Multiple-quantum NMR studies of polymer chain dynamics

Transcription

1 Multiple-quantum NMR studies of polymer chain dynamics Kay Saalwächter 1, 1 Institut für Physik NMR, Martin-Luther-Universität Halle-Wittenberg, D-612 Halle, Germany (Dated: September 1, 216) Multiple-quantum (MQ), specifically the analysis of double-quantum build-up curves, provides quantitative information on proton proton dipolar couplings, or alternatively on deuterium quadrupolar couplings. The study of these interactions, in particular when they are partially averaged by molecular motion, provides valuable information on amplitude and timescale of rotional motions of the related molecular segments. MQ NMR has in recent years evolved as one of the most powerful methods for the study of polymer chain motion in a variety of system, also owing to the fact that the proton-detected variant is readily applicable on cost-efficient low-field instrumentation. This chapter provides an overview of the most relevant technical aspects, recent improvements and applications to different polymer materials, with a focus on work published during the last decade. Particular emphasis is devoted to studies of elastomers and hydrogels, for which the method provides unique structural information. Confined chain motion, found in entangled melts, block-copolymers or grafted systems, is also addressed. Keywords: order parameter, polymer networks, hydrogels, rubber, polymer melts, entanglements, supramolecular polymers, confinement effects, nanocomposites, filled elastomers INTRODUCTION Anisotropic interactions in solid-state NMR provide valuable measures of orientation-dependent phenomena, in particular on rotational dynamics of specific molecular units. In soft materials such as lipids, liquid crystals or in particular polymers, fast anisotropic dynamics of substantial amplitude is common, leading to significant motional averaging and correspondingly low residual interactions. Disorder and inhomogeneities, resulting in interaction distributions, provide another challenge for a quantitative assessment. In this context, multiple-quantum (MQ) NMR as applied to protons or deuterons, providing a robust and quantitative access to residual dipolar [1 3] or quadrupolar [4 6] couplings (RDCs or RQCs, respectively), their distribution, and time-dependent phenomena ( intermediate dynamics), has evolved as the probably most robust spectroscopic tool [7, 8]. The main methodological features of MQ NMR, specifically of double-quantum (DQ) NMR build-up curve analysis, were recently reviewed [8]. The first application to polymer melts is due to Graf et al., who used magicangle spinning (MAS) recoupling DQ NMR to study siteresolved RDCs [1]. MAS DQ NMR in fact provides different modes of extracting the (R)DCs in one- or twodimenstional spectra [8, 9], but the study of DQ build-up curves in static samples is the most robust and quantitative approach [7], provided that site-resolution is not of concern. This is the case for many chemically simple homo- or copolymers. The application of static lowresolution MQ NMR to polymeric soft materials was al- to be submitted to Modern Magnetic Resonance, 2nd ed. ready the subject of a review article in 27 [7]. Here, we focus on more recent methodological developments as well as new applications to different polymerbased materials comprising crosslinked systems (elastomers, gels) and constrained (entangled, end-fixed and nano-confined) chains. This review chapter is exclusively concerned with experiments on rather dynamic systems, but it should be noted that also in rigid polymers DQ NMR also provides unparalleled quantitative access to weak pairwise [1] or multi-spin [11] 13 C 13 C dipolar interactions. These can be used to probe molecular proximities and thus conformations and packing, respectively. For instance, recent work of Miyoshi and coworkers has demonstrated that previously unaccessible information on the chain folding structure in semi-crystalline polymers can be gleaned in this way [11, 12]. POLYMER CHAIN MOTION AND NMR OBSERVABLES The orientation dependence of 1 H 1 H dipolar as well as 2 H quadrupolar couplings follows in both cases the second Legendre polynomial P 2 [cos θ]. For mobile polymer chains far above the glass transition temperature T g, the main difference is that in the former case multiple pair couplings combined with fast intra-segmental motions lead to the appearance of a single effective RDC (D eff ) that is the same for all protons in the monomer unit provided that they do not reside in motionally decoupled side-chains. This is ultimately the consequence of the dipolar truncation effect [8]. The multi-spin situation further justifies the use of the second-moment approximation in theoretical treatments [13, 14]. In the 2 H case, RQCs remain distinguishable for the different labeled positions, which can provide additional informa-

2 2 (a) (b) (c) tensity rel. int constraint DQ excitation DQ 1 C(t).1.1 S 2 b.1.1 DQ reconversion DQ segmental modes ns... s cooperative chain modes s...s entangled melt time network I DQ I ref and tail fit (14% defects) I MQ = I DQ I ref I ndq = I DQ /(I MQ tail) and fit A-l vs. Gauss kernel acq DQ evolution time / ms Figure 1: (a) Orientation fluctuations of a fluctuating polymer chain can be described by a orientation autocorrelation function C(t) of the second Legendre polynomial. In case of longtime stable topological constraints (e.g. chemical crosslinks), this function attains a plateau described by a semilocal dynamic order parameter S b P 2[cos β], which is the norm of the time-averaged segmental ordering tensor S b. Its principal axis is parallel to the end-to-end vector R. (b) Schematic pulse scheme of an MQ NMR experiment. (c) MQ NMR signal functions of a typical rubber sample as a function of pulse sequence time τ DQ: the reference intensity I ref, from which a slowly relaxing defect contribution can be obtained, the DQ intensity build-up I DQ, and the two relevant derived quantities, the MQ sum intensity I ΣMQ and the normalized DQ build-up I ndq. tion or complicate the data analysis. Further, theoretical signal functions are easily calculated, as we have a single-coupling situation. In the following I only refer to RDCs, but imply that the same or at least similar principles apply for RQCs. Unless noted otherwise, all discussed experimental data is from 1 H MQ NMR. The segment-based effective RDC tensor can safely be assumed to