Microscopic observation of the segmental orientation autocorrelation function for entangled and constrained polymer chains

Polymers, due to their chain structure, exhibit dynamics over a wide range of time scales covering easily more than ten decades. The tube model of entangled polymers provides a microscopic description of the chain dynamics within this large time window,^1,2 thereby distinguishing different regimes with characteristic scaling exponents of the segmental mean-square displacement. Different specialized NMR techniques employing field gradients³ or fast field cycling (FFC) relaxometry,^4–6 most recently also simple low-field transverse relaxometry methods,⁷ as well as neutron spin-echo spectroscopy,⁸ were used to assess the segmental displacements in these regimes. Overall convincing adherence to the tube model was reported, while remaining discrepancies are commonly explained^5,9 with additional relaxation processes of the reptation tube, such as contour-length fluctuations (CLFs) or constraint release (CR).

NMR methods which are more readily applicable on standard or even low-field equipment comprise transverse relaxometry^7,10,11 and related techniques,¹² with the probably most quantitative results provided by multiple-quantum (MQ) NMR.^13–16 Provided dominant orientation-dependent intra-segmental proton dipole-dipole couplings (or a focus on deuteron quadrupolar couplings), MQ NMR,^13–16 as well as many other NMR techniques probe the segmental dynamics in terms of the orientation autocorrelation function (OACF) of the second Legendre polynomial $C ( t ) = 5 ⟨ P 2 ( cos ⁡ θ ( t + τ ) ) P 2 ( cos ⁡ θ ( τ ) ) ⟩ τ , e n s$⁠, where $θ ( t )$ is the instantaneous angle between the segmental orientation and the direction of the external magnetic field, and the brackets $⟨ … ⟩ τ , e n s$ represent the ensemble and time average. The OACF describes the probability to find the monomeric segment in the same orientation after some time t. See Fig. 1 for an illustration of C(t) in terms of tube-model regimes,¹² as elucidated previously.^13–15,17

Recent FFC^4,5 and transverse⁷ relaxometry studies have focused on a separation of inter- and intra-segmental couplings by isotope-dilution experiments, providing potentially independent information on segmental translations and rotations, respectively. Notably, good adherence to tube model predictions was found only for the former, in some contrast to in principle equivalent MQ NMR results^14,15,17 as well as different computer simulations.^17,18 The origin of this discrepancy is yet to be elucidated, calling for a concerted multi-method approach.

In Refs. 13, 15, and 16, it was shown how the OACF, characterized by specific time scaling exponents κ (being the slopes in plots of $log ( C ( t ) ∼ t − κ )$ vs. log t), can be constructed as a master curve by suitable time-temperature superposition (TTS) of the normalized double-quantum (DQ) build-up intensity $I nDQ ( τ DQ ) ∝ C ( t = τ DQ ) ⋅ τ DQ 2 ∝ τ DQ 2 − κ$⁠. The observable $I nDQ ( τ DQ )$ is constructed from the data provided by the MQ NMR experiment, and the proportionality to C(t) is only valid for short times. Therefore, because of the short usable time interval, the TTS procedure is not as robust as for instance in rheology² and necessarily relies on a known shift factor. A fit to a single data set $I nDQ ( τ DQ ) ∝ τ DQ 2 − κ$ is possible in favorable cases,¹⁵ but suffers from experimental noise, as I_nDQ is rather small at small $τ DQ$⁠. In fact, spurious signal contributions or sample inhomogeneities can completely hamper such an isothermal assessment of κ. In this contribution, building upon earlier work on polymer networks,¹⁹ we present a new analytical approach based upon a power-law OACF to extract characteristic exponents directly from MQ NMR signal functions. We illustrate its feasibility on the example of theoretical meta-data and demonstrate its superior robustness by the analysis of previous data for entangled polymer melts as well as homogeneous and inhomogeneous (swollen) elastomers.

