We investigate theoretically shear banding in large amplitude oscillatory shear (LAOS) of polymeric and wormlike micellar surfactant fluids. In large amplitude oscillatory shear strain, we observe banding at low frequencies and sufficiently high strain rate amplitudes in fluids for which the underlying stationary constitutive curve of shear stress as a function of shear rate is nonmonotonic. This is the direct (and relatively trivial) analog of quasisteady state banding seen in slow strain rate sweeps along the flow curve. At higher frequencies and sufficiently high strain amplitudes, we report a different but related phenomenon, which we call “elastic” shear banding. This is associated with an overshoot in the elastic (Lissajous–Bowditch) curve of stress as a function of strain and we suggest that it might arise rather widely even in fluids that have a monotonic underlying constitutive curve, and so do not show steady state banding if under a steadily applied shear flow. It is analogous to the elastic banding triggered by stress overshoot in a fast shear startup predicted previously by R. L. Moorcroft and S. M. Fielding [Phys. Rev. Lett. 110(8), 086001 (2013)], but could be more readily observable experimentally in this oscillatory protocol due to its recurrence in each half cycle. In large amplitude oscillatory shear stress, we report shear banding in fluids that shear thin strongly enough to have either a negatively or a weakly positively sloping region in the underlying constitutive curve, noting again that fluids in the latter category do not display steady state banding in a steadily applied flow. This banding is triggered in each half cycle as the stress magnitude transits the region of weak slope in an upward direction such that the fluid effectively yields. It is strongly reminiscent of the transient banding predicted previously in step stress [R. L. Moorcroft and S. M. Fielding, Phys. Rev. Lett. 110(8), 086001 (2013)]. Our numerical calculations are performed in the Rolie-poly model of polymers and wormlike micelles, but we also provide arguments suggesting that our results should apply more widely. Besides banding in the shear strain rate profile, which can be measured by velocimetry, we also predict banding in the shear and normal stress components, measurable by birefringence. As a backdrop to understanding the new results on shear banding in LAOS, we also briefly review earlier work on banding in other time-dependent protocols, focusing in particular on shear startup and step stress.
References
1.
Olmsted
, P. D.
, “Perspectives on shear banding in complex fluids
,” Rheol. Acta
47
(3
), 283
–300
(2008
).2.
Manneville
, S.
, “Recent experimental probes of shear banding
,” Rheol. Acta
47
(3
), 301
–318
(2008
).3.
Fielding
, S. M.
, “Shear banding in soft glassy materials
,” Rep. Prog. Phys.
77
(10
), 102601
(2014
).4.
Divoux
, T.
, M. A.
Fardin
, S.
Manneville
, and S.
Lerouge
, “Shear banding of complex fluids
,” Annu. Rev. Fluid Mech.
48
(1
), 81
–103
(2015
).5.
Britton
, M. M.
, and P. T.
Callaghan
, “Two-phase shear band structures at uniform stress
,” Phys. Rev. Lett.
78
(26
), 4930
–4933
(1997
).6.
Divoux
, T.
, D.
Tamarii
, C.
Barentin
, and S.
Manneville
, “Transient shear banding in a simple yield stress fluid
,” Phys. Rev. Lett.
104
(20
), 208301
(2010
).7.
Martin
, J. D.
, and Y.
Thomas Hu
, “Transient and steady-state shear banding in aging soft glassy materials
,” Soft Matter
8
(26
), 6940
−6949
(2012
).8.
Coussot
, P.
, J.
Raynaud
, F.
Bertrand
, P.
Moucheront
, J.
Guilbaud
, H. T.
Huynh
, S.
Jarny
, and D.
Lesueur
, “Coexistence of liquid and solid phases in flowing soft-glassy materials
,” Phys. Rev. Lett.
88
(21
), 218301
(2002
).9.
Rodts
, S.
, J. C.
Baudez
, and P.
Coussot
, “From “discrete” to “continuum” flow in foams
,” Europhys. Lett.
69
(4
), 636
–642
(2005
).10.
Salmon
, J.-B.
, S.
Manneville
, and A.
