Homogeneous Shear, Wall Slip, and Shear Banding of Entangled

Macromolecules 2011, 44, 183–190

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DOI: 10.1021/ma101223q

Homogeneous Shear, Wall Slip, and Shear Banding of Entangled Polymeric Liquids in Simple-Shear Rheometry: A Roadmap of Nonlinear Rheology Shi-Qing Wang,* S. Ravindranath,† and P. E. Boukany‡ Department of Polymer Science, University of Akron, Akron, Ohio 44325, United States. † Current affiliation: Celanese/Ticona. ‡Current affiliation: The Ohio State University Received June 1, 2010; Revised Manuscript Received December 3, 2010

ABSTRACT: The recent particle-tracking velocimetric (PTV) observations revealed that well-entangled polymer solutions and melts tend to either exhibit wall slip or assume an inhomogeneous state of deformation and flow during nonlinear rheological measurements in simple-shear rheometric setups. Many material parameters and external conditions have been explored since 2006, and a new phenomenological picture has emerged. In this Perspective, we not only point out the challenges to perform reliable rheometric measurements but also discuss the relation between wall slip and internal (bulk) cohesive breakdown and summarize all available findings in terms of a phase diagram. This map specifies the conditions under which shear homogeneity, interfacial slip, and bulk shear inhomogeneity would prevail. The paper is closed by enumerating a number of unresolved questions for future studies.

1. Introduction Rheometry is a widely used experimental technique in polymer science and engineering. The polymer scientist often makes rheometric measurements to characterize the linear viscoelastic properties of his polymer samples. Frequently, linear viscoelastic characteristics have been thought to yield unique information about such molecular information as the molecular weight distribution of a linear polymer melt.1,2 The rheometric measurements usually follow the standard analysis that is based on three key assumptions: (1) no-slip boundary conditions and (2) homogeneous velocity profile during shear as shown in Figure 1a and (3) relaxation upon shear cessation occurring without emergence of macroscopic motions. These assumptions about the state of deformation hold true by definition for linear viscoelastic measurements and have been the basis of most rheometric studies in the literature. The most common rheometric setup3 is a rotational shear device involving either cone-plate (see the middle figure in TOC graphic) or parallel-plate (disk) shear cell, where the typical sample thickness H is macroscopic, in the range 0.1-1 mm. Polymer scientists and engineers have also performed nonlinear rheological studies4 to learn more about the molecular structures including chain architecture.5 In studies involving simple shear, the dominant nonlinearity is strain softening: both the relaxation modulus and steady-shear viscosity of entangled polymers are lower than their values in the linear response regime. In absence of any direct evidence to contradict, in the past we assumed that such strain softening had occurred in a homogeneous manner. To verify whether this is the case, we have recently developed an in situ particle-tracking velocimetric (PTV) method as shown in the middle figure of the TOC graphic.6 The combination of rheometric and PTV measurements revealed a clear violation of each of the three basic assumptions listed in the first paragraph. In other words, shear inhomogeneity was found to take place *Corresponding author. E-mail: [email protected]. r 2011 American Chemical Society

cone-plate shear cell during startup shear7-10 and large-amplitude oscillatory shear.11 Entangled solutions12 and melts13 were not quiescent upon shear cessation from a large step strain. Thus, the nonlinear responses of entangled polymers appear far more complicated than previously revealed. In other words, when using any commercial rheometer for simple shear, we could no longer assume a priori that the states of deformation and flow are homogeneous. In this Perspective, we indicate some guidance for future rheological experiments by showing when, how, and why the three basic assumptions in polymer rheology may be invalid. We first briefly review a primary type of shear inhomogeneity, i.e., wall slip, as schematically illustrated in Figure 1b. We then describe when nonlinear rheological measurements might be complicated by such effects as edge fracture. Subsequently, we enumerate the situations where shear homogeneity prevails before discussing a relation between wall slip and shear banding. We summarize the available results in terms a phase diagram to map out the parameter space and to show where shear homogeneity, wall slip, and shear banding (schematically illustrated in Figure 1c) are “located”. We conclude by indicating what we still do not know and the theoretical implications of the emerging PTV observations. 2. Wall Slip: Primary Example of Shear Inhomogeneity and How To Minimize Entangled linear polymeric liquids can display one unique property that nonentangled liquids do not possess: They are capable of macroscopically measurable wall slip. Phenomena involving polymer wall slip have been reported since the 1950s.14-20 De Gennes21 offered in 1979 the first understanding of why entangled polymers have the ability to display observable wall slip on macroscopic scales. According to de Gennes, slip can be quantified by the extrapolation length b defined in Figure 1b and can be macroscopically large due to the high shear viscosity η originating from chain entanglement. Specifically, it can be Published on Web 01/18/2011