be parallel to the polymer backbone, which means that the additional averaging (lowering) of the RDC reports on chain modes, as illustrated in Fig. 1a. The dipolar interaction frequency relevant for a segment (monomer) that is part of a chain with fixed ends, as for instance found in an elastomer, is ω D (θ) = 3 2 D eff S b P 2 [cos θ], (1) where θ is the orientation of the end-to-end vector with respect to B subject to powder averaging, and S b = 1 [ (3 cos 2 (β) 1) ] = 3 ( ) 2 l R 2 5 R 2. (2) is the segmental dynamic order parameter of the backbone. Here, l is the segmental length, β the segmental orientation with respect to the end-to-end vector R, and... and [...] indicate thermal conformational and structural (segmental) ensemble averages, respectively. Effects of chain deformation (due to macroscopic strain or swelling) enter through the dependence on the end-toend vector R 2 /R, 4 R being the unperturbed dimension [15, 16]. For bulk samples, we have R/R 4 2 = N l 2, leading to a simpler expression for the RDC, D res = D eff S b = D eff 3 5N. (3) This quantity is thus a measure of the length of the network chains in terms of the number of segments N, thus, it is proportional to the density (inverse molecular weigth) of network chains 1/M c in units of mol/kg. The value for D eff is polymer-specific, and values have been reported for a variety of systems [17, 18]. The arguments above imply a pre-averaging due to fast chain modes and the virtual absence of slower processes. This situation is only realized in elastomers at rather high temperatures above T g. Otherwise, ω D (θ) is a random function in time, meaning that only C(t) = 5 P 2 [cos θ(t + τ] P 2 [cos θ(τ)] τ,ens., (4) the orientation autocorrelation function (OACF) of P 2 [cos θ], is a well-defined quantity (θ is now the segmental orientation with respect to B ). This general case is relevant for, e.g., linear entangled chains or chains closer to T g. See again Fig. 1a for sketches of C(t) for different cases. If C(t) does not exhibit a well-defined plateau on the µs to ms timescale (corresponding to the inverse of typical (R)DC values), the apparent (!) RDCs obtained from the data analysis will be a function of temperature, and can be used to conclude on the shape of C(t). A more quantitative theory-based analysis of the signal functions is of course the best approach in this case, see below. Turning to the MQ NMR experiment, Fig. 1b shows the schematic pulse sequence consisting of suitably phasecycled DQ excitation and reconversion periods of equal length τ DQ followed by a z filter and detection after a 9 pulse. See refs. [7, 8, 19] for information on technical details and suitable pulse sequences. A pulse sequence with a DQ Hamiltonian not only creates DQ but also zero-quantum and higher-order coherences (thus MQ

3 3 NMR ), but the analysis can safely be restricted to the dominating DQ coherences [3]. The most relevant technical point is that two phase-cycled controlled signal functions are recorded, namely the DQ build-up I DQ (τ DQ ) and reference decay intensity I ref (τ DQ ). The former contains the desired information on the RDC, while the latter represents an intensity complement subject to the same relaxation effects. Note that given high-resolution conditions (high field, MAS), such data can be obtained for any peak in the spectrum [8], but for simplicity, we restrict this chapter to integral 1 H detection of chemically simple polymers. See Fig. 1c for sample data taken on an elastomer, and information on further data processing steps. Summation provides a relaxation-only full-echo function I ΣMQ (τ DQ ) = I DQ (τ DQ ) + I ref (τ DQ ), which, together with I DQ (τ DQ ) is the basis of further data analyses. Note that I ref (τ DQ ) also contains all signals from isotropically mobile moieties (D res = ), related to network defects (dangling chains, sol) and solvent. These are characterized by a long effective T 2 relaxation time, leading to a long-time tail that can be fitted and subtracted [2]. After tail subtraction, the sum function can be used to relaxation-correct (point-by-point normalize) the DQ build-up, I ndq (τ DQ ) = I DQ (τ DQ )/(I ΣMQ (τ DQ ) tail). (5) As illustrated in Fig. 1c, this function reaches a long-time plateau of 5% of the total intensity, which indicates, simply speaking, the equal partitioning of the signal between modulated longitudinal magnetization (the main contribution to I ref ) and DQ coherences. This is only the case for network chains or other permanently anisotropic situations (e.g. liquid crystals), for which C(t) exhibits a long-time plateau. In this most simple case, the ndq build-up curve is calculated as I ndq (τ DQ ) = sin 2 (k ω D (θ) τ DQ ) θ,ens., (6) where k is a scaling factor that depends on the pulse sequence. For the case of the static Baum-Pines sequence [21], which is the best choice for sufficiently mobile systems with small RDCs [3], k = 2/3, which will be implied in all further equations below. See also below for suitable approximate solutions of eq. (6). Data analysis proceeds differently for the more general situation of chains that exhibit orientation relaxation on the relevant µs ms timescale. The following sections thus treat, first, the case of networks, and then the more general and ultimately more complicated cases of supramolecular networks, entangled polymer melts and confined chains. POLYMER NETWORKS: ELASTOMERS AND SWOLLEN GELS I ndq data measured on an elastomer as shown in Fig. 1c cannot be fitted well to a simple numerical powder average of eq. (6), as the latter exhibits decisive longtime oscillations. Their absence in experimental data is primarily due to multiple RDCs among many protons, which can be accounted for by way of a second-moment approximation [7]. The resulting inverted-gaussian function reads I ndq (τ DQ, D res ) = 1 2 ( 1 exp{.4 D 2 res τdq 2 ). (7) This result also cannot describe actual data on homogeneous rubbers perfectly well, due to a small overshoot related to multiple-spin quantum dynamics [3]. Another factor to consider is that the overall shape of the build-up function is affected by a distribution of RDC values p(d res ) in an inhomogeneous sample, meaning that I ndq (τ DQ ) = I ndq (τ DQ, D res ) p(d res, D res, σ) dd res. (8) The distribution parameters D res and σ (width in terms of standard deviation) can be assessed either by (i) solving the integral analytically based upon an assumed distribution