Colin
, “Shear banding in a lyotropic lamellar phase. I. Time-averaged velocity profiles
,” Phys. Rev. E
68
, 051503
(2003
).11.
Berret
, J.-F.
, and Y.
Séréro
, “Evidence of shear-induced fluid fracture in telechelic polymer networks
,” Phys. Rev. Lett.
87
, 048303
(2001
).12.
Manneville
, S.
, A.
Colin
, G.
Waton
, and F.
Schosseler
, “Wall slip, shear banding, and instability in the flow of a triblock copolymer micellar solution
,” Phys. Rev. E
75
, 061502
(2007
).13.
Rogers
, S. A.
, D.
Vlassopoulos
, and P.
Callaghan
, “Aging, yielding, and shear banding in soft colloidal glasses
,” Phys. Rev. Lett.
100
(12
), 128304
(2008
).14.
Tapadia
, P.
, and S.-Q.
Wang
, “Yieldlike constitutive transition in shear flow of entangled polymeric fluids
,” Phys. Rev. Lett.
91
(19
), 198301
(2003
).15.
Tapadia
, P.
, S.
Ravindranath
, and S.-Q.
Wang
, “Banding in entangled polymer fluids under oscillatory shearing
,” Phys. Rev. Lett.
96
(19
), 196001
(2006
).16.
Ravindranath
, S.
, S.-Q.
Wang
, M.
Olechnowicz
, and R. P.
Quirk
, “Banding in simple steady shear of entangled polymer solutions
,” Macromolecules
41
(7
), 2663
–2670
(2008
).17.
Li
, Y.
, M.
Hu
, G. B.
McKenna
, C. J.
Dimitriou
, G. H.
McKinley
, R. M.
Mick
, D. C.
Venerus
, and L. A.
Archer
, “Flow field visualization of entangled polybutadiene solutions under nonlinear viscoelastic flow conditions
,” J. Rheol.
57
(5
), 1411
−1428 (2013
).18.
Wang
, S.-Q.
, G.
Liu
, S.
Cheng
, P. E.
Boukany
, Y.
Wang
, and X.
Li
, “Letter to the Editor: Sufficiently entangled polymers do show shear strain localization at high enough Weissenberg numbers
,” J. Rheol.
58
(4
), 1059
–1069
(2014
).19.
Li
, Y.
, and G. B.
McKenna
, “Startup shear of a highly entangled polystyrene solution deep into the nonlinear viscoelastic regime
,” Rheol. Acta
54
(9–10
), 771
–777
(2015
).20.
Wang
, S.-Q.
, S.
Ravindranath
, and P. E.
Boukany
, “Homogeneous shear, wall slip, and shear banding of entangled polymeric liquids in simple-shear rheometry: A roadmap of nonlinear rheology
,” Macromolecules
44
(2
), 183
–190
(2011
).21.
Fielding
, S. M.
, and P. D.
Olmsted
, “Flow phase diagrams for concentration-coupled shear banding
,” Soft Matter
11
(1
), 65
–83
(2003
).22.
23.
Cates
, M. E.
, “Nonlinear viscoelasticity of wormlike micelles (and other reversibly breakable polymers)
,” J. Phys. Chem.
94
(1
), 371
–375
(1990
).24.
Yerushalmi
, J.
, S.
Katz
, and R.
Shinnar
, “The stability of steady shear flows of some viscoelastic fluids
,” Chem. Eng. Sci.
25
(12
), 1891
–1902
(1970
).25.
Cates
, M. E.
, and S. M.
Fielding
, “Theoretical rheology of giant micelles
,” in Giant Micelles
, edited by R.
Zana
and E.
Kaler
(Taylor and Francis
, London
, 2007
), Chap. 4, pp. 109
–161
.26.
Berret
, J.-F.
, “Rheology of wormlike micelles: Equilibrium properties and shear banding transitions
,” in Molecular Gels
, edited by R. G.
Weiss
and P.
Terech
(Springer
, Dordrecht
, 2005
), Chap. 19, pp. 667
–720
.27.
Snijkers
, F.
, R.
