Dynamics of a Paradigmatic Linear Polymer: A Proton Field-Cycling

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Dynamics of a Paradigmatic Linear Polymer: A Proton Field-Cycling NMR Relaxometry Study on Poly(ethylene−propylene) M. Hofmann,† B. Kresse,‡ L. Heymann,∥ A. F. Privalov,‡ L. Willner,§ N. Fatkullin,# N. Aksel,∥ F. Fujara,‡ and E. A. Rössler*,† †

Experimentalphysik II, Universität Bayreuth, D-95440 Bayreuth, Germany Institut für Festkörperphysik, TU Darmstadt, D-64289 Darmstadt, Germany ∥ Technische Mechanik und Strömungsmechanik, Universität Bayreuth, D-95440 Bayreuth, Germany § Institute of Complex Systems, Forschungszentrum Jülich, D-52425 Jülich, Germany # Institute of Physics, Kazan Federal University, Kazan 420008, Tatarstan Russia ‡

ABSTRACT: The dynamics of melts of linear poly(ethylene-alt-propylene) (PEP) of different molar masses (M) is investigated by 1H field-cycling (FC) NMR relaxometry. Employing a commercial and a home-built relaxometer the spin-lattice relaxation rate R1(ω) is measured in the frequency range of 200 Hz to 30 MHz and the temperature range of 200−400 K. Transforming the FC NMR relaxation data to the susceptibility representation and applying frequency−temperature superposition, master curves for the dipolar correlation function CDD(t/τα) (containing intra- and intermolecular contributions) are constructed which extend up to six decades in amplitude and eight in time. Here, τα is the time scale of the structural (α-) relaxation, which is obtained over several decades. Comparison with previously reported FC data for polybutadiene (PB) discloses very similar CDD(t). Depending on M, all the five relaxation regimes of a polymer melt are covered: in addition to the α-process (0) and the terminal relaxation (IV), which are immanent to all liquids, three polymer-specific power-law regimes (Rouse, I; constraint Rouse, II; and reptation, III) are found, i.e. CDD(t) ∝ t−ε. The corresponding exponents (εI−III) are close to those predicted by the tube-reptation (TR) model for the segmental translation. In contrast to previous interpretation the intermolecular relaxation dominates CDD(t), in particular in regime II and beyond. The decomposition into intra- (mediated by segmental reorientation) and intermolecular relaxation (mediated by segmental translation) via isotope dilution experiments yields Cinter(t) = Ctrans(t) ∝ t‑0.28±0.05 concerning PEP and Cinter(t) ∝ t‑0.30±0.05 concerning PB for regime II (high-M limit). For the reorientational correlation function Cintra(t) = C2(t) ∝ t−0.50±0.05 (PEP) and C2(t) ∝ t−0.45±0.05 (PB) are obtained. These exponents εIIintra are at variance with εIITR = 0.25 predicted by the TR model. The fact that translation conforms to the TR model, while reorientation does not, now confirmed for the two polymers PEP and PB, challenges de Gennes’ return-to-origin hypothesis which assumes strong translational-rotational coupling in the TR model.

1. INTRODUCTION In addition to the temperature dependence, the frequency dependence (dispersion) of the spin-lattice relaxation rate R1(ω) = T1‑1(ω) provides valuable information about the molecular dynamics in condensed matter. In particular, with the recent availability of a commercial field-cycling (FC) relaxometer, NMR relaxometry received new attention as it is now easily possible to measure R1(ω) in the range of 10 kHz ≤ ν = ω/(2π) ≤ 30 MHz concerning 1H.1−6 Home-built relaxometers may even achieve frequencies down to some 10 Hz.7−9 In FC NMR relaxometry, the magnetic field is produced by an electromagnet and quickly “cycled” from a (high) polarization field down to a (low) relaxation field and after a variable time back again to a (high) detection field. Most of the FC NMR studies employ the nucleus 1H; yet, other nuclei may also be used.2,6,10−12 Among many further applications,2,6,13 the method is well suited to study the slow collective dynamics in polymers3−5,14,15 and dendrimers.16 In such systems the dynamics is governed by frequency− © XXXX American Chemical Society

temperature superposition (FTS), i.e., the spectral shape of the dynamic susceptibility virtually does not change with temperature. Measuring R1(ω) in a broad temperature range and converting the data to the susceptibility representation χ″(ω) ≡ ωR 1 (ω) allows constructing master curves χ″(ωτ).4,5,17−21 Thereby, the still narrow frequency range of the FC technique is effectively enlarged significantly. Here, τ denotes some characteristic time constant, for example, the segmental (“local”) correlation time τα identified with the structural (α-) relaxation time, to which the master curves are scaled. Applying FTS has long been a widespread approach in rheology or dielectric spectroscopy (DS).22−24 In the case of 1H NMR, which is the subject of this work, the relaxation is driven by the fluctuations of the magnetic dipole− dipole interaction. The latter is a many-particle interacReceived: August 30, 2016 Revised: October 22, 2016

