Some consequences of elastic anisotropy

We have so far assumed Kxx=K22=K =K. Real nematics are of course, elastically anisotropic. From the values of the energies of the disclinations calculated for the case where Ki =K22 K22 it follows that the wedge disclinations are more stable than the twist disclinations if A 22 A =Af i=A 33, and vice versa. Anisimov and Dzyaloshin-skii [38] have shown that lines of half-integral strength may be stable against three-dimensional perturbations if the twist elastic constant (22 i/2(A, j-i-A 3 3). More precisely, 

II Defects and Textures in Nematic Main-Chain Liquid Crystalline Polymers 

Real nematics are, of course, elastically anisotropic. In certain situations, as for example at temperatures close to the nematic-smectic transition, the anisotropy becomes very large and certainly cannot be ignored. We shall now investigate some of the consequences of elastic anisotropy on the properties of disclinations. 

As before, we shall begin by considering a planar sample in which the director is confined to the xy plane. In such a case, a wedge disclination involves only splay and bend distortions and we need to take into account only the splay-bend anisotropy (kjj + fejj). 

For j = 1, one obtains the elastically isotropic solution with c = 0 or n/2 to be a solution of (3.5.14) depending upon whether e is negative or positive, and for s = 2 one gets the isotropic solution irrespective of e. 

For a general value of e, the corrections can be evaluated by numerical methods. Fig. 3.5.22 gives the energies of 5 = and 5 = — defects in units oik n R/r. The results of the Nehring-Saupe formula are plotted as squares, and those calculated from a perturbation expansion in the neighbourhood of e = 1 are marked as circles. 

The radial force of interaction between disclination will, of course, be modified because of anisotropy. In addition there will now be an angular component of the force. The physical basis for the angular force can be understood by referring to fig. 3.5.23, which shows the director patterns for two pairs of unlike defects, (+, — ) and ( +1, — 1), each in two different situations. It is seen that there are significant differences in the 

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Most micromechanical theories treat composites where the thermoelastic properties of the matrix and of each filler particle are assumed to be homogeneous and isotropic within each phase domain. Under this simplifying assumption, the elastic properties of the matrix phase and of the filler particles are each described by two independent quantities, usually the Young s modulus E and Poisson s ratio v. The thermal expansion behavior of each constituent of the composite is described by its linear thermal expansion coefficient (3. It is far more complicated to treat composites where the properties of some of the individual components (such as high-modulus aromatic polyamide fibers) are themselves inhomogeneous and/or anisotropic within the individual phase domains, at a level of theory that accounts for the internal inhomogeneities and/or anisotropies of these phase domains. Consequently, there are very few analytical models that can treat such very complicated but not uncommon systems truly adequately. 

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