Rheology of Entangled Polymers Not Far above Glass Transition

Article pubs.acs.org/Macromolecules

Rheology of Entangled Polymers Not Far above Glass Transition Temperature: Transient Elasticity and Intersegmental Viscous Stress Hao Sun,† Gengxin Liu,† Konstantinos Ntetsikas,‡ Apostolos Avgeropoulos,‡ and Shi-Qing Wang†,* †

Morton Institute of Polymer Science and Engineering, University of Akron, Akron, Ohio 44325-3909, United States Department of Materials Science and Engineering, University of Ioannina, University Campus, Ioannina, 45110, Greece

‡

ABSTRACT: This work studies the transient rheological responses of different entangled melts to startup deformation at unconventionally high rates, which can be accessed by working at sufficiently low temperature, e.g., at and below 130 °C for polystyrene. When the rate is as high as the second crossover frequency ωe observed from the small amplitude oscillatory shear (SAOS) data for storage and loss modulus G′ and G″, there is sizable viscous stress that dominates the initial mechanical response. It is shown based on the various polymer melts including PS, PMMA, SBR, PC, and PS mixtures that this viscous component of the stress cannot be neglected when characterizing such fast deformation in either extension or shear. This sizable frictional addition to the rubber elasticity component of the initial stress is founded to be preceded by solid-like deformation. The remarkable transient elasticity can be characterized by a modulus Ginterseg that grows well beyond the magnitude of the melt plateau modulus G0N as the temperature lowers toward the glass transition temperature Tg. In the case of PS, Ginterseg reaches a level of 300 MPa at 110 °C and reduces to 2.0 MPa, i.e., 10G0N, at 130 °C. The initial elasticity is shortlived because the melts quickly transition to a state of viscous flow at a strain of just a few percent. The magnitude of the viscous stress at the solid-to-liquid transition defines a yield stress. This yield stress is found to scale with the applied rate in a power law, agreeing with the dependence of G″ on frequency ω from the SAOS data. Moreover, the intersegmental effect, i.e., the transient elasticity, is shown to also take place in unentangled and barely entangled PS melts as well as branched polyisoprene melts.

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INTRODUCTION Past studies of dynamics and deformation of entangled polymers well above the glass transition temperature Tg have been based on the successful modeling of the rubber elasticity in terms of the entropic conformational changes associated with the affine deformation of a Gaussian chain network.1−7 The classical rubber elasticity theory assumes that the intersegmental friction makes a negligible contribution to the mechanical stress. The agreement of this theory with experiment indicates that the mechanical stress is dominantly intrachain in origin under normal circumstances. The prevailing theoretical framework, based on the tube model for rheology of polymer melts and solutions,8−12 also evaluates the mechanical stress in terms of intrachain retractive forces. By smoothing out the heterogeneous, point-like interchain interactions using a fictitious tube, the tube theory assumes affine deformation of the tube and perceives the primitive chain to undergo Rouse dynamics and meet no entropic barrier against any chain retraction inside the © XXXX American Chemical Society

tube. Thus, by construction that lacks a self-consistent treatment of interchain interactions, it is difficult for the tube theory to describe any structural failure of the entanglement network. Does the chain retraction on Rouse time produce chain disentanglement? How does such retraction modify the “tube”? The tube model cannot answer such questions in a self-consistent manner. On the other hand, the tube theory prescribes the increasing chain orientation as the molecular origin of the stress overshoot under weak shear. However, such a picture is currently being contested by molecular dynamics simulation.13,14 In the past decade, our experimental findings of strain localization in the form of shear banding upon startup shear15 and nonquiescent relaxation after large stepwise deformation of both shear16 and extension17 suggest that nonlinear rheological Received: April 30, 2014 Revised: July 30, 2014

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dx.doi.org/10.1021/ma500899s | Macromolecules XXXX, XXX, XXX−XXX

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Table 1. Chemical and Rheological Characteristics of Different Polymer Melts polymer

Mw (kg/mol)

Me (kg/mol)

G0N (MPa)

PDI

τ = 1/ωc (s)

temp (°C)

Tg (°C)

PS1M PS880K PS163K PS50K PS10K PS4K PC63K SBR161K PMMA125K H−PI150K

1028 880 163 50 10 4 63 161 125 210

13 13 13 13 N/A N/A 1.3 4.3 13 7

0.22 0.22 0.22 0.22 N/A N/A 2.0 0.53 0.42 0.28

1.02 1.02 1.01 1.06 1.01 N/A 1.58 1.05 1.41 1.10

122 000 25 000 166 1.00 N/A N/A 3560 13 200 2250 N/A

130 130 130 130

103 104 101 104 94 N/A 149 −10 113 −11

155 20 135 0

This transient elasticity appears to be different from the previously observed glassy responses52−57 in the vicinity of Tg, as will be explained in the text.

