# American Institute of Mathematical Sciences

March  2011, 15(2): 457-473. doi: 10.3934/dcdsb.2011.15.457

## Shearing the I-N phase transition of liquid crystalline polymers: Long-time memory of defect initial data

 1 Department of Biomedical Engineering, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-7575, United States 2 Departments of Mathematics and Biomedical Engineering, Institute for Advanced Materials, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3250, United States 3 Department of Mathematics, University of South Carolina, Columbia, SC 29208

Received  December 2009 Revised  March 2010 Published  December 2010

Liquid crystalline polymers have been extensively studied in shear starting from an equilibrium nematic phase. In this study, we explore the transient and long-time behavior as a steady shear cell experiment commences during an isotropic-nematic (I-N) phase transition. We initialize a localized Gaussian nematic droplet within an unstable isotropic phase with nematic, vorticity-aligned equilibrium at the walls. In the absence of flow, the simulation converges to a homogeneous nematic phase, but not before passing through quite intricate defect arrays and patterns due to physical anchoring, the dimensions of the shear cell, and transient backflow generated around the defect arrays during the I-N transition. Snapshots of this numerical experiment are then used as initial data for shear cell experiments at controlled shear rates. For homogeneous stable nematic equilibrium initial data, the Leal group [4, 5, 6] and the authors [12] confirm the Larson-Mead experimental observations [7, 8]: stationary 2-D roll cells and defect-free 2-D orientational structure at low shear rates, followed at higher shear rates by an unstable transition to an unsteady 2-D cellular flow and defect-laden attractor. We show at low shear rates that the memory of defect-laden data lasts forever; 2-D steady attractors of [4, 5, 12] emerge for defect free initial data, whereas 1-D unsteady attractors arise for defect-laden initial data.
Citation: Ke Xu, M. Gregory Forest, Xiaofeng Yang. Shearing the I-N phase transition of liquid crystalline polymers: Long-time memory of defect initial data. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 457-473. doi: 10.3934/dcdsb.2011.15.457
##### References:
 [1] M. G. Forest, Q. Wang and R. Zhou, Kinetic structure simulations of nematic polymers in plane Couette cells, II: In-plane structure transitions,, SIAM Multi. Model. Simul., 4 (2005), 1280. doi: 10.1137/040618187. [2] J. J. Feng, J. Tao and L. G. Leal, Roll cells and disclinations in sheared nematic polymers,, J. Fluid Mech., 449 (2001), 179. doi: 10.1017/S0022112001006279. [3] J. J. Feng and L. G. Leal, Simulating complex flows of liquid-crystalline polymers using the Doi theory,, J. Rheol. \textbf{41} (1997), 41 (1997), 1317. doi: 10.1122/1.550872. [4] D. H. Klein, C. J. Garcia-Cervera, H. D. Ceniceros and L. G. Leal, Ericksen number and Deborah number cascade predictions of a model for liquid crystalline polymers for simple shear flow,, Phys. of Fluids, 19 (2007), 023. [5] D. H. Klein, "Dynamics of a Model for Nematic Liquid Crystalline Polymers in Planar Shear and Pressure-Driven Channel Flows,", Ph.D thesis, (2007). [6] D. H. Klein, C. J. Garcia-Cervera, H. D. Ceniceros and L. G. Leal, Three-dimensional shear driven dynamics of polydomain textures and disclination loops in liquid crystalline polymers,, J. Rheol., 52 (2008), 837. doi: 10.1122/1.2890779. [7] R. G. Larson and D. W. Mead, Development of orientation and texture during shearing of liquid-crystalline polymers,, Liq. Cryst., 12 (1992), 751. doi: 10.1080/02678299208029120. [8] R. G. Larson and D. W. Mead, The Ericksen number and Deborah number casades in sheared polymeric nematics,, Liq. Cryst., 15 (1993), 151. doi: 10.1080/02678299308031947. [9] G. de Luca and A. D. Rey, Dynamic interactions between nematic point defects in the extrusion duct of spiders,, Virtual Journal of Biological Physics Research, 124 (2006), 1. [10] J. Shen, Efficient spectral-Galerkin method I. Direct solvers for second and fourth-order equations using Legendre polynomials,, SIAM J. Sci. Comput., 15 (1994), 1489. doi: 10.1137/0915089. [11] T. Tsuji and A. D. Rey, Effect of long range order on sheared liquid crystalline materials : Flow regimes, transitions, and rheological diagrams,, Phys. Rev. E, 62 (2000), 8141. doi: 10.1103/PhysRevE.62.8141. [12] X. Yang, M. G. Forest, Q. Wang and W. M. Mullins, Dynamic defect morphology and hydrodynamics of sheared nematic polymers in two space dimensions,, J. Rheology, 53 (2009), 589. doi: 10.1122/1.3089622. [13] X. Yang, M.G. Forest, Q. Wang and W. M. Mullins, 2-D Lid-driven cavity flow of nematic polymers: an unsteady sea of defects,, Soft Matter, 6 (2010), 1138. doi: 10.1039/b908502e. [14] M. G. Forest and Q. Wang, Monodomain response of finite-aspect-ratio macromolecules in shear and related linear flows,, Rheologica Acta, 42 (2003), 20. doi: 10.1007/s00397-002-0252-0. [15] M. G. Forest, Q. Wang and R. Zhou, The weak shear kinetic phase diagram for nematic polymers,, Rheol. Acta, 43 (2004), 17. doi: 10.1007/s00397-003-0317-8. [16] M.G. Forest, S. Heidenreich, S. Hess, X. Yang and R. Zhou, Robustness of pulsating jet-like layers in sheared nano-rod dispersions,, J. Non-Newtonian Fluid Mech., 155 (2008), 130. doi: 10.1016/j.jnnfm.2008.06.003.

