In 1975 Doi and Edwards predicted that entangled polymer melts and solutions
can have a constitutive instability, signified by a decreasing stress for shear
rates greater than the inverse of the reptation time. Experiments did not
support this, and more sophisticated theories incorporated Marrucci's idea
(1996) of removing constraints by advection; this produced a monotonically
increasing stress and thus stable constitutive behavior. Recent experiments
have suggested that entangled polymer solutions may possess a constitutive
instability after all, and have led some workers to question the validity of
existing constitutive models. In this Letter we use a simple modern
constitutive model for entangled polymers, the non-stretching Rolie-Poly model
with an added solvent viscosity, and show that (1) instability and shear
banding is captured within this simple class of models; (2) shear banding
phenomena is observable for weakly stable fluids in flow geometries that impose
a sufficiently inhomogeneous total shear stress; (3) transient phenomena can
possess inhomogeneities that resemble shear banding, even for weakly stable
fluids. Many of these results are model-independent.Comment: 5 figure

J. M. Adams

M. Doi

P. D. Olmsted

English

arXiv

Recent experiments on entangled polymer solutions may indicate a constitutive instability, and have led some to question the validity of existing constitutive models. We use a modern constitutive model, the Rolie-Poly model plus a solvent viscosity, and show that (i) this simple class of models captures instability, (ii) shear banding phenomena is observable for weakly stable fluids in flow geometries with sufficiently inhomogeneous total stress, and (iii) transient phenomena exhibit inhomogeneities similar to shear banding, even for weakly stable fluids

