Segmental Mean Square Displacement - ACS Publications - American

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Segmental Mean Square Displacement: Field-Cycling 1H Relaxometry vs Neutron Scattering M. Hofmann,† B. Kresse,‡ A. F. Privalov,‡ L. Heymann,∥ L. Willner,§ N. Aksel,∥ N. Fatkullin,⊥ F. Fujara,‡ and E. A. Rössler*,† †

Experimentalphysik II, Universität Bayreuth, D-95440 Bayreuth, Germany Institut für Festkörperphysik, TU Darmstadt, Hochschulstrasse 6, 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: Proton (1H) field-cycling (FC) NMR relaxometry is applied to monitor the crossover in the segmental subdiffusion from the Rouse to the constrained Rouse regime in an entangled linear polymer melt. The method probes the dispersion of the spin−lattice relaxation rate R1(ω). Via Fourier transformation the segmental mean square displacement ⟨r2(t)⟩ is calculated from the intermolecular relaxation contribution R1inter(ω) to the total 1H spin−lattice relaxation dispersion R1(ω). As an example we chose poly(ethylene propylene) (M = 200k), and R1inter(ω) is singled out by performing an isotope dilution experiment. The ⟨r2(t)⟩ data obtained by FC NMR is directly compared to such of neutron scattering (NS) available from the literature. Because of different experimental time windows the NS data is converted to a reference temperature assuming frequency−temperature superposition. Absolute agreement is revealed between FC NMR and NS. The data on ⟨r2(t)⟩ confirm the predictions of the tube-reptation model; i.e., the crossover from Rouse regime to constraint Rouse regime is identified, and the tube diameter is estimated to d ≈ (4.6 ± 0.2) nm. Thus, 1H FC NMR has established itself as an alternative route to access subdiffusion.

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⎧ t 1/2 τ < t < τ Rouse (I) α e ⎪ ⎪ t 1/4 τ < t < τ constraint Rouse (II) ⎪ e R ⟨r 2(t )⟩ ∝ ⎨ ⎪ t 1/2 τR < t < τd reptation (III) ⎪ 1 ⎪t t > τd free diffusion (IV) ⎩

INTRODUCTION

The segmental dynamics in entangled polymer melts is an important subject in soft matter physics. The tube-reptation (TR) model based on the reptation idea of deGennes1 and adapted to polymer melts by Doi and Edwards2 is well established for describing the anomalous segment diffusion of linear, high molar mass (M) chains, i.e. for M > Me, where Me denotes the molar mass between entanglements. The TR model predicts four power-law regimes (I−IV) for the segmental mean-square displacement (msd) ⟨r2(t)⟩, each with a characteristic power-law exponent. Going from short to long times the regime of the local segmental dynamics (0), which actually appears in all kinds of liquids (and is not included in the TR model), is succeeded by the subdiffusive Rouse regime (I). At longer times topological constraints (“entanglements”) confine the lateral segmental motion. Here, the TR model predicts two further subdiffusive power-law regimes, termed the constrained Rouse (II) and the reptation regime (III). At longest times segmental diffusion ultimately becomes normaldiffusive (regime IV). Summarized, the following power-law regimes follow from the TR model:2,3 © XXXX American Chemical Society

(1)

The crossover times are the entanglement time τe initiating the relevance of entanglement effects, the terminal time constant of the Rouse mode spectrum τR, and the disengagement time τd describing the onset of normal (center-of-mass) diffusion. We note that τR and τd display strong M dependence the discussion of which is omitted for the present context as only a single M is studied and regimes III and IV are not reached experimentally. Segmental diffusion is also reflected in the self-part of the dynamic structure factor Sself(q,t) accessible by incoherent quasi-elastic neutron scattering (NS), specifically, by measuring the scattering function Sinc(q,t) of a fully protonated polymer melt with the help of the neutron spin echo (NSE) technique.4 Assuming Gaussian propagation, which describes the initial Received: August 25, 2016 Revised: September 29, 2016

