Temperature dependence Maier-Saupe theory

In their original theory, Maier and Saupe supposed that the molecular interactions responsible for the nematic state are anisotropic van der Waals interactions (discussed in Section 2.3), in which case mms should be temperature-independent. However, it is now recognized that shape anisotropy is also important, even for small-molecule thermotropic nematics. By making mms temperature-dependent, the Maier-Saupe potential can, in principle, accommodate both energetic and entropic effects. In fact, if the function sin(u, u) in the purely entropic Onsager potential Eq. (2-5) is approximated by the expansion 1 — V2 cos (u, u)+. . ., then to lowest order the Maier-Saupe potential (2-7) is obtained with C/ms — Uo bT/S, where we have defined the dimensionless Maier-Saupe energy constant by Uus = ums/ksT, Thus, the Maier-Saupe potential can be used as an approximation to describe orientational order in either lyotropic (solvent-based) or thermotropic nematics. For a thermotropic melt, the Maier-Saupe theory predicts a first-order transition from the isotropic to the nematic phase when mms/ bT = U s — t i.MS = 4.55, and at this transition the scalar order parameter S jumps from zero to 0.43. S increases toward unity with further increases in Uus- The spinodal point at which the isotropic phase is unstable to even small orientational perturbations occurs atU — = 5 for the Maier-... 

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. 

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-... 

These theories all pr>edict a first order nematic-isotropic phase transition, and a weakly temperature dependent order parameter. In rigid rod Maier-Saupe theory, the order parameter is given by the angle of the rod to the direction 0" prefered orientation... 

Notice that this prediction is a universal one. Unlike the Landau-de Geimes theory, there are no parameters that can be adjusted to allow for different order parameter behaviour. Since different substances have slightly different temperature dependences, the Maier-Saupe theory describes only a small portion of them well. 

Similarly, das and dai were found to be 13. 06 and 2 pmA, respectively, for the P(S - alt - M3) films characterized by an order parameter of about 0.3. However, as < 2> increases, so do daa/dai. Clearly for the range 0.3 P2><0.4 one obtains systems on the borderline between isotropic and LC behavior. According to the Maier - Saupe theory for low molar mass nematics, changes discontinuously from zero at the isotropic/nemadc transition to 0.429. However, Luckhurst has shown that the Maier - Saupe theory results in an overestimate of about 25% in ni. In addition, comparison of the temperature dependence of for LC monomers and corresponding side chain polymers has demonstrated that the absolute values of related to the reduced temperature T/Tni are always lower for the polymers. As... 

The temperature and pressure dependencies of q yield important information about the validity of the assumptions of the theories describing the nematic state. In particular, having S values from independent experiments one can check the relation predicted by the Maier-Saupe theory (see Section I.C.l) that q = vS. However, the data on S(T, p) are available for a few LC substances only. " Figure 25 presents the q versus S plots for three substances studied in our lab [the data on 5(T, p) were taken from Ref. 71 for 5CB, Ref. 72 for 6CB, and Ref. 99 for 7PCH]. In the case of two cyanobiphenyls some scatter of points obtained at different experimental conditions are observed, but essentially one can note a proportionality of both these quantities. For 7PCH, however, a nice proportionality was found for the results obtained at p = constant only, whereas at V= constant it is completely failed... 

The Maier-Saupe theory is extremely useful in understanding the spontaneous long-range orientational order and the related properties of the nematic phase. The single-molecule potential Vi(cos0) is given by Eq. (3.19) with e being volume dependent and independent of pressure and temperature. The self-consistency equation for (P2) is... 

Figure 1. The temperature dependence of the order parameter S, at constant pressure, for 4,4 -dimethoxyazobenzene ( ), 4,4 -diethoxyazobenzene (A), anis-aldazine (A), 2,4-nonadienoic acid ( ) and 2,4-undecadienoic acid (O). The curve is predicted by the Maier-Saupe theory (after Luckhurst [23]).

The first theoretical dicussion of the temperature dependenee of elastic constants within the framework of the molecular-statistical Maier-Saupe theory was given by Saupe [241]. He attributed their temperature dependence to changes in the order parameter S and the molar volume with temperature, and he introduced reduced elastic constants... 

To use the model to predict other properties of liquid crystal dimers, for example, the N-I transition temperature and the temperature dependence of the order parameter it is necessary to make an additional approx-imation. This is to relate the strength parameter Xa for a mesogenic group to the orientational order of the nematic mesophase. By analogy with the Maier-Saupe theory [63] and the extension of this to multicomponent mixtures [68] it is assumed that... 

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 ... 

PHIC extrapolate to roughly 6.7, which is close to the value predicted by the Flory theory in the melt. This suggests that even for bulk HPC, the nematic-isotropic transition is driven primarily by excluded-volume, or packing, effects and only secondarily by anisotropic van der Waals interactions. The temperature dependence of the axial ratio could be incorporated into the Maier-Saupe potential by suitably adjusting the temperature dependence of the coefficient 17ms-... 

The Maier and Saupe theory successfully accounts for the observed temperature dependence of the order of nematic phases and correctly predicts the existence of a first order transition at a temperature... 

Judged from the above experimental results the anisotropic dispersion interaction seems to accouht successfully for the average solute orientation at least for aromatic solute molecules. It should be emphasized, however, that the applicability of the dipolar approximation adopted in the simple Maier and Saupe theory is questionable. This approximation has been made responsible for the failure of this simple theory to account in a quantitative way for the observed temperature dependence of the order in nematic phases [120]. In an attempt to overcome these difficulties a number of authors improved the Maier and Saupe theory. 

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