Editing Onsager theory

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, 
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.
==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==
#[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)
[[category: phase transitions]]