Maier-Saupe mean field theory for small molecular mass liquid crystals

Maier-Saupe mean field theory for small molecular mass liquid crystals 

Liquid crystalline polymers can be regarded as a long chain with rods connected in sequence, each rod being, in some sense, equivalent to a small molecular mass liquid crystal. This is the so-called freely-jointed-rod chain, the simplest model of polymers. It is understood that the constituent units — small molecular mass liquid crystals play an essential role in liquid crystalline polymers. Here, we introduce an important theory for small molecular mass liquid crystal — the Maier-Saupe mean field theory (Maier Saupe, 1959, 1960). 

In contrast to the Onsager and Flory theories, the Maier-Saupe theory no longer takes into account molecular steric effects as the basic interaction but instead proposes that the van der Waals interactions between molecules are the basis for forming a liquid crystal phase. The van der Waals interaction depends on molecular orientations. The Maier-Saupe theory adopts a rather simple mathematical treatment and can easily take into account the relationship of system properties to temperature. This theory has been successfully applied to a thermotropic system of small molecular mass liquid crystal. 

In this system, a molecule is subjected to the van der Waals attractions of other molecules in the system which is related to the relative orientations of the pair molecules. Suppose the Z axis is the reference direction. Then, the potential for each molecule is a function of orientation 

Therefore, the mean field potential of a molecule is given by 

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