Note pubs.acs.org/Macromolecules
Inconsistencies in Determining the Entanglement Time of Poly(butadiene) from Rheology and Comparison to Results from Field-Cycling NMR M. Hofmann,† N. Fatkullin,‡ and E. A. Rössler*,§ †
Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803, United States Institute of Physics, Kazan Federal University, Kazan 420008, Tatarstan, Russia § Experimentalphysik II, Universität Bayreuth, D-95440 Bayreuth, Germany ‡
predictions, a common value τe = (3.7 ± 0.93) × 10−7 s was obtained at 298 K. This value is by a factor of 2 larger than that reported by Liu et al. (τe = 1.8 × 10−7 s) previously.14 The latter authors also suggested an experimental method for the determination of τe by taking the cross-point between the extended plateau modulus and the power-law region of G″(ω) ∝ ω0.7 at frequencies above the rubber plateau. Independent of this, the power law G″(ω) ∝ ω0.7 is observed in many polymers;4,14 yet, its exponent is significantly higher than that predicted by the Rouse model (0.5).1 Various other approaches were proposed for determining τe as listed in refs 4 and 14; however, the results for PB scatter widely in the range (0.6−15) × 10−7 s (298−301 K). Recently, we published a study presenting relaxation data on linear PB obtained from FC NMR relaxometry.10 The technique probes the frequency dependence of the spin−lattice relaxation rate R1(ω).6−8 Usually, a frequency range of 10 kHz−30 MHz is covered. As in many studies before, we took recourse to master curve construction after converting the relaxation data measured at different temperature to the susceptibility representation given by χ″(ω) = ωR1(ω).15 If protons are investigated, intra- and intermolecular relaxation contribute to the total spin−lattice relaxation additively, along R1(ω) = R1intra(ω)+ R1inter(ω).6,7 We separated the two contributions by isotope dilution.9 From R1inter(ω) the time dependence of the segmental mean-square displacement (msd) in the Rouse and the constraint Rouse (or incoherent reptation) regime was derived. Adding results from field gradient NMR diffusometry, all four polymer diffusion regimes (I−IV: Rouse, constraint Rouse, coherent reptation, and terminal relaxation or normal diffusion) were revealed in terms of power laws with characteristic exponents, which essentially agreed with the prediction of the TR model (cf. inset of Figure 1).10 This allowed us to directly estimate τe from the crossover of the Rouse (I) to the constraint Rouse (II) regime. In particular, a value τe = 8 × 10−7 s was determined for PB at T = 391 K from the cross-point of the respective subdiffusive power-laws observed in the msd. Here, a comment concerning the determination of the msd via proton FC NMR is necessary. Our NMR measurements included not only high temperatures where polymer-specific dynamics are probed but also low temperatures for which the
he determination of the entanglement time τe is of great importance for testing extensions of the tube reptation (TR) model. Within the original TR model1,2 a polymer chain “reptates” in a virtual tube representing the topological constraints exerted by the neighboring chains. For a quantitative description additional relaxation mechanisms have to be taken into account, most prominently constraint release and contour length fluctuations.1−4 Together with the plateau modulus G0N or the entanglement molar mass Me = ρRT/G0N (ρ denotes the polymer density), and the terminal relaxation time τt, the time τe provides the essential ingredients for a full description of the linear viscoelasticity of linear polymers in terms of the (complex) shear modulus G*(ω). While the plateau modulus G0N and also the terminal relaxation time τt(T,M) can easily be extracted from experiments, extracting τe(T) remained ambiguous. In particular, most of the estimates were derived from some theoretical expressions fitted to experimental viscoelastic data; no direct access was possible. Yet, probing the segmental mean-square displacement (msd), neutron scattering (NS),5 and very recently proton field-cycling (FC) NMR relaxometry6−8 allow to extract the msd in the Rouse and the constraint Rouse regime.9−11 The crossover between these regimes defines the entanglement time τe and thus can directly be accessed.1 In particular, absolute agreement between the msd extracted from NS and FC NMR was reported in the case of poly(ethylene−propylene).11 Concerning the estimates of τe in the case of poly(butadiene) (PB), in the present Note, we compare the τe values reported in the literature analyzing shear-stress relaxation to that of FC NMR.10 As will be demonstrated, in all the cases the literature estimates are significantly shorter than that provided by FC NMR. Moreover, there appear inconsistencies among the rheological data. In a recent publication, Park et al. reanalyzed shear modulus spectra G′(ω) and G″(ω) of different high molar mass PB melts as found in the literature.4 As the usually covered frequency range is narrow, the master curve G*(ωaT) constructed by superposing the results collected from a wide temperature range was discussed.12,13 The authors argued that irrespective of the polymer architecture, linear, star, comb, or hyperbranched, the entanglement time τe should be the same and should be accessible alone from the viscoelastic response at frequencies above those of the plateau regime. Indeed, good agreement among the relaxation spectra of the different PB was found at high frequencies. By subtracting the spectral contributions of the glassy modes and by fitting the Rouse
