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Entangled Linear Polymer Solutions at High Shear: From Strain Softening to Hardening Gengxin Liu* and Shi-Qing Wang* Department of Polymer Science, University of Akron, Akron, Ohio 44325-3909, United States S Supporting Information *

ABSTRACT: The present rheological study reveals for the first time that entangled polymer solutions made of linear polystyrene or poly(methyl methacrylate) can exhibit strain hardening due to non-Gaussian stretching during startup shear at sufficiently high rates and temperatures well above their overall glass transition temperatures: Tg,solute > Texp > Tg,solution. The solutions made of high-Tg polymers only show partial yielding in the sense that both shear and normal stresses grow monotonically in time until a point of rupture, signified by an emergent cusp in the stress vs strain curve and macroscopic breakup along a shear plane. The shear softening-to-hardening transition, which occurs as a function of the applied shear rate, happens at lower equivalent rate with decreasing temperature, violating the time−temperature superposition principle.

softening through chain disentanglement,13 similar to a previous observation26 of strain hardening in a long-chain branched PS solution. Moreover, the borderline between strain softening and hardening, which resides in a narrow window of shear rates, is exploited to show that the time−temperature superposition principle (TTS) breaks down27−33 for fast startup shear of the entangled PS solutions.

1. INTRODUCTION Entangled polymeric liquids display many intriguing rheological features when external deformations are applied at rates much higher than internal molecular relaxation rates.1−3 Such a condition is defined as Wi ≫ 1, where Wi denotes the Weissenberg number, given by the ratio of the deformation rate to the relaxation rate. Among the rich nonlinear rheological phenomenology of entangled polymers, the most essential and fundamental is the stress overshoota transition from initial solid-like elastic response to eventual liquid-like flow during startup deformationand is ubiquitous for all linear entangled polymer solutions.4−11 Stress overshoot and strain softening can be regarded as a sign of the inevitable yielding12 of the entanglement network.12−14 The breakdown of the chain entanglement network after stress overshoot or large stepwise strain could lead to macroscopic inhomogeneous shear or nonquiescent relaxation, as observed by particle-tracking velocimetric (PTV) observations.15−17 However, there are alternative explanations such as constitutive instabilities18−22 for shear banding and over orientation23−25 for stress overshoot. In this work we sheared entangled linear polystyrene (PS) or poly(methyl methacrylate) (PMMA) solutions to explore whether the transition from initial elastic deformation to the eventual shear flow can take a path different from the wellknown phenomena such as smooth stress overshoot and ensuing shear banding. We carried out startup shear in an uncharted territory: lower temperatures and higher rates than common. When the applied rate is high enough, e.g., when the Weissenberg number WiR (the product of the imposed shear rate γ̇ and the Rouse relaxation time τR) is as high as 10, we find that entangled solutions of linear polymers can actually undergo strain hardening and non-Gaussian stretching instead of strain © XXXX American Chemical Society

2. EXPERIMENTAL SECTION 2.1. Samples and Solutions. Entangled linear PS or PMMA solutions were studied well above the overall glass transition temperatures Tg of these solutions. The four PS solutions were prepared with one of the following three types of solvent: tricresyl phosphate (TCP, Aldrich 1330-78-5), a combination of TCP and oligomeric PS of Mw = 6 kg/mol, or a pure oligomeric PS of 2 kg/mol. PS20M5%-TCP containing 5.52% v/v of PS of Mw = 20 000 kg/mol in the solvent of TCP17,34 has a plateau modulus Gpl around 284 Pa. PS400K29%-TCP is made of 28.6% PS of Mw = 400 kg/mol in TCP. PS20M5%-PS2K is a solvent-free PS mixture that involves 5% of PS20 M in 95% of PS2K. PS7M5%(35%6K:60%TCP) has the same volume fraction as that of PS20M5%-TCP and contains a mixed solvent60% TCP and 34.5% of PS6Kto slow down the dynamics. PMMA1M19%-DEP had diethyl phthalate (DEP, Aldrich 84-66-2) as the solvent. The solution properties are detailed in Table 1. 2.2. Rheology. All rheometric measurements were performed using an Anton Paar MCR301 rheometer. A 25 mm diameter cone− plate of 2° cone angle (CP25-2-SN4294) was used for SAOS (smallamplitude oscillatory shear) measurements of PS20M5%-TCP and PS400K29%-TCP. To accommodate higher normal stresses, a custommade 10 mm cone−plate of 4° cone angle (CP10-4) was used in all startup shear tests, unless noted otherwise (CP25-2 for PS400K29%Received: September 20, 2016 Revised: November 23, 2016