responses of entangled polymers are far more complicated and interesting than what can be depicted by the tube model. They informed us that any valid description of entangled polymers under large deformation must address two basic questions simultaneously: (a) What causes chain deformation upon startup deformation? (b) When does the affine deformation of the entanglement network cease? Extensive experimental studies show that the initial response of an entangled polymer to startup deformation was largely elastic, characterized by the rubbery elastic plateau modulus G0N, as if the deformation is affine as found for vulcanized rubbers.6 Then an overshoot in either shear stress18 or tensile force (i.e., engineering stress)19 emerges depending on whether the startup is simple shear or uniaxial extension. This stress peak signifies a transition from elastic deformation to flow (irrecoverable deformation). This is analogous to yielding20 observed in large deformation of solid materials including ductile polymer glasses (i.e., entangled polymers supercooled to below Tg), where the yielding refers to a change from elastic to plastic deformation. To study the molecular mechanism21,22 for the yielding behavior of entangled polymers well above Tg has been a core task of our research in the past decade. The present work explores something entirely different in rheology of entangled polymer melts. Specifically, we carry out standard rheological measurements of several polymer melts at unconventionally low temperatures and high deformation rates to determine whether the approximation to neglect the intersegmental friction is still valid. The earliest studies of viscoelastic properties close to Tg took place many decades ago.23,24 Boyer proposed the idea of “liquid−liquid” transition well above the glass transition temperature Tg.25−31 However, this proposal has been extensively debated,32−37 both experimentally and theoretically, and a modern discussion of the topic can be found in ref 38. It is also relevant to mention that close to Tg the segmental dynamics show different temperature dependence from that of the terminal relaxation, leading to a breakdown of the time−temperature superposition principle.39−51 A simple analysis shows that when the applied rate approaches a time scale characterized by the so-called tubeconfinement time τe, the stress should arise from both the traditional intrachain retraction force and intersegmental friction. Our experiments of startup uniaxial extension and simple shear confirm this conclusion. More interestingly, the intersegmental contributions first appear in the form of transient elasticity associated with local packing of polymer segments. In other words, a Hookean-like solid response precedes a viscous flow response at a strain of only a few percent. The solidlike response involves an elastic modulus that increases well beyond G0N, e.g., over 1000 times G0N for polystyrene at 110 °C.

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EXPERIMENTAL SECTION

A. Materials. This work studies the stretching behaviors of several different polymer melts and solutions at high Hencky rates. PS1M, PS880K, PS163K, PS10K, and PS4K were atactic polystyrene (PS) synthesized by Dr. Hongde Xu at the University of Akron. The PS50K was purchased from PolyScience Inc. and used as received. The PC63K is a bisphenol A polycarbonate (PC), Lexan TM 141 111, acquired from Sabic (GE Plastic). The SBR161 K is a random copolymer of styrene and butadiene (SBR), synthesized by Dr. Xiaorong Wang at Bridgestone Americas Center for Research and Technology and has been characterized before.58 The PMMA125 K is poly(methyl methacrylate) from Plaskolite Inc. and with the item number CA-86. The H-shape 3, 4-polyisoprene (150 K backbone, 25K arm, 63% 3, 4 content) (H-PI) was synthesized at Ioannina, Greece. The chemical and rheological properties of these materials are listed in Table 1. PS mixtures, labeled as PS880 K-10K(8:2) and PS800 K-4K(5:5) respectively, were made by first dissolving the mixture of high (either 80 or 50%) and low molecular weight (either 20% 10K or 50% 4K) PS in toluene. Such solutions were then placed under the hood to evaporate enough solvent before introducing into an aluminum plate to be placed in the vacuum oven for several days at an elevated temperature to remove the remaining solvent. Table 1 gives the terminal relaxation time τ for each sample, evaluated from small amplitude oscillatory shear (SAOS) measurements of the storage and loss moduli G′ and G″ as a function of frequency. The SAOS were carried out with an Advanced Rheometric Expansion System (ARES) rotational rheometer and are shown in Figure 1a−c for PS1M at a reference temperature of 120 °C, PC63K at 155 °C and H−PI at 0 °C respectively. SAOS data for the other samples are presented in Appendix A. From Figure 1a−c, the value of G0N is taken as G′ at the frequency ωmin where the tan δ = G″(ωmin)/ G′(ωmin) shows a minimum. The shaded regime in each plot represents the range of effective rates applied in this study. In passing, we note that the Rouse dynamics59 does not describe the rubber−glass transition regime defined by the second frequency crossover at ωe. In other words, to depict the stress in the mean-field single chain model in terms of intrachain entropic forces and intermolecular friction is clearly inadequate. A proper theory has to account for the nonentropic, intermolecular association and topological chain uncrossability originating from excluded volume interactions. It is a truly many-body problem that at least requires a self-consistent treatment. The molecular weight and PDI were found using GPC, where SEC was done using a Wyatt Dawn Eos multiangle laser light (MALLS) detector in conjunction with Waters Model 2414 differential refractometer concentration detector. This was coupled with Wyatt Astra V 4.73.04 software and three Waters HR styrogel columns using THF at 35 °C flowing at 1 mL/min. Glass transition temperatures were found using a TA Instruments Q2000 DSC ramping at 10 °C/min to well above Tg followed by cooling at the same rate. B