Pasquino
, P. D.
Olmsted
, and D.
Vlassopoulos
, “Perspectives on the viscoelasticity and flow behavior of entangled linear and branched polymers
,” J. Phys.: Condens. Matter
27
(47
), 473002
(2015
).28.
Marrucci
, G.
, “Dynamics of entanglements: A nonlinear model consistent with the Cox-Merz rule
,” J. Non-Newtonian Fluid Mech.
62
(2–3
), 279
–289
(1996
).29.
Ianniruberto
, G.
, and G.
Marrucci
, “Do repeated shear startup runs of polymeric liquids reveal structural changes?
,” ACS Macro Lett.
3
(6
), 552
–555
(2014
).30.
Ianniruberto
, G.
, and G.
Marrucci
, “Convective constraint release (CCR) revisited
,” J. Rheol.
58
(1
), 89
–102 (2014
).31.
Cao
, J.
, and A. E.
Likhtman
, “Shear banding in molecular dynamics of polymer melts
,” Phys. Rev. Lett.
108
(2
), 28302
(2012
).32.
Divoux
, T.
, C.
Barentin
, and S.
Manneville
, “Stress overshoot in a simple yield stress fluid: An extensive study combining rheology and velocimetry
,” Soft Matter
7
(19
), 9335
–9349
(2011
).33.
Boukany
, P. E.
, and S.-Q.
Wang
, “Shear banding or not in entangled {DNA} solutions depending on the level of entanglement
,” J. Rheol.
53
(1
), 73
–83
(2009
).34.
Thomas Hu
, Y.
, L.
Wilen
, A.
Philips
, and A.
Lips
, “Is the constitutive relation for entangled polymers monotonic?
,” J. Rheol.
51
(2
), 275
–295
(2007
).35.
Li
, X.
, and S.-Q.
Wang
, “Elastic yielding after step shear and during LAOS in the absence of meniscus failure
,” Rheol. Acta
49
(10
), 985
–991
(2010
).36.
Boukany
, P. E.
, and S.-Q.
Wang
, “Exploring origins of interfacial yielding and wall slip in entangled linear melts during shear or after shear cessation
,” Macromolecules
42
(6
), 2222
–2228
(2009
).37.
Wang
, S.-Q.
, S.
Ravindranath
, P. E.
Boukany
, M.
Olechnowicz
, R. P.
Quirk
, A.
Halasa
, and J.
Mays
, “Nonquiescent relaxation in entangled polymer liquids after step shear
,” Phys. Rev. Lett.
97
(18
), 187801
(2006
).38.
Fang
, Y.
, G.
Wang
, N.
Tian
, X.
Wang
, X.
Zhu
, P.
Lin
, G.
Ma
, and L.
Li
, “Shear inhomogeneity in poly(ethylene oxide) melts
,” J. Rheol.
55
(5
), 939
–949 (2011
).39.
Ravindranath
, S.
, and S.-Q.
Wang
, “What are the origins of stress relaxation behaviors in step shear of entangled polymer solutions?
,” Macromolecules
40
(22
), 8031
–8039
(2007
).40.
Archer
, L. A.
, Y. L.
Chen
, and R. G.
Larson
, “Delayed slip after step strains in highly entangled polystyrene solutions
,” J. Rheol.
39
(3
), 519
–525 (1995
).41.
Boukany
, P. E.
, and S.-Q.
Wang
, “Use of particle-tracking velocimetry and flow birefringence to study nonlinear flow behavior of entangled wormlike micellar solution: From wall slip, bulk disentanglement to chain scission
,” Macromolecules
41
(4
), 1455
–1464
(2008
).42.
Gibaud
, T.
, D.
Frelat
, and S.
Manneville
, “Heterogeneous yielding dynamics in a colloidal gel
,” Soft Matter
6
(15
), 3482
–3488
(2010
).43.
Divoux
, T.
, C.
Barentin
, and S.
Manneville
, “From stress-induced fluidization processes to Herschel-Bulkley behaviour in simple yield stress fluids
,” Soft Matter
7
(18
), 8409
–8418
(2011
).44.