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© XXXX American Chemical Society
Received: November 24, 2016 Revised: January 26, 2017
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DOI: 10.1021/acs.macromol.6b02546 Macromolecules XXXX, XXX, XXX−XXX
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to do so, temperatures almost as low as Tg are required, which are however, only reached in a few studies. Here, we show results from Colby and co-workers (green circle)21 and from Palade et al. (green square).17 Both groups measured the (complex) shear modulus G*(ω) down to very low temperatures. In the case of the Colby data the master curve G″(ωaT) at the reference temperature T = 298 K was taken, while an unreduced spectrum G″(ω) from Palade et al. recorded at T = 178 K (−95 °C) was analyzed. From the high-frequency relaxation peak in G″(ω) one can directly estimate the structural relaxation time via the condition ωτα ≅ 1 (“peak picking”). The data agree with those from our FC NMR and dielectric spectroscopy results very well. Thus, regarding structural relaxation, NMR, and shear relaxation probe very similar time constants. Here we note that for an exact comparison of the results obtained from different methods it is important to apply the same method for extracting time constants. In the case of FC NMR we fitted a Cole−Davidson distribution to the α-relaxation peak contribution which not necessarily yields the same result as the “peak picking” method. Thus, while in Figure 1 agreement between the τα values of two methods is essentially found on a logarithmic scale, values may still differ by a factor of about 2. In Figure 1 we also included the τe value of 8 × 10−7 s (at 391 K) from our FC experiment (blue cross) analyzing the msd of PB. Together with τα(T = 391 K) = 1.2 × 10−11 s, a ratio τe/ τα ≅ 7 × 104 is found. Next, we added the result of Park et al. for τe(T = 298 K) = 3.7 × 10−7 s (full red star) together with the different values collected by the authors as well as by Liu et al. (open orange circles).3,14 Although derived at much lower temperatures (≈300 K) these τe values are shorter than the NMR value at 391 K. In their publication,4 Park et al. showed master curves G′(ωaT) and G″(ωaT) of linear PB-70k scaled to a reference temperature of 298 K which encompass the contribution of the glassy modes as well, i.e., the local segmental dynamics. The authors extracted data from ref 22 and superposed them to create the master curves. From the corresponding highfrequency maximum in G″(ωaT) one again can directly determine the structural relaxation time (via the condition aT ≅ 1). One finds τα = 3.8 × 10−12 s (open red square), which is actually by a factor of 50 too short compared to the FC NMR result and by a factor of 42 compared to Colby’s value. However, together with τe = 3.7 × 10−7 s almost the same ratio as resulting from our NMR measurements is recovered, τe/τα ≅ 105. Shifting the τe(T = 298 K) value reported by Park et al.4 by a factor of 42, one gets a value (red cross) which virtually lies on an extrapolation of our NMR value by a (shifted) version of the τα(T) line (solid blue curve) (cf. Figure 1). Comparing the data displayed in Park et al.4 with the original one by Colby and co-workers21 or Palade et al.,22 indeed, the master curve reproduced by Park et al. is shifted to much higher frequencies. This discrepancy is currently not understood. However, too short τe values are not only reported by Park et al. but also found in the collection of literature data (open orange circles for linear PB).11 Thus, all reported τe values appear to be significantly too short. In particular, the suggested experimental definition of τe by Liu et al.14 appears to be not adequate. Park et al. obtained the ratio τe/τα by fitting the rheological spectrum G″(ωaT) after eliminating the influence of the structural relaxation by the Likhtman−McLeish model,2 which itself may be contested. In contrast, FC NMR is able to provide the msd from which τe can directly be read off without
Figure 1. Temperature of the structural time constant τα(T) of PB for different M > Me, as obtained from proton FC NMR and dielectric spectroscopy (DS) (different black symbols) interpolated by the black solid line in comparison to rheological data (green solid symbols; see text),21,22 and to the value recalculated from Park et al. (open red square; see text). The entangelement time τe(391 K) as obtained by FC NMR (blue cross) in comparison to rheological data from different sources at room temperature (orange circles);4,14 the value τe = 3.7 × 10−7 s of Park et al. is also given (red star). Values for τe (red cross) and τα by Park et al. corrected by a factor of 42 (arrows; see text). Solid blue line interpolates τe(T) and is parallel to τα(T); dotted blue line: prediction by the TR model. Inset: segmental mean-square displacement of polybutadiene (PB) as a function of time as obtained from different NMR methods.10