A

DOI: 10.1021/acs.macromol.6b01860 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules slope of Sself(q,t) correctly even in the case of the actual nonGaussian character of tube-reptation of the segmental displacement, Sself(q,t) is related to the msd ⟨r2(t)⟩:3,4 ⎡ q2 r 2(t ) ⎤ ⎥ Sself (q , t ) = Sinc(q , t ) = exp⎢ − 6 ⎣ ⎦

over a large temperature interval, PEP provides the best compromise between avoiding crystallization and introducing a minimum of structural complexity. In contrast, rather simple polymers like PE or poly(ethylene oxide) show a strong tendency to crystallize and thus do not allow to probe the dynamics from high temperatures down to Tg. As we will demonstrate very good agreement is found for the msd derived from NMR and NS, and regarding the msd, the TR model is confirmed

(2)

The typical time window of the NSE technique is about 50 ps < t < 0.1 μs; it is suited to probe the slow polymer dynamics provided that systems with a low glass transition temperature Tg are considered.4 Indeed, first experimental evidence for the predicted transition of ⟨r2(t)⟩ from the Rouse (I) to the constrained Rouse (II) regime in the high-M polymers poly(ethylene) (PE) and poly(ethylene-alt-propylene) (PEP) was reported by Richter and co-workers.5 In poly(vinyl ethylene) also the transition from glassy (0) to Rouse dynamics (I) in the msd was observed via NSE.6 Beyond results from simulations,7,8 these NS data provided strong confidence in the validity of the TR model. Field-gradient (FG) NMR is frequently used to measure diffusion in polymer melts at long times, typically between 10 ms and a few seconds.9 Here, the dynamics of polymers is usually in the hydrodynamic limit (regime IV) with a welldefined diffusion coefficient D or in the transition between regions III and IV. It was pointed out that the stimulated echo amplitude measured in FG NMR is formally identical to Sinc(q,t), when the scattering vector q is identified with gγτ, where g denotes the field gradient, γ the gyromagnetic ratio of the considered spin species and τ the evolution time.10,11 The time scales of FG NMR and NS are complementary. Under favorable conditions, FG NMR is able to advance into the regime of anomalous diffusion, specifically into the reptation regime (III).11,12 In ref 13, regimes II, III, and IV of the TR model are claimed to be identified via pulsed FG NMR on poly(styrene) in semidilute solution.14 Recently, it was demonstrated that 1H NMR relaxometry using the field-cycling (FC) technique provides an alternative access to the msd in polymer melts.15 The method probes the fluctuation of the magnetic dipole−dipole interaction, which in the case of 1H contains intra- and intermolecular contributions. Isolating the latter via isotope dilution experiments yields information on translational motion. Furthermore, using frequency−temperature superposition (FTS), i.e., collecting FC data in a broad temperature range and constructing master curves, the msd of poly(dimethylsiloxane) (PDMS) and poly(butadiene) (PB) were determined by our group over a wide time range, covering regimes I and II in high-M melts.16 Later on, the FC results were complemented by such of FG NMR, together encompassing 10 decades in time.17 Doing so, we were able to identify the entire four polymer power-laws predicted by the TR model (cf. eq 1). Very recently, in another type of NMR experiment exploiting the dipolar correlation effect, 1H NMR spectroscopy confirmed the msd data on PB in regime II obtained by FC NMR.18 Concerning reorientation, however, which is also accessible by FC NMR, previous results questioned the TR model,16,19 in particular its return-to-origin hypothesis which assumes a strong rotational-translational coupling.13,20 The present work extends the analysis of segmental subdiffusion via FC 1H NMR to another polymer PEP, which was thoroughly studied by NS previously, as mentioned.4,5,21−25 Thus, for the first time, a direct comparison between FC NMR and NS will be performed. Studying the collective dynamics

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THEORETICAL BACKGROUND Field-cycling NMR relaxometry measures the frequency dependence (dispersion) of the spin−lattice relaxation rate R1(ω).26−29 The technique “cycles” the magnetic field produced by a resistive electromagnet from a (high) polarization field down to a (low) relaxation field and back to a (high) detection field. Concerning 1H the relaxation rate is additively composed by an intra- and an intermolecular contribution along26,30 R1(ω) = R1intra(ω) + R1inter(ω)

(3)

R1intra(ω)