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Figure 1. Small amplitude oscillatory shear measurements of PS1M (a), PS63K (b), and H−PI150 K (c). The full SAOS curves were constructed from measurements at different temperatures and using 120 °C as reference temperature for PS1M and 155 °C for PC63K, respectively. The reference temperature for H−PI is 0 °C. G′ and G″ are also plotted against Dee as the X axis, where Dee = ωτe. The shaded regions indicate the range of applied Hencky rates in startup extension experiments. The two vertical lines in part a correspond to the angular frequencies used in Figure 7a,b. B. Apparatus. The stretching of the polymer samples was carried out at different temperatures above their glass transition temperatures using a first generation SER fixture60,61 mounted on the ARES-LS rotational rheometer equipped with a Rheometric Scientific Oven accurate to 0.1 °C. In order to prepare uniform specimens for the SER fixture, the PS, PS mixtures, SBR, and PMMA were compressionmolded using a CARVER press to make sheets with thickness of 0.5 mm. PC sheets were made in the same way with a smaller thickness of 0.1 mm. Dimensions of 20 mm ×2 mm ×0.5 (0.1) mm were then cut from the large sheets. Ends of such specimens were adhered to the SER fixture using commercially available LOCTITE 486 glue. After the LOCTITE glue dried, the temperature of the oven was increased (or decreased, in the case of SBR161 K) to the desired temperature, and the sample was allowed to equilibrate for at least 5 min. Subsequently, the samples were extended under prescribed conditions. H−PI specimens were made from a dense H−PI/THF solution that slowly evaporated inside a glass ring (3 cm radius) on the Kapton film. The polymer film was placed in a vacuum oven for several days to remove any residual solvent before cut into the same dimensions as other polymer samples. The transient stress−strain curves of SAOS were recorded on an ARES-G2 rotational rheometer from TA Instruments, based on a pair of 8 mm parallel plates. The instrument allows us to collect the transient data during the SAOS cycles by recording the data in the transient mode rather than the correlation mode.

If we only focus on the initial responses, startup extension and shear should provide completely equivalent information. Therefore, most of experiments in this work are based on uniaxial extension of the various polymer melts. To impose sufficiently high rates, we need to carry out the experiment at relatively low temperatures. Figure 2a−e show five sets of stress−strain curves for PS1M and PC63K, each at two different temperatures, and for H−PI at one temperature. Similar data of the other samples are presented in Appendix A. To provide a reference point, the rubber elasticity curve given by σengr = G0N (λ − 1/λ2) is also plotted in each figure. Most of the applied rates fall into the shaded second crossover regime, as indicated in Figure 1a−c, that was usually not probed in previous studies of startup uniaxial extension of entangled polymer melts.58,62 Consequently, the initial stress response well above the rubber elasticity curve has not been seen and studied in the literature, to the best of our knowledge. As the extension grows, the entanglement network starts to produce the familiar rubber-elastic contribution in addition to the viscous stress. A more detailed analysis will be given in the following section to explain the initial sharp peak in the tensile stress at 10 s−1 in Figure 2a. At these low temperatures, the Rouse−Weissenberg number WiR = ε̇τR can be much larger than unity because, for example, Rouse relaxation time τR is 7480 s at 120 °C for PS1M.58 Thus, the entanglement network only partially yields in extension.19,62 In other words, a significant number of entanglements would

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PRELIMINARY RESULTS: NONCHAIN-NETWORK COMPONENT OF STRESS DURING STARTUP It is convenient to characterize transient rheological responses of entangled polymer melts in startup uniaxial extension. C


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Figure 2. Stress−strain curves for PS1M, PC 63K and H-PI150K at different Hencky rates of startup uniaxial extension. Key: (a) PS1M at 120 °C; (b) PS1M at 130 °C; (c) PC63K at 155 °C; (d) PC63K at 160 °C; (e) H−PI at 0 °C; (f) PS1M at 150 °C. Also presented is the rubber elasticity curve as the reference, given by σengr = G0N (λ − 1/λ2).