A

DOI: 10.1021/acs.macromol.6b01906 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules tion.3−5,15,25 As a consequence, there is an intra- R1intra(ω) as well as an intermolecular relaxation contribution R1inter(ω) which reflect reorientational and translational motion, respectively. It is not an ease to attribute both parts, which additively contribute to the total 1H relaxation R1(ω). Yet, at low frequencies the dispersion of the total relaxation is dominated by intermolecular relaxation which is mediated by translational diffusion.26−28 This allows extraction of the diffusion coefficient.29,30 The separation of R1intra(ω) and R1inter(ω) is achieved by the application of the isotope dilution technique.31,32 Diluting protonated molecules in a matrix composed by deuterated analogues supresses the intermolecular relaxation. Not only the diffusion coefficient but also the subdiffusive segmental mean square displacement (msd) ⟨r2(t)⟩ of polymers as a function of time can be extracted after singling out R1inter(ω).32,34 Recently,35 such ⟨r2(t)⟩ data was combined with that of field-gradient NMR, and all four regimes of segmental diffusion of linear entangled chains predicted by the tube-reptation (TR) model36 were identified in polybutadiene (PB) as well as in poly(dimethylsiloxane) (PDMS). Very recently, we reported on the msd of poyl(ethylene-altpropylene) (PEP), where the Rouse (regime I) and the constrained Rouse regime (II) of the TR model could be reproduced by 1H FC NMR, and absolute agreement with data from neutron scattering was revealed.37 Likewise, the intramolecular contribution R1intra(ω) reflecting segmental reorientation displays corresponding power-law dispersions which, however, do not agree with the predictions of the TR model.11,15,21,31,32 Specifically, in the case of PB the second-rank reorientational correlation function C2(t) gained from R1intra(ω) displays an exponent in regime II (constrained Rouse) which is by a factor of 2 larger than predicted by the TR model.38,39 In the Rouse regime (I) agreement was essentially found. Yet, these FC results on C2(t) are at variance with recent coarse-grained computer simulations,40,41 and also with experimental results stemming from 1H double quantum (DQ) NMR spectroscopy.41 In conclusion, while our results on segmental translation essentially confirmed the TR model, the findings on segmental reorientation do not−a result, which in the light of the mentioned controversies, deserves to be confirmed by further experiments on a different polymer which will be presented here. We dwell upon these discrepancies and show results for linear PEP, to clarify the issue. A generalization and reproduction of our previous findings would challenge de Gennes’ “return-to-origin” (RTO) hypothesis, which states an inverse proportionality between ⟨r2(t)⟩ and C2(t) to hold in the entanglement regime of the TR model.38−40 In contrast to the previously investigated polymers, PEP has some advantages. It features a rather simple structure, yet, with only a weak tendency to crystallization. This enables measurements in a broad temperature range 200−400 K, i.e., down as low as the glass transition temperature Tg ≈ 206 K (cf. below). Concerning polymer-specific relaxation PEP was studied extensively by neutron scattering42−45 and also via rheology.46,47 In the present contribution, the local as well as the polymer-specific dynamics are monitored by 1H FC NMR relaxometry employing a commercial as well as a home-built relaxometer.6−8 The latter allows to perform measurements on polymers at frequencies down to some 100 Hz, i.e., 2 decades lower than the commercial relaxometer. Several molar masses 3k ≤ M ≤ 200k were investigated. For PEP with M = 200k we carried out an isotope dilution experiment to isolate the

intramolecular relaxation contribution. Some 2H FC NMR data11 and, as mentioned, also the extraction of the msd35 from the intermolecular relaxation were recently published. As in the case of PB, we will demonstrate that segmental reorientation of PEP is not appropriately described by the TR model with its RTO hypothesis.

2. THEORETICAL BACKGROUND Field-cycling NMR relaxometry probes the frequency dependence of the spin-lattice relaxation rate R1(ω = γB) = T1‑1(ω), where B denotes the applied magnetic relaxation field and γ the gyromagnetic ratio. In the case of 1H, the molecular dynamics modulates the fluctuations of the magnetic dipole−dipole interaction among the proton pairs which comprises two contributions; that acting between protons located on the same molecule, causing intramolecular relaxation, and that located on different molecules, causing intermolecular relaxation. The total relaxation rate measured by 1H FC NMR in fully protonated systems can be written as a sum:2,14,15,25 R1(ω) = R1intra(ω) + R1inter(ω)

(1)

Both contributions follow a Bloembergen, Purcell, and Pound (BPP) equation2,25 R1intra,inter(ω) = K intra,inter[Jintra,inter (ω) + 4Jintra,inter (2ω)] (2) i

where K denotes the corresponding coupling constant. The latter is proportional to the intra- and intermolecular contributions, respectively, to the second moment which reflects the spatial distribution of the spins.25 The Ki values show a weak temperature dependence originating from thermal expansion, the effect of which is neglected here. Regarding the reorientational correlation function of polymers, it was shown that the intersegment contribution can be neglected,31,34 and therefore R1intra(ω) reflects the segmental reorientation in terms of the spectral density Jintra(ω) = J2(ω), which enters eq 2. For isotropic liquids it is given by the Fourier transform of the correlation function of the second Legendre polynomial C2(t):25 C2(t ) ∝

∑ ⟨[3 cos2(ϑij(t )) − 1][3 cos2(ϑij(0)) − 1]⟩ spin pairs

(3)

Here, ϑij(t) denotes the angle of the internuclear vector ri⃗ j(t) between adjacent spins, i and j on the same segment, with respect to the magnetic field. Concerning the intermolecular rate R1inter(ω) the corresponding correlation function Cinter(t) reflects segmental translation in terms of the function3,15,25 C trans(t ) ∝

∑ spin

3 cos2(ϑij(t )) − 1 rij 3(t )

pairs

×

3 cos2(ϑij(0)) − 1 rij 3(0)

(4)

where the sum in this case includes only pairs of spins belonging to different polymers. The corresponding spectral density is denoted as Jinter(ω) = Jtrans(ω). As the intra- and intermolecular rate contribute additively to the measured total relaxation rate R1(ω) (cf. eq 1) also the total (dipolar) B


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Table 1. Predictions for the Different Relaxation regimes of the Tube-Reptation (TR) Model for Quantities Accessible by 1H FC NMRa regime

⟨r2(t)⟩

Rouse (I) constraint Rouse (II) reptation (III) diffusion (IV)

1/2

t t1/4 t1/2 t1

C2(t) −1

t t−1/4 t−1/2 exp(−t/τ1)

R1intra(ω)

Ctrans(t) −3/4

−τ0 ln(ωτ0) ω−3/4 ω−1/2 ω0

t t−3/8 t−3/4 t−3/2

R1inter(ω) −1/4

ω ω−5/8 ω−1/4 R1(0) − cω1/2

ATR(ω)

Aiso(ω)