Adams, JM

Olmsted, PD

White Rose Research Online

This is a repository copy of Nonmonotonic models are not necessary to obtain shear banding phenomena in entangled polymer solutions.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/75661/Article:Adams, JM and Olmsted, PD (2009) Nonmonotonic models are not necessary to obtain shear banding phenomena in entangled polymer solutions. Physical Review Letters, 102 (6). ISSN 0031-9007 https://doi.org/10.1103/PhysRevLett.102.067801eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. Nonmonotonic Models are Not Necessary to Obtain Shear Banding Phenomenain Entangled Polymer SolutionsJ.M. Adams1 and P. D. Olmsted21Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge, CB3 0HE, United Kingdom2School of Physics & Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom(Received 6 May 2008; published 12 February 2009)Recent experiments on entangled polymer solutions may indicate a constitutive instability, and have ledsome to question the validity of existing constitutive models. We use a modern constitutive model, theRolie-Poly model plus a solvent viscosity, and show that (i) this simple class of models capturesinstability, (ii) shear banding phenomena is observable for weakly stable fluids in flow geometries withsufficiently inhomogeneous total stress, and (iii) transient phenomena exhibit inhomogeneities similar toshear banding, even for weakly stable fluids.DOI: 10.1103/PhysRevLett.102.067801 PACS numbers: 61.25.he, 83.50.AxMuch of the rheology of entangled polymer solutionsand melts is captured by the molecular theory of Doi andEdwards (DE) [1], who argued that polymers relax bycurvilinear diffusion (reptation) within a tube of the sur-rounding polymers. The DEmodel has a local maximum inthe constitutive relation (the total shear stress as a functionof shear rate for homogeneous flows). The resulting non-monotonic relation (e.g., the dashed curves in Fig. 2) leadsto an instability that for many years was not observed inexperiments [2], but nonetheless attracted attention [3,4].This stress maximum is predicted to be less pronounced orabsent if the convected constraint release (CCR) of entan-glements due to flow [5–7] is incorporated. A suddenrelease of a constraint can relax both the orientation andconformation of a stretched polymer, which increases theshear stress and, for sufficiently frequent events, eliminatesthe instability. The CCR mechanism also leads to neutronscattering predictions that agree with experiment [8].Similar physics applies to solutions of breakable wormlikemicelles, in which the instability is well documented ex-perimentally and leads to shear banding, in which a fast-flowing oriented state coexists with a more disordered andviscous state along a stress plateau [9]. There, CCR is lesspronounced because of breakage and fails to ameliorate aconstitutive instability [7].Recently, Wang, Hu, and co-workers studied entangledsolutions of a high molecular weight (HMW) polymer inits own oligomer [10–13], or DNA solutions [14], finding anumber of results that may be consistent with instabilityand shear banding after all. In controlled shear rate mode aweakly increasing stress plateau of three decades in shearrate was found, whereas in controlled shear stress mode thesheared solution experienced a jump in the shear rate,together with spatially inhomogeneous birefringence[10]. Local velocimetry revealed spatially inhomogeneousvelocity profiles in both the transient and the steady state[11] regimes, while large amplitude oscillatory shear flow(LAOS) experiments showed an inhomogeneous banding-like shear rate profile at finite frequencies [15]. Similarbehavior was observed in a sliding plate shear cell inmonodisperse solutions [12]. Relaxation after a step straininduced a highly inhomogeneous velocity field with nega-tive local shear rates [13]. Hu et al. found similar inhomo-geneous flow behavior and possible signatures of shearbanding in polymer solutions, and wormlike micelle solu-tions at concentrations where severe shear thinning, but notbanding, might be expected [16].Wang et al. could not reconcile their results with existingtheory, and proposed that the instability is a yield likeeffect due to an unbalanced ‘‘entropic retraction force’’[17]. Here we show that much of the phenomenology ofthese experiments is consistent with the predictions of tubemodels with CCR, perhaps as anticipated in the originaltheory [3], without introducing new physics.Model.—We separate the total stress T into a fastNewtonian (or solvent) stress and a slow viscoelastic stress! (HMW polymer):T ¼ "pIþ 2!DþG!; (1)where I is the identity tensor,D ¼ 12½=vþ ð=vÞT', incom-pressibility determines the pressure p, and ! is the solventviscosity. We define the dimensionless quantity " ¼!