Temperature, Frequency, and Small Static Stress Dependence of the

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Temperature, Frequency, and Small Static Stress Dependence of the Molecular Mobility in Deformed Amorphous Polymers near Their Glass Transition Frantisek Ondreas and Josef Jancar* CEITEC, Brno University of Technology, 616 00 Brno, Czech Republic ABSTRACT: For the first time, we report on using dynamic mechanical analysis (DMA) to directly follow molecular mobility in PMMA and PC deformed by small constant tensile stress (σ) over the temperature interval from Tg − 60 K to Tg + 60 K. At Tα′ > Tg, a pronounced secondary α′-relaxation was observed exhibiting larger frequency shift and stronger σ-dependence compared to the main α-relaxation. After reaching its maximum at approximately the Tg, the strain rate was progressively reduced at the onset of the α′-relaxation, attaining its minimum value at approximately the Tα′. Above the Tα′, the strain rate increased gradually again, suggesting that mobility has again been enhanced. We ascribe the observed variation in mobility to the strain-induced transition of segments from less mobile into more mobile domains, resulting in the alteration of conformational and orientational distribution. The macroscopic deformation (ε) dependence of the transient modulus, defined to characterize the system stiffness between the α and α′ relaxation (Γ*(ε)) and above the α′-relaxation (Ω*(ε)), collapsed to a single linear master curve for Γ*(ε) and the nonlinear Ω*(ε) master curve. Our results seem to indicate that the observed α′-relaxation may be caused by the process of intrabasin jump diffusion of small portions of the backbone chain over the characteristic length scale smaller than or equal to the dynamic Kuhn segment length (lK). The existence of the α′-relaxation may be a manifestation of the large strain-altered chain packing associated with the polyamorphic phase transition as the process accompanying yielding in polymer glasses.

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INTRODUCTION Polymer glasses are generally considered random packing of low and high density nanometer scale domains bound together by intra- and intermolecular forces.1 Because of chain connectivity, local strain affects the degree of freedom of structural units, an effect not observed in molecular glasses.1 Macroscopic deformation and temperature both alter the potential energy landscape and modify structural disorder, segmental mobility, and dynamic heterogeneity in polymer glasses. In a single step isothermal creep photobleaching experiment and MD simulation, it was shown2−4 that deformation-induced segmental mobility correlates with strain rate, increasing at the onset of yielding and decreasing during postyield strain hardening. The reduced mobility during strain hardening was related to the segment scale thermally activated yielding processes,5−9 supposedly coupled with the increase in rate of nonaffine segmental displacements represented by the increase of the effective stiffness length (Kuhn length, lK) with deformation.10 However, using a multistep creep photobleaching experiments, Lee et al.11 found no correlation between strain rate and mobility, suggesting that most likely the connection between any simple mechanical variable and segmental mobility is much more complex than anticipated originally,12,13 and despite significant progress, the understanding of this phenomenon is still far from complete.14−16 In an MD simulation, Lacks and Osborne12 found that high strains push the glass up the potential energy landscape while © XXXX American Chemical Society

small strains can actually push them the opposite direction and that the ratio between these two processes is controlled by the thermal history of the glass. Pushing up the energy landscape means that an external stimulus (T, stress, etc.) supplies energy to the system constituents (segments), thus reducing the energy barrier for their conformational changes (i.e., mobility). This approach, however, does not directly account for cooperativity changes induced by the same stimulus. While T reduces barrier for conformational rearrangements and simultaneously reduces cooperativity, stress or strain may reduce energy barrier and, at the same time, enhance cooperativity; thus, each of these stimuli may affect behavior of the glass in a different way. In their MD simulation, Warren and Rottler17,18 found that distribution of polymer segment hopping frequency scaled with strain, suggesting that the lower mobility segments residing in the denser domains are more accelerated by deformation compared to segments in less dense domains. Liu and Rottler19 found that apparent modification of the aging rate of shear yield stress has its origin in orientation of covalent bonds proportional to strain and logarithmic aging time. It was also shown20 that the relaxation behavior of polymer glass at large strains near the yield point corresponds to that at low strains near the Tg. Received: March 15, 2015 Revised: June 12, 2015

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DOI: 10.1021/acs.macromol.5b00550 Macromolecules XXXX, XXX, XXX−XXX

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syndiotactic form, and 41−52% atactic form, all apparently randomly distributed along each chain The polydispersity of the PMMA used can alter the distribution of relaxation times and the system dynamics resulting in quantitative variations of the results obtained; however, it seems reasonable to assume that it will not affect the existence phenomenon discussed in this paper. Hence, we chose to neglect the influence of polydispersity in this qualitative study. The dried pellets were compression-molded at 190 °C at maximum force 300 kN into 1 mm thick plates. Rectangular 30 × 5 × 1 mm3 specimens for the DMA measurements were cut out from these plates. The quench and wait thermal history of specimens used in this work is schematically depicted in Figure 1. To qualitatively evaluate the effect

Discussing the rejuvenation phenomena, McKenna proposed that tensile strains larger than the yield strain can result in a polyamorphic phase transformation in the glass,13 an effect not fully understood so far. Casalini and Roland21 showed that neither the free volume nor the configurational entropy governs the local dynamics of a polymer glass in any general sense, suggesting that the magnitude of fluctuations in local density controls the nanoscale segmental motions. There seems to be general agreement between experiments,21 theory,9 and MD modeling22−30 on the pivotal role of the amplitude of local density fluctuation, S0, as the primary structural variable in determining the deformation behavior of polymer glasses. The S0 is related to the preparation protocol, thermodynamic, and loading conditions. Varying the mechanically generated disorder alters the S0, thereby affecting the segment-scale relaxation times and the slope of the deformation curve. Hence, the S0 can be altered in a controlled manner by either preparation protocol21 or external deformation. At constant preaging time and temperature in a single relaxation controlled system, the S0 increases with deformation proportionally to the strain rate, and at constant preaging time and strain rate, the S0 increases with deformation proportionally to the temperature. We report on results obtained by subjecting two different polymer glasses to nonisothermal tensile creep with superimposed oscillatory small-amplitude deformation over a wide temperature, frequency, and static stress interval. It was hypothesized that the small oscillations force the segment scale structure evolving in time with gradually increasing strain rate to move locally along the potential energy landscape and the increasing temperature accelerating this process. Reproducibly for both poly(methyl methacrylate) (PMMA) and bisphenol A polycarbonate (PC), two distinct resonance peaks were observed in the DMA spectra of the deforming specimens. The first one was ascribed to the glass transition (main α-relaxation) at Tg, and the second one, observed at Tα′ > Tg, was ascribed to the relaxation of the strain-induced evolving segmental structure of the progressively deformed polymer (α′-relaxation). The α- and α′-relaxations exhibited different frequency ( f) and static load (σ) dependences, suggesting that the underlying microscopic mechanisms involved operate at different length scales. Since our experiments are not restricted to small deformations, the isotropy of the conformational space is most likely to be perturbed; hence, the use of the traditional plateau modulus (G0N) to characterize stiffness above the Tg seems compromised. Thus, we defined transient moduli characterizing the system stiffness between the α- and α′-relaxation (Γ*) and above the α′-relaxation (Ω*), respectively. Both Γ* and Ω* increased with frequency and with static load. When plotted against macroscopic deformation (ε), the experimental data for all the conditions used and fell on a single master curve being linear for Γ*(ε) and nonlinear for Ω*(ε). This stiffening correlated well with the strain induced increase of the Kuhn segment length, lK.