ω−1/4 ω1/8 ω1/4 const. − cω1/2

ω−1/4 ω−1/8 ω−1/4 const. − cω1/2

Segmental mean square displacement ⟨r2(t)⟩,36 second-rank reorientational correlation function C2(t),39 translational correlation function Ctrans(t),15,48 and intramolecular spin-lattice relaxation dispersion R1intra(ω), as well as the intermolecular analogue R1inter(ω). Frequency dependence of the ratio A(ω) (cf. eqs 6 and 7) for the TR model as well as for isotropic polymer models. a

correlation function is a sum of two functions, explicitly CDD(t) = C2(t) + Ctrans(t). The partition of the two relaxation contributions is time dependent and unknown, in general. Yet, there is one important piece of knowledge. For long times, Ctrans(t) is determined by Fickian diffusion leading to Ctrans(t) ∝ t−d/2 at longest times, where d is the spatial dimension.5,26,27As the reorientational function C2(t) essentially decays along a (stretched) exponential the correlation function CDD(t) is consequently dominated by the power-law t−3/2 (d = 3) at long times. Correspondingly, the total relaxation rate in the lowfrequency limit is described by a universal relaxation law, explicitly R1(ω) = R1(0) − cω1/2. From the parameter c, the diffusion coefficient can be derived, an approach which was only recently applied for simple liquids as well as polymer systematically.5,28−30 In the case of polymers with their different subdiffusive regimes, by separating R1inter(ω) from R1(ω) one may even extract the time dependence of the msd, as mentioned.32,34,35 The tube-reptation (TR) model36 is a so-called anisotropic polymer model.14,15,21 Here, in the time interval τe ≤ t ≤ τt spatial displacements of polymer segments are confined inside a fictitious tube, and polymer conformations are temporarily correlated strongly with the polymer chain’s initial conformation. The time τe is the entanglement and τt the terminal (or disengagement) time. The TR model provides four relaxation regimes (I−IV) with characteristic power-laws for C2(t), Ctrans(t), and the segmental msd;14,15 those predictions are listed in Table 1. In addition to these polymer-specific relaxations there exists another regime (0) at shortest times. It is not explicitly included in polymer theories and is determined by the glass transition phenomenon characterized by the segmental time constant τs which is assumed to be identical with the structural relaxation time τα. In the TR regimes II and III associated with entanglement dynamics, the msd and C2(t) are directly linked to each other according to the so-called return-to-origin (RTO) hypothesis38,39 Cl(t ) ∝

ATR (ω) =

(6)

The corresponding power-laws for the different relaxation regimes in A(ω) are included in Table 1. Another class of polymer models is called “isotropic” like the n-renormalized Rouse14 or the polymer mode−mode coupling model.49,50 Also the standard Rouse model (regime I) is isotropic. Here, it is assumed that already at short times t > τs the segmental displacements are not correlated with the polymer chain’s initial conformation. The connection with ⟨r2(t)⟩1/2 is given by15,48 A iso(ω) ∝ ⟨r 2(t = ω−1)⟩+1/2

(7)

Compared to the TR model eq 6, the relation between A(ω) and the msd is inverted in regimes II and III. The corresponding power-laws for A(ω) are also included in Table 1. Thus, the different models of polymer dynamics can be distinguished by measuring A(ω) along the different frequency dependencies. A separation of intra- and intermolecular relaxation contributions is performed by isotopic dilution experiments, in which fully protonated chains are diluted in a deuterated matrix consisting of chains of the same molecular mass (cf. Experimental Section). Approaching the zero concentration limit of the protonated species the intermolecular relaxation contribution is removed and one obtains the intracontribution reflecting segmental reorientation solely.28 We note that also 2 H FC NMR can be applied to probe segmental reorientation.11 In order to construct master curves assuming FTS and to stress the similarity with dielectric or rheological relaxation spectra, the FC NMR dispersion data is converted into in the susceptibility representation χ″(ω) ≡ ωR1(ω).4,5,19−21 In this representation, shifting χ″(ω) measured at different temperatures solely along the frequency axis provides master curves which can be appropriately rescaled to provide χ″(ωτα). As said the temperature dependence of the coupling constants can be ignored. Besides, the temperature dependence τα(T) is gained incidentally.

a0 2 ⟨r 2(t )⟩

R1inter(ω) ∝ ⟨r 2(t = ω−1)⟩−1/2 R1intra(ω)

(5)

3. EXPERIMENTAL SECTION where the index l stands for the order of the reorientational correlation function and a0 for the tube diameter. Thus, checking the relationship between the msd and C2(t) offers an opportunity to test the TR model−and both quantities are accessible from FC NMR relaxometry. An alternative way of testing directly the relation between segmental translation and reorientation is given by considering the ratio A(ω) of the inter- and the intramolecular relaxation rate. Here, the TR model yields for times τe ≤ t ≤ τt (regimes II and III) where entanglement effects are important15,48

Poly(ethylene-propylene) (PEP) with a narrow M distribution (Mw/ Mn < 1.06) was synthesized from poly(isoprene) via hydrogenation.42−45 Batches of fully protonated PEP (PEP-h10) with 3k ≤ M ≤ 200k were provided by the Richter group (Forschungszentrum Jülich, Germany). The critical molar mass for the occurrence of entanglements is known to be about Mc ≅ 3k.47 For the FC NMR measurements the polymers were filled into standard 5 mm NMR tubes and then degassed under vacuum at around 330 K for at least 48 h to remove paramagnetic oxygen. An isotopic mixture composed by 10% (weight) PEP-h10-200k and 90% of PEP-d10-200k (perdeuterated) was blended according to the procedure described in ref 32. The C


