# American Institute of Mathematical Sciences

April  2015, 8(2): 341-379. doi: 10.3934/dcdss.2015.8.341

## Structure formation in sheared polymer-rod nanocomposites

 1 Laboratory of Mathematics and Complex Systems, Ministry of Education and School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China 2 Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, United States 3 School of Mathematics and LPMC, Nankai University, Tianjin 300071

Received  July 2013 Revised  November 2013 Published  July 2014

We develop a hydrodynamic theory for flowing inhomogeneous polymer-nanorod composites (PNCs) coupling the Smoluchowski transport equation for the distribution function of the nanorod dispersed in a polymer matrix and the transport equation for the distribution of the polymer in the host matrix. The polymer molecule phase is modeled by bead-spring Rouse chains while the nanorod phase is modeled as semiflexible rods. The polymer-nanorod surface contact interaction and the conformational entropy of semiflexible nanorods are incorporated, resulting in a coupled system of nonlinear, nonlocal Smoluchowski equations for the polymer and nanorod. We then implement the theory to infer rheological properties and predict mesoscale morphologies in fully coupled plane shear flows. Our numerical study focuses on the mesoscale morphology development with respect to the surface contact interaction due to the pretreated surface properties of the nanorods, extending our studies on monodomain polymer-nanorod composites [16]. We find that surface contact interaction dominates the mesoscopic morphology and thereby corresponding rheological properties. When the nanorod favors parallel alignment with the polymer in the host matrix, the only globally stable state is the flow-aligning steady state. When the nanorod prefers to align orthogonally to the polymer in the matrix, however, spatially inhomogeneous structures, time-dependent homogeneous structures, and various spatial-temporal structures emerge in different regimes of the model parameter space and versus strength of the bulk imposed shear. Effective rheological features of the inhomogeneous morphologies are also predicted by the theory.
Citation: Guanghua Ji, M. Gregory Forest, Qi Wang. Structure formation in sheared polymer-rod nanocomposites. Discrete and Continuous Dynamical Systems - S, 2015, 8 (2) : 341-379. doi: 10.3934/dcdss.2015.8.341
##### References:
 [1] R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, 2nd edition, Wiley-Interscience, 1987. [2] A. V. Bhave, Kinetic Theory for Dilute and Concentrated Polymer Solutions: Study of Nonhomogeneous Effects, Ph. D. Thesis, MIT, 1992. [3] W. Brostow, T. S. Dziemianowicz, M. Hess and R. Kosfeld, Blending of Polymer Liquid Crystals with Engineering Polymers: The importance of Phse Diagrams, in Liquid Crystalline Polymers, (eds. R. A. Weiss and C. K. Ober), American Chemical Society, Washington, DC, (1990), 402-415. [4] G. P. Crawford and S. Zummer, Liquid Crystals in Complex Geometries Formed by Polymer and Porous Networks, Taylor & Francis, London, 1996. [5] P. G. DeGennes, Dynamics of Fluctuations and Spinodal Decomposition in Polymer Blends, J. Chem. Phys., 72 (1980), 4756-4763. doi: 10.1063/1.439809. [6] P. G. DeGennes and J. Prost, The Physics of Liquid Crystals, 2nd edition, Oxford University Press, UK, 1993. [7] J. K. G. Dhont and W. J. Briels, Stresses in inhomogeneous suspensions, J. Chem. Phys., 117 (2002), 3992-3999. doi: 10.1063/1.1495842. [8] J. K. G. Dhont and W. J. Briels, Inhomogeneous suspensions of rigid rods in flow, J. Chem. Phys., 118 (2003), 1466-1478. doi: 10.1063/1.1528912. [9] J. K. G. Dhont and W. J. Briels, Viscoelasticity of suspensions of long, rigid rods, Colloids and Surfaces A: Physicochem Eng. Aspects, 213 (2003), 131-156. doi: 10.1016/S0927-7757(02)00508-3. [10] J. K. G. Dhont, M. P. G. vanBruggen and W. J. Briels, Long-time self-diffusion of rigid rod at low concentration: A variational approach, Macromolecules, 32 (1999), 3809-3816. doi: 10.1021/ma981765i. [11] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, UK, 1986. [12] J. J. Feng, G. Sgalari and