# Numerical study of chain conformation on shear banding using diffusive Rolie-Poly model

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## Abstract

Shear-banding phenomenon in the entangled polymer systems was investigated in a planar Couette cell with the diffusive Rolie-Poly (ROuse LInear Entangled POLYmers) model, a single-mode constitutive model derived from a tube-based molecular theory. The steady-state shear stress *σ* _{s} was constant in the shear gradient direction while the local shear rate changed abruptly, i.e., split into the bands. We focused on the molecular conformation (also calculated from the Rolie-Poly model) around the band boundary. A band was found also for the conformation, but its boundary was much broader than that for the shear rate. Correspondingly, the first normal stress difference (*N* _{1}) gradually changed in this diffuse boundary of the conformational bands (this change of *N* _{1} was compensated by a change of the local pressure). For both shear rate and conformation, the boundary widths were quite insensitive to the macroscopic shear rate but changed with various parameters such as the diffusion constant and the relaxation times (the reptation and the Rouse times). The broadness of the conformational banding, associated by the gradual change of *N* _{1}, was attributed to competition between the molecular diffusion (in the shear gradient direction) and the conformational relaxation under a constraint of constant *σ* _{s}.

## Keywords

Shear banding Rolie-Poly model Velocity band Molecular orientational band Molecular diffusion and relaxation## Notes

### Acknowledgements

This work was partly supported by Grant-in-Aid for Scientific Research on Priority Area “Soft Matter Physics” from the Ministry of Education, Culture, Sports, Science and Technology (grant #18068009). C. Chung thanks a financial support from G-COE program for the stay at ICR.

