Orientational order McMillan theory

The Maier-Saupe theory was developed to accoimt for ordering in the smectic A phase by McMillan [71]. He allowed for the coupling of orientational order to the translational order, by introducing a translational order parameter which depends on an ensemble average of the first harmonic of the density modulation normal to the layers as well as F . This model can accoimt for both first- and second-order nematic-smectic A phase transitions,... 

McMillan s model [71] for transitions to and from the SmA phase (section C2.2.3.21 has been extended to columnar liquid crystal phases formed by discotic molecules [36. 103]. An order parameter that couples translational order to orientational order is again added into a modified Maier-Saupe theory, that provides the orientational order parameter. The coupling order parameter allows for the two-dimensional symmetry of the columnar phase. This theory is able to accormt for stable isotropic, discotic nematic and hexagonal colmnnar phases. 

For the orientational free energy, we employ the conventional molecular field theory of Maier and Saupe [66], or its extension by McMillan [67], which includes both orientational ordering of the mesogenic cores and translational ordering of their centers of mass. It is given by... 

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,... 

Figure 5.19 Predictions of the McMillan theory for the dependence of orientational (P2) and translational (a) order parameters on temperature. At o = 1.1 (where a is defined by Eq. 5.17), there is a first-order transition from the SmA phase to the I phase (at Tai)- At a = 0.85, a first-order transition from SmA to nematic occurs (at Tan) below the N-I transition (at Tni), which is always first order. At a = 0.6, the SmA-N transition is second order...

Garland [10] studied second-order SmA-N and SmC-SmA phase transitions by very precise heat capacity measurements up to 300 MPa. Similar measurements of the critical heat capacity near the SmA-N transition were performed by Kasting et al. [93, 94]. McKee et al. [95] carried out orientational order determinations near a possible SmA-N TCP (p = 289 MPa, 140 °C) by NMR. From McMillan s theory [96] in the case of a TCP of the SmA-N (N ) transition at atmospheric pressure a value of 0.866 (model parameter S = 0) for r(SmA-N)/ T(N-I) follows. Thus a rough test by the corresponding transition temperatures at higher tricritical pressures is possible. 

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. 

---