Phase transitions Maier-Saupe theory

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-... 

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. 

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. 

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-... 

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... 

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. ... 

There are several different theoretical approaches to the problem. The Landau molecular field theory was applied by de Gennes to liquid-crystal phase transitions. (89) The Maier-Saupe theory focuses attention on the role of intermolecular attractive forces.(90) Onsager s classical theory is based on the analysis of the second virial coefficient of very long rodlike particles.(91) This theory was the first to show that a solution of rigid, asymmetric molecules should separate into two phases above a critical concentration that depends on the axial ratio of the solute. One of these phases is isotropic, the other anisotropic. The phase separation is, according to this theory, solely a consequence of shape asymmetry. There is no need to involve the intervention of intermolecular attractive forces. Lattice methods are also well suited for treating solutions, and phase behavior, of asymmetric shaped molecules.(80,92,93)... 

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.)... 

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-... 

It should be noted, however, that the agreement between the Maier-Saupe theory and experiment is not so good if one considers some other parameters of the N-I transition. For example, the discontinuity in entropy is overestimated several times. In partieular, the difference between the transition temperature N-I and the lower limit of stability of the isotropic phase T is strongly overestimated. In the Maier-Saupe theory the parameter is... 

Based on the interaction employed in the Maier-Saupe theory of the nematic state, Preiser [4] was the first to predict the possible existence of an N, phase as an intermediate between two uniaxial nematic phases. Later, a number of other theoretical investigations were carried out [5-7] using various models to predict the possibility of obtaining an liquid crystal. In all these models, a system consisting of hard rectangular plates was considered. These approaches gave the same result, an Nb phase should be obtained between two uniaxial nematics of opposite sign, i.e., those made up of rod-like and plate-like molecules. They also predicted that a transition from a uniaxial nematic to a biaxial nematic would be second order. 

Polymers and liquid ciystals are important materials for various research fields. If the two substances are mixed, novel materials which combine the advantageous properties of both may be formed. 1 began to think about this around 1994. Already at that time, this mixed system had attracted attention as an electro-optical material, but from the perspective of basic physical properties, the center of the liquid crystal research up to that point was the phase transition and Uquid crystal stracture of novel low molecular weight liquid crystals. The physics of liquid crystals was based on the Onsager theory, the Maier-Saupe theory, and the elastic theory by Frank. However, the theoretical study of a liquid crystal mixed with other substances had not yet been developed. So, 1 began to think to build theories of phase separations and phase transitions in mixtures of liquid crystals and other substances. Our first paper on the theory of phase separations in the mixture of a polymer and a liquid crystal was published in 1996 [41]. 1 found at a later date that a paper on the same topic by Prof. Kyu of Akron University had been presented already in 1995 [42]. However, there was a difference between the two theories. Kyu s theory has dealt with low molecular weight liquid crystals in an attractive model, whereas our model considered both attractive and repulsive interactions between rodlike liquid crystal molecules and can handle also long rodlike molecules. After that, I had a variety of discussions with Kyu and it was a valuable experience for my research. 

There are several levels of approximation possible in the consideration of the NA transition. First there is the self-consistent mean field formulation due to Kobayashi and McMillan [8-10]. This is an extension to the smectic-A phase of the self-consistent mean-field formulation for nematics ( Maier-Saupe theory [11]). Kobayashi-McMillan (K-M) theory takes into account the coupling between the nematic order parameter magnitude S with a mean-field smectic order parameter. In Maier-Saupe theory, the key feature of the nematic phase - the spontaneously broken orientational symmetry - is put in by hand by making the pair potential anisotropic. In the same spirit, the K-M formulation puts in by hand a sinusoidal density modulation as well as the nematic-smectic coupling. 

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 ... 

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 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... 

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. 

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. 

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