Environmental Engineering Reference
In-Depth Information
where
U
int
denotes the interaction potential between the
N − m
particles outside the sphere and the
m
particles inside the sphere, and is a function of
r
N−m
and
r
m
. The last integral in Equation 19.38, along
with the coeficient 1/(
h
3
m
m
!), is regarded as the partition function ξ
m
of a cluster, which consists of
m
particles, all conined to
V
S
, and is embedded in the liquid phase.
The coordinate transformation is to be done from a laboratory system to a body coordinate sys-
tem which means that a set of Euler axes is embedded in the object with the origin at the center of
mass
R
c.m
.
and the rotation refers to the rotation of these axes. Denoting by
s
m
−2
and
t
m
−2
the coor-
dinates and the conjugate momenta of the remaining 3
m
−6 degrees of freedom of the embedded
cluster, the partition function ξ
m
is written as [11]
1
⎛
⎜
K
T
⎞
⎟
⎛
⎜
U
k T
⎞
⎟
∫
m
−
2
m
−
2
S
m
ξ
m
=
d
t
d
s
exp
−
exp
−
3
m
−
6
h
m
!
k
B
B
m
r
∈
V
S
ς ς
ς
⎛
⎜
U
k T
⎞
⎟
∫
1 2 3
int
(19.39)
×
d
R
sin
θ
d
θ ϕ ψ
d
d
exp
−
c m
.
.
3
Λ
c
.
m
.
B
m
r
∈
V
S
where
K
S
is the kinetic energy of the
m
particles excluding those due to rigid translation and rotation of
the embedded cluster as a whole
U
m
is the interaction potential among the
m
particles
Euler angles (θ, ϕ, ψ) specify the orientation of the cluster
Λ
c.m
.
and ς
i
are Λ
2
k
B
/ with
M
and
I
i
(
i
= 1, 2, 3) denoting the
mass of the cluster and its principal moments of inertia, respectively
=
h
/ 2
Mk
B
π
and ς =
π
I
h
c m
.
.
i
i
The second integral in Equation 19.39 is regarded as the conigurational partition function
Z
c
of
the embedded cluster due to its translational and rotational degrees of freedom when it is subjected
to the external ield
U
int
. Thus, the conigurational entropy
S
c
associated with these degrees of free-
dom is deined by the following equation:
⎛
⎞
⎟
U
k T
⎛
⎜
U
k T
⎞
⎟
=
⎛
⎜
S
k
B
⎞
⎟
∫
int
int
3
c
Z
=
d
R
sin
θ θ ϕ ψ
d d d
exp
−
δ
exp
−
c
exp
(19.40)
⎜
c
c m
.
.
B
B
m
r
∈
V
S
where 〈 〉
c
denotes the thermal average taken with the Boltzmann weight exp(−
U
int
/
k
B
T
) while
imposing the constraint
r
m
∈
V
S
that the
m
particles are conined to the volume
V
S
.
One should note that the quantity δ is an arbitrary length scale in Equation 19.40 and is intro-
duced to make explicit the dimensionality of various quantities involved. Therefore, the entropy
S
c
thus deined is some “formal entropy.”
Since the coordinates
r
N−m
and
s
m
−2
are ixed when evaluating
Z
c
one can say that the cluster
which is governed by Equation 19.40 is a rigid one in a rigid environment. The idea of Kusaka is that
during the extrusion of such a cluster the modes of luctuation associated with the factor:
ς ς ς
⎛
⎜
S
k
⎞
⎟
1 2 3
3
3
c
δ
exp
(19.41)
Λ
c m
.
.
B
are to be deactivated. Then, the resulting equation for
q
rep
is [11]
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