Chapter 1: Applications of NMR in Polymer Characterization – An Introduction

Characterizing polymer systems in a given state by NMR is generally a challenging task, owing to the coexistence of dynamic processes on vastly different timescales – possibly including the “intermediate” timescale of the NMR experiment itself – and the presence of structural disorder as well as dynamic heterogeneity. In many NMR experiments, in particular in solid samples, molecular reorientations with a given amplitude on certain timescales will lead to potential signal loss and inefficient detection due to low values of specific relaxation times (T₂, T_1ρ) or low polarization transfer efficiencies, e.g., in cross-polarization (CP). This means that, in a structurally and dynamically inhomogeneous sample, one inevitably has to make sure that the NMR method in question provides information that is representative of the whole sample rather than a potentially ill-defined sub-ensemble.

With this caveat in mind, this chapter will give a survey of the broad range of NMR methods that are commonly applied to polymer systems,^1–12  with the aim of either merely confirming the chemical make-up, configuration or conformation, or investigating a specific structural or dynamic property of the material. For a rough classification, Figure 1.1 provides a schematic overview of different morphologies frequently encountered in samples of “simple” homo- or block copolymers submitted to NMR characterization. The wider field of “polymer NMR” covering all depicted states has been the subject of two authoritative textbooks published about 25 years ago,^1,2  and for an impression of the different directions of development in the field, one may consult a more recent edited book based on a conference symposium in 2010.³ 

Figure 1.1(a) represents the typical situation of a polymer in more or less dilute solution, where at least the rotational tumbling of the coils is not significantly altered by the finite concentration. Sufficiently fast segmental motion and overall tumbling leads to well-resolved high-resolution spectra, rendering chemical identification based on common 1D and 2D solution-state NMR methods straightforward. The main source of information is the isotropic chemical shift (mostly of ¹H and ¹³C), which is sensitive to regio-, stereochemical and geometric isomerism, as well as to the conformation of a chain. In copolymers one can further analyse it in terms of the monomer distribution (diads, triads, etc.). Spectral resolution and structural assignment via harnessing neighbourhood relations through J couplings or the nuclear Overhauser effect (NOE) follow the established toolbox of solution-state NMR, on which dedicated polymer-specific presentations were published and updated continuously over the last three decades.^1–5,11 

As far as the dynamics is concerned,¹⁰  one should distinguish between more localized and complex chain motions on the nm scale, i.e., Rouse¹³  or Zimm modes (depending on the importance of hydrodynamic coupling),¹⁴  and the translational diffusion of the whole coil. For NMR purposes, the chain dynamics is described by the segmental orientation autocorrelation function (OACF)

where P₂ is the second Legendre polynomial and θ is the main principal axis orientation. C(t) is the basis of the theoretical description of the vast majority of NMR phenomena that are sensitive to orientation fluctuations of tensorial NMR interactions (note that descriptions based on this simple relation hold for isotropic or uniaxial dynamics and neglect effects of translational motion through intermolecular dipole–dipole coupling). Most prominently, this of course pertains to the various relaxation times, most of which are simply proportional to sums of spectral densities (the Fourier transform of the OACF) at specific frequencies.¹ 

Translational diffusion is a methodologically separated phenomenon, and is in dilute solution directly related to the coil size,¹⁴  enabling pulsed field gradient (PFG) NMR experiments probing the mean-square displacement on the μm scale as a tool for molecular-weight (MW) determination.^10,15,16  Such experiments can be suitably combined with mathematical procedures to reveal the diffusion coefficient and possibly its distribution (and thus the MW distribution) for signals in complex mixtures at different chemical shift in a second spectral dimension, which is referred to as diffusion-ordered spectroscopy (DOSY).¹⁶  This group of methods can serve as a first example of our general caveat, because of transverse relaxation during the PFG NMR pulse sequence. T₂ is generally MW-dependent, leading to potentially biased results for broader MW distributions.¹⁵  The effect seems to be relatively small in dilute solution of flexible chains, but is rather significant in more concentrated or bulk states.

