Maier-Saupe mean field theory

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). 

The Modified Maier-Saupe Mean Field Theory.125... 

The second or alternative mechanism is used by the Maier-Saupe mean field theory in which the stability of the nematic phase is derived from an anisotropic potential. Picken has developed a theory for the nematic phase formation of liquid crystalline polymers, which is based on the Maier-Saupe mean field theory [37, 38], A molecule in a nematic domain, with its axis at an angle

average orientation axis of the domain, is assumed to feel the influence of the surrounding medium only in terms of an anisotropy potential... 

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]... 

Numerical values of the equilibrium order parameter < Pg > for various temperatures between zero and 0.22019 v/k have been found on the computer and are presented in Table 1 as an aid to anyone wishing to perform numerical calculations based on this or the Maier-Saupe mean field theory. 

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]. 

The dielectric spectroscopy of anisotropic fluids started in the 1970s by the extension of the Debye model from isotropic media (described in Appendix D) to uniaxial systems based on statistical mechanical Kubo formalism/ but no quantitative estimates about the critical frequencies or the susceptibilities were obtained. Quantitative estimates were given first on molecules with dipole moments along the long axis/ then for general dipole directions using the rotational Brownian picture in Maier-Saupe mean-field potential. This theory was subsequently refined in the 1990s.i ... 

Contributions to the theory of smectic-A liquid crystals have been made by a number of investigators. " In all cases the treatments are an extension of the Maier-Saupe mean-field model of nematics examined in a previous chapter P Here we essentially follow the development of McMillan. ... 

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,... 

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. 

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. 

Finally, a few words about other liquid-crystal theories the mean-field theory of Maier and Saupe (1959, 1960) has been very successful in describing the behaviour of small-molecule liquid crystals, but it has been much less used for polymeric liquid crystals. Other important theories primarily applied to small-molecule liquid crystals are the Landau theory and its extension, the Landau--de Gennes theory. A detailed presentation of these theories, also including the Maier and Saupe theory, is found in Vertogen and de Jeu (1988). 

The first ten years of this period saw several important developments which escalated interest and research in liquid crystals. Among these, there was the publication by Maier and Saupe [34] of their papers on a mean field theory of the nematic state, focusing attention on London dispersion forces as the attractive interaction amongst molecules and upon the order parameter. This theory must be regarded as the essential starting point for the advances in theoretical treatments of the liquid crystal state which followed over the years. 

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... 

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. ... 

As stated in Sec. 3, the retention of only the first term in Eq. [14] leads to the mean field theory of Maier and Saupe and the equivalent theory of the previous chapter. This version of the theory has been shown to provide a good qualitative picture of the nematic phase and its transition to the isotropic liquid. What, then, is it about the experimental facts that indicate the necessity of higher order terms in Fi ... 

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 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... 

Cotter has examined the postulates underlying the mean field approximation in the light of Widom s analysis of this general problem and has concluded that thermodynamic consistency requires that u should be proportional to V regardless of the nature of the intermolecular pair potential. However, in what follows we have assumed a dependence as in the original formulation of the theory by Maier and Saupe. 

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-... 

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]. 

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