Monodomain response of arbitrary aspect ratio nematic polymers in general linear planar flows

Abstract

Various rheological devices (plane Couette cell, four-roll mill, film tenter) impose approximately linear, planar flow, from which nematic polymers have been characterized across ranges of flow type and strength. Theoretical predictions of monodomain responses of nematic liquids are predominantly for simple shear flows, studied from disparate scale models: continuum theory of Leslie–Ericksen, mesoscopic theory of Landau, deGennes and many others, and kinetic theory of Hess, Doi and Edwards. Our goal here is to illustrate consequences of a monodomain correspondence principle of kinetic and mesoscopic theory for nematic polymers consisting of arbitrary aspect ratio spheroids. The principle states that the bulk rheology of monodisperse nematic polymers for all linear flows in the plane of shear and any concentration can be deduced directly from a model system consisting of pure shear flow and a renormalized molecular aspect ratio parameter. The observation that there is a “trade-off” between molecule shape and flow is not new (cf. [Proc. R. Soc. London, Ser. A 102 (1922) 161; Proc. R. Soc. London, Ser. A. 146 (1934) 501; Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, London, 1974; J. Rheol. 42 (5) (1998) 1095]). Our contribution is to precisely formulate the principle in terms of the solution space of kinetic or mesoscopic models coupled with the relationship between stresses: a two-parameter model generates solutions for an entire four-parameter family of experiments. The wealth of predictions, and available numerical codes, for nematic polymer response in simple shear can be brought to bear on different flow types. We then illustrate a variety of concrete applications. As a primary example, we deduce the monodomain attractors and phase transitions versus flow-type for four-roll mill flow settings of a nematic polymer. We further provide continuous families of planar flow types of different aspect ratio liquids that have identical monodomain dynamics. Finally, we analyze pure planar extension and straining flows as a limit of the correspondence principle.