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| An '''order parameter''' is some observable physical quantity that is able to distinguish between
| | :<math> S_2 = \langle \frac{1}{2} (3 \cos^2 \theta_i -1) \rangle </math> |
| two distinct phases. The choice of order parameter is not necessarily unique.
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| ==Solid-liquid transition==
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| Possible choices:
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| *Fourier transform of the density
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| *Shear modulus
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| ==Isotropic-nematic transition==
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| The '''uniaxial order parameter''' is zero for an isotropic fluid and one for a perfectly aligned system. First one calculates a director vector <ref>[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)]</ref>
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| :<math>Q_{\alpha \beta} | |
| = \frac{1}{N} | |
| \sum_{j=1}^{N} \left( \frac{3}{2}
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| \hat e_{j \alpha} \hat e_{j \beta}
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| -\frac{1}{2} \delta_{\alpha\beta}\right),~~~~~\alpha, \beta = x, y, z,</math>
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| where <math>Q</math> is a second rank tensor, <math>\hat e_{j}</math> is a unit
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| vector along the molecular long
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| axis,
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| and <math>\delta_{\alpha\beta}</math> is the [[Kronecker delta]].
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| Diagonalisation of this tensor
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| gives three eigenvalues <math>\lambda_+</math>, <math>\lambda_0</math> and <math>\lambda_-</math>,
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| and <math>n</math> is the eigenvector associated
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| with the largest eigenvalue (<math>\lambda_+</math>).
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| From this director vector the nematic order
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| parameter is calculated from <ref>[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)]</ref>
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| :<math>S_2 =\frac{d \langle \cos^2 \theta \rangle -1}{d-1}</math>
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| where ''d'' is the dimensionality of the system.
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| i.e. in three dimensions <ref>[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)]</ref>
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| :<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
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| </math>
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| where <math>S_2</math> is known as the uniaxial order parameter.
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| Here <math>P_2</math> is the second order
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| [[Legendre polynomials | Legendre polynomial]],
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| <math>\theta</math> is the angle between a molecular axes and
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| the director <math>n</math>, and the angle brackets
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| indicate an ensemble average.
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| ==Tetrahedral order parameter==
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| <ref>[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)]</ref>
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| ==See also==
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| *[[Landau theory of second-order phase transitions]]
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| ==References== | | ==References== |
| <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)] |
| ;Related reading
| | #[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/10.1103/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/10.1063/1.479982 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/10.1063/1.479982 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)]
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| *[http://dx.doi.org/10.1063/1.3548889 Erik E. Santiso and Bernhardt L. Trout "A general set of order parameters for molecular crystals", Journal of Chemical Physics '''134''' 064109 (2011)]
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| [[category: liquid crystals]]
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