Theories of mesomorphism

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. 

The discontinuous first-order phase transition occurs at a temperature Tc that is slightly higher than T (T is a second-order transition temperature). The first-order nature of the 1 N transition is due to the presence of the odd-order power of S. The associated temperature Tc (equivalent to Td used earlier and in Fig. 5.25) and the order parameter at the I - N transition Sc may be obtained by minimizing g with respect to S (setting (9g/95) = 0)  

As suggested by the thick solid curve in Fig. 5.25, the order parameter changes discontinuously from 5 = 0 to a finite value Sc on lowering the temperature. If one considers the influence of an external field F interacting with the anisotropic molecular susceptibiUty Ax, an additional term — (AAx)5F must be added to the expression for g [24,26]. This external ahgnment influence shows up as a shift in Tc to higher temperatures (the dotted curve in Fig. 5.25). 

The general Landau theory, which was developed by de Gennes to describe critical phenomena in MFCs, has been appUed to elastic networks comprised of PLCs [66]. The Landau formalism also allows one to make contact with the theory used to describe conventional orientation phenomena in nonmesogenic polymer networks. In particular, a mechanical deformation via its associated stress field a influences g (and therefore Tc and 5c) analogously to external magnetic or electric fields. For a small (uniaxial) extension ratio X = e —, where e is the strain, the form of g in Eq. (5.16) is modified by the additional terms as follows  

The Maier-Saupe theory [85] posits a simple potential of mean torque that originates from an average over the interactions a given mesogen experiences because of its (oriented) neighbors - the mean field  

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The theory of mesomorphic phase transitions, developed at Princeton, has been extensively described in the literature (I, 3) and will only be briefly summarized here. (Charvolin and colleagues independently developed similar ideas at about the same time. See reference 13 for a summary.) The basic... 

Molecular theories of mesomorphic polymers may serve to give a better insight into the mechanisms involved in the formation of ordered phases in macromolecular liquids. The importance of parameters such as chain length, chain flexibility... can be explored and the results used in synthesis of new systems with well-defined characteristics. Studies of the influence of solvents on the mesomorphic phase transition and the possibility of ordering by external fields (electric, magnetic, flow) are problems of technological in-ter>est in the spinning of fibers and in the formation of new polymer blends. 

It is well known that water dispersions of amphiphile molecules may undergo different phase transitions when the temperature or composition are varied [e.g. 430,431]. These phase transitions have been studied systematically for some of the systems [e.g. 432,433]. Occurrence of phase transitions in monolayers of amphiphile molecules at the air/water interface [434] and in bilayer lipid membranes [435] has also been reported. The chainmelting phase transition [430,431,434,436] found both for water dispersions and insoluble monolayers of amphiphile molecules is of special interest for biology and medicine. It was shown that foam bilayers (NBF) consist of two mutually adsorbed densely packed monolayers of amphiphile molecules which are in contact with a gas phase. Balmbra et. al. [437J and Sidorova et. al. [438] were among the first to notice the structural correspondence between foam bilayers and lamellar mesomorphic phases. In this respect it is of interest to establsih the thermal transition in amphiphile bilayers. Exerowa et. al. [384] have been the first to report such transitions in foam bilayers from phospholipids and studied them in various aspects [386,387,439-442]. This was made possible by combining the microscopic foam film with the hole-nucleation theory of stability of bilayer of Kashchiev-Exerowa [300,402,403]. Thus, the most suitable dependence for phase transitions in bilayers were established. 

The mesomorphism of polymers has been tentatively related to various parameters of the macromolecule. The early theories of lyotropic mesomorphism (Onsager and Flory) employed a rodlike molecular mode. However, as the data... 

Regular solution theory has been successfully applied to the calculation of binary phase diagrams of mesomorphic materials. Correlation of the parameters required by the theory with simple physical properties of the mixture s components, may offer an empirical way to predict the phase diagrams for mixtures involving members of homologous series. 

The macroscopic theory that takes into accoimt the effect of the orientation order was developed by Ericksen, Leslie and Parodi, and usually is referred as ELP theory. A microscopic theory based on correlation functions, which then were "translated" to macroscopic terms and extended to other mesomorphic phases, was developed by the Harvard group. Although usually tire ELP flieory is accepted, it seems that the two approaches are equivalent. A continuum theory of biaxial nematics was developed by Saupe, who followed the description we give with (4.1)-(4.8). In the uniaxial situation, they reproduce the Leslie-Ericksen and Harvard theories. 

Isotropic polymeric systems as well as particulate systems might also show time-dependent moduli after cessation of flow. As long as the shear does not induce structure growth, the moduli always increase with time after flow. An increase of the moduli upon cessation of flow has also been reported for thermotropic PLCs (18) as well as for lyotropic solutions of hydroxy propyl cellulose in water (19) and in acetic add (20). The possibility of changing in either direction seems to be characteristic for mesomorphic materials. A fundamental theory for describing complex moduli does not exist for such materials. The present results, combined with the information about optical relaxation mentioned above, could be explained on the basis of reorientation of domains or defects. The different domains orient differently, even randomly, at rest whereas flow causes an overall orientation. Depending on the molecular interaction the flow could then cause an increase or decrease in moduli as recently suggested by Larson (21). 

In Section II, we shall briefly recall the statistical mechanics of the helix-coil transition for isolated chains. We shall then present an argument showing that there is little tendency for non-interacting molecules to exhibit substantial mesomorphic fluctuations. A mean field theory for incorporating the interchain interactions will be developed in Section III in the cooperative limit for helix-coil transition. Some final remarks will be made in Section IV. 

Mesomorphic phases can be obtained in various cases (thermotropic, lyotropic in a simple solvent or in a liquid crystal solvent, mixtures of polymers). As predicted, the effect of the molecular weight on phase diagrams and on segregation phenomena is observed. The experimental phase diagrams can be relatively well described by the theory in simple cases. Specific interactions should be included in future work as well as a more detailed examination of diphasic separation ll. Further experimental studies on the biphasic regions are in progress. 

In all of the present theories about the excitation of nematic or cholesteric liquids by an electric field, the mesomorphic material is treated as a continuous elastic anisotropic medium. The Oseen -Frank elastic theory is used to describe the interaction between the applied field and the fluid. The application of an electric field causes the liquid crystal to deform. For a material with a positive dielectric anisotropy, Ae = > 0, the director aligns in the direction of... 

The SCLF method was developed by Koopal and coworkers [46-50] to describe adsorption of surfactant molecules at the solution-solid interface. The method derives from two earlier statistical thermodynamic lattice theories (1) the Flory and Fluggins [51 ] model describing properties of polymers in solution, and (2) the methods of Scheutjens and coworkers [52-55] developed to describe the properties of polymer molecules adsorbed at the solution-solid interface and in associated mesomorphic solution structures such as micelles and vesicles. 

However, there is not yet semicommercial development of cellulose fibers processed via mesomorphic solutions. The statistical physics of lyotropic LCPs has been dealt with in Ref. 8, but a modified Maier-Saupe theory developed by Picken will be discussed in this chapter. 

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