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Figure 1. “Quench and wait” thermal history of specimens used in this work. of thermal history on the α′-relaxation, two different thermal histories were used. Specimens were heated to 190 °C with heating rate of 0.1 K min−1, held at 190 °C for 5 min, and then cooled to 22 °C (the room temperature) with cooling rate of 0.07 K min−1 (referred to as annealed) and/or heated with the same heating rate to and held at 190 °C for 5 min and cooled with cooling rate 40 K min−1 to the room temperature (referred to as quenched). Measurements were performed employing the RSA III G2 dynamic mechanical thermal analyzer (DMA, TA Instruments) with the single cantilever specimen geometry used (Figure 2a). A constant static force (constant static engineering stress), F (σ), ranging from 0.01 to 1 N (0.001 to 0.1 MPa) acting perpendicularly to the sample length was applied over the temperature interval from 313 to 443 K. The temperature dependence of the macroscopic deformation of the sample, ε(T), was determined by measuring the upward displacement of the transducer clamp, D (Figure 2). The nonisothermal mode of the creep experiment was chosen due to the extremely long time for the isothermal creep, performed at the very small stresses allowed by the DMA device used, to reach the steady state secondary creep, especially below Tα. Since a constant heating rate was used, linear relationship exists between the time from starting the test and the temperature reached. Hence, the ε(T) measured in this work is similar to that obtained by connecting points recorded at linearly increasing times on a series of isothermal creep curves at progressively increasing T. During this nonisothermal creep-like experiment, DMA spectra were recorded at constant frequency by applying small oscillation strain, ε′, inducing oscillation stress, σ′, with the phase lag, δ. The storage (E′) and loss (E″) moduli and the loss factor (tan δ) were computed from the raw experimental data using the original sample geometry. The test geometry is shown in Figure 2. All DMA measurements were performed in the linear viscoelastic regime, up to amplitude of deformation, εA = 0.1%, determined by measuring the strain rate sweeps. The relative macroscopic specimen deformation (ε) in the course of acquiring the DMA spectra varied from 0 to 100%. The temperature ramps at 1 Hz with heating rate of 3 K min−1 and the multifrequency temperature sweeps with step 1 K min−1 were used to investigate viscoelastic behavior between glass and terminal zone.

MATERIALS AND METHODS

The materials used in this study were commercial PMMA Plexiglas (Evonik, Germany) with Mn = 50 kg mol−1, Mw/Mn = 2.1, Tg = 113 °C (DSC, 10 K/min) and Mn = 160 kg mol−1, Mw/Mn = 2.7, Tg = 122 °C (DSC, 10 K/min) and commercial PC Makrolon (Bayer, Germany) with Mn = 20 kg mol−1, Mw/Mn = 2.2, Tg = 143 °C (DSC, 10 K/min). According to the NMR analysis provided by the supplier (Evonik, Germany), the PMMA used contains 5−7% of isotactic form, 50−52% B


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Figure 2. (a) Test geometry used in this work showing the independent application of static force and the vibrational deformation. (b) Schematic representation of the nonisothermal creep nature of the experiment. The measured ε(T) at constant σ shown in red is depicted as a curve connecting consequent isochronal jumps between series of isothermal creep curves.

It seems useful to define “transient moduli”, Γ* and Ω*, as the inflection points on the E′(T) dependences between the α- and α′-relaxation (Γ*) and between the α′- and terminal zone (Ω*), respectively (horizontal arrows in Figures 3a and 3b). These terms are intended to emphasize that both their magnitude and dependence on test variables differ from those for the traditionally used plateau modulus, GN0 .15,22,36 More detailed discussion of Γ* and Ω* in terms of their T, f, and σ dependences is presented later in the text. Next, we qualitatively analyze effects of thermal history and/ or molecular weight on the position and strength of the α- and α′-relaxations (Figure 4). Increasing molecular weight resulted in approximately the same shift of both α- and α′-relaxations to higher temperatures (Figure 4a). At the same time, increasing the Mn led to an increase of the α relaxation strength and a reduction of the α′-relaxation strength. Annealing resulted in the expected shift of the α-relaxation to higher T, narrowing of the glass transition region, and slightly increasing the strength of both relaxations (Figure 4b).37,38 The small changes in the position and strength of the α′-relaxation may be considered an indirect evidence that under the conditions of high T and ε the thermal history of the specimen is effectively erased. In the following, we only report on the PMMA with Mn = 50 × 103 g/mol and both PMMA and PC with the “quench and wait” thermal history (Figure 1). The dependences of the loss modulus normalized to its value at Tg, E″/Eg″, on the relative temperature, T/Tg (Figure 5), and the temperature dependence of storage modulus, E′, measured at constant σ = 0.02 MPa for frequencies spanning almost 4 orders of magnitude are shown in Figure 6. The α′-relaxation has longer relaxation times than the α-relaxation and the strength of the α′-relaxation, represented by the maxima of loss modulus, increases with increasing frequency, f, at constant σ (Figures 5a and 5b) and with increasing σ at constant f, while the α-relaxation strength is independent of both σ and f (Figures 5c and 5d). Insets in Figures 5a and 5b show the plots of Tα′ vs Tα, exhibiting a linear correlation with the slope of 2.47 for PMMA and 1.26 for PC. The slope of Tα′ vs Tg larger than unity means stronger frequency dependence of the α′-relaxation

Standard deviation of less than 10% was obtained for all the measurements.

