Chemistry Reference
In-Depth Information
2.4.
MAIER-SAUPE MEAN FIELD APPROACH
2.4.1.
Maier-Saupe mean field theory for small
molecular mass liquid crystals
Liquid crystalline polymers can be regarded as a long chain with rods
connected in sequence, each rod being, in some sense, equivalent to a
small molecular mass liquid crystal. This is the so-called freely-jointed-rod
chain, the simplest model of polymers. It is understood that the con-
stituent units — small molecular mass liquid crystals play an essential role
in liquid crystalline polymers. Here, we introduce an important theory for
small molecular mass liquid crystal — the Maier-Saupe mean field theory
(Maier & Saupe, 1959, 1960).
In contrast to the Onsager and Flory theories, the Maier-Saupe theory
no longer takes into account molecular steric effects as the basic interaction
but instead proposes that the van der Waals interactions between molecules
are the basis for forming a liquid crystal phase. The van der Waals interac-
tion depends on molecular orientations. The Maier-Saupe theory adopts a
rather simple mathematical treatment and can easily take into account the
relationship of system properties to temperature. This theory has been suc-
cessfully applied to a thermotropic system of small molecular mass liquid
crystal.
In this system, a molecule is subjected to the van der Waals attractions
of other molecules in the system which is related to the relative orientations
of the pair molecules. Suppose the
Z
axis is the reference direction. Then,
the potential for each molecule is a function of orientation
U
i
=
−
a
P
2
(cos
θ
ij
)
j
P
2
(cos
θ
i
)
,
(2.68)
where
θ
i
is the angle of the molecular long axis made with respect to the
Z
axis,
θ
ij
is the angle molecule
j
interacting with molecule
i
makes with
respect to the
Z
axis.
P
2
=3
x
2
/
2
−
2
is the second Legend polynomial. The
bracket denotes the statistical average over all the molecules in the system
and according to the definition the averaged quantity is actually the order
parameter.
Therefore, the mean field potential of a molecule is given by
U
=
−
aSP
2
(cos
θ
)
,
(2.69)
where
a
is the constant of a van der Waals type interactions which is asso-
ciated with the mean distance
r
between the molecules, being proportional