above Tg occurs typically at high strains upon chain disentanglement that leads to a structural breakdown of the entanglement network. We replot the initial stress response of PS1M at 120 °C in Figure 3 to examine the characteristics in more detail. The sharp rise in the stress followed by a leveling-off is produced at unconventionally low temperatures and high effective Hencky rates. Since this yield-like behavior occurs at a few percent of deformation, it has nothing to do with yielding of the entanglement network. In other words, the nature of the emerging transient elasticity must be intermolecular, and thus fundamentally different from the rubber elasticity, which is intrachain in origin. The characteristic shown in Figure 3 reminds us of the yielding phenomenon encountered in tensile extension of ductile polymer glasses. All supercooled polymer melts, even the most brittle PS, can yield in uniaxial extension at temperatures below but sufficiently close to Tg. Inoue and co-workers have carried out a systematic series of innovative studies52−54,56,57 on the mechanical behavior of polystyrene at temperatures above but very close to Tg and observed yield-like behavior similar to yielding in ductile polymer glasses. Their studies are actually relevant to the ongoing debate concerning the molecular origin

actually lock in, leading to non-Gaussian stretching of the entanglement network at high strains prior to a window-glass like rupture.58,62 As an additional reference point, we have also carried out startup uniaxial extension at a much higher temperature of 150 °C for PS1M. At this temperature, the available extensional rate is insufficiently high for us to access the shaded regime in Figure 1a. Figure 2f shows that everything went back to “normal”, i.e., the stress response stays below the rubber elasticity reference at the initial stage of extension.

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ADDITIONAL RESULTS, ANALYSIS, AND DISCUSSIONS In order to delineate the remarkable features seen in Figure 2a−e, i.e., (a) the initial elastic response, (b) yielding at a few percent of deformation, and (c) the emergence of viscous contribution to the total stress, we need to carry out detailed analysis and present additional experiments, along with pertinent discussions. 1. Yield Behavior Different from That Observed well above Tg or at Tg. Yielding in nonlinear rheological responses of entangled polymer melts upon startup extension, signaled by an engineering stress overshoot, has been demonstrated before.19 The transition from the initial elastic deformation to flow well D


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Figure 3. Stress−strain curves of PS1M in startup extension at (a) 120 °C and (b) 110 °C, showing the initial stage of the deformation. Here the Young’s modulus Einterseg is defined as the initial slope. The yielding characteristics are specified by the yielding stress σy and yielding strain εy respectively.

Figure 4. (a) Yield stress σy vs Hencky rate ε̇ in diamonds, characterizing the initial response of PS1M at 120 °C to startup uniaxial extension, evaluated according to the definition given in Figure 3, where the smooth curve over the diamonds is the SAOS data of 3G″ vs ω. Also plotted are the yield stresses at two temperatures from Inoue et al.57 that involved uniaxial extension of PS at temperatures much closer to Tg. (b) Yield stress of PC63K against a Weissenberg number defined as Wie = ε̇τe at two temperatures showing power law dependence σy ∼ Wie α where α = 0.37 at 155 °C and 0.74 at 152 °C.

produces α = 0.37 for 155 °C and 0.74 for 152 °C. The exponent is clearly sensitive to the range of Wi and temperature. Thus, the effects observed at these temperatures, still many degrees above Tg, appear different from those probed in the past studies.52−57 However, we note that when the circles and squares in Figure 4a are plotted on double-logarithmic scales they also look approximately linear. 2. Initial Elasticity as a Function of Temperature at Different Rates. Besides the power-law dependence of the yield stress on the rate, another feature of the initial solid-like response is the high modulus Einterseg, where the subscript “interseg” indicates that the initial elasticity should be intersegemental in origin. On the basis of data such as those in Figure 3, we can describe how Ginterseg = Einterseg/3 changes with the applied rate at different temperatures and compare with the melt plateau modulus G0N. Figure 5a shows that Ginterseg is over 10 MPa for PS1M at 120 °C. At comparable rates, as shown in terms of a new Weissenberg number Wie = ε̇τe, PS1M shows only a rubbery elastic modulus G0N = 0.2 MPa at 150 °C. We note that the same phenomenon occurs for PC63K as shown in Figure 5b. In both Figure 5a and 5b, as a reference, we add the SAOS data, plotting G′ as a function of a new Deborah number Dee = ωτe. Here the choice of the dimensionless Wie or Dee is to emphasize that the observed initial elasticity is only observed for Wie close to unity. In other words, it is not informative to characterize the condition for the initial elasticity using the conventional

