A Guide for Hydrodynamic Reinforcement Effect in Nanoparticle-filled Polymers

References

• D. Feldman, Elastomer nanocomposite; properties, J. Macromole. Sci. Part A 49, 784–793 (2012).
   Web of Science ®Google Scholar
• R. Sengupta, S. Chakraborty, S. Bandyopadhyay, S. Dasgupta, R. Mukhopadhyay, K. Auddy, and A. S. Deuri, A short review on rubber/clay nanocomposites with emphasis on mechanical properties, Polym. Eng. Sci. 47, 1956–1974 (2007).
   Web of Science ®Google Scholar
• J. Karger-Kocsis and C. M. Wu, Thermoset rubber/layered silicate nanocomposites. Status and future trends, Polym. Eng. Sci. 44, 1083–1093 (2004).
   Web of Science ®Google Scholar
• R. S. Sinha and M. Okamoto, Polymer/layered silicate nanocomposites: a review from preparation to processing, Prog. Polym. Sci. 28, 1539–1641 (2003).
   Web of Science ®Google Scholar
• G. Allegra, G. Raos, and M. Vacatello, Theories and simulations of polymer-based nanocomposites: From chain statistics to reinforcement, Prog. Polym. Sci. 33, 683–731 (2008).
   Web of Science ®Google Scholar
• Y. Zhang, Y. Song, and Q. Zheng, Mechanical and thermal properties of nanosized titanium dioxide filled rigid poly(vinyl chloride), Chin. J. Polym. Sci. 31, 325–332 (2013).
   Web of Science ®Google Scholar
• Z. Liu, J. Wu, H. Yang, Y. Song, and Q. Zheng, Melt compounding and mechanical properties of polyphenylene sulfide/low melting-temperature glass composites, Acta Polym. Sin. 141–149 (2014).
   Web of Science ®Google Scholar
• A. A. Gavrilov, A. V. Chertovich, P. G. Khalatur, and A. R. Khokhlov, Study of the mechanisms of filler reinforcement in elastomer nanocomposites, Macromolecules 47, 5400–5408 (2014).
   Web of Science ®Google Scholar
• V. Mansard and A. Colin, Local and non local rheology of concentrated particles, Soft Matt. 8, 4025–4043 (2012).
   Web of Science ®Google Scholar
• N. J. Balmforth, I. A. Frigaard, and G. Ovarlez, Yielding to stress: Recent developments in viscoplastic fluid mechanics, Annu. Rev. Fluid. Mech. 46, 121–146 (2014).
   Web of Science ®Google Scholar
• R. Krishnamoorti and K. Yurekli, Rheology of polymer layered silicate nanocomposites, Curr. Opin. Colloid In 6, 464–470 (2001).
   Web of Science ®Google Scholar
• R. S. Sinha, Rheology of polymer/layered silicate nanocomposites, J. Ind. Eng. Chem. 12, 811–842 (2006).
   Web of Science ®Google Scholar
• H. J. Choi and S. S. Ray, A review on melt-state viscoelastic properties of polymer nanocomposites, J. Nanosci. Nanotechnol. 11, 8421–8449 (2011).
   PubMed Web of Science ®Google Scholar
• F. J. Galindo-Rosales, P. Moldenaers, and J. Vermant, Assessment of the dispersion quality in polymer nanocomposites by rheological methods, Macromol. Mater. Eng. 296, 331–340 (2011).
   Web of Science ®Google Scholar
• R. Hill, A self-consistent mechanics of composite materials, J. Mech. Phys. Solids 13, 213–222 (1965).
   Web of Science ®Google Scholar
• J. S. Chong, E. B. Christiansen, and A. D. Baer, Rheology of concentrated suspensions, J. Appl. Polym. Sci. 15, 2007–2021 (1971).
   Web of Science ®Google Scholar
• J. Bicerano, J. F. Douglas, and D. A. Brune, Model for the viscosity of particle dispersions, J. Macromole. Sci. Part C 39, 561–642 (1999).
   Google Scholar
• A. I. Leonov, On the rheology of filled polymers, J. Rheol. 34, 1039–1068 (1990).
   Web of Science ®Google Scholar
• X. Chateau, G. Ovarlez, and K. L. Trung, Homogenization approach to the behavior of suspensions of noncolloidal particles in yield stress fluids, J. Rheol. 52, 489–506 (2008).
   Web of Science ®Google Scholar
• S. Jiang, G. Li, Y. Lu, and Z. Wu, Advances in the rheology of polymer-based clay nanocomposite, Polym. Bull. 15, 51–53 (2002).
   Google Scholar
• K. Saalwächter, Microstructure and molecular dynamics of elastomers as studied by advanced low-resolution nuclear magnetic resonance methods, Rubber Chem. Technol. 85, 350–386 (2012).
   Web of Science ®Google Scholar
• S. Bredeau, S. Peeterbroeck, D. Bonduel, M. Alexandre, and P. Dubois, From carbon nanotube coatings to high-performance polymer nanocomposites, Polym. Int. 57, 547–553 (2008).
   Web of Science ®Google Scholar
• K. Chrissopoulou and S. H. Anastasiadis, Polyolefin/layered silicate nanocomposites with functional compatibilizers, Eur. Polym. J. 47, 600–613 (2011).
   Web of Science ®Google Scholar
• X.-L. Xie, Y.-W. Mai, and X.-P. Zhou, Dispersion and alignment of carbon nanotubes in polymer matrix: A review, Mater. Sci. Eng. R: Rep. 49, 89–112 (2005).
   Web of Science ®Google Scholar
• K.-t. Lau, C. Gu, and D. Hui, A critical review on nanotube and nanotube/nanoclay related polymer composite materials, Comps. Part B: Eng. 37, 425–436 (2006).
   Web of Science ®Google Scholar
• L. Bokobza, Multiwall carbon nanotube elastomeric composites: A review, Polymer 48, 4907–4920 (2007).
   Web of Science ®Google Scholar
• J. N. Coleman, U. Khan, W. J. Blau, and Y. K. Gun'ko, Small but strong: A review of the mechanical properties of carbon nanotube–polymer composites, Carbon 44, 1624–1652 (2006).
   Web of Science ®Google Scholar
• S. C. Tjong, Structural and mechanical properties of polymer nanocomposites, Mater. Sci. Eng. R: Rep. 53, 73–197 (2006).
   Web of Science ®Google Scholar
• I. R. Rutgers, Relative viscosity and concentration, Rheol Acta 2, 305–348 (1962).
   Google Scholar
• D. G. Thomas, Transport characteristics of suspension: VIII. A note on the viscosity of Newtonian suspensions of uniform spherical particles, J. Colloid Sci. 20, 267–277 (1965).
   Google Scholar
• D. J. Jeffrey and A. Acrivos, The rheological properties of suspensions of rigid particles, AlChE J. 22, 417–432 (1976).
   Web of Science ®Google Scholar
• J. C. H. Affdl and J. L. Kardos, The Halpin-Tsai equations: A review, Polym. Eng. Sci. 16, 344–352 (1976).
   Web of Science ®Google Scholar
• W. B. Russel, Review of the role of colloidal forces in the rheology of suspensions, J. Rheol. 24, 287–317 (1980).
   Web of Science ®Google Scholar
• A. B. Metzner, Rheology of suspensions in polymeric liquids, J. Rheol. 29, 739–775 (1985).
   Web of Science ®Google Scholar
• A. Y. Malkin, Rheology of filled polymers, Adv. Polym. Sci. 96, 69–97 (1990).
   Web of Science ®Google Scholar
• L. B. Kandyrin and V. N. Kuleznev, Dependence of viscosity on the composition of concentrated dispersions and the free volume concept of disperse systems. In Free Radical Copolimerization Dispersions Glassy State Relaxation, Vol. 103, Springer, Berlin Heidelberg (1992); pp. 103–147.
   Google Scholar
• L. Karásek and M. Sumita, Characterization of dispersion state of filler and polymer-filler interactions in rubber-carbon black composites, J. Mater. Sci. 31, 281–289 (1996).
   Web of Science ®Google Scholar
• P. R. Hornsby, Rheology, compounding and processing of filled thermoplastics, Min. Fillers Thermoplast. 139, 155–217 (1999).
   Google Scholar
• C. L. Tucker and E. Liang, Stiffness predictions for unidirectional short-fiber composites: Review and evaluation, Compos. Sci. Technol. 59, 655–671 (1999).
   Web of Science ®Google Scholar
• M. Klüppel and J. Schramm, A generalized tube model of rubber elasticity and stress softening of filler reinforced elastomer systems, Macromol. Theor. Simul, 9, 742–754 (2000).
   Web of Science ®Google Scholar
• D. Quemada and C. Berli, Energy of interaction in colloids and its implications in rheological modeling, Adv. Colloid Interf. Sci. 98, 51–85 (2002).
   PubMed Web of Science ®Google Scholar
• D. J. Kohls and G. Beaucage, Rational design of reinforced rubber, Curr. Opin. Solid State Mater. Sci. 6, 183–194 (2002).
