Microscopic-Macroscopic Simulations of Rigid-Rod Polymer Hydrodynamics: Heterogeneity and Rheochaos

Abstract

Rheochaos is a remarkable phenomenon of nematic (rigid-rod) polymers in steady shear, with sustained chaotic fluctuations of the orientational distribution of the rod ensemble. For monodomain dynamics, imposing spatial homogeneity and linear shear, rheochaos is a hallmark prediction of the Doi–Hess theory [M. Doi, J. Polym. Sci. Polym. Phys. Ed., 19 (1981), pp. 229–243; M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, London, New York, 1986; S. Hess, Z. Naturforsch., 31 (1976), pp. 1034–1037]. The model behavior is robust, captured by second-moment tensor approximations [G. Rienäcker, M. Kröger, and S. Hess, Phys. Rev. E (3), 66 (2002), 040702; G. Rienäcker, M. Kröger, and S. Hess, Phys. A, 315 (2002), pp. 537–568; M. G. Forest and Q. Wang, Rheol. Acta, 42 (2003), pp. 20–46] and high-order Galerkin simulations of the Smoluchowski equation for the orientational probability distribution function (PDF) [M. Grosso, R. Keunings, S. Crescitelli, and P. L. Maffettone, Phys. Rev. Lett., 86 (2001), pp. 3184–3187; M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 43 (2004), pp. 17–37; M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 44 (2004), pp. 80–93], and persistent up to critical thresholds of coplanar extensional flow [M. G. Forest, R. Zhou, and Q. Wang, Phys. Rev. Lett., 93 (2004), 088301; M. G. Forest, Q. Wang, R. Zhou, and E. Choate, J. Non-Newt. Fluid Mech., 118 (2004), pp. 17–31; S. Heidenreich, P. Ilg, and S. Hess, Phys. Rev. E (3), 73 (2006), 061710] and magnetic fields [M. G. Forest, Q. Wang, H. Zhou, and R. Zhou, J. Rheol., 48 (2004), pp. 175–192], as well as fluctuating shear rates [S. Heidenreich, P. Ilg, and S. Hess, Phys. Rev. E (3), 73 (2006), 061710]. To be experimentally relevant, rheochaos of the Doi–Hess theory must persist amid heterogeneity observed in birefringence patterns [Z. Tan and G. C. Berry, J. Rheol., 47 (2003), pp. 73–104]. Modeling can further shed light on shear bands produced by hydrodynamic feedback which have thus far eluded measurement. Some numerical evidence supports persistence: a one-dimensional (1D) study [B. Chakrabarti, M. Das, C. Dasgupta, S. Ramaswamy, and A. K. Sood, Phys. Rev. Lett., 92 (2004), 188301] with a second-moment tensor model and imposed simple shear; and a two-dimensional (2D) study [A. Furukawa and A. Onuki, Phys. D, 205 (2005), pp. 195–206] with a second-moment tensor model and flow feedback. Here we stage the micro-macro (Smoluchowski and Navier–Stokes) system so that monodomain rheochaos is embedded in a 1D simulation [R. Zhou, M. G. Forest, and Q. Wang, Multiscale Model. Simul., 3 (2005), pp. 853–870] of a planar shear cell experiment with distortional elasticity. Longtime simulations reveal (i) heterogeneous rheochaos marked by chaotic time series in the PDF, normal and shear stresses, and velocity field at each interior gap height; (ii) coherent spatial morphology in the PDF and stress profiles across the shear gap and weakly nonlinear shear bands in each snapshot; and (iii) consistency between heterogeneous and monodomain rheochaos as measured by Lyapunov exponents and pointwise orbits of the peak orientation of the PDF but with enhancement rather than reduction in Lyapunov exponent values in the flow coupled, heterogeneous system.