L. G. Leal, A Theory for Flowing Nematic Polymers with Orientational Distortion, J. Rheol., 44 (2000), 1085-1101. doi: 10.1122/1.1289278. [13] Y. Dzenis, MATERIALS SCIENCE: Structural nanocomposites, Science, 319 (2008), 419-420. doi: 10.1126/science.1151434. [14] W. E and P. Palffy-Muhoray, Phase Separation in Incompressible e Systems, Phys. Rev. E, 55 (1997), 3844-3846. [15] H. Eslami, M. Grmela and M. Bousmina, A mesoscopic rheological model of polymer/layered silicate nanocomposites, J. Rhol., 51 (2007), 1189-1222. doi: 10.1122/1.2790461. [16] M. G. Forest, Q. Liao and Q. Wang, A 2-D Kinetic Theory for Flows of Monodomain Polymer-Rod Nanocomposites, Commun. Comput. Phys., 7 (2009), 250-282. doi: 10.4208/cicp.2009.08.204. [17] M. G. Forest and Q. Wang, Monodomain response of finite-aspect-ratio macromolecules in shear and related linear flows, Rheol. Acta, 42 (2003), 20-46. [18] M. G. Forest, R. Zhou and Q. Wang, Chaotic boundaries of nematic polymers in mixed shear and extensional flows, Phys. Rev. Lett., 93 (2004), 088301-088305. [19] M. G. Forest, Q. Wang and R. Zhou, The flow-phase diagram of Doi-Hess theory for sheared nematic polymers II: Finite shear rates, Rheol. Acta, 44 (2004), 80-93. doi: 10.1007/s00397-004-0380-9. [20] G. Forest and Q. Wang, Hydrodynamic theories for mixture of polymers and rodlike liquid crystalline polymers, Phys. Rev. E, 72 (2005), 041805. doi: 10.1103/PhysRevE.72.041805. [21] A. R. Khokhlov and A. N. Semenov, Liquid-crystalline ordering in solutions of semiflexible macromolecules with rotational-isomeric flexibility, Macromolecules, 17 (1984), 2678-2685. doi: 10.1021/ma00142a040. [22] R. G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, 1988. [23] L. C. Polymers, Report of the Committee on Liquid Crystalline Polymers, National Academic Press, 1990. [24] A. J. Liu and G. H. Fredrickson, Phase separation kinetics of rod/coil mixtures, Macromolecules, 29 (1996), 8000-8009. doi: 10.1021/ma960796f. [25] D. Long and D. C. Morse, A rouse-like model of liquid crystalline polymer melts: Director dynamics and linear viscoelasticity, J. Rheol., 46 (2002), 49-92. doi: 10.1122/1.1423313. [26] T. C. Lubensky and P. M. Chaikin, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, 1995. [27] P. A. Mirau, J. L. Serres, D. Jacobs, P. H. Garrett and R. A. Vaia, Structure and dynamics of surfactant interfaces in organically modified clays, J. Phys. Chem. B, 112 (2008), 10544-10551. doi: 10.1021/jp801479h. [28] C. Muratov and W. E, Theory of phase separation kinetics in polymer-liquid crystal system, J. Chem. Phys., 116 (2002), 4723-4734. doi: 10.1063/1.1426411. [29] M. Rajabian, C. Dubois and M. Grmela, Suspensions of semiflexible fibers in polymeric fluids: Rheology and thermodynamics, Rheol. Acta, 44 (2005), 521-535. doi: 10.1007/s00397-005-0434-7. [30] A. V. Richard and H. D. Wagner, Framework for nanocomposites, Materials Today, 7 (2004), 32-37. [31] R. A. Vaia, Polymer Nanocomposites Open a New Dimension for Plastics and Composites, DTIC Report, 2005. [32] R. A. Vaia, Nanocomposites: Remote-controlled actuators, Nature Materials, 4 (2005), 429-430. doi: 10.1038/nmat1400. [33] H. D. Wagner and R. A. Vaia, Nanocomposites: Issues at the interface, Materials Today, 7 (2004), 38-42. doi: 10.1016/S1369-7021(04)00507-3. [34] Q. Wang, A hydrodynamic theory of nematic liquid crystalline polymers of different configurations, J. Chem. Phys., 116 (2002), 9120-9136. [35] Q. Wang, W. E, C. Liu and P. Zhang, Kinetic theories for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential, Phys. Rev. E, 65 (2002), 051504. doi: 10.1103/PhysRevE.65.051504. [36] K. I. Winey and R. A. Vaia, Polymer nanocomposites, MRS bulletin, 32 (2007), 314-322. doi: 10.1557/mrs2007.229. [37] D. Wu, C. Zhou, Z. Hong, D. Mao and Z. Bian, Study on rheological behaviour of poly(butylene terephthalate)/montmorillonite nanocomposites, Eur. Polym. J., 41 (2005), 2199-2207. doi: 10.1016/j.eurpolymj.2005.03.005. [38] J. Zhao, A. B. Morgan and J. D. Harris, Rheological characterization of polystyreneclay nanocomposites to compare the degree of exfoliation and dispersion, Polymer, 46 (2005), 86418660.