   Web of Science ®Google Scholar
• G. Heinrich and M. Kluppel, Recent advances in the theory of filler networking in elastomers, Filled Elastom. Drug Deliv. Syst. 160, 1–44 (2002).
   Google Scholar
• G. Heinrich, M. Kluppel, and T. A. Vilgis, Reinforcement of elastomers, Curr. Opin. Solid St. M 6, 195–203 (2002).
   Web of Science ®Google Scholar
• J. L. Leblanc, Rubber–filler interactions and rheological properties in filled compounds, Prog. Polym. Sci. 27, 627–687 (2002).
   Web of Science ®Google Scholar
• W. Pabst, Fundamental considerations on suspension rheology, Ceram-silikaty 48, 6–13 (2004).
   Web of Science ®Google Scholar
• G. M. Brown and F. Ellyin, Assessing the predictive capability of two-phase models for the mechanical behavior of alumina/epoxy nanocomposites, J. Appl. Polym. Sci. 98, 869–879 (2005).
   Web of Science ®Google Scholar
• J. E. Mark, R. Abou-Hussein, T. Z. Sen, and A. Kloczkowski, Some simulations on filler reinf,orcement in elastomers, Polymer 46, 8894–8904 (2005).
   Web of Science ®Google Scholar
• V. A. Buryachenko, A. Roy, K. Lafdi, K. L. Anderson, and S. Chellapilla, Multi-scale mechanics of nanocomposites including interface: Experimental and numerical investigation, Compos. Sci. Technol. 65, 2435–2465 (2005).
   Web of Science ®Google Scholar
• G. Lin, Y. Wu, X. Zhang, Y. Qian, Q. Jia, and L. Zhang, Change of structure and morphology of nanofiller/rubber system during storage period II. Factors and their influences on properties, China Synth. Rubber Industr., 28, 1–7 (2005).
   Google Scholar
• J. Oberdisse, Aggregation of colloidal nanoparticles in polymer matrices, Soft Matt. 2, 29–36 (2006).
   Web of Science ®Google Scholar
• S. Z. Wu and J. E. Mark, Some simulations and theoretical studies on poly(dimethylsiloxane), Polym. Rev. 47, 463–485 (2007).
   Web of Science ®Google Scholar
• X.-Q. Wang and A. S. Mujumdar, A review on nanofluids - part I: theoretical and numerical investigations, Braz. J. Chem. Eng. 25, 613–630 (2008).
   Web of Science ®Google Scholar
• S.-Y. Fu, X.-Q. Feng, B. Lauke, and Y.-W. Mai, Effects of particle size, particle/matrix interface adhesion and particle loading on mechanical properties of particulate–polymer composites, Comps. Part B: Eng. 39, 933–961 (2008).
   Web of Science ®Google Scholar
• J. C. Halpin, Stiffness and expansion estimates for oriented short fiber composites, J. Compos. Mater. 3, 732–734 (1969).
   Web of Science ®Google Scholar
• E. H. Kerner, The elastic and thermo-elastic properties of composite media, Proc. Phys. Soc. London Sect. B 69, 808 (1956).
   Google Scholar
• L. E. Nielsen, Generalized equation for the elastic moduli of composite materials, J. Appl. Phys. 41, 4626–4627 (1970).
   Web of Science ®Google Scholar
• M. Takayanagi, T. Ogata, M. Morikawa, and T. Kai, Polymer composites of rigid and flexible molecules: System of wholly aromatic and aliphatic polyamides, J. Macromol. Sci.-Part B 17, 591–615 (1980).
   Web of Science ®Google Scholar
• L. Bokobza, New developments in rubber reinforcement, Kgk-Kautschuk Gummi Kunststoffe 62, 23–27 (2009).
   Web of Science ®Google Scholar
• B. A. Horri, P. Ranganathan, C. Selomulya, and H. Wang, A new empirical viscosity model for ceramic suspensions, Chem. Eng. Sci. 66, 2798–2806 (2011).
   Web of Science ®Google Scholar
• D. B. Genovese, Shear rheology of hard-sphere, dispersed, and aggregated suspensions, and filler-matrix composites, Adv. Colloid Interf. Sci. 171–172, 1–16 (2012).
   PubMed Web of Science ®Google Scholar
• J. Liu, L. Zhang, D. Cao, J. Shen, and Y. Gao, Computational simulation of elastomer nanocomposites: current progress and future challenges, Rubber Chem. Technol. 85, 450–481 (2012).
   Web of Science ®Google Scholar
• E. Hesami, D. Jelagin, N. Kringos, and B. Birgisson, An empirical framework for determining asphalt mastic viscosity as a function of mineral filler concentration, Constr. Build. Mater. 35, 23–29 (2012).
   Web of Science ®Google Scholar
• A. Papon, T. C. L. Guy, K. Saalwachter, J. Oberdisse, S. Merabia, D. Long, P. Sotta, H. H. Frielinghaus, A. Radulescu, B. Deme, L. Noirez, H. Montes, and F. Lequeux, Studying Model samples to understand mechanical Properties of filled Elastomers, Kgk-Kautschuk Gummi Kunststoffe 66, 52–58 (2013).
   Web of Science ®Google Scholar
• K. Jana, M. Patel, D. Rana, and P. Maiti, Nonlinear viscoelasticity of one dimensional filler reinforced elastomer composites. In Non-Linear Viscoelasticity of Rubber Composites and Nanocomposites, Vol. 264, D. Ponnamma and S. Thomas, Eds., Springer International Publishing (2014); pp. 15–41.
   Google Scholar
• Y. Song and Q. Zheng, Linear rheology of nanofilled polymers, J. Rheol. 59, 155–191 (2015).
   Web of Science ®Google Scholar
• S. Faroughi and C. Huber, A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number, Rheol Acta 54, 85–108 (2015).
   Web of Science ®Google Scholar
• R. Tuinier, T.-H. Fan, and T. Taniguchi, Depletion and the dynamics in colloid–polymer mixtures, Curr. Opin. Colloid In 20, 66–70 (2015).
   Web of Science ®Google Scholar
• C. I. Mendoza and I. Santamaría-Holek, The rheology of hard sphere suspensions at arbitrary volume fractions: An improved differential viscosity model, J. Chem. Phys. 130, 044904 (2009).
   Google Scholar
• C. Mobuchon, P. J. Carreau, and M.-C. Heuzey, Structural analysis of non-aqueous layered silicate suspensions subjected to shear flow, J. Rheol. 53, 1025–1048 (2009).
   Web of Science ®Google Scholar
• M. Soutrenon, V. Michaud, and J. A. E. Manson, Influence of processing and storage on the shear Thickening Properties of Highly Concentrated Monodisperse Silica Particles in Polyethylene Glycol, Appl. Rheol. 23, 287–295 (2013).
   Google Scholar
• A. Mukherjee, D. Sharma, S. S. Chauhan, and H. Singh, Time-dependent and shear-dependent transient viscosity of an alumina suspension, J. Dispersion Sci. Technol. 36, 951–969 (2014).
   Web of Science ®Google Scholar
• M. Van Den Tempel, Rheology of concentrated suspensions, J. Colloid Interf. Sci. 71, 18–20 (1979).
   Web of Science ®Google Scholar
• V. Pryamitsyn and V. Ganesan, Origins of linear viscoelastic behavior of polymer-nanoparticle composites, Macromolecules 39, 844–856 (2006).
   Web of Science ®Google Scholar
• Z. Y. Zhu, T. Thompson, S. Q. Wang, E. D. von Meerwall, and A. Halasa, Investigating linear and nonlinear viscoelastic behavior using model silica-particle-filled polybutadiene, Macromolecules 38, 8816–8824 (2005).
   Web of Science ®Google Scholar
• V. Prasad, V. Trappe, A. D. Dinsmore, P. N. Segre, L. Cipelletti, and D. A. Weitz, Universal features of the fluid to solid transition for attractive colloidal particles, Faraday Discuss. 123, 1–12 (2003).
   PubMed Web of Science ®Google Scholar
• V. Trappe and D. A. Weitz, Scaling of the viscoelasticity of weakly attractive particles, Phys. Rev. Lett. 85, 449–452 (2000).
   PubMed Web of Science ®Google Scholar
• G. Romeo, G. Filippone, A. Fernandez-Nieves, P. Russo, and D. Acierno, Elasticity and dynamics of particle gels in non-Newtonian melts, Rheol. Acta 47, 989–997 (2008).
   Web of Science ®Google Scholar
• Y. Kantor and I. Webman, Elastic properties of random percolating systems, Phys. Rev. Lett. 52, 1891–1894 (1984).
   Web of Science ®Google Scholar
• S. C. Feng and M. Sahimi, Position-space renormalization for elastic percolation networks with bond-bending forces, Phys. Rev. B 31, 1671–1673 (1985).
   PubMed Web of Science ®Google Scholar
• S. Arbabi and M. Sahimi, Elastic properties of three-dimensional percolation networks with stretching and bond-bending forces, Phys. Rev. B 38, 7173–7176 (1988).