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RESULTS AND DISCUSSION We begin by discussing results for the primary measurements of interestE′, E″, and tan δfor all systems and test conditions. Characteristic DMA spectra of the PMMA and PC measured under the external static load, σ, of approximately 0.02 MPa are plotted in Figures 3a and 3b. Even though the applied load is rather small to cause any significant deformation below the Tg, the deformation starts to increase substantially for T approaching Tg (blue curve in Figures 3a and 3b). In addition to the main α-relaxation manifesting itself at the glass transition, a pronounced secondary α′-relaxation has been observed at Tα′ > Tg for both PMMA and PC. For the stressed polymers, the Tα was found at 110 °C for PMMA and 144 °C for PC, and the Tα′ was found at 123 °C for PMMA and 152 °C for PC. For the unstressed PMMA (Figures 3c to 3e), the Tα was found at 111 °C, in agreement with published results.31 The α′-relaxation was observed at approximately 10 K (polymer and frequency dependent) above the Tg, i.e., at 1.04Tg for PMMA and 1.02Tg for PC. The fundamental difference between the DMA spectra of the stressed and unstressed specimens is shown in Figures 3c to 3e, providing convincing evidence that the presence of the α′-relaxation is indeed induced by the external static stress. Temperature dependences of the storage and loss modulus of deformed and undeformed specimen start to diverge slightly above the Tg. The external load tends to split the single tan δ peak into the α- and α′-relaxations with simultaneous reduction of the damping strength. To provide further qualitative support for the deformation origin of the α′-relaxation, the undeformed polymers were subject to DSC experiments at four different heating rates (Figures 3f and 3g). As expected, a single minimum corresponding to the α-relaxation shifting to higher T with increasing heating rate for both PMMA and PC was observed. DSC measurements yielded the Tg = 111 °C at 3 K/min for unstressed PMMA and Tg = 142 °C at 3 K/min for the unstressed PC; no further analysis of the DSC measurements was performed. C


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Figure 3. Characteristic temperature dependences of storage modulus, E′, loss modulus, E″, loss factor, tan δ, and macroscopic deformation, ε, of (a) PMMA and (b) PC under static external load, σ, of 0.02 MPa. Vertical arrows show the positions of the α- and α′-relaxations. The blue line in (a, b) represents the deformation, ε, of the specimen induced by the static load. Comparison of (c) the E′(T), (d) the E″(T), and (e) the tan δ(T) for stressed and unstressed PMMA. Horizontal arrows show the value of the traditional plateau modulus GN0 and the introduced transient moduli, Γ* and Ω* (a−c). DSC traces of unstressed (f) PMMA and (g) PC measured at four heating rates.

compared to the α-relaxation and may reflect their different molecular origin.

Increasing the f at constant σ results in the expected shift of the Tα to higher temperatures and even larger shift of the Tα′ D


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Figure 4. (a) Temperature dependence of the loss factor, tan δ, for PMMA with Mn = 50 × 103 g/mol (black) and 160 × 103 g/mol (red) prepared with the thermal history depicted in Figure 1. (b) Temperature dependence of tan δ for PMMA with Mn = 50 × 103 g/mol for for quenched (red) and annealed (black) PMMA. Data obtained at σ = 0.02 MPa, f = 1 Hz, and heating rate of 3 K min−1.

Figure 5. Dependence of the reduced loss modulus, E″/Eg″, on the relative temperature, T/Tg, obtained for oscillatory load frequency, f, ranging from 0.05 to 10 Hz at constant external static load of 0.02 MPa and at constant heating rate of 3 K min−1 for (a) PMMA and (b) PC, respectively. Insets: plots of Tα′ vs Tg. Dependence of the reduced loss modulus (filled symbols) and macroscopic deformation (open symbols) on temperature obtained at constant heating rate of 3 K min−1 for (c) PMMA and (d) PC at 1 Hz and various external loads, respectively.

(Figures 5a and 5b). At constant f, increasing the σ shifts the ε(T) toward smaller reduced temperature (increasing the strain

at any given T), thus causing the α-relaxation to occur in the progressively higher strain region with slightly reduced strain E


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Figure 6. Dependence of storage modulus on the temperature obtained for oscillatory load frequency, f, ranging from 0.05 to 10 Hz at constant external static load of 0.02 MPa for (a) PMMA and (b) PC. Temperature dependences of the storage modulus (filled symbols) and macroscopic deformation (open symbols) for (c) PMMA and (d) PC at 1 Hz and various σ, showing the increase of both Γ* and Ω* with increasing static load, σ, and shortening of the terminal zone with increasing σ at constant f.

along the ε(T) curve. At small frequencies up to 1 Hz, the α-relaxation coincides with the position of the first maximum strain rate and α′ appears at the minimum strain rate (Figure 7). At higher frequencies, the E″/Eg″ maxima for both relaxation were shifted further along the ε(T) curve, the α into region where the strain rate starts to decrease and the α′ to the region when strain rate becomes increasing again. This observation corresponds with the f = 1 Hz as the break point of the f dependences of both transient moduli (Figure 9). Since the T increases linearly with time, the dε/dT has the physical meaning of strain rate (Figure 7). After the initial period of almost constant strain rate for T < Tg, a pronounced nonlinear increase of the strain rate was observed at the onset of the α-relaxation reaching a maximum at Tg. This effect was related to the enhanced segmental mobility similarly to that observed at the onset of plastic deformation in isothermal creep below Tg by Lee et al.2,3 Qualitatively, the strain-enhanced segmental mobility at Tg has been ascribed to the trans−gauche rearrangements increasing the structural disorder S0. After reaching its maximum at approximately the Tg, the strain rate is progressively reduced at the onset of the α′-relaxation, attaining its minimum value at approximately the Tα′. Above the Tα′, the strain rate increases gradually again, suggesting that mobility has again been enhanced. The observed strain rate minimum is

rate, especially for PMMA (Figures 5c and 5d). On the other hand, the α′-relaxation appears approximately in the region of the ε(T) curve with apparently the same strain rate. At constant σ, the temperature increasing linearly with time supplies increasing amount of thermal energy changing the strain rate. At any T, the prestrained specimen is then a subject of small-amplitude oscillatory deformation. This oscillatory excitation forces the prestrained local segment scale structures to adapt to the changing strain state with resonance occurring at Tα′. Obtaining the DMA spectra for different σ then allows one to investigate strain-induced relaxation processes not present in the undeformed specimens. The experimentally obtained data show (Figure 5) that the Tα does not change with σ. The α-relaxation strength is independent of σ while the strength of the α′-relaxation, represented by the maxima of loss modulus, gradually increases with σ. Moreover, at T > Tg, the storage modulus, E′(T), increased with σ (Figures 6c and 6d) similarly to its increase with f (Figures 6a and 6b). While frequency at constant σ does not affect the ε(T) (Figures 6a and 6b), increasing σ at constant f leads to a significant increase of ε for any given T (Figures 6c and 6d). Since, at constant σ, the ε(T) is not affected by f (Figures 6a and 6b) while the position of both α- and α′-relaxation is shifted to higher T, the relaxations appear at different positions F


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Figure 7. Dependence of the strain rate on (a) temperature and (b) reduced temperature for PMMA and PC (σ = 0.02 MPA, f = 1 Hz, heating rate 3 K min−1). Arrows show positions of the α- and α′-relaxations.