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Shi-Qing Wang is Professor of Polymer Science at the University of Akron. He obtained a BS in Physics from Wuhan University in China before receiving his Ph.D. in Physics from the University of Chicago in 1987 and did postdoctoral studies in the Department of Chemistry at UCLA before joining the Department of Macromolecular Science and Engineering at Case Western Reserve University in Fall 1989. Dr. Wang was elected an APS Fellow in 1997. Since Fall 2000, he has been in his current position. His research interests are in polymer physics, particularly interfaces, dynamics, and rheology of entangled polymeric materials. Since summer 2006, Dr. Wang has made at least one visit each year to different academic institutions in China, where he has been made a guest professor in Beijing University of Chemical Technology, Nanjing University, University of Science and Technology of China, and Changchun Institute of Applied Chemistry of the Chinese Academy of Sciences.

Sham Ravindranath obtained his B.E. in Polymer Science and Technology from Sri Jayachamarajendra College of Engineering, Visvesvaraya Technological University, India. He received his Ph.D in Polymer Science from University of Akron. His doctoral work focused on understanding the nonlinear rheology of entangled polymer solutions in Professor Shi-Qing Wang’s group. He is currently working as Analytical Scientist in Celanese/Ticona.

readily shown that21,22 b ¼ ðη=ηi Þa

ð1Þ

where η is the shear viscosity in the bulk, ηi is a local viscosity at the polymer/wall interface, and a is the interfacial slip-layer thickness of a molecular dimension. This viscosity ratio can be exceedingly large, allowing b to attain a macroscopic magnitude when the value of ηi diminishes as a consequence of either chain desorption or disengagement of adsorbed chain from the bulk entanglement network.22 Subsequent to de Gennes’ analysis,21 a number of studies were devoted to theoretical23-25 and experimental accounts26-34 of steady-state wall slip.22,35 Since wall slip is such an important rheological character of entangled polymers, we describe it in some detail in Appendix A. In particular, we

Pouyan E. Boukany obtained his B.S and M.S degree in Textile Engineering with major in Textile Chemistry and Fiber Science from the Isfahan University of Technology, Iran. He received his Ph.D in Polymer Science from University of Akron. His doctoral work explored various aspects of nonlinear responses of entangled liquids ranging from DNA fluids to entangled polymer melts in Professor Shi-Qing Wang’s group. He is currently a Postdoctoral Research Associate working with Professor L. James Lee in the Center for Affordable Nanoengineering of Polymeric Biomedical Devices at The Ohio State University.

illustrate in Appendix A why both entangled solutions and melts could exhibit significant wall slip. In Appendix B, we present a list of nomenclatures to ease the reading of this paper. Because polymeric materials, particularly entangled polymers, are prone to wall slip, rheometric measurements of nonlinear behavior can be difficult to interpret. Many of the nonlinear rheological measurements of entangled polymer solutions and melts in the literature might have involved wall slip. This may be particularly true in the large step strain studies36-41 of well-entangled polymer solutions made with small organic liquids as solvents. In other words, wall slip42 is likely the cause28 for the observed ultra strain softening37-40 in stress relaxation from step strain that deviated from the prediction43 of the Doi-Edwards tube model.44 Even when the experimental data agree with the model calculation,38 interfacial failure can be present after shear cessation as revealed recently in a PTV study.12 More discussion on the development of the tube theory in the context of continuous shear is provided in Appendix C. For entangled solutions, eq A.4 in Appendix A explains why wall slip could be present in previous studies of entangled polymer solutions in both step shear36-40 and continuous shear45 due to use of small molecular organic solvents with a water-like viscosity ηs. Because of the low ηs in the dominator of eq A.4, bmax can be comparable to or larger than the sample thickness H. Solutions with higher concentration c or polymer molecular weight Mw are more affected by wall slip in their nonlinear rheological measurements. This explains why there is more strain softening in a step strain test when cMw is higher.37-40 For entangled polymer solutions that show significant wall slip,46,47 we have confirmed two ways to minimize slip. An effective way to get rid of wall slip stems from our understanding of what parameters control the magnitude of wall slip. It has been shown that the degree of wall slip can be reduced by using an entangled polymeric solvent.12,48 In other words, eq A.4 for bmax not only explains why polymer solutions can undergo strong wall slip but also points to an effective way to minimize wall slip. According to a previous study,49 the solution viscosity η does not increase proportionally with the solvent viscosity ηs when the polymeric solvent’s molecular weight increases. Thus, bmax of eq A.4 can be considerably reduced by using a polymeric