=ðG#dÞ, where #d is the reptation time, and study thecreeping flow limit = ( T ¼ 0. We use the Rolie-Poly (RP)model [6], a simplified tube model that incorporates CCR[7], for the dynamics of !ðr; tÞ:ð@t þ v ( =Þ!" ð=vÞ (!"! ( ð=vÞT þ1#d! ¼ 2D"2#Rð1" AÞ½Iþ!ð1þ $AÞ' þD=2!; (2)where A ¼ ð1þ tr!=3Þ"1=2 and the Rouse time #R governs chain stretch. The stress ‘‘diffusion’’ termD=2! describes thePRL 102, 067801 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending13 FEBRUARY 20090031-9007=09=102(6)=067801(4) 067801-1 ! 2009 The American Physical Societyresponse to an inhomogeneous viscoelastic stress; whilenot in the original RP model, it can arise due to diffusion orfinite persistence length [18–20]. We specify Neumannboundary conditions (=! ¼ 0) [20,21].From experimental values of the plateau modulus G)6* 102 Pa, reptation time #d ) 20 s, and solvent viscosity!) 1 Pa s [10], we use " ¼ 10"5. Here we use #d=#R )103, which is consistent with the length of the stress-shearrate plateau reported in [10]. The parameter $ controls theefficiency of CCR, and its value is not yet agreed upon.Reference [6] chose $ ¼ 1 to fit steady state data inpolymer melts, and used multiple modes with $ ¼ 0:5 tofit experimental transient data. Here we tune between twoqualitatively different types of constitutive curve, either anonmonotonic (0.65) or a monotonic (0.728) constitutivecurve with a broad plateau (Figs. 1 and 2).Equation (2) was solved in one spatial dimension usingthe Crank-Nicolson algorithm [18], for unidirectionalCouette flow vðr; tÞ%̂ between cylinders of radii R1 andR2 parameterized by q + lnðR2=R1Þ. In this geometry thetotal shear stress Tr% ) 1=r2, so that the stress differenceacross the flow cell is ! lnTr% ¼ 2q. Cone and plate flowwith cone angles of % ¼ ð4,; 1,Þ has been reported [11,16],so we use consistent values of stress difference correspond-ing to q ’ !R=R ¼ ð2* 10"3; 2* 10"4Þ [20]. Stressesare measured in units of G, shear rates in units of #"1d ,and velocities in units of qR1=#d - !R=#d for small q. Toplot numerical data we use ", the dimensionless specifictorque (per height per radian) on the inner cylinder. Thediffusion constant used was D#d=ðR1qÞ2 ¼ 4* 10"4.Flow curves.—To calculate the steady state flow curves astep shear rate was applied from rest and evolved for 500#dwith time step 10"5#d, after which subsequent shear ratesteps and time evolutions were applied to scan up anddown in shear rate (Fig. 2). For nonmonotonic constitutivecurves ($ ¼ 0:65) shear banding always occurs, with hys-teresis and a stress ‘‘plateau.’’ For the monotonic case($ ¼ 0:728) shear banding could be inferred in the morehighly curved geometry with the larger stress difference(larger q), since the flow curve no longer follows theconstitutive curve, butwithout hysteresis. Crudely, a mono-tonic flow curve exhibits bandinglike flows when most ofthe shear rate in the gap occurs over a small range ofstresses; i.e., the slope of the plateau must be much smallerthan the apparent slope specified by the flow geometry:d"d _&!!!!!!!!C:C:."ðR1Þ " "ðR2Þ! _&!!!!!!!!g)eq " 1; (3)where ‘‘C.C.’’ denotes the flat portion of the constitutivecurve (dashed in Fig. 2) and ‘‘g’’ refers to the range oftorques and shear rates specified by the flow geometry.The steady state velocity profiles are shown in Fig. 3 assolid (red) lines. The nonmonotonic flow curves ($ ¼0:65) lead to a pronounced kink in the velocity profile, asignature of shear banding. The monotonic case does notshear band in the flatter geometry (small q), but for a morecurved geometry (larger q) more shear rates are accessibleand the resulting smooth velocity profile could easily beinterpreted as banding [11]; certainly the constitutive curveis not followed (Fig. 2). Similar smooth profiles werereported in [11,16], in a flow geometry with q ’0:004–0:02. A slightly increasing stress plateau over sev-eral decades in shear rates (as in [10]) would thus lead toapparently banding (inhomogenous) flow in geometrieswith small stress gradients, but homogeneous flow obtainsfor sufficiently small q [Eq. (3)].Start-up transients.