Time-Domain NMR Observation of Entangled Polymer Dynamics

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Time-Domain NMR Observation of Entangled Polymer Dynamics: Focus on All Tube-Model Regimes, Chain Center, and Matrix Effects Marie-Luise Trutschel,† Anton Mordvinkin,† Filipe Furtado,† Lutz Willner,‡ and Kay Saalwac̈ hter*,† †

Institut für Physik − NMR, Martin-Luther-Universität Halle-Wittenberg, Betty-Heimann-Str. 7, D-06120 Halle, Germany Institute of Complex Systems, Forschungszentrum Jülich, D-52425 Jülich, Germany

‡

ABSTRACT: Proton multiple-quantum time-domain NMR combined with time−temperature superposition is a powerful method to study entangled chain dynamics. Overcoming the previous limitation to regimes II−IV of the tube model, this study extends the method to regime I (localized Rouse motions) by use of a pulse sequence adapted to shorter times, thus covering all relevant regimes for the model case of poly(butadiene) with molecular weights (M) between 10 and 200 kDa. We determine a value for the entanglement time that is consistent with current rheological results and confirm a value below 1 for the time scaling exponent of the segmental orientation autocorrelation function (OACF) in regime I previously observed by other NMR techniques. The origins of deviations from tube-model predictions are assessed by focusing on the dynamics of the chain centers in end-chain deuterated triblock samples and by dilution of probe chains to 15% in a deuterated matrix with M of 2 MDa. The study is complemented by self-diffusion coefficients measured by pulsed-gradient NMR. Our OACF-based results for the terminal time reinforce the current consensus that below a chain length of 30−50 entanglements matrix effects cannot explain the nonideal M-scaling exponent of 3.4 but are responsible for an M-independent slowdown. The protracted approach of the pure local-reptation scaling in regime II is found to be only somewhat reduced for both chain centers and chains in a long matrix, confirming its generic intrachain origin. These microscopic insights could be compared with results from large-scale computer simulations and provide a gauge for theoretical approaches such as those dealing with constraint release and contour-length fluctuations. observables such as the time-dependent modulus G(t), τd, or the long-time translational self-diffusion coefficient of the center of mass (Dcm). For instance, Doi demonstrated that an additional process related to chain-end dynamics termed contour length fluctuations (CLF) can explain the wellknown exponent in M3.4 deviating from the predicted M3 dependence of τd as well as the zero-shear viscosity.8 Constraint release (CR), accounting for stress relaxation beyond a “fixedtube” assumption through dynamics of the chains surrounding a given one, is another process that has been invoked early on to improve tube-model predictions.9 It can be readily studied in mixtures of shorter chains with very long matrix chains.10 For instance, CR has been shown to be able to account for the M−2.3 rather than M−2 dependence of Dcm.11−13 Considering that disentanglement and viscosity are related to self-diffusion, it is reasonable to assume that CR also plays a role in the M3.4 scaling of the viscosity as well as τd. To illustrate the state of discussion, no change in the M scaling exponent away from 3.4 was first observed for τd of

I. INTRODUCTION The melt-phase dynamics of entangled polymer chains with molecular weight (M) much larger than the entanglement molecular weight Me extends over a large time range with regimes characteristic of rather different types of motional processes within and between the polymer chains. In the limit of short times, where constraints exerted by neighboring chains are not yet relevant, the localized segmental motions can be successfully described by the well-known Rouse model.1 With the onset of interchain restriction effects, a modified picture is necessary, incorporating the constraints and the resulting cooperativity in the bulk melt. This was addressed in the theory of Doi and Edwards,2−5 which incorporates the pioneering tube model and reptation concepts of Edwards6 and de Gennes,7 resulting in four dynamic regimes separated by three regime transition times: I (Rouse)−τe (entanglement time)−II (constrained Rouse/local reptation)−τR (Rouse time)−III (reptation)−τd (disentanglement time)−IV (terminal dynamics/free diffusion). While capturing the microscopic dynamics and the macroscopic behavior qualitatively well, the Doi−Edwards model in its simplest form has shortcomings when it comes to quantitative predictions, prominently the M dependence of © XXXX American Chemical Society