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Figure 1. (a) 3-D view of homogeneous shear at an apparent shear rate γ_ app = V/H in absence of wall slip. (b) Side view: velocity profile in presence of wall slip on two identical shearing surfaces, when γ_ appτ > 1, with τ being the dominant chain relaxation time. When a slip velocity Vs is present, the _ (c) Side view: bulk shear banding amount of slip can be quantified by the extrapolation length b defined in the figure involving the bulk shear rate γ. along with a small amount of wall slip is a characteristic response of well-entangled linear polymer solutions during startup shear at high rates when wall slip alone can no longer prevent the sample bulk from shearing at rates higher than the dominant relaxation rate 1/τ.

solvent of sufficiently high molecular weight.12 Surface roughness has been demonstrated to be effective in reducing wall slip.32 We have also used sandpapers as the shear surfaces to produce shear banding in a solution that would only exhibit interfacial failure on smooth surfaces.48,50 Well-entangled melts undergo even greater amount of wall slip:21,22,34,51 The maximum value bmax is inherently dictated by the degree of chain entanglement, as expressed in eq A.1. Thus, for well-entangled linear polybutadiene melts with a large bmax, large step strain tests could only demonstrate ultra strain softening,52 resulting from wall slip. We must have some means for the shear surfaces to grab polymer melts so that sufficient shear deformation can be exerted on the sample to probe its intrinsic nonlinear rheological responses. For certain melts such as styrene-butadiene rubber (SBR) with high vinyl content, we have demonstrated that various superglues can effectively fix the samples onto solid surfaces so that simple shear can take place in sliding plate without wall slip.13 3. Meniscus Instability or Edge Fracture Simple shear in rotational device, e.g., cone-plate or paralleldisk or circular Couette (coconcentric cylinders), can in principle reach a steady rheological state. In reality, such geometries leave a rim of meniscus exposed to the ambient air. Since the free surface is cohesively weaker, highly viscoelastic materials such as strongly entangled polymer melts and solutions are often found to undergo edge or meniscus instability upon significant strains. This finite size effect severely constrains rheometric studies of large deformation behavior. The intrinsic difficulty of meniscus instability in rotational rheometry can be successfully dealt with by using a cone-partitioned plate (CPP) setup to decouple rheometric measurements from any disturbance due to the edge fracture. In CCP, there is a rotating cone as in a conventional cone-plate setup, and the stationary surface is made of two pieces: an inner disk linked to the torque transducer and an outer ring unattached to the inner disk. Because the inner disk and the ring are not physically connected, any edge fracture or instability occurring over the ring region would not affect the rheometric measurements. Figure 1 of ref 10 depicted such a CPP device coupled with particle-tracking velocimetric (PTV) capability. For entangled polymer solutions, CPP appears to ensure reliable characterization during (a) steady-state continuous shear,10 (b) large-amplitude oscillatory shear of linear polymer melts,53 and (c) nonquiescent relaxation after a large step shear.54 4. Particle-Tracking Velocimetry: A Necessary Tool in Nonlinear Rheometry In complicated setups such as channel flow, particle tracking has been applied to visualize the state of polymer flow since 1950s.55-57 In most cases, capillary flow in the entrance and channel flow in die land have been observed using either particle-tracking55-59 or laser Doppler velocimetry.60,61 Such