function (e.g. Gaussian), (ii) by numerical integration as part of the fitting procedure, or (iii) by numerical integral inversion procedures using eq. (7) as the Kernel function. In any case, errors due to small misfits at longer τ DQ due to eq. (7) being an approximation can be minimized by focusing on the initial-rise region [3, 22]. In order to be able to fit the whole range of reliable I ndq (τ DQ ) data, we have more recently used data taken on elastomers that were known to be very homogeneous (negligible σ) to derive an empirical build-up function that is suitable for any elastomer with protons restricted to chemical groups close to the main chain, I ndq (τ DQ, D res ) =.5 ( 1 exp { (.378 D res τ DQ ) 1.5} cos(.583 D res τ DQ )). (9) Owing to its similarity to the function describing an on-resonance 1 H free-induction decay (FID) of a rigid dipolar-coupled solid as popularized by Abragam [23], it is referred to as Abragam-like (A-l) function. This function can now be used as Kernel build-up function in the different implementations of eq. (8). The data in Fig. 1c demonstrate that the use of this function leads to a near-perfect fit. With the current level of methodological development, MQ NMR is ideally suited to address relevant scientific questions in the field of elastomers and gels. By way

4 4 of the analysis of the relevant signal functions, the content of non-elastic defects, the average crosslink density in terms of D res (the angular brackets are often, and will henceforth be, omitted) and the width or shape of its distribution (the latter quantified by σ) are straightforwardly accessible. The method thus provides information on network micorstructure and topology that is often physically more relevant than the analogous but more qualitative insights from traditional Hahn-echo 1 H T 2 relaxometry or the purely chemical information (e.g. on crosslink structure) obtained from 13 C MAS NMR [24]. To illustrate this point, Fig. 2a shows D res distributions of three different types of elastomers. It is immediately seen that the curing system (sulfur- vs. peroxidebased) affects the microstructure significantly [25], and that copolymer-based rubbers such as EPDM may be unexpectedly inhomogeneous [26], up to a level at which the applicability of the classical theories of rubber elasticity may be questioned [26, 27]. Swollen elastomers always exhibit rather broad D res distributions arising from inhomogeneous swelling and the resulting variations in local chain stretching [28]. Such inhomogeneities can only be avoided in rather special hydrogel model systems synthesized in the swollen state [29, 3]. In previous related studies, it was shown that in the absence of spatial inhomogeneities it is even possible to distinguish between regular network chains and different types of connectivity defects, see Fig. 2b. These are identified by largely different associated D res values of the monomers in these respective structures. The wide range of D res values in this system (varying over 2 decades) poses a challenge to the normalization procedure discussed above. This should be elaborated upon due to its exemplary character: Instead of increasing step-wise to the value of.5 (the heights of the steps reflecting the relative amplitudes), the ndq build-up curve shows distinct maxima and a long-time decay. This is due to the fact that DQ intensity normalization, see eq. (5), assumes that all components of an inhomogeneous system have the same relaxation decay as described by I ΣMQ. This condition is often fulfilled well enough to obtain an analyzable ndq signal function (exhibiting the expected long-time plateau) if the overall decay of I ΣMQ occurs on a comparably long τ DQ timescale. But if I ΣMQ differs among the components and is significant already during the DQ intensity buildup, normalization will fail. Then, one needs to resort to a simultaneous multi-component fit of the constituting signal functions [29]: I DQ (τ DQ ) = I ΣMQ (τ DQ ) = n i=1 ] a i I ndq (τ DQ, D res) (i) exp [ (τ DQ /τ i ) βi (1) 3 ] a i exp [ (τ DQ /τ i ) βi. (11) i=1 (a) / a.u. probability density / (b) norm. intensity p.5 NR (sulfur vulcanization) NR (peroxide x-linking) EPDM (peroxide x-linking) D res /2 / khz regular links (~65%) I MQ I DQ I MQ I DQ I ndq double links (~25%) higher defects (~1%) DQ evolution time / ms Figure 2: (a) D res distributions characterizing the extent of spatial crosslinking inhomogeneities in different types of rubber. The homogeneous case of sulfur-vulcanized natural rubber (NR) is compared to the less well-defined cases of peroxide-based radical crosslinking of NR [25] and ethylene propylene diene terpolymer (EPDM) rubber [26]. (b) MQ NMR data and multi-component fits revealing quantized defect structures in hydrogels based upon end-linking of star precursors. The structural complexity requires in this case a joint fit to the raw signal functions [29]. Here, a i are the amplitudes of the different components, τ i and β i their transverse relaxation parameters (assuming a stretched or compressed exponential shape), and D res (i) the corresponding RDCs. Such a multi-parameter fit must be carefully checked for stability, using e.g. a simulated-annealing algorithm, and is probably not feasible for systems with more than n = 3 distinct components. The blue solid lines in Fig. 2b represent an example for a 3-component fit. Microstructure studies of bulk elastomers In addition to the exemplary data discussed above (Fig. 2), many more applications related to the microstructure characterization of elastomers by low-field 1 H MQ NMR have appeared in the last years. Chinn, Maxwell and colleagues have used it extensively to study for instance thermal degradation [31], radiation-induced

