The tip of iceberg in nonlinear polymer rheology: Entangled liquids are “solids”


  • Shi-Qing Wang

    Corresponding author
    1. Department of Polymer Science, University of Akron, Akron, Ohio 44325-3909
    2. Laboratoire de Physicochimie des Polymères et Milieux Dispersés, UMR 7615/UPMC/CNRS/ESPCI, 10 rue Vauquelin, 75231 Paris cedex 05, France
    • Department of Polymer Science, University of Akron, Akron, Ohio 44325-3909
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It is very timely to be asked to provide a personal perspective in my field of research—nonlinear rheology of entangled polymers that is at a cross-road and may undergo profound transformation if one road proves to lead us much further as judged by evidence-based experimentation but not necessarily by the conventional criterion of obtaining quantitative agreement between experiment and theory. Although it may be very early to draw convincing conclusions, a vivid phenomenological picture has emerged.

Polymer dynamics and rheology,1, 2 along with the subjects of polymer crystallization and polymer glass formation, is a core study in polymer physics. A central theme in the past four decades has been to describe chain entanglement, respectively, in absence and presence of external deformation.3 Pioneers in the field include Beuche, Lodge, Ferry, Graessley, Edwards, de Gennes, and Doi, to name a just few most familiar. Since 1970s, the focus has shifted to exploring how chain entanglement gives rise to sluggish polymer dynamics far above the glass transition temperature. The first idea to account for the origin of chain entanglement in terms of a packing concept emerged almost simultaneously between 1985 and 1987 from Rault, Heymans, Lin, Kavassalis, and Noolandi. A competing notion of percolating network was subsequently introduced by Wool in 1993. The readers can get a comprehensive review of the relevant literature in a recent publication.4 A great number of experiments have accumulated for more than three decades to indicate that the de Gennes-Doi-Edwards tube model3, 5 is very successful in describing polymer dynamics in the linear response regime, including diffusion and linear viscoelastic behavior in bulk and at surfaces, and that much is reliably known about linear viscoelasticity at a phenomenological level, although some recent theoretical activities indicated6–8 that the tube theory may have serious limitations even in the simplest case of stress relaxation after small step strain.

In sharp contrast, phenomenology of nonlinear polymer rheology has been elusive and largely assumptive although large deformation and flow behavior of polymeric liquids have been rheometrically studied for several decades. Entangled polymer solutions and melts have been regarded as liquids, capable of undergoing homogeneous flow even during sudden large deformation.9 Various theoretical descriptions3, 10, 11 of constitutive behavior of entangled polymers have compared with experiment on the basis of homogenous shear. Standard rheological methods that work well for linear viscoelastic measurements have been extended to explore steady-state nonlinear rheological behavior, recognizing that entangled polymer solutions and melts are liquids at long times.

Based on an effective application of a particle-tracking velocimetric (PTV) method12 shown in Figure 1, a recent series of new experiments on entangled polymer solutions and melts have indicated that these liquids respond to sudden large deformation more like breakable solids. In particular, PTV observations of step strain tests reveal macroscopic motions in shear13, 14 and nonuniform extension,15 after cessation of the imposed deformation for both solutions13 and melts,14, 15 leading to a phenomenological suggestion that the entanglement network is a fragile temporary solid of finite cohesive strength16 and cannot escape structural failure.17 Apart from these surprising “strain adjustments,” shear banding was observed in sudden startup shear of entangled solutions18, 19 and melts.14 The familiar stress overshoot has been suggested to signify a yield point16 where specific scaling laws emerge in both shear20, 21 and extensional15 deformations.

Figure 1.

Sketch of an effective particle tracking velocimetric setup, suitable for any commercial rotational rheometers.

