McMillan’s model of smectic

Extension of McMillan s model of smectic A to discotic liquid... 

Extension of McMillan s Model of Smectic A Phases to Discotic Liquid... 

Ghose et al. [59] have extended this theory using a variational principle to solve the problem with the full potential rather than the one truncated to the first Fourier component. This method, which is closely analogous to the extension of McMillan s model of the smectic A phase by Lee et al. [60], leads to some qualitative improvements in the phase diagram for a homologous series. More recently, Ghose et al. [61] have presented a simpler mean field version of their theory, which yields essentially similar results. 

The Maier-Saupe theory can also be extended to describe the smectic A-nematic transition in what is called McMillan s model. Two order parameters are introduced into the mean-field potential energy function, the usual orientational order parameter S and an order parameter a related to the amplitude of the density wave describing the smectic A layers,... 

A direct method of studying the translational order (or the amplitude of the density wave) is by measuring the intensity of the Bragg scattering from the smectic planes. McMillan s experimental results on cholesteryl myr-istate are shown in fig. 5.2.6 and as can be seen there is excellent agreement with the refined model. The X-ray intensities reveal an appreciable pretransitional smectic-like behaviour in the cholesteric (nematic) phase. This aspect of the problem will be dealt with in a later section. 

In the McMillan model, the smectic A-nematic transition can be continuous or discontinuous. If a is less than 0.7, then o decreases to zero continuously and S is continuous at the smectic A- nematic transition. If a is between 0.7 and 0.98, then a jumps to zero discontinuously and S has a small discontinuity at the smectic A-nematic transition. When a is greater than 0.98, the smectic phase transforms directly into the isotropic phase with discontinuities in both order parameters. So just as in the extended Landau-de Geimes theory for the smectic A phase, a tricritical point is predicted at a=0.7, which corresponds to a ratio in the smectic A—nematic transition temperature to the nematic-isotropic transition temperature of 0.87. A great deal of experimental work has been done on the smectic A-nematic transition, and the results seem to indicate that the tricritical point occurs when the ratio of the two transition temperatures is significantly larger than 0.87. 

The question of whether the normally first-order smectic-A to nematic phase transition can be second order in some materials is somewhat controversial. All of the published mean-field theories " of the smectic-A phase do exhibit second-order phase changes for certain values of the potential parameters. In both McMillan s theory and that of Lee et al, the second-order transition is predicted to occur at that end of homologous series having short chain lengths. More specifically, these models predict the second-order changes to occur when the ratio of transition temperatures Tan/Tni (or Tac/Tci) is at or below about 0.88 (see Fig. 5). 

Fig. 5.2.1. Order parameters s and a, entropy S and specific heat versus reduced temperature T/Q.12Sil predicted by the model for a = 1.1 showing a first order smectic A-i otropic transition. S and are expressed in terms of R , the gas constant. (After McMillan. )...

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