Comment on “New Experiments for Improved Theoretical Description

Comment on “New Experiments for Improved Theoretical Description of Nonlinear Rheology of Entangled Polymers”. Richard S. Graham* ,. School of ...
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Comment on “New Experiments for Improved Theoretical Description of Nonlinear Rheology of Entangled Polymers” Richard S. Graham* School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, U.K.

Macromolecules 2013, 46, 3147−3159. DOI: 10.1021/ma300398x

Ewan P. Henry and Peter D. Olmsted Soft Matter Physics Group, School of Physics, Astronomy, University of Leeds, Leeds LS2 9JT, U.K.

Macromolecules 2013, 46, 3147−3159. DOI: 10.1021/ma300398x S Supporting Information *

1. INTRODUCTION The tube model1 is the most successful microscopically based theory for describing the nonlinear flow of entangled polymers.2 It postulates that the complicated multibody interactions that dominate entangled polymer dynamics can be represented by a tube surrounding the test chain. This mean field approximation explains numerous nonlinear flow phenomena; for example shear thinning, shear stress overshoots and extension hardening. Furthermore, a specific formulation of the tube model3 leads to successful predictions for mechanical stresses4 and neutron scattering measurements5−9 for linear polymers under rapid flow in a variety of geometries. Over the past decade, Shi-Qing Wang and co-workers have published numerous experiments describing inhomogeneous flow in sheared polymer fluids.10,11 These experiments have suggested that inhomogeneous flows such as shear banding may occur in strong flows, which is contrary to the usual modeling approaches that have assumed homogeneous flows. Wang has further interpreted this inhomogeneous flow as evidence that the tube model is fundamentally flawed, necessitating a total reevaluation of our modeling approach.12,13 However, more detailed calculations of the inhomogeneous responses predicted by tube models show that shear banding or transient inhomogeneities can be explained and predicted by tube models, for various realistic ranges of parameters.14−19 Some results that have seemingly been at odds with the tube model, such as some kinds of type C response after a step strain,20,21 can be understood in terms of an inhomogeneous response.14,19 Most recently, a round-robin study on nonlinear shear of entangled melts was conducted by several world-leading experimental groups.22 This work concluded that observed shear banding may be strongly influenced and potentially induced by edge effects rather than due to the bulk instability that is predicted by tube models with certain parameter ranges (namely weak convected constraint release). Thus, the current state of theoretical and experimental understanding of tube models in strong nonlinear flows remains incompletely understood.23 There is therefore an urgent need for careful experiments that probe the boundaries of our understanding,22 and a recent work by Wang et al. (2013)24 claims to address this need. In © 2013 American Chemical Society

this paper, Wang et al. extend their previous criticisms of the tube model12,13 to flow conditions where homogeneous flow is claimed to be assured and so edge effects are not an issue. They presented a range of unusual flow experiments, which they claimed contradict the tube picture and, instead, support an entirely different theory. Here we demonstrate that, of the measurements from Wang et al. (2013), all but one can be predicted by standard tube models. The single feature that the tube model fails to predict amounts to a ∼ 10% disagreement between the model and experiments. We will also highlight existing literature measurements that directly contradict Wang’s theoretical interpretation. In combination, these strongly refute Wang’s theoretical picture and indicate, instead, that Wang et al.’s24 experiments support the standard tube model. 1.1. Wang’s Theoretical Interpretation. Wang interprets his prior shear banding experiments and more recent homogeneous flow data in a entirely different way to the tube model. He claims that a fundamental force, important to nonlinear polymer dynamics, is missing from the tube model. This force is claimed to arise from strongly localized intermolecular interactions and gives a finite static opposition to retraction in nonlinear response. This would mean that chain retraction, rather than occurring at a rate determined by the local friction, must overcome a finite force through the buildup of chain stretching before any relaxation can occur. Wang calls this force the “intermolecular gripping force” or IGF. Once the chain tension due to stretching exceeds the IGF, the cohesion of the entanglement network is broken, leading to a huge loss of entanglement and a large stress decline. Up until this yielding/disentanglement event, no retraction is possible. This picture assigns a cohesive strength to the melt, which, once overcome, leads to yielding and flow. 1.2. Wang’s Homogeneous Flow Experiments. Table 1 summarizes the material used in each of Wang et al.’s experiments. Received: Revised: Accepted: Published: 9849

