American Institute of Mathematical Sciences

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

Structure formation in sheared polymer-rod nanocomposites

Guanghua Ji 1, , M. Gregory Forest 2, and  Qi Wang 3,

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.
Keywords: Shear flows, structure formation, finite difference methods., polymer-rod nanocomposites.
Mathematics Subject Classification: Primary: 76A05, 76T20; Secondary: 74E30, 74S2.
Citation: Guanghua Ji, M. Gregory Forest, Qi Wang. Structure formation in sheared polymer-rod nanocomposites. Discrete & 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, (1987).   Google Scholar

[2]

A. V. Bhave, Kinetic Theory for Dilute and Concentrated Polymer Solutions: Study of Nonhomogeneous Effects,, Ph. D. Thesis, (1992).   Google Scholar

[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, (1990), 402.   Google Scholar

[4]

G. P. Crawford and S. Zummer, Liquid Crystals in Complex Geometries Formed by Polymer and Porous Networks,, Taylor & Francis, (1996).   Google Scholar

[5]

P. G. DeGennes, Dynamics of Fluctuations and Spinodal Decomposition in Polymer Blends,, J. Chem. Phys., 72 (1980), 4756.  doi: 10.1063/1.439809.  Google Scholar

[6]

P. G. DeGennes and J. Prost, The Physics of Liquid Crystals,, 2nd edition, (1993).   Google Scholar

[7]

J. K. G. Dhont and W. J. Briels, Stresses in inhomogeneous suspensions,, J. Chem. Phys., 117 (2002), 3992.  doi: 10.1063/1.1495842.  Google Scholar

[8]

J. K. G. Dhont and W. J. Briels, Inhomogeneous suspensions of rigid rods in flow,, J. Chem. Phys., 118 (2003), 1466.  doi: 10.1063/1.1528912.  Google Scholar

[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.  doi: 10.1016/S0927-7757(02)00508-3.  Google Scholar

[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.  doi: 10.1021/ma981765i.  Google Scholar

[11]

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics,, Oxford University Press, (1986).   Google Scholar

[12]

J. J. Feng, G. Sgalari and L. G. Leal, A Theory for Flowing Nematic Polymers with Orientational Distortion,, J. Rheol., 44 (2000), 1085.  doi: 10.1122/1.1289278.  Google Scholar

[13]

Y. Dzenis, MATERIALS SCIENCE: Structural nanocomposites,, Science, 319 (2008), 419.  doi: 10.1126/science.1151434.  Google Scholar

[14]

W. E and P. Palffy-Muhoray, Phase Separation in Incompressible e Systems,, Phys. Rev. E, 55 (1997), 3844.   Google Scholar

[15]

H. Eslami, M. Grmela and M. Bousmina, A mesoscopic rheological model of polymer/layered silicate nanocomposites,, J. Rhol., 51 (2007), 1189.  doi: 10.1122/1.2790461.  Google Scholar

[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.  doi: 10.4208/cicp.2009.08.204.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s00397-004-0380-9.  Google Scholar

[20]

G. Forest and Q. Wang, Hydrodynamic theories for mixture of polymers and rodlike liquid crystalline polymers,, Phys. Rev. E, 72 (2005).  doi: 10.1103/PhysRevE.72.041805.  Google Scholar

[21]

A. R. Khokhlov and A. N. Semenov, Liquid-crystalline ordering in solutions of semiflexible macromolecules with rotational-isomeric flexibility,, Macromolecules, 17 (1984), 2678.  doi: 10.1021/ma00142a040.  Google Scholar

[22]

R. G. Larson, Constitutive Equations for Polymer Melts and Solutions,, Butterworths, (1988).   Google Scholar

[23]

L. C. Polymers, Report of the Committee on Liquid Crystalline Polymers,, National Academic Press, (1990).   Google Scholar

[24]

A. J. Liu and G. H. Fredrickson, Phase separation kinetics of rod/coil mixtures,, Macromolecules, 29 (1996), 8000.  doi: 10.1021/ma960796f.  Google Scholar

