The modified Maier-Saupe mean field theory

1 The Modified Maier-Saupe Mean Field Theory 

Two mechanisms can lead to the formation of a nematic phase in a liquid solution. The first is based on the traditional Onsager or Flory type of approach.This treatment predicts that at a certain concentration the molecular asymmetry alone is sufficient to create an ordered phase without any attractive interactions. The second approach shows that a stable mesophase is formed by an anisotropic potential. 

The excluded-volume theories of Onsager and Flory show that, depending on the axial ratio of the rod-like particles, there is a critical concentration above which a nematic phase is formed. This concentration does not depend on the temperature of the system as these theories are essentially athermal, i.e., the part of the free energy leading to the anisotropy is an entropy term. 

A later version of the Flory theory describes a system of semiflexible particles. The persistence length is used to determine an effective axial ratio for the particles. A temperature-dependent persistence length thus leads to a thermotropic type of behavior where the phase transition is governed by temperature and concentration. 

The second or alternative mechanism is used by the Maier-Saupe mean field theory in which the stability of the nematic phase is derived from an anisotropic potential. Picken has developed a theory for the nematic phase formation of liquid-crystalline polymers which is based on the Maier-Saupe mean field theory. A molecule in a nematic domain, with its axis at an angle p with respect to the director or average orientation axis of the domain, is assumed to feel the influence of the surrounding medium only in terms of an anisotropic potential 

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