## References

- Adams JM, Olmsted PD (2009) Nonmonotonic models are not necessary to obtain shear banding phenomena in entangled polymer solutions. Phys Rev Lett 102(6):067801CrossRefGoogle Scholar
- Adams JM, Fielding SM, Olmsted PD (2008) The interplay between boundary conditions and flow geometries in shear banding: hysteresis, band configurations, and surface transitions. J Non-Newton Fluid Mech 151(1–3):101–118CrossRefGoogle Scholar
- Archer LA, Larson RG, Chen YL (1995) Direct measurements of slip in sheared polymer solutions. J Fluid Mech 301:133–151CrossRefGoogle Scholar
- Baaijens FPT (1998) Mixed finite element methods for viscoelastic flow analysis: a review. J Non-Newton Fluid Mech 79(2–3):361–385CrossRefGoogle Scholar
- Berret JF, Roux DC, Porte G, Lindner P (1994) Shear-induced isotropic-to-nematic phase transition in equilibrium polymers. Europhys Lett 25(7):521–526CrossRefGoogle Scholar
- Boukany PE, Wang SQ (2009a) Exploring the transition from wall slip to bulk shearing banding in well-entangled DNA solutions. Soft Matter 5(4):780–789CrossRefGoogle Scholar
- Boukany PE, Wang SQ (2009b) Shear banding or not in entangled DNA solutions depending on the level of entanglement. J Rheol 53(1):73–83CrossRefGoogle Scholar
- Boukany PE, Hu YT, Wang SQ (2008) Observations of wall slip and shear banding in an entangled DNA solution. Macromolecules 41(7):2644–2650CrossRefGoogle Scholar
- Brooks AN, Hughes TJR (1982) Streamline upwind Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 32(1–3):199–259CrossRefGoogle Scholar
- Cappelaere E, Berret JF, Decruppe JP, Cressely R, Lindner P (1997) Rheology, birefringence, and small-angle neutron scattering in a charged micellar system: evidence of a shear-induced phase transition. Phys Rev E 56(2):1869–1878CrossRefGoogle Scholar
- Cates ME, Mcleish TCB, Marrucci G (1993) The rheology of entangled polymers at very high shear rates. Europhys Lett 21(4):451–456CrossRefGoogle Scholar
- Chung C, Hulsen MA, Kim JM, Ahn KH, Lee SJ (2008) Numerical study on the effect of viscoelasticity on drop deformation in simple shear and 5:1:5 planar contraction/expansion microchannel. J Non-Newton Fluid Mech 155:80–93CrossRefGoogle Scholar
- Doi M, Edwards SF (1989) The theory of polymer dynamics. Clarendon, OxfordGoogle Scholar
- Douglas JF, Hubbard JB (1991) Semiempirical theory of relaxation: concentrated polymer solution dynamics. Macromolecules 24(11):3163–3177CrossRefGoogle Scholar
- Fielding SM (2005) Linear instability of planar shear banded flow. Phys Rev Lett 95(13):134501CrossRefGoogle Scholar
- Fielding SM, Olmsted PD (2003a) Early stage kinetics in a unified model of shear-induced demixing and mechanical shear banding instabilities. Phys Rev Lett 90(22):224501CrossRefGoogle Scholar
- Fielding SM, Olmsted PD (2003b) Kinetics of the shear banding instability in startup flows. Phys Rev E 68(3):036313CrossRefGoogle Scholar
- Fielding SM, Olmsted PD (2006) Nonlinear dynamics of an interface between shear bands. Phys Rev Lett 96(10):104502CrossRefGoogle Scholar
- Fischer E, Callaghan PT (2001) Shear banding and the isotropic-to-nematic transition in wormlike micelles. Phys Rev E 6401(1):011501CrossRefGoogle Scholar
- Furukawa A, Onuki A (2005) Spatio-temporal structures in sheared polymer systems. Physica D 205(1–4):195–206CrossRefGoogle Scholar
- Graham RS, Likhtman AE, McLeish TCB, Milner ST (2003) Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release. J Rheol 47(5):1171–1200CrossRefGoogle Scholar
- Hu YT, Lips A (2005) Kinetics and mechanism of shear banding in an entangled micellar solution. J Rheol 49(5):1001–1027CrossRefGoogle Scholar
- Hulsen MA, Fattal R, Kupferman R (2005) Flow of viscoelastic fluids past a cylinder at high Weissenberg number: stabilized simulations using matrix logarithms. J Non-Newton Fluid Mech 127(1):27–39CrossRefGoogle Scholar
- Jupp L, Yuan XF (2004) Dynamic phase separation of a binary polymer liquid with asymmetric composition under rheometric flow. J Non-Newton Fluid Mech 124(1–3):93–101CrossRefGoogle Scholar
- Kim JM, Kim C, Ahn KH, Lee SJ (2004) An efficient iterative solver and high-resolution computations of the Oldroyd-B fluid flow past a confined cylinder. J Non-Newton Fluid Mech 123(2–3):161–173CrossRefGoogle Scholar
- Lerouge S, Decruppe JP, Olmsted P (2004) Birefringence banding in a micellar solution or the complexity of heterogeneous flows. Langmuir 20(26):11355–11365CrossRefGoogle Scholar
- Liberatore MW, Nettesheim F, Wagner NJ, Porcar L (2006) Spatially resolved small-angle neutron scattering in the 1-2 plane: a study of shear-induced phase-separating wormlike micelles. Phys Rev E 73(2):020504CrossRefGoogle Scholar
- Likhtman AE, Graham RS (2003) Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie-Poly equation. J Non-Newton Fluid Mech 114(1):1–12CrossRefGoogle Scholar
- Liu AW, Bornside DE, Armstrong RC, Brown RA (1998) Viscoelastic flow of polymer solutions around a periodic, linear array of cylinders: comparisons of predictions for microstructure and flow fields. J Non-Newton Fluid Mech 77(3):153–190CrossRefGoogle Scholar
- Lodge TP (1999) Reconciliation of the molecular weight dependence of diffusion and viscosity in entangled polymers. Phys Rev Lett 83(16):3218–3221CrossRefGoogle Scholar
- Lu CYD, Olmsted PD, Ball RC (2000) Effects of nonlocal stress on the determination of shear banding flow. Phys Rev Lett 84(4):642–645CrossRefGoogle Scholar