[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.  doi: 10.1122/1.1423313.  Google Scholar

[26]

T. C. Lubensky and P. M. Chaikin, Principles of Condensed Matter Physics,, Cambridge University Press, (1995).   Google Scholar

[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.  doi: 10.1021/jp801479h.  Google Scholar

[28]

C. Muratov and W. E, Theory of phase separation kinetics in polymer-liquid crystal system,, J. Chem. Phys., 116 (2002), 4723.  doi: 10.1063/1.1426411.  Google Scholar

[29]

M. Rajabian, C. Dubois and M. Grmela, Suspensions of semiflexible fibers in polymeric fluids: Rheology and thermodynamics,, Rheol. Acta, 44 (2005), 521.  doi: 10.1007/s00397-005-0434-7.  Google Scholar

[30]

A. V. Richard and H. D. Wagner, Framework for nanocomposites,, Materials Today, 7 (2004), 32.   Google Scholar

[31]

R. A. Vaia, Polymer Nanocomposites Open a New Dimension for Plastics and Composites,, DTIC Report, (2005).   Google Scholar

[32]

R. A. Vaia, Nanocomposites: Remote-controlled actuators,, Nature Materials, 4 (2005), 429.  doi: 10.1038/nmat1400.  Google Scholar

[33]

H. D. Wagner and R. A. Vaia, Nanocomposites: Issues at the interface,, Materials Today, 7 (2004), 38.  doi: 10.1016/S1369-7021(04)00507-3.  Google Scholar

[34]

Q. Wang, A hydrodynamic theory of nematic liquid crystalline polymers of different configurations,, J. Chem. Phys., 116 (2002), 9120.   Google Scholar

[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).  doi: 10.1103/PhysRevE.65.051504.  Google Scholar

[36]

K. I. Winey and R. A. Vaia, Polymer nanocomposites,, MRS bulletin, 32 (2007), 314.  doi: 10.1557/mrs2007.229.  Google Scholar

[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.  doi: 10.1016/j.eurpolymj.2005.03.005.  Google Scholar

[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).   Google Scholar

show all references

References:
[1]

R. B. Bird, R. C. Armstrong and O. Hassager, Dynamics of Polymeric Liquids,, 2nd edition, (1987).   Google Scholar

[2]

A. V. Bhave, Kinetic Theory for Dilute and Concentrated Polymer Solutions: Study of Nonhomogeneous Effects,, Ph. D. Thesis, (1992).   Google Scholar

[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, (1990), 402.   Google Scholar

[4]

G. P. Crawford and S. Zummer, Liquid Crystals in Complex Geometries Formed by Polymer and Porous Networks,, Taylor & Francis, (1996).   Google Scholar

[5]

P. G. DeGennes, Dynamics of Fluctuations and Spinodal Decomposition in Polymer Blends,, J. Chem. Phys., 72 (1980), 4756.  doi: 10.1063/1.439809.  Google Scholar

[6]

P. G. DeGennes and J. Prost, The Physics of Liquid Crystals,, 2nd edition, (1993).   Google Scholar

[7]

J. K. G. Dhont and W. J. Briels, Stresses in inhomogeneous suspensions,, J. Chem. Phys., 117 (2002), 3992.  doi: 10.1063/1.1495842.  Google Scholar

[8]

J. K. G. Dhont and W. J. Briels, Inhomogeneous suspensions of rigid rods in flow,, J. Chem. Phys., 118 (2003), 1466.  doi: 10.1063/1.1528912.  Google Scholar

[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.  doi: 10.1016/S0927-7757(02)00508-3.  Google Scholar

[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.  doi: 10.1021/ma981765i.  Google Scholar

[11]

M. Doi and S. F. Edwards, The Theory of Polymer Dynamics,, Oxford University Press, (1986).   Google Scholar

[12]