of mechanical stress produced by large deformation of polymer glasses in the postyield “strain hardening” regime63−69 where the mechanical stress increases with the growing external deformation. They showed that the stress associated with rubber elasticity is inconsequentially low,57 i.e., much smaller than the total stress. In other words, even at many degrees above Tg, a glassy component is the dominant contribution to the mechanical stress. To determine whether the yield-like behavior in Figure 2a,b has the same physical origin as that studied by Inoue et al. and Muller et al.,52−57 we read the onset stress for yielding from Figure 3 and plot the yield stress as a function of the applied extensional rate in Figure 4a. For polymer glasses, i.e., below Tg, the yield stress typically scales logarithmically with the deformation rate.64 On the other hand, we also note that the theoretical description of yielding of polymer glasses does appear to show stronger than logarithmic dependence.70−73 The comparison of circles and squares with diamonds indicates that the transient solid-like response of PS at 120 °C may be different from the typical response of polymer glasses. Instead of logarithmic dependence, the relationship between σy and ε̇ can be depicted by a power law σy ∼ ε̇α, with α = 0.58, as shown by the smooth curve in the semilog plot in Figure 4a. This dependence actually tracks the curve of 3G″ vs ω from the SAOS data of PS at 120 °C, as shown in the following Figure 11. Similar power law scaling is observed for PC63K at two temperatures as shown in Figure 4b where the fitting E


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Figure 5. Initial solid modulus Ginterseg vs Wie, evaluated from raw data such as those in Figure 3. Also included is G′ vs Dee from the SAOS data. Key: (a) PS1M at 150, 130, 120, and 110 °C; (b) PC63K at 152, 155, 160, and 170 °C. Also presented in semifilled squares are the initial modulus Gi from the transient SAOS measurements given in subsequent Figure 7. Note that for PS1M τe involved in the definition of Wie is longer at 110 °C than its value at 120 °C by a factor of 60.

Weissenberg number Wi that is molecular weight dependent (whereas Wie is not). A couple of comments on Figure 5 are in order. First, Ginterseg shows a transition, i.e., a jump from a level of the elastic plateau modulus G0N, to a level that depends on temperature, as a threshold rate is approached. Second, the threshold value of Wie for the transition is higher at a higher temperature, suggesting that this phenomenon of strong initial elasticity is temperature sensitive and is not dictated by the normalized rate, i.e., Wie, alone. The temperature dependence of Ginterseg can be summarized as shown in Figure 6. There appears to be a cutoff temperature,

we have recorded the initial response during SAOS as shown in Figure 7a for PS1M at 120 °C. The initial data of SAOS clearly point to a much stiffer sample, revealing an elastic modulus Gi on the order of 9 MPa. The structure responsible for the initial elasticity soon yields apparently, around 0.1 s after the start of the SAOS as indicated in Figure 7a. In other words, the transient structure of PS at 120 °C yields in SAOS just as it does in startup extension. The long-time SAOS behavior reflects an elastic modulus as low as 0.6 MPa at this frequency. Thus, in Figure 5a we have included additional data based on such SAOS tests, as shown in the semifilled squares. As a reference, Figure 7b shows that no transient stiff response occurs at 150 °C. In passing, we note that the initial behavior of the SAOS reminds of a similar figure observed of a melt comprised of close-packed nanosized polybutadiene particles,75 which was also capable of yielding at low strains. We note that the emergence of the transient elasticity and subsequent yielding may be regarded as nonlinear response. It can even be probed with the so-called SAOS measurements, which we tend to identify as a method to characterize linear response. What is important is that the transient elasticity and accompanying yielding by definition cannot be depicted by any experimental data associated with the steady state, including the steady-state SAOS data. 3. Universality of Transient Elasticity. Since the initial elasticity is not associated with intrachain conformational deformation, we anticipate the behavior to be independent of molecular weight. Figure 8 presents the same curves in open symbols for three different molecular weights of entangled PS. Moreover, such transient elasticity due to intersegmental association should show up independent of the mode of deformation. In other words, startup shear should generate a similar mechanical response. Indeed, PS1M reveals the same transient elasticity: From the raw data of startup shear given in Appendix B, we extract the information (filled small circles) to compare with the data from startup uniaxial extension, as shown in Figure 8. Startup shear also allows us to examine this transient behavior for polymer melts that are unentangled or barely entangled. Solid squares and diamonds in Figure 8 show the Ginterseg at comparable applied rates, obtained from the raw data in Appendix B. The lower values of Ginterseg for PS10K and PS50K suggest that the presence of chain entanglement induces a tighter formation of the transient short-ranged structure among the polymer segments. In other words, although the intersegmental process (leading to the sudden buildup of elastic

