Environmental Engineering Reference
In-Depth Information
−
1
⎛
⎜
⎞
⎟
1
ς ς ς
⎛
⎜
S
k
⎞
⎟
1 2 3
3
3
c
=
δ
exp
(19.42)
q
Λ
.
rep
c m
.
B
l
where 〈 〉
l
indicates a thermal average taken in the bulk liquid held at constant (
T
,
P
l
,
N
).
On the other hand, according to Kusaka, the activation of the free translation within the volume
V
and the free rotation of the extruded cluster, when averaged over all possible values of
m
and
internal conigurations
s
m
−2
, leads to the factor
ς ς ς
1 2 3
3
2
Q Q
rot
=
8
π
V
(19.43)
tr
Λ
c m
.
.
l
Upon the extrusion, the
m
-sized cluster loses its interaction
U
int
with the surroundings, acquires
n−m
particles from the vapor phase, and then undergoes structural relaxation; as in the original
Lothe-Pound prescription Kusaka assumes that the reversible work associated with these processes
is included fully in σ
S
. Uniting Equations 19.42 and 19.43 Kusaka arrives inally at the expression for
factor Φ
K
(which we refer to as the Kusaka factor) designated to substitute the Lothe-Pound factor:
−
1
⎛
⎜
⎞
⎟
ς ς ς
ς ς ς
⎛
⎜
S
k
⎞
⎟
1 2 3
3
2
1 2 3
3
3
c
Φ
K
=
8
π
V
δ
exp
(19.44)
Λ
Λ
c m
.
.
c m
.
.
B
l
l
Kusaka [11] has numerically calculated the correction factor for the Lennard-Jones system. The
calculated values ranged from 10
9
to 10
13
which were considerably higher than the Reiss correction
factor and lesser than the Lothe-Pound one. However, it is hardly possible to numerically calculate
the correction factor for real systems like water, organic species, metals, etc. Therefore, in this
section we propose an analytical formula applicable to real systems. Later, in Section 19.7 we will
compare the estimations of this formula with Kusaka's numerical simulation results [11].
To derive such a formula we will analyze the key equation in Kusaka's theory (Equation 19.40).
Actually for all the
m
-sized clusters that happened to be inside the volume
V
S
the cluster's volume
v
m
<
V
S
. Thus, there is some volume
V
S
−
v
m
accessible for the rigid cluster motion. Hence, there is
some volume Δ
V
c.m
.
accessible for the motion of the cluster's center of mass. Note that Δ
V
c.m
.
is the
volume over which the integration is made in Equation 19.40. The estimations made in Ref. [38]
have shown that the size of the integration region is small enough, that is, the variation of
R
c.m.
when
taking the integral in Equation 19.40 occurs in a very narrow region. Therefore, the variation of the
integrand function
U
int
is rather weak during the integration over
d
R
c.m
.
. Thus, we can consider the
interaction potential
U
int
as independent of
R
c.m
.
. Then the integral in Equation 19.40 can be written
as the product of two integrals and we arrive at
⎛
⎜
⎞
⎟
U
k T
⎛
⎜
⎞
⎟
⎛
⎜
U
k T
⎞
⎟
S
k
ʹ
∫
int
3
int
3
c
Z
=
d
exp
−
sin
θ θ φ ψ
d d d
=
d
exp
exp
−
c
(19.45)
c
B
B
B
m
r
∈
V
S
where
d
3
= Δ
V
c.m
.
.
The integral in Equation 19.45 may be regarded as the conigurational partition function of the
embedded cluster due to its rotational degrees of freedom only, when it is subjected to the external
ield
U
int
(θ, φ, ψ). As a consequence, the conigurational entropy
�
S
c
is associated here with these
degrees of freedom, that is, it has a deinite physical sense of the entropy linked with the cluster's
rotational motion (in contrast to the formal entropy in Equation 19.40). The entropy �
S
c
has a deinite
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