Orientational order theories

Emsiey J W, Luckhust G R and Stockley C P 1982 A theory of orientational ordering in uniaxial liquid-crystals composed of molecules with alkyl chains Proc. R. See. A 381 117-28... 

Analytical approaches to understanding the effect of molecular flexibility on orientational order have concentrated on both the isotropic-nematic and the nematic-smectic transition [61, 62] and mean field theory has shown that cholesteric pitch appears not to depend on the flexibility of the molecule [63]. 

As seen in this chapter, the theory and procedures for orientation measurements are well established, including for quantitative characterization. These methods can provide very accurate and useful information in the fields of synthetic, natural, and bio-inspired macromolecules. To this aim, researchers can make use of a wide range of techniques, each having its advantages and limitations. As judged from the recent literature, the studies devoted to the quantification and characterization of molecular orientation still represent a very dynamic research field and advances still continue to emerge. Further progresses in the development of new methods and new techniques to characterize orientational order are thus expected in the future. 

In the present book, we aim at the unified description of ground states and collective excitations in orientationally structured adsorbates based on the theory of two-dimensional dipole systems. Chapter 2 is concerned with the discussion of orientation ordering in the systems of adsorbed molecules. In Section 2.1, we present a concise review on basic experimental evidence to date which demonstrate a variety of structures occurring in two-dimensional molecular lattices on crystalline dielectric substrates and interactions governing this occurrence. 

The book thus embraces an extended study on a variety of issues within the theory of orientational ordering and phase transitions in two-dimensional systems as well as the theory of anharmonic vibrations in low-dimensional crystals and dynamic subsystems interacting with a phonon thermostat. For the sake of readability, the main theoretical approaches involved are either presented in separate sections of the corresponding chapters or thoroughly scrutinized in appendices. The latter contain the basic formulae of the theory of local and resonance states for a system of bound harmonic oscillators (Appendix 1), the theory of thermally activated reorientations and tunnel relaxation of orientational... 

The polarization properties of the evanescent wave(93) can be used to excite selected orientations of fluorophores, for example, fluorescent-labeled phosphatidylethanolamine embedded in lecithin monolayers on hydrophobic glass. When interpreted according to an approximate theory, the total fluorescence gathered by a high-aperture objective for different evanescent polarizations gives a measure of the probe s orientational order. The polarization properties of the emission field itself, expressed in a properly normalized theory,(94) can also be used to determine features of the orientational distribution of fluorophores near a surface. 

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. 

Fig. 12a-c. Polymer concentration dependence of the orientational order parameters S for three liquid-crystalline polymer systems a PBLG-DMF [92,93] b PHIC-toluene [94] c PYPt-TCE [33], Marks experimental data solid curves, theoretical values calculated from the scaled particle theory. The left end of each curve gives the phase boundary concentration cA... 

The electrostatic part, Wg(ft), can be evaluated with the reaction field model. The short-range term, i/r(Tl), could in principle be derived from the pair interactions between molecules [21-23], This kind of approach, which can be very cumbersome, may be necessary in some cases, e.g. for a thorough analysis of the thermodynamic properties of liquid crystals. However, a lower level of detail can be sufficient to predict orientational order parameters. Very effective approaches have been developed, in the sense that they are capable of providing a good account of the anisotropy of short-range intermolecular interactions, at low computational cost [6,22], These are phenomenological models, essentially in the spirit of the popular Maier-Saupe theory [24], wherein the mean-field potential is parameterized in terms of the anisometry of the molecular surface. They rely on the physical insight that the anisotropy of steric and dispersion interactions reflects the molecular shape. 

To summarize this part of the chapter, we have constructed a consistent theory of linear and cubic dynamic susceptibilities of a noninteracting superparamagnetic system with uniaxial particle anisotropy. The scheme developed was specified for consideration of the assemblies with random axis distribution but may be easily extended for any other type of the orientational order imposed on the particle anisotropy axes. A proposed simple approximation is shown to be capable of successful replacement of the results of numerical calculations. 

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 ... 

However, it must be recalled that the Lifshitz theory was originally formulated23 25 for the model of beads (see Fig. 7 a). In this model, each monomer is represented as a material point thus, this model cannot be used for the description of the intramolecular liquid-crystalline phase. The description of the orientational ordering, requires the generalization of the Lifshitz consideration for the models, in which the state of an elementary monomer is defined not only by its spatial position but also by its orientation (see, for example, the models of Fig. 7 b-db Such a generalization will be our first aim in this section. 

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-... 

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. 

Rather than attempting to compute and a a from molecular theory, it is more convenient to obtain them from fits to the viscosity data. Consider first the viscosity of the liquid at temperatures above Tni, at which the liquid is in the isotropic state and has no orientational order, so that S2 = S4 = 0. We then find from Eqs. (10-20a)-(10-20f) that only o 4 is nonzero in the isotropic state from Eq. (10-10) we find that the Newtonian viscosity of this isotropic liquid is given by rjiso = o 4/2. From Eq. (10-20d) we then have... 

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