McMillan mean field approximation

The temperature dependence of SX depends on both and The theory has been worked out in detail by McMillan using the mean field approximation and by Brochard assuming dynamical scaling laws. The critical exponents for the divergence of the visosity as predicted by the two theories are different ... 

The results of several molecular theories that describe the smectic ordering in a system of hard spherocylinders enable us to conclude that the contribution from hardcore repulsion can be described by the smoothed-density approximation. On the other hand, a realistic theory of thermotropic smectics can only be developed if the intermolecular attraction is taken into account, The interplay between hard-core repulsion and attraction in smectic A liquid crystals has been considered by Kloczkow-ski and Stecki [17] using a very simple model of hard spherocylinders with an ad-ditonal attractive r potential. Using the Onsager approximation, the authors have obtained equations for the order parameters that are very similar to the ones found in the McMillan theory but with explicit expressions for the model parameters. The more general analysis has been performed by Me-deros and Sullivan [76] who have treated the anisotropic attraction interaction by the mean-field approximation while the hardcore repulsion has been taken into account using the nonlocal density functional approach proposed by Somoza and Tarazona. 

The Maier-Saupe mean field theory of nematics can be extended to smectic A liquid crystals following the development of McMillan [3.24]. The smectic A phase has a unique axis (the director) like the nematic phase, but it also possesses a one-dimensional translational periodicity. The centers of mass of the molecules tend to lie on planes normal to the director. The interplanar distance, d, is approximately a molecular length, twice the molecular length or in between these two length scales. There is no positional ordering of the centers of mass of the molecules within each plane. The single-molecule potential may be deduced from the Kobayashi s pair interaction potential [3.25]... 

There are several levels of approximation possible in the consideration of the NA transition. First there is the self-consistent mean field formulation due to Kobayashi and McMillan [8-10]. This is an extension to the smectic-A phase of the self-consistent mean-field formulation for nematics ( Maier-Saupe theory [11]). Kobayashi-McMillan (K-M) theory takes into account the coupling between the nematic order parameter magnitude S with a mean-field smectic order parameter. In Maier-Saupe theory, the key feature of the nematic phase - the spontaneously broken orientational symmetry - is put in by hand by making the pair potential anisotropic. In the same spirit, the K-M formulation puts in by hand a sinusoidal density modulation as well as the nematic-smectic coupling. 

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