Alignment and rheo-oscillator criteria for sheared nematic polymer films in the monolayer limit

Monolayer films of liquid crystalline polymers (LCPs) are modelled with a mesoscopic two-dimensional (2D) analogue of the Doi-Hess (1981, 1976) rigid rod model. One aim is to establish a more complete solution to the classical problem of how orientational degeneracy of quiescent nematic equilibria breaks in weak shear. We exploit the simplicity of 2D liquids to extend results of Kuzuu and Doi (1983,1984), Marrucci and Maffetone (1989-1991), See, Doi and Larson (1990), Forest et al. (2003-2004). We recall the distinction between two versus three dimensional quiescent phase diagrams and the isotropic-nematic phase transition, then analyze the deformation of these respective bifurcation diagrams in shear flow. We give a simple proof that limit cycles, known as tumbling orbits, must arise beyond the parameter boundary for the steady-unsteady transition. Finally, we show the shear-perturbed 2D phase diagram is significantly more robust to closure approximations than the 3D system.