   PubMed Web of Science ®Google Scholar
• M. Sahimi and S. Arbabi, Mechanics of disordered solids. II. Percolation on elastic networks with bond-bending forces, Phys. Rev. B 47, 703–712 (1993).
   PubMed Web of Science ®Google Scholar
• L. Benguigui, Lattice and continuum percolation transport exponents - experiments in 2-dimensions, Phys. Rev. B 34, 8176–8178 (1986).
   PubMed Web of Science ®Google Scholar
• L. Li and T. Masuda, Effect of dispersion of particles on viscoelasticity of CaCO3-filled polypropylene melts Polym. Eng. Sci. 30, 841–847 (1990).
   Web of Science ®Google Scholar
• J. X. Ren, A. S. Silva, and R. Krishnamoorti, Linear viscoelasticity of disordered polystyrene-polyisoprene block copolymer based layered-silicate nanocomposites, Macromolecules 33, 3739–3746 (2000).
   Web of Science ®Google Scholar
• M. I. Aranguren, E. Mora, J. V. DeGroot, and C. W. Macosko, Effect of reinforcing fillers on the rheology of polymer melts, J. Rheol. 36, 1165–1182 (1992).
   Web of Science ®Google Scholar
• S. S. Sternstein and A. J. Zhu, Reinforcement mechanism of nanofilled polymer melts as elucidated by nonlinear viscoelastic behavior, Macromolecules 35, 7262–7273 (2002).
   Web of Science ®Google Scholar
• M. A. Osman and A. Atallah, Effect of the particle size on the viscoelastic properties of filled polyethylene, Polymer 47, 2357–2368 (2006).
   Web of Science ®Google Scholar
• A. Yim, R. S. Chahal, and L. E. St. Pierre, The effect of polymer-filler interaction energy on the Tg of filled polymers, J. Colloid Interf. Sci. 43, 583–590 (1973).
   Web of Science ®Google Scholar
• A. S. Sarvestani and E. Jabbari, Modeling the viscoelastic response of suspension of particles in polymer solution: The effect of polymer-particle interactions, Macromol. Theor. Simul. 16, 378–385 (2007).
   Web of Science ®Google Scholar
• R. Krishnamoorti and E. P. Giannelis, Rheology of end-tethered polymer layered silicate nanocomposites, Macromolecules 30, 4097–4102 (1997).
   Web of Science ®Google Scholar
• P. Cassagnau, Melt rheology of organoclay and fumed silica nanocomposites, Polymer 49, 2183–2196 (2008).
   Web of Science ®Google Scholar
• J. Happel and H. Brenner, The viscosity of particulate systems. In Low Reynolds Number Hydrodynamics, Vol. 1, Springer, The Netherlands (1983); pp. 431–473.
   Google Scholar
• S. Ahmed and F. R. Jones, A review of particulate reinforcement theories for polymer composites, J. Mater. Sci. 25, 4933–4942 (1990).
   Web of Science ®Google Scholar
• G. Kraus, Reinforcement of elastomers by carbon black, Adv. Polym. Sci. 8, 155–237 (1971).
   Google Scholar
• M. A. Osman, A. Atallah, T. Schweizer, and H. C. Ottinger, Particle-particle and particle-matrix interactions in calcite filled high-density polyethylene-steady shear, J. Rheol. 48, 1167–1184 (2004).
   Web of Science ®Google Scholar
• A. Einstein, On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat, An.n Phys. 17, 549–560 (1905).
   Web of Science ®Google Scholar
• A. Einstein, Berichtigung zu meiner Arbeit: “Eine neue Bestimmung der Moleküldimensionen”, Ann. Phys. 34, 591–592 (1911).
   Google Scholar
• H. M. Smallwood, Limiting law of the reinforcement of rubber, J. Appl. Phys. 15, 758–766 (1944).
   Google Scholar
• R. D. Sudduth, Modulus evaluation of particulate composites using generalized viscosity model for solutions with suspended particles, J. Appl. Polym. Sci. 54, 1243–1262 (1994).
   Web of Science ®Google Scholar
• Y. Song, Q. Zheng, and Q. Cao, On time-temperature-concentration superposition principle for dynamic rheology of carbon black filled polymers, J. Rheol. 53, 1379–1388 (2009).
   Web of Science ®Google Scholar
• A. Acrivos, Shear-induced particle diffusion in concentrated suspensions of noncolloidal particles, J. Rheol. 39, 813–826 (1995).
   Web of Science ®Google Scholar
• I. Santamaria-Holek and C. I. Mendoza, The rheology of concentrated suspensions of arbitrarily-shaped particles, J. Colloid Interf. Sci. 346, 118–126 (2010).
   PubMed Web of Science ®Google Scholar
• A. M. Wierenga and A. P. Philipse, Low-shear viscosity of isotropic dispersions of (Brownian) rods and fibres; a review of theory and experiments, Colloids Surf. A 137, 355–372 (1998).
   Web of Science ®Google Scholar
• E. Guth, Theory of filler reinforcement, J. Appl. Phys. 16, 20–25 (1945).
   Web of Science ®Google Scholar
• S. Mueller, E. W. Llewellin, and H. M. Mader, The rheology of suspensions of solid particles, Proc. Roy. Soc. London Ser. A Mathem. Phys. Sci. 466, 1201–1228 (2010).
   Web of Science ®Google Scholar
• I. Pliskin and N. Tokita, Bound rubber in elastomers - Analysis of elastomer-filler interaction and its effect on viscosity and modulus of composite systems, J. Appl. Polym. Sci. 16, 473–492 (1972).
   Web of Science ®Google Scholar
• H. A. Barnes, Shear‐thickening (“dilatancy”) in suspensions of nonaggregating solid particles dispersed in newtonian liquids, J. Rheol. 33, 329–366 (1989).
   Web of Science ®Google Scholar
• T. A. Witten, M. Rubinstein, and R. H. Colby, Reinforcement of rubber by fractal aggregates, J. Phys. II France 3, 367–383 (1993).
   Google Scholar
• F. Booth, The electroviscous effect for suspensions of solid spherical particles, Proc. Roy. Soc. London Ser. A Mathem. Phys. Sci. 203, 533–551 (1950).
   Web of Science ®Google Scholar
• H. X. Zhou, Calculation of translational friction and intrinsic viscosity. I. General formulation for arbitrarily shaped particles, Biophys. J. 69, 2286–2297 (1995).
   PubMed Web of Science ®Google Scholar
• C.-J. Lin, J. H. Peery, and W. R. Schowalter, Simple shear flow round a rigid sphere: inertial effects and suspension rheology, J. Fluid Mech. 44, 1–17 (1970).
   Web of Science ®Google Scholar
• P. Cassagnau, Linear viscoelasticity and dynamics of suspensions and molten polymers filled with nanoparticles of different aspect ratios, Polymer 54, 4762–4775 (2013).
   Web of Science ®Google Scholar
• S. Y. Kim and C. F. Zukoski, Super- and sub-Einstein intrinsic viscosities of spherical nanoparticles in concentrated low molecular weight polymer solutions, Soft Matt. 8, 1801–1810 (2012).
   Web of Science ®Google Scholar
• M. Wang and R. J. Hill, Anomalous bulk viscosity of polymer-nanocomposite melts, Soft Matt. 5, 3940–3953 (2009).
   Web of Science ®Google Scholar
• K. Nusser, G. J. Schneider, W. Pyckhout-Hintzen, and D. Richter, Viscosity decrease and reinforcement in polymer–silsesquioxane composites, Macromolecules 44, 7820–7830 (2011).
   Web of Science ®Google Scholar
• A. Tuteja, M. E. Mackay, C. J. Hawker, and B. Van Horn, Effect of ideal, organic nanoparticles on the flow properties of linear polymers:  Non-einstein-like behavior, Macromolecules 38, 8000–8011 (2005).
   Web of Science ®Google Scholar
• M. E. Mackay, T. T. Dao, A. Tuteja, D. L. Ho, B. Van Horn, H.-C. Kim, and C. J. Hawker, Nanoscale effects leading to non-Einstein-like decrease in viscosity, Nat. Mater. 2, 762–766 (2003).
   PubMed Web of Science ®Google Scholar
• R. G. Schmidt, G. V. Gordon, C. c. A. Dreiss, T. Cosgrove, V. J. Krukonis, K. Williams, and P. M. Wetmore, A critical size ratio for viscosity reduction in poly(dimethylsiloxane)-polysilicate nanocomposites, Macromolecules 43, 10143–10151 (2010).
   Web of Science ®Google Scholar
• J. T. Kalathi, G. S. Grest, and S. K. Kumar, Universal viscosity behavior of polymer nanocomposites, Phys. Rev. Lett. 109, 198301 (2012).
   PubMed Web of Science ®Google Scholar
• H. Goldansaz, F. Goharpey, F. Afshar-Taromi, I. Kim, F. J. Stadler, E. van Ruymbeke, and V. Karimkhani, Anomalous rheological behavior of dendritic nanoparticle/linear polymer nanocomposites, Macromolecules 48, 3368–3375 (2015).
