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approaching cylinder's center-of-mass cannot enter (due to the hard-rod repulsion
between the two idealized objects). Thus, this angular arrangement sees a decrease
in the net positional entropy of the approaching cylinder (there are fewer states avail-
able to it).
It may be asserted that the fundamental reason arises from the fact that, while par-
allel arrangements of anisotropic objects lead to a decrease in orientational entropy,
there is an increase in positional entropy. Thus, in some cases, greater positional
order will be entropically favorable. This theory therefore predicts that a solution of
rod-shaped objects will undergo a phase transition at sufficient concentration into a
nematic phase. Recently, this theory has been used to observe the phase transition
between nematic and smectic-A at very high concentration (Hanif et al.). Although
this model is conceptually helpful, its mathematical formulation makes several
assumptions that limit its applicability to real systems.
This statistical theory includes contributions from an attractive intermolecular
potential. The anisotropic attraction stabilizes the parallel alignment of neighboring
molecules, and the theory then considers a mean-field average of the interaction.
Solved self-consistently, this theory predicts thermotropic phase transitions consis-
tent with the experiment.
According to one theoretical model, it can be assumed that the liquid crystal
material is a continuum. It has been suggested that three types of distortions can take
place in these structures:
1. Twists of the material, where neighboring molecules are forced to be angled
with respect to one another
2. Nonlinearity (or splay) of the material, where bending occurs perpendicular
to the director
3. Bend of the material, where the distortion is parallel to the director and
mesogen axis
All three types of distortions incur an energy penalty, and are defects that often
occur near domain walls or boundaries of the enclosing container. The response of
the material can then be decomposed into terms based on the elastic constants cor-
responding to the three types of distortions.
It has been mentioned in the literature that chiral mesogens usually produce
chiral mesophases. For molecular mesogens, this means that the molecule must
possess some form of asymmetry, usually a stereogenic center. An additional
requirement is that the system not be racemic; a mixture of right- and left-handed
versions of the mesogen will cancel the chiral effect. Because of the cooperative
nature of LC ordering, however, a small amount of chiral dopant in an otherwise
achiral mesophase is often enough to select out one domain handedness, making
the system overall chiral.
Chiral phases usually have a helical twisting of the mesogens. If the pitch of this
twist is on the order of the wavelength of visible light, then interesting optical inter-
ference effects can be observed. The chiral twisting that occurs in chiral LC phases
also makes the system respond differently to right- and left-handed circularly polar-
ized light. These materials can therefore be used as polarization filters.
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