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at about the Rouse time, due to chain retraction. Wang et al. interpreted this lack of stress drop at τR as evidence that chain retraction is prevented by the intermolecular gripping force. Recoverable Strain−Shear (Z = 18). In these experiments, Wang et al. performed a start-up shear experiment for a variety of strains and strain rates, allowed a waiting time of tw, and then measured the recoverable strain. There were a number of striking results from these experiments. First, the recoverable strain was essentially the imposed strain (i.e., recovery was complete), even for waiting times of several multiples of the chain Rouse time. Second, for tw = 0 elastic recovery was also complete for larger strains; the full recovery persists up to the strain corresponding to the maximum stress in the shear stress transient, even for WiR < 1 where full chain retraction occurs during the flow. Wang et al. argued that chain retraction should significantly reduce the recoverable strain and, therefore, that these data support the intermolecular gripping force (IGF) picture. Recoverable Strain−Extension (Z = 98). Similarly to their shear experiments, Wang et al. measured the elastically recoverable strain, following a uniaxial extensional strain, at a rate below the inverse Rouse time (WiR = 0.12). The elastic recovery was effectively complete up to some critical strain. This critical strain corresponded to the maximum in the engineering stress, defined as the nominal extensional stress that does not account the reduction is cross-sectional area during the flow. Wang et al. concluded that, in analogy to their shear experiments, the maximum in the engineering stress indicates a yielding event and that the full elastic recovery demonstrates that no retraction occurs prior to this critical strain, supporting the IGF picture. 1.3. Summary. A central tenet of Wang et al.’s paper is that, for each of their experiments, the tube model cannot predict their observed qualitative features. They further argue that their IGF and related ideas provide a better explanation of their data. However, Wang et al. do not directly compare any tube model to their data. Instead they use ideas from the tube model to estimate its qualitative predictions. In contrast, we will show below that direct calculations with standard versions of the tube model describe all of the above data, except for a single measurement. This single anomaly amounts to a ∼10%

Table 1. Materials Used in Each of Wang et al.’s Experimentsa experiment

label

shear recovery step shear relaxation

1M5%-10K SBR160 K

recoverable strain (shear) recoverable strain (extension)

1M5%-10K SBR241K

material

Z

polybutadiene solution styrene−butadiene copolymer melt polybutadiene solution

18 33

styrene−butadiene copolymer melt

98

18

a

Z is the number of entanglements. In each case, the material is a monodisperse linear polymer.

Shear Recovery (Z = 18). Wang et al. (2013) imposed a start up shear at a rate of WiR = 0.0875 to a strain of γ = 1.5, where WiR = γ̇τR is the Weissenberg number based on the chain Rouse time τR. This was followed by a shear-free waiting period of tw, then resumption of shear at a much higher rate of WiR = 5.25. During the second shear phase, the shear stress achieved a transient maximum σxy,max whose magnitude depended on tw. See parts a and b of Figure 1 for further details. Wang et al. (2013) claimed that, as no stretching occurs during the first shear, the tube model must predict that the oriented state relaxes to equilibrium monotonically. Hence, for longer waiting time tw the entanglement network would be more fully healed, leading to a larger stress maximum σxy,max, with σxy,max a monotonically increasing function of the tw. The experiments show a clear nonmonotonic response for σxy,max(tw), which Wang et al. interpreted as evidence that further damage or disentanglement occurs after the first shear through their molecular elastic yielding mechanism. Stress Relaxation after Step Shear (Z = 33). Wang et al. performed step shear experiments at WiR ≫ 1 for a small strain (γ = 0.1) and a slightly larger strain (γ = 0.7). These strains were achieved in 0.035 and 0.04 s, respectively, which is much shorter than the Rouse time of 13 s. They showed that, after applying an ad hoc 4% shift to the larger strain data, the subsequent relaxation modulus (σ(t)/γ) was identical for the two strains. The damping function in the original Doi−Edwards model predicts the same modulus until the relaxation curve from the larger strain shows a noticeable drop in stress (∼14%)

Figure 1. Rolie-Poly calculations of Wang et al.’s shear recovery measurements. Part a shows the transient stress produced by the shear flow protocol in part b. Part c: black line (left axis) shows the Rolie-Poly shear stress transient for a strain of γ = 1.5 at γ̇1τd = 4.75 (WiR = 0.0875) followed by relaxation with no shear. Blue squares (right axis) show the shear stress maxima (σxy,max) achieved by a shear rate γ̇2τd = 285 (WiR = 5.25) applied after waiting times tw. Here the x-coordinate of each σxy,max point is the time γ̇2 begins. 9850

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Figure 2. GLaMM model calculations of the relaxation modulus following a rapid step strain (WiR ≫ 1) for shear strains of 0.1 and 0.7(a). The larger strain shows a small but noticeable stress drop around t = τR due to chain retraction, which is absent from Wang et al.’s experiments. (b) Wang et al.’s experimental data. Reprinted from ref 24. Copyright 2013 American Chemical Society.