show all references

##### References:
 [1] R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids, 2nd edition, Wiley-Interscience, 1987. [2] A. V. Bhave, Kinetic Theory for Dilute and Concentrated Polymer Solutions: Study of Nonhomogeneous Effects, Ph. D. Thesis, MIT, 1992. [3] W. Brostow, T. S. Dziemianowicz, M. Hess and R. Kosfeld, Blending of Polymer Liquid Crystals with Engineering Polymers: The importance of Phse Diagrams, in Liquid Crystalline Polymers, (eds. R. A. Weiss and C. K. Ober), American Chemical Society, Washington, DC, (1990), 402-415. [4] G. P. Crawford and S. Zummer, Liquid Crystals in Complex Geometries Formed by Polymer and Porous Networks, Taylor & Francis, London, 1996. [5] P. G. DeGennes, Dynamics of Fluctuations and Spinodal Decomposition in Polymer Blends, J. Chem. Phys., 72 (1980), 4756-4763. doi: 10.1063/1.439809. [6] P. G. DeGennes and J. Prost, The Physics of Liquid Crystals, 2nd edition, Oxford University Press, UK, 1993. [7] J. K. G. Dhont and W. J. Briels, Stresses in inhomogeneous suspensions, J. Chem. Phys., 117 (2002), 3992-3999. doi: 10.1063/1.1495842. [8] J. K. G. Dhont and W. J. Briels, Inhomogeneous suspensions of rigid rods in flow, J. Chem. Phys., 118 (2003), 1466-1478. doi: 10.1063/1.1528912. [9] J. K. G. Dhont and W. J. Briels, Viscoelasticity of suspensions of long, rigid rods, Colloids and Surfaces A: Physicochem Eng. Aspects, 213 (2003), 131-156. doi: 10.1016/S0927-7757(02)00508-3. [10] J. K. G. Dhont, M. P. G. vanBruggen and W. J. Briels, Long-time self-diffusion of rigid rod at low concentration: A variational approach, Macromolecules, 32 (1999), 3809-3816. doi: 10.1021/ma981765i. [11] M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, UK, 1986. [12] J. J. Feng, G. Sgalari and L. G. Leal, A Theory for Flowing Nematic Polymers with Orientational Distortion, J. Rheol., 44 (2000), 1085-1101. doi: 10.1122/1.1289278. [13] Y. Dzenis, MATERIALS SCIENCE: Structural nanocomposites, Science, 319 (2008), 419-420. doi: 10.1126/science.1151434. [14] W. E and P. Palffy-Muhoray, Phase Separation in Incompressible e Systems, Phys. Rev. E, 55 (1997), 3844-3846. [15] H. Eslami, M. Grmela and M. Bousmina, A mesoscopic rheological model of polymer/layered silicate nanocomposites, J. Rhol., 51 (2007), 1189-1222. doi: 10.1122/1.2790461. [16] M. G. Forest, Q. Liao and Q. Wang, A 2-D Kinetic Theory for Flows of Monodomain Polymer-Rod Nanocomposites, Commun. Comput. Phys., 7 (2009), 250-282. doi: 10.4208/cicp.2009.08.204. [17] M. G. Forest and Q. Wang, Monodomain response of finite-aspect-ratio macromolecules in shear and related linear flows, Rheol. Acta, 42 (2003), 20-46. [18] M. G. Forest, R. Zhou and Q. Wang, Chaotic boundaries of nematic polymers in mixed shear and extensional flows, Phys. Rev. Lett., 93 (2004), 088301-088305. [19] M. G. Forest, Q. Wang and R. Zhou, The flow-phase diagram of Doi-Hess theory for sheared nematic polymers II: Finite shear rates, Rheol. Acta, 44 (2004), 80-93. doi: 