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

Abstract 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 [Arch. Rational Mech. Anal. 4 (1960) 231; Arch. Rational Mech. Anal. 28 (1968) 265], 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 [Phys. Soc. Jpn. 52 (1983) 3486; Phys. Soc. Jpn. 53 (1984) 1031; J. Chem. Phys. 116 (2002) 9120; A hydrodynamic theory for solutions of non-homogeneous nematic liquid crystalline polymers with density variations, in: Proceedings of ASME International Mechanical Engineering Congress, IMECE2002-32189, N.O., LA, 17-22 November 2002]. At mesoscales, orientation tensor models [J. Chem. Phys. 103 (1995) 3108; The Physics of Liquid Crystals, Oxford University Press, Oxford, 1993; Thermodynamics of Flowing Systems with Internal Microstructure, Oxford Science Publications, Oxford, 1994; J. Chem. Phys. 103 (1995) 807; The Structure and Rheology of Complex Fluids, Oxford University Press, Oxford, 1998] apply at all concentrations and shear rates; explicit “Leslie angle” formulas exist only in the weak shear limit [Macromolecules 23 (1990) 4446; J. Rheol. 37 (1993) 413; J. Rheol. 37 (1993) 289; J. Rheol. 41 (1997) 943; Phys. A 267 (1999) 294; J. Non-Newtonian Fluid Mech. 90 (2000) 283; Rheol. Acta 42 (2003) 20; J. Rheol. 47 (2003) 105], inheriting hysteresis in alignment properties from the quiescent isotropic–nematic transition. Since the fundamental results of Onsager [Ann. N.Y. Acad. Sci. 51 (1949) 627] 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 [Phys. Soc. Jpn. 52 (1983) 3486; Phys. Soc. Jpn. 53 (1984) 1031] and Semenov [Sov. Phys. JETP 66 (1983) 321] by weak flow asymptotic analysis, and by Marrucci and Maffettone [Macromolecules 22 (1989) 4076] 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 See et al. [J. Chem. Phys. 92 (1990) 792] analyzed by a singular perturbation analysis. We then show how to bridge our method with that of See et al. [J. Chem. Phys. 92 (1990) 792], thereby completing the existence, construction, and stability of shear-perturbed isotropic equilibria for all concentrations. We verify the formulas both by numerical simulations and by comparison with mesoscopic model predictions [Rheol. Acta 42 (2003) 20; J. Rheol. 47 (2003) 105; J. Chem. Phys. 92 (1990) 792].