Abstract.

 Semi-dilute nano-rod dispersions interact nonlocally and
nonlinearly through excluded-volume and distortional elasticity
potentials. When driven by steady shear with con¯nement boundary
conditions, remarkable behavior of the rod orientational distribu-
tion ensues: strong anisotropy; steady and unsteady responses; and
gradient structure on (thus far) unpredictable lengthscales. Extreme
variability and sensitivity of these fea- tures 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 im- pose 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 com-
munity. Next, we present the ¯rst study of numerical stability (for
all attractors in the phase diagrams) to 2-d perturbations in the
°ow-gradient and vorticity directions. The key ¯ndings 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 attrac- tors in 0 and 1 space dimension are
unstable to coherent 2-d morphology, but remain chaotic; as De
increases, chaotic dynamics becomes regularized, ¯rst to periodic and
then to steady structure attractors, along with a return to robust 1-d
morphology; and ¯nally, logrolling (vorticity-aligned) anchoring
selects the most distinct attractors and De cascade with respect to
other anchoring conditions.