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.
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[edit] Solid-liquid transition
Possible choices:
- Fourier transform of the density
- Shear modulus
[edit] 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
\[Q_{\alpha \beta} = \frac{1}{N} \sum_{j=1}^{N} \left( \frac{3}{2} \hat e_{j \alpha} \hat e_{j \beta} -\frac{1}{2} \delta_{\alpha\beta}\right),~~~~~\alpha, \beta = x, y, z,\]
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
where d is the dimensionality of the system.
i.e. in three dimensions
\[S_2 = \lambda _{+}= \langle P_2( n \cdot e)\rangle = \langle P_2(\cos\theta )\rangle =\langle \frac{3}{2} \cos^{2} \theta - \frac{1}{2} \rangle \]
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.
[edit] Tetrahedral order parameter
[edit] See also
[edit] References
- ↑ 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)
