Editing 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.
==Solid-liquid transition==
Possible choices:
*Fourier transform of the density
*Shear modulus
==Isotropic-nematic transition==
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)

:<math>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,</math>

where <math>Q</math> is a second rank tensor,  <math>\hat e_{j}</math> is a unit
vector along the molecular long
axis,
and <math>\delta_{\alpha\beta}</math> is the [[Kronecker delta]].
Diagonalisation of this tensor
gives three eigenvalues <math>\lambda_+</math>, <math>\lambda_0</math> and <math>\lambda_-</math>,
and  <math>n</math> is the eigenvector associated
with the largest eigenvalue (<math>\lambda_+</math>).
From this director vector the nematic order
parameter is calculated from (Ref. 5)

:<math>S_2 =\frac{d \langle \cos^2 \theta \rangle -1}{d-1}</math>

where ''d'' is the dimensionality of the system.

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

:<math>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
</math>

where <math>S_2</math> is known as the uniaxial order parameter.
Here <math>P_2</math> is the second order
[[Legendre polynomials | Legendre polynomial]],
<math>\theta</math> is the angle between a molecular axes and
the director <math>n</math>, and the angle brackets
indicate an ensemble average.
==Tetrahedral order parameter==
*[http://dx.doi.org/10.1080/002689798169195 P. -L. Chau and A. J. Hardwick "A new order parameter for tetrahedral configurations", Molecular Physics '''93''' pp. 511-518 (1998)]
==See also==
*[[Landau theory of second-order phase transitions]]
==References==
#[http://dx.doi.org/10.1103/PhysRevA.10.1881   Joseph P. Straley "Ordered phases of a liquid of biaxial particles", Physical Review A '''10''' pp. 1881 - 1887 (1974)]
#[http://dx.doi.org/10.1080/00268978400101951 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)]
#[http://dx.doi.org/10.1016/0167-7322(95)00918-3 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)]
#[http://dx.doi.org/10.1063/1.479982      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)]
#[http://dx.doi.org/10.1002/mats.1992.040010402 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)]
[[category: liquid crystals]]