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##### References:
 [1] M. G. Forest, Q. Wang and R. Zhou, Kinetic structure simulations of nematic polymers in plane Couette cells, II: In-plane structure transitions,, SIAM Multi. Model. Simul., 4 (2005), 1280. doi: 10.1137/040618187. [2] J. J. Feng, J. Tao and L. G. Leal, Roll cells and disclinations in sheared nematic polymers,, J. Fluid Mech., 449 (2001), 179. doi: 10.1017/S0022112001006279. [3] J. J. Feng and L. G. Leal, Simulating complex flows of liquid-crystalline polymers using the Doi theory,, J. Rheol. \textbf{41} (1997), 41 (1997), 1317. doi: 10.1122/1.550872. [4] D. H. Klein, C. J. Garcia-Cervera, H. D. Ceniceros and L. G. Leal, Ericksen number and Deborah number cascade predictions of a model for liquid crystalline polymers for simple shear flow,, Phys. of Fluids, 19 (2007), 023. [5] D. H. Klein, "Dynamics of a Model for Nematic Liquid Crystalline Polymers in Planar Shear and Pressure-Driven Channel Flows,", Ph.D thesis, (2007). [6] D. H. Klein, C. J. Garcia-Cervera, H. D. Ceniceros and L. G. Leal, Three-dimensional shear driven dynamics of polydomain textures and disclination loops in liquid crystalline polymers,, J. Rheol., 52 (2008), 837. doi: 10.1122/1.2890779. [7] R. G. Larson and D. W. Mead, Development of orientation and texture during shearing of liquid-crystalline polymers,, Liq. Cryst., 12 (1992), 751. doi: 10.1080/02678299208029120. [8] R. G. Larson and D. W. Mead, The Ericksen number and Deborah number casades in sheared polymeric nematics,, Liq. Cryst., 15 (1993), 151. doi: 10.1080/02678299308031947. [9] G. de Luca and A. D. Rey, Dynamic interactions between nematic point defects in the extrusion duct of spiders,, Virtual Journal of Biological Physics Research, 124 (2006), 1. [10] J. Shen, Efficient spectral-Galerkin method I. Direct solvers for second and fourth-order equations using Legendre polynomials,, SIAM J. Sci. Comput., 15 (1994), 1489. doi: 10.1137/0915089. [11] T. Tsuji and A. D. Rey, Effect of long range order on sheared liquid crystalline materials : Flow regimes, transitions, and rheological diagrams,, Phys. Rev. E, 62 (2000), 8141. doi: 10.1103/PhysRevE.62.8141. [12] X. Yang, M. G. Forest, Q. Wang and W. M. Mullins, Dynamic defect morphology and hydrodynamics of sheared nematic polymers in two space dimensions,, J. Rheology, 53 (2009), 589. doi: 10.1122/1.3089622. [13] X. Yang, M.G. Forest, Q. Wang and W. M. Mullins, 2-D Lid-driven cavity flow of nematic polymers: an unsteady sea of defects,, Soft Matter, 6 (2010), 1138. doi: 10.1039/b908502e. [14] M. G. Forest and Q. Wang, Monodomain response of finite-aspect-ratio macromolecules in shear and related linear flows,, Rheologica Acta, 42 (2003), 20. doi: 10.1007/s00397-002-0252-0. [15] M. G. Forest, Q. Wang and R. Zhou, The weak shear kinetic phase diagram for nematic polymers,, Rheol. Acta, 43 (2004), 17. doi: 10.1007/s00397-003-0317-8. [16] M.G. Forest, S. Heidenreich, S. Hess, X. Yang and R. Zhou, Robustness of pulsating jet-like layers in sheared nano-rod dispersions,, J. Non-Newtonian Fluid Mech., 155 (2008), 130. doi: 10.1016/j.jnnfm.2008.06.003.
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