Thomas Hu
, Y.
, C.
Palla
, and A.
Lips
, “Comparison between shear banding and shear thinning in entangled micellar solutions
,” J. Rheol.
52
(2
), 379
−400 (2008
).45.
Thomas Hu
, Y.
, and A.
Lips
, “Kinetics and mechanism of shear banding in an entangled micellar solution
,” J. Rheol.
49
(5
), 1001
–1027 (2005
).46.
Thomas Hu
, Y.
, “Steady-state shear banding in entangled polymers?
” J. Rheol.
54
(6
), 1307
–1323 (2010
).47.
Moorcroft
, R. L.
, and S. M.
Fielding
, “Criteria for shear banding in time-dependent flows of complex fluids
,” Phys. Rev. Lett.
110
(8
), 086001
(2013
).48.
Moorcroft
, R. L.
, and S. M.
Fielding
, “Shear banding in time-dependent flows of polymers and wormlike micelles
,” J. Rheol.
58
(1
), 103
–147 (2014
).49.
Moorcroft
, R. L.
, M. E.
Cates
, and S. M.
Fielding
, “Age-dependent transient shear banding in soft glasses
,” Phys. Rev. Lett.
106
(5
), 055502
(2011
).50.
Adams
, J. M.
, S. M.
Fielding
, and P. D.
Olmsted
, “Transient shear banding in entangled polymers: A study using the Rolie-Poly model
,” J. Rheol.
55
(5
), 1007
–1032 (2011
).51.
Manning
, M. L.
, E. G.
Daub
, J. S.
Langer
, and J. M.
Carlson
, “Rate-dependent shear bands in a shear-transformation-zone model of amorphous solids
,” Phys. Rev. E
79
(1
), 16110
(2009
).52.
Marrucci
, G.
, and N.
Grizzuti
, “The free energy function of the Doi-Edwards theory: Analysis of the instabilities in stress relaxation
,” J. Rheol.
27
(5
), 433
–450 (1983
).53.
Fielding
, S. M.
, R. L.
Moorcroft
, R. G.
Larson
, and M. E.
Cates
, “Modeling the relaxation of polymer glasses under shear and elongational loads
,” J. Chem. Phys.
138
(12
), 12A504
(2013
).54.
Jagla
, E. A.
, “Shear band dynamics from a mesoscopic modeling of plasticity
,” J. Stat. Mech.
2010
(12
), P12025
.55.
Ravindranath
, S.
, and S.-Q.
Wang
, “Steady state measurements in stress plateau region of entangled polymer solutions: Controlled-rate and controlled-stress modes
,” J. Rheol.
52
(4
), 957
–980
(2008
).56.
Boukany
, P. E.
, S.-Q.
Wang
, and X.
Wang
, “Step shear of entangled linear polymer melts: New experimental evidence for elastic yielding
,” Macromolecules
42
(16
), 6261
–6269
(2009
).57.
Boukany
, P. E.
, and S.-Q.
Wang
, “Exploring the transition from wall slip to bulk shearing banding in well-entangled DNA solutions
,” Soft Matter
5
(4
), 780
–789
(2009
).58.
Hyun
, K.
, M.
Wilhelm
, C. O.
Klein
, K. S.
Cho
, J. G.
Nam
, K. H.
Ahn
, S. J.
Lee
, R. H.
Ewoldt
, and G. H.
McKinley
, “A review of nonlinear oscillatory shear tests: Analysis and application of large amplitude oscillatory shear (LAOS)
,” Prog. Polym. Sci.
36
(12
), 1697
–1753
(2011
).59.
Cohen
, I.
, B.
Davidovitch
, A. B.
Schofield
, M. P.
Brenner
, and D. A.
Weitz
, “Slip, yield, and bands in colloidal crystals under oscillatory shear
,” Phys. Rev. Lett.
97
(21
), 215502
(2006
).60.
Dimitriou
, C. J.
, L.
Casanellas
, T. J.
Ober
, and G. H.
McKinley
, “Rheo-PIV of a shear-banding wormlike micellar solution under large amplitude oscillatory shear
,” Rheol. Acta
51
(5
), 395
–411
(2012
).61.
