Theory of the Liquid Crystalline State

Finally, the third group, which will be primarily considered in the present chapter, includes rigid- and semirigid-chain polymers of linear structure which exhibit the properties of liquid crystals due to the pronounced geometric asymmetry of the macromolecules. According to the theory of the liquid-crystalline state, these polymers, described in detail in Chapter 1, pass into an ordered state with a certain (critical) concentration of polymer in the solution. 

The effect of anisotropic interactions, orientation-dependent interactions in particular, which is responsible for the stability of the nematic phase to some degree, is prevalent in all mixtures. This question has been assigned an important place in the theory of low-molecular-weight liquid crystals of Melw and Saupe [49]. In the review of Flory s woric in [30], it was emphasized that although orientation-dependent interactions in polymers containing phenylene units, for example, can cause stabilization of the liquid-crystalline state, the asymmetry of the molecular shape is undoubtedly the dominant molecular characteristic responsible for the liquid-crystalline state in such systems. 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

Tins concept of a liquid as an iin wr)eci crystal requires that the molecules ill a liquid are packed sufficiently loosely for comparatively free movement, i.e. the energy required to move a molecule from a lattice site to a vacant space is not large compared with thermal energies. Under these conditions, shear (low ol the liquid resembles closely Ihe high temperature creep of crystalline solids. A number of theories of the liquid state have this concept as their Mailing point. 

Maier and Saupe [13] developed a statistical theory to describe the liquid crystalline state and the molecular ordering for the nematic phase. In analogy to the treatment of ordering phenomena in ferromagnetics or ferroelectrics, this theory describes the intermolecular orientational forces by a mean field method. Each individual molecule feels a nematic potential D = f (0, S, V) which depends on the momentaneous angle 6 between its long axis and the optic axis, the order parameter S and the molar volume V. S is then given by... 

The possibility of the transition into the liquid-crystalline state for such samples (glassy amorphous and crystallized) is realized in the presence of solvents. The transition to the liquid-crystalline state can take place in two ways. The first is dissolution of the crystallized or dilution of the glassy amorphous polymer by the solvent. A concentrated solution of polymer is formed, and the mobility of the macromolecules in the solution is sufficient for the equilibrium state to be established and consequently for the transition into the liquid-crystalline phase, which naturally occurs if the critical concentration below which the system enters the region of the isotropic state, according to the theory, is not exceeded during dilution. 

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

Both theoretical approaches qualitatively describe the "thermotropic" and "lyotropic" liquid crystalline state of rod-like molecules ( see also D.B. DuPre, R. Parthasarathy, this book). Combination of both theories (Flory, Ronca)(7) slightly improves the predictions compared to the experimental findings. Anisotropic dispersion interactions and/or anisometric molecular shape can thus be the basis for explaining theoretically the appearance of "lyotropic" and "thermotropic" liquid crystalline phases. 

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