# Order parameters

An **order parameter** is some observable physical quantity that is able to distinguish between
two distinct phases. The choice of order parameter is not necessarily unique.

## Contents

## Solid-liquid transition[edit]

Possible choices:

- Fourier transform of the density
- Shear modulus

## Isotropic-nematic transition[edit]

The **uniaxial order parameter** is zero for an isotropic fluid and one for a perfectly aligned system. First one calculates a director vector ^{[1]}

where is a second rank tensor, is a unit
vector along the molecular long
axis,
and is the Kronecker delta.
Diagonalisation of this tensor
gives three eigenvalues , and ,
and is the eigenvector associated
with the largest eigenvalue ().
From this director vector the nematic order
parameter is calculated from ^{[2]}

where *d* is the dimensionality of the system.

i.e. in three dimensions ^{[3]}

where is known as the uniaxial order parameter. Here is the second order Legendre polynomial, is the angle between a molecular axes and the director , and the angle brackets indicate an ensemble average.

## Tetrahedral order parameter[edit]

^{[4]}

## See also[edit]

## References[edit]

- ↑ 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) - ↑ 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) - ↑ 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) - ↑ P. -L. Chau and A. J. Hardwick "A new order parameter for tetrahedral configurations", Molecular Physics
**93**pp. 511-518 (1998)

- Related reading

- Joseph P. Straley "Ordered phases of a liquid of biaxial particles", Physical Review A
**10**pp. 1881 - 1887 (1974) - 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) - Erik E. Santiso and Bernhardt L. Trout "A general set of order parameters for molecular crystals", Journal of Chemical Physics
**134**064109 (2011)