It retains deformation information along the chain contour and so provides the most detailed stress predictions, in addition to enabling predictions of neutron scattering patterns. For each experiment, we used the number of entanglements corresponding to the experimental material, as summarized in Table 1. Shear Recovery (Z = 18). Wang et al.’s experiments correspond to dimensionless shear rates of γ̇1τd = 4.75 (WiR = 0.0875) and γ̇2τd = 2855 (WiR = 5.25), for the first and second shear rates, respectively. To match the experimental material we took Z = 18. We ran a start-up shear calculation using the Rolie-Poly model at γ̇1 up to a strain of 1.5, held the strain fixed for a period tw, resumed shear at γ̇2 and recorded the maximum shear stress σxy,max from the ensuing stress transient. Plots of σxy,max against tw are shown in Figure 1, which should be compared to Figure 6 in Wang et al. In close agreement with Wang et al.’s measurement, σxy,max initially falls with increased tw but eventually grows with tw. We can understand this behavior by considering the response of normal and shear stresses. For nonlinear flows these components of the polymeric stress will have different relaxation functions. A detailed calculation using the Rolie-Poly model shows that, upon cessation of the applied shear strain, the shear stress relaxes monotonically back to equilibrium, while σyy continues to reduce for a short time period ∼τR. This postflow reduction of σyy is due to retraction of the small amount of stretching that accumulates during the flow. Only after a delay time does σyy begin to increase in time back toward equilibrium. The nonmonotonic response in σxy,max(tw) is due to this delayed relaxation of the normal stress. Interestingly, this nonmonotonic relaxation perpendicular to the flow direction due to postflow retraction has previously been confirmed by neutron scattering measurements and is in agreement with GLaMM model predictions.6 Thus, the RoliePoly model can predict and explain this nonmonotonicity using standard tube model concepts of stretch and orientation, without needing to evoke yielding, disentanglement or barriers to chain retraction. Relaxation after Step Shear (Z = 33). In these measurements, Wang et al.’s conclusions are dependent on a small disagreement between the Doi−Edwards model and their step shear measurements. Thus, we used the most detailed tube model, the GLaMM model.3 We took cν = 0.1 for the CCR parameter, as a comparison with a wide range of experiments3−6 has established that this value. Changes in cν generally have a small effect on the predicted damping function.25 We imposed a shear strain in a time of 0.1τe (the Rouse time of an

disagreement between the tube model and Wang et al.’s measurements. We do not dispute the fact that the tube model may be incomplete or may yet require severe modifications for very strong flows. However, in our view the data of Wang et al. provide very few, if any, experimental clues in this direction.

2. TUBE MODEL PREDICTIONS We model the experiments from Wang et al. using two standard versions of the tube model. In accord with Wang et al.’s experiments all of our computations are for spatially homogeneous flows. The first model is the Rolie-Poly model.25 This is a single mode approximation to the tube model, which includes the three key relaxation mechanisms of reptation, retraction and convective constraint release (CCR). The Rolie-Poly model is given by dW 1 = κ·W + W·κ T − (W − I) dt τd ⎞ ⎛ tr W ⎞δ 2(1 − 3/tr W ) ⎛ ⎟ (W − I)⎟ ⎜⎜W + β ⎜ − ⎟ ⎝ 3 ⎠ τR ⎝ ⎠

(1)

where W is the polymer stress and κ is the velocity gradient tensor, and the ratio of the reptation and Rouse times is τd/τR = 3Z, where Z is the number of entanglements, β sets the CCR strength and δ controls the suppression of CCR with chain stretch. The stress predictions for this model have been shown to agree well with both more detailed implementations of the tube model25 and experimental data,5,16,26 with the CCR suppression parameter δ = −1/2 (which we use throughout). We take this as the only physical value, as it was shown in the original Rolie-Poly paper that δ = −1/2 corresponds to the same physics as implemented in the more detailed GLaMM model and also leads to the closest agreement between these two models. We took β = 0.1 for our calculations, although we tested β on the range 0−1 and obtained comparable results. In general, as the value of β is varied the qualitative features seen in our calculations remain but the features become less pronounced as β is increased. For example, the calculations in Figure 1b, were repeated for β = 0−1, yielding in all cases an observable undershoot comparable in size to Wang’s measurements. The second model is the GLaMM (Graham, Likhtman and Milner, McLeish) model,3 which is a detailed implementation of all of the standard relaxation mechanisms in the tube model. 9851

