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)

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_{\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 (Ref. 5)

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

where d is the dimensionality of the system.

i.e. in three dimensions (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.

Tetrahedral order parameter

• 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

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)
5. 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)