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Figure 1. (a) Total 1H spin-lattice relaxation dispersion of PEP with M = 200k in the susceptibility representation χ″DD(ω) = ωR1(ω), measured at different temperatures as indicated. In addition to the α-peak representing regime 0, two power-law regimes (I, II) are observed reflecting polymerspecific relaxation (dashed lines). (b) Master curves χ″DD(ωτα) for different M in the range 3k ≤ M ≤ 200k of PEP. The curve of the liquid dipropylene glycol (di-PG) is included for reference. concentration was controlled by thorough weighing. Because of dilution by the deuterated matrix, the intermolecular relaxation is (mostly) suppressed. We note that highly diluting a protonated polymer with its deuterated counterpart still leaves the heteronuclear (1H−2H) intermolecular relaxation contribution which, however, is negligible.31,34 Measuring such a sample allows for a separation of intra- and intermolecular relaxation contributions which are studied separately. The obtained intramolecular NMR susceptibility χ″intra(ωτα) was subtracted from the total dipolar susceptibility χ″DD(ωτα) to get the pure intercontribution χ″inter(ωτα). The measurements were performed in a broad temperature range of about 400 K down to about 220 K. In all experiments the temperature was controlled better than ±1 K. Two different relaxometers were employed. A FFC 2000 device from STELAR s.r.l. (Mede, Italy) is operated at the University of Bayreuth and provides a frequency range of 10 kHz ≤ ω/2π ≤ 20 MHz concerning 1H.2,11 The home-built apparatus located at the Technische Universität Darmstadt is equipped with unique low-field technology. It allows for dispersion measurements in a range of 200 Hz ≤ ω/2π ≤ 30 MHz, i.e., fields well below the Earth’s are achieved.6−8 Besides the field range, the switching time between polarization and relaxation field as well as that between relaxation and detection field, respectively, poses a limitation: the largest relaxation rates R1 measurable are about 1000/s. The magnetization built-up/recovery curves were monoexponential over at least one decade in magnitude. Thus, flip-flop processes are always efficient enough in establishing a common spin-temperature.

In the next step master curves were constructed from the susceptibility data. For that purpose frequency shift factors aT were applied to the χ″DD(ω) data sets recorded at the different temperatures to achieve best overlap. No vertical scaling was done as the NMR coupling constant K (cf. eq 2) is virtually temperature independent. By fitting a Cole−Davidson (CD)51 function to the peak of the master curve χ″DD(ωaT) the frequency axis is rescaled to yield χDD″(ωτα). Thereby, the temperature dependence τα(T) is revealed; it is displayed in Figure 7 and will be addressed below (Discussion). We anticipate that τα(T) is independent of M in the investigated range. In Figure 1b the master curves χ″DD(ωτα) belonging to PEP with different M values are shown, in comparison to that of the simple liquid dipropylene glycol (di-PG, black crosses). The α-process (regime 0) dominates all spectra. In the case of di-PG it is actually the only relaxation feature. As said, the αpeak can be well interpolated by a CD function. For the PEP polymers the α-peak is succeeded by different power-law regimes at ωτα ≪ 1. The higher M the more excess intensity with respect to the simple liquid limit for all M is apparent (dashed line). At even lower frequencies a crossover to a another power-law regime reflecting entanglement dynamics is observed in Figure 1b with an exponent decreasing with growing M, i.e., the χ″DD(ωτα) curves become flatter with a final exponent of 0.4. The terminal relaxation, where χ″DD(ω) ∝ ω1 holds approximately, is only observed for rather low M up to 29k. As discussed (cf. Theoretical Background), at lowest frequencies, the total relaxation is always dominated by the intermolecular relaxation, which is mediated by translational diffusion.26−30 As a consequence χ″DD(ω) = ωR1(0) − Cω3/2 holds at low frequencies, which is actually difficult to distinguish from χ″DD(ω) ∝ ω1. Via their corresponding spectral densities, the χ″DD(ωτα) curves presented in Figure 1b can be transformed into the time domain applying a numerical Fourier transformation algorithm based on Filon’s method.52 This provides the dipolar correlation function CDD(t/τα), which is shown in Figure 2a for all M, again together with that of di-PG. For comparison, Figure 2b shows equivalent curves of PB taken from our previous publications.9,21 The correlation functions are probed over six decades in amplitude and eight decades in time. This

4. RESULTS Total Relaxation. As an example the total dispersion R1(ω) comprising intra- as well as intermolecular relaxation contributions of protonated PEP-200k is shown in Figure 1a in the susceptibility representation, χ″DD(ω) = ωR1(ω). In the Appendix (Figure 8), we provide the corresponding original R1(ω) data. Qualitatively, the dominating peak in χ″DD(ω) reflects the segmental or α-relaxation (regime 0). The peak shifts toward lower frequency when temperature is decreased. At the lowest temperature (219 K) χ″DD(ω) tends toward a plateau indicating the onset of a secondary (β-) relaxation. Upon increasing the temperature the α−peak is shifted out of the frequency window and two power-law regimes are recognized (dashed lines in Figure 1a), which reflect collective chain dynamics typical of high-M polymers, specifically Rouse (I) and entanglement dynamics (II).4,5,14,17−21 D


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For shorter chains the exponent εII = εII(M) increases with decreasing M. In Figure 3, we compare the exponent εII(M/Mc)

Figure 3. Power-law exponent εII in regime II as a function of the reduced molar mass M/Mc obtained from the dipolar correlation function CDD(t) (black) as well as from the reorientational correlation function Cintra(t) = C2(t) (red) and the intermolecular one Cinter(t) = Ctrans(t) (magenta) for PEP and PB. Prediction of the tube-reptation model for regime II is indicated (dotted line). Concerning C2(t) data from isotope dilution (intramolecular relaxation, red) as well as from 2 H FC NMR11 (quadrupolar relaxation, denoted by “Q” and shown in green) are displayed. Entanglement molar masses Mc = 2k for PB and Mc = 3k for PEP were used for scaling. Dashed lines: guide to the eye.