While reflects reorientational motion of segments, which is of less interest for the present work, R1inter(ω) is sensitive to translational motion of segments from different macromolecules. A separation is achieved in isotopic blends, where fully protonated chains are highly diluted in a matrix of fully deuterated ones (cf. Experimental Section). Then, the intermolecular contribution is suppressed while the intramolecular one prevails.15−17,19,28 From the latter and the total relaxation the intermolecular relaxation rate R1inter(ω) is extracted. We note that highly diluting a protonated polymer with its deuterated counterpart still leaves the heteronuclear (1H−2H) intermolecular relaxation contribution which, however, can be neglected.15,31 The Bloembergen, Purcell and Pound equation connects the relaxation rates R1i(ω) with a spectral density J2i(ω)30 R1i(ω) = K i[J2i (ω) + 4J2i (2ω)]

(4)

i

Here, K denotes the corresponding coupling constant and the superscript distinguishes intra- and intermolecular relaxation. In the case of the intermolecular relaxation here of relevance, the spectral density J2inter(ω) is given by the Fourier transform of the correlation function C inter(t ) = C trans(t ) ∝

∑ spin

3 cos2(ϑij(t )) − 1 rij 3(t )

pairs

×

3 cos2(ϑij(0)) − 1 rij 3(0)

(5)

where the sum includes only pairs of spins with internuclear distance rij and orientation ϑij (at time t and time zero) belonging to different polymers. In the case of intramolecular relaxation not considered here, the sum runs over all spins located on the same polymer segment. In order to allow for the construction of master curves it is convenient to introduce the susceptibility representation of the relaxation data, 16,27,28 explicitly we write χ″ inter(ω) = ωR1inter(ω) which leads to the master curve χ″inter(ωτα) by B


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Figure 1. (a) Spin−lattice relaxation dispersion of neat (protonated) PEP-h10 with M = 200k in the susceptibility representation χ″DD(ω) = ωR1(ω), measured at different temperatures. (b) Susceptibility data χ″intra(ω) of neat 10% PEP-h10 in fully deuterated PEP-d10.

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shifting the individual χ″inter(ω) data sets measured at different temperatures solely along the frequency axis to provide best overlap. Here, τα specifies the structural correlation time controlled by the glass transition (α-process). A relation between the absolute msd ⟨r2(t)⟩ and the intermolecular relaxation rate R1inter(ω) was derived.15−17,32 Expressed in terms of the susceptibility representation it reads: ⟨r 2(t /τα)⟩ =

2 ⎡ 1 ⎢ 5 ⎛ 4π ⎞ 8 1 1 ⎜⎜ ⎟⎟ 2 ⎢⎣ 4 ⎝ μ0 ⎠ 3π 3 γH 4ℏ2ns 1 + 225/16

∫0

∞

⎤−2/3 d(ln ω) cos(ωt )χ ″inter (ωτα)⎥ ⎥ ⎦

EXPERIMENTAL SECTION

For this work, two different FC relaxometers were employed: a commercially available STELAR FFC 2000 machine operated at the University of Bayreuth provides a frequency range of 10 kHz ≤ ω/2π ≤ 20 MHz, and a self-built apparatus called FC-1 and located at the Technical University Darmstadt covers an even broader range of 100 Hz ≤ ω/2π ≤ 30 MHz. For further technical details concerning the FC-1 relaxometer, we refer to previous publications.29,34,35 The intermolecular relaxation rate was obtained via an isotope dilution experiment. An isotopic mixture composed by 10% fully protonated PEP-h10 and 90% of fully deuterated PEP-d10 was investigated via FC 1H NMR and compared to neat, fully protonated PEP-h10. It is assumed that for this low concentration the intermolecular relaxation is essentially suppressed, i.e., Rintra 1 (ω) ≅ (ω). The intermolecular relaxation rate is obtained from Rintra R10% 1 1 (ω) intra = Rtotal 1 (ω) − R1 (ω). The dispersion R1(ω) of neat PEP-h10 as well as that of the isotopic blend was measured in a broad temperature range 400 K > T > 220 K and transformed into the susceptibility representation χ″(ω) = ωR1(ω).16,17,19,27,28 Subsequently, frequency− temperature superposition (FTS) is exploited to cover an effectively larger frequency window compared to a single experiment, a procedure well-known from rheology, for example.36,37 As the segmental relaxation (α-process) is covered by our NMR experiments the susceptibility master curve is scaled according to χ″(ωτα) and τα(T) values were obtained. Protonated and deuterated PEP were provided by the Richter group (Forschungszentrum Jülich, Germany). It is synthesized with a narrow M distribution from poly(isoprene) via hydrogenation.22,23 Both PEP samples had M = 200k and are in the entangled state, as the critical molar mass Mc ≈ 3 k is exceeded significantly.38 Homogeneous mixing of the isotopic blends was ensured by preparing solutions with chloroform and subsequent removal of the solvent under vacuum for several days. Phase separation would yield nonexponential magnetization recovery which was never observed. The spin density ns = nHρNA/M = 6.9 × 1028 m−3, needed for applying eq 4, results from the number of spins per chain nH ≈ 2900 and from the mass density ρ ≈ 0.79 g/cm−3 of PEP.38