A

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Macromolecules Table 1. Basic Characteristics of Five Entangled Polymer Solutions Tg °C < Tref °C < Tg °C

a

label

polymer

Mw (kg/mol)

PDI

solvent

c (% v/v)

Z

solution

exp

pure solute

PS20M5%-TCP PS20M5%-PS2K PS400K29%-TCP PMMA1M19%-DEP PS7M5% (35%6K:60%TCP)

20M PSa 20M PSa 400K PSb 1M PMMAd 7.1M PSb

20000 20000 400 996 7100

1.2 1.2 1.1 n.a 1.25

TCP PS2K TCP DEP 60%(TCP) 34.5%(6Kc)

5.52 5 28.6 19.2 5.5

43 40 6.5 10.5 15

−60 60 −45 −60 −40

0 110 0 −10 24 R.T.

110 110 105 105 110

τ (s)

τR (s)

474

3.66

54.3 116 1.4 × 104

2.77 10

Polymer Standards Service. bPolysciences, Inc. cPolymer Source Inc. dAldrich 182265.

°C for PS20M5%-TCP and PS400K29%-TCP, respectively. At the lowest rates, a familiar shear stress overshoot shows up. At intermediate rates, the strain softening after the stress maximum is rather weak. Further increasing shear rates, a new phenomenon emerges when the stress vs strain curve shows divergence and displays a cusp as shown in Figures 2a and 3a, and the peak stress moves to lower strains with increasing applied rate, as summarized in Figure 1. Figure 3a indicates that a high level of entanglement is not necessary as Z is only 6.5 for PS400K29%-TCP. Since the final peak stresses at the cusp can well exceed the rubber elasticity limit given by σ = γGpl, as shown in Figure 2a, the behavior reveals strain hardening. For rates equal to or higher than 4 s−1, σ eventually increases even more strongly with γ than linearly. The normal stress also exhibits cusp-like response at the same strains of the shear stress cusp, as shown Figure 2b. The corresponding transient viscosity arises above the zero-shear envelope of linear response as shown in Figure 2c. This remarkable strain hardening shows up not only in the two PS solutions but also in PMMA1M19%-DEP, as shown in Figure 3b. To further demonstrate the universality of strain hardening, we show in Figure 3d that a pure PS mixture, PS20M5%-PS2K as characterized by Figure 3c, sheared at 110 °C, can also display strong strain hardening at shear rates of 0.6 and 2 s−1, respectively. 3.2. Shear Softening-to-Hardening (SSH) Transition. Below the strain hardening regime, i.e., at lower rates, the peak of normal stress N1 lags behind that of σ, as is the case for PS20M5%-TCP at γ̇ equal to or lower than 3 s−1. All available data in the literature on startup shear of entangled polymers show such delayed peaking of N1. In contrast, the normal stress N1 peaks at the same moment when σ shows a cusp in the strain hardening regime. Figure 4a illustrates such different correlations between N1 and σ at three rates of 3, 3.5, and 5 s−1. To better illustrate the shear softening-to-hardening (SSH) transition at a critical rate, we replot the emergent N1 as a function of σ in Figure 4b. Marching along the curve, upon reaching σmax, N1 moves upward for 3 s−1 but downward for 3.5 s−1 as the curve turns “backward” (to the left) along the X-axis, indicating a transition between 3.0 and 3.5 s−1. 3.3. WiR at Onset of Strain Hardening: Temperature Dependence and TTS Breakdown. The shear softening-tohardening (SSH) transition occurs at noticeably different values of WiR at the different temperatures, at least 60 °C above Tg of the solution. Using the WLF shift factor aT from the construction of the master curves for G′ and G″ in Figure 1, we investigate the responses of PS20M5%-TCP to startup shear at various equivalent rates aTγ̇ at three different temperatures. Figure 5a shows that the time−temperature superposition (TTS) breaks down at higher WiR since the three curves involving T = −10, 0, and 20 °C begin to split for WiR > 4. The