obtained in regime II of CDD(t) for both PEP and PB. The scaling by the reduced molar mass M/Mc was chosen to take the different crossover molar masses (Mc = 3k for PEP vs 2k for PB) into account. A common decrease of εII(M) is observed. This broad crossover from a high εII value close to εI ≈ 0.9−0.8 at small M (Rouse regime) down to a plateau at around 0.3 at highest M, previously called “protracted transition” to the final entanglement exponent9 also occurs in other polymers20 and was monitored by 1H DQ NMR, too.41,53 Recently, the effect was exposed by FC NMR as a finite-length effect which disappears when merely the center-sections of long chains are regarded, i.e., in polymers with deuterated chain ends.21 Thus, the protracted transition to the plateau value of εII ≅ 0.3 occurs within a much narrower M interval close to Mc, when solely the inner segments are investigated, a fact which was also concluded from computer simulations.40,54,55 For PEP-80k, PEP-50k, PB-24k, and PB-56k, even a third regime III may be anticipated in Figure 2 with an exponent being close to εIII = 0.75. This regime III apparently does not show up in PEP of M values lower than 50k. Otherwise, in PEP-200k, it is already outside the accessible time window, hence only regime II is observable at lowest frequencies here. In any case the correlation ultimately decays to zero along εIV = 1.5, which is observed in the experimental window for M ≤ 30k (dashed lines). This terminal power-law is immediately understood: it reflects the regime governed by intermolecular relaxation which is mediated by Fickian diffusion at longest times, a fact well-known since long26,27 but only recently exploited systematically,28−30 as mentioned above. For higher M the value of the terminal time τt is estimated from shearstress measurements which are exemplarily presented for PEP80k in the Appendix (Figure 9) and will be discussed in detail

Figure 2. (a) Dipolar correlation function CDD(t/τα) of PEP with 3k ≤ M ≤ 200k, as well as of dipropylene glycol (di-PG). Relaxation regimes (0−IV) and corresponding power-laws with exponents are indicated (dashed lines). The Rouse time τR explicitly belongs to PEP-80k (shown in green). The terminal time τt of PEP-80k is estimated from shear-stress measurements (cf. Appendix). (b) For comparison CDD(t/ τα) of selected M of PB taken from ref 9. Here, τR and τt belong to PB24k (marked in green). Concerning PB-56k (red) τR is also shown. In regime II the exponent depends on M. (c) Predictions of the TR model for the translational correlation function Ctrans(t) (cf. Table 1).

time domain representation of the FC NMR data is most appropriate for discussing the different relaxation regimes, in particular, the corresponding power-law exponents which can be compared to those expected for the different polymer theories. For reasons which will be clarified below Figure 2c shows the prediction of Ctrans(t) resulting from the TR model (cf. Table 1). Clearly, both sets of curves belonging to PEP and PB, respectively, highly resemble each other. Depending on M, several power-law regimes CDD(t)∝ t−ε are disclosed beyond the initial decay of the α-process (regime 0), i.e., at t ≫ τα. In regime I for PEP and PB one observes exponent values of about εI = 0.8 and 0.9, respectively, independent of M. Beyond the entanglement time τe, i.e., in regime II, at highest M one finds εII = 0.38 ± 0.03 for PEP-200k and 0.31 ± 0.03 for PB-441k. E


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Macromolecules in a forthcoming publication (part III). The rheological measurements allow projecting the further progress of CDD(t) of PEP-80k, as indicated in Figure 2a by the solid green lines. In the simple liquid di-PG the α-process is immediately succeeded by the terminal power-law CDD(t)∝ t‑3/2 as expected, due to the absence of polymer dynamics. Attributing the exponents in regimes I−III to those predicted by the TR model, for example, is not straightforward. Previously,17−20 in the case of PB, we discussed the exponents of CDD(t) assuming that it is dominated by the intramolecular pathway, i.e., by segmental reorientation. This was suggested by the fact that the high-M limit εII = 0.31 ± 0.03 is not so far from the TR prediction εIITR = 0.25. Furthermore, the experimental exponent εI in the Rouse regime is actually close to 1, as expected concerning C2(t) (cf. Table 1).38,39 Later on in refs 32 and 33, after separating intra- and intercontribution of PB, it became clear that the exponent εIIintra of C2(t) extracted from R1intra(ωτα) which reflects only segmental reorientation is much higher (0.45 ± 0.05) than the TR prediction. Thus, it appeared that the TR model does not apply, at least for segmental reorientation, and the actual exponent of the total correlation function CDD(t) is significantly influenced by the intermolecular relaxation contribution.11,21,32 Actually, the exponents of CDD(t) for both, PB and PEP are close to those forecast by the TR model for the intermolecular correlation function Ctrans(t). Figure 2c shows the prediction for Ctrans(t) resulting from the TR model (cf. Table 1). Explictly, Ctrans(t) ∝ t‑0.75 is predicted by the Rouse model (regime I), and Ctrans(t) ∝ t‑0.375 for the constrained Rouse regime (II), which is in between the experimentally found εII values of high-M PEP (0.38) and PB (0.31) in CDD(t). Likewise, the exponent in regime III for PEP-80k (green colored in Figure 2a), PB-24k (green colored in Figure 2b), and PB-56k (red colored in Figure 2b) is compatible with εIII = 0.75, as predicted by the TR model for Ctrans(t) in the reptation regime. Besides regime II, regime III is the second power-law regime characteristic for entanglement dynamics (cf. Figure 2c and Theoretical Background). Tentatively interpreting CDD(t) in terms of the TR model, the terminal Rouse time τR/τα ≈ 5 × 106 of PEP-80k (shown in green in Figure 2a) is estimated from the power-law intersection between regimes II and III. For PB one estimates τR/τα ≈ 1 × 106 (PB-24k) and τR/τα ≈ 7 × 106 (PB-56k) from Figure 2b. Isotope Dilution Experiment. In the case of PEP-200k, deuterated material was available, offering the opportunity to perform an isotope dilution experiment as it was done for PB previously.11,21,32 This enables a separation between intra- and intermolecular 1H relaxation by suppressing the latter. An intramolecular NMR susceptibility master curve χ″intra(ωτα) was constructed from FC measurements on the isotopic blend at different temperatures in the same way as for the fully protonated PEP systems. In the Appendix (Figure 8b), the curves χ″intra(ωτα) and χ″inter(ωτα) = χ″DD(ωτα) − χ″intra(ωτα) are shown, in comparison to χ″DD(ωτα) obtained from its undiluted, fully protonated counterpart. Both χ″intra(ωτα) and χ″inter(ωτα), were transformed into the time domain. Figure 4 displays the resulting correlation functions Cintra(t/τα) = C2(t/ τα), as well as Cinter(t/τα) = Ctrans(t/τα) of PEP-200k. In regime I, Cinter(t) closely follows ∝ t‑0.75 (black dashed line), as predicted by the Rouse model for the relaxation mediated by segmental translation (cf. Table 1 and Figure 2c), while C2(t) ∝ t‑0.9 results from the intramolecular relaxation mediated by reorientation. The latter behavior is close to C2(t) ∝ t‑1