visualization is informative and necessary because the flow field is unknown a priori. Particle tracking velocimetry has been applied to visualize complex flow behavior of other fluids.62-66 In simple-shear rheometry we have often conveniently assumed that the state of deformation or flow would be homogeneous independent of the constitutive properties of entangled polymers. Typically, rheometric studies of entangled polymers have been carried out on the basis of shear homogeneity, and no commercial rheometers have been designed to directly probe the velocity field in such shear cells as cone-plate or parallel disk. We have recently recognized the need to independently measure the states of deformation and flow during startup shear, largeamplitude oscillatory shear, and step shear. Specifically, an effective PTV method6 has been implemented to determine whether entangled polymer solutions (made with polymeric solvents) would undergo inhomogeneous shear instead of, or in addition, to wall slip. Because of the revelation of wall slip and shear banding through the PTV observations,6 the PTV measurements should now be regarded as an indispensable part of any nonlinear rheological study of highly viscoelastic materials including entangled polymers. It is worth noting in passing that the time resolution of our PTV method is only limited by the camera speed, whereas the previous attempts placed a CCD camera along the velocity gradient direction and could only image the shear-vorticity plane one layer at a time.28,67 In fact, because of the poor vertical resolution, this arrangement may only be effective to monitor wall slip behavior at the sample/plate interface. 5. The Roadmap for Nonlinear Rheology of Entangled Polymers In this section we begin by indicating the situations where rapid external shearing fields would not result in inhomogeneous deformation and flow. Then we discuss the relation between wall slip and shear banding. We close by presenting a phase diagram to summarize what can be expected in startup continuous shear. 5.1. Homogeneous Shear with Wiapp > 1 and Quiescent Relaxation after Step Shear. We enumerate several conditions, under which homogeneous shear and quiescent relaxation prevail so that rheometric measurements may receive straightforward interpretation. When the apparent Weissenberg number Wiapp = γ_ appτ > 1, shear homogeneity can still prevail under the following circumstances. (a) When an entangled polymer solution is not strongly entangled, we have bmax/H , 1. Consequently, neither wall slip nor shear banding is observable. In other words, the steady-state profile tends to be homogeneous although transient shear inhomogeneity may occur.9,68,69 (b) Even when the entanglement level is high, shear inhomogeneity can be suppressed10,48 during startup shear by reducing the extrapolation length bmax using a polymeric solvent of sufficiently high molecular weight to produce a large ηs in eq A.4.

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Figure 2. Phase diagrams in the parameter space of Wiapp vs 2bmax/H (double log scales) to map out the different responses in terms of the state of deformation for (a) a fixed sample thickness so that bmax/H* simply denotes the level of chain entanglement and (b) a given sample so that b*max/H increases with lowering H until the upper bound is reached at H = 2lent when only wall slip is possible, moving along the red vertical line. (c) Regions of the phase diagram where PTV studies have been carried out, including the small bmax/H regime inside the rectangle, and six solutions of different 2bmax/H equal to 22, 26, 136, 140, 259, and 1470. The open symbols indicate approximate shear homogeneity. Table 1. Five PTV Studies of Entangled Solutions of Different Ability To Undergo Slipa references

samples

solution solvent viscosity (kPa) viscosity (Pa) 2bmax/H

ref 9

0.7M(5%)-2K 21 97 0.01 0.7M(10%)-9K 57 10 0.2 1M(15%)-9K 510 10 1.5 ref 10 1M(10%)-1.5K 50 0.7 3 1M(10%)-15K 250 36 0.27 ref 72 (diamond symbols 1% DNA solution 190 0.001 136 in Figure 2c) water based ref 50 (square symbols in Figure 2) water based 5.8 0.001 1470 ethylene glycol based 13 0.023 142 glycerol based 58 0.7 22 ref 75 at a low gap of H = 50 μm 1M(13%)-1.5K 130 0.7 259 (circle symbols in Figure 2c) 1M(13%)-10K 600 14 26 a H = 800-900 μm for the first four papers. The last column indicates available PTV results.

(c) When b/H , 1, macroscopic motions after a large step strain may not be observable. In other words, with b/H , 1, no macroscopic motions take place during the relaxation.48 (d) Preliminary evidence based on both entangled DNA70 and polymer71 solutions indicates that shear banding observed upon sudden startup shear can be avoided if the shear rate is gradually approached from the Newtonian regime. (e) Shear homogeneity has been observed to return at sufficiently high rates where no entanglement network structure could survive.9,69 This condition corresponding to a Weissenberg number Widis is discussed in some detail in Appendix A. 5.2. Relationship between Wall Slip and Bulk Shear Banding. Wall slip usually precedes any bulk inhomogeneous response to an external shearing field because polymer/wall interface typically has weaker adhesion than the cohesion of entangled polymers. Because of wall slip sketched in Figure 1b, the sample bulk can avoid high shear and stay in the Newtonian regime (Wiapp < 1) until an apparent rate γ_ app, corresponding to a critical Weissenberg number50,72 Wiws-sb ¼ 1 þ 2bmax =H

ð2Þ

where the subscript “ws-sb” denotes this critical shearing condition for the transition from wall slip to (bulk) shear banding. To estimate this critical condition, we have equated the bulk shear rate γ_ c at the borderline between the Newtonian and shear thinning regimes to the reciprocal relaxa-

