Dimensional Robustness & Instability of Sheared, Semi-dilute, Nano-rod Dispersions

Semi-dilute nano-rod dispersions interact nonlocally and nonlinearly through excluded-volume and distortional elasticity potentials. When driven by steady shear with confinement boundary conditions, remarkable behavior of the rod orientational distribution ensues: strong anisotropy; steady and unsteady responses; and gradient structure on (thus far) unpredictable lengthscales. Extreme variability and sensitivity of these features to experimental controls, coupled with nano-rod measurement limitations, continue to confound materials processing strategies. Thus, modeling and simulation play a critical role. In this paper, we present a hierarchy of 0-d, 1-d and 2-d physical space simulations of steady parallel-plate shear experiments, using a mesoscopic tensor model for the rod orientational distribution [45, 8] and a spectral-Galerkin numerical algorithm [52]. We impose steady shear to focus on the orientational response of the nano-rod ensemble to two experimental controls: the Deborah number (De), or normalized imposed shear rate; and physical plate anchoring conditions on the rod ensemble. Our results yield dimensional robustness versus instability of sheared, semi-dilute, nano-rod dispersions: To begin, we present 0-d and 1-d phase diagrams that are consistent with results of the modeling community. Next, we present the first study of numerical stability (for all attractors in the phase diagrams) to 2-d perturbations in the flow-gradient and vorticity directions. The key findings are: time-periodic 1-d structure attractors at low-to-moderate De are robust to 2-d perturbations; period-doubling transitions at intermediate De to chaotic attractors in 0 and 1 space dimension are unstable to coherent 2-d morphology, but remain chaotic; as De increases, chaotic dynamics becomes regularized, first to periodic and then to steady structure attractors, along with a return to robust 1-d morphology; and finally, logrolling (vorticity-aligned) anchoring selects the most distinct attractors and De cascade with respect to other anchoring conditions.