Received: March 1, 2018 Revised: May 14, 2018

A

DOI: 10.1021/acs.macromol.8b00443 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Mainly, two different classes of mostly 1H NMR experiments have been used in this context: first, fast field-cycling (FFC) relaxometry at different Larmor frequencies ωL probing the Fourier transform of the segmental OACF21−23 and timedomain techniques, most prominently multiple- or doublequantum (MQ or DQ) NMR, probing the OACF directly.24−28 More recently, the use of even simpler Hahn- and solid-echobased techniques has also been demonstrated.29,30 FFC NMR, as limited to Larmor frequencies of at least several kilohertz on commercial instruments, is most readily applied to probe unentangled and moderatly entangled chains, while timedomain techniques can probe τd up to Z of around 100. Both methods combined were demonstrated to provide a measure of the OACF coveringand thus confirmingall tube-model regimes.25,26 This includes the signature of CR effects, which unexpectedly showed up already in the time scaling exponent of the OACF in regime II.20,25 This early favorable comparison of two NMR techniques turned out to be somewhat fortuitous in that FFC NMR was shown to be strongly affected by interchain dipole−dipole couplings. This opened avenues to extract mean-square displacements (and thus segmental Dcm values in the subdiffusive regimes I−III) in the low-ωL limit from isotopic dilution experiments in FFC NMR22,31,32 as well as timedomain techniques29,30 and to essentially confirm tube-model predictions over vast length and time scale ranges. For reasons not clear to date, however, the intramolecular part of the OACF also extracted from FFC NMR and isotope dilution (as well as 2 H-based results focusing on quadrupolar coupling) provides results that are at variance with tube-model predictions in terms of too large time-scaling exponents in regime II.22,23 In contrast, as also assessed by isotope dilution and 2H-based experiments, MQ NMR provides a faithful measure of the shape of the segmental OACF28 that is in tune with tube-model predictions and is further supported by computer simulations at rather different levels of coarse graining.20,28 As the interchain coupling effect appears to only affect our results through an adjustable prefactor of the OACF,28 we will not discuss it further in the present study. In this contribution, taking PB as a model polymer studied extensively in earlier NMR works,21−28,32 we first address a shortcoming of MQ NMR as it was up to now limited to the study of dynamics in regimes II−IV. As the amplitude of the OACF is a measure of the overall strength of the measured dipole−dipole couplings, it was not possible to probe the stronger couplings in regime I with the pulse sequence used so far. Here, we introduce the use of an even simplified MQ NMR experiment to probe also regime I and thus confirm previous findings from FFC NMR on the Rouse dynamics in regime I. Our main focus are the changes of the dynamics when a probe chain of largely variable M is diluted in very long and “invisible”, i.e., deuterated, matrix chains, or when the detectable protonated part of the chain is restricted to the chain center. Our results complement preliminary results from MQ NMR25,28 and FFC NMR23 and provide a comprehensive picture of the chain dynamics in these situations, covering all regimes of the tube model and highlighting the role of matrix effects (CR).

short chains in mixtures with long matrix chains as extracted from rheology experiments,14 while later work of Liu et al. using such “probe rheology”10 as well as measurements of Dcm in long matrices11−13,15 seem to agree on lower exponents and thus on the relevance of CR for the M scaling of both quantities. A closer analysis has revealed that the CR effect on Dcm is most pronounced due to the absence of CLF effects on this quantity,13 and it has been argued that the 3.4 exponent of viscosity or τd has a relatively larger contribution from CLF at lower molecular weights with around Z = M/Me ≈ 10 entangled units.13 Yet, this conclusion has been questioned since the “probe rheology” findings10 support a dominance of CR, with τd ∼ M3.1 even in this range. Besides pulsed-gradient NMR to measure Dcm, methods focusing on the microscopic relaxations of the actual chains play an important role in gaining deeper insights and in evaluating theoretical models. For instance, neutron spin-echo experiments were instrumental in proving that CLF of matrix chains causes the CR effect on a test chain,16 demonstrating the subtle interplay between the two processes. Using dielectric spectroscopy (DS) of a polymer displaying a normal mode related to τd (polyisoprene, PI) diluted in a “neutral” polybutadiene (PB) matrix, Adachi and co-workers were the first to show that the M scaling of τd depends on matrix length, reaching a value of 3 in the long-matrix limit. To address potential shortcomings related to the miscibility of different polymers, Watanabe and coworkers have more recently revisited these findings by studying bidisperse PI−PI blends by both DS and rheology,17 demonstrating that (i) CR leads to an overall acceleration of chain dynamics that is most pronounced at lower molecular weight and (ii) the lack of CR processes in long matrices enables the observation of a transition from M3.4 to M3 scaling when Zprobe ≈ 10. They have further demonstrated that after closer inspection the data of Liu et al.10 is in agreement with their findings. In conclusion, CLF is also identified as the leading contribution to the 3.4 exponent in moderately entangled melts. Similar conclusions have somewhat earlier been drawn from detailed comparisons of dielectric and rheological mode distributions measured on homopolymers.18 Clearly, only with an extremely long matrix of the same polymer as compared to the test chain can one investigate a sufficiently large molecular weight range of test chains with fully suppressed CR. The combination of rheology and normal-mode DS is appealing because the normal mode reflects the reorientation of the chain’s end-to-end vector, which is closely associated with the return-to-origin and tube-survival probabilities (ΨRTO(t) and Ψtube(t), respectively) of a Rouse chain undergoing curvilinear diffusion, as calculated in all tube-reptation models and being the basis of the prediction of rheological behavior.5 As early on noted by de Gennes,7 NMR spectroscopy is ideally suited to study details of the reptation model on a molecular level. It is sensitive to the segmental orientation autocorrelation function (OACF) through the dominance of orientationdependent tensorial interactions (mostly dipole−dipole couplings for 1H NMR) and thus directly probes19 ΨRTO(t) (regime II) and ΨRTO(t)×Ψtube(t) (regimes III and IV). Simulation work has revealed that the second-order tensorial quantity probed by NMR, in contrast to the vectorial quantity probed by DS, is particularly sensitive to CR effects.20 Consequently, NMR spectroscopy (excluding the abovementioned pulsed-gradient diffusometry) provides an ideal approach to “molecular rheology”.