   Web of Science ®Google Scholar
• G. D. Smith, D. Bedrov, L. Li, and O. Byutner, A molecular dynamics simulation study of the viscoelastic properties of polymer nanocomposites, J. Chem. Phys. 117, 9478–9489 (2002).
   Web of Science ®Google Scholar
• J. Mewis and N. J. Wagner, Current trends in suspension rheology, J. Non-Newtonian Fluid Mech. 157, 147–150 (2009).
   Web of Science ®Google Scholar
• W. B. Russel and A. P. Gast, Nonequilibrium statistical mechanics of concentrated colloidal dispersions: Hard spheres in weak flows, J. Chem. Phys. 84, 1815–1826 (1986).
   Web of Science ®Google Scholar
• B. Cichocki, M. L. Ekiel-Jeżewska, and E. Wajnryb, Three-particle contribution to effective viscosity of hard-sphere suspensions, J. Chem. Phys. 119, 606–619 (2003).
   Web of Science ®Google Scholar
• C. U. Thomas and M. Muthukumar, Three‐body hydrodynamic effects on viscosity of suspensions of spheres, J. Chem. Phys. 94, 5180–5189 (1991).
   Web of Science ®Google Scholar
• B. Cichocki and B. U. Felderhof, Linear viscoelasticity of semidilute hard-sphere suspensions, Phys. Rev. A 43, 5405–5411 (1991).
   PubMed Web of Science ®Google Scholar
• B. Cichocki and B. U. Felderhof, Linear viscoelasticity of colloidal suspensions, Phys. Rev. A 46, 7723–7732 (1992).
   PubMed Web of Science ®Google Scholar
• G. K. Batchelor and J. T. Green, The determination of the bulk stress in a suspension of spherical particles to order c2, J. Fluid. Mech. 56, 401–427 (1972).
   Web of Science ®Google Scholar
• R. Pasquino, N. Grizzuti, P. L. Maffettone, and F. Greco, Rheology of dilute and semidilute noncolloidal hard sphere suspensions, J. Rheol. 52, 1369–1384 (2008).
   Web of Science ®Google Scholar
• R. S. Simha, A treatment of the viscosity of concentrated suspensions, J. Appl. Phys. 23, 1020–1024 (1952).
   Web of Science ®Google Scholar
• E. Guth and O. Gold, On the hydrodynamical theory of the viscosity of suspensions, Phys. Rev. 53, 322 (1938).
   Google Scholar
• N. Saitô, Concentration dependence of the viscosity of high polymer solutions. I, J. Phys. Soc. Jpn. 5, 4–8 (1950).
   Google Scholar
• M. Schneider, J. Claverie, C. Graillat, and T. F. McKenna, High solids content emulsions. I. A study of the influence of the particle size distribution and polymer concentration on viscosity, J. Appl. Polym. Sci. 84, 1878–1896 (2002).
   Web of Science ®Google Scholar
• D. Quemada, Rheology of concentrated disperse systems and minimum energy dissipation principle, Rheol. Acta 16, 82–94 (1977).
   Web of Science ®Google Scholar
• C. G. de Kruif, E. M. F. van Iersel, A. Vrij, and W. B. Russel, Hard sphere colloidal dispersions: Viscosity as a function of shear rate and volume fraction, J. Chem. Phys. 83, 4717–4725 (1985).
   Web of Science ®Google Scholar
• H. Eggers and P. Schummer, Reinforcement mechanisms in carbon black and silica loaded rubber melts at low stresses, Rubb. Chem. Technol. 69, 253–265 (1996).
   Web of Science ®Google Scholar
• J. G. Oldroyd, The elastic and viscous properties of emulsions and suspensions, Proc. Roy. Soc. London Ser. A. Mathemat. Phys. Sci. 218, 122–132 (1953).
   Web of Science ®Google Scholar
• J. M. Peterson and M. Fixman, Viscosity of polymer solutions, J. Chem. Phys. 39, 2516–2523 (1963).
   Web of Science ®Google Scholar
• J. W. Ju and T. M. Chen, Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities, Acta Mech. 103, 123–144 (1994).
   Web of Science ®Google Scholar
• D. Bedeaux, R. Kapral, and P. Mazur, The effective shear viscosity of a uniform suspension of spheres, Physica A 88, 88–121 (1977).
   Web of Science ®Google Scholar
• C. W. J. Beenakker, The effective viscosity of a concentrated suspension of spheres (and its relation to diffusion), Physica A 128, 48–81 (1984).
   Web of Science ®Google Scholar
• K. F. Freed and M. Muthukumar, Cluster theory for concentration dependence of shear viscosity for suspensions of interacting spheres. I, J. Chem. Phys. 76, 6186–6194 (1982).
   Web of Science ®Google Scholar
• C. U. Thomas and M. Muthukumar, Convergence of multiple scattering series for two‐body hydrodynamic effects on shear viscosity of suspensions of spheres, J. Chem. Phys. 94, 4557–4567 (1991).
   Web of Science ®Google Scholar
• H.-S. Chen and A. Andreas, The effective elastic moduli of composite materials containing spherical inclusions at non–dilute concentrations, Int. J. Solids Struct. 14, 349–364 (1978).
   Web of Science ®Google Scholar
• G. K. Batchelor, The effect of Brownian motion on the bulk stress in a suspension of spherical particles, J. Fluid Mech. 83, 97–117 (1977).
   Web of Science ®Google Scholar
• J. Bergenholtz and N. J. Wagner, The Huggins coefficient for the square-well colloidal fluid, Ind. Eng. Chem. Res. 33, 2391–2397 (1994).
   Web of Science ®Google Scholar
• R. Verberg, I. M. de Schepper, and E. G. D. Cohen, Viscosity of colloidal suspensions, Phys. Rev. E 55, 3143–3158 (1997).
   Web of Science ®Google Scholar
• V. Vand, Viscosity of solutions and suspensions. I. Theory, J. Phys. Colloid Chem. 52, 277–299 (1948).
   PubMedGoogle Scholar
• G. J. Kynch, The effective viscosity of suspensions of spherical particles, Proc. Roy. Soc. London. Ser. A Mathemat. Phys. Sci. 237, 90–116 (1956).
   Web of Science ®Google Scholar
• A. J. Banchio and G. Nägele, Short-time transport properties in dense suspensions: From neutral to charge-stabilized colloidal spheres, J. Chem. Phys. 128, 104903 (2008).
   PubMed Web of Science ®Google Scholar
• E. Guth and O. Gold, On the hydrodynamical theory of the viscosity of suspensions, Phys. Rev. 53, 322 (1938).
   Google Scholar
• S. J. Choi and W. R. Schowalter, Rheological properties of nondilute suspensions of deformable particles, Phys. Fluids, 18, 420–427 (1975).
   Web of Science ®Google Scholar
• R. A. Lionberger, Viscosity of bimodal and polydisperse colloidal suspensions, Phys. Rev. E 65, 061408 (2002).
   Google Scholar
• B. Cichocki and B. U. Felderhof, Diffusion coefficients and effective viscosity of suspensions of sticky hard spheres with hydrodynamic interactions, J. Chem. Phys. 93, 4427–4432 (1990).
   Web of Science ®Google Scholar
• J. Bergenholtz and N. J. Wagner, The Huggins coefficient for the square-well colloidal fluid, Ind. Eng. Chem. Res. 33, 2391–2397 (1994).
   Web of Science ®Google Scholar
• A. T. J. M. Woutersen, J. Mellema, C. Blom, and C. G. de Kruif, Linear viscoelasticity in dispersions of adhesive hard spheres, J. Chem. Phys. 101, 542–553 (1994).
   Web of Science ®Google Scholar
• A. T. J. M. Woutersen and C. G. de Kruif, The viscosity of semidilute, bidisperse suspensions of hard spheres, J. Rheol. 37, 681–693 (1993).
   Web of Science ®Google Scholar
• W. B. Russel, The Huggins coefficient as a means for characterizing suspended particles, J. Chem. Soc. Faraday Trans. 2: Molecul. Chem. Phys. 80, 31–41 (1984).
   Web of Science ®Google Scholar
• R. C. Ball and P. Richmond, Dynamics of colloidal dispersions, Phys. Chem. Liq. 9, 99–116 (1980).
   Web of Science ®Google Scholar
• S. Arrhenius, The viscosity of solutions, Biochem. J. 11, 112–133 (1917).
   PubMedGoogle Scholar
• A. Leviton and A. Leighton, Viscosity relationships in emulsions containing milk fat, J. Phys. Chem. 40, 71–80 (1935).
   Google Scholar
• D. A. G. Bruggeman, Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen, Ann. Phys. 416, 636–664 (1935).
   Google Scholar
• C. H. Hsueh and W. C. J. Wei, Analyses of effective viscosity of suspensions with deformable polydispersed spheres, J. Phys. D: Appl. Phys. 42, 075503 (2009).