segmental or structural relaxation (related to the time constant τα) dominates the relaxation rate R1(ω). Given that broad temperature range (200−400 K), susceptibility master curves χ″inter(ωτα) were constructed covering more than 9 decades in (reduced) frequency and displaying all four regimes of polymer dynamics as well as the glassy dynamics at highest frequencies. Actually, the temperature interval of 200−400 K is still well above Tg ≅ 174 K of PB,16 and as the FC method covered a frequency window of 300 Hz−30 MHz the structural relaxation probed at low temperatures is in the MHz regime. Thus, possible artifacts due to the breakdown of the FTS are not expected. Usually, significant decoupling of structural (or segmental) is observed at so low temperatures for which τα > 10−7−10−5 s.17,18 The segmental msd = msd(t/τα) results from χ″inter(ωτα) after Fourier transformation.10 The master curve construction yields the structural relaxation time τα(T) (assumed to be identical with the segmental relaxation time) instead of frequency shift factors aT(T) as encountered in many rheological studies. Typically, a range of 10−12 s < τα < 10−7 s is covered by FC NMR, as demonstrated in Figure 1 for PB. In addition, we included results from dielectric spectroscopy (DS) for different M values (M > Me) which accessed τα(T) in the range 10−5 s < τα < 104 s.10 As shown in Figure 1 (black symbols), the complete temperature dependence of τα is covered, between Tg = 174 K16 and about 400 K, and interpolated by a four-parameter function described in ref 19. The data refer to M > 2k where τα is independent of M.15,20 Having determined τα(T), we transformed the segmental msd to the reference temperature 391 K, as displayed in the inset of Figure 1. The structural time constant τα(T) can be compared to the corresponding results from rheological measurements. In order B
DOI: 10.1021/acs.macromol.6b02546 Macromolecules XXXX, XXX, XXX−XXX
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ACKNOWLEDGMENTS The work was financially supported by the Deutsche Forschungsgemeinschaft (DFG) through the grants RO 907/ 17.
applying any model. Of course, given the limited accuracy of the msd data and the smooth transition between the power laws I and II (cf. Figure 1, inset), the extracted τe value shows a large error bar of about a factor 2. For further check, we are not aware of a τe estimate of PB reported by NS. Comparing the results from different experiments not done at similar temperatures implies the construction of master curves based on the assumption that FTS holds. Thus, any error in the shift factors will influence the result. In particular, in the case of the rheological data low temperatures close to Tg are covered, and here segmental and chain relaxation may not show the same temperature dependence, as discussed.17,18 Furthermore, while FC NMR provides an experimentally well-defined estimate for τe, in the case of the viscoelastic spectra direct access appears not to be possible; i.e., theoretical models are needed to extract τe. For example, given that τe is defined as the Rouse time of an entanglement segment, then this is bound to be shorter than the observed crossover time in the msd, as at t = τe the influence of the confinement is still marginal. What is actually needed is to derive a coefficient which relates the msd crossover time τemsd to the predicted τerheo from rheologya future task. So far, “many authors. . . feel justified in adjusting τe to obtain the best agreement with linear viscoelastic data for entangled polymers.”23 We compare the experimental ratio τe/τα = 7 × 104 (from our NMR results) with the estimate of the simple Doi− Edwards TR model which states τeDE(T) = τs(T)Ne2.1 With Ne = Me/M0 and taking the segment molal mass M0 = 0.105 K12 together with Me = 1.930 K,12 one gets a ratio τe/τα = 324, respectively, which is much smaller than all the experimental values (cf. dashed blue line in Figure 1).9 Here, we assumed that τα = τs. The discrepancy between the experimental and the forecast ratio τe/τα may be removed if instead of the molar mass associated with the Kuhn length the mass of the “random step” related to the size of a “Rouse unit” is introduced, as pointed out by Ding and Sokolov.24 It may also be possible that τα and the segmental time τs are not identical, i.e., that τs > τα holds. Finally, some remarks on our theoretical understanding of the relaxation behavior of entangled polymer melts are worth to be given. The present state of the TR model, even with all its modifications is analytically incomplete because we have not any resolvable equation of motions, even on a phenomenological level, which would be able to analytically describe the dynamical transitions from region I to regime II (τs < t < τe) or from regime II to regime III (reptation). The existing predictions were actually derived only on the level of scaling arguments; i.e., they can predict characteristic quantities only with an accuracy of a numerical coefficient. In particular, this is the case for the entanglement time τe. This is why it is important to discuss experimental ways for its unambiguous determination, and FC NMR together with NS offers this.
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REFERENCES
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected] (E.A.R.). ORCID
E. A. Rössler: 0000-0001-5586-973X Notes
The authors declare no competing financial interest. C
DOI: 10.1021/acs.macromol.6b02546 Macromolecules XXXX, XXX, XXX−XXX
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DOI: 10.1021/acs.macromol.6b02546 Macromolecules XXXX, XXX, XXX−XXX