II. MATERIALS AND METHODS Samples. Narrowly dispersed 1,4-(cis/trans)-PB samples with a vinyl content between about 5 and 8% were purchased from Polymer Standards Service GmbH (PSS, Mainz, Germany). The sample names B


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Macromolecules Table 1. Characteristics of the Investigated Samples (References Are Given in Parentheses) sample name PB 10k PB 18k PB 24k PB 30k (PB 35k) (PB 47k) PB 55k PB 87k PB 196k PB 2M d-PB 24k d-PB 30k d-PB 50k d-PB 2M DHD 5k/20k/4k DHD 13k/6k/10k DHD 26k/4k/26k

Mw (kDa) 9.5 18.0 23.6 30.7 34.4 47.0 55.2 87.5 196 2190 22.8 32.9 50.3 2000 4.7 + 19.6 + 4.0 = 28.3 13.2 + 6.2 + 10.1 = 29.5 26.4 + 3.5 + 26.2 = 56.1

PDI

Z= Mw/Me

1.02 1.05 1.01 1.01 1.03 1.04 1.03 1.05 1.02 1.04 1.02 1.06 1.02 1.4 1.02

4.9 9.3 12.2 15.9 17.8 24.4 28.6 45.3 102 1135 11.8 17.0 26.1 1036 14.7

1.04

15.3

1.03

29.1

Tg (±1 K) 17623 17723 182 18723 183 184 181

(e.g., PB 10k) encode the weight-average molecular weight Mw in kDa, as collected in Table 1, where also the number of entangled units Z = Mw/Me with Me = 1.93 kg/mol is indicated. Deuterated samples d-PB 30k and 2 M of similar microstructure were purchased from Polymer Source, Inc. (Dorval, Canada). The three triblock polymers with deuterated end blocks (DHD triblock samples) were custom-prepared at the Forschungszentrum Jülich (see refs 23 and 33 for details). The microstructure was characterized by high-resolution 1H NMR34 using a Bruker 600 MHz Avance II instrument using d-THF (most h-PB samples) and d-chloroform (PB 196k and d-PB) as solvents; d-PB samples were characterized by 2H NMR with proton decoupling. Most samples contain about 40% 1,4-cis and 55% 1,4-trans units. Only PB 24k and PB 2M have about 50% 1,4-cis and 45% 1,4-trans units. For some samples the glass transition temperature was measured by DSC on 4−8 mg of sample in aluminum pans using a Netzsch DSC 204F1 Phoenix instrument at a heating rate of 10 K/min (see also Table 1). See ref 28 for a discussion of small Tg variations, specifically also for deuterated samples, and their minor effect on the time− temperature superposition (TTS) procedure described below. A few samples (bracketed in Table 1) had a higher vinyl content and correspondingly increased Tg, causing potential problems with TTS. These samples were thus not studied in bulk. Mixtures of the h-PB samples at 15% in 85% d-PB 2M (and in similar M in some cases) were prepared by dissolving 7.5 and 42.5 mg, respectively, in toluene (Roth, Germany). The solvent was then evaporated under nitrogen gas flow over 3 days. The mixtures were finally dried for 3 days under reduced pressure (0.5 bar) in an oven at at 50 °C. The samples were finally placed in 5 mm NMR tubes and flame-sealed under vacuum. Time-Domain NMR. The chain dynamics was characterized by multiple-quantum (MQ) NMR following previously published procedures.25,26,28,35 As opposed to our earlier works using low-field NMR,25,26 in order to account for the lower sensitivity of 1H isotopediluted samples, the experiments were performed on Bruker Avance III spectrometers at 1H Larmor frequencies of 200 and 400 MHz (B0 = 4.7 and 9.5 T, respectively) using static probes providing 90° pulses of 2−3 μs length, with a temperature control based upon a heated or cooled air or nitrogen flow in range between 223 and 400 K with an accuracy of about 0.5 K. The MQ experiment used earlier was based on the Baum−Pines (BP) double-quantum (DQ) excitation/reconversion pulse sequence segment36 having a minimum cycle time of about 0.1 ms. Using appropriate four-step phase cycles, the experiment yields an intensity buildup curve dominated by DQ coherences (IDQ) and reference decay

