Liquid Crystals

Introduction

Homogeneous, isotropic liquids have an average structure that is invariant under arbitrary rotations and translations.It has no long range order, and has the highest possible symmetry with maximum possible entropy.The crystalline state has long range translational and rotational order, with the lowest possible symmetry consistent with a regular filling of space. Between these two, there are systems which exhibit short range correlations in some directions and long range in others, and have symmetries intermediate between between those of liquids and crystals.

Among the materials that show intermediate order, the most widely studied are liquid crystals.Liquid crystals are usually made of strongly anisotropic organic molecules, either elongated (calamitic, rod-like molecules) or disk-like (discotic molecules). As a rule, the inner part of mesogenic molecules is rigid (e.g. phenyl groups) and the outer part flexible (aliphatic chains). This double character explains altogether the existence of steric interactions (between rod-like or disk-like cores) yielding orientational order and the fluidity of the mesomorphic phases. Typical examples are cyanobiphenyls and MBBA. These produce thermotropic mesophases,i.e. phases with a single component, whose phase transitions can be induced by a change in temperature.The other broad LC class is constituted by the lyotropic mesophases: they occur when anisotropic amphiphilic molecules (soaps, phospholipids, various types of surfactant molecules and biomolecules) are added to a solvent. Because amphiphiles have two distinct parts, a polar head and a non-polar tail,the building units of lyotropic phases are usually aggregates of many molecules(micelles) rather than single molecules. This microphase separation dominating the lyotropic behavior is partly present also in thermotropic LC, as for example in the smectic phases, where polar and non polar portions of the molecules form distinct alternatinig planes in the system.A typical example of lyotropics is a water solution of SDS, sodium dodecyl sulphate. For concentrations above the critical micellar concentration, cmc, these molecules form aggregates of different shapes, spherical or cylindrical micelles,bilayers, inverse cylinders, and inverse micelles.[1,2.3]

Classification of LC phases

LCs show many possible structures, which can belong to the same compound (polymorphism). There are four basic types of liquid crystalline phases, classified accordingly to the dimensionality of the translational correlations of building units: nematic (no translational correlations), smectic (1D correlation),columnar (2D) and various 3D-correlated structures, such as cubic phases.

Isotropic, nematic and cholesteric phases

Structure Factor

A lot of information about the bulk structure of LCDs can be obtained via scattering of X-rays. let $|{\boldsymbol {k}}\rangle$ and $|{\boldsymbol {k}}^{'}\rangle$ be the incident and final plane wave state of the scattered particle with respective momenta $\hbar {\boldsymbol {k}}$ and $\hbar {\boldsymbol {k}}^{'}$ If the scattered particle interacts weakly with the scaterring medium via a sufficiently short-ranged interaction $U$, then by Fermi's Golden rule, the transition rate between $|{\boldsymbol {k}}\rangle$ and $|{\boldsymbol {k}}^{'}\rangle$ is proportional to the square of the matrix element,

$M_{{\boldsymbol {k}},{\boldsymbol {k^{'}}}}=\langle {\boldsymbol {k}}|U|{\boldsymbol {k^{'}}}\rangle =\int d^{d}xe^{-i{\boldsymbol {k}}.{\boldsymbol {x}}}U({\boldsymbol {x}})e^{i{\boldsymbol {k^{'}}}.{\boldsymbol {x}}}$

where $U({\boldsymbol {x}})$ is the scattering potential in the coordinate representation of the scattered particle, and our plane wave states are unnormalized.

In multiparticle systems, the scattering potential is the sum of terms from individual atoms in the material:

$U({\boldsymbol {x}})=\sum _{\alpha }U_{\alpha }({\boldsymbol {x}}-{\boldsymbol {x^{'}}})$