Order parameters - Revision history

Carl McBride: Added a recent publication

2011-02-15T10:47:40Z

Added a recent publication

← Older revision		Revision as of 12:47, 15 February 2011
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	*Shear modulus		*Shear modulus
	==Isotropic-nematic transition==		==Isotropic-nematic transition==
	The '''uniaxial order parameter''' is zero for an isotropic fluid and one for		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>
	a perfectly aligned system.
	First one calculates a director
	vector (~~see Ref. 2~~)

	:<math>Q_{\alpha \beta}		:<math>Q_{\alpha \beta}
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	with the largest eigenvalue (<math>\lambda_+</math>).		with the largest eigenvalue (<math>\lambda_+</math>).
	From this director vector the nematic order		From this director vector the nematic order
	parameter is calculated from (~~Ref. 5~~)		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>

	:<math>S_2 =\frac{d \langle \cos^2 \theta \rangle -1}{d-1}</math>		:<math>S_2 =\frac{d \langle \cos^2 \theta \rangle -1}{d-1}</math>

	where ''d'' is the dimensionality of the system.		where ''d'' is the dimensionality of the system.

	i.e. in three dimensions (~~see Ref~~. 3)		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>

	:<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>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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	indicate an ensemble average.		indicate an ensemble average.
	==Tetrahedral order parameter==		==Tetrahedral order parameter==
	*[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>[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>
	==See also==		==See also==
	*[[Landau theory of second-order phase transitions]]		*[[Landau theory of second-order phase transitions]]
	==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)]		<references/>
	#[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)]		;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)]
	#[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~~)]		*[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)]
	[[category: liquid crystals]]		[[category: liquid crystals]]

Nice and Tidy: Started section on the tetrahedral order parameter.

2009-04-01T13:05:35Z

Started section on the tetrahedral order parameter.

← Older revision		Revision as of 15:05, 1 April 2009
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	the director <math>n</math>, and the angle brackets		the director <math>n</math>, and the angle brackets
	indicate an ensemble average.		indicate an ensemble average.
			==Tetrahedral order parameter==
			*[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)]
	==See also==		==See also==
	*[[Landau theory of second-order phase transitions]]		*[[Landau theory of second-order phase transitions]]

Carl McBride: Added short intro.

2008-02-27T10:50:29Z

Added short intro.

← Older revision		Revision as of 12:50, 27 February 2008
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	==~~Liquid crystals~~==		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		The '''uniaxial order parameter''' is zero for an isotropic fluid and one for
	a perfectly aligned system.		a perfectly aligned system.

Carl McBride at 11:21, 26 November 2007

2007-11-26T11:21:36Z

← Older revision		Revision as of 13:21, 26 November 2007
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			==Liquid crystals==
	The '''uniaxial order parameter''' is zero for an isotropic fluid and one for		The '''uniaxial order parameter''' is zero for an isotropic fluid and one for
	a perfectly aligned system.		a perfectly aligned system.
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	indicate an ensemble average.		indicate an ensemble average.

			==See also==
			*[[Landau theory of second-order phase transitions]]
	==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)]		#[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)]

Carl McBride at 10:16, 22 June 2007

2007-06-22T10:16:58Z

← Older revision		Revision as of 12:16, 22 June 2007
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	with the largest eigenvalue (<math>\lambda_+</math>).		with the largest eigenvalue (<math>\lambda_+</math>).
	From this director vector the nematic order		From this director vector the nematic order
	parameter is calculated from (see Ref. 3)		parameter is calculated from (Ref. 5)

			:<math>S_2 =\frac{d \langle \cos^2 \theta \rangle -1}{d-1}</math>

			where ''d'' is the dimensionality of the system.

			i.e. in three dimensions (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>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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	#[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/10.1016/0167-7322(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)]		#[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/10.1002/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)]
	[[category: liquid crystals]]		[[category: liquid crystals]]

Carl McBride at 17:08, 30 May 2007

2007-05-30T17:08:18Z

← Older revision		Revision as of 19:08, 30 May 2007
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	where <math>S_2</math> is known as the uniaxial order parameter.		where <math>S_2</math> is known as the uniaxial order parameter.
	Here <math>P_2</math> is the second order		Here <math>P_2</math> is the second order
	[[Legendre ~~polynomial~~ \| Legendre polynomial]],		[[Legendre polynomials \| Legendre polynomial]],
	<math>\theta</math> is the angle between a molecular axes and		<math>\theta</math> is the angle between a molecular axes and
	the director <math>n</math>, and the angle brackets		the director <math>n</math>, and the angle brackets
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	#[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/10.1016/0167-7322(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)]		#[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)]
			[[category: liquid crystals]]

Carl McBride at 17:07, 30 May 2007

2007-05-30T17:07:40Z

← Older revision		Revision as of 19:07, 30 May 2007
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	where <math>S_2</math> is known as the uniaxial order parameter.		where <math>S_2</math> is known as the uniaxial order parameter.
	Here <math>P_2</math> is the second order		Here <math>P_2</math> is the second order
	[[Legendre polynomial]],		[[Legendre polynomial \| Legendre polynomial]],
	<math>\theta</math> is the angle between a molecular axes and		<math>\theta</math> is the angle between a molecular axes and
	the director <math>n</math>, and the angle brackets		the director <math>n</math>, and the angle brackets

Carl McBride at 16:52, 23 April 2007

2007-04-23T16:52:17Z

← Older revision		Revision as of 18:52, 23 April 2007
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	a perfectly aligned system.		a perfectly aligned system.
	First one calculates a director		First one calculates a director
	vector (see Ref. 1)		vector (see Ref. 2)

	:<math>Q_{\alpha \beta}		:<math>Q_{\alpha \beta}

Carl McBride at 16:51, 23 April 2007

2007-04-23T16:51:57Z

@@ 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
 ==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/10.1016/0167-7322(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)]

Carl McBride at 16:31, 23 April 2007

2007-04-23T16:31:03Z

← Older revision		Revision as of 18:31, 23 April 2007
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	:<math> S_2 = \langle \frac{1}{2} (3 \cos^2 \theta_i -1) \rangle </math>		:<math> S_2 = \langle \frac{1}{2} (3 \cos^2 \theta_i -1) \rangle </math>
			==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/10.1016/0167-7322(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)]