Molecular Theories of Uniaxial Phases

The interactions between rod-like molecules in nematics are highly anisotropic. The forces between molecules depend not only on their separation but also on their relative orientations. Unfortunately, the precise form of the pair potential is not known. However, it is possible to proceed with a perfectly general pair potential U12. Three Eulerian angles (0,0,-0) are needed to specify the orientation of a rigid particle. In particular, if the particle is a rigid cylindrical rod, the angle is unimportant. Thus, the pair potential depends on five coordinates  

The appearance of a single m index in Eq. (3.20) is due to the fact that V12 depends on the difference of two (p angles. The above expansion 

FIGURE 3.2. The coordinate system required to describe the interaction between two asymmetric molecules, (a) The upper diagram illustrates the ri2 frame in which the intermolecular vector ri2 is the mutual polar axis, while (b) the lower diagram illustrates the director frame in which the director is the mutual polar axis. 

To get a single-molecule potential, first the average over all orientations of the intermolecular vector f 2 is obtained, then the average over all orientations of molecule 2, and finally the average over the intermolecular separation ri2- If a spherical distribution of the f 2 vector is assumed for the moment, the average over all orientations of ri2 involves only the Wigner rotation matrices 

Now Eq. (3.23) is an approximation since, in reality, the fi2 has cylindrical symmetry in nematics. This point is deferred [see Eq. (3.57)], but note that additional terms [3.10] are needed in the above equation for (U12). The average over the orientations of molecule 2 only influences and requires an orientation distribution function /( 2) molecule 2. Since there is no (f) dependence in nematics because of uniaxial symmetry, the integral over (f) vanishes unless m is zero and L is even. Thus, 

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The majority of the existing molecular theories of nematic liquid crystals are based on simple uniaxial molecular models like sphe-rocylinders. At the same time typical mes-ogenic molecules are obviously biaxial. (For example, the biaxiality of the phenyl ring is determined by its breadth-to-thick-ness ratio which is of the order of two.) If this biaxiality is important, even a very good statistical theory may result in a poor agreement with experiment when the biaxiality is ignored. Several authors have suggested that even a small deviation from uniaxial symmetry can account for important features of the N-I transition [29, 42, 47, 48], In the uniaxial nematic phase composed of biaxial molecules the orientational distri-... 

Realistic intermolecular interaction potentials for mesogenic molecules can be very complex and are generally unknown. At the same time molecular theories are often based on simple model potentials. This is justified when the theory is used to describe some general properties of liquid crystal phases that are not sensitive to the details on the interaction. Model potentials are constructed in order to represent only the qualitative mathematical form of the actual interaction energy in the simplest possible way. It is interesting to note that most of the popular model potentials correspond to the first terms in various expansion series. For example, the well known Maier-Saupe potential JP2 (Sfli )) is just the first nonpolar term in the Legendre polynomial expansion of an arbitrary interaction potential between two uniaxial molecules, averaged over the intermolecular vector r,-, ... 

The molecular theory is similar to Cauchy s description of the elastic theory of solids [1] and utilizes additive local molecular pair interactions to describe elasticity. The latter approach was taken by Oseen [2], who was the first to establish an elastic theory of anisotropic fluids. Oseen assumed short-range intermolecular forces to be the reason for the elastic properties, and he derived eight elastic constants in the expression for the elastic free energy density of uniaxial nematic phases. Finally, he retained only five of them, which enter the Euler-Lagrange equations describing equilibrium deformation states of the nematic mesophase, and omitted the other three. 

The first molecular field theory of biaxial nematics was presented by Preiser [14] indeed it was his prediction which stimulated the hunt for thermotropic biaxial nematics. An alternative vision of the theory was then given by Straley [17] and, although not primarily concerned with liquid crystals, Boccara et al [33] have presented a theory applicable to uniaxial and biaxial nematic phases. The key feature of these theories is the potential of mean torque which is written using a second rank interaction as... 

The non-collective motions include the rotational and translational self-diffusion of molecules as in normal liquids. Molecular reorientations under the influence of a potential of mean torque set up by the neighbours have been described by the small step rotational diffusion model.118 124 The roto-translational diffusion of molecules in uniaxial smectic phases has also been theoretically treated.125,126 This theory has only been tested by a spin relaxation study of a solute in a smectic phase.127 Translational self-diffusion (TD)29 is an intermolecular relaxation mechanism, and is important when proton is used to probe spin relaxation in LC. TD also enters indirectly in the treatment of spin relaxation by DF. Theories for TD in isotropic liquids and cubic solids128 130 have been extended to LC in the nematic (N),131 smectic A (SmA),132 and smectic B (SmB)133 phases. In addition to the overall motion of the molecule, internal bond rotations within the flexible chain(s) of a meso-genic molecule can also cause spin relaxation. The conformational transitions in the side chain are usually much faster than the rotational diffusive motion of the molecular core. 

Molecular order in the smectic-A phase was probed recently via a proton NMR study of three aromatic solutes in the liquid crystal 8CB [35]. The results were analyzed in the context of a simple modification of K-M theory for dissolved non-uniaxial solutes. The smectic solute Hamiltonian was written in the form ... 

Due to the uniaxial or symmetry of a nematic phase, the dielectric permittivity of a nematic is represented by a second rank tensor with two principal elements, 8 and 8 The component 8 is parallel to the macroscopic symmetry axis, which is along the director, and is perpendicular to this. According to a molecular field theory, they are approximated by... 

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