Extension dynamics of discotic nematics of variable order : geodesic flow and viscoelastic relaxation

Variational methods are used to develop the governing equations that describe the flow of spatially invariant uniaxial discotic nematic liquid crystals of variable order ; since the equations are based on a phenomenological truncated expansion of the entropy production, the equations are approximations. Restrictions in the phenomenological parameters appearing in the governing equations are imposed taking into account the ordering of the discotic phase. Numerical and analytical solutions of the director n and alignment S are presented for a given uniaxial extensional start-up flow. The unit sphere description of the director is used to discuss and analyze the sensitivity of the director trajectories and the coupled alignment relaxation to the initial conditions (${f n}_0, S_0$) and to the alignment Deborah number (De). The numerical results are used to characterize the relaxation of the tensor order parameter Q and to compute the steady flow birefringence. When the poles of the unit sphere are along the extension axis and the equator lies in the compression plane of the flow, it is found that the director trajectories belong to the meridians (great circles through the poles) and the dynamics follows a geodesic flow ; when subjected to flow the director follows the shortest path that connects the initial orientation ${f n}_0$ and the equator (compression plane). As typical of geodesic flows, there is a strong sensitivity to initial conditions : when ${f n}_0$ lies on the poles no predictions on the eventually steady director orientation are possible. If the prior to flow orientation is close to the poles the coupled alignment relaxation along the geodesics is nonmonotonic and for large De the discotic may become temporarily isotropic. The couplings between n and S are captured by the tensor order parameter relaxation. At steady state, the director lies on the equator, and the alignment and birefringence increase with increasing De.