from the glass transition for both PMMA and PC with the rate of the increase being greater for PC (Figure 8c). For both polymers, the J increase is greatly accelerated above the Tα′. Most probably, the onset of the fast growth of J, occurring above Tα′ being closer to the Tg for PC compared to PMMA, may be related to the difference in conformational statistics between the relatively low Mn worm-like-chain PC and the PMMA exhibiting Pincus blob chain statistics.37,38 Finally, to ascertain the possible underlying molecular processes occurring in the deformed polymer above the Tg, frequency (f), stress (σ), and strain (ε) dependences of the transient moduli, Γ* and Ω*, defined earlier (see Figure 3), were investigated. The Γ*( f) and Ω*(f) dependences are all bilinear, exhibiting vertical shift and slope increase with increasing σ over the entire frequency interval investigated (Figure 9). For PC, the first transient modulus Γ*( f) is only about 3/4 of that for PMMA (Figures 9a and 9b). On the other hand, the second transient modulus Ω*( f) is for PC almost twice that for the PMMA (Figures 9c and 9d). It seems evident from Figure 9 that all the Γ*( f) and Ω*( f) lines change their slope near f ≈ 1 Hz for all the σ used. Since only two data points were obtained above frequency of 1 Hz, the functional form of the Γ*( f) and Ω*( f) will be very speculative; hence, these data points are shown separately as points in Figure 9. The σ dependence of the first transient modulus, Γ*(σ) (Figures 10a and 10b), is clearly nonlinear for both PMMA and PC; it is shifted vertically with increasing frequency, and moreover, the extent of the vertical shift is increasing with σ. However, plotting the Γ* against the macroscopic deformation, ε, instead of σ results in all the data for all the frequencies to fall onto a single linear master curve for both PMMA and PC (Figure 10c). This suggests that in the temperature region investigated strain may be a more relevant variable than stress, since at a given small constant stress, the transient moduli at varying frequency are measured at different strain level (Figure 2b). The σ dependences of the second transient modulus, Ω*(σ) (Figures 11a and 11b), are clearly nonlinear for both PMMA and PC and are shifted vertically. The extent of the vertical shift with increasing f grows with σ for PMMA while being almost constant for PC. Plotting the Ω* against the ε instead of σ results in all the data for all the frequencies to fall onto two nonlinear master curves differing for PMMA and PC (Figure 11c).

a result of at least two competing processes. We assume that the observed drop in the strain rate for PMMA consists most probably of strain-induced segmental gauche−trans rearrangements and is accompanied by the reduction of the amplitude of density fluctuations, S0, similarly to strain hardening. Further increase of temperature attempts to reverse this process by amplifying the trans−gauche rearrangement which first occurred at Tα and caused the initial the mobility increase. Near Tα′, these processes are balanced, resulting in the observed minimum of the T dependence of the strain rate. For any given reduced temperature, the strain rate induced by the same static stress is significantly greater for PC compared to PMMA. To relate our findings obtained under complex deformation conditions to the current understanding of the deformation response of polymer glasses, let us first recall the constant strain rate and creep deformation behavior of a common polymer glass well below Tg.14,15 At constant strain rate, increasing strain makes the solid polymer softer represented by the nonlinear viscoelastic behavior followed by the yield point. Depending on the thermal history and aging, strain softening may appear followed by the f low region15 and the strain hardening region.36 In an isothermal creep experiment,2,3 after the instantaneous elastic response and a short primary creep region with progressively decreasing strain rate, the secondary (stationary) creep with constant strain rate occurs followed by the tertiary creep with progressively increasing strain rate terminated by fracture. Increasing the T causes increase in the strain rate at the stationary creep and shortening its length and vice versa, at a constant T, increasing the σ leads to enhanced strain rate and shortening of the secondary creep region.57 Transition between the secondary and tertiary creep is accompanied by substantial increase of segmental mobility.2,3 Considering the experimental protocol used a nonisothermal creep with superimposed small oscillatory deformation, we can assign the positions of the individual relaxations observed in the DMA spectra to specific stages of the creep process with progressively increasing strain rate. First, plotting the strain− stress isotherms (Figures 8a and 8b) for temperatures near both Tα and Tα′, linear dependences are obtained with the slopes representing the system compliance, J. Each of the strain−stress curves depicted in Figures 8a and 8b is an isothermal cut of the creep curves shown in Figures 6c and 6d at a given T, respectively. Above the Tα, the J increases with the distance G


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Figure 8. Strain−stress isotherms for (a) PMMA and (b) PC over interval of temperatures near its Tg. Insets show the dependence of the compliance, J(T), on the distance from the glass transition. (c) Comparison of the J(T/Tg) dependences for PMMA and PC showing onset of substantial PC softening occurring closer to Tg compared to PMMA.

rearrangements of the mobile segments. Since the α′-relaxation acts against the α increased mobility, one possible explanation is that the α-relaxation released mobility allowed alteration of the segmental packing and dynamic heterogeneity in the strained solid by redistributing segments from less mobile denser domains into the more mobile less dense ones.17,18 As a result, the enhanced anisotropy of conformations results in a suppressed density fluctuation. This seems in line with the model envisioning plastic deformation in PC to nucleation of localized disordered zones acting as quasi-point defects corresponding to local density fluctuations.36 The different functional form of the Γ*(ε) (Figure 10c) and Ω*(ε) (Figure 11c) suggests apparently different length scales for their underlying molecular processes. While the Γ*(ε) dependence is linear, the Ω*(ε) exhibits pronounced nonlinearity. One way to interpret the magnitude of the transient moduli in comparison to the entropic plateau modulus G0N is to assume strain-induced transition from Pincus blob dynamics38 to worm-like chain dynamics.39 The nonlinearity of the Ω*(ε) at large strains is caused by conjecture of strain-induced chain stiffening due to segmental orientation and the onset of local plastic flow heterogeneously distributed throughout the specimen.

Most probably, the observed larger scatter of the Ω*(ε) data for PMMA obtained at 5 and 10 Hz is caused by the change in the slope of the Ω*(f) above 1 Hz (see Figure 9c). For an unstrained linear amorphous polymer solid, standard DMA spectra consist of a substantial drop of storage modulus and a maximum of the loss modulus induced by the segmental mobility released by the α-relaxation at Tg, followed by the plateau zone characterized by plateau modulus, G0N, and the terminal zone representing onset of viscous flow (Figures 3e, 3f, and 3g). Under constant static stress, mobility is greatly enhanced upon onset of plastic yielding.2,3 In an analogy to the flow region observed in constant strain rate experiment, the portion of the E′(T) dependence observed in our experiments between the α- and α′-relaxation regions and characterized by Γ* encompasses both the “strain sof tening” caused by the enhanced segmental mobility induced by the α-relaxation and the “strain hardening” accompanied by the reduction of the segmental mobility induced by the onset of the α′-relaxation. The latter is gradually prevailing resulting in the minimum strain rate attained near the Tα′. Hence, under the experimental conditions used here, the α-relaxation can be considered a precursor of the α′-relaxation by allowing strain-induced H


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Figure 9. Frequency dependence of the transient modulus Γ* for various static stresses, σ, for (a) PMMA and (b) PC. Frequency dependence of the transient modulus Ω* on the various static stresses, σ, for (c) PMMA and (d) PC. Lines serve as the guide for the eyes only.