5 5 crosslinking [32], and the relation between experimentally accessible and computer-simulated microstructure [33] of silicone-based elastomers. Valentín et al. focused on open questions concerning natural rubber (NR), including the study of prevulcanization of dispersed latex [34] and a comparison of NMR and dielectric spectroscopy results highlighting the microstructural differences between sulfur- and peroxide-based curing and their relation to the segmental dynamics [35]. The latter work is in fact a follow-up of the earlier results shown in Fig. 2a, which stressed that peroxide-based crosslinking leads, in particular in case of NR, to a high level of non-elastic defects and pronounced crosslinking inhomogeneities [25]. Finally, Simonutti et al. demonstrated the sensitivity of MQ NMR to small structural changes during the late stage of rubber vulcanization, comparing different bulk and filled diene elastomers [36]. This work relied on our previous work on the linear correlation between crosslink densities of bulk rubbers determined by NMR and by the Flory-Rehner swelling method, which in fact allowed for a critical evaluation of data analysis procedures involved in the swelling technique [37]. The effect of filler particles [38] deserves closer attention, and is deferred to the section dealing with composites. Swollen elastomers and gels Pioneering work concerning NMR studies of networks swollen in solvent is due to Cohen-Addad, who has very early on realized the decisive role of local chain deformation upon swelling, the relevance of topological interactions (entanglements) and the influence of solvent quality [41 44]. As is obvious from eq. (2), D res S b R 2 is sensitive to the state of chain stretching upon swelling. In our earlier work, relying on the unique sensitivity of MQ NMR to D res distributions, we could demonstrate the first direct NMR observation of swelling inhomogeneities, a phenomenon so far characterized only by scattering experiments [28]. Later, benefiting from a more quantitative data analysis on the basis of eq. (9) and improved experimental data [15, 4], we were able to complement the earlier work of Cohen-Addad in two main directions. Our starting point was the insight that the unperturbed dimension R of the network chain, relevant for eq. (2), is modified in the swollen (= semi-dilute ) state when using a good solvent. Under these conditions, the chains are subject to additional expansion due to excluded-volume interactions, and the point to consider is that the size scaling behavior is different below and above the so-called correlation length ζ corresponding to the mesh size. An in-depth treatment of S b measured in the equilibriumswollen state using scaling theory [45] provides a universal power-law dependence on the degree of volume (a) R 2 order parameter S eq ~ (b) D res /2 / khz NR peroxide-cured NR sulfur-cured PDMS end-linked PDMS randomly x-linked equilibrium swelling degree Q eq D res D res (Q=1) D res,sw (Q eq ).5 D res,prn = D res,sw Q -2/3 (affine range) desinter- phantom reference spersion network state swelling degree Q Figure 3: (a) Segmental order parameters derived from average D res values measured in different equilibrium-swollen networks (good solvent case) as a function of the equilibrium degree of volume swelling Q eq = V eq/v [15]. The universal scaling law is in agreement with theoretical predictions and emphasizes the role of thermodynamic effects on the NMR observable [15, 16, 39]. (b) D res values as a function of partial below-equilibrium swelling. The late-stage data proves affine deformation of the network chains, and allows for the definition of a phantom reference state value D res,prn via backextrapolation, which quantifies the actual chemical crosslinks without entanglement effects [4]. swelling Q = V/V in good solvent, see Fig. 3a [15]. The same result has been found earlier by Cohen-Addad, but was mistaken as a proof of de Gennes influential c - theorem [43]. We found that also a suitably modified classic Flory-Rehner treatment can also explain the result, providing new insights into the phenomenon of gel swelling. In a follow-up study, we extended our experiments to the poor solvent range, establishing the sensitivity of the NMR observables to thermodynamic effects [16, 39]. A second, even more relevant finding related to the microstructural basis of rubber elasticity was the insight that the dependence of D res (Q) below swelling equilibrium is in agreement with simple affine deformation of the chains, D res (Q) Q 2/3 according to eq. (2), once an initial desinterspersion stage is overcome (Fig. 3b). Backextrapolation of the affine region to Q = 1 provides the so-called phantom reference network value D res,prn, which quantifies the actual chemical crosslinks without

6 6 entanglement effects. In this way, by comparison with theoretical models describing entangled rubber elasticity [4], we could establish for the first time a clear relation between microstructure (connectivity vs. topology), and macroscopic behavior [27]. An in-depth comparison of crosslink densities obtained by NMR vs. equilibrium swelling, including the quantitative consideration of non-elastic defects in experiment and theory, allowed us to clarify the role of latter and and prove the validity of simple statistical models of the crosslinking process [46]. Finally, MQ NMR proved also useful to characterize the crosslink density and structural inhomogeneities in swollen superabsorber gels [47], which are often inhomogeneous-by-design. The microstructural characterization provided by MQ NMR can thus aid in improving such materials. Uniaxial deformation The concepts that apply for elastomer swelling are easily transferred to the study of mechanically deformed samples, providing another platform to put elasticity theories to a test. Also in this case we rely on the dependence of D res S b R 2 on the distance between constraints, see eq. (2). The main difference is that while free swelling corresponds to isotropic dilation, stretched rubber samples are anisotropic, introducing a dependence of the measured data on the orientation Ω of the main stretching direction with respect to B (for the uniaxial case). This offers new opportunities for data fitting, but also poses some new challenges. Early insights into stretched elastomers were gleanded from static 2 H NMR spectra of either deuterated lowmolecular probe molecules [48] or deuterated network chains [49]. In the former case, only diffusion-averaged mean-field information is obtained, while in the latter case full microscopic information on local deformation is in principle within reach. The early lineshape analyses relied on partially critical model assumptions, and because of this, the application of MQ NMR without the need for isotope labeling appeared promising [5, 51]. Our first application was concerned with quantifying the so-called overstrain of the network matrix for the case of filled elastomers [5], followed by a quantitative assessment of microscopic deformation in bulk elastomers [51]. In Fig. 4a I ndq build-up curves are compared for an undeformed vs. a uniaxially strained unfilled rubber sample. In the latter case we observe significantly increased D res values and a clear orientation dependence. At that time it was not possible to model the strained-network build-up data, so we introduced an artificial powder average by superposition of Ω-dependent data [5, 51]. In this way, eq. (9), which relies on an isotropic sample, can be used to extract D res distributions and averages. The (a).6 ty ndq intensit (b) (c) sity ndq inten = 1 = 3 = = 54 = DQ evolution time / ms rel. DQ Intensity 2 1 = 3 >1.u. pro ob. density / a. affine D res /D res, =1 phantom iso. tube sample orientation / 1. eq.