(6)

Here, γH denotes the gyromagnetic ratio of the proton. The spin density ns = nHρNA/M depends on the number of spins per chain nH and on the mass density ρ. For the derivation of eq 6 it is assumed that chains move independently of each other and that the propagation of displacements is again Gaussian, as done in NS experiments to obtain eq 2. However, in the entanglement regime dynamics is actually non-Gaussian, yet, the non-Gaussian corrections are minor (cf. Appendix). In practice a numerical integration algorithm based on Filon’s method33 was implemented for carrying out the cosine transformation of eq 6 on a logarithmic time scale. As for integral transformations the full functional form is required, the missing α-peak in χ″inter(ωτα) was extrapolated using the total relaxation data (represented by the dashed line in Figure 2).

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RESULTS Figure 1a displays the dipolar susceptibility data χ″DD(ω) = ωR1(ω) comprising the dispersion of the total 1H relaxation (including intra− and intermolecular contributions) of neat PEP-h10 at different temperatures. Similar data was measured for the isotopic blend with 10% protonated PEP, from which it is assumed to reflect intramolecular relaxation χ″intra(ω) solely (cf. Figure 1b). The original relaxation rates are shown in Figure 5 (Appendix). Inspecting Figure 1a, a maximum is observed at around T = 277 K representing the α-process. At higher temperatures two power-law regimes characteristic for polymer dynamics are apparent and referred to Rouse (regime I) and entanglement dynamics (II). Qualitatively, the same

Figure 2. Susceptibility master curves of neat PEP-h10 (χDD″(ωτα)), of PEP-h10 diluted to 10% in PEP-d10 (χintra″(ωτα)), and of the intermolecular contribution (χinter″(ωτα)). Dynamic regimes (0, I, II) are indicated. C


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NMR requires a common reference temperature with respect to which the time axis is scaled. We chose Tref = 393 K, where we measured R1(ω) down to lowest frequencies and the slowest polymer dynamics is revealed (cf. Figure 1a). The value τα(492 K) is projected using an Arrhenius function fitted to our high-T data (red dashed line in Figure 3). The NSE data on ⟨r2(t)⟩ was horizontally rescaled by the factor τα(393 K)/ τα(492 K) ≈ 12. It is emphasized that no vertical scaling was applied to the NSE data. As no M-dependence in τα(T) is observed (cf. Figure 3) the somewhat different M of PEP-200k (FC NMR) and PEP-80k (NSE), respectively, can be directly compared. Figure 4 displays the msd as obtained by NSE (ref 5) and from our FC measurements after applying eq 6 to χ″inter(ωτα) and scaling the outcome to the reference temperature of Tref = 393 K.

features are apparent in the isotopic blend (Figure 1b). Only at the highest T and at lowest frequencies do differences occur (cf. below). Assuming FTS master curves were constructed by applying temperature dependent frequency shift factors. Figure 2 displays the master curves χ″DD(ωτα), χ″intra(ωτα) and χ″inter(ωτα) = χ″DD(ωτα) − χ″intra(ωτα), after scaling the frequency axis with the time constant τα; the latter was obtained from a fit of the relaxation maximum at T = 277 K with a Cole−Davidson function.39 With respect to measurements at a single temperature the effective frequency window is strongly widened to almost 10 decades and the temperature dependence τα(T) is gained (cf. Figure 3). The master curves of

Figure 3. Temperature dependence of the time constant τα(T) of PEP of different M, as indicated. The data was obtained from FC 1H NMR (black symbols) and shear rheology (open colored symbols). Red star: value of τα(492 K) using an Arrhenius law for extrapolating the hightemperature data (red dashed line). Black solid line: guide to the eye.

