Scaling behavior of kinetic orientational distributions for dilute nematic polymers in weak shear

Analytical descriptions of shear-aligned nematic monodomains have a long history across continuum, mesoscopic and mean-field kinetic models. Their value lies in the prediction of explicit scaling properties of the orientational distribution and of normal and shear stresses, with respect to molecular and flow parameters. At the coarsest scale of continuum Leslie-Ericksen theory [1, 2], an explicit macroscopic alignment angle formula always exists in terms of bulk Miesowicz viscosities; the theory applies only to nematic (highly concentrated) liquids with low molecular weight and in the weak flow limit [3, 4, 5, 6]. At mesoscales, orientation tensor models [7, 8, 9, 10, 11] apply at all concentrations and shear rates; explicit ”Leslie angle” formulas exist only in the weak shear limit [12, 13, 14, 15, 16, 17, 18, 19], inheriting hysteresis in alignment properties from the quiescent isotropic-nematic transition. Since the fundamental results of Onsager [20] for quiescent nematic equilibria, exact probability distribution functions (PDFs) of kinetic theory have proven elusive for flow-driven nematic states. In their stead, flow-aligned and unsteady PDF formulas, elegant though implicit, have been derived by Kuzuu-Doi [3, 4] and Semenov [21] by weak flow asymptotic analysis, and by Marrucci and Maffettone [22] by restriction to two dimensions. A simpler problem concerns the dilute concentration regime where the quiescent equilibrium is isotropic, non-degenerate, and stable, which is the focus of this paper. Using weak-flow asymptotics, we explicitly construct, and establish stability of, stationary, shear-perturbed PDFs for dilute concentrations; our formulas prove unsteady tumbling of perturbed isotropic states cannot occur. Exact scaling properties are predicted, including explicit Leslie alignment angle and degree-of-alignment (birefringence) formulas, as well as normal and shear stresses, in terms of molecular parameters and normalized shear rate for dilute nematic polymers. Our formulas apply except in a neighborhood of the isotropic instability transition, which