The α′-relaxation changes the interchain segment packing manifesting itself by the values of the transient modulus Ω* being 2 orders of magnitude larger than the G0N commonly used to characterize its response above the Tg22,40,41 and 2 orders of magnitude larger than the strain hardening modulus.15,20 This may be considered another experimental evidence for the large strain-induced change in segment scale packing proposed above and suggested by McKenna as the polyamorphic transition accompanying the onset of yielding in polymer glasses.13 In the temperature interval investigated, the supercooled liquid changes its behavior from the Vogel−Fulcher−Tammann (VFT) to the Arrhenius one.42 Hence, we used both the VFT (eq 1) and Arrhenius (eq 3) models to analyze relaxations occurring in the vicinity of the Tg phenomenologically: ⎞ ⎛ B f = f0 exp⎜ ⎟ ⎝ T − TVFT ⎠

temperature, TVFT the VFT divergence temperature, and B is material specific parameter related to activation energy.43 Both the value of Tg at which τ = 100 s and the parameter B for both PMMA and PC are listed in Table 1. The apparent activation energies Eg(α) and Eα′(α′) were calculated using the expression EA =

RBTg 2 (Tg − TVFT)2

(2)

where EA is the apparent activation energy, R is the universal gas constant, Tg is the temperature at which τ = 100 s, and TVFT is the VFT divergence temperature determined as Tg − C2 of the WLF equation (C2 = 51.6 K). Similarly, the modified Arrhenius equation was used to determine apparent activation energies, EA, for both relaxations:

(1)

⎛ E ⎞ f = f0 exp⎜ − A ⎟ ⎝ RT ⎠

In eq 1, τ is the relaxation time, τ0 the pre-exponential factor (or relaxation time at infinitely high temperature), T the absolute I

(3) DOI: 10.1021/acs.macromol.5b00550 Macromolecules XXXX, XXX, XXX−XXX

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Figure 10. Dependence of the transient modulus Γ* on the external static load σ at various oscillatory load frequency, f, for (a) PMMA and (b) PC. Dependence of the transient moduli, Γ*, on the deformation at various oscillatory load frequency, f, for (c) PMMA and (d) for PC. Lines serve as the guide for the eyes only.

α′-relaxation in the PMMA. The EA(α′) increases with σ, suggesting a non-Eyring character of the process underlying the α′-relaxation. The experimental data also suggest that the strength of the α′-relaxation increases with increasing σ (Figure 5). In other words, increasing the σ requires larger apparent activation energies for the microscopic deformation events to occur. Our experimental results for the stress dependence of the α′-relaxation activation energy are in a qualitative agreement with the work of Shelby et al. reporting similar increase of the activation enthalpy of the α relaxation with the stretch ratio in the PC. This increase is interpreted as an enlargement of the size of cooperatively relaxing regions due to increasing segmental orientation with increasing stress. The difference between our work and the work of Shelby et al. is the fact that orientation in our experimental protocol was formed in situ during the measurement while the PC studied by Shelby et al.75 was hot-drawn at T above the Tg prior to the DMA measurements. In our case, drawing was initiated at Tg as evidenced by the steep increase in both the strain and strain rate (see Figure 8). By extrapolating the dependence in Figure 12 to σ = 0, the EA(0) = 220 kJ mol−1 is obtained similar to the activation

where f is the frequency, f 0 the pre-exponential factor (or the frequency at infinite temperature), EA the apparent activation energy, R the universal gas constant, and T the absolute temperature. The EA values obtained for the main α-relaxation of both PMMA and PC (Table 1) are of the same order as the data published by McKenna et al.43 and others.44,56 The α-relaxation activation energy calculated using either the VFT or the Arrhenius model differs between PMMA and PC by only about 10% (Table 1). On the other hand, the activation energy of the secondary α′-relaxation for PC is almost double that for the PMMA regardless of the model used. Moreover, for PC, the α and α′ activation energies are almost equal, while for PMMA, the α′ activation energy is approximately half of that for the α-relaxation. In the framework of the Eyring model,68,69 the fundamental microscopic deformation event is a thermally activated hopping of a segment or an ensemble of segments over the energy barrier, EA, in the direction of the applied stress. Hence, by increasing σ, a reduction of the EA(α′) and tilting of the energy landscape should result preferring the hopping in the direction of the load. Figure 12 shows the σ-dependence of the EA for the J


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Figure 11. Dependence of the transient modulus Ω* on the external static load σ at various oscillatory load frequency, f, for (a) PMMA and (b) PC dependence of the transient modulus Ω* on the deformation at various oscillatory load frequency, f, for (c) PMMA and (d) for PC. Lines serve as the guide for the eyes only.

Table 1. Glass Transition Temperature at τ = 100 s, Tα, Material Specific Parameter from VFT Equation, B, Apparent Activation Energies from VFT Analysis, E, Calculated from Eq 1, and Apparent Activation Energies from Modified Arhenius Analysis, EA, Calculated from Eq 3 for Both α and α′ Relaxation of PMMA and PC, Respectively VFT behavior PMMA PC a

Arrhenius behavior

Tα (K)

EA(α)a (kJ mol−1)

Bα (103 K)

Bα′ (103 K)

EA(α) (kJ mol−1)

EA(α′) (kJ mol−1)

EA(α) (kJ mol−1)

EA(α′) (kJ mol−1)

379 412

724 1070

2,11 2,06

1,41 2,18

945 1091

633 1153

802 882

349 729

Data taken from ref 43.

energy determined for unstrained PMMA using DSC44 or the apparent activation energy for the contribution of the α-process to the plastic deformation of PMMA.15,36 This observation may be related to the hypothesized13 strain-induced polyamorphic phase transition changing both segment scale conformational and orientational distribution. One striking difference between PMMA and PC is that while for PMMA both models (eqs 1 and 3) yielded significantly lower α′ relaxation activation energy, EA(α′), compared to the α relaxation, the EA(α), for the PC, approximately equal energies were obtained for both relaxations. For the PMMA, the lower EA(α′) compared to the EA(α) is apparently

contradicting the common wisdom assuming that relaxations occurring at higher temperature require supply of higher amount of thermal energy. This common wisdom, however, does not consider change in the relaxation mechanism due to T-induced shallowing of the potential energy landscape (possible high-T shoulder of the β relaxation in PMMA) and its concurrent strain-induced tilting. The potential energy landscape funnels have a hierarchical structure. On short time scales, the system undergoes a back and forth motion between a limited number of basins forming a cluster, also called a metabasin,45 while on longer time scales, the system performs an irreversible Brownian diffusion between metabasins. This K