10.1007/s00397-004-0380-9. [20] G. Forest and Q. Wang, Hydrodynamic theories for mixture of polymers and rodlike liquid crystalline polymers, Phys. Rev. E, 72 (2005), 041805. doi: 10.1103/PhysRevE.72.041805. [21] A. R. Khokhlov and A. N. Semenov, Liquid-crystalline ordering in solutions of semiflexible macromolecules with rotational-isomeric flexibility, Macromolecules, 17 (1984), 2678-2685. doi: 10.1021/ma00142a040. [22] R. G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, 1988. [23] L. C. Polymers, Report of the Committee on Liquid Crystalline Polymers, National Academic Press, 1990. [24] A. J. Liu and G. H. Fredrickson, Phase separation kinetics of rod/coil mixtures, Macromolecules, 29 (1996), 8000-8009. doi: 10.1021/ma960796f. [25] D. Long and D. C. Morse, A rouse-like model of liquid crystalline polymer melts: Director dynamics and linear viscoelasticity, J. Rheol., 46 (2002), 49-92. doi: 10.1122/1.1423313. [26] T. C. Lubensky and P. M. Chaikin, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, 1995. [27] P. A. Mirau, J. L. Serres, D. Jacobs, P. H. Garrett and R. A. Vaia, Structure and dynamics of surfactant interfaces in organically modified clays, J. Phys. Chem. B, 112 (2008), 10544-10551. doi: 10.1021/jp801479h. [28] C. Muratov and W. E, Theory of phase separation kinetics in polymer-liquid crystal system, J. Chem. Phys., 116 (2002), 4723-4734. doi: 10.1063/1.1426411. [29] M. Rajabian, C. Dubois and M. Grmela, Suspensions of semiflexible fibers in polymeric fluids: Rheology and thermodynamics, Rheol. Acta, 44 (2005), 521-535. doi: 10.1007/s00397-005-0434-7. [30] A. V. Richard and H. D. Wagner, Framework for nanocomposites, Materials Today, 7 (2004), 32-37. [31] R. A. Vaia, Polymer Nanocomposites Open a New Dimension for Plastics and Composites, DTIC Report, 2005. [32] R. A. Vaia, Nanocomposites: Remote-controlled actuators, Nature Materials, 4 (2005), 429-430. doi: 10.1038/nmat1400. [33] H. D. Wagner and R. A. Vaia, Nanocomposites: Issues at the interface, Materials Today, 7 (2004), 38-42. doi: 10.1016/S1369-7021(04)00507-3. [34] Q. Wang, A hydrodynamic theory of nematic liquid crystalline polymers of different configurations, J. Chem. Phys., 116 (2002), 9120-9136. [35] Q. Wang, W. E, C. Liu and P. Zhang, Kinetic theories for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential, Phys. Rev. E, 65 (2002), 051504. doi: 10.1103/PhysRevE.65.051504. [36] K. I. Winey and R. A. Vaia, Polymer nanocomposites, MRS bulletin, 32 (2007), 314-322. doi: 10.1557/mrs2007.229. [37] D. Wu, C. Zhou, Z. Hong, D. Mao and Z. Bian, Study on rheological behaviour of poly(butylene terephthalate)/montmorillonite nanocomposites, Eur. Polym. J., 41 (2005), 2199-2207. doi: 10.1016/j.eurpolymj.2005.03.005. [38] J. Zhao, A. B. Morgan and J. D. Harris, Rheological characterization of polystyreneclay nanocomposites to compare the degree of exfoliation and dispersion, Polymer, 46 (2005), 86418660.
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