   Google Scholar
• H. C. Brinkman, The viscosity of concentrated suspensions and solutions, J. Chem. Phys. 20, 571–571 (1952).
   Web of Science ®Google Scholar
• N.-S. Cheng and A. W.-K. Law, Exponential formula for computing effective viscosity, Powder Technol. 129, 156–160 (2003).
   Web of Science ®Google Scholar
• W. Pabst and E. Gregorová, Derivation of the simplest exponential and power-law relations for the effective tensile modulus of porous ceramics via functional equations, J. Mater. Sci. Lett. 22, 1673–1675 (2003).
   Google Scholar
• J. F. Brady, The rheological behavior of concentrated colloidal dispersions, J. Chem. Phys. 99, 567–581 (1993).
   Web of Science ®Google Scholar
• R. Roscoe, The viscosity of suspensions of rigid spheres, Br. J. Appl. Phys. 3, 267 (1952).
   Web of Science ®Google Scholar
• R. D. Sudduth, A generalized model to predict the viscosity of solutions with suspended particles. I, J. Appl. Polym. Sci. 48, 25–36 (1993).
   Web of Science ®Google Scholar
• R. D. Sudduth, A generalized model to predict the viscosity of solutions with suspended particles. III. Effects of particle interaction and particle size distribution, J. Appl. Polym. Sci. 50, 123–147 (1993).
   Web of Science ®Google Scholar
• R. D. Sudduth, A new method to predict the maximum packing fraction and the viscosity of solutions with a size distribution of suspended particles. II, J. Appl. Polym. Sci. 48, 37–55 (1993).
   Web of Science ®Google Scholar
• R. D. Sudduth, A generalized model to predict the viscosity of solutions with suspended particles. IV. Determination of optimum particle-by-particle volume fractions, J. Appl. Polym. Sci. 52, 985–996 (1994).
   Web of Science ®Google Scholar
• A. J. Poslinski, M. E. Ryan, R. K. Gupta, S. G. Seshadri, and F. J. Frechette, Rheological behavior of filled polymeric systems II. The effect of a bimodal size distribution of particulates, J. Rheol. 32, 751–771 (1988).
   Web of Science ®Google Scholar
• I. M. Krieger and T. J. Dougherty, A mechanism for non-newtonian flow in suspensions of rigid spheres, Trans. Soc. Rheol. 3, 137–152 (1959).
   Web of Science ®Google Scholar
• M. Mooney, The viscosity of a concentrated suspension of spherical particles, J. Colloid Sci. 6, 162–170 (1951).
   Google Scholar
• V. Vand, Viscosity of solutions and suspensions. II. Experimental determination of the viscosity–concentration function of spherical suspensions, J. Phys. Colloid Chem. 52, 300–314 (1948).
   PubMedGoogle Scholar
• J. G. Brodnyan, The concentration dependence of the newtonian viscosity of prolate ellipsoids, Trans. Soc. Rheol. 3, 61–68 (1959).
   Google Scholar
• S. Ahmed and F. R. Jones, A review of particulate reinforcement theories for polymer composites, J. Mater. Sci. 25, 4933–4942 (1990).
   Web of Science ®Google Scholar
• E. Barnea and J. Mizrahi, On the “effective” viscosity of liquid-liquid dispersions, Ind. Eng. Chem. Fundam. 15, 120–125 (1976).
   Web of Science ®Google Scholar
• D. He and N. N. Ekere, Viscosity of concentrated noncolloidal bidisperse suspensions, Rheol. Acta 40, 591–598 (2001).
   Web of Science ®Google Scholar
• T. Hao, Viscosities of liquids, colloidal suspensions, and polymeric systems under zero or non-zero electric field, Adv. Colloid Interf. Sci. 142, 1–19 (2008).
   PubMed Web of Science ®Google Scholar
• E. Ruiz-Reina, F. Carrique, F. J. Rubio-Hernández, A. I. Gómez-Merino, and P. García-Sánchez, Electroviscous effect of moderately concentrated colloidal suspensions, J. Phys. Chem. B, 107, 9528–9534 (2003).
   Web of Science ®Google Scholar
• A. Graham, On the viscosity of suspensions of solid spheres, Appl. Sci. Res. 37, 275–286 (1981).
   Google Scholar
• N. A. Frankel and A. Acrivos, On the viscosity of a concentrated suspension of solid spheres, Chem. Eng. Sci. 22, 847–853 (1967).
   Web of Science ®Google Scholar
• C. I. Mendoza, Effective static and high-frequency viscosities of concentrated suspensions of soft particles, J. Chem. Phys. 135, 054904 (2011).
   Google Scholar
• A. Chougnet, A. Audibert, and M. Moan, Linear and non-linear rheological behaviour of cement and silica suspensions. Effect of polymer addition, Rheol. Acta 46, 793–802 (2007).
   Web of Science ®Google Scholar
• C. Tsenoglou, Scaling concepts in suspension rheology, J. Rheol. 34, 15–24 (1990).
   Web of Science ®Google Scholar
• A. L. Graham, R. D. Steele, and R. B. Bird, Particle clusters in concentrated suspensions. 3. Prediction of suspension viscosity, Ind. Eng. Chem. Fundam. 23, 420–425 (1984).
   Web of Science ®Google Scholar
• T. Gillespie, The effect of aggregation and liquid penetration on the viscosity of dilute suspensions of spherical particles, J. Colloid Sci. 18, 32–40 (1963).
   Google Scholar
• D. I. Lee, The viscosity of concentrated suspensions, Trans. Soc. Rheol. 13, 273–288 (1969).
   Google Scholar
• J. W. Bullard, A. T. Pauli, E. J. Garboczi, and N. S. Martys, A comparison of viscosity–concentration relationships for emulsions, J. Colloid Interf. Sci. 330, 186–193 (2009).
   PubMed Web of Science ®Google Scholar
• T. Dabak and O. Yucel, Shear viscosity behavior of highly concentrated suspensions at low and high shear-rates, Rheol. Acta 25, 527–533 (1986).
   Web of Science ®Google Scholar
• A. Ogawa, H. Yamada, S. Matsuda, K. Okajima and M. Doi, Viscosity equation for concentrated suspensions of charged colloidal particles, J. Rheol. 41, 769–785 (1997).
   Web of Science ®Google Scholar
• H.-S. Chen, Y. Ding, Y. He, and C. Tan, Rheological behaviour of ethylene glycol based titania nanofluids, Chem. Phys. Lett. 444, 333–337 (2007).
   Web of Science ®Google Scholar
• H.-S. Chen, Y. Ding, and C. Tan, Rheological behaviour of nanofluids, New J. Phys. 9, 367 (2007).
   Web of Science ®Google Scholar
• H. J. H. Brouwers, Viscosity of a concentrated suspension of rigid monosized particles, Phys. Rev. E 81, 051402 (2010).
   Google Scholar
• R. S. Farr, Simple heuristic for the viscosity of polydisperse hard spheres, J. Chem. Phys. 141, 214503 (2014).
   PubMed Web of Science ®Google Scholar
• H. Eilers, Die Viskosität von Emulsionen hochviskoser Stoffe als Funktion der Konzentration, Kolloid-Zeitschrift, 97, 313–321 (1941).
   Google Scholar
• R. F. Fedors, Viscosity of Newtonian suspensions, Polymer 16, 305–306 (1975).
   Web of Science ®Google Scholar
• D. Leighton and A. Acrivos, Viscous resuspension, Chem. Eng. Sci. 41, 1377–1384 (1986).
   Web of Science ®Google Scholar
• I. R. Collins, On the viscosity of concentrated aggregated suspensions, J. Colloid Interf. Sci. 178, 361–363 (1996).
   Web of Science ®Google Scholar
• A. Tsutsumi, K. Yoshida, M. Yui, S. Kanamori, and K. Shibata, Shear viscosity behavior of flocculated suspensions, Powder Technol. 78, 165–172 (1994).
   Web of Science ®Google Scholar
• P. Gondret and L. Petit, Dynamic viscosity of macroscopic suspensions of bimodal sized solid spheres, J. Rheol. 41, 1261–1274 (1997).
   Web of Science ®Google Scholar
• B. Dames, B. R. Morrison, and N. Willenbacher, An empirical model predicting the viscosity of highly concentrated, bimodal dispersions with colloidal interactions, Rheol. Acta 40, 434–440 (2001).
   Web of Science ®Google Scholar
• D. Quemada, Rheology of concentrated disperse systems II. A model for non-newtonian shear viscosity in steady flows, Rheol. Acta 17, 632–642 (1978).
   Web of Science ®Google Scholar
• P. Mills and P. Snabre, Apparent viscosity and particle pressure of a concentrated suspension of non-Brownian hard spheres near the jamming transition, Europ. Phys. J. E 30, 309–316 (2009).
   PubMed Web of Science ®Google Scholar
• D.-M. Liu, Particle packing and rheological property of highly-concentrated ceramic suspensions: ϕm determination and viscosity prediction, J. Mater. Sci. 35, 5503–5507 (2000).
