Saupe theory, nematics

Luckhurst G R and Zannoni C 1977 Why is the Maier-Saupe theory of nematic liquid crystals so successful Nature 267 412-14... 

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-... 

This approximate expression, using the Maier-Saupe theory for S2 and 54 and taking R(p) 1, agrees reasonably well with measurements of X for a variety of liquid crystals (see Fig. 10-10), as long as there is no transition to a smectic phase near the temperature range considered. When a smectic-A phase is nearby, as is the case for 8CB, then smecticlike fluctuations of the nematic state can significantly reduce A. For 8CB, for example, A drops to around 0.3-0.4 when T — 34°C (Kneppe et al. 1981 Mather et al. 1995), which is around 0.7°C above the transition to the smectic-A phase. 

The application of the Maier-Saupe theory to the polymer system results in the nematic to isotropic (N-I) transition temperature, Tc, the order... 

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... 

One can obtain the free energy as a function of S for various values of kBT/U from the solutions of Eqs. (19) and (17). For high values of kBT/U, the minimum in the free energy is found for S = 0 corresponding to the isotropic phase. As the value of kBT/U falls below 4.55, the minimum in the free energy is found for a nonzero value of S that is, the nematic phase becomes stable. For this critical value of kBT/U = 4.55, there is a discontinuous change in the order parameter from S = 0 to S 0.44. The Maier-Saupe theory thus predicts a first-order transition from the isotropic to the nematic phase. 

We consider first the Maier-Saupe theory and its variants. In its original formulation, this theory assumed that orientational order in nematic liquid crystals arises from long-range dispersion forces which are weakly anisotropic [60. 61 and 62]. However, it has been pointed out [63] that the form of the Maier-Saupe potential is equivalent to one in... 

The Maier-Saupe theory was developed to accoimt for ordering in the smectic A phase by McMillan [71]. He allowed for the coupling of orientational order to the translational order, by introducing a translational order parameter which depends on an ensemble average of the first harmonic of the density modulation normal to the layers as well as F . This model can accoimt for both first- and second-order nematic-smectic A phase transitions,... 

McMillan s model [71] for transitions to and from the SmA phase (section C2.2.3.21 has been extended to columnar liquid crystal phases formed by discotic molecules [36. 103]. An order parameter that couples translational order to orientational order is again added into a modified Maier-Saupe theory, that provides the orientational order parameter. The coupling order parameter allows for the two-dimensional symmetry of the columnar phase. This theory is able to accormt for stable isotropic, discotic nematic and hexagonal colmnnar phases. 

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-... 

Before closing Section 7.3 we would like to mention briefly the optical anisotropy of the nematic conformation of the spacer. The parameter A, representing the strength of the anisotropic interaction in theMaier-Saupe theory [Eq. (7.1)], should be related... 

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... 

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 theory is very useful in considering liquid crystal systems consisting of more than one type of molecules, such as mixtures of nematic liquid crystals and dichroic dyes. The interactions between different molecules are different and the eonstituent molecules have different order parameters. 

The Maier-Saupe theory can also be extended to describe the smectic A-nematic transition in what is called McMillan s model. Two order parameters are introduced into the mean-field potential energy function, the usual orientational order parameter S and an order parameter a related to the amplitude of the density wave describing the smectic A layers,... 

Recently, phase behaviour of mixtures consisting of a polydisperse polymer (polystyrene) and nematic liquid crystals (p-ethoxy-benzylidene-p-n-butylani-line) was calculated and determined experimentally. The former used a semi-empirical model based on the extended Flory-Huggins model in the framework of continuous thermodynamics and predicted the nematic-isotropic transition. The model was improved with a modified double-lattice model including Maier-Saupe theory for anisotropic ordering and able to describe isotropic mixing. ... 

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 starting point for a theory of the anisotropic intermolecular interaction in liquid crystals is the Maierand Saupe theory [114,115,116,118].This theory is based on the assumption that the intermolecular interaction potential in nematic liquid crystals is determined primarily by Lx)ndon dispersion forces. The effective anisotropic potential U of a molecule C in the anisotropic dispersion field generated by its oriented neighbors s is calculated by averaging the pair potential between two molecules C and s over all orientations of the solvent molecules s and over all... 

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. 

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... 

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