The mode of action of the used MQ NMR experiment and the pulse sequence is detailed elsewhere.^20,21 In short, the MQ experiment is based on the pulse sequence developed by Baum and Pines,²² consisting of excitation and reconversion evolution blocks of variable duration $τ DQ$ (DQ evolution time) each and yielding two types of signals: (i) the DQ signal and (ii) the reference signal.

The most relevant analyzed quantity is the DQ build-up signal (I_DQ), which originates from all segments that move anisotropically, i.e., which exhibit residual dipole-dipole (or in the case of ²H MQ NMR residual quadrupole) couplings. It decays at long DQ evolution times due to relaxation effects caused by motions either significantly faster than, or on the time scale of, $τ DQ$⁠. The faster motions mainly pertain to regimes 0 and I, since the experiment is mostly used to probe regime II-IV dynamics (an extension into probing regime-I dynamics is possible and the subject of ongoing work, but poses some experimental challenges and is beyond the present scope).

The reference signal (I_ref) is an intensity complement, i.e., it comprises all signals that have not evolved into DQ coherences. The second relevant quantity for analysis is the sum of the two measured signal functions, i.e., the “MQ sum” signal $I Σ MQ$⁠, which is a refocused relaxation-only function with additional contributions from components characterized by motions that become isotropic at times below $τ DQ$ (chain ends, loops, etc.). These components appear in the $I Σ MQ$ curve in the form of a slowly decaying tail, which can be fitted and must be subtracted. Then, a point-by-point normalization of the DQ signal by the tail-corrected ΣMQ signal removes the mentioned relaxation effects due to fast motions and results in the normalized DQ build-up curve (I_nDQ).

For a dipole-dipole coupled spin pair in ¹H MQ NMR, or for a single ²H nucleus subject to quadrupolar interaction, the aforementioned time- and ensemble-averaged signal functions read

where $ϕ 1 = ϕ ( 0 , τ DQ )$ and $ϕ 2 = ϕ ( τ DQ , 2 τ DQ )$ are the evolution phases for the excitation and reconversion blocks, respectively. These are calculated as

where D_eff represents the intra-segmental static-limit apparent dipole-dipole coupling constant and $| t b − t a | = τ DQ$⁠. From now on, we restrict our arguments to the ¹H case, where multiple couplings render D_eff an effective, 2nd-moment-type quantity. The coupling distribution in such a multi-spin system justifies the use of the Anderson-Weiss (AW or 2nd-moment) approximation,²³ which assumes that the interaction frequency is normally distributed. Eqs. (1) and (2) can thus be simplified and become^19,24

with

and

where $M 2 eff = 9 20 D eff 2$⁠. Different analytical forms of C(t) can now be pre-supposed and implemented to calculate the MQ NMR signal functions. Notably, in our previous work we have used the experimental C(t) master curve obtained by TTS of short-time $I nDQ ( τ DQ )$ data in a suitably interpolated analytical form (mostly piecewise power laws). It was thus possible to quantitatively predict the $I DQ ( τ DQ )$ and $I Σ MQ ( τ DQ )$ signal functions even for longer $τ DQ$⁠,¹⁶ thus validating both the C(t) master curve as well as the AW theory. The question at hand is whether it is really necessary to know the full C(t), see Fig. 1, in order to perform meaningful fits to the signal functions.

