McMillan theory

Similar behaviour is found for the singlet translational distribution function, p z), in the two smectic phases. According to the McMillan theory... 

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...

P2 (Eq. 5.11) and a (Eq. 5.16) for three different values of a is shown in Eig. 5.19. Eor large a (for example a = 1.1) d is large and smectic ordering is favoured. There is thus a first-order transition on heating from the SmA phase to the isotropic phase. However, as a is lowered, a nematic phase is formed between smectic and isotropic phases. In the case o = 0.85, the transition between SmA and N phases is first order, whereas at lower o, for example a = 0.6, it is continuous (second order), as shown by the continuous decrease of a to zero (P2 also varies continuously, but there is a change of slope with respect to temperature at the transition). The McMillan theory predicts that the crossover from a first-order to a second-order transition (called a tricritical point) occurs ata = 0.98, which corresponds in the model to a ratio of phase transition temperatures 7an/ i = 0.870. 

The simple theory of the nematic-smectic A transition has been proposed by McMillan [59] (and independently by Kobayashi [60]) by extending the Maier-Saupe approach to include the possibility of translational ordering. The McMillan theory is a classical mean-field theory and therefore the free energy is given by the general Eq. (34). For the smectic A phase it can be rewritten as... 

In general the McMillan theory provides a good qualitative and sometimes even quantitative description of the nematie-smectic A phase transition. The theory accounts successfully for the decrease in the transition entropy with the breadth of the ne-... 

Figure 5. Phase diagram of a liquid crystal system according to the McMillan theory. Inset typical phase diagram for a homologous series of compounds (after McMillan [591).

The McMillan theory has been further refined by several authors [62-64] to improve the quantitative agreement with experiment. However, the basic structure of the theory remains the same. This theory presents another example of a successful application of a simple mean-field approach. On the other hand, there are several limitations of the McMillan theory that cannot be ignored. Firstly, the theory is based on the semi phenomenological potential that does not allow determination of the smectic period in a self-consistent way. Secondly, the model poten-... 

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 interaction potentials (Eqs. 85 and 90) essentially depend on a coupling between the molecular orientation and the intermolecular vector. We note that this coupling could be neglected in the first approximation in the theory of the nematic-smectic A transition, as it is done, for example, in the McMillan theory. At the same time this coupling just determines the effect in the theory of transition into the smectic C phase. 

MBBA/Canada balsam, textures 438 2MBCB, atoim stic simulations 83 McMillan theory 61,284 f... 

Mean field theories can also be extended to the phase behavior of smectic A liquid crystal and flexible polymer blends [63, 70-74] by combining the Flory-Huggins theory for isotropic mixing and Kobayashi-McMillan theory [32, 75] for smectic A ordering of liquid crystals. 

A-nematic phase transition (SNT) is of second order, while the NIT is of first order. In Figure 2.16b, the smectic order parameter discontinuously drops to zero with increasing <)> and the SNT is of first order. In Figure 2.16c, the nematic phase disappears and we only have the first-order smectic A-isotropic phase transition (SIT). Owing to McMillan theory for a pure nematogen (<)> = 0), the SNT should be second order for Tsf /T j < 0.87 and first order for larger values of 7 /T j, where shows the SNT temperature of a pure nematogen. 

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