Onsager theory - Revision history

Nice and Tidy at 13:04, 19 February 2008

2008-02-19T13:04:10Z

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	The '''Onsager theory''' for the isotropic-[[nematic phase \|nematic]] phase transition was developed by [[Lars Onsager]] (Ref. 1).		The '''Onsager theory''' for the isotropic-[[nematic phase \|nematic]] phase transition was developed by [[Lars Onsager]] (Ref. 1).
	In a 3-dimensional gas of [[hard rods]] there are two contributions to the [[entropy]]: a part due to translation and a part due to orientation. These two contributions are coupled. From the point of view of the translational component of the entropy,		In a 3-dimensional gas of [[3-dimensional hard rods \|hard rods]] there are two contributions to the [[entropy]]: a part due to translation and a part due to orientation. These two contributions are coupled. From the point of view of the translational component of the entropy,
	a parallel configuration of the rods is favored. This is because the excluded volume is effectively zero. However, from an		a parallel configuration of the rods is favored. This is because the excluded volume is effectively zero. However, from an
	orientational point of view a gas of perfectly aligned rods is a very low entropy configuration.		orientational point of view a gas of perfectly aligned rods is a very low entropy configuration.

Carl McBride at 13:39, 13 August 2007

2007-08-13T13:39:21Z

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	#[http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x Lars Onsager "The effects of shape on the interaction of colloidal particles", Annals of the New York Academy of Sciences '''51''' pp. 627-659 (1949)]		#[http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x Lars Onsager "The effects of shape on the interaction of colloidal particles", Annals of the New York Academy of Sciences '''51''' pp. 627-659 (1949)]
	#Ping Sheng "Hard rod model of the nematic-isotropic phase transition", RCA Review '''35''' pp. 132-143 (1974)		#Ping Sheng "Hard rod model of the nematic-isotropic phase transition", RCA Review '''35''' pp. 132-143 (1974)
			[[category: phase transitions]]

Carl McBride at 12:15, 24 July 2007

2007-07-24T12:15:18Z

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	These are the Onsager equations.		These are the Onsager equations.
	==References==		==References==
	#Lars Onsager "The effects of shape on the interaction of colloidal particles", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)		#[http://dx.doi.org/10.1111/j.1749-6632.1949.tb27296.x Lars Onsager "The effects of shape on the interaction of colloidal particles", Annals of the New York Academy of Sciences '''51''' pp. 627-659 (1949)]
	#Ping Sheng "Hard rod model of the nematic-isotropic phase transition", RCA Review '''35''' pp. 132-143 (1974)		#Ping Sheng "Hard rod model of the nematic-isotropic phase transition", RCA Review '''35''' pp. 132-143 (1974)

Carl McBride at 14:56, 23 July 2007

2007-07-23T14:56:44Z

← Older revision		Revision as of 16:56, 23 July 2007
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	These are the Onsager equations.		These are the Onsager equations.
	==References==		==References==
	#L. Onsager "~~THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES~~", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)		#Lars Onsager "The effects of shape on the interaction of colloidal particles", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)
			#Ping Sheng "Hard rod model of the nematic-isotropic phase transition", RCA Review '''35''' pp. 132-143 (1974)

Carl McBride at 14:52, 23 July 2007

2007-07-23T14:52:56Z

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	that a perfectly aligned system is the most favorable. Thus at some point a transition must take place between		that a perfectly aligned system is the most favorable. Thus at some point a transition must take place between
	the isotropic and nematic phases.		the isotropic and nematic phases.
			==Onsager equations==

			:<math>\frac{1}{N} \ln Z (\{N_a\}, N, V, k_BT) = -\ln \eta \lambda^3 \rho ~- \int f(\Omega) \ln f(\Omega) d\Omega ~- \frac{\rho}{2}\iint f(\Omega) f(\Omega') V^{\rm excluded} (\Omega - \Omega') d\Omega d\Omega'</math>

			The second term on the right hand side is the contribution due to orientational entropy, and the third term is the effect of the excluded volume. One also has

			:<math>\ln f(\Omega) +1 + \nu + \rho\int f(\Omega')V^{\rm excluded} (\Omega - \Omega') d\Omega' = 0</math>.

			These are the Onsager equations.
	==References==		==References==
	#L. Onsager "THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)		#L. Onsager "THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)

Carl McBride at 14:39, 23 July 2007

2007-07-23T14:39:29Z

← Older revision		Revision as of 16:39, 23 July 2007
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	The '''Onsager theory''' for the isotropic-[[nematic phase \|nematic]] phase transition was developed by [[Lars Onsager]] (Ref. 1).		The '''Onsager theory''' for the isotropic-[[nematic phase \|nematic]] phase transition was developed by [[Lars Onsager]] (Ref. 1).
	In a 3-dimensional gas of [[hard rods]] there are two contributions to the [[entropy]]: a part due to translation and a part due to orientation. These two contributions are coupled.		In a 3-dimensional gas of [[hard rods]] there are two contributions to the [[entropy]]: a part due to translation and a part due to orientation. These two contributions are coupled. From the point of view of the translational component of the entropy,
			a parallel configuration of the rods is favored. This is because the excluded volume is effectively zero. However, from an
			orientational point of view a gas of perfectly aligned rods is a very low entropy configuration.

			At very low densities the orientational term ''wins'', i.e. there is very little gain in entropy to be made by
			reducing the excluded volume via parallel alignments. However, at very high densities it is clear
			that a perfectly aligned system is the most favorable. Thus at some point a transition must take place between
			the isotropic and nematic phases.


	==References==		==References==
	#L. Onsager "THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)		#L. Onsager "THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)

Carl McBride at 14:26, 23 July 2007

2007-07-23T14:26:23Z

← Older revision		Revision as of 16:26, 23 July 2007
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	[[Lars Onsager]]		The '''Onsager theory''' for the isotropic-[[nematic phase \|nematic]] phase transition was developed by [[Lars Onsager]] (Ref. 1).
			In a 3-dimensional gas of [[hard rods]] there are two contributions to the [[entropy]]: a part due to translation and a part due to orientation. These two contributions are coupled.
	==References==		==References==
	#L. Onsager "THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)		#L. Onsager "THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)

Carl McBride: New page: Lars Onsager ==References== #L. Onsager "THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)

2007-04-17T11:29:31Z

New page: Lars Onsager ==References== #L. Onsager "THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)

New page

[[Lars Onsager]]
==References==
#L. Onsager "THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES", Annals of the New York Academy of Sciences '''51''' pp. 627- (1949)