Dimensional Robustness and Instability of Sheared, Semidilute, Nanorod Dispersions

Abstract

Semidilute, nanorod dispersions interact nonlocally and nonlinearly through excluded-volume and distortional elasticity potentials. When driven by steady shear with confinement boundary conditions, remarkable rod orientational distribution behavior 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 nanorod measurement limitations, continue to confound materials processing strategies. Thus, modeling and simulation play a critical role. In this paper, we present a hierarchy of zero-dimensional (0-d), one-dimensional (1-d), and two-dimensional (2-d) physical space simulations of steady parallel-plate shear experiments using a mesoscopic tensor model for the rod orientational distribution [E. H. MacMillan, A Theory of Anisotropic Fluid, Ph.D. thesis, Department of Mechanics, University of Minnesota, Minneapolis, MN, 1987, Z. Cui and Q. Wang, J. Non-Newtonian Fluid Mech., 138 (2006), pp. 44–61] and a spectral-Galerkin numerical algorithm [J. Shen, SIAM J. Sci. Comput., 15 (1994), pp. 1489–1505]. We impose steady shear to focus on the orientational response of the nanorod 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, semidilute, nanorod 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 the following: 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 dimensions 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; finally, logrolling (vorticity-aligned) anchoring selects the most distinct attractors and De cascade with respect to other anchoring conditions.