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

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

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

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

where d is the dimensionality of the system.

i.e. in three dimensions

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

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.

See also

• Landau theory of second-order phase transitions

References

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)