Monodomain response of finite-aspect-ratio macromolecules in shear and related linear flows

Abstract.

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, parameterized by molecular aspect ratio. In this paper we explore phenomena associated with finite vs 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 around 3:1 or 1:3, similar to full kinetic simulations by Grosso et al. [Grosso M, Keunings R, Crescitelli S, Maffettone PL (2001), Prediction of chaotic dynamics in sheared liquid crystalline polymers. Preprint (2001) and lecture, Society of Rheology Annual Meeting, Hilton Head, SC, February 2001] for infinite aspect ratios. We develop symmetries of mesoscopic tensor models robust to closure approximations but specific to linear flow fields, and analytical methods to determine:

• The entire monodomain phase diagram of a finite-aspect-ratio nematic fluid in a linear flow field is equivalent to the phase diagram of an infinite-aspect-ratio fluid (thin rods or discs) in a related linear velocity field.

• Rod-like and discotic macromolecules with reciprocal aspect ratios have equivalent bulk shear response, related by a simple director transformation.

• Out-of-plane, shear-induced monodomains (steady and transient) either are symmetric about the shearing plane (e.g., logrolling and kayaking modes), or occur in pairs mirror-symmetric about the shearing plane (out-of-plane steady and periodic "tilted kayaking" modes), revealing a symmetry mechanism for bi-stability.

• A tensor analog of the Leslie alignment vs tumbling criterion, which is developed and applied to predict the multiplicity, stability, and steady or transient property of shear-induced monodomains.

Simulations highlight the degree to which scaling properties of Leslie-Ericksen theory are violated. By varying molecular aspect ratio, 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.

Appendix. Viscosity coefficients

The results of Jeffery [74], Batchelor [9] as well as Hinch and Leal [69, 70] on ellipsoidal suspensions in a viscous solvent are utilized in the derivation of the viscous stress [140]:

$$\matrix{ {\eta _s = \eta + 3/2\nu kT\zeta _3 ,} \hfill \cr {\zeta _3 = {{\zeta ^{\left( 0 \right)} } \over {I_1 }},\;\;\zeta _1 = \zeta ^{\left( 0 \right)} \left( {{1 \over {I_3 }} - {1 \over {I_1 }}} \right),\;\;\zeta _2 = \zeta ^{\left( 0 \right)} \left[ {{{J_{\rm I} } \over {I_1 J_3 }} + {1 \over {I_1 }} - {2 \over {I_3 }}} \right],} \hfill \cr {I_1 = 2r\int_0^ \propto {{{dx} \over {\sqrt {\left( {r^2 + x} \right)\left( {1 + x} \right)^3 } }},\;\;I_3 = r\left( {r^2 + 1} \right)\int_0^\infty {{{dx} \over {\sqrt {\left( {r^2 + x} \right)\left( {1 + x} \right)^2 \left( {r^2 + x} \right)} }}} ,} } \hfill \cr {J_1 = r\int_0^\infty {{{xdx} \over {\sqrt {\left( {r^2 + x} \right)\left( {1 + x} \right)^3 } }},\;\;J_3 = r\int_0^\infty {{{xdx} \over {\sqrt {\left( {r^2 + x} \right)\left( {1 + x} \right)^2 \left( {r^2 + x} \right)} }}.} } } \hfill \cr } $$

(A1)