The phase transition of rigid rods

The right-hand side takes the value unity when all the polymers are parallel to n, and zero when their direction is completely random. Thus S represents how perfectly the polymers are orient along m and is called the orientational order parameter. 

If there is no external field, the director m in nematics is entirely arbitrary. Thus the equilibrium state of nematics is not unique and can be changed by infinitesimal perturbation. This property, generally called broken symmetry in statistical mechanics, necessitates a special treatment in the mathematical handling of the kinetic equation, and introduces a new type of constitutive equation, unique to the ordered fluid. 

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Rod—coil block copolymers have both rigid rod and block copolymer characteristics. The formation of liquid crystalline nematic phase is characteristic of rigid rod, and the formation of various nanosized structures is a block copolymer characteristic. A theory for the nematic ordering of rigid rods in a solution has been initiated by Onsager and Flory,28-29 and the fundamentals of liquid crystals have been reviewed in books.30 31 The theoretical study of coil-coil block copolymer was initiated by Meier,32 and the various geometries of microdomains and micro phase transitions are now fully understood. A phase diagram for a structurally symmetric coil—coil block copolymer has been theoretically predicted as a... 

On the basis of the molecular theory combining both the excluded volume interactions and the orientation-depen-dent attractive interactions between liquid crystals, we have theoretically studied the phase transitions of mixtures of a flexible polymer and a liquid crystal. We have found that (1) our theory can qualitatively explain the observed phase diagrams not only in the polymer-liquid crystal systems but also in the solutions of rigid rod-like molecules, (2) gel immersed in a liquid-crystal solvent can undergo the second-order volume transition due to the nematic ordering of the external solvent. 

In order to analyze the dependence of the liquid crystalline transition properties on temperature (i.e. on the solvent quality), it is necessary to introduce the attraction of rods parallel to their steric repulsion. This has been done by Rory9 . The classical phase diagram of Rory for the solution of rods (see Fig. 2) agrees well with experimental results from the qualitative point of view1 . However, the Rory theory cannot give adequate answers to all the questions connected with the orientational ordering in the system of rigid rods. Indeed ... 

As is for low mass liquid crystals, incorporation of kinked moieties will result in destructive effects on the liquid crystallinity of polymers (Figure 3.4). 2,2-diphenylpropane, diphenylmethane, diphenyl ether, diphenyl ketone, 1,2-phenylene, 1,3-phenylene, and 1,2-naphthalene are examples of kinked moieties used in the modification of liquid crystalline polymers. They are very effective in destroying the linearity of rigid rods. Polymers with kinked units have less crystallinity and lower phase transition temperatures. Appropriate use of kinked units is thus of help from case to case. However, the type and amount of kinked units should be carefully determined so as to maintain desirable liquid crystallinity. 

The mesomorphic behavior of a solution of rigid rods has been studied by several authors.i-iZ-Ll In particular, for long rods of length b and radius a, OnsagerL has shown that the effective excluded volume per rod in the isotropic phase is of order b a. The actual volume occupied per rod is ira b. A lyotropic transition to a nematic phase occurs at a rod concentration O given by... 

The principal goal of this work is to pursue the concept of induced rigidity. Our current investigations include extensions to the cases of arbitrary values of the cooperativity parameter 1 > E > 0, more accurate treatments of the rod-rod excluded volume interaction,— and the inclusion of some attractive dispersion forces. While we refer to the phase transition as lyotropic, in fact we do expect that it also occurs as a function of temperature (as well as pH, ionic strength, etc.) because of the dependence of s on these parameters. We must further emphasize the semi-quantitative nature of our results. Indeed under most actual circumstances, s and c will not be completely independent variables. Nevertheless we believe that the physical picture presented here has some relation to reality. 

We will discuss nematic ordering in polymer systems and we start with solutions of rigid rods as the simplest system in which isotropic-nematic transition occurs. Solutions of a flexible polymer and a nematic low molecular liquid crystal display at low polymer content, when cooled down from the isotropic phase, segregation into a nematic and an isotropic phase. At higher polymer content, the solution decays first in two isotropic phases, one rich in polymer, and the other poor in polymer. Further cooling leads to separation of the latter in a nematic phase very poor in polymer and the isotropic phase rich in polymer. This is sketched schematically in Figure 19. Phase behavior of the indicated type was observed in EBBA (/ -ethoxy benzylidene- w-4- -butylaniline) mixed with polystyrene (Ballauf, 1986 Lee et al., 1994) and with poly(ethylene oxide) (Kronberg et al., 1978). 

Usually the conditions for liquid crystal formation are best met when the molecules have at least some portion of their structure in the form of rods or disks. However, this is not an absolute requirement, and a surprisingly large number of polymers are now being found to exhibit liquid crystalline phases. While truly rigid-rod polymers ordinarily do not have a glass transition, if some degree of flexibility is built into the polymer chain, there may be one. 

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