Kate Gurnon
, A.
, and N. J.
Wagner
, “Large amplitude oscillatory shear (LAOS) measurements to obtain constitutive equation model parameters: Giesekus model of banding and nonbanding wormlike micelles
,” J. Rheol.
56
(2
), 333
–351 (2012
).62.
Kate Gurnon
, A.
, C. R.
Lopez-Barron
, A. P. R.
Eberle
, L.
Porcar
, and N. J.
Wagner
, “Spatiotemporal stress and structure evolution in dynamically sheared polymer-like micellar solutions
,” Soft Matter
10
(16
), 2889
–2898
(2014
).63.
Wilhelm
, M.
, “Fourier-transform rheology
,” Macromol. Mater. Eng.
287
(2
), 83
–105
(2002
).64.
Tee
, T.-T.
, and J. M.
Dealy
, “Nonlinear viscoelasticity of polymer melts
,” J. Rheol.
19
(4
), 595
–615 (1975
).65.
Klein
, C. O.
, H. W.
Spiess
, A.
Calin
, C.
Balan
, and M.
Wilhelm
, “Separation of the nonlinear oscillatory response into a superposition of linear, strain hardening, strain softening, and wall slip response
,” Macromolecules
40
(12
), 4250
–4259
(2007
).66.
Klein
, C.
, P.
Venema
, L.
Sagis
, and E.
van der Linden
, “Rheological discrimination and characterization of carrageenans and starches by Fourier transform-rheology in the non-linear viscous regime
,” J. Non-Newtonian Fluid Mech.
151
(1–3
), 145
–150
(2008
).67.
Cho
, K. S.
, K.
Hyun
, K. H.
Ahn
, and S. J.
Lee
, “A geometrical interpretation of large amplitude oscillatory shear response
,” J. Rheol.
49
(3
), 747
–758 (2005
).68.
Ewoldt
, R. H.
, A. E.
Hosoi
, and G. H.
McKinley
, “New measures for characterizing nonlinear viscoelasticity in large amplitude oscillatory shear
,” J. Rheol.
52
(6
), 1427
–1458 (2008
).69.
Rogers
, S. A.
, and M.
Paul Lettinga
, “A sequence of physical processes determined and quantified in large-amplitude oscillatory shear (LAOS): Application to theoretical nonlinear models
,” J. Rheol.
56
(1
), 1
–25 (2012
).70.
Rogers
, S. A.
, B. M.
Erwin
, D.
Vlassopoulos
, and M.
Cloitre
, “A sequence of physical processes determined and quantified in LAOS: Application to a yield stress fluid
,” J. Rheol.
55
(2
), 435
−458 (2011
).71.
Zhou
, Lin
, Randy H.
Ewoldt
, L.
Pamela Cook
, and Gareth H.
McKinley
, “Probing Shear-Banding Transitions of Entangled Liquids Using Large Amplitude Oscillatory Shearing (LAOS) Deformations
,” AIP Conf. Proc.
1027
(1
), 189
−191 (2008
).72.
Zhou
, L.
, P. A.
Vasquez
, L.
Pamela Cook
, and G. H.
McKinley
, “Modeling the inhomogeneous response and formation of shear bands in steady and transient flows of entangled liquids
,” J. Rheol.
52
(2
), 591
−623 (2008
).73.
Adams
, J. M.
, and P. D.
Olmsted
, “Nonmonotonic models are not necessary to obtain shear banding phenomena in entangled polymer solutions
,” Phys. Rev. Lett.
102
(6
), 067801
(2009
).74.
Likhtman
, A. E.
, and R. S.
Graham
, “Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie-Poly equation
,” J. Non-Newtonian Fluid Mech.
114
(1
), 1
–12
(2003
).75.
Graham
, R. S.
, A. E.
Likhtman
, T. C. B.
McLeish
, and S. T.
Milner
. “Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release
,” J. Rheol.
47
(5
), 1171
–1200
(2003
).76.
Lu
, C.-Y. D.
, P. D.
