The Molecular Potential of Smectogens

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] 

The expansion coefficients involve not only purely orientational (Ax o). nd translational (Aon) order parameters, the Pl os6)) and (cos(27rn /d)), but also mixed-order parameters Pl(pos0) cos 2 Knz/d)). Therefore, Alu describe coupling between the degrees of orientational and translational order. The three order parameters of lowest degree in Eq. (3.61) have been given special symbols in the literature  

In the isotropic phase, 77 = r = a = 0 in the nematic phase, 77 0, r = cr = 0 in the smectic A phase, all three order parameters are non-zero. The molecular theory of smectic A liquid crystals should predict temperature dependences of these order parameters. 

FIGURE 3.5. Temperature dependence of the order parameter for 6 = 0 and several values of a (after Ref. [3.24]). 

In addition to the smectic A, nematic, and isotropic solutions, Eq. (3.64) yields various less physical solutions. Now the free energy is given by 

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