Figure 1. (a) Pulse sequence for MQ NMR based on (apart from the phase cycling) identical pulse sequence blocks for DQ excitation and reconversion, showing different excitation schemes. The gray and white boxes represent 90° and 180° pulses, respectively. (b) Comparison of measured DQ buildup and the reference intensities using the 3- and 2-pulse segments. curve (Iref) as a function of pulse sequence time τDQ, which is incremented in small steps. Here, in order to access shorter times and thus regime I dynamics, we extend a previous idea28,37 of using a less well compensated but shorter pulse sequence (see Figure 1a). Previously, mainly to access higher coupling values, we used the simpler three-pulse segment for DQ excitation and reconversion,38 consisting only of the DQ evolution delay τDQ flanked by two 90° pulses, with a refocusing 180° pulse in the center for chemical shift and offset compensation. Here, working on-resonance with well-shimmed samples, it turned out to be possible to even skip the 180° pulse, further minimizing the smallest accessible τDQ value down to 2 μs. The remaining two pulses have opposite phases, and on top of the DQ/ref phase cycle an additional two-step inversion phase cycle was applied for the final read-out pulse. The z-storage delay before it was set to 1 ms, long enough to dephase unwanted coherences and short enough to prevent magnetization buildup by T1 relaxation. The performances of the two simplified experiments are compared in Figure 1b, where the lack of shift compensation is seen to mainly affect the long-time decay, as expected. The experiment based upon the 2-pulse segment was used to probe regime I and the transition to regime II, while longer times were covered by the BP sequence. Matching the τDQ-based time axes requires a few considerations. Any specific pulse sequence for DQ excitation is characterized by a specific scaling factor for the probed dipolar coupling, which is proportional to the square root of the OACF of the dipole−dipole coupling tensor. For the BP sequence it was calculated by Average Hamiltonian theory (AHT) to be a(tp) < 1, depending on the 90° pulse length tp, and since it is not far from unity, it is always absorbed into the time axis.39 As to the potential interference of the time scale of chain motion with the variable C


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Figure 2. Construction of the OACF. Left panel: InDQ measured for PB 55k for three temperatures using the 2-pulse segment, including regime I at 223 K and the regime I/II transition at 248 K. Middle panel: same data divided by τ2DQ (the gray points are omitted). Right panel: OACF through time−temperature-superposition of the same data (here the gray points are from additional temperatures). Note that longer times are covered by the BP sequence. In order to probe the complete C(t) over many decades in time, time−temperature superposition (TTS) must be applied. This is achieved by dividing τDQ by the entanglement time, t/τe = τDQ/τe(T), where

interpulse spacings in the BP sequence with a given cycle time, we have demonstrated earlier that the results are largely independent of the cycle time, proving the validity of the AHT treatment.40 Since AHT applies only for cyclic pulse sequences, we have assessed the exact relation between pulse separation Δ in the 2-pulse segment and the true τDQ on the basis of spin dynamics simulations using the SIMPSON software.41 It turned out that each pulse contributes half its length to the effective evolution time, as also tested by experiments on rigid dipolar solids, comparing DQ buildup and time-domain analysis of the free-induction decay.42 In previous work concerned with (quasi)static (residual) dipolar couplings in networks,37 we have absorbed another factor of 3/2 in τDQ of the 2-pulse segment. This takes into account for the 1.5 times higher efficiency of the 2-pulse segment lacking the scaling factor of 2/3 of the DQ Hamiltonian of the BP sequence35 and allowed us to represent results from both experiments on a common time axis and to use the same fitting functions. This is not possible in the present case of ongoing dynamics in real time. We have thus matched the data from the two experiments by adjusting the y intensity scale (see below). The NMR experiments provide a direct time-domain measure of C(t), the segmental OACF24−28

τe(T ) ≃

(2)

ξ(T) = ξ0 exp{[α(T − TV)]−1} is the monomeric friction coefficient (α = 7.1 × 10−4 K−1, ξ0 = 1 × 10−14 N s/m44), b = 0.96 nm is the Kuhn segment length, κ is a model-dependent numerical factor (we use κ = π), Ne = 18 is the number of segments in an entangled strand, and TV = 126 K is the Vogel temperature.44 These data analysis steps, starting from normalized nDQ buildup curves measured at different temperatures, are illustrated in Figure 2. Pulse-Gradient NMR Diffusometry. 1H pulsed-gradient stimulated-echo (PGSTE) NMR experiments45−47 have been conducted at 70 °C on a Bruker Avance II 400 MHz instrument equipped with a DIFF60 gradient probe for samples in 5 mm tubes, capable of delivering gradients up to 3000 G/cm. The length δ of the variableamplitude gradient pulses was always between 5 and 7 ms, and the fixed diffusion time Δ was varied between 200 and 700 ms to ensure consistent results. For a single diffusing species, the PGSTE intensity decay follows I ∼ exp{−DcmΘ}, where Dcm is the self-diffusion coefficient and Θ = g2γH2δ2(Δ − δ/3) the diffusion function with g the gradient strength and γH the proton magnetogyric ratio.47 We always observed a fast initial decay in the range of 10−30% of the whole signal associated with a very small amount of low-molecular-weight species (e.g., residual solvent), which is considerably overemphasized by the very large T2-related losses of the polymer signal during encoding. The main part of the measured intensity decays did not deviate significantly from single exponentials and were fitted accordingly.