(12) eq.(12) 9.8 eq.(9) A-l eq.(6) sin DQ evolution time / ms Figure 4: (a) I ndq build-up curves for a uniaxially deformed NR sample (λ = 3), comparing three orientations Ω (stretching direction vs. B, symbols) and the artifical powder average (lines) constructed by weighted superposition, also compared to the unstretched case. The right panel shows the derived powder-averaged D res distributions. (b) Relative ndq intensity at fixed τ DQ =.3 ms (λ = 3) as a function of Ω along with network model predictions. (c) New angle-dependent generic ndq build-up function (two orientations) whose powder average corresponds to eq. (9), to be used in future modeling of ndq build-up curves measured on anisotropic samples, compared with a powder-averaged sin 2 - build-up, eq. (6). Data adapted from refs. [51, 52]. inset of Fig. 4a demonstrates that deformation not only leads to an increase, but also to a considerable broadening of the D res distribution, reflecting an inhomogeneous ensemble of chains in different stretching states. Such a distribution is in fact expected from simple modeling, see the inset of Fig. 4b. Depending on the orientations of the different network chains with respect to the strain (represented by a spokeswheel ), chains are either stretched or even compressed upon deformation. A comparison with model predictions demonstrated that the classical models of rubber elasticity (affine, phantom) cannot describe the observations, thus, more elaborate tube models have to be employed, which take into account effects of topological entanglements. The

7 7 mentioned comparison was possible for D res distributions (Fig. 4a) and for ndq intensities measured at short τ DQ as a function of Ω (Fig. 4b), and did not yet provide a fully consistent picture, as different tube models are able to capture certain features differently well [51]. So far, the analysis of whole I ndq build-up curves was precluded by the lack of a suitable single-orientation build-up function representative of a protonated polymer segment. For 2 H NMR, the situation is simple in that eq. (6) can be used. For 1 H NMR, a single-angle build-up function taking into account the multiple dipolar couplings was not known. In principle, spin dynamics simulations can be used for that purpose [17], but the number of spins necessary to obtain a converged results is still unfeasibly large. In a recent effort, we have thus resorted to simple reverse fitting of an empirical function with limited number of fitting parameters, formally similar and optimized to fit eq. (9) upon powder averaging. The result is I ndq (τ DQ, D res, θ) =.5 ( 1 exp { (.25 D res P 2 [cos θ]τ DQ ) 2} cos(.839 D res P 2 [cos θ]τ DQ )). (12) Fig. 4c shows this ndq build-up for two canonical orientations along with its powder average, eq. (9). We are currently exploring the use of eq. (12) in analyses of stretched-network data, and expect further uses in other cases where polymer chains move in macroscopically anisotropic environments. Supramolecular and phase-separated networks In supramolecular networks chemical crosslinks are replaced by physically interacting entities, e.g. hydrogenbonded units, phase-separated regions of chemically distinct and matrix-incompatible moieties, or ionic clusters. Consequently, the operating temperature plays a very prominent role in that thermal motion can render such physical links labile. Such materials can thus flow or even self-heal [53] at elevated temperatures, but are permanently elastic at lower temperatures. In terms of the segmental OACF C(t) (eq. (4), see also Fig. 1a), we can thus expect a decay at longer times beyond the rubbery plateau, leading to an apparently temperature dependent RDC, D res (T ). While a quantitative understanding for such a situation has been achieved for entangled polymer melts (see below), we here summarize the results of some studies based upon this qualitative observation. A first, recent study of a supramolecular self-healing rubber material consisting of low-molecular hydrogenbonded components, combining MQ NMR with thermal and mechanical studies, demonstrated the use of MQ NMR to reveal aging phenomena in the investigated system. Under aging conditions actual chemically linked network chains are formed in addition to purely supramolecular interactions [53]. For a phaseseparated poly(urethane) elastomer with chemically reversible links based upon the Diels-Alder reaction, I ndq build-up curves, which are found to be temperaturedependent only above a certain critical temperature, could directly reveal the presence of thermally labile links [54]. Ionomeric rubbers, obtained for instance by ionization of carboxylic acid groups pending from a olefinic rubber main chain, are chemically rather simple, but reveal a surprisingly complex microstructure and morphology, related to the formation of ionic cluster structures. A significant amount of main-chain segments is entrapped and immobilized in such structures [19]. Consequently, such samples exhibit wide D res distributions, requiring the combination of different DQ pulse sequences (specifically, the three-pulse segment and the Baum-Pines sequence) to capture the full coupling range [19, 54]. The high-d res component reflects the clusterimmobilized segments, while the low-d res component is related to entropically elastic chains. Increasing the temperatures mainly reduces the fraction associated with the former, corresponding to the softening of the linking regions, and decreases the absolute value of the latter, reflecting the loss of elasticity and the onset of long-time flow [19]. A similar combination of MQ NMR methods has also been used to monitor the reverse situation of curing an epoxy resin, where increasing temperature induces cross-linking and the appearance of high-d res components [55]. ENTANGLED