—Transients were studied by evolv-ing from rest using a time step of 10"5#d (Fig. 3). In allcases shown here strongly inhomogeneous flow developsafter the stress overshoot, leading to a sharply bandedtransient state, with a negative velocity and shear rate inthe less viscous band. In the monotonic case the velocityprofile eventually smooths out. For a narrower stress pla-FIG. 1. (a) Parameters ($, ") where the constitutive curve ofthe Rolie-Poly model is nonmonotonic (shaded). (b) Sectionthrough Couette rheometer showing the flow field and totalstress components, T.FIG. 2 (color online). Flow curves (solid black) and constitu-tive curves (dashed red) for a range of stress gradients q andCCR values $. The shaded area shows the size of the torquedifference !log10" ¼ log10e2q. Circles indicate applied shearrates at which the transient state responses are shown in Fig. 3.PRL 102, 067801 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending13 FEBRUARY 2009067801-2teau (e.g., $ ¼ 0:3, not shown) the overshoot has a lesspronounced kink and typically a positive shear rate. Wehave found inhomogeneous transients with negative veloc-ities with stress differences corresponding to a cone angle% ¼ 0:003,, while for % ¼ 0:001, the inhomogeneousstrain rate is no longer negative, and for % ¼ 0, the flowremains homogeneous. With perturbed initial conditionsthen the inhomogeneous transient behavior returns. Thetransient for a monotonic model ($ ¼ 0:728) in whichspatial gradients are artificially prohibited exhibits a slowerdecrease after the stress overshoot than in the spatiallyresolved model (Fig. 3); hence, inhomogeneities are im-portant when using transient data to help differentiatecandidate constitutive models [1,22].Large amplitude oscillatory shear (LAOS).—A sinusoi-dal spatially averaged shear rate was applied with fre-quency # and maximum shear rate _&*, and evolvedfrom rest (zero stress) until initial transients decayed. Wecharacterize the dynamics by the Deborah (De ¼ ##d)and Weissenberg (Wi ¼ _&*#d) numbers. Low De (fre-quency) should lead to some features of the steady statebehavior, such as transient banding for Wi roughly withinthe nonmonotonic part of the flow curve, while higherfrequencies (De) should produce sharper profiles similarto the transient behavior in Fig. 3, since the system cannotrelax before flow reversal. At the highest frequencies thefast reversing dynamics should prohibit an inhomogenousstate.Figure 4(a) shows this behavior on a ‘‘Pipkin diagram’’of Wi vs De, for a monotonic flow curve ($ ¼ 0:728) in aslightly curved geometry. The inhomogeneous profiles inthe banding regime [Fig. 4(d)] can be represented para-metrically in terms of shear rate and torque, ð _&ð*Þ;"ð*ÞÞ[Fig. 4(b)]. At the high stress regions of the cycle a portionof the sample enters the high shear rate band as reportedexperimentally [22]. In these calculations the position */ ofthe interface at a given strain _&0=# ¼ 3 varied with De as*/ ) ðDeÞ', where ') 0:4–0:6, unlike the fixed positionreported in Ref. [15]. We suspect that these experimentsdid not attain steady state. The torque overshoot is typicalof polymer solutions, like that found in [16] (Fig. 4). Atlow frequencies the system has time to find a selectedstress, which remains constant while the shear band growsinto the cell. At high frequencies the fluid cannot relax orshear band, which leads to a sinusoidal response and anearly affine spatial profile.Step strain.—Some step strain experiments found astrong inhomogeneous recoil that developed a negativevelocity gradient [13]. We illustrate this for a monotonicconstitutive curve ($ ¼ 0:728), Fig. 5. As in Fig. 5 of [13],an inhomogeneous velocity profile develops and the veloc-ity becomes negative as the system recoils. Experi-mentally, the total displacement after recoil is of the order(a)(b)(c)(d)55544433222113FIG. 4 (color online). (a) Pipkin diagram (Wi vs De) for $ ¼0:728 and q ¼ 2* 10"3, showing regions with homogeneous(0) and inhomogeneous profiles (d). (b) Parametric shearrate—torque profiles ð _&ð*Þ;"ð*ÞÞ (dashed lines: 3,4) overlayingthe constitutive curve (solid). (c) Torque evolution for differentDe noted on (a). (d) Velocity profiles for ii); profiles (1–5) are atdifferent times in the cycle for De ’ 3 (ii) [0 in (c)].FIG. 3 (color online). Velocity as a function of position * ¼q"1 lnðr=R1Þ for different ($, q), at shear rate log10 _& ¼ 1:2 (0in Fig. 2). The solid (red) lines indicate the steady state profile,and dashed lines are transient profiles at the times shown. Thetransient torque response "ðtÞ in the spatially uniform model isshown for the monotonic case ($ ¼ 0:728) by a dashed line.PRL 102, 067801 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending13 FEBRUARY 2009067801-3of a tenth of the gap size which is comparable with thatobserved here. Thus, inhomogeneous flow must be consid-ered when using step strain data to help discriminateamong candidate constitutive models, as when using theDE damping function [1,22].Summary.