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Figure 3. Rolie-Poly calculations for recoverable strain γr, following a shear strain of γ. (a) Recovery from a rapid shear strain (WiR = 299 ≫ 1) and a postflow waiting time tw. For these calculations Z = 18 and therefore tw/τd ranges from 0.093 (dark blue triangles) to 2.96 (black squares). (b) Recovery from shear at rates from nonstretching (WiR = 0.1) to rapid (WiR > 1), with no waiting time. Here, the strain is expressed as a fraction of γmax which is the strain at which the shear stress transient achieves a maximum.

is independent of εmod because the recovery period ≪ τR. We use a value εmod = 10−4. Our recoverable strain calculations are shown in Figure 3, which should be compared to Figures 8(a) and 9(b) in Wang et al.. All features observed by Wang et al. can be predicted by the Rolie-Poly model, without modification. Most importantly, the strain is fully recovered for γ < 1 even for waiting times of up to 10 Rouse times (Figure 3a); and the critical strain for the onset of incomplete recovery is γmax, even for shear rates below the inverse Rouse time. Here, γmax is defined as the strain at which the shear stress transient achieves a maximum. We are able to predict all of these features using the standard tube picture of barrier-free chain retraction at a rate determined by the Rouse time. The almost complete lack of influence of chain retraction on recoverable strain can be explained by a simple analytic calculation. As the recovery period is very fast compared to the chain relaxation times the strain recovery period is dominated by the upper convected Maxwell (UCM) derivative in the Rolie-Poly model (ie chain relaxation is negligible). Because of the linear coupling between dσxy/dt, shear rate and σyy in the UCM derivative then 1/σyy determines how effectively shear stress can be converted into recovered strain. Hence it can be shown that, γr = σxy(0)/σyy(0), where σxy(0) and σyy(0) are the stresses at the start of the recovery period. Here σxy is the shear stress available to drive the recovery of strain and σyy defines the effective resistance to this recovery. As retraction reduces equally all components of σ it has no effect on the ratio = σxy/σyy and hence the recoverable strain. This explains why Wang et al. observed full elastic recovery even in experiments lasting much longer than τR: retraction still occurs, as expected, but it causes an almost equal reduction in σxy and σyy, leaving their ratio unaffected. Retraction does indirectly reduce recoverable strain through the CCR mechanism. However, CCR’s influence is weak because, for unstretched chains, the total amount of CCR prior to γmax is small and, for stretched chains, CCR is suppressed.3 Hence, increasing β in our calculations softens the transition from full recovery to loss, but does not change our overall conclusion. Recoverable Strain−Extension (Z = 98). Similarly to recovery in shear, we model extension recovery by setting the total stress to zero and allowing the extension rate to be determined by a balance of the polymer stress and the

entanglement segment), which is much shorter than the chain Rouse time. Strains of 0.1 and 0.7 were used to correspond to the experiments. From the subsequent shear stress relaxation we obtained the relaxation modulus G(t) = σxy(t)/γ, which is plotted in Figure 2. As anticipated by Wang et al., the model predicts that the larger strain shows a small but noticeable stress drop around t = τR due to chain retraction, which is not seen experimentally. We note here that Wang et al.’s data only report one nonlinear strain; the two calculated relaxation moduli disagree by only ∼14%; Wang et al. required an ad hoc 4% upward shift to the larger strain data to superpose their data, indicating some quantitative experimental issues; and there is a large body of experimental data that support the tube model’s damping function predictions for entangled polymers with Z < 50.27 Equally, there are numerous damping function measurements that contradict the tube model.20,21 This is important when assessing whether significant new conclusions can be drawn from this single step shear measurement. Nevertheless, there is a noticeable difference in shape between the predicted relaxation function and Wang et al.’s measurement. Thus, we conclude that this discrepancy deserves further experimental investigation to determine how robust it is to variations in strain and molecular weight. Recoverable Strain−Shear (Z = 18). To compute the recoverable strain we apply zero total shear stress to our system after the initial imposed shear strain. This corresponds to setting the stress on the top plate of the rheometer to zero, as in experiments. Thus, we obtain Txy = σxy + ηNγ̇ = 0, where T is the total stress, σ is the polymer stress and ηN is the Newtonian viscosity. The stress is always a superposition of such separated fast and slow modes, as discussed in Doi and Edwards (section 7.2.1). In melts one usually ignores the dissipative term because it is so small. However, it is necessary (and also entirely physical) in order to describe the observed recoil. Hence the shear recovery becomes a balance of the slow relaxing polymer stress with the dissipative stress due to the current deformation rate. Substituting σ = GW, where W is the dimensionless polymer stress and G is the modulus, and rearranging gives γ̇τd = −(Gτd/ηN)Wxy. Thus, we numerically compute the shear rate from the polymer shear stress at each time step and allow the model to run to steady state, to compute the recoverable strain. Typical values for the modulus ratio εmod = ηN/(Gτd) are 10−3− 10−7.10,19 At these low values the computed recoverable strain 9852

