A lattice model of liquid crystalline microstructure has been developed. It provides the basis for the three-dimensional solution of the Frank elasticity equations for given boundary conditions while, in addition, providing a mechanistic representation of the development of texture as the microstructure relaxes with time. It is also able to represent disclination motion and the processes associated with their interaction. In particular, it has been used to study (s = ± 1/2) disclination loops, both those described by a single rotation vector, 17, and those in which 17 has a constant angular relationship with the loop line and are equivalent to a point singularity at a distance much larger than the loop radius. The application of the model to disclinations of unit strength, which are unstable both energetically and topologically, has shown that the decomposition into two 1/2 strength lines of lower total energy occurs much more readily than topological escape in the third dimension. The implication for structures observed in capillary tubes is discussed. The influence on microstructure of a splay constant much higher than that of twist or bend is explored in the context of main-chain liquid crystalline polymers, in particular, the stabilization of tangential +1 lines under such conditions is predicted in accord with observed microstructural features.
Approximation of the axisymmetric elasticity equations