—We have shown that behavior reminiscent ofshear banding, as reported recently, can be reproducedusing the Rolie-Poly model supplemented by a term toaccommodate spatial gradients. The RP model containsan unknown parameter $, which controls the efficacy ofconvected constraint release. Even for $ large enough toyield a stable (monotonic) constitutive curve, shear band-ing signatures can appear if the ‘‘stress plateau’’ is flatenough: (i) a geometry with a high stress gradient caninduce a flow profile that could be mistaken for banding;(ii) sharp bandinglike profiles can appear in start-up tran-sients even though the steady state is nonbanded;(iii) LAOS can trap these sharp transient profiles; and(iv) relaxation after a large step strain can be very inho-mogeneous, sometimes with a negative shear rate recoil.Several recent experiments, particularly on polydispersepolymer solutions, may fall into this category [16]. Awideplateau is believed to accompany very highly entangledsystems [7], and the larger number of relaxation times arelikely to render polydisperse systems intrinsically morestable than monodisperse systems [3], as was noted inrecent experiments [16]. Our results are not specific tothe RP model; Zhou et al. recently studied a differenttwo-fluid model of shear banding (with a nonmonotonicconstitutive relation), and found qualitative results similarto some of ours [23].We thank S.-Q. Wang, R. Graham, T. McLeish, O.Radulescu, and S. Fielding for lively discussions. Thiswork was supported by the Royal Commission of 1851.[1] M. Doi and S. F. Edwards, The Theory of PolymerDynamics (Clarendon, Oxford, 1989).[2] R. A. Stratton, J. Colloid Interface Sci. 22, 517 (1966);E. V. Menezes and W.W. Graessley, J. Polym. Sci., Polym.Phys. Ed. 20, 1817 (1982); C.A. Hieber and H.H. Chiang,Rheol. Acta 28, 321 (1989); C. Pattamaprom and R.Larson, Macromolecules 34, 5229 (2001).[3] M. Doi and S. Edwards, J. Chem. Soc., Faraday Trans. 75,38 (1979).[4] T. C. B. McLeish and R. C. Ball, J. Poly. Sci., B, Polym.Phys. 24, 1735 (1986); T. C. B. McLeish, J. Poly. Sci., B,Polym. Phys. 25, 2253 (1987).[5] G. Marrucci, J. Non-Newtonian Fluid Mech. 62, 279(1996); D.W. Mead, R. G. Larson, and M. Doi,Macromolecules 31, 7895 (1998); G. Ianniruberto andG. Marrucci, J. Non-Newtonian Fluid Mech. 95, 363(2000).[6] A. E. Likhtman and R. S. Graham, J. Non-NewtonianFluid Mech. 114, 1 (2003).[7] S. T. Milner, T. C. B. McLeish, and A. E. Likhtman,J. Rheol. (N.Y.) 45, 539 (2001).[8] J. Bent, L. R. Hutchings, R.W. Richards, T. Gough,R. Spares, P. D. Coates, I. Grillo, O. G. Harlen,D. J. Read, and R. S. Graham, et al., Science 301, 1691(2003).[9] N. A. Spenley, M. E. Cates, and T. C. B. McLeish, Phys.Rev. Lett. 71, 939 (1993); M. E. Cates and S.M. Fielding,Adv. Phys. 55, 799 (2006).[10] P. Tapadia and S. Q. Wang, Phys. Rev. Lett. 91, 198301(2003).[11] P. Tapadia and S. Q. Wang, Phys. Rev. Lett. 96, 016001(2006).[12] P. E. Boukany and S.-Q. Wang, J. Rheol. (N.Y.) 51, 217(2007).[13] S.-Q. Wang, S. Ravindranath, P. Boukany, M.Olechnowicz, R. Quirk, A. Halasa, and J. Mays, Phys.Rev. Lett. 97, 187801 (2006).[14] P. E. Boukany, Y. T. Hu, and S.Q. Wang, Macromolecules41, No. 7, 2644 (2008).[15] P. Tapadia, S. Ravindranath, and S. Q. Wang, Phys. Rev.Lett. 96, 196001 (2006).[16] Y. T. Hu, L. Wilen, A. Philips, and A. Lips, J. Rheol.(N.Y.) 51, 275 (2007); Y. T. Hu, C. Palla, and A. Lips,J. Rheol. (N.Y.) 52, 379 (2008).[17] S.-Q. Wang, S. Ravindranath, Y. Wang, and P. Boukany,J. Chem. Phys. 127, 064903 (2007); Y. Wang, P. Boukany,S.-Q.Wang, and X. Wang, Phys. Rev. Lett. 99, 237801(2007).[18] P. D. Olmsted, O. Radulescu, and C.-Y. D. Lu, J. Rheol.(N.Y.) 44, 257 (2000).[19] A.W. El-Kareh and L.G. Leal, J. Non-Newtonian FluidMech. 33, 257 (1989).[20] J.M. Adams, S.M. Fielding, and P. D. Olmsted, J. Non-Newtonian Fluid Mech. 151, 101 (2008).[21] A. V. Bhave, R. C. Armstrong, and R.A. Brown, J. Chem.Phys. 95, 2988 (1991).[22] S. Ravindranath and S. Q. Wang, Macromolecules 40,8031 (2007).[23] L. Zhou, P. A. Vasquez, L. P. Cook, and G.A. McKinley,J. Rheol. (N.Y.) 52, 591 (2008).FIG. 5. Relaxation of a step strain of & ¼ 8, applied using_&#d ¼ 40 for t ¼ 0:2#d. (Left) Velocity before the shear ratestops and (right) snap shots of recoil velocities.PRL 102, 067801 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending13 FEBRUARY 2009067801-4

Nonmonotonic models are not necessary to obtain shear banding phenomena in entangled polymer solutions

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Nonmonotonic Models are Not Necessary to Obtain Shear Banding Phenomena in Entangled Polymer Solutions

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