Figure 4. Segmental mean squared displacement ⟨r2(t)⟩ at Tref = 393 K of PEP 200k obtained by FC 1H NMR. Corresponding NSE data from Wischnewski et al.5 on PEP-80k is included (shifted horizontally by applying FTS, see text). Relaxation regimes (I, II) are indicated, as well as the forecast of the TR model (dashed lines, cf. eq 1).

the total (neat PEP-h10) and of the intramolecular relaxation (isotopic blend) display similar shapes, except at lowest (reduced) frequencies, where χ″DD(ωτα) features a higher amplitude than χ″intra(ωτα). Around the α-peak the interdata is very noisy and thus omitted as the relative contribution of inter-relaxation becomes very small. Assuming FTS, the absolute msd can be obtained for any reference temperature in the range where the temperature dependence τα(T) is known. Field-cycling NMR master curves were not only constructed for M = 200k but also for the lower M (80k and 3k) and the corresponding τα(T) was gained (cf. Figure 3 and ref 40). In addition, for all M oscillatory shear measurements were carried out as well, down to temperatures close to Tg; details will be reported in another future work. Thus, τα(T) data are also available from rheology and are included in Figure 3 (colored symbols). The results of the two methods, rheology and FC NMR, perfectly overlap over about eight decades, demonstrating the reliability of the τα(T) data as obtained from FC NMR. Super-Arrhenius temperature dependence is found, as typical for glass forming liquids, and no Mdependence of τα is observed. Altogether, the slowdown of the α-process during the glass transition is probed over 15 decades in time, and Tg is determined to about 206 K.41 For the isotopic blend of PEP-200k, τα(T) was also determined by FC NMR for T > 260 K. These data are included in Figure 3 and also agree with the data for the fully protonated melts; i.e., no isotope effect is observed. Applying NSE, Wischnewski et al. measured ⟨r2(t)⟩ of PEP 80k at 492 K, which exceeds the temperature range of our measurements.5 To compare the NSE data with those of FC

Almost perfect agreement is found between the msd data of FC NMR and of NSE in the Rouse regime (I). Even if the Arrhenius extrapolation used in Figure 3 misses the real value of τα(492 K) by a factor of, shall we say two, would not spoil this good agreement among the methods. The regime I exponent ε = 0.5 matches that of the TR model (cf. eq 1). While NSE is restricted to regime I, FC 1H NMR reveals even the constrained Rouse regime (II), as before in the case for high-M PB and PDMS.17 The exponent in regime II is slightly lower than forecast, ⟨r2(t)⟩ ∝ t0.2 rather than ∝ t0.25, a fact which was also observed before in PB.17 We note that for PE, NSE experiments reached regime II, too.5 The almost perfect correspondence between the msd derived from NSE and from FC 1H NMR highly supports the reliability of the FC measurements on the isotopic blend of PEP, and moreover, the whole concept of determining the msd from 1H spin−lattice relaxation dispersion. FC 1H NMR is able to cover the transition from Rouse to entanglement dynamics in polymer melts. Together with analyzing the intramolecular16,19 1

H as well as 2H relaxation dispersion41 yielding information on the segmental reorientation and the close correspondence with rheological results,42 FC NMR becomes a powerful tool of molecular rheology.

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DISCUSSION Within the error margin, both methods, NMR and NS, yield the same msd as a function of time. From the value of the msd at the crossover between Rouse and entanglement dynamics D


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Figure 5. a) Proton relaxation dispersion R1(ν = ω/2π) of fully protonated PEP-h10-200k. (b) Proton relaxation dispersion of the isotopic blend.

