Order parameters

The uniaxial order parameter is zero for an isotropic fluid and one for a perfectly aligned system. First one calculates a director vector (see Ref. 2)

$Q_{\alpha \beta }={\frac {1}{N}}\sum _{j=1}^{N}\left({\frac {3}{2}}{\hat {e}}_{j\alpha }{\hat {e}}_{j\beta }-{\frac {1}{2}}\delta _{\alpha \beta }\right),~~~~~\alpha ,\beta =x,y,z,$

where $Q$ is a second rank tensor, ${\hat {e}}_{j}$ is a unit vector along the molecular long axis, and $\delta _{\alpha \beta }$ is the Kronecker delta. Diagonalisation of this tensor gives three eigenvalues $\lambda _{+}$, $\lambda _{0}$ and $\lambda _{-}$, and $n$ is the eigenvector associated with the largest eigenvalue ($\lambda _{+}$). From this director vector the nematic order parameter is calculated from (Ref. 5)

$S_{2}={\frac {d\langle \cos ^{2}\theta \rangle -1}{d-1}}$

where d is the dimensionality of the system.

i.e. in three dimensions (see Ref. 3)

$S_{2}=\lambda _{+}=\langle P_{2}(n\cdot e)\rangle =\langle P_{2}(\cos \theta )\rangle =\langle {\frac {3}{2}}\cos ^{2}\theta -{\frac {1}{2}}\rangle$

where $S_{2}$ is known as the uniaxial order parameter. Here $P_{2}$ is the second order Legendre polynomial, $\theta$ is the angle between a molecular axes and the director $n$, and the angle brackets indicate an ensemble average.

References

1. Joseph P. Straley "Ordered phases of a liquid of biaxial particles", Physical Review A 10 pp. 1881 - 1887 (1974)
2. R. Eppenga and D. Frenkel "Monte Carlo study of the isotropic and nematic phases of infinitely thin hard platelets", Molecular Physics 52 pp. 1303-1334 (1984)
3. Mark R. Wilson "Determination of order parameters in realistic atom-based models of liquid crystal systems", Journal of Molecular Liquids 68 pp. 23-31 (1996)
4. Denis Merlet, James W. Emsley, Philippe Lesot and Jacques Courtieu "The relationship between molecular symmetry and second-rank orientational order parameters for molecules in chiral liquid crystalline solvents", Journal of Chemical Physics 111 pp. 6890-6896 (1999)
5. Anna A. Mercurieva, Tatyana M. Birshtein "Liquid-crystalline ordering in two-dimensional systems with discrete symmetry", Die Makromolekulare Chemie, Theory and Simulations 1 pp. 205 - 214 (1992)