,
56
,
58
]
J
D
4a.R
s
/
2
N
o
f
00
.m;R
0
/Œf.m;R
0
/
1=2
exp
"
#
16
cf
2
3kTŒkT
.1
C
/
2
f.m;R
0
/
B
(7.32)
ln
with
cf
kT
1=2
B
D
.C
1
/
2
4Dˇ
kink
(7.33)
f
00
.m;R
0
/
D
1
C
.1
R
0
m/=
w
(7.34)
2
and
f
00
.m;R
0
/
D
f
00
.m/
D
1
R
0
>> 1
2
.1
m/
at
(7.35)
where
B
is the kinetic constant and
N
o
denotes the number density of the substrates
(or “seeds”). The growth of nuclei is subject to the effective collision and incorpo-
ration of growth units into the surfaces of the nuclei (cf. Fig.
7.3
c). In the case of
homogeneous nucleation, the growth units can be incorporated into the nuclei from
all directions. However, for heterogeneous nucleation, the presence of the substrate
will block the collision path of the growth units to the surfaces of these nuclei from
the side of the substrate (cf. Fig.
7.3
c). This is comparable to the “shadow” of the
substrate cast on the surface of nuclei.
f
(
m,R
0
) in the pre-exponential factor, which
is the ratio between the average effective collision in the presence of substrates and
that of homogeneous nucleation (i.e., in the absence of a substrate), describes this
effect.
Both
f
(
m,R
0
)and
f
00
(
m,R
0
) are functions of
m
and
R
0
.When
R
0
!
00
0or
m
!
-1,
f
(
m,R
0
),
f
(
m
,
R
0
)
1. This is equivalent to the case of homogeneous nucleation.
In the case where
m
00
D
1, one has
f
(
m,R
0
),
f
(
m,R
0
)
!
1and
R
00
D
0. Normally,