POLYMER DYNAMICS Understanding the dynamics of entangled linear chains with high molecular weight (MW) in the melt is a key quest in polymer physics, and a rather mature theoretical understanding has been reached [45, 56, 57]. Chainbased theories can represent the complex macroscopic mechancial behavior rather well, but many features of these theories are yet to be confirmed on a microscopic level. The main feature is that long chains can relax terminally only by reptation motion, i.e., one-dimensional (1D) diffusion along their own contour [45, 58], which explains the extraordinarily high viscosity of high polymer melts. Later refinements include the modeling of the constraints posed by neighboring chains by a confining tube with certain properties [56]. De Gennes in his famous paper [58] has in fact already foreseen the possibility to predict the signature of reptation by NMR measurement of the dipolar OACF, C(t) (eq. (4), see also Fig. 1a), in terms of power-law regimes with time exponents 1/4 and 1/2 for tube-constrained local fluctuations (regime II) and larger-scale reptation (regime III), respectively

8 8 [59]. Based upon groundwork of Ball, Callaghan and Samulski [59], the first experimental observation of these scaling exponents is due to Graf, Heuer and Spiess [1], who used DQ MAS NMR to directly probe C(t). At that time, only a single point in time was accessible due to pulse sequence restrictions, meaning that the actual C(t) had to be constructed by time-temperature superposition [57]. Our more recent work represents an elaboration and quantification of this early work, providing full quantitative information on tube-confined chain motion [6, 6 62]. Static MQ NMR is perfectly suited, as one can in principle access C(t) over a continuum of times represented by τ DQ, and thus directly probe the shape of C(t). MQ NMR thus complements traditional relaxation time measurements, lineshape analyses or real-time exchange NMR [63, 64] by providing direct time domain access to C(t) in the range of about 1 µs and a few ms, i.e., the range of typical τ DQ. A non-constant C(t) means that the signal functions are to be calculated as follows [14, 59, 62]: I DQ (τ DQ ) = sin φ 1 sin φ 2 (13) sinh φ 1 φ 2 exp{ φ 2 1}, (14) I ΣMQ (τ DQ ) = sin φ 1 sin φ 2 + cos φ 1 cos φ 2 (15) exp φ 1 φ 2 exp{ φ 2 1 }. (16) The brackets denote an ensemble average. The key physics is encoded in the phases φ i, obtained as φ i (t a, t b ) = D eff tb t a P 2 [cos θ(t)] dt, (17) with the interval t b t a = τ DQ. With this, φ(, τ DQ ) and φ(τ DQ, 2τ DQ ) are identified with φ 1 and φ 2, representing the excitation and reconversion periods, respectively. This treatment neglects pulse sequency details by assuming the continuous action of a well-defined average Hamiltonian. Discrete-interval effects have previously been addressed and turned out to be minor [14]. Importantly, P 2 [cos θ(t)] is a random function in time. The (within the spin pair limit) exact results eqs. (13,15) can for instance be evaluated on the basis of dynamic simulation models supplying trajectories θ(t) over sufficiently long times, evaluating eq. (17) via a finite-interval summation. This approach has been followed in previous work, relying on a simple 1D random-walk model of reptation motion along a tube, which in itself is represented by a random walk structure in 3D [65]. For a more theory-based evaluation, the Anderson- Weiss (second-moment) approximation [66, 67] can be employed to arrive at eqs. (14,16), where an ensemble and time average is now to be taken over products of the individual phases, which is a rather straightforward exercise [59, 67]. The final signal functions are calculated by simple time integrals over C(t), eq. (4), namely and φ 2 1 = 4 9 M 2eff 2 τ φ 1 φ 2 = 4 [ τ 9 M 2eff t C(t ) dt + 2τ τ (τ t ) C(t ) dt (18) (2τ t ) C(t ) dt ]. (19) The dipolar second moment follows the common definition M 2eff = 9 2 D2 eff ; note that the two equations include a correction factor (2/3) 2 = 4/9 taking into account the specific scaling factor of the most-used Baum-Pines DQ pulse sequence. A simple way to link the experimental observables with the target quantity C(t) is a direct time-domain analysis of I DQ or actually better, I ndq. Using the latter mainly corrects for transverse relaxation effects arising from dynamics on timescales t < τ DQ and improves the validity range of the inherent approximations. I ndq data for a very high MW polymer (with >1 entanglements per chain) are shown in Fig. 5a. Temperature variation demonstrates that an apparent backbone order parameter S b (T ) = D res (T )/D ref, (2) as encoded in the initial rise, decreases with temperature. In addition, relaxation at long times unaccounted for by normalization prevails, and both phenomena demonstrate that C(t=τ DQ ) is not constant in time. An approximate numerical value for the apparent RDC, D res (T ), can be obtained by fitting I ndq (τ DQ ) up to τdq max (where it reaches its highest value) to eq. (7), combined with an empirical exponential relaxation term exp{ 2τ DQ /T 2,app }, see the dashed line in Fig. 5a. This is only a good strategy if I ndq reaches values exceeding.3. Otherwise, one can use a fit to a short-time expansion of eq. (7), I ndq (τ DQ ).2 D res (T ) 2 τdq 2 without the use of a relaxation term. In this way, one can quantify C(t) at a time corresponding the center of the fitted interval, C(.5τDQ max, T ) S b(t ) 2. For a more detailed analysis, we rely on a short-time approximation first published by Graf et al. [1, 69]: Expanding I ndq (τ DQ ) = sin φ 1 sin φ 2 for small φ 1,2 and assuming C(t) to be constant for t < τ DQ, one can show that I ndq (τ DQ ) k τ 2 DQ C(τ DQ ), (21) which we found to be a good approximation for either I ndq.1 or up to half the τ DQ it takes to reach the I ndq maximum if it is significantly lower than.1. See ref. [61] on the relation between the numerical constant k in eq. (21) and S b (T ) obtained from eq. (2). Typically,

