Chemistry Reference
In-Depth Information
to
r
−
6
. The coupling constant
a
depends on the molecular properties. For
the nematic phase, which has no positional order and is uniaxially symmet-
rical, the potential
U
is independent of the position of the molecular masses
and the azimuthal of the molecular orientations.
The orientational distribution function is thus expressed by
f
(Ω) =
exp(
−
βU
)
Z
exp[
βaSP
2
(cos
θ
)]
exp[
βaSP
2
(cos
θ
)]
d
Ω
,
=
(2.70)
where the denominator,
Z
, is the partition function. The order parameter
S
can be evaluated in terms of the orientational function
S
=
1
[
P
2
(cos
θ
)] exp[
βaSP
2
(cos
θ
)] sin
θdθ
1
−
1
−
1
(2.71)
exp[
βaSP
2
(cos
θ
)] sin
θdθ
Equation 2.71 is the self-consistency equation for
S
.
S
depends on the
combination
k
B
T/a
only, shown in Figure 2.15, and thus is a function of
temperature and the coupling constant
a
.As
k
B
T/a
=0
.
22019, a nematic
to isotropic transition occurs. The temperature
T
c
is the N-I transition
temperature at which
S
jumps from 0.4289 to zero.
Figure 2.15. The order parameter
vs.
reduced temperature for small molecular mass
liquid crystals. (Modified from Maier & Saupe, 1959.)