Figure 1. Master curves of G′ and G″ for PS20M5%-TCP (green open circles and squares) and PS400K29%-TCP (blue open upper and lower-pointed triangles), referring the left Y-axis, based on SAOS measurements at −10, 0, 20, and 50 °C. Also plotted is the strain at the peak of the shear stress, γmax, at various values of γ̇ at T = 0 °C, referring to the right Y-axis. The X-axis is either DeR or WiR. Red ◆ depicts γmax of PS400K29%-TCP as a function of WiR, whereas orange ● is for PS20M5%-TCP. ◁ replots Figure 9 in ref 35 with a shifting factor of τd/τR = 36, while ▷ is directly from Figure 3 in ref 11. The straight lines mark scaling slope of 1/3 and 1. TCP at shear rates lower than 1 s−1, 8 mm parallel plates for PS20M5%-PS2K). The precision of this fixture has been verified by comparing SAOS data of PS20M5%-TCP from CP25-2. Each startup shear was done either with a fresh loading or after relaxation at elevated temperatures for at least 104τ, to ensure that the terminal crossover frequency returns to the equilibrium value according to the SAOS tests.

3. RESULTS 3.1. Strain Hardening in Solutions of Linear Polymers. The SAOS data of PS20M5%-TCP and PS400K29%-TCP are given in Figure 1 in terms of the Rouse−Deborah number DeR = ωτR. Against the reference curves of G′ and G″ vs DeR, Figure 1 also shows how the critical strain γmax at the peak of the shear stress changes with WiR, obtained from stress vs strain curves based on such raw data as those in Figures 2a and 3a. The dependence of γmax on WiR collapses onto the same master curve for the two solutions as well as the literature data involving a linear PS melt35 of Mw = 182 kg/mol with Z = 12 and a PS2.89M9.5%-TCP solution11 with Z = 13. In a small range of WiR, the master curve exhibits 1/3 scaling behavior, which has previously been shown to occur in entangled polybutadiene solutions over a wide range of rates.35,36 At higher values of WiR, γmax increases with WiR as strongly as linearly, with the eventual approach to a critical point (γc as discussed in section 4.1) being even stronger than linear. Figures 2a and 3a summarize the stress responses to startup shear as a function of strain γ at different shear rates γ̇ at T = 0 B

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Figure 2. Shear stress σ vs strain γ during startup shear of PS20M5%-TCP at various shear rates γ̇ and 0 °C. The rubbery elasticity formula of σ = Gplγ for Gaussian chain network deformation is also drawn as a reference. (b) Corresponding normal stress N1 versus γ. (c) Transient shear viscosity vs time can grow above the zero-rate envelope given by |η*(1/ω)|.

Figure 3. Shear stress σ versus strain γ during startup shear at various shear rates γ̇ of (a) PS400K29%-TCP at 0 °C and (b) PMMA1M19%-DEP at −10 °C. (c) G′ and G″ curves based on SAOS measurements of PS20M5%-PS2K involving temperatures ranging from 160 to 110 °C, where the reference temperature is 110 °C. (d) Stress σ versus strain γ of PS205%-PS2K during startup shear at three rates of 0.1, 0.6, and 2.0 s−1.

at −10 °C, 4 s−1 at 0 °C, and 33 s−1 at 20 °C as shown in Figure 5c. In other words, the solution responds in an ordinary strainsoftening manner at 20 °C, sharply strain hardens at −10 °C, and is moderately strain hardening at the intermediate 0 °C.