Figure 6. Initial elastic modulus Ginterseg as the function of temperature. Here, Ginterseg is determined for Wie = 1.8 (110 °C), Wie = 1 (120 and 130 °C), and Wie = 0.0075 (150 °C) respectively. As a reference, G′ is plotted in open squares from the steady state SAOS data involving Dee = 1.8 (110 °C) and 1 (120 and 130 °C) respectively.

around 150 °C, above which Ginterseg is comparable to the melt plateau modulus G0N = 0.2 MPa. Between 110 and 150 °C, Ginterseg increases by 3 orders of magnitude. It is remarkable that Ginterseg ≫ G0N shows up in startup deformation whereas the SAOS data, explored on the same time scale, i.e., Dee ∼ Wie, only reveals elasticity at the level consistent with the melt rubbery elasticity. The data of Figure 6 remind us of the theoretical description74 of a growing elastic modulus as a function of temperature when a polymer melt is cooled toward Tg. We recall that typical SAOS data are presented from the steady-state rheometric measurements. To find out whether any initial elasticity would also show up in a SAOS experiment, F


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Figure 7. Transient SAOS data for PS1M at specific frequencies at (a) 120 and (b) 150 °C, respectively. The testing conditions are included in each plot, and the two angular frequencies used here were also indicated in Figure 1a. The data contain the first 20 cycles. The determination of initial modulus Gi and steady-state modulus G were included in part a. Also indicated is the time scale ty, beyond which the response softens.

Figure 8. Initial elastic modulus Ginterseg of PS for different molecular weights. The solid small circles are obtained from startup shear of PS1M at Wie = 1 (120 and 130 °C), matching the same effective rates for extension in open circles, and Wie = 1.8 (110 °C); Wie = 0.0075 (150 °C). The open squares represent PS880 K at Wie = 0.6 (110 °C) and Wie = 0.2 (120 °C). The open diamonds are Ginterseg of PS163 K at Wie = 0.6 (110 °C); Wie = 1 (120 °C); Wie = 0.23 (130 °C); Wie = 0.023 (150 °C). The solid diamonds involve startup shear of PS10K at 110 °C: 1.0 s−1; 115 °C: 3.0 s−1. 120 °C: 3.0 s−1. The solid squares designate Ginterseg obtained from startup shear of PS50K at Wie = 1 (120 and 125 °C); Wie = 0.3 (130 °C).

Figure 9. Yield strain εy, as defined in Figure 3, for PS1M at 120 °C as a function of Wie. The inset shows how the corresponding time ty for yielding as a function of Wie.

comprised of two components after the yielding of the transient intersegmental association: σ = σnetwork + σv. We apply the Edward−Vilgis model81,82 to represent the σnetwork so that we can extract and quantify the viscous component σv from the data presented in Figures 2a-d. For example, Figure 10 shows the “long time” behavior of the non-networking component of the tensile stress for PS1M at

stress) plausibly involves length scales within the entanglement spacing, the entanglement appears to enhance the transient intersegmental association. 4. Additional Characteristics of Yielding of Intersegmental Association. Figure 3 shows that the transition to viscous flow occurs at a higher yield strain εy for a higher rate. Approximately, we see εy =ε̇ty to be proportional to Wie, or ty to be nearly constant as shown in Figure 9 and its inset, respectively. Specifically, Figure 3 indicates that the transient structure has a lifetime close to ty ∼ 0.1 s at 120 °C. According to the literature,76 the local segmental relaxation process takes place on a time scale of τα ∼ 0.001 s at 120 °C, as determined by dielectric relaxation spectrum (DRS) measurements,76−80 which is shorter than ty by 2 orders of magnitude. Since it is well-known that α-relaxation time τα from mechanical experiments and from dielectric experiments can differ significantly, the observed yielding could still be related to α-relaxation processes. 5. “Long Time” Dissipative Flow. At such high rates, the entanglement network is initially undergoing affine deformation, contributing to the stress as if the melts are a cross-linked rubber. Thus, we can treat the “long time” overall stress σ as

Figure 10. Viscous component σv of the tensile stress as the function of stretching ratio λ at 120 °C for three Hencky rates of startup extension, where the apparent extensional viscosity ηa nearly matches the extensional dynamic viscosity 3η′ read from Figure 1a.

