# Onsager theory

The **Onsager theory** for the isotropic-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[edit]

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

- .

These are the Onsager equations.

## References[edit]

- 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)