Nematic-isotropic transition 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-... 

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

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. 

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. 

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... 

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... 

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... 

Using the Maier-Saupe theory with the parameters kT/u2 = 0.2203 and = 0.429, determine the value of P4 = cos P — cos + ) at the nematic-isotropic phase transition temperature. (You will need to evaluate the appropriate equations numerically.)... 

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 ... 

Crystalline or orientational orderings are mostly controlled by repulsive forces, such as excluded-volume forces. The crystalline transitions in ideal hard-sphere fluids and the nematic liquid crystalline transitions in hard-rod suspensions are convenient simple models of corresponding transitions in fluids composed of uncharged spherical or elongated molecules or particles. The transition from the isotropic to the nematic state can be described theoretically using the Onsager, Maier-Saupe, or Rory theories. 

Brochard and de Gennes [67] discussed theoretically a flow-induced isotropic-mesophase transition in a polydisperse polymer system occurring through spinodal decomposition. Following Maier-Saupe s [50] theory of the nematic phase, the orientation-dependent interaction energy was taken 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 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]. 

Four principal theories describing the transformation of the isotropic fluid into a spontaneously organized nematic fluid, the I N transition, have been developed (1) Onsager s density expansion of the free energy of anisometric particles, (2) Rory s estimate of the insertion probability for a rod-like (multisite) solute into a lattice, (3) Maier and Saupe s construction of apotential of mean torque experienced by mesogens (or solutes) in a nematic environment, and (4) de Gennes transposition of Landau theory to the I N transition. We briefly examine each of these in reverse chronological order because each is relevant to more-recent theoretical descriptions of polymer mesophases. 

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 ... 

Such molecrrlar-stmcture-based approaches are clearly extremely complex and often tend to yield contradictory predictions, because of the wide variation in the molecrrlar electronic stmctures and intermolecirlar interactions present. In order to explain the phase transition and the behavior of the order parameter in the vicinity of the phase transition temperature, some simpler physical models have been employed. For the nematic phase, a simple but qirite successful approach was introduced by Maier and Saupe. The liquid crystal molecirles are treated as rigid rods, which are correlated (described by a long-range order parameter) with one another by Corrlomb interactions. For the isotropic phase, deGennes introduced a Landau type of phase transition theory, which is based on a short-range order parameter. 

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