For entangled polymeric liquids, plastic deformation is an unfamiliar concept.22 It is typically assumed that one-to-one correspondence between stress and strain rate would exist even in plastic flow,1 making it feasible to formulate both analytical and numerical depiction of polymer flow.23 Doi and Edwards3 extrapolated the de Gennes reptation picture for polymer diffusion24 to a full molecular theory in an attempt to develop a constitutive description for flow behavior of entangled polymers. The consensus is that the revised DE theory25 would be one allowing most experimental observations to be understood either quantitatively or at least qualitatively and that shear banding would not occur. Thus, our recent experimental observations of shear banding in entangled polymers26 upon startup flow and “strain adjustment” after step strain have come as a surprise.

The original DE tube theory27 produced a nonmonotonic flow curve for steady shear. McLeish and Ball carried out a first study28 to follow up a specific suggestion of Doi and Edwards, taking the interfacial spurt phenomenon29, 30 in capillary flow as the experimental evidence for shear banding. In the subsequent decade, no shear banding was ever found in entangled polymers except in a special complex fluid of wormlike micellar solutions31, 32 that nevertheless attracted extensive theoretical studies.33–35 This situation led to the 1996 proposal of the convective constraint release (CCR) concept by Marrucci36 to remove the stress maximum in the original DE theory. Recent reports of shear banding in entangled polymer melts14 and solutions18 beyond the point of the stress overshoot suddenly revived strong interest in the original nonmonotonic character of the DE theory. However, it appears to us that the stress decline beyond overshoot signifies structural yielding17, 37 that has not been addressed by any version of a tube model and has only been speculatively suggested by Matsuoka.22

The most widely known prediction of the DE theory is the damping function for large step strain38 that shows agreement with experiment in Figure 7.16 of ref.3. This comparison was carried out by assuming quiescent relaxation.39 Replotting the damping function, one sees a nonmonotonic character as shown in Figure 2. The PTV observations reveal that macroscopic motions take place in the sample interior rather than at polymer/wall interfaces during the stress drop13 depicted in the left inset of Figure 2. For samples capable of massive wall slip, delayed slip40 occurs to produce the stress drop and type C behavior.13 We reckon that we have seen elastic yielding of the strained entanglement network.16 Some might wish to take the appearance of nonmonotonicity in Figure 2 literally to suggest an elastic instability41, 42 that would cause strain adjustment as indicated in the right inset of Figure 2. But how could moduli of different magnitude arise if there was no structural failure? It turns out that this particular nonmonotonic curve in Figure 2 has little real significance because most data do not fall onto this curve.13(b) From our viewpoint, the “strain softening” depicted by the damping function simply reflected the fact that the cohesion of the entanglement network has been overcome by the elastic retraction force.16 The single chain picture of the tube model perhaps oversimplifies the yielding process: the chain retraction was predicted to occur on Rouse relaxation time τR in a tube model, leading to a decline of shear stress because the stress was assumed to directly correlate with the contour length of a primitive chain. Emerging experimental evidence shows14 that the shear stress remains high long (e.g., 10 τR) after a sizable step strain until decohesion or elastic yielding through chain disentanglement causes steep stress relaxation.

Figure 2.

The DE tube theory predicts lower long-time shear stress for a step strain of higher magnitude when the strain amplitude γ0 exceeds γc ∼ 2.2. This nonmonotonicity is actually rather arbitrary numerically according to experiment, that is, experimental data can be on either side of the DE prediction.13(b) It is not obvious how to arrive at the upturn shown in blue. The left inset sketches the actual stress at different strains where a fast relaxation process appears at high strains depicted by purple and red lines, and the right inset depicts a perceived elastic instability.

In passing, we point out that if macroscopic motions observed13 after a sudden large step shear could be prevented, we would have a chance to meaningfully quantify stress relaxation dynamics at large strains. In principle, a macroscopic sample could be forced to undergo quiescent relaxation if a simple-sheared sample (to strain γ) is rigidly constrained not only by the two shearing surfaces but also by the two tilted surfaces whose normal are at an angle θ = cotan−1(γ) to the shearing direction. It is unclear, under this condition of no shape change, at what critical strain γc any nonlinear behavior would first take place. Could elastic yielding occur homogeneously under such a constraint of no shape change? This is one focus of our current research.