   Web of Science ®Google Scholar
• S. H. Maron and P. E. Pierce, Application of ree-eyring generalized flow theory to suspensions of spherical particles, J. Colloid Sci. 11, 80–95 (1956).
   Google Scholar
• T. Kitano, T. Kataoka, and T. Shirota, An empirical equation of the relative viscosity of polymer melts filled with various inorganic fillers, Rheol. Acta 20, 207–209 (1981).
   Web of Science ®Google Scholar
• V. Pryamitsyn and V. Ganesan, A coarse-grained explicit solvent simulation of rheology of colloidal suspensions, J. Chem. Phys. 122, 104906 (2005).
   PubMed Web of Science ®Google Scholar
• A. Dörr, A. Sadiki, and A. Mehdizadeh, A discrete model for the apparent viscosity of polydisperse suspensions including maximum packing fraction, J. Rheol. 57, 743–765 (2013).
   Web of Science ®Google Scholar
• P. Mills, Non-Newtonian behaviour of flocculated suspensions, J. Phys. Lett. 46, 301–309 (1985).
   Google Scholar
• P. Snabre and P. Mills, I. Rheology of weakly flocculated suspensions of rigid particles, J. Phys. III France 6, 1811–1834 (1996).
   Google Scholar
• A. J. Hughes, The Einstein relation between relative viscosity and volume concentration of suspensions of spheres, Nature 173, 1089–1090 (1954).
   Web of Science ®Google Scholar
• D. R. Oliver and S. G. Ward, Relationship between relative viscosity and volume concentration of stable suspensions of spherical particles, Nature 171, 396–397 (1953).
   Web of Science ®Google Scholar
• T. F. Ford, Viscosity-concentration and fluidity-concentration relationships for suspensions of spherical particles in Newtonian liquids, J. Phys. Chem. 64, 1168–1174 (1960).
   Web of Science ®Google Scholar
• T. S. Lundgren, Slow flow through stationary random beds and suspensions of spheres, J. Fluid Mech. 51, 273–299 (1972).
   Web of Science ®Google Scholar
• D. Bedeaux, The effective shear viscosity for two-phase flow, Physica A 121, 345–361 (1983).
   Web of Science ®Google Scholar
• J. Mellema and M. W. M. Willemse, Effective viscosity of dispersions approached by a statistical continuum method, Phys. A 122, 286–312 (1983).
   Web of Science ®Google Scholar
• B. Abedian and M. Kachanov, On the effective viscosity of suspensions, Int. J. Eng. Sci. 48, 962–965 (2010).
   Web of Science ®Google Scholar
• B. Budiansky, On the elastic moduli of some heterogeneous materials, J. Mech. Phys. Solids 13, 223–227 (1965).
   Web of Science ®Google Scholar
• H. J. H. Clercx and P. P. J. M. Schram, High-frequency effective viscosity of hard-sphere suspensions, Phys. Rev. A 45, 860–872 (1992).
   PubMed Web of Science ®Google Scholar
• C. H. Hsueh and W. C. J. Wei, Effective viscosity of semidilute suspensions of rigid ellipsoids, J. Appl. Phys. 107, 024905 (2010).
   Google Scholar
• D. Bedeaux, The effective viscosity for a suspension of spheres, J. Colloid Interf. Sci. 118, 80–86 (1987).
   Web of Science ®Google Scholar
• J. Smeets, G. J. M. Koper, J. P. M. van der Ploeg, and D. Bedeaux, Viscosity of droplet-phase water/AOT/isooctane microemulsions: Solid sphere behavior and aggregation, Langmuir 10, 1387–1392 (1994).
   Web of Science ®Google Scholar
• R. A. Lionberger and W. B. Russel, High frequency modulus of hard sphere colloids, J. Rheol. 38, 1885–1908 (1994).
   Web of Science ®Google Scholar
• J. C. van der Werff, C. G. de Kruif, C. Blom, and J. Mellema, Linear viscoelastic behavior of dense hard-sphere dispersions, Phys. Rev. A 39, 795–807 (1989).
   PubMed Web of Science ®Google Scholar
• A. J. C. Ladd, Hydrodynamic transport coefficients of random dispersions of hard spheres, J. Chem. Phys. 93, 3484–3494 (1990).
   Web of Science ®Google Scholar
• C.-H. Hsueh and P. F. Becher, Effective viscosity of suspensions of spheres, J. Am. Ceram. Soc. 88, 1046–1049 (2005).
   Web of Science ®Google Scholar
• H. Furuse, Viscosity of concentrated solution, Jpn. J. Appl. Phys. 11, 1537 (1972).
   Web of Science ®Google Scholar
• K. Toda and H. Furuse, Extension of Einstein's viscosity equation to that for concentrated dispersions of solutes and particles, J. Biosci. Bioeng. 102, 524–528 (2006).
   PubMed Web of Science ®Google Scholar
• T. Gillespie, The effect of aggregation and particle size distribution on the viscosity of Newtonian suspensions, J. Colloid Interf. Sci. 94, 166–173 (1983).
   Web of Science ®Google Scholar
• S. Hess, Non-Newtonian viscosity and normal pressure differences of simple liquids, Phys. Rev. A. 25, 614–616 (1982).
   Web of Science ®Google Scholar
• A. De Noni Jr, D. E. Garcia, and D. Hotza, A modified model for the viscosity of ceramic suspensions, Ceram. Int. 28, 731–735 (2002).
   Web of Science ®Google Scholar
• I. E. Zarraga, D. A. Hill, and D. T. Leighton, The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids, J. Rheol. 44, 185–220 (2000).
   Web of Science ®Google Scholar
• B. H. A. A. van den Brule and R. J. J. Jongschaap, Modeling of concentrated suspensions, J. Stat. Phys. 62, 1225–1237 (1991).
   Web of Science ®Google Scholar
• G. S. Khodakov, On suspension rheology, Theor. Found. Chem. Eng. 38, 430–439 (2004).
   Web of Science ®Google Scholar
• H. L. Cox, The elasticity and strength of paper and other fibrous materials, Br. J. Appl. Phys. 3, 72–79 (1952).
   Web of Science ®Google Scholar
• B. Paul, Prediction of elastic constants of multiphase materials, Trans. Am. Instit. Min. Metallurg. Eng. 218, 36–41 (1960).
   Web of Science ®Google Scholar
• S.-Y. Fu, X. Hu, and C.-Y. Yue, A new model for the transverse modulus of unidirectional fiber composites, J. Mater. Sci. 33, 4953–4960 (1998).
   Web of Science ®Google Scholar
• S.-Y. Fu, G. Xu, and Y.-W. Mai, On the elastic modulus of hybrid particle/short-fiber/polymer composites, Comps. Part B: Eng. 33, 291–299 (2002).
   Web of Science ®Google Scholar
• Z. Hashin and S. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials, J. Mech. Phys. Solids 11, 127–140 (1963).
   Web of Science ®Google Scholar
• U. J. Counto, The effect of the elastic modulus of the aggregate on the elastic modulus, creep and creep recovery of concrete, Mag. Concr. Res. 16, 129–138 (1964).
   Google Scholar
• O. Ishai and L. J. Cohen, Elastic properties of filled and porous epoxy composites, Int. J. Mech. Sci. 9, 539–546 (1967).
   Google Scholar
• J. Lee and A. F. Yee, Fracture behavior of glass bead filled epoxies: Cleaning process of glass beads, J. Appl. Polym. Sci. 79, 1371–1383 (2001).
   Web of Science ®Google Scholar
• N. Jouault, F. Dalmas, F. Boué, and J. Jestin, Multiscale characterization of filler dispersion and origins of mechanical reinforcement in model nanocomposites, Polymer 53, 761–775 (2012).
   Web of Science ®Google Scholar
• S. S. Sternstein, S. Amanuel, and M. L. Shofner, Reinforcement mechanisms in nanofilled polymer melts and elastomers, Rubb. Chem. Technol. 83, 181–198 (2010).
   Web of Science ®Google Scholar
• R. Buscall, I. J. McGowan, and C. A. Mumme-Young, Rheology of weakly interacting colloidal particles at high concentration, Faraday Discuss. Chem. Soc. 90, 115–127 (1990).
   Google Scholar
• C. Prestidge and T. F. Tadros, Rheological investigation of depletion flocculation of concentrated sterically stabilised polystyrene latex dispersions, Colloids Surf. 31, 325–346 (1988).
   Google Scholar
• W. B. Russel, The rheology of suspensions of charged rigid spheres, J. Fluid Mech. 85, 209–232 (1978).
   Web of Science ®Google Scholar
• R. Buscall, Effect of long-range repulsive forces on the viscosity of concentrated latices: comparison of experimental data with an effective hard-sphere model, J. Chem. Soc. Faraday Trans. 87, 1365–1370 (1991).
   Google Scholar
• R. Buscall, Comment on the interaction forces of polymer-coated surfaces, Langmuir 8, 2077–2079 (1992).