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Macromolecules

⟨R2⟩ has to increase upon deformation. Under these assumptions and at the large strain, the PMMA adopts the shape of chain of the Pincus blobs while PC exhibits conformational statistics of a worm-like chain.37,38 Finally, we make an attempt to assign the observed changes in molecular dynamics during the nonisothermal creep experiments to the molecular processes for both polymers investigated. For PMMA, increasing the macroscopic deformation induces changes in orientational58 and conformational59 statistics of the chains, and increasing T results in growing population of gauche conformations resulting in enhanced chain flexibility. By exerting large strain at T below Tg, the average conformational state of a chain at the onset of yield corresponds to that obtained at T > Tg for an unstressed chain.71 At the same time, the rate at which the trans−trans backbone conformations, prevailing below Tg in the undeformed PMMA, change to the various gauche conformations72 depends on the degree of chain stereoregularity. The PMMA used isa stereocopolymer of a-PMMA and s-PMMA with few percent of i-PMMA sequences. The cooperativity between the backbone chain and side groups mobility also depends on the degree of stereoregularity29,30,60 and, moreover, on its molecular weight and molecular weight distribution.61 The enhanced intrachain cooperativity is pronounced the most in the temperature range corresponding to the high-temperature region of the β-relaxation involving ester group π-flips coupling with the backbone chain torsion.71 Above the first release of segmental mobility at Tg, the amount of gauche conformations in the strained PMMA is reduced again due to deformation induced segmental orientation favoring the trans conformations.73 This is expressed by the reduction of the strain rate at the onset of the α′-relaxation (Figure 7). The high-T enhanced segmental mobility partially counteracts the strain induced chain stiffening, corresponding to the reduction of the lK above the α′-relaxation (Figures 13c and 13d) to about one-half of the values obtained immediately after the main α-relaxation (Figures 13a and 13b). Ube et al.62,63 showed that the microscopic strain in a single chain is smaller than the macroscopic strain. This deviation increases with increasing macroscopic deformation, suggesting significant slipping of the chains through the entanglements. SANS experiments showed64 that for high molecular weight PMMA the deformation is affine on the scale of the radius of gyration; however, it is markedly nonaffine for short chains and suggested that pronounced nonaffine modes of plastic deformation operate on the length scale below 2.5 nm. On the basis of our experimental results and published MD simulations29,30,65 for PMMA, we propose that the α′-relaxation is caused by the −CH2− groups exploring local metabasins by the process of jump diffusion over the characteristic length scale of the lK. Both the change of conformational statistics toward more flexible conformations and reduction of segmental cooperativity70 above the Tα′ result in the experimentally observed increase in chain flexibility and mobility (Figures 7 and 8) expressing itself in the progressively reduced rate of lK increase with ε. The carbonate group in the PC is planar and, below Tg for an undeformed polymer, preferring the trans−trans conformation over the trans−cis.71 Below the Tg, the conformational distribution was found to be unaffected by the deformation on a larger scale, whereas small changes in the local packing of molecular segments could be detected66 resulting in very localized softening.71 The phenyl ring planes exhibit a tendency of orienting parallel

Figure 12. Plot of the apparent activation energy EA of the α′-relaxation as the function of the static stress.

may explain the change in the slope of the frequency dependences of transient moduli in the vicinity of 1 Hz. Needless to say, the effects related to physical aging were not accounted for in this work. Although polymers act as linear entropic springs for small extensions, their finite length causes the retractive force to become nonlinear as the polymer is extended farther. If the polymer is at equilibrium, the fluctuations of the end points will be small and the force can be expanded in a Taylor series. Thus, the Rouse equations of motion do not change as a function of extension; only the value of the spring constant varies.47 Force versus extension curves have been calculated and measured experimentally on single polymers. For a simple freely jointed chain F(ε) = (kBT/lK)L−1(ε/Rmax), where the Langevin function L(α) = [coth(α) − 1/α], kB is the Boltzmann constant, T is the temperature, lK is the Kuhn length of the polymer, and Rmax is the fully extended chain size.48 For a wormlike chain such as PC, the force is typically approximated by the function F(ε) = (2kBT/lK)[1/4(1 − ε/Rmax) − 1/4 + ε/Rmax].49 Our results suggest that the α′-relaxation may have its origin in segment scale events in time and length scales between the local segmental and the Rouse relaxation modes.50 Hence, in the following, we analyze the experimental results in terms on the dynamic Kuhn segment length, lK (Figure 13), with direct relationship to strain enhanced chain stiffness.51 One has to keep in mind that local interactions determine the packing and dynamics but not the generic properties37,52 and that in the disordered matter the stress−strain nonlocality may become an issue.53 The Kuhn length, lK, is commonly defined as37 lK =

Cnl ⟨R2⟩ = R max cos(Θ/2)

(4)

where ⟨R ⟩ is the mean-square end-to-end distance and Rmax = nl cos(θ) for a linear chain consisting of n bonds of length l and Cn is the chain characteristic ratio. Each Kuhn segment consists of Cn/cos2(θ) bonds. The Cn for PMMA is equal to 8.22,40 and that for PC ranges from 1.7446 to 3.0 nm;55 the molecular weight per entanglement strand, Me, at Tg + 10 K is equal to 15 × 103 g/mol for PMMA41 and 2.5 × 103 g/mol for PC.41 The Kuhn length for the unstressed PMMA is equal to 1.4 and 5.3 nm for the PC.43 The bond length and bond angle are not likely to change upon deformation; hence, in this simple model, the Cn has to increase, or in other words, the 2

L


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Figure 13. (a) Macroscopic deformation (ε) dependence of the dynamic Kuhn segment length (lK) for PMMA over the range of vibrational frequencies used calculated according to ref 54 from the Γ* data. (b) Dependence of the dynamic Kuhn segment length (lK) on deformation for PC over the range of vibrational frequencies used calculated according to ref 54 from the Γ* data. (c) Macroscopic deformation (ε) dependence of the dynamic Kuhn segment length (lK) for PMMA over the range of vibrational frequencies used calculated according to ref 54 from the Ω* data. (d) Dependence of the dynamic Kuhn segment length (lK) on deformation for PC over the range of vibrational frequencies used calculated according to ref 54 from the Ω* data.

to emphasize is that the current microscopic models of deformation behavior of polymer glasses9 do not account for possible structural heterogeneity and homogenize the local behavior when extending the predictions to macroscopic scale. This approach may be closer to reality for PC exhibiting the low-temperature β-relaxation at much greater distance from the Tg than the PMMA; hence, strain-induced relaxation decoupling may be much weaker.