120 °C. It is important to note that such a viscous stress arises because of the unconventionally high applied Wi close to τ/τe, i.e., Wie ∼ 1. In other words, it can be argued that σv is G


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negligibly small in most previous rheometric measurements that do not reach the “second crossover regime”. The polymer segments relax on time scales shorter than τe, and their dissipative motions produce a “local viscosity” η1 < ηe, where ηe ∼ G0Nτe is the Rouse viscosity of a melt with the entanglement molecular weight Me. When Wi ≫ 1 but still satisfied Wi ≪ τ/τe, i.e., Wie ≪ 1, the dissipative stress is σv ∼ η1ε̇ < G0NWie ≪ G0N. On the other hand, σv can approach the level of G0N, which is the level of the elastic stress σnetwork from the entanglement network upon an appreciable strain, when Wie = Wi(τe/τ) ∼ 1 or >1. We expect σv to show up at Wie ∼ 1 independent of temperature although it has been practically difficult to realize Wie > 1 in experiment without going to the unconventionally low temperatures as done in the present study. The rate dependence of σv can be expected to occur in this second crossover regime because the segmental orientation is expected to increase with ε̇. Such “shear thinning” behavior can be characterized by plotting σv against ε̇, along with 3G″ vs ω, as shown in Figure 11. Here we summarized the information

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CONCLUSIONS

In this paper, we have explored a regime of polymer dynamics that has rarely been studied in the literature. Specifically, we applied startup deformation at unconventionally high rates where intersegmental association/interaction comes into play. Such high effective rates are only accessible in commercial rheometric instruments by conducting the startup experiment at unconventionally low temperatures, i.e., not very far above Tg. Both startup extension and shear experiments show that there is a sizable viscous component σv in the overall stress when the applied rate is close to the reciprocal tubeconfinement time τe. The existence of σv appears universal, i.e., σv emerges in all five different polymer melts studied in this work. Since σv depends on the applied rate in the same manner as the loss modulus G″ varies with ω, we observe power-law scaling of the effective viscosity with the rate. Such “shear thinning” behavior suggests that the orientation of chain segments increases with the applied rate. Our results have the following implications. At sufficiently high rates of deformation, the mechanical stress has contributions from both the wellknown rubbery elasticity due to the entanglement network and intermolecular viscous friction that we can neglect at low rates. We expect the viscous contribution to be present in this unconventionally high rate regime (Wie ∼ 1) even at high temperatures although we are unable to verify this assertion in the present study. For cross-linked polymers, we can expect the same rate-dependent viscous contribution to correct the classical rubber elasticity formula. A more interesting outcome of our investigation is the revelation that a transient elastic response shows up at the beginning of the startup extension and shear. This solidlike deformation can be characterized by an elastic modulus Einterseg = 3Ginterseg that increases sharply with lowering temperature and can be more than 2 orders of magnitude higher than the melt plateau modulus at 120 °C for PS. The transient elasticity is present as long as the experimental temperature is sufficiently low and the applied rate is above a threshold value that can be still much lower than 1/τe. The transient intermolecular association has a finite lifetime ty, beyond which it transitions to viscous flow. Such yield-like behavior, which does disappear at high temperatures, e.g., at 150 °C for PS, is rather different from the typical yielding of a glassy polymer although it must also be an activated process. To our knowledge, this transient elasticity has not been reported in the literature before and thus deserves to be further studied in the future. The emergence of such transient elasticity well above the rubbery elasticity in a temperature range of T/Tg ∼ 1.1 is perhaps consistent with the literature reports25−31,38,70−73 of a liquid−liquid transition in polymer melts.

Figure 11. Viscous stress σv vs Hencky rate ε̇ in filled symbols for seven melts and mixtures undergoing startup extension. Also presented are 3G″ vs ω from the corresponding SAOS data of these samples in open symbols.

from similar startup extension experiments on all the polymer melts and mixtures studied in the present work, including three relatively monodisperse PS, a relatively monodisperse SBR, commercial PC and PMMA, as well as two PS mixtures based on data in Figure 2a−d and those in Appendix A. The relationship between σv and rate is insensitive to the molecular weight and its distribution, as expected, because the viscous process responsible for σv occurs on segmental scales. In passing, we note that the magnitude of σv matches that of the yield stress σy because there is hardly any overshoot as shown in Figures 3 and 11. Consequently, in Figure 4a, the yield stress represented by the diamonds approximately match in the smooth curve that represents 3G″ vs ω. Finally we explain the sharp cusp in the curve of 10 s−1 in Figure 2a. At such a high rate, σv is substantial. When expressed in terms of the engineering stress as given in Figure 2a, this contribution σv‑engr = σv/λ naturally decreases as the crosssectional area shrinks. Thus, the emergence of the cusp underscores that the viscous flow, after yielding from the initial elastic deformation, is indeed dominant. The subsequent increase in σengr stems from the extension of the entanglement network that can be initially characterized by the classical rubber elasticity formula of σengr = G0N (λ − 1/λ2).