   Web of Science ®Google Scholar
• R. Buscall, An effective hard-sphere model of the non-Newtonian viscosity of stable colloidal dispersions: Comparison with further data for sterically stabilised latices and with data for microgel particles, Colloids Surf. A 83, 33–42 (1994).
   Web of Science ®Google Scholar
• J. Jancar, J. F. Douglas, F. W. Starr, S. K. Kumar, P. Cassagnau, A. J. Lesser, S. S. Sternstein, and M. J. Buehler, Current issues in research on structure-property relationships in polymer nanocomposites, Polymer 51, 3321–3343 (2010).
   Web of Science ®Google Scholar
• J. J. Brennan, T. E. Jermyn, and B. B. Boonstra, Carbon black-polymer interaction: A measure of reinforcement, J. Appl. Polym. Sci. 8, 2687–2706 (1964).
   Google Scholar
• J. L. Leblanc, Elastomer–filler interactions and the rheology of filled rubber compounds, J. Appl. Polym. Sci. 78, 1541–1550 (2000).
   Web of Science ®Google Scholar
• J. L. Leblanc, Rubber-filler interactions and rheological properties in filled compounds, Prog. Polym. Sci. 27, 627–687 (2002).
   Web of Science ®Google Scholar
• A. Banc, A.-C. Genix, M. Chirat, C. Dupas, S. Caillol, M. Sztucki, and J. Oberdisse, Tuning structure and rheology of silica-latex nanocomposites with the molecular weight of matrix chains: A coupled SAXS-TEM-simulation approach, Macromolecules 47, 3219–3230 (2014).
   Web of Science ®Google Scholar
• J. L. White and J. W. Crowder, The influence of carbon black on the extrusion characteristics and rheological properties of elastomers: Polybutadiene and butadiene–styrene copolymer, J. Appl. Polym. Sci. 18, 1013–1038 (1974).
   Web of Science ®Google Scholar
• K. Qin and A. A. Zaman, Viscosity of concentrated colloidal suspensions: comparison of bidisperse models, J. Colloid Interf. Sci. 266, 461–467 (2003).
   PubMed Web of Science ®Google Scholar
• Z. Zhou, P. J. Scales and D. V. Boger, Chemical and physical control of the rheology of concentrated metal oxide suspensions, Chem Eng Sci, 56, 2901–2920 (2001).
   Web of Science ®Google Scholar
• A. Zohrevand, A. Ajji, and F. Mighri, Relationship between rheological and electrical percolation in a polymer nanocomposite with semiconductor inclusions, Rheol. Acta 53, 235–254 (2014).
   Web of Science ®Google Scholar
• Y. Fukahori, A. A. Hon, V. Jha, and J. J. C. Busfield, Modified guth–gold equation for carbon black–filled rubbers, Rubb. Chem. Technol. 86, 218–232 (2013).
   Web of Science ®Google Scholar
• M. Bailly, M. Kontopoulou, and K. El Mabrouk, Effect of polymer/filler interactions on the structure and rheological properties of ethylene-octene copolymer/nanosilica composites, Polymer 51, 5506–5515 (2010).
   Web of Science ®Google Scholar
• M. Kluppel, The role of disorder in filler reinforcement of elastomers on various length scales, Adv. Polym. Sci. 164, 1–86 (2003).
   Web of Science ®Google Scholar
• A. I. Medalia, Effective degree of immobilization of rubber occluded within carbon black aggregates, Rubb. Chem. Technol. 45, 1171–1194 (1972).
   Google Scholar
• J.-C. Majesté and F. Vincent, A kinetic model for silica–filled rubber reinforcement, J. Rheol. 59, 405–427 (2015).
   Web of Science ®Google Scholar
• R. Scotti, L. Conzatti, M. D'Arienzo, B. Di Credico, L. Giannini, T. Hanel, P. Stagnaro, A. Susanna, L. Tadiello, and F. Morazzoni, Shape controlled spherical (0D) and rod-like (1D) silica nanoparticles in silica/styrene butadiene rubber nanocomposites: Role of the particle morphology on the filler reinforcing effect, Polymer 55, 1497–1506 (2014).
   Web of Science ®Google Scholar
• A. I. Medalia, Morphology of aggregates: I. Calculation of shape and bulkiness factors; application to computer-simulated random flocs, J. Colloid Interf. Sci. 24, 393–404 (1967).
   Web of Science ®Google Scholar
• J. R. Potts, O. Shankar, L. Du, and R. S. Ruoff, Processing-morphology-property relationships and composite theory analysis of reduced graphene oxide/natural rubber nanocomposites, Macromolecules 45, 6045–6055 (2012).
   Web of Science ®Google Scholar
• A. K. Van Dyk, T. Chatterjee, V. V. Ginzburg, and A. I. Nakatani, Shear-dependent interactions in hydrophobically modified ethylene oxide urethane (HEUR) based coatings: Mesoscale structure and viscosity, Macromolecules 48, 1866–1882 (2015).
   Web of Science ®Google Scholar
• L. Mullins and N. R. Tobin, Stress softening in rubber vulcanizates. I. Use of a strain-amplification factor to describe the elastic behavior of filler-reinforced vulcanized rubber, J. Appl. Polym. Sci. 9, 2993–3009 (1965).
   Web of Science ®Google Scholar
• A. I. Medalia, Effect of carbon black on dynamic properties of rubber vulcanizates, Rubb. Chem. Technol. 51, 437–523 (1978).
   Web of Science ®Google Scholar
• V. M. Litvinov and H. W. Spiess, Molecular mobility in the adsorption layer and chain orientation in strained poly(dimethylsiloxane) networks by 2H NMR, Die Makromolekulare Chemie 193, 1181–1194 (1992).
   Web of Science ®Google Scholar
• V. M. Litvinov and P. A. M. Steeman, EPDM-carbon black interactions and the reinforcement mechanisms, as studied by low-resolution 1H NMR, Macromolecules 32, 8476–8490 (1999).
   Web of Science ®Google Scholar
• S. Westermann, M. Kreitschmann, W. PyckhoutHintzen, D. Richter, and E. Straube, Strain amplification effects in polymer networks, Physica B, 234, 306–307 (1997).
   Web of Science ®Google Scholar
• S. Westermann, M. Kreitschmann, W. Pyckhout-Hintzen, D. Richter, E. Straube, B. Farago, and G. Goerigk, Matrix chain deformation in reinforced networks: a SANS approach, Macromolecules 32, 5793–5802 (1999).
   Web of Science ®Google Scholar
• A. Botti, W. Pyckhout-Hintzen, D. Richter, and E. Straube, Filled elastomers: polymer chain and filler characterization by a SANS–SAXS approach, Physica A 304, 230–234 (2002).
   Web of Science ®Google Scholar
• A. Botti, W. Pyckhout-Hintzen, D. Richter, V. Urban, and E. Straube, A microscopic look at the reinforcement of silica-filled rubbers, J. Chem. Phys. 124, 174908 (2006).
   PubMed Web of Science ®Google Scholar
• G. D. Hattemer and G. Arya, Viscoelastic properties of polymer-grafted nanoparticle composites from molecular dynamics simulations, Macromolecules 48, 1240–1255 (2015).
   Web of Science ®Google Scholar
• E. Voyiatzis, M. Rahimi, F. Müller-Plathe, and M. C. Böhm, How thick is the polymer interphase in nanocomposites? Probing it by local stress anisotropy and gas solubility, Macromolecules 47, 7878–7889 (2014).
   Web of Science ®Google Scholar
• W. Gleissle and B. Hochstein, Validity of the Cox-Merz rule for concentrated suspensions, J. Rheol. 47, 897–910 (2003).
   Web of Science ®Google Scholar
• L. A. Faitel'son and É. É. Yakobson, Rheology of filled polymers. Steady-state shear flow and periodic deformation. 1. Relaxation time spectra, viscosity, Mech. Compos. Mater. Struct. 13, 898–906 (1977).
   Google Scholar
• R. Pal, Complex shear modulus of concentrated suspensions of solid spherical particles, J. Colloid Interf. Sci. 245, 171–177 (2002).
   PubMed Web of Science ®Google Scholar
• J. Domurath, M. Saphiannikova, G. Ausias, and G. Heinrich, Modelling of stress and strain amplification effects in filled polymer melts, J. Non-Newtonian Fluid Mech. 171, 8–16 (2012).
   Web of Science ®Google Scholar
• J. Domurath, M. Saphiannikova, and G. Heinrich, Non-linear viscoelasticity of filled polymer melts: Stress and strain amplification approach, Macromol. Symp. 338, 54–61 (2014).
   Google Scholar
• S. Govindjee, An evaluation of strain amplification concepts via monte carlo simulations of an ideal composite, Rubb. Chem. Technol. 70, 25–37 (1997).
   Web of Science ®Google Scholar
• Y. Merckel, J. Diani, M. Brieu, P. Gilormini, and J. Caillard, Effect of the microstructure parameters on the mullins softening in carbon-black filled styrene–butadiene rubbers, J. Appl. Polym. Sci. 123, 1153–1161 (2012).