to one another upon deformation, and the pair correlation functions between the molecular segments increase in amplitude at short separations. At the Tg, the π-flips of the phenyl rings become possible causing torsion of the backbone chain by approximately 15°, resulting in intrachain rearrangements on the 1 nm scale and interchain rearrangements on a 0.7 nm scale.71 Both these processes result in enhancement of the backbone chain mobility. The results of MD simmulations67 indicate that high segmental mobility and low local density enable the nucleation of highly deformed nanometer sized regions that grow to form plastic defects, and the effect of chemical structure was found to dominate the actual deformation mechanism on the molecular scale.71 Our results are in a qualitative agreement with interpretation of the large strain deformation behavior of oriented PS74 and plastically cycled PC.36 In the framework of recent microscopic models, the reduced mobility occurring at the α′ relaxation can be related to the segment scale thermally activated deformation processes, supposedly coupled with the increase in rate of nonaffine segmental displacements represented by the increase of the effective stiffness length, lK, with deformation. Hence, at larger strains a higher rate of plastic events in terms of nonaffine displacements is necessary to produce this effect. Another point

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CONCLUSIONS The dynamic mechanical analysis was used to investigate temperature, frequency, and tensile stress effects on the molecular mobility in PMMA and PC progressively deformed with small constant static stress in the temperature ranging from glassy to terminal zone. Above the main transition denoted as the α relaxation, an additional pronounced strain induced α′ relaxation was observed at Tα′ > Tg. The frequency and stress dependences of both Tα′ and the α′ relaxation strength differed from those observed for the α relaxation fundamentally, suggesting their underlying microscopic processes operating at different length scales. While for the PMMA both VFT and Arrhenius models yielded significantly lower α′ relaxation activation energy compared to the α relaxation, for the PC, M


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(14) Tervoort, T. A.; Klompen, E. T. J.; Govaert, L. E. J. Rheol. 1996, 40, 779−797. (15) Govaert, L. E.; Engels, T. A. P.; Wendlandt, M.; Tervoort, T. A.; Suter, U. W. J. Polym. Sci., Part B: Polym. Phys. 2008, 46, 2475−2481. (16) Toepperwein, G. N.; Schweizer, K. S.; Riggleman, R. A.; de Pablo, J. J. Macromolecules 2012, 45, 8467−8481. (17) Warren, M.; Rottler, J. Phys. Rev. Lett. 2010, 104, 205501. (18) Warren, M.; Rottler, J. J. Chem. Phys. 2010, 133, 164513. (19) Liu, A. Y.-H.; Rottler, J. Soft Matter 2010, 6, 4858−4862. (20) Jancar, J.; Hoy, R. S.; Jancarova, E.; Lesser, A. J.; Zidek, J. Macromolecules 2013, 46, 9409. (21) Casalini, R.; Roland, C. M. J. Chem. Phys. 2009, 131, 114501. (22) Hoy, R. S.; Robbins, M. O. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, 3487−3500. (23) Lyulin, A.; Michels, M. A. J. Phys. Rev. Lett. 2007, 99, 085504. (24) Siline, M.; Leonov, A. I. Polymer 2002, 5521−5525. (25) Ding, Y.; Sokolov, A. P. J. Polym. Sci., Part B: Polym. Phys. 2004, 42, 3505−3511. (26) Utz, M.; Debenedetti, P. G.; Stillinger, F. H. Phys. Rev. Lett. 2000, 84, 1471−1474. (27) Ogura, I.; Yamamoto, T. Polymer 1995, 36, 1375−1381. (28) Robertson, C. G.; Palade, L. I. J. Non-Cryst. Solids 2006, 352, 342−348. (29) Soldera, A.; Metatla, N. Composites, Part A 2005, 36, 521−530. (30) Soldera, A.; Grohens, Y. Polymer 2004, 45, 1307−1311. (31) Kuebler, S. C.; Schaefer, D. J.; Boeffel, C.; Pawalzik, U.; Spiess, H. W. Macromolecules 1997, 30, 6597. (32) Ngai, K. L. J. Phys.: Condens. Matter 2003, 15, 1107. (33) Eisenberg, A.; Hird, B.; Moore, R. B. Macromolecules 1990, 23, 4098. (34) Tsagaropoulos, G.; Eisenberg, A. Macromolecules 1994, 28, 396−398. (35) Dudowicz, J.; Freed, K. F.; Douglas, J. F. Adv. Chem. Phys. 2008, 137, 125−222. (36) Ǵ Sell, C.; El Bari, H.; Perez, J.; Cavaille, J. Y.; Johari, G. P. Mater. Sci. Eng. 1989, A110, 223−229. (37) Kremer, K. Entangled polymers: From universal aspects to structure property relationships. In Computational Soft Matter: From Synthetic Polymers to Proteins; Attig, N., Binder, K., Grubmüller, H., Kremer, K., Eds.; NIC Series Vol. 23, 2004; pp 141−168. (38) Latinwo, F.; Schroeder, C. M. Soft Matter 2011, DOI: 10.1039/ c1sm05298e. (39) Everaers, R.; Sukumaran, S. K.; Grest, G. S.; Svaneborg, C.; Sivasubramanian, A.; Kremer, K. Science 2004, 303, 823−826. (40) Fetters L. J.; Lohse D. J.; Colby R. H. In Physical Properties of Polymers Handbook; Mark J. E., Ed.; Springer: Berlin, 2007; Chapter 25, p 445. (41) Krishnamoorti, R.; Graessley, W. W.; Zirkel, A.; Richter, D.; Hadjichristidis, H.; Lewis, N.; Fetters, J.; Lohse, D. J. J. Polym. Sci., Part B: Polym. Phys. 2002, 40, 1768−1776. (42) Dudowicz, J.; Freed, K. F.; Douglas, J. F. Adv. Chem. Phys. 2008, 137, 125−222. (43) Qin, Q.; McKenna, G. B. J. Non-Cryst. Solids 2006, 352, 2977− 2985. (44) Avramov, I.; Vassilev, T. Polym. Polym. Compos. 1994, 2, 241− 246. (45) Rodney, D.; Tanguy, A.; Vandembroucq, D. Modell. Simul. Mater. Sci. Eng. 2011, 19, 083001. (46) Sundararajan, P. R. Macromolecules 1989, 22, 2149−2154. (47) Hatfield, J. W.; Quake, S. R. Phys. Rev. Lett. 1999, 82, 3548. (48) Flory, P. J. Statistical Mechanics of Chain Molecules; Hanser Publishers: New York, 1989. (49) Marko, J. F.; Siggia, E. D. Macromolecules 1995, 28, 8759. (50) Wu, X.; Wang, H.; Liu Ch; Zhu, Z. Solid State Phenom. 2012, 184, 52−59. (51) Wendlandt, M.; Terwoort, T. A.; van Beek, J. D.; Suter, U. W. J. Mech. Phys. Solids 2006, 54, 589−610. (52) Padding, J. T.; Briels, W. J. Soft Matter 2011. (53) Maranganti, R.; Sharma, P. Phys. Rev. Lett. 2007, 98, 195504.

