Order parameters: Difference between revisions

Revision as of 17:51, 23 April 2007

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. 1)

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 (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 $P_{2}$ 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

@@ Line 1: / Line 1: @@
-:<math> S_2 =  \langle \frac{1}{2} (3 \cos^2 \theta_i -1) \rangle </math>
+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. 1)
+:<math>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,</math>
+where <math>Q</math> is a second rank tensor,  <math>\hat e_{j}</math> is a unit
+vector along the molecular long
+axis,
+and <math>\delta_{\alpha\beta}</math> is the [[Kronecker delta]].
+Diagonalisation of this tensor
+gives three eigenvalues <math>\lambda_+</math>, <math>\lambda_0</math> and <math>\lambda_-</math>,
+and  <math>n</math> is the eigenvector associated
+with the largest eigenvalue (<math>\lambda_+</math>).
+From this director vector the nematic order
+parameter is calculated from (see Ref. 3)
+:<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
+</math>
+where <math>S_2</math> is known as the uniaxial order parameter.
+Here <math>P_2</math> is the second order
+[[Legendre polynomial]],
+<math>\theta</math> is the angle between a molecular axes and
+the director <math>n</math>, and the angle brackets
+indicate an ensemble average.
 ==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)]
+#[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)]
 #[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/10.1063/1.479982      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)]

Order parameters: Difference between revisions

Revision as of 17:51, 23 April 2007

References

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