C(t ) = 5⟨P2(cos θ(t + τ ))P2(cos θ(τ ))⟩τ ,ens 2 = AI nDQ (t = τDQ )/τDQ

ξ(T )b2Ne 2 = τs(T )Ne 2 κkBT

(1)

where θ is the segmental orientation relative to the external magnetic field. Here, InDQ(τDQ) = IDQ(τDQ)/[IDQ(τDQ) + Iref(τDQ)] is a normalized signal function that is directly derived from the experimental signal functions, being corrected for incoherent relaxation effects. The second line of eq 1 represents a short-time approximation,24 the validity of which was tested previously.27 The prefactor A depends quadratically on the efficiency of the specific pulse sequence and the effective intrasegmental dipolar coupling value.26 In previous work, a slowly decaying tail contribution from isotropically mobile parts (mostly chain ends) was always subtracted from Iref(τDQ) prior to calculation of InDQ(τDQ).26 Since the long-time performance of the experiment based upon the 2-pulse segment is not good enough for reliable tail subtraction, we have in this work refrained from doing this correction. This means that the results represent an average over the whole chain. The good agreement of the present results with those from previous work26 demonstrates the small influence of the tail correction. The only significant change was a reduction of the extracted apparent disentanglement times by a factor of about 2, without appreciable change in the M dependence. In agreement with earlier work of Cohen-Addad,43 even the DHD triblock samples exhibited a significant tail fraction, indicating its nontrivial origin, being not only due to the unentangled chain ends (see ref 26 for a detailed discussion).

III. RESULTS AND DISCUSSION Self-Diffusion in the Terminal Regime. Long-time selfdiffusion coefficients Dcm have been measured with the help of 1 H pulsed-gradient stimulated-echo (PGSTE) experiments,45−47 which are well-established for the study the largescale translational dynamics of entangled polymers on the micrometer scale.15,48 Here, we mainly seek to characterize our samples and ensure that we can reproduce the well-known scaling relations Dcm ∼ M−2.3 measured in bulk homopolymers5,15 and Dcm ∼ M−2 as predicted by the tube model5 and confirmed experimentally for chains in a long matrix, i.e., excluding CR effects.5,15 The data in Figure 3 for the bulk melts are in rough agreement with earlier results.15,48 The M dependencies also D


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Figure 3. NMR-based long-time self-diffusion coefficients of PB in bulk and in the long matrix at 70 °C vs the molecular weight in terms of the number of entangled units Z in double-log representation. Note that the slopes (power-law exponents) are consistent with the established values of −2.315,49 and −2,5,15 respectively, after consideration of polydispersity-related bias effects.50

essentially confirm the previous results in terms of a notable slowdown of diffusion and weaker M-scaling in the long matrix, especially for shorter tracer chains. Yet, our scaling exponents are systematically too large by about 0.3. On the one hand, such deviations can find their origin in systematic variations in Tg, requiring the comparison of data measured at adapted temperatures to ensure “isofrictional” conditions.51 This effect should be weak for our sample series. On the other hand, the likely most relevant source of deviation is a polydispersity effect50 that was neglected in our data analysis. It leads to a bias toward too high Dcm values especially for samples with lower M. Shorter chains have a larger fraction of signal components with longer T2 relaxation time, which are commonly associated with chain ends.26 These preferentially survive the gradient encoding acting as a T2 filter, resulting in a signal that is entirely dominated by these signal components, resulting in a strong bias toward the shortest chains. Segmental Orientation Dynamics across All TubeModel Regimes. Using the combination of the short 2-pulse segment and the BP sequence, we could measure C(t/τe) covering all regimes of the tube model; see Figure 4 for representative results of the investigated sample series. In the now observed regime I, we can confirm a time scaling exponent whose absolute value is somewhat lower than unity as predicted for a Rouse chain,20 in harmony with previous observations by FFC NMR.21−23 A part of the deviation is likely due to chain stiffness and the correspondingly limited number of Rouse modes, as confirmed by computer simulations.20 The data for the center-labeled triblock samples in Figure 4a appear to blend in well with their fully protonated counterparts, and the subtle differences will be discussed below. Effects of dilution in the long matrix can be appreciated in the comparison of bulk and diluted samples in Figure 4b, where the most obvious change is an increase in the terminal relaxation time τd, as also discussed in more detail below. In regime I, quite some scatter among the samples is observed, which we attribute to uncertainties related to fluctuations in experimental temperature and small T g variations (in view of the small nonsystematic variations related to differences in the microstructure, we assumed a constant literature value for TV in the shifting procedure). Since τe(T) used for shifting varies more strongly toward lower T, the systematic errors are correspondingly larger. Also, one has to