Wiapp

PTV observations of steady state

10-150 15-50 45-400 16-110 14-140 14-14000

no slip, no banding only transient banding banding slip, banding no slip, no banding slip, banding

0.5-220 0.9, 750 950-22800 12-240

slip banding banding slip

20-390

slip, banding

tion time 1/τ, i.e., γ_ cτ = 1. The expression bmax is given by eq A.1 for entangled melts and by eq A.4 for entangled solutions in Appendix A. According to eq 2, the higher the value of bmax/H, the wider range of the apparent rate is for the wall slip to be the dominant response. A more recent study involved only this slip regime because Wiapp , Wiws-sb in their experiments.73 When Wiapp > Wiws-sb, shear banding may be expected to emerge for well-entangled polymers. Moreover, bulk shear banding could emerge at a much lower value of Wiapp than Wiws-sb when rough shearing surfaces are used to remove wall slip. For example, surfaces made of sandpaper can effectively reduce wall slip so that shear banding takes its place.48,50 5.3. The “Phase Diagram”. Clearly, bmax/H is the key parameter to control the nonlinear rheology of entangled polymers, dictating when and what type of shear inhomogeneity may occur for a given apparent Weissenberg number Wiapp. Here we summarize what we know about the nonlinear rheological responses to startup shear in terms of a phase diagram as shown in Figure 2a, where for a fixed gap distance H* the X axis of 2bmax/H* is controlled by bmax. For linear-chain melts, eq A.1 shows how the molecular weight dictates the magnitude of bmax. For entangled solutions made with polymeric solvents, besides the molecular weight of the solute, the solvent viscosity ηs can also alter bmax according to eq A.4. For a given polymeric system, i.e., for a fixed b*max, varying H would have the consequence shown in Figure 2b.

Perspective

Figure 3. PTV observations of steady-state shear in a gap of H = 50 μm at two different rates of 0.3 and 3.0 s-1 for an entangled solution of 13 wt % polybutadiene (Mw ∼ 103 kg/mol) in a polybutadiene solvent (Mw ∼ 10 kg/mol).

In both (a) and (b) of Figure 2, at small values of bmax/H, we have the vertical “homogeneous” strips to the left of the vertical white lines. The inclined white borderline between wall slip and shear banding is given by eq 2. The boundary between “uniform disentanglement” and “shear banding” is depicted by eq A.3 or eq A.6, which is inclined in Figure 2a and horizontal in Figure 2b. Actually, Figure 2a is somewhat analogous to Figure 6 of ref 6. Both (a) and (b) of Figure 2 serve as a roadmap for the steady-state behavior of wellentangled polymers in startup shear. We can summarize the available results from the literature74 in Figure 2c as well as in Table 1 that lists five PTVbased studies9,10,50,72,75 on 11 entangled solutions. Here the specification of the various 1,4-polybutadiene (PB) solutions follows the following convention: 0.7M(5%)-2K means 5 wt % of 1,4-PB with molecular weight equal to 0.7 106 g/mol in a PB solvent of molecular weight equal to 2 kg/mol. The first two papers in Table 1 focused on the phase space circled by the (red) rectangle in Figure 2c in the region of 2bmax/H < 1. We specifically mark the other regions corresponding to the other six solutions listed at the bottom of Table 1, having respectively 2bmax/H equal to 136, 1470, 142, 22, 259 and 26. Here both axes, i.e., Wi and 2b/H, are implicitly on logarithmic scales. Particularly worth mentioning is the system with 2bmax/H=26 represented by the five circles that shows both slip dominant and shear banding prevailing features as the applied shear rate increases. Also special about this study75 is the significantly reduced sample thickness from a conventional rheometric dimension of ca. 1000 to 50 μm. Reducing H broadens the window for wall-slip-dominated rheological responses, leading some to suggest76 that wall slip would be the only characteristic rheological response of entangled polymers. Indeed, when we reduce H from 1 mm to 50 μm, a sample that would not show significant wall slip “suddenly” does. However, Figure 3 shows that the wall-slip dominant response switches to bulk shear banding behavior as the applied rate crosses the predicted borderline given by eq 2, represented by the two (light blue) circles in Figure 2c. This switch would always occur unless the sample thickness H becomes comparable to the entanglement spacing lent. Figure 2b shows that the shear banding regime shrinks to zero only at the red vertical line at H=lent/2. The window for shear banding is given by Widis/ Wiws-sb = H/2lent, as indicated in Figure 2a,b. 6. Unresolved Questions in Both Shear and Extensional Rheometry We close by listing a number of important issues that we have either not resolved or only obtained partial, preliminary answers. Our PTV method has its natural constraints. In particular,


