Orientational order Maier—Saupe theory

The electrostatic part, Wg(ft), can be evaluated with the reaction field model. The short-range term, i/r(Tl), could in principle be derived from the pair interactions between molecules [21-23], This kind of approach, which can be very cumbersome, may be necessary in some cases, e.g. for a thorough analysis of the thermodynamic properties of liquid crystals. However, a lower level of detail can be sufficient to predict orientational order parameters. Very effective approaches have been developed, in the sense that they are capable of providing a good account of the anisotropy of short-range intermolecular interactions, at low computational cost [6,22], These are phenomenological models, essentially in the spirit of the popular Maier-Saupe theory [24], wherein the mean-field potential is parameterized in terms of the anisometry of the molecular surface. They rely on the physical insight that the anisotropy of steric and dispersion interactions reflects the molecular shape. 

This conclusion was reached, tentatively, by Frenkel, Shaltyko and Elyashevich A phenomenological analysis presented by Pincus and de Gennes predicted a first-order phase transition even in the absence of cooperativity in the conformational transition. These authors relied on the Maier-Saupe theory for representation of the interactions between rodlike particles. Orientation-dependent interactions of this type are attenuated by dilution in lyotropic systems generally. In the case of a-helical polypeptides they should be negligible owing to the small anisotropy of the polarizability of the peptide unit (cf. seq.). Moreover, the universally important steric interactions between the helices, regarded as hard rods, are not included in the Maier-... 

In their original theory, Maier and Saupe supposed that the molecular interactions responsible for the nematic state are anisotropic van der Waals interactions (discussed in Section 2.3), in which case mms should be temperature-independent. However, it is now recognized that shape anisotropy is also important, even for small-molecule thermotropic nematics. By making mms temperature-dependent, the Maier-Saupe potential can, in principle, accommodate both energetic and entropic effects. In fact, if the function sin(u, u) in the purely entropic Onsager potential Eq. (2-5) is approximated by the expansion 1 — V2 cos (u, u)+. . ., then to lowest order the Maier-Saupe potential (2-7) is obtained with C/ms — Uo bT/S, where we have defined the dimensionless Maier-Saupe energy constant by Uus = ums/ksT, Thus, the Maier-Saupe potential can be used as an approximation to describe orientational order in either lyotropic (solvent-based) or thermotropic nematics. For a thermotropic melt, the Maier-Saupe theory predicts a first-order transition from the isotropic to the nematic phase when mms/ bT = U s — t i.MS = 4.55, and at this transition the scalar order parameter S jumps from zero to 0.43. S increases toward unity with further increases in Uus- The spinodal point at which the isotropic phase is unstable to even small orientational perturbations occurs atU — = 5 for the Maier-... 

The Maier-Saupe theory of nematic liquid crystals is founded on a mean field treatment of long-range contributions to the intermolecular potential and ignores the short-range forces [88, 89]. With the assumption of a cylindrically symmetrical distribution function for the description of orientation of the molecules and a nonpolar preferred axis of orientation, an appropriate order parameter for a system of cylindrically symmetrical molecules is... 

We consider first the Maier-Saupe theory and its variants. In its original formulation, this theory assumed that orientational order in nematic liquid crystals arises from long-range dispersion forces which are weakly anisotropic [60. 61 and 62]. However, it has been pointed out [63] that the form of the Maier-Saupe potential is equivalent to one in... 

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. 

These theories all pr>edict a first order nematic-isotropic phase transition, and a weakly temperature dependent order parameter. In rigid rod Maier-Saupe theory, the order parameter is given by the angle of the rod to the direction 0" prefered orientation... 

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,... 

The Maier-Saupe theory is extremely useful in understanding the spontaneous long-range orientational order and the related properties of the nematic phase. The single-molecule potential Vi(cos0) is given by Eq. (3.19) with e being volume dependent and independent of pressure and temperature. The self-consistency equation for (P2) is... 

Figure 5.17 Orientational order parameter P2 versus reduced temperature. Experimental data for the nematogen PAA (Fig. 5.1). Open circles data from neutron diffraction experiments closed circles data from NMR experiments line Maier-Saupe theory. Reprinted from I. W. Hamley et al,J. Chem. Phys., 104,10046-10054 Oriental Ordering... Copyright (1996), American Institute of Physics...

To use the model to predict other properties of liquid crystal dimers, for example, the N-I transition temperature and the temperature dependence of the order parameter it is necessary to make an additional approx-imation. This is to relate the strength parameter Xa for a mesogenic group to the orientational order of the nematic mesophase. By analogy with the Maier-Saupe theory [63] and the extension of this to multicomponent mixtures [68] it is assumed that... 

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. 

The major information available experimentally concerning the orientational order of nematics is the second rank order parameter which is given by the Maier-Saupe theory as... 

FIGURE 3 Orientational order parameters determined for PAA (points) compared with the predictions of the Maier-Saupe theory (line). 

The first attempt to develop a theory for non-rigid mesogens was made by Marcelja who extended the Maier-Saupe theory for nematics composed of cylindrically symmetric particles to include molecular flexibility. The advent of studies of the variation of the orientational order... 

Orientational energy from this source provides the basis for the Maier and Saupe theory asymmetry of molecular shape and its influence on molecular packing in the fluid are not taken into account. Consequently, in order to reconcile this theory with experiments, it has been necessary to postulate unreasonably large orientation-dependent energies and, by implication, excessively large value of Aa... 

Crystalline or orientational orderings are mostly controlled by repulsive forces, such as excluded-volume forces. The crystalline transitions in ideal hard-sphere fluids and the nematic liquid crystalline transitions in hard-rod suspensions are convenient simple models of corresponding transitions in fluids composed of uncharged spherical or elongated molecules or particles. The transition from the isotropic to the nematic state can be described theoretically using the Onsager, Maier-Saupe, or Rory theories. 

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... 

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