- Manneville S, Salmon JB, Becu L, Colin A, Molino F (2004) Inhomogeneous flows in sheared complex fluids. Rheol Acta 43(5):408–416CrossRefGoogle Scholar
- McLeish TCB (2002) Tube theory of entangled polymer dynamics. Adv Phys 51(6):1379–1527CrossRefGoogle Scholar
- McLeish TCB, Ball RC (1986) A molecular approach to the spurt effect in polymer melt flow. J Polym Sci Polym Phys 24(8):1735–1745CrossRefGoogle Scholar
- Miller E, Rothstein JP (2007) Transient evolution of shear-banding wormlike micellar solutions. J Non-Newton Fluid Mech 143(1):22–37CrossRefGoogle Scholar
- Oberhauser JP, Leal LG, Mead DW (1998) The response of entangled polymer solutions to step changes of shear rate: signatures of segmental stretch? J Polym Sci Polym Phys 36(2):265–280CrossRefGoogle Scholar
- Olmsted PD (2008) Perspectives on shear banding in complex fluids. Rheol Acta 47(3):283–300CrossRefGoogle Scholar
- Olmsted PD, Radulescu O, Lu CYD (2000) Johnson-Segalman model with a diffusion term in cylindrical Couette flow. J Rheol 44(2):257–275CrossRefGoogle Scholar
- Onuki A (2002) Phase Transition Dynamics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Radulescu O, Olmsted PD (2000) Matched asymptotic solutions for the steady banded flow of the diffusive Johnson-Segalman model in various geometries. J Non-Newton Fluid Mech 91(2–3):143–164CrossRefGoogle Scholar
- Radulescu O, Olmsted PD, Lu CYD (1999) Shear banding in reaction-diffusion models. Rheol Acta 38(6):606–613CrossRefGoogle Scholar
- Radulescu O, Olmsted PD, Decruppe JP, Lerouge S, Berret JF, Porte G (2003) Time scales in shear banding of wormlike micelles. Europhys Lett 62(2):230–236CrossRefGoogle Scholar
- Ramirez J, Laso M (2005) Size reduction methods for the implicit time-dependent simulation of micro-macro viscoelastic flow problems. J Non-Newton Fluid Mech 127(1):41–49CrossRefGoogle Scholar
- Ravindranath S, Wang SQ (2007) What are the origins of stress relaxation behaviors in step shear of entangled polymer solutions? Macromolecules 40(22):8031–8039CrossRefGoogle Scholar
- Ravindranath S, Wang SQ, Ofechnowicz M, Quirk RP (2008) Banding in simple steady shear of entangled polymer solutions. Macromolecules 41(7):2663–2670CrossRefGoogle Scholar
- Salmon JB, Colin A, Manneville S, Molino F (2003) Velocity profiles in shear-banding wormlike micelles. Phys Rev Lett 90(22):228303CrossRefGoogle Scholar
- Sato K, Yuan XF, Kawakatsu T (2010) Why does shear banding behave like first-order phase transition? Deviation of a potention from a mechanical constitutive model. Eur Phys J E 31:135–144CrossRefGoogle Scholar
- Schmitt V, Marques CM, Lequeux F (1995) Shear-induced phase separation of complex fluids: the role of flow-concentration coupling. Phys Rev E 52(4):4009–4015CrossRefGoogle Scholar
- Takenaka M, Nishitsuji S, Taniguchi T, Yamaguchi M, Tada K, Hashimoto T (2006) Computer simulation study on the shear-induced phase separation in semidilute polymer solutions in 3-dimensional space. Polymer 47(22):7846-7852CrossRefGoogle Scholar
- Tao H, Lodge TP, von Meerwall ED (2000) Diffusivity and viscosity of concentrated hydrogenated polybutadiene solutions. Macromolecules 33(5):1747–1758CrossRefGoogle Scholar
- Tapadia P, Wang SQ (2003) Yieldlike constitutive transition in shear flow of entangled polymeric fluids. Phys Rev Lett 91(19):198301CrossRefGoogle Scholar
- Tapadia P, Wang SQ (2004) Nonlinear flow behavior of entangled polymer solutions: yieldlike entanglement-disentanglement transition. Macromolecules 37(24):9083–9095CrossRefGoogle Scholar
- Tapadia P, Wang SQ (2006) Direct visualization of continuous simple shear in non-newtonian polymeric fluids. Phys Rev Lett 96(1):016001CrossRefGoogle Scholar
- van den Noort A, Briels WJ (2007) Coarse-grained simulations of elongational viscosities, superposition rheology and shear banding in model core-shell systems. Macromol Theory Simul 16(8):742–754CrossRefGoogle Scholar
- Wang SQ (2003) Chain dynamics in entangled polymers: diffusion versus rheology and their comparison. J Polym Sci B: Polym Phys 41(14):1589–1604CrossRefGoogle Scholar
- Wang SQ (2008) The tip of lceberg in nonlinear polymer rheology: entangled liquids are “solids”. J Polym Sci B: Polym Phys 46:2660–2665CrossRefGoogle Scholar
- Wang SQ, Ravindranath S, Wang YY, Boukany PY (2007) New theoretical considerations in polymer rheology: elastic breakdown of chain entanglement network. J Chem Phys 127(6):064903CrossRefGoogle Scholar
- Wheeler LM, Lodge TP (1989) Tracer diffusion of linear polystyrenes in dilute, semidilute, and concentrated poly(vinyl methyl ether) solutions. Macromolecules 22(8):3399–3408CrossRefGoogle Scholar
- Wilson HJ, Fielding SM (2006) Linear instability of planar shear banded flow of both diffusive and non-diffusive Johnson-Segalman fluids. J Non-Newton Fluid Mech 138(2–3):181–196CrossRefGoogle Scholar
- Yerushalmi J, Katz S, Shinnar R (1970) Stability of steady shear flows of some viscoelastic fluids. Chem Eng Sci 25(12):1891–1902CrossRefGoogle Scholar
- Yesilata B, Clasen C, McKinley GH (2006) Nonlinear shear and extensional flow dynamics of wormlike surfactant solutions. J Non-Newton Fluid Mech 133(2–3):73–90CrossRefGoogle Scholar
- Yuan XF (1999) Dynamics of a mechanical interface in shear-banded flow. Europhys Lett 46(4):542–548CrossRefGoogle Scholar
- Zhou L, Vasquez PA, Cook LP, McKinley GH (2008) Modeling the inhomogeneous response and formation of shear bands in steady and transient flows of entangled liquids. J Rheol 52(2):591–623CrossRefGoogle Scholar