Olmsted
, and R. C.
Ball
, “Effects of nonlocal stress on the determination of shear banding flow
,” Phys. Rev. Lett.
84
(4
), 642
–645 (2000
).77.
Graham
, R. S.
, E. P.
Henry
, and P. D.
Olmsted
, “Comment on “New experiments for improved theoretical description of nonlinear rheology of entangled polymers”
,” Macromolecules
46
(24
), 9849
–9854
(2013
).78.
Agimelen
, O. S.
, and P. D.
Olmsted
, “Apparent fracture in polymeric fluids under step shear
,” Phys. Rev. Lett.
110
(20
), 204503
(2013
).79.
Larson
, R. G.
, Constitutive Equations for Polymer Melts and Solutions
(Butterwirth
, Stoneham, MA
, 1988
).80.
Press
, W. H.
, S. A.
Teukolsky
, W. T.
Vetterling
, and B. P.
Flannery
, Numerical Recipes in C
(Cambridge University
, Cambridge
, 1992
).81.
Fujii
, S.
, H.
Morikawa
, M.
Ito
, and T.
Takahashi
, “Transient behavior of stress in a wormlike micellar solution under oscillatory shear
,” Colloid Polym. Sci.
293
(11
), 3237
–3248
(2015
).82.
Ewoldt
, R. H.
, and G. H.
McKinley
, “On secondary loops in LAOS via self-intersection of Lissajous-Bowditch curves
,” Rheol. Acta
49
(2
), 213
–219
(2009
).83.
Grand
, C.
, J.
Arrault
, and M. E.
Cates
, “Slow transients and metastability in wormlike micelle rheology
,” J. Phys. II
7
(8
), 1071
–1086
(1997
).84.
Fardin
, M. A.
, C.
Perge
, L.
Casanellas
, T.
Hollis
, N.
Taberlet
, J.
Ortín
, S.
Lerouge
, and S.
Manneville
, “Flow instabilities in large amplitude oscillatory shear: A cautionary tale
,” Rheol. Acta
53
(12
), 885
–898
(2014
).85.
Nghe
, P.
, S. M.
Fielding
, P.
Tabeling
, and A.
Ajdari
, “Interfacially driven instability in the microchannel flow of a shear-banding fluid
,” Phys. Rev. Lett.
104
(24
), 248303
(2010
).86.
Fielding
, S. M.
, “Linear instability of planar shear banded flow
,” Phys. Rev. Lett.
95
(13
), 134501
(2005
).87.
Fielding
, S. M.
, “Viscoelastic Taylor-Couette instability of shear banded flow
,” Phys. Rev. Lett.
104
(19
), 198303
(2010
).88.
Milner
, S. T.
, “Dynamical theory of concentration fluctuations in polymer solutions under shear
,” Phys. Rev. E
48
(5
), 3674
–3691
(1993
).89.
Schmitt
, V.
, C. M.
Marques
, and F.
Lequeux
, “Shear-induced phase separation of complex fluids: The role of flow-concentration coupling
,” Phys. Rev. E
52
(4
), 4009
–4015
(1995
).90.
Fielding
, S. M.
, and P. D.
Olmsted
, “Kinetics of the shear banding instability in startup flows
,” Phys. Rev. E
68
(3
), 036313
(2003
).91.
Fielding
, S. M.
, and P. D.
Olmsted
, “Early stage kinetics in a unified model of shear-induced demixing and mechanical shear banding instabilities
,” Phys. Rev. Lett.
90
(22
), 224501
(2003
).92.
Skorski
, S.
, and P. D.
Olmsted
, “Loss of solutions in shear banding fluids driven by second normal stress differences
,” J. Rheol.
55
(6
), 1219
–1246 (2011
).93.
Berret
, J.-F.
, G.
Porte
, and J.-P.
Decruppe
, “Inhomogeneous shear flows of wormlike micelles:mA master dynamic phase diagram
,” Phys. Rev. E
55
(2
), 1668
–1676
(1997
).© 2016 The Society of Rheology.
2016
The Society of Rheology
You do not currently have access to this content.
