Maier-Saupe form

However, since the same function is obtained from the simple distribution in Eq. (35) as for the complete orientational distribution in Eq. (36) we see that even in the high order limit the Maier-Saupe form will still give a good fit to the simulation result. 

Further details of the model, including the effect of allowing a small attractive force between the rods through the introduction of a non-zero Flory-Huggins parameter, the prediction of phase diagrams and the effect of introducing an orientation-dependent interaction of the Maier-Saupe form are given in reference (4) of section 12.5. 

From a suitable expansion of/(j8), one can then extract estimates for not only the second rank orientational order parameter (P2). but also one or even more of the higher rank terms (P4), Pf) etc. On the other hand, if a Maier-Saupe form can be assumed for f(P), then a simplified form of Eq. (21) results [ 124], and the order parameter (P2) can... 

If Eq. (11-3) is multiplied by uu and integrated over the unit sphere, one obtains an evolution equation for the second moment tensor S (Doi 1980 Doi and Edwards 1986). In this evolution equation, the fourth moment tensor (uuuu) appears, but no higher moments, if one uses the Maier-Saupe potential to describe the nematic interactions. Doi suggested using a closure approximation, in which (uuuu) is replaced by (uu) (uu), thereby yielding a closed-form equation for S, namely. 

In contrast to the Onsager and Flory theories, the Maier-Saupe theory no longer takes into account molecular steric effects as the basic interaction but instead proposes that the van der Waals interactions between molecules are the basis for forming a liquid crystal phase. The van der Waals interaction depends on molecular orientations. The Maier-Saupe theory adopts a rather simple mathematical treatment and can easily take into account the relationship of system properties to temperature. This theory has been successfully applied to a thermotropic system of small molecular mass liquid crystal. 

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... 

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. 

This integral relation is very tedious to invert and the most practical approach there is simply to postulate a convenient form for the distribution function f(jS). A very convenient, reasonable and widespread choice is the Maier-Saupe distribution function [13] of the form... 

The resulting distribution function is similar to that in the Maier-Saupe theory, except that the coefficient of the potential has the form [(,Vip/k T) + A(p)], i.e., a temperature dependent attractive part and an athermal part as given by the scaled particle theory. A similar result can be obtained using the Andrews model as well. These last two approaches appear to be promising for example, calculations show that y 4 for l/b 2 without violating Cotter s thermodynamic consistency condition that the mean field potential should be proportional to p. Further the transition parameters and the properties of the nematic phase are in reasonably good agreement with the experimental values for PAA. Gen-... 

The first attempt to develop a statistical model of the cholesteric phase was by Goossens who extended the Maier-Saupe theory to take into account the chiral nature of the intermolecular coupling and showed that the second order perturbation energy due to the dipole-quadrupolar interaction must be included to explain the helicity. However, a diflUculty with this and some of the other models that have since been proposed is that in their present form they do not give a satisfactory explanation of the fact that in most cholesterics the pitch decreases with rise of temperature. 

A similar analysis may be applied to the partially ordered nematic fluids composed of molecules comprising the mesogenic unit and flexible chain segments. In the LC state, one must consider the orientation-dependent interactions in addition to those of the isotropic nature. As mentioned earlier, the volume dependence (1/V ) incorporated in the Maier-Saupe expression may be replaced by MV. In its modified form, Maier-Saupe potential can easily be accommodated by introducing an additional term in the conventional van der Waals expression ... 

From Table 1.3 it can be seen that there is a definite correlation between the increase in polarizability of the substituent (Aa) and the increase in Tni- Such a dependence would, generally speaking, be expected from the Maier-Saupe molecular-statistical theory if the role of steric factors were not allowed for. In the case of reasonably bulky substituents these factors prove to be significant, and, in particular, they can explain the fall-out points for CH3O and C2H5O groups. This correlation is also confirmed for compounds of the form (l.xx), where X, Y = —COO—, —OCO, —N=N—, or a simple bond [63]. 

The Maier-Saupe theory assmnes high symmetry for molecules forming liquid crystals. In reahty, this is usually not the case and the theory has been extended [3.18] to lath-like molecules. The order parameter tensor S is given by Eq. (3.8) for a biaxial molecule in a uniaxial phase. In the principal axis x y z) system of 5, only two order parameters, Szz and D = Sxx — Syy, are needed, which are related to the Wigner matrices according to Eq. (2.43) ... 

For cylindrical molecules in mesophases of Dooh symmetry, a truncated form of the pseudo-potential (Maier-Saupe potential) may be used,... 

Equation (27) presents a simple anisotropic attraction potential that favors nematic ordering. This potential has been used in the original Maier-Saupe theory [11, 12]. We note that the interaction energy (Eq. 27) is proportional to the anisotropy of the molecular polarizability Aa. Thus, this anisotropic interaction is expected to be very weak for molecules with low dielectric anisotropy. Such molecules, therefore, are not supposed to form the nematic phase. This conclusion, however, is in conflict with experimental results. Indeed, there exist a number of materials (for example, cyclo-... 

Realistic intermolecular interaction potentials for mesogenic molecules can be very complex and are generally unknown. At the same time molecular theories are often based on simple model potentials. This is justified when the theory is used to describe some general properties of liquid crystal phases that are not sensitive to the details on the interaction. Model potentials are constructed in order to represent only the qualitative mathematical form of the actual interaction energy in the simplest possible way. It is interesting to note that most of the popular model potentials correspond to the first terms in various expansion series. For example, the well known Maier-Saupe potential JP2 (Sfli )) is just the first nonpolar term in the Legendre polynomial expansion of an arbitrary interaction potential between two uniaxial molecules, averaged over the intermolecular vector r,-, ... 

In principle, bonding in thermotropic systems such as the compound in Fig. 20B and the polymer in Fig. 2A should be enhanced by the formation of the nematic phase, just as in the case of the lyotropic systems discussed below. Bla-don and Griffin [7] have indeed theoretically shown that growth is expected for a wormlike chain forming a nematic phase stabilized by soft interactions of the Maier-Saupe type. The modest effect expected would not be easily revealed by... 

Polymers and liquid ciystals are important materials for various research fields. If the two substances are mixed, novel materials which combine the advantageous properties of both may be formed. 1 began to think about this around 1994. Already at that time, this mixed system had attracted attention as an electro-optical material, but from the perspective of basic physical properties, the center of the liquid crystal research up to that point was the phase transition and Uquid crystal stracture of novel low molecular weight liquid crystals. The physics of liquid crystals was based on the Onsager theory, the Maier-Saupe theory, and the elastic theory by Frank. However, the theoretical study of a liquid crystal mixed with other substances had not yet been developed. So, 1 began to think to build theories of phase separations and phase transitions in mixtures of liquid crystals and other substances. Our first paper on the theory of phase separations in the mixture of a polymer and a liquid crystal was published in 1996 [41]. 1 found at a later date that a paper on the same topic by Prof. Kyu of Akron University had been presented already in 1995 [42]. However, there was a difference between the two theories. Kyu s theory has dealt with low molecular weight liquid crystals in an attractive model, whereas our model considered both attractive and repulsive interactions between rodlike liquid crystal molecules and can handle also long rodlike molecules. After that, I had a variety of discussions with Kyu and it was a valuable experience for my research. 

The Maier-Saupe theory has been extended to include the influence of molecular biaxiality in flie potential of mean torque which now takes the form [18]... 

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