In the present work, we seek an improved method to estimate an assumed power-law exponent of the OACF, $C ( t ) ∼ t − κ$⁠. As mentioned in the Introduction, the previous approach based upon an initial-slope analysis, $I nDQ ( τ DQ ) ∝ C ( t = τ DQ ) × τ DQ 2 ∝ τ DQ 2 − κ$⁠, is subject to uncertainties related to small overall signal and the limited data range, requiring TTS to obtain reliable results. Thus, we use Eqs. (5) and (6) in combination with the following specific form of C(t):

where $S b 2$ is the amplitude of the normalized C(t) (i.e., the square of the dynamic order parameter of the backbone) describing the residual anisotropy level at t₀. The result reads¹⁹ 

where D_res = D_effS_b describes the residual dipolar coupling corresponding to the C(t) plateau up to t₀. These equations are valid for $κ > 0$⁠, $κ ≠ 1,2$⁠, and $t 0 < τ DQmin$ and can now be used in a simultaneous fit to separate I_DQ and IΣMQ data sets over a wider time range. The value of t₀ is to be fixed just below the first experimental data point $τ DQmin$⁠. The term $e − 2 τ DQ / T 2$ was additionally introduced in order to account for the effect of the neglected fast molecular motions taking place on time scales below t₀. T₂ is thus the transverse relaxation time associated with these fast modes.

The above result is not new, but rather a modified version of a previous test of AW theory, Eqs. (3)–(6), as applied to the special case of elastomers.¹⁹ Such polymer networks have the special property that the polymer chains are end-fixed, leading to a plateau value $C ( t → ∞ ) > 0$⁠, see Fig. 1. Among the many models brought to test in Ref. 19, a power-law model $C ( t ≥ τ 0 ) = ( 1 − S b , net 2 ) ( t / τ 0 ) − κ + S b , net 2$ was used, assuming $C ( t < τ 0 ) = 1$ (i.e., the action of a time-stable $D eff 2$ for $t < τ 0$⁠). In networks, the long-time plateau corresponds to a residual dipole-dipole coupling D_res = S_b,netD_eff, with the segmental dynamic order parameter of the backbone S_b,net = 3/(5N) providing the link to the microstructure, i.e., the number N of statistical segments between crosslinks.^25,26

OACFs corresponding to such a more specific power-law model including a long-time plateau, which turned out to describe the MQ signal functions and the chain dynamics in networks in the most consistent way, are plotted in Fig. 2(a). For the enlarged transition zone in Fig. 2(b), it is observed that the negative slope in the given log-log plot becomes lower (i.e., the effective power-law exponent decreases) upon approaching the C(t) plateau. The leading idea of the present work is that one could force-fit experimental or analytical MQ NMR results with the analytical solution corresponding to a simple power-law model without a long-time plateau, Eqs. (8) and (9), in order to characterize the average power-law exponent characteristic for the fitted time range.

It is noted that a studied interval of the MQ data comprising $τ DQmin$ and $τ DQmax$ covers a corresponding real-time interval of C(t) that is flanked by twice these values, see Fig. 2(b). This arises from the mere definition of $τ DQ$⁠, being the length of the individual excitation or reconversion pulse sequence block. The second main assumption is that details of dynamics occurring below t₀ can be neglected, i.e., we assume a short-time plateau characterized by a generalized $S b = D res / D eff$⁠, while effects of fast motions are subsumed in an empirical exponential decay term.

The new power-law-model-based signal functions were first tested on simulated meta-data. In Fig. 2(a), five correlation functions were constructed according to the power-law model with plateau specified in Ref. 19, with varying $τ 0$⁠, fixed $κ = 0.9$ (regime I), and the plateau value defined by $D res / 2 π = 0.2$ kHz. The corresponding DQ and ΣMQ signal functions were then generated within the usual Baum/Pines pulse sequence time window between 0.1 and 10 ms, and fitted simultaneously to Eqs. (8) and (9), see Fig. 2(c). For comparison, approximate average exponents and $D res ( t 0 )$ values were also obtained by direct simple power-law fits to the C(t) model curves between $2 τ DQ min$ and $2 τ DQ max$⁠, see Fig. 2(b). It can be seen that upon increasing $τ 0$ the accessed time window starting from 2 × 0.1 ms covers different parts of the C(t) curves. At $τ 0$ = 10⁻⁷ and 10⁻⁶ ms the C(t) curves are virtually completely in the plateau region, at $τ 0 = 10 − 5$ ms the C(t) curve contains a contribution of the transition from regime I, and at $τ 0$ equal to 10⁻⁴ and 10⁻³ ms almost the whole relevant region of the C(t) curve is in regime I.

