Module: liquid_crystal

Description	Usage
nematic and ferronematic order parameters for distinguishing isotropic, nematic and ferronematic phases of liquid crystals
Authors: Alexander Humeniuk

Details

The nematic and ferronematic order parameters can be used to distinguish the isotropic, nematic and ferronematic phases of liquid crystals.

The nematic order parameter S characterizes the orientational order of molecules and ranges from S=0 (isotropic) to S=1 (all molecular axes are perfectly parallel). Most liquids are isotropic, as there is no preferred direction, and have an order parameter close to 0. In liquid crystals, membranes and solids, molecules tend to align giving rise to order parameters closer to 1.

$S$ is calculated from the distribution of the angles between the molecular axes ( $\hat{u}_i$ for $i=1,\ldots,N$ ) and the nematic director $\hat{n}$ , $S = \frac{1}{N} \sum_{i=1}^N \left(\frac{3}{2} \cos^2(\theta_i) - \frac{1}{2} \right),$ with $\cos(\theta_i) = \hat{n} \cdot \hat{u}_i$ .

The ferronematic order parameter P depends on the relative orientations of the molecular axes. If the axes all point into the same direction, giving rise to a net polarization if the molecules have permanent dipole moments, P is close to 1. If the molecular axes are oriented isotropically or are aligned antiparallel so that there is no net polarization, P is close 0.

$P$ is length of the average of the molecular axes ( $\hat{u}_i$ for $i=1,\ldots,N$ ) $P = \vert \frac{1}{N} \sum_{i=1}^N \hat{u}_i \vert$ Since the molecular axes are unit vectors, P ranges from a minimum of 0 to a maximum of 1.

By adding a bias to the (ferro)nematic order parameter, one can drive a liquid crystal from the isotropic to the (ferro)nematic phase.

Actions

The following actions are part of this module

Name	Description	Tags
FERRONEMATIC_ORDER	Calculate the ferronematic order parameter.	COLVAR
NEMATIC_ORDER	Calculate the nematic order parameter.	COLVAR

References

More information about this module is available in the following articles:

R. Eppenga, D. Frenkel, Monte Carlo study of the isotropic and nematic phases of infinitely thin hard platelets. Molecular Physics. 52, 1303–1334 (1984)