occurring at the entanglement time τe the tube diameter d emerging in the TR model can be estimated. Specifically, the characteristic length, where the transition from Rouse dynamics 2 1/2 to entanglement dynamics occurs is a0 = (⟨r (te)⟩) ≈ (2.8 ± 0.2) nm; we extracted it from determining the intersection of the two power-laws (I, II) in Figure 4. A polymer chain of N Kuhn segments with length b performing Rouse dynamics would have a thermal displacement at a time τe = τsNe2 of about

spins from different macromolecules W(ri′⃗ j;ri⃗ j,t).15,17,32 By definition this quantity reflects the probability density that two spins with numbers i and j are separated by a radius vector r′i⃗ j after time t has elapsed, if they were initially separated (t = 0) by the vector ri⃗ j.15 The first approximation is that the propagator merely depends merely on the relative displacement ∼ r ⃗ ij:= ri′⃗ j − ri⃗ j of two spins from different macromolecules during t, hence we substitute W(r′i⃗ j;ri⃗ j,t) = W̃ (r′i⃗ j − ri⃗ j,t).For times t > τs the intermolecular correlation function for isotropic systems takes the asymptotic form:15,17,32

⟨r 2(τe)⟩1/2 = [2b2 /π 3/2(τe/τα)1/2 ]1/2 = 2/π 3/2 d ;2 therefore, the tube diameter d can be estimated to

d = π 3/2/2 ⟨r 2(τe)⟩ ≈ 4.6 nm , which is in good agreement with 4.9 nm derived by Fetters et al. from rheological data at the somewhat higher temperature 413 K.38 Here we introduced the number of Kuhn segments Ne between the entanglements and assumed τs ≅ τα.

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C trans(t ) = ns

4π ̃ W (0, t ) 9

(A1)

∼ Importantly, only the value r ⃗ ij = 0 is relevant; i.e., translational correlation is conserved if the relative displacement remains unchanged or is regained after time t. Using the Gaussian approximation ∼ ∼ W̃ ( rij⃗ , t ) ≅ WG̃ ( rij⃗ , t )

CONCLUSION 1

Proton ( H) field-cycling (FC) NMR relaxometry was applied to monitor the crossover in the msd from the Rouse to the constrained Rouse regime in an entangled poly(ethylene propylene) melt. The segmental msd ⟨r2(t)⟩ was calculated

⎧ 3 r 2̃ ⎫ ⎬ = (2π /3⟨r 2̃ (t )⟩)−3/2 exp⎨− ⎩ 2 ⟨r 2̃ (t )⟩ ⎭

from the intermolecular relaxation contribution to the total 1H spin−lattice relaxation dispersion. The latter was obtained from an isotope dilution experiment which allows to disentangle intra- and intermolecular relaxation contributions. The msd extracted from the NMR data is compared to that of neutron spin echo (NSE) spectroscopy available from the literature. Because of different experimental windows the NSE data is converted to a reference temperature assuming frequency− temperature superposition. Absolute agreement is revealed. The data on ⟨r2(t)⟩ confirm the predictions of the tubereptation model; in particular, the crossover from Rouse regime to constraint Rouse regime is identified. Thus, 1H FC NMR has established itself as an alternative route to access subdiffusion.

(A2)

yields WG̃ (0, t ) = (2π /3⟨r 2̃ (t )⟩)−3/2

(A3)

used for the derivation of eqs 2 and 6. However, the more general quantity W̃ (0,t) can be exactly calculated within the TR model for times τe ≪ t ≪ τt, if one assumes that the single chain moves independently inside the tube and obeys Rouse dynamics.43 Then, the one-dimensional segmental displacement s(t) along the curvilinear contour of the tube (primitive path) is describable by the one-dimensional Gaussian propagator2 ⎧ 1 s2 ⎫ ⎬ Wtube(s , t ) = (2π ⟨s 2(t )⟩)−1/2 exp⎨− ⎩ 2 ⟨s 2(t )⟩ ⎭

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APPENDIX For determining the segmental msd in polymer melts from the intermolecular 1H rate R1inter(ω), the propagator for describing the relative motions of different polymer segments is approximated by a Gaussian (cf. Introduction). However, it is well-known that the Gaussian assumption is not adequate for all times in modeling the single chain motion in the framework of the TR model.43 A key quantity for the derivation of the eq 6, is the probability density which describes relative motions of two

(A4)

The three-dimensional msd is connected to the onedimensional displacement along the primitive path via ⟨r̃2(t)⟩ = 2d ⟨|s(t)|⟩.2 Using eq A4, after some calculation the propagator W̃ rep(r̃,t) in real space differs from eq A3 by a numerical coefficient λ, i.e. W̃ rep(0, t ) = λW̃ G(0, t )

(A5)