The results of the fits are listed in Table I. For all cases, except for the data with $τ 0 = 10 − 3$ ms, t₀ was fixed to 0.09 ms. The MQ data with $τ 0 = 10 − 3$ ms, being virtually completely in the regime I, manifest a strong effective dipolar coupling. This leads to a fast DQ build-up, which necessitates data points at shorter times. This is why, in this case, the DQ/ΣMQ data were generated and fitted starting from $τ DQmin =$ 0.01 ms and, therefore, t₀ was fixed at 0.005 ms. Experimentally, this would mean that a DQ pulse sequence with a shorter cycle time must be used to access the regime-I (and ultimately regime-0) dynamics at lower temperatures. This will be the subject of a forthcoming publication.

Looking at the results in Table I, the fitted $D res / 2 π$ values increase with $τ 0$ upon approaching regime I because the motional averaging is increasingly incomplete in the fitted time interval. This is expected because the fitted D_res by definition reflect the amplitude of C(t) at the time t = t₀. The values are seen to be in good agreement with what is expected from the direct power-law fits of C(t). Due to the same reason the T₂ values reflecting the fast-motion contributions decrease with $τ 0$⁠, because the absolute value of C(t) at t₀ increases and its dispersion around this time is becoming stronger. On the other extreme, if C(t) reaches the plateau region well before $τ DQmin$⁠, the corresponding T₂ relaxation contribution becomes less significant. It is noted that we have also tested a stretched exponential relaxation decay $∼ e − ( 2 τ DQ / T 2 ) β$ in Eqs. (8) and (9). The results for β in Table I demonstrate that the fast-motion contribution is indeed always exponential.

The fitted exponents κ are also in rather good agreement with the average exponents obtained by the power-law fits to the C(t) curves. For the DQ/ΣMQ data set at $τ 0 = 10 − 5$ ms, the fits were carried out up to $τ DQmax =$1 ms and also over the whole time range up to $τ DQmax =$ 10 ms. It can be seen that in this transition region of the C(t) curve without a well-defined power-law exponent, the fitted κ becomes lower when the fitted interval is longer, as expected. In such a way, one can probe the shape of the OACF qualitatively as follows: If an increase of the fitting interval causes a reduction of κ, then the actual C(t) undergoes a transition from a dynamic regime with a larger exponent to a regime with a smaller exponent, e.g., in the present case a transition from regime I to a plateau. On the opposite, if an increase of the fitting interval leads to an increase in κ, then one has a transition from, e.g., regime II to regime III of a polymer melt.

The new approach was further tested on reference samples with known power-law exponents as obtained from the corresponding OACF master curves. These are poly(butadiene) (PB) polymer melt samples of different molecular weights with low polydispersity, which were studied in Ref. 15, where also details on the OACF master curve construction by TTS can be found. In Fig. 3, the new fitting approach can be appreciated on the example of PB 35 kDa in regime II (constrained Rouse) of the tube model. Fig. 3(a) shows the full OACF master curve, and the power-law exponent derived from it is compared to the value obtained from the initial-rise analysis of I_nDQ (see the inset). For the former, the indicated slopes can be obtained by means of piecewise fits of the master curve¹⁵ or a smoothed derivative analysis,¹⁷ which both require the combination of data taken at two to three temperatures for sufficient precision. The initial-rise analysis of I_nDQ was performed approximately for the first 10% of signal intensity.

In Fig. 3(b), the fit curves resulting from the new power-law model are demonstrated to describe the experimental DQ/ΣMQ signal functions rather well. The simultaneous fit was restricted to the time interval until $τ DQ$ close to the I_DQ maximum, in order to benefit from a larger time range but to not overestimate the power-law exponent, as C(t) approaches the transition to reptation at the given temperature. It can be seen that for this nearly monodisperse and hence structurally and dynamically homogeneous polymer melt all approaches give similar results within the indicated error.

