Uniaxial Order

For a uniaxial phase (nematic, discotic nematic, SmA, SmB, etc.) with the symmetry axis along z, all properties along x and y are the same and Xn= X22 X33- The corresponding matrix 

In principle, this tensor might be used as orientational order parameter for a uniaxial phase, however, its dimensionless form would be more preferable. Therefore, we normalize anisotropy Xz. to the maximum possible anisotropy corresponding to the ideal molecular alignment as in a solid crystal at absolute zero temperamre. Then we arrive at the order parameter tensor [16].  

Here S = scalar modulus of the order parameter dependent on the 

The experimental values of the orientational order parameter found macroscop-ically for conventitMial nematics from the magnetic or optical anisotropy are in good agreement with those calculated from microscopic data (NMR, Raman 

Order parameter tensor can be written using the director comprments (ct=x,y,z). 

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The doublet appears in various other polymer matrices, which shows that the uniaxial ordering of chain segments is perhaps a quite general phenomenon. For instance, PB networks with chains deuterated in 1,4 positions (Figure 15.8) have been studied [18-21]. Their behaviour is essentially the same, even though the linewidth is much broader... 

Almost all of these examples involve diffusion of a chemical species measuring diffusion rates has long been a specialty of NMR spectroscopy. The studies of KBr and drawn polyethylene produced unique information in the latter case, the known orientation of the deuterium electric field gradient in C-D bonds is used to determine the orientation, with respect to the magnetic field, of a polymer chain of a uniaxially ordered polyethylene fiber. The real time imaging of the polymerization of methyl methacrylate is very interesting and may represent a major direction for NMR imaging applications to polymer science. 

The continuous-phase morphology is firmly established for copolymers of HBA and ET, mentioned above, by the recent work of Windle and collaborators They observed birefringent domains in sections of oriented pellets obtained by extrusion of these copolymers. The combined techniques of X-ray diffraction and optical birefringence revealed biaxial order within the domains, instead of the uniaxial order... 

The general features of this phase diagram have been well confirmed by experiments, especially those of Nakajima and of Miller and their co-workers. The emergence of a concentrated, well-ordered anisotropic phase when % exceeds a small positive value (that depends on the axial ratio x) is readily understood as the consequence of interactions that are effectively attractive between the rodlike particles. The dense anisotropic phase may be regarded as the prototype of a quasi-crystalline state with uniaxial order only. Comprehension of the three-dimensional order characteristic of the crystalline state is, of course, beyond the scope of the model, which does not... 

Table 14.2. Structural parameters (degrees) of the ordered components for uniaxially ordered PET samples determined from CP NMR ( f was assumed to be 90°)...

The tendency to form lamellar structures can be understood as the result of dipole-dipole repulsions between the uniaxially ordered amphiphile molecules theoretical analyses of the pattern formation based on this principle have been carried out. It appears that the lamellar thickness is defined by the equilibrium thermodynamic conditions (Fig. 23) but that the shape and density of the patterns are determined by the dynamics of the process by which they are formed. 

Anisotropic composites, characterized by uniaxially ordered filler particles... 

Fig. 4 Preparation of uniaxially ordered polymer composite under uniform magnetic field...

Optical second-harmonic generation experiments give a more detailed description of the anchoring at a microscopic scale [68,69 see also Chapter 5]. The molecule/surface interaction determines the orientational distribution in a thin surface layer extending up to 1 nm. The bulk uniaxial order develops on top of this layer via a transition layer of thickness which is well described by the usual mean-field theory, possibly including non-uniaxial components of the tensor order parameter. 

Fig. 3.8 Device configurations of back-gated oriented (a) and random CNT-TFTs (b) with Au source/drain electrodes, c AFM topographical image of a CNT-TFT with uniaxially ordered micropatterned CNT array, d AFM topographical image of a CNT-TFT with randomly oriented CNT surface film. Reproduced with permission from [85]. Copy right 2006 American Chemical Society...

For the purpose of this article, we focus our attention on the nematic mesophase smectic orders are more crystal-like and thus are beyond our scope. Typical nematic liquid crystals are characterized by a uniaxial order, though imperfect, along the preferred axis of the domain. No such long-range order exists in directions transverse to the domain axis. In most examples, low molar mass (monomer) liquid crystals carry flexible tails. Conformational ordering of these tails in the mesophase has been extensively studied in relation to the odd-even character of the phase behavior with the number of constituent atoms of the pendant chain. Various statistical models and theories have been presented [52-57]. In most cases, however, the ordering of the tail is relatively weak [58,59]. 

As a result of the chain structure of the macromolecules, they may exhibit not only uniaxial order observed in the nematic mesophase but also the orientational polar order. The physical value directly characterizing the orientational polar order in a chain molecule is its dipole moment. If the monomer unit exhibits a dipole moment directed along the chain, then the molecule as a whole in a given conformation is characterized by the dipole moment y ... 

Cooper (1962) has derived a more complex expression for the case of dysprosium with uniaxial ordering in the basal plane below 85 K ... 

For the simplest model of pofetA uniaxial order it is assumed that the polymer chains are all oriented parallel to the draw direction, and that the transition moments associated with the vibrations of the absorbing groups lie in a cone with a semiangle and the draw direction as axis. The dichroic ratio may then be expressed by... 

Not only are refractive indices needed to optimise device and materials design, but also they provide a simple route to order parameters. Extrapolation methods can be used to determine the uniaxial order parameter 5, if contributions from molecular bi-axiality (D) are ignored [18-20], For molecules having axial symmetry au=cx ) Eq, (9) can be written as ... 

At extremely high fields the isotropic phase should be indistinguishable from the nematic one, even well above the zero field transition temperature, since the uniaxial order induced by a magnetic or ac electric field in the isotropic phase will be comparable with the nematic orientational order. However, such fields are hardly accessible, even with the pulse technique. Much stronger changes in order parameter may be achieved with ferroelectric transitions (see below). 

In Ihis expression X and Xl are the susceptibility along the director and normal to it. They relate to the average molecular susceptibility Xmu (unit m /kg), the mass-density p, and the uniaxial order parameter S as ... 

In addition to the uniaxial order which is obsmred in the nematic meso-... 

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