Maier-Saupe model

On the theoretical side, Marcelja [26] was first to account explicitly for flexible tail chains in nematic ordering, using the Maier-Saupe model potential (Eq. 1) for each segment of the molecule. More complex models were proposed by Samulski et al. [27] and Emsley et al. [28]. In these approaches alkyl chains are assumed to exist in a discrete set of conformers described by... 

It has been the merit of Picken (1989, 1990) having modified the Maier-Saupe mean field theory successfully for application to LCPs. He derived the stability of the nematic mesophase from an anisotropic potential, thereby making use of a coupling constant that determines the strength of the orientation potential. He also incorporated influences of concentration and molecular weight in the Maier-Saupe model. Moreover, he used Ciferri s equation to take into account the temperature dependence of the persistence length. In this way he found a relationship between clearing temperature (i.e. the temperature of transition from the nematic to the isotropic phase) and concentration ... 

The self-consistent Maier-Saupe model, as extended by Van der Vorst and Picken [157, 158] (MSVP model). 

Details of the experimental method are given in Ref. 65. Figure 2 shows the anisotropy of the polarizability (at optical frequencies) as a function of the temperature measured for an anisotropic PpBAT solution. The continous curves results from Picken s theory, whereas the dot-dashed curves comes from the standard Maier-Saupe model. 

Fig. 2. Anisotropy of the dielectric constant and , as a function of the relative temperature (T-T j) for M = 8000. The continous curve is from Picken s theory. The dot-dashed curve is from the traditional Maier-Saupe model...

In this section we consider a general model that has broad applicability to phase transitions in soft materials the Landau theory, which is based on an expansion of the free energy in a power series of an order parameter. The Landau theory describes the ordering at the mesoscopic, not molecular, level. Molecular mean field theories include the Maier-Saupe model, discussed in detail in Section 5.5.2. This describes the orientation of an arbitrary molecule surrounded by all others (Fig. 1.5), which set up an average anisotropic interaction potential, which is the mean field in this case. In polymer physics, the Flory-Huggins theory is a powerful mean field model for a polymer-solvent or polymer-polymer mixture. It is outlined in Section 2.5.6. 

A generalization of the Maier-Saupe model to include forces of other symmetry has been given by Freisner, i Chandrasekhar and Madhusudana. ... 

The Maier-Saupe tlieory was developed to account for ordering in tlie smectic A phase by McMillan [71]. He allowed for tlie coupling of orientational order to tlie translational order, by introducing a translational order parameter which depends on an ensemble average of tlie first haniionic of tlie density modulation noniial to tlie layers as well as / i. This model can account for botli first- and second-order nematic-smectic A phase transitions, as observed experimentally. 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

Smectic A and C phases are characterized by a translational order in one dimension and a liquid-like positional order in two others. In the smectic A phase the molecules are oriented on average in the direction perpendicular to the layers, whereas in the smectic C phase the director is tilted with respect to the layer normal. A simple model of the smectic A phase has been proposed by McMillan [8] and Kobayashi [9] by extending the Maier-Saupe approach for the case of one-dimensional density modulation. The corresponding mean field, single particle potential can be expanded in a Fourier series retaining only the leading term ... 

The molecular theory of Doi [63,166] has been successfully applied to the description of many nonlinear rheological phenomena in PLCs. This theory assumes an un-textured monodomain and describes the molecular scale orientation of rigid rod molecules subject to the combined influence of hydrodynamic and Brownian torques, along with a potential of interaction (a Maier-Saupe potential is used) to account for the tendency for nematic alignment of the molecules. This theory is able to predict shear thinning viscosity, as well as predictions of the Leslie viscosity coefficients used in the LE theory. The original calculations by Doi for this model employed a preaveraging approximation that was later... 

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. 

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. 

The Maier-Saupe potential is a phenomenological model originally proposed for thermotropic small-molecule LCs. It is obtained by replacing the excluded-volume interaction in Eq. (9) by... 

Liquid crystalline polymers can be regarded as a long chain with rods connected in sequence, each rod being, in some sense, equivalent to a small molecular mass liquid crystal. This is the so-called freely-jointed-rod chain, the simplest model of polymers. It is understood that the constituent units — small molecular mass liquid crystals play an essential role in liquid crystalline polymers. Here, we introduce an important theory for small molecular mass liquid crystal — the Maier-Saupe mean field theory (Maier Saupe, 1959, 1960). 

A realistic theory of nematics should, of course, incorporate the attractive potential between the molecules as well as their hard rod features. There have been several attempts to develop such hybrid models. Equations of state have been derived based on the Percus-Yevick and BBGKY approximations for spherical molecules subject to an attractive Maier-Saupe potential.However, a drawback with these models is that they lead to y = 1 (see (2.3.18)). 

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. 

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. 

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