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Newtonian viscosity, giving ε̇τd = −(Wxx − Wyy)/(3εmod). Our Rolie-Poly calculations for WiR = 0.12, shown in Figure 4,

Furthermore, Baig et al.’s results for Rg2 show only very weak deformation for these Weissenberg numbers. Admittedly, Baig et al. studied steady state flow, whereas Lu et al.’s results are for a transient flow. Nevertheless, it seems unlikely that Lu et al.’s simulations could give a steady state value of Rg that is consistent with Baig et al.’s results.

3. DISCUSSION AND CONCLUSIONS We have used standard implementations of the tube model to capture recent experiments by Wang et al.24 All of these recent experiments are predicted by the tube model, except for a single measurement. In particular, the model predicts that retraction has a weak influence on the recoverable strain in both shear and extension. Additionally, in an interrupted shear flow, the model successfully predicts the nonmonotonic dependence of the shear stress peak during the second flow period. These effects can be predicted and explained using standard tube model relaxation mechanisms and their effect on chain stretch and alignment. The single feature that the tube model fails to predict amounts to a ∼10% disagreement between the tube model and Wang et al.’s stress relaxation measurements. Wang et al. concluded that their results were evidence of a need to substantially revise our understanding of the nonlinear dynamics of entangled polymers. They claimed that their results could not be reproduced by the tube model and that, instead, the intermolecular gripping force (IGF) was necessary to explain their results. However, the success of our tube model calculations herein show that Wang et al.’s measurements cannot justify this radical departure from standard tube model concepts. There is also compelling microscopic evidence for barrierfree retraction in the literature, which is dismissed or ignored by Wang et al. Blanchard et al.6 performed small angle neutron scattering (SANS) measurements on an entangled linear polymer following a modest uniaxial extension deformation. Their data show a nonmonotonic relaxation of chain deformation perpendicular to the flow direction Rperp g . This distinctive signature of chain retraction occurs at t = τR, precisely the time scale predicted by the tube model. The IGF picture would predict monotonic relaxation of Rperp g , which occurs on the much longer time scale of the reptation time. Furthermore, SANS measurements in a contraction flow5 at WiR < 1 are well described by the tube model. The IGF picture would predict no retraction for these measurements and so would substantially overpredict the degree of molecular deformation. Finally, barrier free retraction, both during and after shear flow, has been directed observed in entangled DNA fluids using molecular staining.30 Overall we conclude that Wang et al.’s measurements do not provide convincing experimental evidence for the intermolecular gripping force. Instead, they further support the standard tube model picture of barrier-free retraction controlled by the chain Rouse time. This supplements the great deal of mechanical stress data that support the barrier-free retraction process. These data are further supported by microscopic probes such as neutron scattering and direct observation of barrier-free retraction in DNA. Nonetheless, apparent contradiction remain between the round-robin results22 and the earlier experiments of Wang et al.11 We share Wang et al.’s frustration that the current tube model can describe both sets of experiments, albeit with different parameters. Furthermore, theoretical work shows that flow is expected to be homogeneous either as long as the strain is small enough

Figure 4. Rolie-Poly predictions for recoverable strain after an extension flow of WiR = 0.12 and Z = 98. The imposed stretching ratio λ is normalized by λmax = 2.1, which is the extension ratio at which the stress transient achieves a maximum in the engineering stress, σeng = (σxx − σyy)/λ.