Figure 4. Inter- and intramolecular correlation functions Cinter(t/τα) = Ctrans(t/τα) (black) and Cintra(t/τα) = C2(t/τα) (red), respectively, resulting from 1H FC NMR on isotopically diluted PEP-200k. C2(t/τα) obtained from 2H FC NMR is also included (green triangles).11 Dynamic regimes are indicated. Dotted lines: apparent power-laws in the Rouse regime. Dashed lines: predictions of the TR model for Ctrans(t) and C2(t) (cf. Table 1).

predicted by the Rouse model (red dashed line; cf. Table 1). Regarding Cinter(t) in regime II, εIIinter = 0.30 ± 0.05 is found, which is somewhat smaller than the corresponding TR prediction of Ctrans(t) ∝ t‑0.375 (black dashed line at long times). A similar value of εIIinter = 0.28 ± 0.05 was observed before for PB.32 The exponents referring to Ctrans(t) can also be found in Figure 3 for the systems of PEP and PB, for which isotopic dilution experiments have been performed. Once again, it is seen that εII of the total relaxation function CDD(t) is close to that of Ctrans(t), indicating that intermolecular relaxation determines the shape of CDD(t). Otherwise, for the intramolecular correlation function Cintra(t) reflecting reorientation, an exponent εII = 0.50 ± 0.05 is observed for PEP, which is clearly steeper than C2 ∝ t‑0.25 predicted by the TR model, just as it was found before for PB21,32 (0.45 ± 0.05) and mentioned above (cf. Figure 3). Also the quadrupolar relaxation dispersion R1(ω) of per-deuterated PEP-200k was measured via 2H FC NMR.11 The corresponding time domain data is included in Figure 4 (green triangles). The quadrupolar relaxation inherently monitors C2(t), fits to the intramolecular data Cintra(t) obtained from the isotope dilution experiment well, and thus it confirms the exponent value: εIIintra = 0.50 ± 0.05 ≈ εIIQ ≈ 0.48 ± 0.05 for PEP-200k. The exponent values εIIintra and εIIQ are also included in Figure 3. For all M values the intra exponents lie significantly above those of the inter as well as the total exponent. Instead of comparing C inter (t) and C intra (t) to the corresponding model predictions for Ctrans(t) and C2(t), respectively, one can consider the frequency dependence of the ratio A(ω) = R1inter(ω)/R1intra(ω), i.e., the original relaxation rates are directly used at a single temperature for a given system; no FTS is applied. As discussed previously15 and in the Theoretical Background, this ratio offers the possibility to test polymer models on a fundamental level. Specifically, it may discriminate between the isotropic and anisotropic polymer models. Comprising results for three different polymers, the ratio is plotted in Figure 5, where the frequency axis is scaled along ωτe for each polymer. The entanglement times were estimated from the respective power-law intersections between F


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the intermolecular relaxation than it was perceived before.19 Similar results were also reported in the case of PEO.31,34 The separation of intra- and inter-relaxation by performing an isotope dilution experiment yields for the reorientational correlation function Cintra(t) = C2(t) ∝ t−0.50±0.05 (PEP) and C2(t) ∝ t−0.45±0.05 (PB) in regime II. Clearly, those exponent values disagree with the prediction of the TR model for segmental reorientation (εIITR = 0.25). In the case of translation, the exponents εIIinter = 0.30 ± 0.05 (PEP) and εIIinter = 0.28 ± 0.05 (PB) are not so far from εIITR = 0.375. Here a note of caution is given: Determining power-law exponents may be ambiguous as in the present case not always sufficiently large intervals in the log−log plots are covered. The values obtained also depend on whether they are taken in the frequency or time domain (after Fourier transformation of the measured data). The present values were always derived from the time domain data and are meant as an estimate and a way of characterizing a complex decay curve.56 Independent of this, we emphasize that the disagreement with TR model is directly reflected in the failure of qualitatively reproducing the frequency dependence of the ratio A(ω) of the inter- and the intramolecular relaxation rate (cf. Figure 5). As concluded before from NS 57 and FC NMR 32,35 experiments as well as from simulations40,54 regarding segmental subdiffusion the TR model is essentially confirmed. In the case of segmental reorientation, however, the TR model fails for both, PB and PEP. The situation is depicted in Figure 6, where the product of the measured functions C2(t) and

Figure 5. Ratio A(ω) = R1inter/R1intra vs ωτe of PB-24k, PB-196k, PEP200k, and PEO-180k. The latter data were taken from ref 31. The prediction for Rouse dynamics (Aiso(ω) ∝ ω‑0.25, solid line) as well as for isotropic (Aiso(ω) ∝ ω‑0.125, solid line) and anisotropic (TR model, ATR(ω) ∝ ω0.125, dashed line) entanglement dynamics is indicated.