For completeness, all three approaches were applied to data covering a wide temperature range of PB samples with different molecular weights, see Fig. 4 for a comparison of the results for κ. The results demonstrate that all three strategies give consistent results, in particular in regime II, which is characterized by a well-defined power-law exponent ϵ. Note that the abrupt change in slope around the regime II/III transition assumed in the model is of course not reflected in the actual data due to an expected smooth transition.

A central result of previous work^5,14,15,18 was that ϵ changes little within regime II but decreases with increasing molecular weight, which was explained by CR and CLF effects leading to an apparently protracted transition to “pure” regime-II dynamics characterized by $ϵ = 1 / 4$⁠. This is confirmed by the new analyses. It should be stressed here that the I_nDQ initial-rise analysis is subject to the largest errors since it relies on a few data points of rather low intensity. The derivative analysis of the OACF master curve overcomes this problem by relying on several data sets upon master curve construction, thus providing a smooth interpolation. Yet, these data are certainly subject to uncertainties related to the choice of the shift factor, i.e., $τ e ( T )$⁠.

Important limitations are illustrated by the data taken for PB 35 kDa and 87 kDa in regime III (Figs. 4(a) and 4(b)), which are in theory characterized by a smooth transition from an initial power law with an exponent of −0.5 to a multi-exponential decay, see Fig. 3(a) and Refs. 12 and 15. This means that the power-law model can only provide a force-fitted average exponent, which is on average larger than what is obtained by initial-rise analysis. This is expected because of the downturn of the OACF associated with the exponential decay in regimes III/IV and the different relevant time ranges. We have illustrated this by showing for comparison interpolated results (lines) derived from the appropriate analytical combined regime-III/IV formula¹² as fitted to the C(t) master curves.¹⁵ The solid lines correspond to instantaneous power-law exponents, i.e., tangent slopes in log-log plots evaluated at $2 τ DQmin / τ e ( T )$ (using the known temperature dependence of $τ e ( T )$⁠), while the dashed line is based upon a forced power-law fit of the whole interval $2 τ DQmin … 2 τ DQmax$⁠. As expected, the latter describes the DQ/ΣMQ-based fitting results better. Ultimately of course, DQ/ΣMQ data from the later regime III and regime IV may be better approximated by an analytical function based upon an exponential OACF, as published in Ref. 19.

The final test and proof of principle focus on elastomer samples, which exhibit an OACF plateau when chains between crosslinks sweep all the possible conformations, which is the case at sufficiently high experimental temperatures, thus ensuring a sufficient separation from the regime-I decay. Clearly, a plateau implies a power-law exponent equal to zero, which makes rubbers perfect reference samples. In parts (a) and (b) of Fig. 5, the results from the I_nDQ initial-rise analysis and the power-law model applied to DQ/ΣMQ signal functions, respectively, are compared for the cases of homogeneous and inhomogeneous rubber samples. Natural rubber (NR) crosslinked with 0.5 phr dicumyl peroxide (DCP) served as a homogeneous sample.²⁷ The relative width of the crosslink density distribution, a measure of the structural inhomogeneity (spatial distribution of crosslinks and thus chain lengths between them) as defined by the ratio of the standard deviation of the assumed Gaussian distribution (σ) of D_res to the mean value of $D res$⁠, was found to be around $σ / D res$ = 0.13. This corresponds to a rather homogeneous sample. Both fitting approaches gave in this case power-law exponents very close to zero.