   Web of Science ®Google Scholar
• J. Rault, J. Marchal, P. Judeinstein, and P. A. Albouy, Chain orientation in natural rubber, Part II: 2H-NMR study, Europ. Phys. J. E 21, 243–261 (2006).
   PubMed Web of Science ®Google Scholar
• J. Rault, J. Marchal, P. Judeinstein, and P. A. Albouy, Stress-induced crystallization and reinforcement in filled natural rubbers: ; 2H NMR study, Macromolecules 39, 8356–8368 (2006).
   Web of Science ®Google Scholar
• S. Trabelsi, P. A. Albouy, and J. Rault, Effective local deformation in stretched filled rubber, Macromolecules 36, 9093–9099 (2003).
   Web of Science ®Google Scholar
• Y. Aoki, A. Hatano, T. Tanaka, and H. Watanabe, Nonlinear stress relaxation of ABS polymers in the molten state, Macromolecules 34, 3100–3107 (2001).
   Web of Science ®Google Scholar
• J. S. Bergstrom and M. C. Boyce, Mechanical behavior of particle filled elastomers, Rubb. Chem. Technol. 72, 633–656 (1999).
   Web of Science ®Google Scholar
• A. P. Ting and R. H. Luebbers, Viscosity of suspensions of spherical and other isodimensional particles in liquids, AlChE J. 3, 111–116 (1957).
   Web of Science ®Google Scholar
• F. Bueche, Mullins effect and rubber-filler interaction, J. Appl. Polym. Sci. 5, 271–281 (1961).
   Google Scholar
• F. Bueche, Molecular basis for the mullins effect, J. Appl. Polym. Sci. 4, 107–114 (1960).
   Google Scholar
• E. Hatschek, The general theory of viscosity of two-phase systems, Trans. Faraday Soc. 9, 80–92 (1913).
   Google Scholar
• E. Hatschek, Die Viskositat der Dispersoide, Kolloid-Zeitschrift 8, 34–39 (1911).
   Google Scholar
• L. E. Nielsen, Simple theory of stress-strain properties of filled polymers, J. Appl. Polym. Sci. 10, 97–103 (1966).
   Web of Science ®Google Scholar
• M. Narkis, L. Nicolais, and E. Joseph, The elastic modulus of particulate-filled polymers, J. Appl. Polym. Sci. 22, 2391–2394 (1978).
   Web of Science ®Google Scholar
• S. Govindjee and J. Simo, A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating Mullins' effect, J. Mech. Phys. Solids 39, 87–112 (1991).
   Web of Science ®Google Scholar
• S. Govindjee and J. Simo, Transition from micro-mechanics to computationally efficient phenomenology: Carbon black filled rubbers incorporating mullins' effect, J. Mech. Phys. Solids 40, 213–233 (1992).
   Web of Science ®Google Scholar
• A. S. Ermilov and E. M. Nurullaev, Mechanical properties of elastomers filled with solid particles, Mech. Compos. Mater. 48, 243–252 (2012).
   Web of Science ®Google Scholar
• A. S. Ermilov and E. M. Nurullaev, Energy of the mechanical destruction of an elastomer filled with solid particles, Mech. Compos. Mater. 50, 757–762 (2015).
   Web of Science ®Google Scholar
• T. L. Smith, Volume changes and dewetting in glass bead‐polyvinyl chloride elastomeric composites under large deformations, Trans. Soc. Rheol. 3, 113–136 (1959).
   Web of Science ®Google Scholar
• G. Huber and T. A. Vilgis, Universal properties of filled rubbers: Mechanisms for reinforcement on different length scales, Kaut. Gummi. Kunstst. 52, 102–107 (1999).
   Web of Science ®Google Scholar
• D. Y. Perera, Effect of pigmentation on organic coating characteristics, Prog. Org. Coat 50, 247–262 (2004).
   Web of Science ®Google Scholar
• R. Pérez-Aparicio, M. Schiewek, J. L. Valentín, H. Schneider, D. R. Long, M. Saphiannikova, P. Sotta, K. Saalwächter, and M. Ott, Local chain deformation and overstrain in reinforced elastomers: An NMR study, Macromolecules 46, 5549–5560 (2013).
   Web of Science ®Google Scholar
• R. Pérez-Aparicio, A. Vieyres, P.-A. Albouy, O. Sanséau, L. Vanel, D. R. Long, and P. Sotta, Reinforcement in natural rubber elastomer nanocomposites: Breakdown of entropic elasticity, Macromolecules 46, 8964–8972 (2013).
   Web of Science ®Google Scholar
• S. Dupres, D. R. Long, P.-A. Albouy, and P. Sotta, Local deformation in carbon black-filled polyisoprene rubbers studied by NMR and X-ray diffraction, Macromolecules 42, 2634–2644 (2009).
   Web of Science ®Google Scholar
• A. Crié, C. Baritaud, R. Valette, and B. Vergnes, Rheological behavior of uncured styrene-butadiene rubber at low temperatures, pure and filled with carbon black, Polym. Eng. Sci. DOI: 10.1002/pen.24090 (2015).
   Web of Science ®Google Scholar
• S. B. Ross-Murphy and S. Todd, Ultimate tensile measurements of filled gelatin gels, Polymer 24, 481–486 (1983).
   Web of Science ®Google Scholar
• K. R. Langley and M. L. Green, Compression strength and fracture properties of model particulate food composites in relation to their microstructure and particle-matrix interaction, J. Texture. Stud. 20, 191–207 (1989).
   Web of Science ®Google Scholar
• B. Turcsányi, B. Pukánszky, and F. Tüdõs, Composition dependence of tensile yield stress in filled polymers, J. Mater. Sci. Lett. 7, 160–162 (1988).
   Google Scholar
• X. Li, Z. Li, and Y. Xia, Test and calculation of the carbon black reinforcement effect on the hyper-elastic properties of tire rubbers, Rubb. Chem. Technol. 88, 98–116 (2015).
   Web of Science ®Google Scholar
• Y. Song and Q. Zheng, Linear viscoelasticity of polymer melts filled with nano-sized fillers, Polymer 51, 3262–3268 (2010).
   Web of Science ®Google Scholar
• P. Cassagnau, Payne effect and shear elasticity of silica-filled polymers in concentrated solutions and in molten state, Polymer 44, 2455–2462 (2003).
   Web of Science ®Google Scholar
• Y. Song and Q. Zheng, Application of two phase model to linear viscoelasticity of reinforced rubbers, Polymer 52, 593–596 (2011).
   Web of Science ®Google Scholar
• Y. Song and Q. Zheng, Application of two phase model to linear dynamic rheology of filled polymer melts, Polymer 52, 6173–6179 (2011).
   Web of Science ®Google Scholar
• Y. Tan, X. Yin, M. Chen, Y. Song, and Q. Zheng, Influence of annealing on linear viscoelasticity of carbon black filled polystyrene and low-density polyethylene, J. Rheol. 55, 965–979 (2011).
   Web of Science ®Google Scholar
• Q. Cao, Y. Song, Y. Tan, and Q. Zheng, Thermal-induced percolation in high-density polyethylene/carbon black composites, Polymer 50, 6350–6356 (2009).
   Web of Science ®Google Scholar
• Q. Cao, Y. Song, Y. Tan, and Q. Zheng, Conductive and viscoelastic behaviors of carbon black filled polystyrene during annealing, Carbon 48, 4268–4275 (2010).
   Web of Science ®Google Scholar
• Y. Song, Q. Cao, and Q. Zheng, Annealing-induced rheological and electric resistance variations in carbon black-filled polymer melts, Colloid Polym. Sci. 290, 1837–1842 (2012).
   Web of Science ®Google Scholar
• A. Guan, Y. Song, Y. Tan, and Q. Zheng, Time-dependent dynamic modulus percolation of multi-walled carbon nanotubes filled polystyrene melts during annealing, Acta Polym. Sin. 11–17 (2013).
   Web of Science ®Google Scholar
• M. Chen, Y. Tan, and Y. Song, Rheological behavior of carbon black filled polystyrene melts, Chem. J. Chinese. U. 34, 2228–2232 (2013).
   Web of Science ®Google Scholar
• A. Guan, Y. Song, Y. Tan, and Q. Zheng, Dynamic rheological properties of polystyrene/multi-walled carbon nanotubes composites, Acta Polym. Sin. 224–229 (2011).
   Web of Science ®Google Scholar
• Y. Song and Q. Zheng, Dynamic rheology of vapor grown carbon nanofiber filled polystyrene melts, Acta Polym. Sin. 21, 1383–1388 (2012).
   Google Scholar
• L. Zhao, H. Yang, Y. Song, G. Hu, and Q. Zheng, Effect of mixing condition on electrical percolation and dynamic rheological behavior for vapor grown carbon fiber filled polystyrene composites, Acta Polym. Sin. 1305–1310 (2011).
   Web of Science ®Google Scholar