correctly predicts Wang et al.’s experiments (see Figure 10 in Wang et al.). In particular, λmax is the critical strain required to induce incomplete recovery. Similarly to shear, in the limit of rapid recovery the recoverable strain is given by εr = (1/3) ln(σxx/σyy). Thus, a reduction in σxx due to retraction is compensated for by a corresponding reduction in σyy, leaving the recoverable strain unaffected. The connection between λmax and incomplete recovery is explained because they both correspond to the saturation of σxx with strain, meaning that further strain does not translate to increased recoverable strain. We note here two minor weaknesses of our calculations. First, the Rolie-Poly model predicts λmax = 2.1, whereas the experimental value is closer to 3.0. Generally, a multimode version of the Rolie-Poly model is required for quantitative predictions such as this. However, as our conclusions are based on qualitative features they are unaffected. Second, Wang et al.’s experiments show a very abrupt transition to incomplete recovery beyond λmax whereas our Rolie-Poly calculations show a smoother transition. We note that that abruptness of the transition in Wang et al.’s data is supported only by a single data point. Recent Simulations by Lu et al. Recently published simulation results on entangled polymers by Lu et al.28 purport to have observed the lack of chain retraction as predicted by the IGF picture. In these simulations the total radius of gyration, Rg deforms significantly even at WiR = 1/6. Lu et al. assert that, in contrast, the tube model predicts that, for WiR < 1, Rg remains at its equilibrium value. However, in the Supporting Information we demonstrate that the GLaMM model3 is able to account for the degree of deformation of Rg seen in Lu et al.’s simulations via orientation effects alone. The GLaMM model does not account for the low strain at which the plateau value of Rg is achieved in the simulations. There are also some issues with the simulations of Lu et al.,28 as discussed in the Supporting Information. Baig et al.29 have performed simulations that are free of these issues. Their results for chains with 12 entanglements under steady shear show essentially no change in the primitive path contour length (chain stretching) for γ̇τd ≲ 2, in accord with the tube model. 9853

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that stress maximum has not been reached,18,19 or perhaps earlier, in the case of nonstretching flows.31 In the latter case the detection of inhomogeneities requires strong enough growth of incipient instabilities.18 It seems entirely likely that new ideas will be required to understand the very strongly aligned regime, where the nature of entanglements, if they can be defined, is very different from the interlocking loops usually envisioned in isotropic melts. This will require new experiments that can distinguish these effects from those predicted by the tube model. Unfortunately the manuscript from Wang et al. provides only one such unexplained experiment, which will required more detailed (and independent) study before its ramifications can safely be considered.

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(19) Agimelen, O. S.; Olmsted, P. D. Phys. Rev. Lett. 2013, 110, 204503. (20) Osaki, K.; Kurata, M. Macromolecules 1980, 13, 671. (21) Vrentas, C. M.; Graessley, W. W. J. Rheol. 1982, 26, 359. (22) Li, Y.; Hu, M.; McKenna, G. B.; Dimitriou, C. J.; McKinley, G.; Mick, R. M.; Venerus, D. C.; Archer, L. A. J. Rheol. 2013, 57, 1411. (23) Likhtman, A. E. J. Non-Newtonian Fluid Mech. 2009, 157, 158. (24) Wang, S.-Q.; Wang, Y.; Cheng, S.; Li, X.; Zhu, X.; Sun, H. Macromolecules 2013, 46, 3147. (25) Likhtman, A. E.; Graham, R. S. J. Non-Newtonian Fluid Mech. 2003, 114, 1. (26) Collis, M. W.; Lele, A. K.; Mackley, M. R.; Graham, R. S.; Groves, D. J.; Likhtman, A. E.; Nicholson, T. M.; Harlen, O. G.; McLeish, T. C. B.; Hutchings, L. R.; Fernyhough, C. M.; Young, R. N. J. Rheol. 2005, 49, 501. (27) Osaki, K. Rheol. Acta 1993, 32, 429. (28) Lu, Y.; An, L.; Wang, S.-Q.; Wang, Z.-G. Macro Lett. 2013, 2, 561. (29) Baig, C.; Mavrantzas, V. G.; Kröger, M. Macromolecules 2010, 43, 6886. (30) Teixeira, R. E.; Dambal, A. K.; Richter, D. H.; Shaqfeh, E. S. G.; Chu, S. Macromolecules 2007, 40, 2461. (31) Moorcroft, R. L.; Fielding, S. M. Phys. Rev. Lett. 2013, 110, 086001.

ASSOCIATED CONTENT

S Supporting Information *

Calculations of the transient radius of gyration for the GLaMM model3 and further discussion of the simulations by Lu et al.28 This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: (R.S.G.) [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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