Figure 4. Segmental OACF master curves, C(t/τe), of (a) selected bulk h-PB samples and the three DHD triblocks samples and (b) 2 bulk h-PB samples compared with the same polymers diluted to 15% in a long matrix. The dotted vertical lines indicate the transition from regime I to II at t/τe = τDQ/τe(T) = 0.56 (see below). Following previous arguments,25,26 a model-based estimate for the prefactor A in the second line of eq 1 involving intrasegmental averaging of intramonomer couplings52 has been included so as to roughly normalize C(t/τe=1) to a value of 0.002. Absolute-value differences arising from isotope dilution (inter- vs intramolecular dipolar couplings) are also accounted for.28

keep in mind that C(t) ∝ InDQ(τDQ)/τDQ2, so small errors in InDQ, e.g., related to background signal, are much amplified at the small τDQ in regime I. To further analyze the shape of C(t/τe) and to compare with model predictions and previous findings, we focus on apparent power-law exponents. These are obtained by working in logarithmic units and employing a derivative analysis after appropriate smoothing using the Origin 7.5 software (see Figure 5 for sample data). The regime I exponents α represent averages over about half a decade in time during which the

Figure 5. Exemplary analysis of the segmental OACF in terms of power-law exponents (= log−log slopes) α (regime I) and ε (regime II) for the two samples with lowest M and thus only marginally developed regime II. An experimental value for τe can be estimated on the basis of the intercept when both slopes are known. E


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Macromolecules derivative values are minimal, while ε for regime II is averaged over 0.5−1.5 decades where the slope is maximal (least negative), depending on M. With α and ε fixed, the intercept of the two power laws (lines in log−log units) is taken to represent an experimental value for log(τe,exp/τe(T)). The normalized logarithmic disentanglement time log(τd/τe) was determined by fitting the final decade of the OACF to the theoretical prediction19 (see also ref 26). For some samples not all regimes exist or are experimentally accessible. For the two low-M samples highlighted in Figure 5 there is no well-developed regime III due to early onset of terminal relaxation and transitional effects, which is exemplified by an apparent regime II exponent larger than 0.5, i.e., already larger than the pure-reptation value of 0.5 characteristic of a well-developed regime III. Note that these effects are implicit in the theoretical prediction combining regimes III and IV.19 For PB 196k in the long matrix as well as PB 2M, regime IV was not accessible within our temperature range, and a fit to the regime III/IV theory formula was too uncertain (note the large error bar for bulk PB-196k, see Figure 6). For PB 2M we did not even reach regime III, since the terminal slope only reaches 0.35. Across the whole regime I, we find an α value of around 0.8 as opposed to the Rouse value of unity, as noted above (see Figure 4). Avoiding bias effect related to a flatting of C(t) toward short times, the steepest part of the curve obtained from derivative analysis provides α = 0.81 ± 0.07 across the whole sample series, in good agreement with a value of 0.83 ± 0.01 found in coarse-grained Monte Carlo simulations28 as well as a value of 0.85 measured by FFC NMR.21 Coarse-grained MD simulations provided values of 0.86 for flexible chains and 0.62 for stiffer chains,20 supporting an explantion in terms of a too limited number of Rouse modes in regime I. Despite the high (apparent) values of ε for samples with lower M, the regime I/II transition is always well discernible in terms of a clear increase in the log−log slope (see the dotted vertical line in Figure 5). From simulation work20 it was concluded that the smooth crossover from α in regime I to a variable ε in regime II can impose uncertainties in determining the crossover at τe. We find rather consistent values for all samples with a weakly significant trend toward increasing values upon increasing the molecular weight. This is compatible with crossover issues, but we can conclude that the related

uncertainty is not too large. The experimental mean value of log(τe,exp/τe) for bulk PB and in the long matrix is −0.25 ± 0.2, which is determined at a nominal temperature of −25 °C, where the experimental time interval (the τDQ interval) encompasses τe(T) (see Figure 2). Using eq 2, our result for τe,exp/τe = 10−0.25 = 0.56 can be converted to a value of τe,exp = 5.7 × 10−7 s at 25 °C, which is not significantly larger than the well-informed rheology-based value of τe = (3.7 ± 0.93) × 10−7 s for PB of several architectures and M between 76 and 130 kDa.53 Our microscopic value is thus in better ageement with rheological data than the roughly 40 times higher value estimated by FFC NMR.32,54 This adds a new perspective to the discussion on discrepancies betweeen macroscopic and microscopic approaches.54 In Figure 6 the scaled disentanglement times τd are plotted against the molecular weight. The M scaling exponent of the bulk samples is 3.27 ± 0.2, well in line with our previous studies25,26 and the literature consensus value of about 3.4 derived from rheological studies. The scaling exponent for the mixture of the reference PB in the long matrix is not appreciably different, but a trend toward an overall Mindependent slowdown by about half a decade is readily apparent. Only the highest-M data point supports a trend toward a lower M-scaling in the long matrix and thus matching values beyond about 50 entangled units, as illustrated by the green dash-dotted line. Our finding thus agrees perfectly with recent probe rheology results of different groups10,14,17 as well as a corresponding study of the dielectric normal mode.17 The data are thus compatible with an almost unchanged exponent well above 3 in the lower M range (Z = 5−50) indicating CLF as dominant source of the nonideal scaling,8,13 and CR leading to a speed-up of terminal relaxation independent of M,14 but of course dependent on matrix length. In the higher M range, both data sets seem to converge, and above about Z = 30 a transition to pure-reptation scaling is expected and compatible with the data.55 We can thus contribute in consolidating this state of knowledge with an independent technique. As to the center-labeled triblock samples, the data in Figure 6 demonstrate that the isotropization of the chain center in the two lower M samples occurs with about the same τd as in the full-length polymers for the two 15-fold entangled samples. Only for the longer triblock with Z = 29, having also the shortest labeled center section (6% of all monomers, as opposed to 70% and 21%), do we observe a slight slowdown. According to the theory of the OACF in regimes IV, C(t) is proportional to the product ΨRTO(t) × Ψtube(t) of return-toorigin and tube-survival probabilities,19 where the latter depends on the position in the chain.5 Our fitting approach is based upon the position-averaged result for Ψtube(t)5,19 and is thus expected to provide a larger value when only the very chain center is probed. Because of limited data quality, we refrain from a more detailed data analysis. Finally, we focus on subtle changes in the shape of the segmental OACF in terms of the regime I/II scaling exponents as plotted in Figure 7 as a function of molecular weight in terms of Z. In panel (a) the fit results for the Rouse regime exponent α are collected, demonstrating the considerable scatter addressed above, but clearly confirming values significantly lower than unity. We can notice a weak trend toward lower values for longer chains, which may originate from transitional effects into an increasingly well-developed regime II but also

