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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} is a second rank tensor, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat e_{j}} is a unit vector along the molecular long axis, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_{\alpha\beta}} is the Kronecker delta. Diagonalisation of this tensor gives three eigenvalues Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_+} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_0} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_-} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is the eigenvector associated with the largest eigenvalue (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda_+} ). From this director vector the nematic order parameter is calculated from (see Ref. 3)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_2} is known as the uniaxial order parameter. Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_2} is the second order Legendre polynomial, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} is the angle between a molecular axes and the director Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)