TTS only holds at lower rates, e.g., at WiR = 3.66 when the responses are comparable as shown in Figure 5b. The failure of TTS shows up in a dramatic fashion when the solution is examined at WiR = 14.6, corresponding to γ̇ equal to 0.924 s−1 C

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level of chain (terminal) dynamics or in the “plateau and terminal zones”.30 Since nonlinear rheological behavior has been thought to be dictated by the chain dynamics, e.g., by the value of Wi or WiR,3,37 the TTS has been expected to apply. Violation of the TTS principle is usually reported for linear viscoelastic properties of certain polymers when terminal dynamics and segmental dynamics do not vary with temperature in the same way, as shown by Plazek27−29 and others.31−33 At sufficiently high temperatures, namely well above Tg, the TTS principle is restored. In contrast, the present observation is different: The failure of the TTS is found during transient nonlinear responses to startup shear well above the overall Tg of the sample. Thus, the observed failure of the TTS indicates that (a) segmental dynamics control the transient nonlinear responses and (b) the temperature dependence of the segmental dynamics cannot be predicted according to the temperature dependence of the terminal dynamics. 3.4. Rupture at Cusp by PTV Observation. The particletracking velocimetric observations (PTV) were carried out to examine the macroscopic strain field before and after the cusp using PS7M5% (35%6K:60%TCP). The PTV component consists of a custom-made aluminum (Al) cone (cone angle of 5° and truncation of 130 μm) bottom plate with a glass window at 6−7 mm from the center. The top plate was a circular glass coverslip glued on an Al frame. A sheet of laser illuminates the PTV particles in the solution at an angle, with camera also positioned at an angle as shown in Figure 6a. The applied shear rate was calibrated by the movie in the Supporting Information. The illuminated path has a length of about 0.7 mm and corresponds to 400 au, as shown in Figure 6b. Incorporation of PS6K is necessary to slow down the polymer segmental dynamics with a reasonable long terminal relaxation time of τ = 1.4 × 104 s at room temperature. This solution can readily undergo strain hardening at rates amenable to PTV measurements at room temperature. Figure 6 and the PTV movie in the Supporting Information shows that the rupture takes place right at the cusp of shear stress, leading to transient macroscopic shear recoil. Prior to the cusp, the shear field is homogeneous, as indicated by the PTV movie and straight blue line in Figure 6b. This rupture is analogous to the

Figure 4. (a) Contrasting σ (a group of three curves below 104 Pa) and N1 (a group of three curves above 104 Pa) as a function of γ to show that the peak of N1 lags behind the peak of σ at 3 s−1. At 3.5 and 5 s−1, N1 peaks at the same strains as σ does. (b) Replot of the same data from (a) as N1 vs σ. Also included is data from the startup shear measurement at a rate of 1 s−1.

It is necessary to point out that the observed breakdown of the TTS is not an artifact due to any failure to maintain isothermal condition. Given the low stress level involved to shear this 5% PS solution, even if all of the accumulated mechanical energy converts to internal kinetic energy, the temperature rise in the sheared sample is still negligible (i.e., < 0.1 °C) according to (T − T0) = (1/ρcp)∫ γ0dγ′σ(γ′) using cp ∼ 1.9 kJ/(kg K). The TTS principle is known to work well at the

Figure 5. (a) γmax versus aTγ̇ and WiR curves from a series of startup shear of PS20M5%-TCP carried out at −10, 0, and 20 °C with aT being 4.33, 1, and 0.12, respectively. The inset shows the WLF shift factors aT and bT as a function of temperature, where the reference temperature is 0 °C. (b) Stress vs strain curves at the same WiR = 3.66 were almost identical, obeying the TTS. (c) Shear stress responses are markedly different at the same WiR = 14.6 during startup shear of PS20M5%-TCP at −10, 0, and 20 °C, where the actual rates are indicated in the parentheses. The inset shows the corresponding curves of N1 vs σ. D

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solutions have marginal states approximately at the respective rates of 3.5 (0 °C), 0.3 (−10 °C), and 4 (0 °C) s−1. The corresponding γc can be read from Figures 4a, 3b, and 3a respectively to be 100, 45, and 35, revealing γc ∼ ϕ−0.65, as shown in Figure 7. The estimate is evidently crude, especially for PMMA19% solution in Figure 3b where the curve of 0.3 s−1 is obviously already about the SHH transition. Despite the uncertainty, the scaling γc ∼ ϕ−0.65 and γc ∼ lent(ϕ) is plausible.

Figure 7. Scaling of stretchability expressed in terms of γc ∼ ϕ−0.65 for the three solutions at concentrations of 5, 19, and 29%.