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startup continuous shear over a large amount of strain units always requires wrapping a film around the meniscus to prevent sample loss. Sometimes, air bubbles are trapped during sample loading. Small bubbles are difficult to remove under vacuum. The role of bubbles remains to be clarified. Moreover, many questions remain as listed in no particular order that future studies must deal with: (1) Shear banding appears not to be a steady-state property for entangled polymers. At least for solutions, we were able to achieve homogeneous shear by very gradual rate ramp-up instead of a sudden startup shear.70,71 It remains to be seen whether the same is true for entangled melts. (2) Only a handful of studies exist in the literature on transient rheological behavior of melts during startup shear.13,77,78 Rheological measurements along with PTV observations of steady shear behavior of entangled melts remain to be carried out. (3) During startup shear, shear banding emerges after the shear stress maximum where the high shear band is usually adjacent to the moving surface and low shear band stays next to the stationary surface. The origin of this symmetry breaking is elusive and could be related to the presence of polymer/wall interfaces although the initial elastic deformation is homogeneous. (4) The cohesive breakdown, i.e., elastic yielding after a large step shear, appears to occur within the sample thickness at unpredictable locations for entangled solutions although styrene-butadiene rubbers show failure in the midplane when they are well adhered to the shear surfaces by superglue.13 More data are required to show whether the failure location is truly random. (5) Macroscopic shear inhomogeneity has been observed in various entangled polymeric liquids. However, the PTV technique cannot detect any microscopic structural inhomogeneity on micrometer to submicrometer length scales. Systematic smallangle neutron scattering experiments should be carried out during shear banding to probe any microscopic structural inhomogeneity as a consequence of chain disentanglement. Preliminary experiments found some submicrometer structural development during shear.79 (6) Other complementary methods including optical birefringence and small-angle X-ray scattering along the vorticity direction would help gathering more information in a spatial-resolved manner about the nature of the shear inhomogeneity. (7) So far, most studies focused on monodisperse samples. It remains to be demonstrated how shear inhomogeneity weakens with increasing polydispersity in the molecular weight distribution.8 PTV studies have yet to be carried out for miscible and immiscible blends in the nonlinear response regimes. 7. Summary: Localized Yielding in Entangled Polymers In such standard rheometric tests as startup shear and step shear, the PTV observations offer fresh insight into the nature of rheological responses of entangled polymers. A sudden startup simple-shear produces the well-known stress overshoot, which is an indication of yielding80 with universal scaling characteristics:81,82 At the shear stress maximum, the system reaches a force imbalance between the growing elastic (intrachain) retraction force and intermolecular gripping force,83 beyond which further deformation of the entanglement network is no longer possible and chains mutually slide past one another to approach a state of flow.84 This transformation is a cohesive failure that can occur in a spatially localized manner in simple-shear apparatuses. In a large step strain test, instead of quiescent relaxation as perceived in previous experimental and theoretical studies, macroscopic motions were observed after shear cessation.12,13 For systems with a sizeable bmax/H, quiescent relaxation only occurs when the amplitude of step strain is below one strain unity, implying that the entanglement network has a characteristic level of cohesion on shorter time scales than the terminal relaxation

