Maier—Saupe theory

An alternative nematic potential due to Maier and Saupe (1958, 1959, 1960) is perhaps more appropriate for thermotropic nematics. The Maier-Saupe potential is given by 

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- 

Saupe potential, and U Uq = 10.19 for the Onsager potential. The two potentials are significantly different in their predictions of the dependence of the order parameter on U. For a given value of U/ U, the Onsager potential predicts a higher-order parameter than does the Maier-Saupe potential. 

From the orientational distribution function we can calculate the order parameter  

From Equation (1.48) we have ln[f[0)]=-ViSyksT-InZ, and therefore En=— V + MfllnZ and the free energy is 

Although the second term of the above equation looks abnormal, this equation is correct and can be checked by calculating the derivative of F with respect to 5  

In the Maier-Saupe theory there are no fitting parameters. The predicted order parameter as a function of temperature is universal, and agrees qualitatively - but not quantitatively - with 

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. 

With these assumptions, we give the mean-field potential in the following 

The fact that the system must be in thermodynamic equilibrium demands that the probability of a molecule being oriented at an angle ft from the director is given by the Boltzmann factor. 

The order parameter can be expressed with the help of this probability fimction as  

Since LZ, itself contains S, we have a self-consistent equation involving S, T and V. With the help of this equation, and after some algebra, we can get the temperature dependence of the order parameter as follows. 

A convenient way to see how S depends on T is to choose a value for m, and using tabulated values of the Dawson integral or, these days, rather a computer, one can find S from (3.29). Now knowing the pair m and S and using the definition of m from (3.27), we get the corresponding temperature. 

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

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. 

This conclusion was reached, tentatively, by Frenkel, Shaltyko and Elyashevich A phenomenological analysis presented by Pincus and de Gennes predicted a first-order phase transition even in the absence of cooperativity in the conformational transition. These authors relied on the Maier-Saupe theory for representation of the interactions between rodlike particles. Orientation-dependent interactions of this type are attenuated by dilution in lyotropic systems generally. In the case of a-helical polypeptides they should be negligible owing to the small anisotropy of the polarizability of the peptide unit (cf. seq.). Moreover, the universally important steric interactions between the helices, regarded as hard rods, are not included in the Maier-... 

Figure 10.4 Measured values of the order parameter S — S2 versus temperature for MBBA using Raman scattering ([J), NMR (A), birefringence (A), and diamagnetic anisotropy (V), as described in Deloche et al. (1971). The solid line is the prediction of the Maier-Saupe theory the dashed line is a modification of the Maier-Saupe theory by Humphries, James, and Luckhurst (1972). (From de Gennes and Frost, Copyright 1993, by Oxford University Press, Inc. Used by permission of Oxford University Press, Inc.)...

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. 

Problem 10.2 (Worked Example) Using the Maier-Saupe theory described in Section 2.2.2.2, derive an implicit equation relating the order parameter 5 as a function to the Maier-Saupe... 

Later Flory further took the two soft interactions between the molecules into account. The anisotropic interaction is associated with molecular orientations while the isotropic one is irrelevant of the molecular orientation. In fact, the anisotropic interaction was the basis of another well-known theory in liquid crystals — Maier-Saupe theory (Maier Saupe, 1959). Flory successfully captured the essence of the theory. 

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. 

It is worthwhile to point out that the Maier-Saupe theory has been successful in analyzing the behavior of small molecular mass liquid crystals at transition, such as the temperature change of the order parameter. The jump of the order parameter at transition, Sc = 0.43 is in reasonable agreement with most experiments. The Onsager and Flory theories, which take into account the steric effects predict a higher critical order parameter. 

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

It is known that the persistence length is normally about 2-10 times that of the repeat units for polymers, depending on the real structure of polymers. It is seen that the latent entropy is no longer a universal value, as in the Maier-Saupe theory. Some experimental results has been reviewed by Luckhurst (1986). 

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. 

Fig. 2.3.2. Variation of the free energy with the order parameter calculated from the Maier-Saupe theory for different values of A/k TV. The minima in the curves occur at values of s which fulfil the consistency relation (2.3.13). ...

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