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APPENDIX A TRANSIENT EXTENSIONAL RHEOLOGICAL BEHAVIOR OF PS MIXTURES, PMMA, AND SBR To show the universality of the phenomena reported in this paper, in Figure 12, we present the transient rheological responses to startup uniaxial extension of PS mixtures, i.e., PS880−10K (8:2) and PS880−4K (5:5), a PMMA melt and a SBR melt, each at a temperature close to Tg with several different rates close to the high-frequency crossover ωe. The phenomena involving the PS mixtures, PMMA melt and SBR melt are consistent with what was observed for PS and PC melts as shown in Figure 2. Again, such initial stress response H


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Figure 12. SAOS data and corresponding stress−strain curves for different startup extensional rates of the PS880 K-10K(8:2) mixture at 120 °C (a, b), the PS880 K-4K(5:5) mixture at 120 °C (c, d), PMMA125 K at 130 °C (e, f), and SBR161 K at 0 °C (g, h). Shaded regions in SAOS correspond to the range of applied rates in startup extensions.

with a large Ginterseg has not been observed in our previous study of such materials at “normal” temperatures.58

Therefore, we carried out startup shear experiment. The initial stage of startup shear of PS1M at 120 °C is shown here in Figure 13 as an example. The Ginterseg from shear data has been summarized in Figure 8. Besides, we found the yield stress σy shows a power law dependence on rate and follows the dependence of G″ on ω, which is also consistent with Figure 4a.

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APPENDIX B. TRANSIENT ELASTICITY IN STARTUP SHEAR The transient elasticity due to intersegmental association should show up independent of the mode of deformation. I


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Figure 13. (a) Stress−strain curves from startup shear of PS1M at 120 °C for three different shear rates. (b) Yield stress σy and G″ vs the shear rate and frequency, respectively.

Figure 14. SAOS data of (a) PS10K at 110 °C and (b) PS50K at 125 °C, where the shaded region shows the range of applied shear rates. Stress− strain curves from startup shear of (c) PS10K at 110 °C and (d) PS50K at 125 °C.

We also performed measurements on a Rouse melt PS10K, whose SAOS is shown in Figure 14a, as well as a barely entangled PS50K, whose SAOS is given in Figure 14b. We found that these two PS melts, at temperatures sufficiently close to Tg, also display a large initial modulus, as shown in Figure 14c and 14d. From Figure 14c,d, we evaluate Ginterseg and present the data in Figure 8.

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APPENDIX C TRANSIENT ELASTICITY INDEPENDENT OF CHAIN ARCHITECTURE The transient elasticity also shows up in H−PI150 K melt above its Tg = −11 °C, indicating that the phenomenon is insensitive to chain architecture. Reading from data such as those in Figure 2e, we obtain the modulus Ginterseg characterizing the transient elasticity in Figure 15 that is similar to Figure 5. We see that the transient elasticity also vanishes at high temperatures, i.e., 23 °C for the present H−PI.

Figure 15. Transient elasticity given by Ginterseg against Wie, evaluated from raw data such as those in Figure 2e for H−PI melt at 23 and 0 °C. Also included is G′ vs Dee from the SAOS data in Figure 1c. J


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APPENDIX D EXAMINATION OF TIME−TEMPERATURE SUPERPOSITION Time−temperature superposition principle has been widely used to study the temperature dependence of rheological properties and to expand the frequency (rate) range under a specific temperature.83−85 In order to better understand the dynamics of the intersegmental friction, we examined its temperature dependence. From Figure 5 and 6, we noticed at the same effective rate, Ginterseg increases as temperature decreases. In other words, time−temperature superposition failed to capture the dynamics associated with Ginterseg since it shows different values at the same effective rate. On the other hand, the temperature dependence of the “long time” dissipative stress obeys the time−temperature equivalence principle. When we write the viscous stress σv in the form of σv = ε̇ηa, we note that ηa at different temperatures but the same effective rate (Wi) follows the temperature dependence of the dynamic viscosity η′: Since σv(ε̇) ∼ 3G″(ω) according to Figure 11, we have that ηa(ε̇) ∼ 3η′(ω) at different temperatures, as shown in Figure 16.

Figure 16. Apparent viscosity ηa and dynamic extensional viscosity 3η′ of PS1M, showing the time−temperature superposition. Here ηa and η′ involve Wi = De = 1.6 × 106 at 120 °C and Wi = De = 106 at 110 °C.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (S.Q.W.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is, in part, supported by the NSF (DMR-1105135). The authors thank the reviewers for helpful comments and would like to thank Dr. David Simmons for discussions.

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Rheology of Entangled Polymers Not Far above Glass Transition

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