Maier-Saupe Mean Field Approach

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]. 

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 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. 

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 nature of the nematic-isotropic transition of low molar mass liquid crystals is described by two main theories. The Maier and Saupe (MS) approach uses a mean field theory with a quadrupo-lar interaction between adjacent molecules. It gives a first order transition, usually in agreement with experiments. Nevertheless, it... 

However, the number of liquid crystals that have been studied under pressure is very limited. In most cases neither the equation of state nor the pressure dependence of the order parameter is known. Only the mean-field theory of Maier and Saupe was extended to explain the dielectric properties of liquid crystalline phases. However, a recent approach by Photinos et al. analyzed the nematic reentrance and phase stability based on the variational cluster method. The lack of a full theoretical description as well as insufficient experimental data should stimulate further high-pressure investigations in this field. 

Theoretical treatments of liquid crystals such as nematics have proved a great challenge since the early models by Onsager and the influential theory of Maier and Saupe [34] mentioned before. Many people have worked on the problems involved and on the development of the continuum theory, the statistical mechanical approaches of the mean field theory and the role of repulsive, as well as attractive forces. The contributions of many theoreticians, physical scientists, and mathematicians over the years has been great - notably of de Gennes (for example, the Landau-de Gennes theory of phase transitions), McMillan (the nematic-smectic A transition), Leslie (viscosity coefficients, flow, and elasticity). Cotter (hard rod models), Luckhurst (extensions of the Maier-Saupe theory and the role of flexibility in real molecules), and Chandrasekhar, Madhusudana, and Shashidhar (pre-transitional effects and near-neighbor correlations), to mention but some. The devel-... 

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... 

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 results obtained here and in the previous several sections are completely equivalent to the mean field theory derived by Maier and Saupe Eqs. [2], [3], [5], and [8] are identical to those presented in this classic series of papers. Their approach is, of course, more systematic than presented here, and the volume dependence of the... 

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