Figure 6. A log−log plot of the scaled disentanglement times τd/τe obtained from fits to the terminal parts of the segmental OACFs in Figure 4 vs the number of entanglements Z for h-PB, 15% h-PB in 85% long or similar-M matrix, and the center-labeled DHD triblocks. The value for PB 196k (Z = 102) was excluded from the fit due to its large uncertainty. The green dash-dotted line illustrates an expectation based upon literature data.10,14,17 F


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are now put into perspective by using a very long, essentially CR-free matrix. At higher Z, the observed ε are noticeably smaller than the ε of the bulk reference, but the overall approach of the matrix value is still protracted. The earlier approach of ideal behavior is obviously a true effect of “switching off” the CR effect and is in line with computer simulations,20 which have demonstrated the specific sensitivity of the second-order correlation function (as opposed to the first-order one detected by, e.g., dielectric spectroscopy) to CR effects. Lastly addressing the DHD triblock samples, we observe about the same ε values for the labeled chain centers. This is also true for the samples isotope-diluted in a deuterated matrix of similar M. This stands in some contrast to FFC NMR results showing an increase by as much as 0.2 when comparing with fully protonated chains at comparably high isotope dilution to ensure observation of intrachain dynamics.23 Notably, however, on an absolute-value scale our ε results for the chain centers agree within error with those from FFC NMR! For an illustration of this surprising finding, we compare in Figure 8 correlation functions measured by MQ and FFC NMR directly. MQ NMR always provides similar results (up to a factor that depends approximately linearly22,28,56 on the degree of isotope dilution), irrespective of whether a protonated chain center (isotope-diluted intrinsically to about 20%), a bulk, or an isotope-diluted system is probed. In contrast, FFC NMR results match with our ones for protonated bulk samples and the chain center but deviate significantly for the isotope-diluted case in both the regime II slope ε and the apparent terminal time. This leads us to speculate that the FFC NMR results of fully protonated chains (even when isotope diluted) may be significantly biased either toward the more mobile chain ends with their higher ε or through a more subtle averaging effect related to the regime II/III transition19 and the variation of the tube-survival probability Ψtube(t) along a chain.5 The good agreement for the bulk samples may be coincidental, as the lowfrequency (long-time) FFC results are known to be dominated by intermolecular dipolar couplings.22 These options may offer potential explanations for the puzzling discrepancies of FFC results when compared to MQ NMR as well as various kinds of

Figure 7. Time-scaling exponents of the segmental OACFs in Figure 4 for (a) regime I (α) and (b) regime II (ε) as a function of the number of entangled units Z for h-PB, 15% h-PB in 85% long similar-M matrix, and the center-labeled DHD triblocks. The solid and dashed trend lines merely guide the eye; in (a) they are linear fits. Panel (b) includes data from 2H MQ NMR on deuterated PB,28 demonstrating negligible effects from interchain couplings.

