Monodomain response of finite-aspect-ratio macromolecules in shear and related linear flows
M. Gregory Forest, Qi Wang
Recent extensions of the Doi kinetic theory for monodisperse nematic liquids describe rigid, axisymmetric, ellipsoidal macromolecules with finite aspect ratio. Averaging and presumed linear flow fields provide tensor dynamical systems for mesoscopic, bulk orientation response, parametrized by molecular aspect ratio. In this paper we explore phenomena associated with finite versus infinite aspect ratios, which alter the most basic features of monodomain attractors: steady vs. unsteady, in-plane vs. out-of-plane, multiplicity of attracting states, and shear-induced transitions. For example, the Doi moment-closure model predicts a period-doubling cascade in simple shear to a chaotic monodomain attractor for aspect ratios nearby $3:1$ or $1:3$, similar to full kinetic simulations by Grosso et al. (2001) for infinite aspect ratios. We develop theoretical properties first, independent of closure approximation but specific to linear flow fields, which imply:
1. the entire monodomain phase diagram of a finite-aspect-ratio nematic fluid in linear flow fields is equivalent to the phase diagram of an infinite-aspect-ratio fluid (thin rods or discs) with a related linear velocity field;
2. rod-like and discotic macromolecules with reciprocal aspect ratios have equivalent bulk shear response, related by a simple director transformation;
3. all shear-induced monodomains respect symmetries relative to the shearing plane, e.g., major director (kayaking) motions that rotate around an axis between the vorticity axis and shearing plane are always accompanied by a symmetric response reflected through the shear plane. This provides a symmetry mechanism for bi-stable attractors.
A tensor analog of the Leslie alignment vs. tumbling criterion is developed and applied. Simulations highlight the degree to which scaling properties of Leslie-Ericksen theory (e.g., shear response can be scaled in terms of strain units, monodomain transitions are independent of shear rate) are violated. With finite aspect ratios, any shear-induced monodomain is reproducible among the well-known closure approximations, yet no single closure rule suffices to capture all known attractors and transition scenarios.