9 9 (a) 4.4 DQ intensity norm. (b) C(t/ e ) DQ C(t) ~ DQ absolute-value fit 243 K 33 K 363 K validity limit of eq. (21) PB2k (N/N e = 11) DQ evolution time / ms 1 t S e2 ~ 1/N 2 e C C C s I e II t.29 III PB23k IV PB2k PB87k PB35k (c) t/ e 1. PB87k (N/N e = 45).8 MQ DQ K eqs. (23,24).4 full C(t) 273 K 2.2 normlized intensity DQ evolution time / ms Figure 5: (a) I ndq build-up curves of highly entangled polybutadiene (PB; MW given in in the sample name) at different temperatures along with fits as described in the text. (b) C(t) for a wide range of PB samples covering all regimes of the tube model [56], combining data from MQ NMR (solid symbols) covering the high-mw range [6] and T 1 field-cycling relaxometry (open symbols) covering the low-mw range [68]. (c) I ΣMQ and I DQ signal functions for PB at two different temperatures [61], along with a comparison to theory [62]. The dotted lines mark the upper end of the fitting interval to eqs. (23,24). N in the captions is the degree of polymerization, and N e the number of monomers per entangled stand. a power law C(t) τ ɛ DQ represents a good approximation over a sufficiently short time. The data shown in Fig. 5a representing the highly entangled limit is compatible with ɛ.29 for all temperatures. This value is close to de Gennes prediction of 1/4 for the tube model regime II ( local reptation ). For shorter chains, a cross-over to tube model regime III (actual reptation) and, beyond the terminal time τ d, into the free-diffusion regime IV is expected. This could be confirmed quantitatively for several polymers over large MW ranges by constructing C(t=τ DQ ) I ndq (τ DQ )/τdq 2, measured piecewise over short time intervals at different temperatures, and combining the data to master curves as a function of a master variable t/τ e (T ) [61]. The reference τ e (T ) [56] is the MW-independent entanglement time that separates regime II from the Rouse regime I (locally unconstrained chain modes), and its temperature dependence is known. The data in Fig. 5b presents on overview of results obtained, combining data from MQ NMR [6] with data derived from T 1 field-cycling relaxometry (FCR) covering the range of shorter times [68], i.e. the glassy regime (segmental fluctuations) and the Rouse regime I. Typical polymer timescales, prominently the disentanglement time τ d (M) marking the transition between the reptation regime III and the free-diffusion regime IV, could be extracted from the MQ data and were shown to be in very good agreement with expectations from tube theory and other experiments [61]. It was in fact possible to highlight small but relevant deviations from the tube model predictions, in particular of the exponent ɛ mentioned above [6, 61], providing new evidence for additional chain relaxation mechanisms unaccounted for in the simple tube model [56]. Note that recent work has shown that the apparently very good fit of results from MQ NMR and FCR is partially coincidental, as the two methods exhibit different sensitivities to inter-chain dipole-dipole couplings [6, 7]. Notably, MQ NMR results provide a direct time-domain measure of C(t), with comparable results for 1 H and 2 H NMR, the latter probing purely intra-chain dynamics, which are further in good qualitative agreement with different polymer simulation studies [6, 71]. As C(t) for entangled polymers could thus be determined in a model-free way (Fig. 5b), one can use a piecewise analytical representation of it and feed it into eqs. (18,19), to be plugged into eqs. (14,16) for a theoretical prediction of the signal functions [62]. The results shown in Fig. 5c provide a first-time confirmation of an analytical theory for MQ NMR signal functions of entangled polymer melts. The only free parameter in the associated fits was M 2eff, which assumed reasonable values expected for the static limit. It is noted that the same formalism can also be used to quantitatively describe experimental Hahn-echo transverse relaxation curves [62]. The theory using the full C(t) is ultimately straightforward but algebraically cumbersome, calling for possible simplifications. Closer scrutiny reveals that motions occurring at times shorter than the shortest experimentally accessible τ DQ only lead to a multiplicative exponential decay exp{ 2τ DQ /T 2f } for both signal functions, where the transverse relaxation time related to fast motions T 2f can just as well be fitted instead of being predicted. The part of C(t) relevant for the measured data is then only the real-time interval between 2τ DQmin and

10 1 2τ DQmax. A reasonable proxy is to just assume a power law [72], { S C(t) = b 2 for t < τ Sb 2 (τ / t ) ɛ, (22) for t τ with constant time-scaling exponent ɛ. On this basis, fits to I DQ/ΣMQ raw signal functions should be possible as long as this assumption holds. If the actual C(t) deviates from the assumption, a forced best-fit of it is obtained, with ɛ still providing valuable qualitative insight. Solving the AW expressions eqs. (14,16,18,19) using eq. (22), one obtains [14] { I DQ (τ DQ ) = e 2τ DQ 4 T 2f 9 exp M 2res (ɛ 2)(ɛ 1) ) } +(2ɛ 2 4ɛ)τ DQ τ + 2τ 2 ɛ DQ τ ɛ sinh I ΣMQ (τ DQ ) = e 2τ DQ T 2f ( (ɛ ɛ 2 )τ 2 { 4 M ( 9 2res (ɛ 2 ɛ)τ 2 2(ɛ 2)(ɛ 1) +(2 3 ɛ 4)τ 2 ɛ DQ τ ɛ o) }, (23) ( { exp 4 9 M 2res (ɛ 2)(ɛ 1) 3 2 (ɛ ɛ2 )τ 2 + (2ɛ 2 4ɛ)τ DQ τ + (4 2 2 ɛ )τ 2 ɛ DQ τ ɛ )}. (24) These relations are valid under the conditions ɛ >, ɛ 1, 2, and τ < τ DQ. Practically, τ is fixed at a value just below τ DQmin. The residual second moment reflecting the RDC (thus the segmental order parameter) at τ is M 2res = Sb 2M 2eff = 9 2 D2 res. A demonstration of these fitting functions on the example of data from a well-entangled polymer at two different temperatures is shown in Fig. 5c, and the results are seen to be virtually indistinguishable from the full theory prediction. The fit results comprise ɛ values (.34±.4 and.8±.4 for 273 K and 353 K, resp.) that are in good agreement with values expected for regime II and late-stage regime III, respectively. This fitting approach should be of use for investigations of a variety of more complex systems, such as supramolecular networks with dynamic crosslinks, or branched systems. Specifically, we are currently pursuing studies where differences between different parts of the branched structures, accessible by suitable 1 H vs. 2 H isotropic labeling, are in the focus. GRAFTED CHAINS, (NANO)COMPOSITES AND CONFINEMENT EFFECTS The dynamics of polymer chains becomes significantly altered in the presence of internal solid surfaces, which (a) (b) sd rel. DQ intensity low MW r high MW free PEO adsorbed PEO constant layer thickness hybrid hydrogels sd DQ evolution time / ms high MW, high silica content entropy-elastic bridging chains Figure 6: (a) Model of PEO/silica nanocomposites based upon NMR results, where free and surface-immobilized chains can be distinguished and quantified [76]. MQ NMR provided the characterization of