The same comparison was performed for a rather inhomogeneous sample, namely, NR crosslinked with 3.1 phr sulfur swollen to a volume swelling degree Q = V/V₀ = 3.11. Details on sample preparation and NMR experiments on swollen elastomers can be found elsewhere.²⁸ The swollen sample showed $σ / D res$ = 0.65, which is attributed to spatially inhomogeneous stretching deformations of network chains upon swelling. In such a case, the I_nDQ build-up curve exhibits significant deviations from the predicted parabolic initial rise due to the distribution of residual couplings.²⁰ As can be seen in Fig. 5, the I_nDQ initial-rise analysis provides an apparent power-law exponent of 0.45, whereas the analytical power-law model still gives the expected value close to zero. Note that in this case the fitting quality deteriorates somewhat, as expected from an inhomogeneous sample. Nevertheless, the power-law model is overall weakly sensitive to structural inhomogeneities and thus provides true dynamic information.

Additionally, in the presence of inhomogeneities, the power-law model is much more robust in terms of the effect of the fitting interval on the obtained result. The variation in κ for a range of fitting intervals from well before to much beyond the I_DQ maximum was never larger than 0.05. In contrast, the apparent κ from the I_nDQ initial-rise analysis varied by as much as ±0.2, depending on the chosen fitting interval.

Finally, it is noted that the results so far pertained to ¹H MQ NMR data, where the applicability of the AW approximation is justified by the multi-spin coupling situation. However, Eqs. (8) and (9) should with some compromise on the prediction of long-time behavior also be valid for ²H MQ NMR data, thus approximating the Pake distribution of the anisotropic quadrupolar frequency by a Gaussian. Therefore, additional tests were performed for ²H MQ NMR data taken on a fully deuterated entangled PB sample.¹⁷ Again, the power-law exponents extracted by the described approaches were found to be in a good mutual agreement. Expectedly though, power-law-model fits to DQ/ΣMQ data had to be restricted to shorter times (usually to before the maximum of I_DQ) in order to obtain consistent results. Thus, the strategy to study the variation in κ upon prolongation of the $τ DQ$ fitting interval to assess the OACF shape is not applicable in the ²H case.

In summary, a new analytical fitting approach based upon a simple power-law model for the segmental orientation autocorrelation function (OACF) was implemented and applied to various ¹H (as well as ²H) MQ NMR data sets measured on polymer melt and elastomer samples in order to extract dynamic information on the shape of the OACF, specifically, its amplitude and approximate power-law exponent. The new approach was found to be applicable to structurally homogeneous systems (narrow residual coupling distributions) equally well as the established smoothed derivative analysis of OACF master curves constructed by time-temperature superposition (TTS) and the initial-rise analysis of normalized DQ build-up curves.

Clear advantages of the new approach are its independence of assumptions inherent to TTS and its robustness related to the fit of a larger data range as compared to the noise-challenged initial-rise analysis. Moreover, it was shown that the power-law model analysis delivers consistent and true dynamic information for structurally inhomogeneous systems featuring a significant distribution of residual couplings, for which the other approaches clearly fail. Further, a variation of the fitted data range (time interval) in the power-law model allows to characterize transition regions, e.g., when the OACF switches between different power-law regimes. Ultimately of course, the approach should not be used when independent information is available on the shape of the OACF, which is often well approximated by an exponential. Analytical solutions for the latter case exist and may then be more suitable.

We expect the new approach to be of particular use in the study of more complex materials such as supramolecular networks, where thermally labile physical junctions such as hydrogen-bonded units or ionic clusters preclude the use of TTS due to the coexistence of dynamic processes with different activation barriers, i.e., segmental fluctuations and opening/closing of supramolecular links. In this way, we hope to be able to obtain a consistent picture of the chain dynamics subject to such physical junctions.

We thank Juan López Valentín and Anna Naumova for providing and measuring, respectively, the NR samples. Funding was provided by the Deutsche Forschungsgemeinschaft (DFG) in the framework of Project No. SA 982/3-2 and the Priority Programme No. SPP 1568 “Self-Healing Materials” Project No. SA 982/9-1.