Chemistry Reference
In-Depth Information
Figure 2.5. The partition function
vs.
off-orientation degree of a rod system of axial
ratio
x
= 100. (Modified from Flory, 1956.)
As
φ>φ
∗
there is a maximum
Z
max
with
y
.If
Z
max
is greater than
Z
0
,
the value for
y
=
x
, then
Z
max
gives the equilibrium state; otherwise, it
is the meta-equilibrium state.
Another conclusion from Figure 2.5 is that the
y
value at
Z
=
Z
max
is much less than
x
/2, even for
φ
=
φ
∗
.As
φ
increases,
y
at
Z
=
Z
max
decreases. It is illustrated that when the system transforms from the dis-
ordered state (
y
=
x
) to the liquid crystal state,
y
changes abruptly
to
x
/2.
If
φ
and
x
are great enough, there are two extremes in
Z
. The maximal
value is associated with the stable (or meta-stable) states, in which
y
is less
than that at the minimum
Z
. Differentiate
F
in Equation 2.39 with respect
to
y
and make d(
−
ln
Z
)
/
d
y
equal to zero. The
y
for the two extremes are
given by
φ
=
x
(
x
1
−
exp
−
2
y
.
(2.41)
−
y
)
There are two solutions for
y
. The smaller solution corresponds to the stable
or meta-stable state in which
Z
is at a maximum, as shown in Figure 2.5.
The dependence of
φ
on
y
can be obtained numerically.