elastic bridging chains with welldefined but low D res, which explain the elastic properties. (b) DQ build-up curve characterizing the adsorbed polymer fraction in nanocomposites of PEO with laponite platelets in water. Figure reproduced from ref. [77] with permission of the American Chemical Society. are for instance present in mixtures with (nano)particles. For hydrophilic surfaces such as OH- and H 2 O-covered oxides and polymers with polar moieties, H-bonding interactions usually lead to strong adsorption and immobilization within a layer of 1 to a few nm thickness. The polymer in this layer is largely immobilized, thus features high dipolar couplings, and is easily quantified by 1 H line shape or time-domain FID analyses [73, 74]. In contrast, the chain parts emanating from this layer are still highly mobile but their dynamics is constrained by the end fixation. Further, if the chains are long enough and the nanoparticle content is high enough, bridging chains are formed. These network-like chains provide connectivity and are responsible for the formation of an elastic gel. MQ NMR is ideally suited to quantify the relative amount and the relative length of these confined chains through their RDC, as demonstrated on mixtures of poly(dimenthyl siloxane), PDMS [75], and poly(ethylene oxide), PEO [76], with silica spheres. A corresponding structural model is shown in Fig. 6a. The dynamics within the highly immobilized adsorption layer can be studied by 1 H time-domain techniques based upon the magic-sandwich echo [73, 74, 78]. In another recent study [77], Lorthioir et al. demonstrated the use of (non-normalized) DQ build-up curves based upon

11 11 the 5-pulse sequence, suited to study stronger dipolar couplings, to characterize the rather constrained internal mobility within the adsorption layer of PEO on clay mineral sheets (laponite), see Fig. 6b. A semi-quantitative picture could be derived by comparison of data taken on PEO composites with PEO in a fully amorphous model system not too far above its T g. This is in agreement with the notion that polymers in the adsorption layer exhibit an effectively increased T g, in fact, in-depth investigations confirmed the presence of T g gradient [74]. Similar phenomena have been revealed for an unpolar rubbery polymer (EPDM) filled with carbon black (CB), using DQ-filtered 1 H FID and echo signals, and spindiffusion experiments [79]. CB exhibits favorable interactions in particular to polymers with double bonds, yet the adsorption layer is usually thinner, i.e., of the order of.5 nm. Additional crosslinking of nanoparticle-filled polymers leads to high-performance gels or elastomers, which are mechanically reinforced mainly via the formation of a second mechanical network formed by the filler particles [8]. The RDC of the network chains easily overrides the effect of the surface bonding, as its area density is usually low [79]. Only in cases where efficient surface grafting agents are used can MQ NMR provide evidence of a locally increased crosslink density close to the surface [73]. The most common observation is in fact that the crosslink density is reduced by filler addition, due to the partial inactivation of the vulcanization system by adsorption to the high-surface filler [38, 8]. Note that this insight can only be gleaned from MQ NMR, which provides localized information, while mechanical properties are dominated by the filler network, precluding quantitative conclusions on the properties of the rubber matrix. Even rather large filler particles have recently been shown to lead to enhanced aging (mostly inhomogeneous chain scission) effects, as again quantified in detail by MQ NMR [81]. Less dense surface bonding to nanoparticles can, however, be detected indirectly, by comparison of results from MQ NMR with equilibrium swelling experiments [38]. As already discussed above, both the RDC quantified by MQ NMR as well as the equilibrium degree of swelling provide measures of the network chain density, with near-perfect linear correlation between the results of the different methods, termed the master line. If the rubber chains exhibit permanent bonding to the filler particles, the equilibrium degree of swelling is reduced due to the filler particles acting as giant crosslinks (if no bonds are present, the rubber just swells away from the filler forming a solvent-filled vacuole). This leads to apparently high network chain densities from swelling and to deviations from the master line, the extent of which characterizes the rubber-filler bonds. This method has been established for NR containing different types of fillers [38], and has in the meantime been applied to filled (a) (b) I nd.1 Q(.5ms).15.5 B sample tube 9 AAO polymer. 1 1 MW 1 kg/mol fast exchange slow exchange h s bulk surfaceinduced orientation ti Figure 7: (a) Schematic sample set-up for orientationdependent NMR measurements of polymers infiltrated into nanopores in stacked AAO membranes. (b) Short-time singlepoint ndq intensities at τ DQ=.5 ms, reflecting the average apparent RDC, as a function of MW for PB confined to 2 nm pores at 383K and and 9 nominal orientation, and corresponding model explaining the two MW domains with qualitatively different angle dependence. Figure reproduced from ref. [65]. styrene-butadiene rubber [82], and other diene elastomers [36]. Block copolymers, with phase-separated nanometerscale soft and hard domains [83], the latter formed by e.g. glassy poly(styrene), represent another case of confined motion of grafted mobile chains in the soft domain. For the case of a diblock copolymer, with only one end of the mobile chain fixed, one does not necessarily expect anisotropic chain motion. However, anisotropy arises from the ordered environment represented by the, e.g. lamellar, nanodomain. In addition, it is known that the chains in such systems are stretched along the interface normal [83]. These phenomena have recently been studied by 2 H DQ NMR [5, 84], revealing a substantial RDC distribution for the mobile chains in a lamellar diblock copolymer. Notably, a clearly detectable fraction of isotropically mobile parts, attributed to a location close to the lamelar mid plane, was identified for a fully deuterated [5] and an end-deuterated mobile block [84]. These findings are mutually consistent and emphasize that the mobile chain ends are distributed throughout the soft domain [84]. Finally, we address the influence of a neutral, non-