Biomedical Engineering Reference
In-Depth Information
(for all
n
;
n
D
2,3,4,
:::
) with the effective total number of “molecules” per unit
volume:
Let
n
be the rate of molecule addition. That is,
n
D
ˇ
kink
K
n
(7.25)
where
K
n
denotes the collision rate of monomers with an
n
-sized cluster and
ˇ
kink
n
is the conversion probability . Also let
be the rate at which the cluster loses
molecules. Obviously, at the equilibrium state, one has the detailed balance between
the growth and disintegration of clusters,
n
C
n
nC1
C
nC1
D
0:
(7.26)
n
*
,
C
n
displays a minimum at the
critical nucleus size. The increasing nonphysical branch of
C
n
at
n
G
heter
./
Since
has a maximum at
n
D
n
*
>
reflects the
fact that the mother phase is saturated.
7.3.2.2
Stationary (or Steady) State
The stationary state is a state at which the cluster size distribution does not change
with time. Normally, this state is so far the most important state and occurs in the
middle stage of nucleation and can be treated quantitatively. In the stationary state,
d
Z
n
/d
t
Z
n
Z
n
is the steady-state cluster
size distribution. The stationary nucleation rate for homogeneous nucleation is given
by the Becker-Doering formula [
56
]
D
0. Because
J
n
D
constant
D
J
n
D
J
,
exp
K
m
G
homo
kT
z
J
D
(7.27)
with
D
Z
0
n
C
n
Z
0
n
C1
z
(7.28)
C
n
C1
K
where
z
is the so-called Zeldovich factor [
48
,
59
],
D
K
n
is the frequency
of monomer attachment to the critical nucleus,
m
denotes the average volume of
structural units in the ambient phase, and
C
n
is the equilibrium concentration of
n
-sized clusters given by[
49
],
C
n
Š
C
1
exp
.
G
n
=kT/
(7.29)
Based on the definition of
J
n
, one has
J
n
D
k
n
Z
n
(7.30)
