274 15 Liquid Crystals and Other Anisotropic Fluids
15.1 Remarks on Nomenclature and Notations.
Liquid crystals are substances which can flow, and thus are liquid-like and, at the same
time, exhibit anisotropic properties which are typical for crystalline solids. Around
1880, Otto Lehmann coined the words “flowing crystal” and “liquid crystal” to refer
to same types ofanisotropic fluids. Some contemporary scientists opposed to his
ideas that fluid and crystalline properties can occur in a homogeneous substance and
that various liquid-crystalline states are phases in the sense of thermodynamics, just
asgas, isotropic liquid and crystalline solid. Nowadays the notionliquid crystalis
well accepted, in particular “LCD”, i.e. “Liquid Crystal Display” became a household
word.
Liquid crystals are composed of non-spherical particles which have an orienta-
tional degree of freedom. Prototypes for effectively axisymmetric or uniaxial parti-
cles have a rod-like or a disc-like shape. They are referred to asprolateandoblate
particles.
The main types of liquid crystals are callednematic,cholestericandsmecticliquid
crystals.
The termsthermotropic liquid crystalandlyotropic liquid crystalare used to
indicate that the change of the temperature or of the concentration in a solution
drives the phase transition from an isotropic liquid to a liquid crystalline state.
15.1.1 Nematic and Cholesteric Phases, Blue Phases.
Innematics,themainaxisoftheparticleshavealongrangepreferentialorderwhereas
the positions of their centers of mass have no long range order. Usually, it is under-
stood that nematics are composed of prolate particles, otherwise the notion “discotic
nematic” is used for fluids composed of oblate particles. Symmetry considerations,
however, apply to both types of nematics: they are characterized by anorder para-
meterwhich is a symmetric traceless second rank tensor.
In ordinary nematics, the phase has uniaxial symmetry, even when its constituents
are biaxial particles. Fluids with an overall biaxial symmetry, but without any long
ranged positional order are referred to asbiaxial nematics. One has to distinguish
between the symmetry of the particles and the symmetry of the phase. In principle,
it is possible to have a biaxial phase composed of uniaxial particles and a uniaxial
phase composed of biaxial particles. Typically, however, the rare biaxial nematic
phase is found for substances composed of biaxial particles.
A cartoon of the orientation of ‘particles’ in the nematic phase is shown in
Fig.15.1. Although ‘up and down’ or ‘head and tail’ can be distinguished for each
particle, the average orientation hashead-tail symmetry.
Acholesteric liquid crystalis essentially a nematic, where the preferential direc-
tion of the phase is twisted in space, with a twist axis pointing in a certain direction.
Cholesterics possess a spontaneously formed helix, which is geometrically similar
to the steps of a spiral staircase. The twist axis is also calledhelical axis.