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time τ. The nonquiescent relaxation indicates that sufficient residual elastic retraction force can overcome the network’s cohesion, locally transforming this elastically strained “solid” into a liquid. When this elastic yielding occurs inhomogeneously in space, macroscopic motions naturally arise: a solid region can sustain stress but a liquid region cannot. This elastic yielding upon step strain83 is a unique nonlinear rheological response of entangled polymers and can be most straightforwardly demonstrated using PTV without any experimental difficulty such as edge fracture because of the small amount of strain involved. Finally, a word is due about the theoretical situation. Is there any theory to account for the observed shear banding? If shear banding exists in steady state, then one needs a constitutive theory that allows the shear rate to be double-valued for a given stress. But if this shear banding can be avoided by a slow rate ramp-up, then we have to resort to a different theory that prescribes a oneto-one correspondence between stress and rate simple shear. However, such a monotonic constitutive theory would not permit any shear banding to prevail upon a sudden startup shear. Thus, if the conclusions of refs 70 and 71 are true, we would not be able to resort to any version of the tube model for a proper theoretical description of the experimental observations. Acknowledgment. This work is supported, in part, by a grant (DMR-0821697) from the US National Science Foundation. Appendix A. Account of Wall Slip and Ultimate Homogeneous Shear When the apparent shear rate V/H indicated in Figure 1a is below the slowest relaxation rate 1/τ of an entangled polymer, i.e., (V/H)τ < 1, the amount of external shear deformation is lower than unity within the relaxation time τ. As a consequence, the entanglement network does not deform sufficiently to break up elastically. In other words, under this condition, there is no chain disentanglement at either the polymer/wall interface or in the bulk. Homogeneous shear prevails as shown in Figure 1a. The viscosity ratio in eq 1 is on the order of unity, and b is only molecularly small. When the apparent Weissenberg number Wiapp = (V/H)τ is much higher than unity, the adsorbed chains cannot remain engaged with the bulk chain entanglement network indefinitely during startup simple shear. Such interfacial yielding34 produces an interfacial layer (of thickness a) with a greatly reduced viscosity ηi. Consequently, b can be millions of times a, i.e., macroscopically large according to eq 1. For an entangled linear polymer melt with (weight-average) molecular weight Mw, the maximum value of b can be estimated,22 upon complete interfacial chain disengagement by conducting a controlled-stress experiment to produce a stick-slip transition (that occurs near the end of the Newtonian regime),51 to be related to the Newtonian viscosity η0 bmax ¼ ðη0 =ηe Þlent ¼ ðMw =Me Þ3:4 lent ,

for entangled melts

ðA.1Þ

when ηi in eq 1 turns into ηe, which is that of a melt with entanglement molecular weight Me. Equation A.1 assumes the slip layer thickness to be the entanglement spacing lent, related to Me as85,86 (Me/FNAp)1/2, where F is the polymer mass density, p is the packing length, and NA is the Avogadro constant. Here we adopted the empirical exponent 3.4 to depict the molecular weight dependence of η0. The viscosity ratio in eq A.1 can readily approach 106, making bmax macroscopically large, for polymer melts, especially polyethylene, polybutadiene, and polyisoprene that are usually highly entangled.

Industrial polymer processing always occurs in the shear thinning regime. Under a typical processing condition the limit of homogeneous shear at Widis, which is discussed in eq A.3, is not accessible for well-entangled polymers. Even for a sample with Mw ∼ 10Me, it requires Widis ∼ 103.4 = 2512 to reach this regime. On the other hand, since wall slip occurs whenever the Weissenberg number Wi is greater than unity, wall slip is highly relevant for processing. However, the magnitude of wall slip, as characterized by bmax of eq A.1, is usually not large for polymers such as polystyrene (PS) because the processable PS typically only have 10-20 entanglements per chain, corresponding to a molecular weight range of 130-260 kg/mol; i.e., bmax would not exceed 7.7 203.4 = 0.2 mm, where the entanglement spacing is 7.7 nm for PS. This value is smaller than the typical dimensions of processing equipment. On the other hand, polymers such as linear polyethylenes (PE) of both high and low density and polybutadiene (PB) suffer significant wall slip because the number of entanglement per chain readily exceeds 100 in these polymers, with the entanglement molecular weight being as low as 0.9 and 1.6 kg/mol for PE and 1,4-PB, respectively. Thus, the magnitude of bmax can easily exceed 10 mm. For monodisperse well-entangled polymer, there may exist a rather broad stress plateau whose width, depicted in terms of the range of the apparent rate, is bounded on the low end by γ_ c∼ 1/τ and the on the high end, we assert, by the local shear rate γ_ dis found in the entanglement-free slip layer upon maximum slip. At the onset of shear thinning, we have shear stress η0γ_ c in the bulk and ηeγ_ dis in the slip layer. Thus, we arrive at γ_ dis ¼ ðη0 =ηe Þγ_ c ¼ ðγ_ c τÞ=τe ∼1=τe

ðA.2Þ

where η0/ηe = τ/τe follows from the tube model,44 with the confinement time τe is related to the dominant relaxation (i.e., reptation) time τ as τe = τ(Me/Mw)3.4. The corresponding Weissenberg number Widis is given by Widis ¼ γ_ dis τ ¼ τ=τe ¼ bmax =lent