Figure 6. (a) Scheme of the particle tracking velocimetry setup. (b) Velocity profiles at different stages upon startup shear of the PS7M5% solution at 2.5 s−1 and room temperature according to the PTV observations. The PTV video shows a homogeneous shear field from the beginning of the startup shear up to the point of rupture that terminates the strain hardening behavior. The rupture takes place as an abrupt break in a middle plane inside the sample, leading to macroscopic recoil.

4.2. Chain Scission. To examine the effect of the rupture shown in Figure 8a on the chain dynamics, SAOS measurements were carried out (after relaxation at an elevated temperature for 104τ) as shown in Figure 8b. There is a discernible increase in the crossover frequency relative to that of the equilibrium sample. Multiple strain-hardening runs were necessary to produce a 10% change of τ in PS400K27%-TCP. Figure 9a shows a systematic shift of the crossover frequency ωc = 1/τ to higher values as the sample underwent more repeats of the startup shear past the point of rupture at T = 0 °C and rate of 6 s−1. There are more noticeable changes in the stress responses as shown in Figure 9b. To identify the observed change in τ, we carried out GPC measurements of the postrupture PS400K27%-TCP. Figure 9c indeed shows a slight decrease of molecular weight. Thus, rheometric data, PTV observations, and GPC result are consistent with the picture that at sufficiently high rates entanglement tightens up, causing the solution to undergo non-Gaussian stretchinguntil a moment when chain scission takes place to induce a localized collapse of the entanglement network, propagating into macroscopic rupture and causing rapid stress decline. This strain hardening response to startup simple shear is highly unexpected for linear polymers.26 Although true strain hardening leading to melt rupture is commonly observed in uniaxial extension for Wi > Wirupture,40 the same melt usually only shows strain softening at the same Wi upon startup shear.43,44 4.3. On the Origin of Non-Gaussian Stretching. The shear strain hardening has so far only been found in entangled solutions where the solute polymer is of high Tg, with Tg > Texp as shown in Table 1. For comparison, we examined the case of Texp > Tg of solute polymer (for solutions of polybutadiene and polyisoprene, for PS melt and solution) and Texp > Tg of solvent, at WiR > 10. Stress overshoots all appear to be normal at comparable or even high WiR for strain hardening, as shown in any previous literature results,25,36,45 or on solution of 4 M polybutadiene (4000 kg/mol, Tg = −100 °C) 1% in 1.8K oligobutadiene at 20 °C (shown in Figure 10 for comparison) or on

previous rheometric38 and PTV39 observations of shear rupture in entangled worm-like micellar solutions.

4. DISCUSSION 4.1. Stretchability and Mesh Size of Entangled Network. It is worthwhile to discuss the behavior at the shear-softening-to-hardening transition. For the PS20M5%TPC solution, the critical shear rate is 3.5 s−1. At this rate the shear stress stays saturated around 1900 Pa between γ of 25 and 105 whereas N1 keeps increasing as shown in Figure 4a. The continuous monotonic growth of the normal stress N1 confirms that the state between γ = 20 and 105 is not steady flow but plausibly at a delicate balance between further stretching of the residual entanglement strands and continuing loss of entanglements. We term this critical condition the “marginal state”. At this “marginal state”, the strain γc at the breakdown of the solution (at the stress cusp) appears to scale with concentration in the same way as the mesh size of the entanglement network, γc ∼ lent(ϕ). The entanglement network of the three solutions can be viewed to get denser and less stretchable with increasing concentration ϕ from 5% to 29%, as in PS20M5%-TCP, PMMA1M19%-DEP, and PS400K29%-TCP. The mesh size of the entanglement network in pure melt is given by lent0 = Ne(ϕ = 1) lK with lK being the Kuhn length; the mesh size in solution of concentration ϕ is greater and scales with ϕ as40−42 lent(ϕ) = Ne(ϕ) lK = lent0ϕ−0.65. (Ne(ϕ) = Ne(ϕ = 1)ϕ−1.3, Ne(ϕ = 1) is the number of Kuhn segments in an entanglement strand in pure melt, 22 for PS and 21 for PMMA, so we can omit this difference in scaling.) On the other hand, these three E

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Figure 8. (a) Stress vs strain curve of PS20M5%-TCP upon startup shear at 3 s−1 and −10 °C. (b) SAOS data at 60 °C before and after the startup shear depicted in (a) shows a noticeable change of the crossover frequency. The sample was in the stress-free state at T = 60 °C for 104τ.

