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.

Density Correlation and 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:

$U({\boldsymbol {x}})=\sum _{\alpha }U_{\alpha }({\boldsymbol {x}}-{\boldsymbol {x^{'}}})$ where ${\boldsymbol {x^{'}}}$ is the position of the atom labeled $\alpha$. the matrix element, therefore, is

$\langle {\boldsymbol {k}}|U|{\boldsymbol {k^{'}}}\rangle =\sum _{\alpha }\int d^{d}xe^{-i{\boldsymbol {k}}.{\boldsymbol {x}}}U_{\alpha }({\boldsymbol {x}}-{\boldsymbol {x_{\alpha }}})e^{i{\boldsymbol {k^{'}}}.{\boldsymbol {x}}}$

To seperate the potential interaction and the interatomic correlation factors, we shift our centers to each $x_{\alpha }$. Let ${\boldsymbol {R}}_{\alpha }={\boldsymbol {x}}-{\boldsymbol {x_{\alpha }}}$.

$\langle {\boldsymbol {k}}|U|{\boldsymbol {k^{'}}}\rangle =\sum _{\alpha }\int d^{d}xe^{-i{\boldsymbol {k}}.({\boldsymbol {x}}_{\alpha }+{\boldsymbol {R}}_{\alpha })}U_{\alpha }({\boldsymbol {R}}_{\alpha })e^{i{\boldsymbol {k^{'}}}.({\boldsymbol {x}}_{\alpha }+{\boldsymbol {R}}_{\alpha })}$ $=\sum _{\alpha }[\int d^{d}xe^{-i{\boldsymbol {q}}.{\boldsymbol {R}}_{\alpha }}U_{\alpha }({\boldsymbol {R}}_{\alpha })]e^{-i{\boldsymbol {q}}.{\boldsymbol {x}}_{\alpha }}$ $=\sum _{\alpha }U_{\alpha }({\boldsymbol {q}})e^{-i{\boldsymbol {q}}.{\boldsymbol {x}}_{\alpha }}$ Here the scattering wave vector is ${\boldsymbol {q}}={\boldsymbol {k}}-{\boldsymbol {k}}^{'}$ and $U_{\alpha }({\boldsymbol {q}})$ is the atomic form factor, which is nothing but the Fourier transform of the atomic potential. The differential cross-section is proportional to the matrix element squared: $|\langle {\boldsymbol {k}}|U|{\boldsymbol {k^{'}}}\rangle |^{2}=\sum _{\alpha ,\alpha ^{'}}U_{\alpha }({\boldsymbol {q}})U_{\alpha ^{'}}^{*}({\boldsymbol {q}})e^{-i{\boldsymbol {q}}.{\boldsymbol {x}}_{\alpha }}e^{i{\boldsymbol {q}}.{\boldsymbol {x}}_{\alpha ^{'}}}$

If the positions of atoms are rigidly fixed,then this expression gives the exact answer. However, in real materials, the particles move around,probing large regions of phase space consistent with statistical mechanics, and we need an ensemble average of the ideal cross-section. Under ergodic hypothesis and assuming the particles are identical,

${\frac {\mathrm {d} ^{2}\sigma }{\mathrm {d} \Omega }}\sim |U_{\alpha }({\boldsymbol {q}})|^{2}{\boldsymbol {I}}({\boldsymbol {q}})$ where the function

${\boldsymbol {I}}({\boldsymbol {q}})=\left\langle \sum _{\alpha ,\alpha ^{'}}e^{-i{\boldsymbol {q}}.({\boldsymbol {x}}_{\alpha }-{\boldsymbol {x_{\alpha ^{'}}}})}\right\rangle$ is called the structure function. As intensive version of the structure function is called the structure factor.

$S({\boldsymbol {q}})={\frac {{\boldsymbol {I}}({\boldsymbol {q}})}{N}}$ or $S({\boldsymbol {q}})={\frac {{\boldsymbol {I}}({\boldsymbol {q}})}{V}}$

Two point density correlation function and its relation with the structure factor

The number density operator is defined as $n({\boldsymbol {x}})=\sum _{\alpha }\delta ({\boldsymbol {x}}-{\boldsymbol {x}}_{\alpha })$

The two point density density correlation function is defined as $C_{nn}({\boldsymbol {x}}_{1},{\boldsymbol {x}}_{2})=\left\langle n({\boldsymbol {x}}_{1})n({\boldsymbol {x}}_{2})\right\rangle$ ) $=\left\langle \sum _{\alpha ,\alpha ^{'}}\delta ({\boldsymbol {x}}_{1}-{\boldsymbol {x}}_{\alpha })\delta ({\boldsymbol {x}}_{2}-{\boldsymbol {x}}_{\alpha ^{'}})\right\rangle$

The structure function is simply a Fourier transform of this function:

${\boldsymbol {I}}({\boldsymbol {q}})=\left\langle n({\boldsymbol {q}})n(-{\boldsymbol {q}})\right\rangle$ where $n({\boldsymbol {q}})=\int d^{d}xe^{-i{\boldsymbol {k}}.{\boldsymbol {x}}}n({\boldsymbol {x}}=\sum _{\alpha }e^{-i{\boldsymbol {k}}.{\boldsymbol {x}}}$ is the Fourier transform of the density. In a statistical ensemble, $\left\langle n({\boldsymbol {q}})\right\rangle =\int d^{d}xe^{-i{\boldsymbol {k}}.{\boldsymbol {x}}}\left\langle n({\boldsymbol {x}})\right\rangle =\left\langle \sum _{\alpha }e^{-i{\boldsymbol {k}}.{\boldsymbol {x}}}\right\rangle$ So scattering measures density-density correlation.

Isotropic, nematic and cholesteric phases[4]

In isotropic liquids, $\left\langle n({\boldsymbol {x}})\right\rangle =\left\langle n\right\rangle$, a constant, so $\left\langle n({\boldsymbol {q}}\right\rangle =V\left\langle n\right\rangle \delta _{q,0}$ is nonzero only for ${\boldsymbol {q}}=0$, and there is a contribution to the structure factor only for forward scattering.

In a regularly periodic crystal, $\left\langle n({\boldsymbol {x}})\right\rangle =\sum _{\boldsymbol {G}}\left\langle n_{\boldsymbol {G}}\right\rangle e^{-i{\boldsymbol {G}}.{\boldsymbol {x}}}$ So there is a peak corresponding to each wavevector ${\boldsymbol {G}}$ in the reciprocal space.

In an Isotropic LCD, the structure factor is isotropic, but shows liquid-like rings at wave numbers corresponding to the two characteristic lengths of the individual molecules-their length l and diameter a. When the isotropic liquid is cooled, the first phase rhat condenses in the nematic (N)phase in which long molecules align so that they are on an average parallel to a particular direction specified by a unit vector ${\boldsymbol {n}}$ called the directorThe positions of the molecular centers of mass remains randomly distributed, like in an isotropic fluid. So the nematic phase breaks rotational invariance but maintains translational invariance. It is easy to visualize that in a statistical ensemble, rotations about ${\boldsymbol {n}}$ leaves the phase unchanged, while rotation in a plane perpendicular to ${\boldsymbol {n}}$ does not.