Uniaxial extension is a second insightful way to probe rheological behavior of entangled polymers. Yielding has recently been observed both during continuous extension and after a step extension.15, 43 A signature of structural yielding may be appearance of a maximum in the tensile force. Beyond the force maximum, analogous to startup shear, the initial elastic deformation transitions to plastic flow during rapid extension. In other words, the decline in the measured tensile force has the same origin as in simple shear, that is, yielding that leads to specimen failure. In other words, during sufficiently rapid stretching a well-entangled polymeric liquid suffers nonuniform extension due to structural failure,17 making it difficult to attain steady flow state, contrary to the consensus that steady flow is possible in fast uniaxial uniform extension of well-entangled polymers. A tensile force maximum also emerges during uniaxial extension at rates below the terminal relaxation rate. Because of the slow rate, elongational flow can eventually take place smoothly through directed molecular diffusion allowing uniform extension in steady state. Thus, a tensile force maximum is not necessarily a sufficient condition for sample failure. Conversely, the Doi-Edwards calculation and conclusion published in ref.27 may not be taken as having anticipated and predicted any necking instability in fast stretching of well-entangled polymers, nor may it be used to explain the origins of the yielding behavior observed15, 43 during both continual stretching and after step extension.

In short, the DE tube theory is built with a view that plastic yielding of the entanglement network would not occur because the tube is inherently present. For the same reason, it could not handle the phenomenology of wall slip that involves disentanglement of interfacial chains from the bulk chains. Brochard and de Gennes envisioned such disentanglement as due to coil-stretch transition in the tethered chains in their theoretical treatment of wall slip.44 Moreover, de Gennes did not think45 that the results in ref.13 could be accounted for within the framework of the DE tube theory and had to contemplate other experimental factors such as edge fracture or slip between the polymer sample and tracking particles as the cause for the observed macroscopic motions after step shear. The tube theory indeed does not encompass any notion of finite cohesion that an entangled polymeric liquid would possess at short times relative to its overall relaxation time. Therefore, it does not anticipate that a rapid startup flow would result in yielding of the entanglement structure. On the other hand, a small step strain experiment readily illustrates existence of transient cohesion for entangled polymeric liquids. Existence of finite cohesion, which can be overcome after a large step strain13 by the elastic retraction force to produce elastic yielding,16 provides us a common ground on which to treat both shear and extension in a unified manner and to ask a new set of questions related to plasticity of entangled polymeric liquids. For example, analogous to shear banding produced by sudden startup simple shear of entangled melts,14 we found the same samples undergo nonuniform extension beyond the yield point during startup uniaxial extension.15

We close by making a list of questions at different levels and providing some corresponding remarks. First, concerning the observed scaling laws15, 20, 21 associated with the yield point, we wonder

  • Where do the scaling laws originate from? How does affine deformation give in to molecular rearrangement beyond the force maximum? Clearly, the classical assumption of affine deformation needs to be replaced by a yield criterion. Because the structural yielding17 is a many-body problem, it is naturally beyond the scope of the standard tube model that could at best offer a single-chain mean-field description of homogeneous steady state. It appears to be a formidable task to explicitly depict how the intermolecular interactions eventually fail to cause further chain deformation and allow molecular slippage among different entangling chains. Nevertheless, it is possible46 to understand conceptually why yielding occurs and perhaps even feasible to propose how intermolecular interactions may depend on the imposed external rate and on the level of chain deformation, at least phenomenologically.

  • Yielding and plasticity are familiar subjects in solid mechanics of structured matters including glassy, colloidal, and granular materials. A great deal of modeling has been carried out in the past. Could some treatments of the existing classical theories such as the shear-transformation-zone theory47–49 be borrowed to provide a basic framework for plastic flow of entangled polymers although a highly deformable entangled polymer is a very different system, with its yield stress increasing20 strongly with the deformation rate?