regimes I and II of CDD(t) shown in Figure 2 to τe/τα = 7 × 103 (PEP) and 6 × 103 (PB). Concerning poly(ethylene oxide) (PEO), data is taken from Kehr et al.31 As τe of PEO is unknown the corresponding A(ω) curve was horizontally shifted to coincide with that of PB-196k in regime II. Because of the division of two quantities the ratio A(ω) is noisy, especially at high frequencies, where R1intra(ω) is large and R1inter(ω) small. In the Rouse regime (I) as well as in regime (II) the ratio grows with decreasing frequencies. In other words, the lower the frequency, the higher the relative contribution of the intermolecular contribution to the total relaxation becomes. At ωτe < 0.01 the intermolecular relaxation rate even overtakes. The behavior in regime II clearly contradicts the prediction of the TR model (dashed black line, and Table 1), which forecasts an increase of the ratio with frequency, yet, our results (regime II) are in qualitative agreement with that of the isotropic models (solid lines) like the renormalized Rouse14,15 or the polymer mode coupling theory.48,50 As reported,41 in the case of DQ 1H NMR, interand intramolecular relaxation turned out to show the same frequency dependence. This would result in a ratio A(ω) being frequency independent in regime II, a behavior not predicted by any polymer model.

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Figure 6. Scaled (see text) product of the segmental msd ⟨r2(t)⟩ and the reorientational correlation function C2(t) for PB-24k, PB-196k, and PEP-200k. The predictions of the TR model for regimes I and II are indicated (dashed lines).

DISCUSSION Transforming FC 1H relaxation data to the susceptibility representation and applying FTS the dipolar correlation function CDD(t) is obtained for PEP of different M after Fourier transformation. All the five relaxation regimes of a polymer melt are discovered. Similar results were observed for PB 9 which were also confirmed by 1 H DQ NMR quantitatively.53 In addition to the α-process (0) and the terminal relaxation (IV), which are immanent to all liquids including polymers, three polymer specific regimes (I−III) are found, depending on M. By its very nature, the CDD(t) function contains contributions stemming from segmental reorientation as well as from segmental translation. Regarding the different power-law exponents, they are actually rather close to those predicted by the TR model for the segmental translation. Thus, the shape of CDD(t) in polymers appears to be more affected by

⟨r2(t)⟩ is plotted for PB-24k, PB-196k, and PEP-200k vs t/τα. The PB data was collected from our previous publications32,35 and vertically scaled along C2 × ⟨r2⟩(t = τα) = 1 to compensate the slightly different msd of PB and PEP. The corresponding predictions following from the TR model are represented by dashed lines. While in the Rouse regime the prediction (C2 × ⟨r2⟩ ∝ t‑0.5) (cf. Table 1) is verified, the product clearly decays in time in regime II (t > τe) instead of being constant as anticipated from by the TR model. As mentioned, the TR model assumes strong rotational-translational coupling in terms of deGennes’ RTO hypothesis which predicts C2(t) ∝ ⟨r2(t)⟩−1. Consequently, our results on PEP and PB show that in real G


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dielectric spectroscopy (unpublished data) measurements (crosses). We also added values of τα(T) derived from shear rheology (colored solid symbols) (cf. Appendix and Figure 9). Altogether, the super-Arrhenius slowdown of the α-process undergoing the glass transition is consistently probed by the different methods over 15 decades. The glass transition temperature is determined to be Tg ≅ 206 K via the condition τα(Tg) = 100 s. As noted before, τα(T) and also Tg do not change with M, at least beyond molar masses M = 3k. The combined τα(T) data is interpolated well by a four-parameter function (black solid line), the parameters of which are given in the Appendix.58 The terminal relaxation time τt(T) for PEP-29k and for PEP80k extracted from rheological relaxation spectra recorded at high temperatures (cf. Appendix) are included in Figure 7 (open stars). Versions of the interpolation function of τα(T) are vertically shifted (dashed lines) to intersect those τt values. A satisfactory interpolation of the τt(T) data of both, PEP-29k and PEP-80k is revealed. Concerning PEP-29k, we added the τt value obtained by FC NMR, specifically from the intersection point between the entanglement and the terminal regime observed in CDD(t) (cf. Figure 2a). This τt value (solid green star) agrees well with the extrapolation of the corresponding rheological data toward high temperatures via the green dashed line. The fact that τα and τt feature the same temperature dependence proves the assumption of FTS, meaning that the αprocess reflecting the glass transition, also drives the collective, nonlocal polymer dynamics and even the terminal relaxation. Finally, we turn to the entanglement time τe(T), which describes the time scale when the chain “feels” the tube. Its value for PEP is estimated to τe/τα = 7 × 103, or equivalently to τe(T = 393 K) ≅ 0.2 μs (or τe(T = 298 K) ≅ 20 μs), from the crossover in CDD(t) between the power-law regimes I and II observed in Figure 2a. From NS, a value of τe(T = 492 K) = 0.008 μs was reported,57 which is included in Figure 7 (open square). Again, when vertically shifted, the temperature dependence of τα(T) (black dashed line) intersects the two data points of τe(T). In the case of PB, we find a ratio τe/τα = 6 × 103 which is about one decade smaller than that reported in ref 35, or equivalently to τe(T = 298 K) ≅ 2 μs laying at the upper margin of the values discussed by Larson and co-workers reviewing rheological data in the frequency domain.59

polymer melts the RTO hypothesis does not hold. In contrast, generic simulations appear to be in agreement with the RTO hypothesis.40,41 Also the discrepancy to the intramolecular DQ NMR results on PB,41 showing the same time dependence as the total correlation function, is not settled. Atomistic simulations are needed for a deeper test. With the current computational power, however, it is still difficult to reach the regime of fully established entanglement dynamics. Kimmich and Fatkullin state that “The topological constraints restricting translational segment diffusion are not necessarily the same as those being responsible for limited rotational reorientations”.3 Given the experimental evidence, this statement is confirmed by our findings. Next, in Figure 7, we discuss the temperature dependences of the time constant τα(T), the terminal relaxation time τt(T), and

