Chemistry Reference
In-Depth Information
in which rods can move freely decreases. Therefore, the collision between
particles increases and the entropy decreases, reducing the stability of the
system. Inversely, if
y
decreases, the degree of orientation along the direc-
tor becomes high, hence
Z
comb
becomes great and the contribution to the
energy reduction accordingly becomes important.
In fact, the above-mentioned equation is valid only for the perfectly
ordered case,
i.e.
, the rods are all aligned in parallel. This illustrates that
the Flory theory works well for concentrated solutions.
Another contribution to the total partition function of the system arises
from the orientation,
i.e
.,
Z
orient
Z
orient
=
y
ω
y
n
p
n
py
n
py
,
(2.35)
where
n
py
is the number of rods with off-orientation degree
y, ω
y
is the
solid angle fraction associated with
y,
and
n
py
/n
p
represents the orientation
distribution function. The average of
y
is given by
y
=
y
yn
py
n
p
(2.36)
It is illustrated from Equation 2.35 that if the system is in a perfectly
ordered state,
y
=1;thus
n
py
=
n
p
,
Z
orient
becomes very small. Otherwise,
the system is in disorder (
y
=
x
) then
ω
y
=
n
py
/n
p
and
Z
orient
=1.
According to the Flory's (1956) approximation, when the orientational
order is high,
n
py
/n
p
is important only in the range
θ
θ
. Assume that
n
py
/n
p
is uniform within the range. When
θ>θ
,
n
py
/n
p
is zero. In the
range
θ
θ
the solid angle becomes approximately (
y/x
)
2
. Therefore,
≤
(
y/x
)
2
n
p
.
Z
orient
≈
(2.37)
The fact that the orientational partition function
Z
orient
increases as
y
increases can be understood. Suppose the next neighbor of each cell in a
lattice is six. If the orientation is random each basic unit of a particle has
five ways and hence the particle of
x
units has 5
x
ways to enter into the
lattice and thus the contribution to the entropy of the system is
k
B
x
ln 5.
In the perfectly ordered state, after the first unit is put into the lattice the
remaining units enter the lattice via the same direction. The contribution
to the system entropy is about zero, and thus is not favored, taking only
the orientational entropy into account. Therefore, the fundamental reason