Plastic deformation takes place as the entanglement network gives in to continuing external deformation. Upon plastic flow, the imposed strain is no longer 100% recoverable. In contrast to flow taking place in the terminal regime where homogeneous flow is possible due to molecular diffusion, the plastic flow resulting from structural failure17 appears to occur unevenly under certain conditions.

  • Does chain disentanglement (in the sense used to describe wall slip) really take place to allow plastic flow? What direct evidence is there? Why and how does inhomogeneous deformation occur either during continuing external deformation or after step strain?

Admittedly, the physics required to explain why the yielding produces subsequent inhomogeneity is currently completely missing. We speculate that some nucleation and growth process produce percolation of entanglement-free regions. This naturally occurs in a localized way leading to the observed inhomogeneity, in analogy to heterogeneous nucleation phenomena. More experiments must be carried out to delineate the nature of these recent observations,12–21 and, in particular, to find direct evidence for or against the notion that the entanglement structure yields during and after large external deformation. In other words, if an entangled polymer can be strained without any finite-size effects that are present in standard rheological setups, would we still encounter elastic yielding after step strain and shear banding during shear?

  • How can we derive such a kinetic mechanism of chain disentanglement? It would seem that a first step is to describe the localization of entanglement-free regions for the simplest case of an elastically strained entanglement network, that is, to delineate the inhomogeneous dynamic structural collapse17 after a large step strain.13 Currently, we have no idea how to link the yielding criterion to the phenomenon of localized failure. Work is ongoing to verify that the reported phenomenon is free of any complications due to edge instability.

  • To describe an inhomogeneous structural collapse of such an entanglement network clearly requires a many-body dynamical treatment. How can we hope to treat such a problem except in computer simulations? When will computing power be adequate to carry out full-blown molecular dynamics simulations of well-entangled polymers in presence of large deformation?

Depending on how the external deformation is imposed, responses of an entanglement network may be different. Shear banding is typically created by a sudden continual deformation of high rate that causes a well-entanglement network to be torn apart. For well-entangled polymers, the structural yielding17 can take place in a remarkably inhomogeneous manner, permitting the subsequent flow to capitalize the uneven structural failure and to retain inhomogeneous states of entanglement. Can one insist that steady state must have not been reached whenever shear inhomogeneity is still present? What benefit does such a criterion for steady state bring in the practical world where polymers usually undergo only a finite amount of strain during processing?

  • Some constitutive models claim to depict shear banding33, 35, 50 based on existence of a nonmonotonic relationship between stress and rate. But how realistic are these models since rheological hysteresis was never observed experimentally in simple shear?

  • What would be a structural theory that can predict whether shear banding is present or not based on how external deformation is imposed?

Currently, we have little idea about what a suitable theoretical framework should look like, within which the origin of various scaling behavior can be accounted for, and inhomogeneous flow can be depicted as a function of external conditions. If the observed inhomogeneity can be shown to be indeed structural in origin free of experimental artifacts, then how could we apply a conventional continuum mechanical treatment? If entangled polymers under large deformation may no longer be a continuum, we have to resort to unfamiliar methods that can depict such structural variation in a dynamic sense. Finally, to what extent can the tube model be remedied? Is it at least alright to apply tube model along with the convective constraint release concept when we know from experiment that the steady flow is homogeneous? Clearly, enormous theoretical and experimental tasks await us, and we have to rise to the challenges.


This presentation is based on the insights derived from the research results of former and current students, P. Tapadia, P. Boukany, S. Ravindranath, Y. Wang, and X. Li. The author gratefully acknowledges helpful communications with Xiaorong Wang, Tom McLeish, Peter Olmsted, Ken Schweizer, Steve Granick, and Costantino Creton. This research is supported, in part, by a small exploratory research grant (DMR-0603951) and standard grant (DMR-0821697) from the National Science Foundation. The author thanks Costantino Creton for his hospitality at ESPCI where the manuscript was originally drafted.