EXPLICIT FLOW-ALIGNED ORIENTATIONAL DISTRIBUTION FUNCTIONS FOR DILUTE NEMATIC POLYMERS IN WEAK SHEAR

摘要

Flow-alignment of sheared nematic polymers occurs in various flow-concentration regimes. Analytical descriptions of shear-aligned nematic monodomains have a long history across continuum, mesoscopic and mean-field kinetic models, sacrificing precision at each finer scale. Continuum Leslie-Ericksen theory applies to highly concentrated, weak flows of small molecular weight polymers, giving an explicit macroscopic alignment angle formula dependent only on Miesowicz viscosities. Mesoscopic tensor models apply at all concentrations and shear rates, but explicit "Leslie angle" formulas exist only in the weak shear limit (Cocchini et. al, 90; Bhave et. al, 93; Wang, 97; Rien-acker and Hess, 99; Maffettone et. al, 00; Forest and Wang, 02; Forest et, al, 02c; Grecov and Rey, 02), with distinct behavior in dilute versus concentrated regimes. Exact probability distribution functions (pdf's) of kinetic theory do not exist for highly concentrated nematic states, even without flow, although appealing flow-aligned approximations have been derived (Kuzuu and Doi, 83; Kuzuu and Doi, 84; Semenov, 83; Semenov, 86; Archer and Larson, 95; Kroger and Seller, 95), which offer a molecular theory basis for the Leslie alignment angle. A simpler problem concerns the dilute concentration regime where the unique quiescent equilibrium is isotropic, corresponding to a constant pdf, and whose weak shear deformation is robust to mesoscopic closure approximation (Forest and Wang, 02; Forest et. al, 02c): steady, flow-aligning, weakly anisotropic, and biaxial. The purpose of this paper is to explicitly construct the weakly anisotropic branch of stationary pdf's by a weak-shear asymptotic expansion of kinetic theory. A second-moment pdf projection confirms mesoscopic model predictions, and further yields explicit Leslie angle and degree of alignment formulas in terms of molecular parameters and normalized shear rate.