J. J. Feng, G. Sgalari and L. G. Leal, A Theory for Flowing Nematic Polymers with Orientational Distortion,, J. Rheol., 44 (2000), 1085.  doi: 10.1122/1.1289278.  Google Scholar

[13]

Y. Dzenis, MATERIALS SCIENCE: Structural nanocomposites,, Science, 319 (2008), 419.  doi: 10.1126/science.1151434.  Google Scholar

[14]

W. E and P. Palffy-Muhoray, Phase Separation in Incompressible e Systems,, Phys. Rev. E, 55 (1997), 3844.   Google Scholar

[15]

H. Eslami, M. Grmela and M. Bousmina, A mesoscopic rheological model of polymer/layered silicate nanocomposites,, J. Rhol., 51 (2007), 1189.  doi: 10.1122/1.2790461.  Google Scholar

[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.  doi: 10.4208/cicp.2009.08.204.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/s00397-004-0380-9.  Google Scholar

[20]

G. Forest and Q. Wang, Hydrodynamic theories for mixture of polymers and rodlike liquid crystalline polymers,, Phys. Rev. E, 72 (2005).  doi: 10.1103/PhysRevE.72.041805.  Google Scholar

[21]

A. R. Khokhlov and A. N. Semenov, Liquid-crystalline ordering in solutions of semiflexible macromolecules with rotational-isomeric flexibility,, Macromolecules, 17 (1984), 2678.  doi: 10.1021/ma00142a040.  Google Scholar

[22]

R. G. Larson, Constitutive Equations for Polymer Melts and Solutions,, Butterworths, (1988).   Google Scholar

[23]

L. C. Polymers, Report of the Committee on Liquid Crystalline Polymers,, National Academic Press, (1990).   Google Scholar

[24]

A. J. Liu and G. H. Fredrickson, Phase separation kinetics of rod/coil mixtures,, Macromolecules, 29 (1996), 8000.  doi: 10.1021/ma960796f.  Google Scholar

[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.  doi: 10.1122/1.1423313.  Google Scholar

[26]

T. C. Lubensky and P. M. Chaikin, Principles of Condensed Matter Physics,, Cambridge University Press, (1995).   Google Scholar

[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.  doi: 10.1021/jp801479h.  Google Scholar

[28]

C. Muratov and W. E, Theory of phase separation kinetics in polymer-liquid crystal system,, J. Chem. Phys., 116 (2002), 4723.  doi: 10.1063/1.1426411.  Google Scholar

[29]

M. Rajabian, C. Dubois and M. Grmela, Suspensions of semiflexible fibers in polymeric fluids: Rheology and thermodynamics,, Rheol. Acta, 44 (2005), 521.  doi: 10.1007/s00397-005-0434-7.  Google Scholar

[30]

A. V. Richard and H. D. Wagner, Framework for nanocomposites,, Materials Today, 7 (2004), 32.   Google Scholar

[31]

R. A. Vaia, Polymer Nanocomposites Open a New Dimension for Plastics and Composites,, DTIC Report, (2005).   Google Scholar

[32]

R. A. Vaia, Nanocomposites: Remote-controlled actuators,, Nature Materials, 4 (2005), 429.  doi: 10.1038/nmat1400.  Google Scholar

[33]

H. D. Wagner and R. A. Vaia, Nanocomposites: Issues at the interface,, Materials Today, 7 (2004), 38.  doi: 10.1016/S1369-7021(04)00507-3.  Google Scholar

[34]

Q. Wang, A hydrodynamic theory of nematic liquid crystalline polymers of different configurations,, J. Chem. Phys., 116 (2002), 9120.   Google Scholar

[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).  doi: 10.1103/PhysRevE.65.051504.  Google Scholar

[36]

K. I. Winey and R. A. Vaia, Polymer nanocomposites,, MRS bulletin, 32 (2007), 314.  doi: 10.1557/mrs2007.229.  Google Scholar

[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.  doi: 10.1016/j.eurpolymj.2005.03.005.  Google Scholar

[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).   Google Scholar

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