Molecular field theories Maier-Saupe theory

The nematic phase being the liquid crystal of highest symmetry, its condensation from the isotropic liquid should be the simplest to describe. Indeed, molecular theories convincingly explain the natural onset of nematic ordering in a population of anisotropic molecules with excluded volume interaction (Onsager) or in mean field theory (Maier-Saupe). Regarding the effect of symmetry on the isotropic to nematic (I-N) phase transition, the phenomenological approach is useful too. 

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. 

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... 

There are several different theoretical approaches to the problem. The Landau molecular field theory was applied by de Gennes to liquid-crystal phase transitions. (89) The Maier-Saupe theory focuses attention on the role of intermolecular attractive forces.(90) Onsager s classical theory is based on the analysis of the second virial coefficient of very long rodlike particles.(91) This theory was the first to show that a solution of rigid, asymmetric molecules should separate into two phases above a critical concentration that depends on the axial ratio of the solute. One of these phases is isotropic, the other anisotropic. The phase separation is, according to this theory, solely a consequence of shape asymmetry. There is no need to involve the intervention of intermolecular attractive forces. Lattice methods are also well suited for treating solutions, and phase behavior, of asymmetric shaped molecules.(80,92,93)... 

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 seminal molecular field theory of nematics was developed by Maier and Saupe [20] who showed that the potential of mean torque for a uniaxial molecule in a nematic is given 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 ... 

Maier-Saupe mean field theory for small molecular mass liquid crystals... 

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). 

It is of interest to examine the relationship between the phenomenological model that we have just discussed and the molecular statistical theory of Maier and Saupe. The free energy of the weakly ordered isotropic phase in the presence of an external magnetic field is, according to the mean field theory,... 

Maier and Saupe, in their well-known molecular-statistical theory, described the intermolecular orientational forces by a mean field method. The Maier-Saupe theory successfully predicts the relationship between the molecular orientation parameter S and the nematic potential D as a function of temperature [10,14]. 

The Maier-Saupe mean field theory of nematics can be extended to smectic A liquid crystals following the development of McMillan [3.24]. The smectic A phase has a unique axis (the director) like the nematic phase, but it also possesses a one-dimensional translational periodicity. The centers of mass of the molecules tend to lie on planes normal to the director. The interplanar distance, d, is approximately a molecular length, twice the molecular length or in between these two length scales. There is no positional ordering of the centers of mass of the molecules within each plane. The single-molecule potential may be deduced from the Kobayashi s pair interaction potential [3.25]... 

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. 

In the previous chapter we examined a simple version of the molecular theory of nematic liquid crystals. The problem was treated as an order-disorder phenomenon with the solution based on a phenomenologically derived single-molecule orientational potential. The results of this development were found to be equivalent to the well known mean field theory of Maier and Saupe. ... 

The third problem is the possible effect of stress or external field on isotropic-nematic phase transition. In equilibrium, this phase transition is usually described by the well-known Landau phenomenology or more specifically (however, less reliably because of large fluctuations) by the Maier-Saupe mean field theory [2] (see also Refs [30,31 ]). The assumption that the transition behavior of nematic elastomers is independent of stress was roughly confirmed while testing the LCE theory [3], where the parameters of anisotropy were assumed to be independent of stress. The possible dependences of scalar/tensor order parameter on stress/extemal field have been considered in molecular Doi theory [9, 11] or phenomenological approach by Ericksen [41]. 

Maier and Saupe [13] developed a statistical theory to describe the liquid crystalline state and the molecular ordering for the nematic phase. In analogy to the treatment of ordering phenomena in ferromagnetics or ferroelectrics, this theory describes the intermolecular orientational forces by a mean field method. Each individual molecule feels a nematic potential D = f (0, S, V) which depends on the momentaneous angle 6 between its long axis and the optic axis, the order parameter S and the molar volume V. S is then given by... 

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