Chemistry Reference
In-Depth Information
x
x/y
Ψ
x
(a)
(b)
Figure 2.4. A rigid rod in the Flory lattice. (a) a rod making an angle Ψ to the director;
(b) the rod divided into
y
sub-particles.
particle is aligned off the director the value of
y
increases. Therefore,
y
can
be regarded as a measure of a rod's deviation off the director.
y
is called
the off-orientation degree or disorder degree.
Assume that the total number of cells in the system is
n
0
and (
j
−
1)
rods have been placed in the lattice. They have occupied
x
(
j
−
1) lattice
cells and hence
n
0
−
x
(
j
−
1) lattice cells remain unoccupied. In this case,
there are
ν
j
ways to put the
j
-th rod into the lattice
1)]
N
(
x
−
y
j
)
j
P
(
y
j
−
1)
j
ν
j
=[
n
−
x
(
j
−
,
(2.29)
where
y
j
is the number of the sub-particles of the rod; the first term
represents the number of ways of putting the first basic unit of the first
sub-particle into the lattice which is the number of unoccupied cells.
P
j
is
the ways of putting the first unit of remaining (
y
−
1) sub-particles;
N
j
is
the number of ways of placing the remaining (
x
−
y
j
) units entering into
the lattice.
First we will work out
P
j
. Once the first sub-particle's position is deter-
mined each of other sub-particles must be the closest neighbor to the
preceding sub-particle shown in Figure 2.4. In addition, the first unit must
be immediately next to the last unit of the preceding sub-particle. The
probability of such an arrangement is the volume fraction of unoccupied
cells in the system, so that
P
j
=
[
n
−
x
(
j
−
1)]
.
(2.30)
n
All units of each sub-particle must be in same row of the cell. Once the
first unit has been put into the lattice (the cell must be unoccupied and is
allowed to put in) each of remaining units must be positioned immediately
next to preceding unit (the cell must be unoccupied). There are two pos-
sibilities: the cell may be unoccupied and is allowed to enter in; the other
possibility is that it has been occupied by the first unit of a sub-particle