both models provided approximately equal activation energies for α and α′ relaxations. This suggests that different local processes operate during plastic deformation of PMMA and PC. Despite these significantly different molecular mechanisms of deformation, the α′ relaxation exists in both polymers exhibiting qualitatively the same frequency and static stress dependence, suggesting its more general nature. The transient moduli, characterizing system stiffness between Tg and Tα′ (Γ*) and above Tα′ (Ω*), were 2 orders of magnitude larger than the rubbery plateau modulus G0N for PMMA and PC used. Both Γ* and Ω* increased linearly with frequency while nonlinear increase with σ was observed for both PMMA and PC. Plotting the transient moduli against macroscopic deformation (ε) resulted in all the data for all the conditions to fall onto a linear master curve for Γ*(ε) and onto a nonlinear master curve for Ω*(ε). The different functional form of the Γ*(ε) and Ω*(ε) suggests the apparently different length scales of the underlying segmental processes operating at greatly different strain levels at which Γ*(ε) and Ω*(ε) are determined. This is reflected by the obtained strain dependence of the size of the dynamic Kuhn length calculated from Γ*(ε) and Ω*(ε). Our results seem to support the hypothesis considering large deformation in polymer glass a polyamorphic phase transition nucleated in localized disordered zones acting as quasi-point defects. In order to elucidate its general features and its relationship to the deformation behavior of polymer glasses well below Tg, investigation of the effects of molecular weight, molecular weight distribution, and inherent chain flexibility on the α′ relaxation has to be conducted.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (J.J.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Funding under the Project 205/15/18495S from the Czech Grant Agency is greatly appreciated. Kenneth S. Schweizer, Jack F. Douglas, and Gregory B. McKenna are acknowledged for motivating discussions and criticism.

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REFERENCES

(1) Perez, J.; Cavaillé, J. Y. J. Non-Cryst. Solids 1994, 172−174, 1028− 1036. (2) Lee, H.-N.; Paeng, K.; Swallen, S. F.; Ediger, M. D. Science 2009, 323, 231−234. (3) Lee, H.-N.; Paeng, K.; Swallen, S. F.; Ediger, M. D.; Stamm, R. A.; Medvedev, G. A.; Caruthers, J. M. J. Polym. Sci., Part B: Polym. Phys. 2009, 47, 1713−1727. (4) Riggleman, R. A.; Lee, H.-N.; Ediger, M. D.; de Pablo, J. J. Soft Matter 2010, 6, 287−291. (5) Chen, K.; Schweizer, K. S. EPL 2007, 79 (2), 26006. (6) Chen, K.; Schweizer, K. S. Macromolecules 2008, 41, 5908−5918. (7) Chen, K.; Schweizer, K. S. Phys. Rev. Lett. 2009, 102 (3), 240401. (8) Chen, K.; Saltzman, E. J.; Schweizer, K. S. Annu. Rev. Condens. Matter Phys. 2010, 1, 277−300. (9) Chen, K.; Schweizer, K. S. Macromolecules 2011, 44, 3988−4000. (10) Vorselaars, B.; Lyulin, A. V.; Michels, M. A. J. Macromolecules 2009, 42, 5829−5842. (11) Lee, H.-N.; Riggleman, R. A.; de Pablo, J. J.; Ediger, M. D. Macromolecules 2009, 42, 4328−4336. (12) Lacks, D. J.; Osborne, M. J. Phys. Rev. Lett. 2004, 93, 255501. (13) McKenna, G. B. J. Phys.: Condens Matter 2003, 15, S737−S763. N


Article

Macromolecules (54) Fetters, L. J.; Lohse, D. J.; Richter, D.; Witten, T. A.; Zirkel, A. Macromolecules 1994, 27, 4639−4647. (55) Akashi, K.; Nakamura, Y.; Norisuye, T. Polymer 1998, 39, 5209−5213. (56) Mark, J. E., Ed.; Physical Properties of Polymers Handbook; Springer: Berlin, 2007; Chapter 13, pp 220−221. (57) Achereiner, F.; Engelsing, K.; Bastian, M.; Heidemeyer, P. Polym. Test. 2013, 32, 447−454. (58) Min, K.; Silberstein, M.; Aluru, N. R. J. Polym. Sci., Part B: Polym. Phys. 2014, 52, 444−449. (59) Grohens, Y.; Carriere, P.; Spevacek, J.; Schultz, J. Polymer 1999, 40, 7033−7043. (60) Wübbenhorst, M.; Murray, C. A.; Forrest, J. A.; Dutcher, J. R. Proc 11t Int Symp on Electrects; IEEE: New York, 2002; pp 401−406. (61) Aoki, H.; Anryu, M.; Ito, S. Polymer 2005, 46, 5896−5902. (62) Ube, T.; Aoki, H.; Ito, S.; Horinaka, J.; Takigawa, T. Polymer 2007, 48, 6221−6225. (63) Ube, T.; Aoki, H.; Ito, S.; Horinaka, J.; Takigawa, T.; Masuda, T. Polymer 2009, 50, 3016−3021. (64) Dettenmaier, M.; Macconachie, A.; Higgins, J. S.; Kausch, H. H.; Nguyen, T. Q. Macromolecules 1986, 19, 773−778. (65) Kim, Y.; Lin, K.-H.; Strachan, A. Modell. Simul. Mater. Sci. Eng. 2013, 21, 065010. (66) Utz, M.; Robyr, P.; Suter, U. W. Macromolecules 2000, 33, 6808−6814. (67) Shenogin, S.; Ozisik, R. Polymer 2005, 46, 4397−4404. (68) Eyring, H. J. Chem. Phys. 1936, 4, 283. (69) Rottler, J.; Robbins, M. O. Phys. Rev. E 2003, 68, 011507. (70) Shavit, A.; Douglas, J. F.; Riggleman, R. A. J. Chem. Phys. 2013, 138, 12A528. (71) Monnerie, L.; Halary, J. L.; Kausch, H.-H. Adv. Polym. Sci. 2005, 187, 215−364. (72) Kirschner, K. N.; Heikamp, K.; Reith, D. Mol. Simul. 2010, 36, 1253−1264. (73) Theodorou, M.; Jasse, B.; Monnerie, L. J. Polym. Sci., Part B: Polym. Phys. 1985, 23, 445. (74) De Focatiis, D. S. A.; Embery, J.; Buckley, C. P. J. Polym. Sci., Part B: Polym. Phys. 2010, 48, 1449−1463. (75) Shelby, M. D.; Hill, A. J.; Burgar, M. I.; Wilkes, G. L. J. Polym. Sci., Part B: Polym. Phys. 2001, 39, 32−46.

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Temperature, Frequency, and Small Static Stress Dependence of the

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