Figure 7. Relaxation map of PEP: τα(T) determined by 1H FC NMR (black closed symbols) and shear-stress rheology (colored closed symbols) for various M, as indicated. For PEP-50k, τα(T) data from dielectric spectroscopy (unpublished data) and solid-state 2H NMR11 are included (crosses); All the τα(T) data is interpolated by a fourparameter function58 (solid black line). The terminal relaxation time τt(T) of PEP-29k and PEP-80k was determined from shear-stress measurements (open stars). A τe value from FC NMR (black open circle) and one from neutron scattering (NS, black open square)56 are also included. Dashed lines: interpolation function of τα(T) vertically shifted to intersect the τt and τe data points.

the entanglement time τe(T). First, we attend to τα(T), which is gained from the construction of the master curves χ″DD(ωτα) for all M of PEP. The FC NMR data (black solid symbols) is complemented by such from solid-state 2H NMR11 and from

Figure 8. (a) Dispersion of the spin-lattice relaxation rate of PEP-200k as measured at temperatures indicated. (b) Susceptibility master curves χ″intra(ωτα) and χ″inter(ωτα) of PEP-200k obtained from χ″DD(ωτα) applying an isotope dilution experiment. H


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Figure 9. Storage modulus G′(ν) (a) and loss modulus G″(ν) (b) of PEP-80k measured within 213 K ≤ T ≤ 323 K. In part b, the intersection points between G″ and G′ (dashed lines) used to determine τα (at 213 K) and τt (at 273 K), respectively, are indicated, as an example.

6. CONCLUSIONS Transforming the 1H FC NMR relaxation data of PEP of different M to the susceptibility representation and applying FTS, master curves of the dipolar correlation function CDD(t) were constructed which extend up to six decades in amplitude and eight in time. Depending on M, they cover all the five relaxation regimes of a polymer melt. In addition to the αprocess (0) and the terminal relaxation (IV), which are immanent to all polymers (or liquids), three polymer specific regimes (Rouse, constraint Rouse, and reptation) are found. Very similar correlation functions are observed for PB studied previously. The corresponding power-law exponents are rather close to those predicted by the TR model for the segmental translation, i.e., intermolecular relaxation largely dictates the shape of the total correlation function CDD(t), in particular in regime II (constraint Rouse) and beyond. The separation of intra- and intermolecular relaxation by carrying out an isotope dilution experiment for PEP yields C2(t) ∝ t‑0.50±0.05 for the reorientational correlation function in regime II and Ctrans(t) ∝ t‑0.30±0.05 for the translational one. The exponent of C2(t) is at variance with the prediction of the TR model (εIITR= 0.25). The fact that translation appears to conform to the TR model, while segmental reorientation does not, now confirmed for the two polymers PEP and PB, challenges de Gennes’ return-toorigin hypothesis which assumes strong rotational-coupling in a polymer melt in the frame of TR model.

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The polymers PEP-29k and PEP-80k were also investigated by oscillatory shear-stress experiments, i.e., the complex shear modulus G*(ω) was measured using an Anton Paar MCR-500 rheometer at the University of Bayreuth (Lehrstuhl für Technische Mechanik and Strömungsmechanik). It provides a frequency window of 0.003 Hz ≤ ν = ω/2π ≤ 30 Hz. A broad temperature range was covered from high temperatures where the terminal relaxation is observed down to temperatures as low as the glass transition temperature Tg. Figure 9 gives an example showing the storage G′(ν) (Figure 9a) and the loss modulus G″(ν) (Figure 9b) of PEP-80k. We determined τα from the intersection point between G′(ν) and G″(ν) occurring at lowest T. This point is indicated in Figure 9b for T = 213 K as an example, for which both, G″ as well as G′ (dashed lines) are shown. Likewise, the low-frequency intersection between the G′(ν) and G″(ν) data observed at high T provides a measure for the terminal relaxation time τt.22 This is exemplified in Figure 9b for T = 273 K. The extracted temperature dependences τα(T) and τt(T) are included in Figure 7 and compared to data from FC NMR, solid state 2H NMR, and dielectric spectroscopy (cf. below). Finally we present the four-parameter formula58 which was applied to interpolate the temperature dependence of the time constant τα(T). ⎧ ⎫ ⎡ ⎛ ⎛T ⎞ ⎞ ⎤⎪ ⎪E τα(T ) = τ∞ exp⎨ ∞ ⎢1 + exp⎜⎜ −λ⎜ − 1⎟⎟⎟⎥⎬ ⎪ T ⎢ ⎪ ⎠⎠⎥⎦⎭ ⎝ ⎝ Ta ⎩ ⎣

APPENDIX

( τs ) = −16.08, E

with the parameter referring to τα(T): log

Figure 8a shows the original spin-lattice relaxation rate of PEP200k as a function of frequency for different temperatures. The majority of the data was collected in Bayreuth with a FFC 2000 relaxometer. In some cases the dispersion experiments were extended to lower frequencies employing a home-built relaxometer operated in Darmstadt (cf. Experimental Section). Figure 8b compares the different relaxation contributions in terms of the master curves for the susceptibilities χ″DD(ωτα), χ″intra(ωτα) and χ″inter(ωτα). In regime 0, the intra- and the intermolecular contribution are barely distinguishable and the noisy interdata is not shown here. As for Fourier transformations the full functional form is required, the missing αpeak in χ″inter(ωτα) was extrapolated using the total relaxation data (dashed line in Figure 8b).

= 5078 K, λ = 9.43, and Ta = 200.6 K.

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AUTHOR INFORMATION

Corresponding Author

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

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors express their gratitude to D. Richter (Forschungszentrum Jülich, Germany) for providing the sample material of PEP. The work was financially supported by the I


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Deutsche Forschungsgemeinschaft (DFG) through Grants RO 907/17 and FU 308/17.

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