indicate weak effects of entanglement constraints on the more local Rouse modes. This could be corroborated by the (again weakly significant) trend toward lower values for short chains when embedded in the long matrix. The latter could of course also have to do with an overall lower level of interchain dipole− dipole couplings. The regime II exponents ε plotted in Figure 7b show less scatter, and to consolidate our statement on the measurement of true orientation fluctuations of a given chain and negligible effects of interchain couplings, we include results for samples isotope diluted in a similar-M matrix and data from 2H MQ NMR on deuterated chains,28 which are free of interchain coupling effects. We essentially confirm previous results on a rather protracted approach of the pure local-reptation scaling exponent of 0.25,21−23,25,26,28 the latter having been first predicted for the NMR observable by de Gennes.7 Recent computer simulations have identified this to be mainly an intrinsic consequence of the specific observable C(t) being a second-order tensorial correlation function; the trend shows that the return-to-origin argument employed in its theoretical treatment is only valid approximately,20 in a way that the predicted value is only reached when regime II covers more than a decade in time. This is obviously the case only for sufficiently long chains with longer Rouse time. In our earliest publication, we have already reported on results of diluting a given sample (PB 55k, Z = 29) in deuterated longer (Z = 102) and shorter (Z = 12) matrices, leading to a decrease and an increase of ε by about 0.05, respectively, resulting in an absolute value of that of the matrix in each case. We interpreted this as a CR effect and supposed that ε could be essentially slaved by the matrix. These results

Figure 8. Direct comparison of segmental OACF master curves from MQ and FFC NMR for samples of similar M: a center-protonated DHD triblock, a bulk, and an isotope-diluted sample, the latter two shifted along y by a decade each for clarity. The FFC data taken from refs 22 and 23 transformed into the time domain and shifted along x by a factor of τe/τs = 7000 to account for the different time references were kindly provided by M. Hofmann. The y amplitudes were scaled to match at C(t/τe=1) = 0.002. G


Macromolecules

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simulations.20,28 More work along these lines is clearly necessary.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (K.S.).

IV. SUMMARY AND CONCLUSIONS Using different pulse sequence blocks for double-quantum excitation/reconversion, adapted to probing larger and smaller motion-averaged dipole−dipole coupling tensors, we were able to extend the range of proton multiple-quantum NMR to probing all regimes of the tube model for entangled polymer melts. On the basis of time−temperature superposition, we have constructed the dipolar orientation autocorrelation function covering segmental fluctuations over up to 8 decades in time and have thus analyzed the microscopic chain dynamics in terms of regime transition times and time scaling exponents, focusing in particular on the effect of dilution in a deuterated long matrix free of constraint-release (CR) effects. In agreement with earlier studies, we find for the Rouse regime a time scaling exponent of around 0.8, i.e., lower than the theory prediction of unity. A weak trend toward lower values for more strongly entangled samples suggests a small influence of the tube constraint also on faster Rouse modes. We further obtain a consistent value for the entanglement time τe in good agreement with previous results from rheology,53 as compared to a 40 times higher value from field-cycling NMR.54 The time scaling exponent in the constrained-Rouse (localreptation) regime approaches de Gennes’ prediction of 0.25 only in a rather protracted way. This trend, in the meantime found to be a specificity of the probed tensorial correlation function,20 persists in the matrix-dilute case, where the expected value is reached somewhat earlier, confirming the sensitivity of this quantity to CR effects.20 The terminal times scale with the molecular weight with an exponent well above 3 in both the bulk and CR-free cases and switching off CR is demonstrated to mainly lead to an overall slowdown. Though, our data are consistent with the terminal times only converging at higher molecular weight, which is in line with recent findings from probe rheology and dielectric spectroscopy, reinforcing the notion that the well-established exponent of 3.4 is mainly due to contour length fluctuations.13,17 In specifically labeled samples, we find that the chain center moves in almost all cases in a similar way as the average over all monomers in a full chain. The only exception is a trend toward slower terminal relaxation of the center of rather long chains, which is expected on the basis of the tube theory of our observable.5,19 Notably, the scaling exponent of the constrained-Rouse regime for the chain center is almost identical to that found by field-cycling NMR in the same samples,23 while for fully labeled chains the latter method gives much higher values beyond the range consistent with tube-model predictions, our results, and different kinds of simulations.20,28 This may indicate a strong bias of field-cycling NMR toward the more mobile chain ends, the origin of which remains to be clarified. In conclusion, we expect that our findings will help to gauge theoretical approaches of rheological behavior with respect to molecular details. They should be especially useful to be compared with data extracted from coarse-grained computer simulations, where recent progress enables the study of segmental fluctuations in highly entangled chains over all relevant time scales.57 We are currently extending our studies to controlled branched structures such as stars and combs with specific isotope labeling of different chain sections.

ORCID

Kay Saalwächter: 0000-0002-6246-4770 Present Address

F.F.: CeNTICentre for Nanotechnology and Smart Materials, Rua Fernando Mesquita, 2785, 4760-034 Vila Nova de Famalicão, Portugal. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank the DFG (grants SA982/3, /9, and /11) for funding of this work and Sophie Reimann (Institut für Chemie, MLU Halle) for conducting the DSC experiments. We acknowledge many stimulating discussions over the years with Ernst Rössler, Axel Herrmann, and Marius Hofmann. We further thank the latter for helpful comments on the manuscript and for providing the data in Figure 8. We acknowledge D. Richter for making the partially deuterated triblock PB samples available to us.

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REFERENCES

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Time-Domain NMR Observation of Entangled Polymer Dynamics

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