Figure 10. An example of solution with Texp > Tg,solute at WiR > 10 shows no sign of strain hardening. (a) G′ and G″ master curves of PB4M1%-oligoB at 20 °C based on SAOS measurements from 40 to 0 °C. (b) Adjusted stress vs strain during startup shear at rates from 0.01 to 10 s−1. Shear stress was adjusted by minus contribution from solvent viscosity of 265 Pa·s.

solution (data not shown) of 900 kg/mol polyisoprene (Tg ∼ −15 °C, high vinyl) 8% in C18 oil at −15 °C. When solute of an entangled solution is high-Tg polymer, perhaps entanglements under fast shear are much slower than the overall dynamics revealed by the terminal relaxation time. It is plausible through the self-concentration effect46 or segmental chain friction arises during fast shear.47 The local “effective” Tg for the solute polymer chains might be much closer to Texp although Texp is much higher than the overall Tg of the solution. This difference between local chain dynamics and solution dynamics may also cause the observed violation of the TTS principle.

There are other scenarios that are less plausible for causing strain hardening in shear. (A) To explain the discrepancy in the rheological responses to uniaxial extension between entangled melts and solutions,41,42,48 the idea of monomeric friction reduction has been proposed.49,50 Since there would be more friction reduction in a melt than in a solution, according to these studies, solutions should be more prone to strain hardening than melts. However, this idea cannot explain why polybutadiene and polyisoprene solutions do not show strain hardening. (B) The observed shear strain hardening is unlikely to be due to any concentration fluctuations. A previous light

Figure 9. PS400K27%-TCP undergoes multiple (28) cycles of (1) startup shear at 0 °C and rate of 6 s−1 to a strain of 45, (2) resting at 50 °C for over 104τ, and (3) SAOS at 50 °C. (a) Terminal crossover frequency ωc from the SAOS at 50 °C as a function of the number of the startup runs. (b) Selected stress vs strain curves after various numbers of run where the stress level has been corrected for any sample loss due to edge instability after the cusp. (c) GPC curves for the pristine sample and the sample after 28 runs of startup shear. F

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Macromolecules scattering study51 showed that shear-induced structures and concentration fluctuation52−55 need prolonged shear in steady state. Structural changes on chain level leading to concentration fluctuation would only take place after the entanglement network has suffered a collapse, long after stress overshoot. The observed strain hardening occurs because the network is unable to fully yield.