However, the correction factor λ E


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25/2 [ π 7/4

∫0

π /2

(16) Herrmann, A.; Kresse, B.; Wohlfahrt, M.; Bauer, I.; Privalov, A. F.; Kruk, D.; Fatkullin, N.; Fujara, F.; Rössler, E. A. Mean Square Displacement and Reorientational Correlation Function in Entangled

[cos(ϕ) + sin(ϕ)]−3/2 dϕ] ≈ 0.8507 (A6)

1

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Combining Field-Cycling and Field-Gradient 1H NMR. Macromolecules 2015, 48, 4491−4502. (18) Lozovoi, A.; Mattea, C.; Herrmann, A.; Rössler, E. A.; Stapf, S.; Fatkullin, N. Communication: Proton NMR dipolar-correlation effect as a method for investigating segmental diffusion in polymer melts. J. Chem. Phys. 2016, 144, 241101. (19) Hofmann, M.; Kresse, B.; Privalov, A. F.; Willner, L.; Fatkullin, N.; Fujara, F.; Rössler, E. A. Field-Cycling NMR Relaxometry Probing the Microscopic Dynamics in Polymer Melts. Macromolecules 2014, 47, 7917−7929. (20) Ball, R. V.; Callaghan, P. T.; Samulski, E. T. A simplified approach to the interpretation of nuclear spin correlations in entangled polymeric liquids. J. Chem. Phys. 1997, 106, 7352−7361. (21) Richter, D.; Farago, B.; Fetters, L. J.; Huang, J. S.; Ewen, B.; Lartigue, C. Direct Observation of the Entanglement Distance in a Polymer Melt. Phys. Rev. Lett. 1990, 64, 1389−1392. (22) Richter, D.; Butera, R.; Fetters, L. J.; Huang, J. S.; Farago, B.; Ewen, B. Entanglement Constraints in Polymer Melts. A Neutron Spin Echo Study. Macromolecules 1992, 25, 6156−6164. (23) Pérez-Aparicio, R.; Arbe, A.; Frick, B.; Colmenero, J.; Willner, L.; Richter, D.; Fetters, L. J. Quasielastic Neutron Scattering Study on the Effect of Blending on the Dynamics of Head-to-Head Poly(propylene) and Poly(ethylene−propylene). Macromolecules 2006, 39, 1060−1072. (24) Pérez-Aparicio, R.; Arbe, A.; Alvarez, F.; Colmenero, J.; Willner, L. Quasielastic Neutron Scattering and Molecular Dynamics Simulation Study on the Structure Factor of Poly(ethylene-altpropylene). Macromolecules 2009, 42, 8271−8285. (25) Pérez-Aparicio, R.; Alvarez, F.; Arbe, A.; Willner, L.; Richter, D.; Falus, P.; Colmenero, J. Chain Dynamics of Unentangled Poly(ethylene-alt-propylene) Melts by Means of Neutron Scattering and Fully Atomistic Molecular Dynamics Simulations. Macromolecules 2011, 44, 3129−3129. (26) Kimmich, R.; Anoardo, E. Field-cycling NMR relaxometry. Prog. Nucl. Magn. Reson. Spectrosc. 2004, 44, 257−320. (27) Kruk, D.; Herrmann, A.; Rössler, E. A. Field-cycling NMR relaxometry of viscous liquids and polymers. Prog. Nucl. Magn. Reson. Spectrosc. 2012, 63, 33−64. (28) Meier, R.; Kruk, D.; Rössler, E. A. Intermolecular Spin Relaxation and Translation Diffusion in Liquids and Polymer Melts:

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (E.A.R.). 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 Deutsche Forschungsgemeinschaft (DFG) through Grants RO 907/17 and FU 308/14.

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Polymer Melts Revealed by Field Cycling H and H NMR Relaxometry. Macromolecules 2012, 45, 6516−6525. (17) Kresse, B.; Hofmann, M.; Privalov, A. F.; Fatkullin, N.; Fujara, F.; Rössler, E. A. All Polymer Diffusion Regimes Covered by

poses a negligible correction. Finally, in Figure 5, we show the relaxation dispersions R1(ω) of fully protonated PEP-h10-200k and of the isotopic blend. We assume that the latter is of purely intramolecular origin.

REFERENCES

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