ðA.3Þ

which is given by (Mw/Me)3.4 for melts according eq A.1, where the last equality follows from eq A.1. For an entangled polymer solution with polymer weight fraction φ, complete slip corresponds to having ηi in eq 1 reduced to the solvent viscosity ηs, where a may also be thought of as the enlarged entanglement spacing49,87 lent(φ) = lentφ-0.6, so that we rewrite eq 1 as bmax ðφÞ ¼ ½η0 ðφÞ=ηs lent ðφÞ,

for entangled polymer solutions ðA.4Þ

-3

Since ηs can be as low as 10 Pa 3 s (i.e., that of water), entangled solutions could also display significant wall slip. Similar to eq A.2, the local shear rate in the slip layer upon complete slip is given by γ_ dis ¼ ½η0 ðφÞ=ηs γ_ c

ðA.5Þ

which can be rewritten in terms of the corresponding Weissenberg number Widis as Widis ¼ γ_ dis τðφÞ ¼ ½γ_ c τðφÞ½η0 ðφÞ=ηs ∼ ηðφÞ=ηs ¼ bmax ðφÞ=lent ðφÞ

ðA.6Þ

where the last equality follows from eq A.4, and the product γ_ cτ is on the order of unity. Note that eq A.6 is of the same form as eq A.3. Homogeneous shear should occur upon complete chain disentanglement when the imposed rate is as high as the local shear rate γ_ dis in the slip layer during maximum wall slip. This critical shear rate is given by eq A.2 for melts and by eq A.5 for solutions corresponding to a Weissenberg number Widis that is related to the maximum slip length bmax as Widis=bmax/lent as shown in eqs A.3 and A.6.

Perspective

Appendix B. Nomenclature Glossary a interfacial slip layer thickness b, bmax extrapolation length or slip length, and its maximum value, for melts and solutions φ polymer concentration in terms of the weight fraction γ_ app apparent shear rate or nominal shear rate shear rate to reach homogeneous fully disentangled γ_ dis state onset shear rate for shear thinning, equal to 1/τ γ_ c lent(φ) entanglement spacing of solution at concentration φ entanglement spacing of melt lent slip velocity Vs τ dominant or overall relaxation time of polymer melt τ(φ) longest relaxation time of solution at φ entanglement confinement time τe H typical dimension of a rheometric setup such as the gap distance zero-shear melt viscosity η0 zero-shear solution viscosity at polymer concentration φ η0(φ) ηi viscosity at the polymer/wall interface zero-shear viscosity of a Rouse melt at Me ηe ηs solvent viscosity Wiapp Weissenberg number equal to γ_ appτ Weissenberg number corresponding to γ_ dis Widis Wiws-sb critical Weissenberg number, beyond which wall slip is expected to evolve to bulk shear banding Appendix C. Brief Survey of the Tube Theory Concerning Shear Banding The original 1978–1979 tube model44 of Doi and Edwards revealed a nonmonotonic constitutive relation in its calculation. The nonmonotonicity occurs because of excessive chain alignment in the shearing direction. It has the same physical origin as the shear stress overshoot that the tube theory is able to produce.88 They and subsequently others related such a feature to a flow instability (known as the stick-slip transition22,51) observed in capillary rheometry.89,90 Subsequently, it was proposed in a 1993 theoretical study91 that shear banding could occur in entangled polymer solutions under simple shear. The experimental report for shear banding, inspired by this theoretical analysis, was made in 1996 for wormlike micellar solutions.92,93 Because the wormlike micelles are capable of adjusting their sizes in shear and may even suffer breakage by shear,100,101,102 shear banding in wormlike micellar solutions94-103 was not taken to imply that conventional noncharged, nonassociative polymers would also shear inhomogeneously. In the decade between 1996 and 2005, a couple of studies reported a hint of shear banding in semidilute polyacrylamide/water solutions where chain-chain associations occur through hydrogen bonding.104,105 Apart from this special case, the important premise of shear homogeneity had never been observed to break down for entangled polymer liquids. In absence of a perfect stress plateau in the steady state flow curve and any direct evidence of shear inhomogeneity, major modifications106,107 including the introduction of convective constraint release108 were made to make sure that the tube model does not prescribe a nonmonotonic constitutive relation for entangled polymers. References and Notes (1) Wasserman, S. H.; Graessley, W. W. J. Rheol. 1992, 36, 543. (2) Mead, D. W. J. Rheol. 1994, 38, 1797. (3) Macosko, C. W. Rheology: Principles, Measurements and Applications; VCH: New York, 1994.


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Homogeneous Shear, Wall Slip, and Shear Banding of Entangled

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