(8) Wagner, M. H.; Meissner, J. Network disentanglement and timedependent flow behaviour of polymer melts. Makromol. Chem. 1980, 181, 1533−1550. (9) Menezes, E. V.; Graessley, W. W. Nonlinear rheological behavior of polymer systems for several shear-flow histories. J. Polym. Sci., Polym. Phys. Ed. 1982, 20, 1817−1833. (10) Osaki, K.; Inoue, T.; Isomura, T. Stress overshoot of polymer solutions at high rates of shear. J. Polym. Sci., Part B: Polym. Phys. 2000, 38, 1917−1925. (11) Osaki, K.; Inoue, T.; Uematsu, T. Stress overshoot of polymer solutions at high rates of shear: Semidilute polystyrene solutions with and without chain entanglement. J. Polym. Sci., Part B: Polym. Phys. 2000, 38, 3271−3276. (12) Maxwell, B.; Nguyen, M. Measurement of the elastic properties of polymer melts. Polym. Eng. Sci. 1979, 19, 1140−1150. (13) Wang, S. Q.; Ravindranath, S.; Wang, Y.; Boukany, P. New theoretical considerations in polymer rheology: Elastic breakdown of chain entanglement network. J. Chem. Phys. 2007, 127, 064903. (14) Mohagheghi, M.; Khomami, B. Elucidating the flow-microstructure coupling in entangled polymer melts. Part II: Molecular mechanism of shear banding. J. Rheol. 2016, 60, 861−872. (15) Tapadia, P.; Wang, S. Q. Direct visualization of continuous simple shear in non-newtonian polymeric fluids. Phys. Rev. Lett. 2006, 96, 016001. (16) Wang, S. Q.; Ravindranath, S.; Boukany, P. E. Homogeneous Shear, Wall Slip, and Shear Banding of Entangled Polymeric Liquids in Simple-Shear Rheometry: A Roadmap of Nonlinear Rheology. Macromolecules 2011, 44, 183−190. (17) Liu, G.; Wang, S. Q. A Particle Tracking Velocimetric Study of Stress Relaxation Behavior of Entangled Polystyrene Solutions after Stepwise Shear. Macromolecules 2012, 45, 6741−6747. (18) Lu, C. Y. D.; Olmsted, P. D.; Ball, R. C. Effects of Nonlocal Stress on the Determination of Shear Banding Flow. Phys. Rev. Lett. 2000, 84, 642−645. (19) Olmsted, P. Perspectives on shear banding in complex fluids. Rheol. Acta 2008, 47, 283−300. (20) Adams, J. M.; Olmsted, P. D. Nonmonotonic Models are Not Necessary to Obtain Shear Banding Phenomena in Entangled Polymer Solutions. Phys. Rev. Lett. 2009, 102, 067801. (21) Agimelen, O. S.; Olmsted, P. D. Apparent Fracture in Polymeric Fluids Under Step Shear. Phys. Rev. Lett. 2013, 110, 204503. (22) Moorcroft, R. L.; Fielding, S. M. Shear banding in timedependent flows of polymers and wormlike micelles. J. Rheol. 2014, 58, 103−147. (23) Masubuchi, Y.; Watanabe, H. Origin of Stress Overshoot under Start-up Shear in Primitive Chain Network Simulation. ACS Macro Lett. 2014, 3, 1183−1186. (24) Cao, J.; Likhtman, A. E. Simulating Startup Shear of Entangled Polymer Melts. ACS Macro Lett. 2015, 4, 1376−1381. (25) Costanzo, S.; Huang, Q.; Ianniruberto, G.; Marrucci, G.; Hassager, O.; Vlassopoulos, D. Shear and Extensional Rheology of Polystyrene Melts and Solutions with the Same Number of Entanglements. Macromolecules 2016, 49, 3925−3935. (26) Liu, G.; Cheng, S.; Lee, H.; Ma, H.; Xu, H.; Chang, T.; Quirk, R. P.; Wang, S. Q. Strain Hardening in Startup Shear of Long-Chain Branched Polymer Solutions. Phys. Rev. Lett. 2013, 111, 068302. (27) Plazek, D. J. Temperature Dependence of the Viscoelastic Behavior of Polystyrene. J. Phys. Chem. 1965, 69, 3480−3487. (28) Plazek, D. J.; Chay, I. C.; Ngai, K. L.; Roland, C. M. Viscoelastic properties of polymers. 4. Thermorheological complexity of the softening dispersion in polyisobutylene. Macromolecules 1995, 28, 6432−6436. (29) Plazek, D. J. 1995 Bingham Medal Address: Oh, thermorheological simplicity, wherefore art thou? J. Rheol. 1996, 40, 987−1014. (30) Dealy, J. M.; Plazek, D. J. Time-temperature superpositiona users guide. Rheol. Bull. 2009, 78, 16−31. (31) Ngai, K. L.; Casalini, R.; Roland, C. M. Volume and Temperature Dependences of the Global and Segmental Dynamics

5. CONCLUSION In conclusion, entangled solutions made of high-Tg linear polymers (PS and PMMA, Tg,solute > Texp > Tg,solution) exhibit a shear softening-to-hardening transition at sufficiently high rates.26 The observed strain hardening implies that upon startup simple shear chain entanglement can get locked up to prevent complete yielding of the entanglement network, leading to non-Gaussian chain stretching. A collapse of the entanglement network, leading to the observed macroscopic sample rupture, is initiated presumably by a few events of chain scission. The strain hardening response is more severe at lower temperatures, at a given effective shear rate. This trend indicates a failure of the TTS.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.6b02053. Particle-tracking velocimetric observations (MPG)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (G.L.). *E-mail [email protected] (S.-Q.W.). ORCID

Gengxin Liu: 0000-0002-2998-8572 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported, in part, by the National Science Foundation (DMR-1105135). REFERENCES

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DOI: 10.1021/acs.macromol.6b02053 Macromolecules XXXX, XXX, XXX−XXX