Recent applications of different solution-state NMR methods to (bio)polymer systems of current interest comprise the study of silk proteins highlighted in Chapter 18 by Guo and Yarger, peptide-nanoparticle interactions in aqueous suspension in Chapter 2 by Suzuki and carbohydrates in ionic liquids in Chapter 3 by Ezzawam and Ries. The latter also addresses applications of DOSY (PFG-NMR), which is the main subject of Chapter 4 by Alam and Chapter 5 by Ute et al. The former focuses on its combination with magic-angle spinning (MAS) to improve spectral resolution, and the latter highlights its use in combination with cryoprobe technology.

Being rather open random coils, the polymer chains start to overlap upon increasing the concentration, ultimately forming a dense melt as depicted in Figure 1.1(b). This leads to a significant slowdown of translational diffusion and correspondingly large, qualitative MW-dependent changes in the relaxation spectrum related to chain modes.^10,14,17  Probably the most suitable method to obtain information on the shape of the segmental OACF, eqn (1.1), and thus on the relaxation spectrum of the chain, is field-cycling (FC) NMR relaxometry, which probes T₁ of protons as a function of Larmor frequency.^10,17  Chapter 6 of this book by Hofmann, Flämig and Rössler is devoted to an up-to-date presentation of applications of this technique. Section 1.2 of this chapter will also demonstrate the use of instrumentally less demanding methods based upon transverse spin evolution to also probe the segmental OACF. Notably, Rössler and coworkers have recently established FC NMR, combined with isolating the effect of inter-segmental ¹H dipole–dipole couplings, as a powerful technique to probe translational diffusion on length scales much below those accessible by PFG NMR.¹⁸ 

Depending on the specific polymer and temperature, solvent removal may further entail the crossing of T_g and thus vitrification. The segmental and sub-segmental (local conformational) dynamics in polymer glasses around and below T_g are characterized by wide correlation time distributions and spatial variations (dynamic heterogeneity),^8,12,19,20  which are intimately related to the mechanical properties of the materials.¹⁹  Therefore, in simple homopolymer melts and glasses one is already confronted with the noted wide range of physical and NMR phenomena related to the coexistence of dynamics covering vast timescale ranges.

As a second example of our initial caveat concerning the detection of potentially non-representative sub-ensembles we present and discuss in Figure 1.2 an early example of static ²H wide-line NMR in glassy poly(carbonate), PC, focusing on the phenyl ring flip process in the glassy state.²¹  Figure 1.2(a) demonstrates the significant intensity loss during the short solid echo, which is needed to solve the dead-time problem by refocusing the dominant quadrupolar interaction and thus to acquire well-phased spectra. The wide flip rate distribution covers the whole range from the rigid to the fast-motion limit. When, at a given temperature, its maximum is right in the intermediate motional regime (minimum¹ of the refocused T₂), most of the relevant spectral intensity is lost and the observed spectrum appears to be a “two-phase” superposition of the limiting cases. Any attempt to fit and interpret such a spectrum neglecting the T₂-related losses will result in physically incorrect, even meaningless, conclusions on the dynamics in the sample. In the opinion of the author, such effects have been largely underestimated in many other contexts involving, e.g., ¹H and ¹³C MAS spectroscopy based upon pulse sequences that differ from a simple 90° excitation pulse. More examples will be given below.

Given such dynamic complexity, it is thus no surprise that NMR studies of the typical states of matter in polymers (and the richer micro- and nanostructures addressed below) require the whole toolbox of solid-state NMR. The most relevant textbooks to date were again published two to three decades ago.^6–9  The text of Schneider and Fedotov presents a comprehensive treatment of relaxation phenomena, with particular emphasis on dynamics coexisting on different timescales.⁶  McBrierty and Packer have a focus on applications of the standard suite of 1D ¹H and ¹³C solid-state NMR methods, e.g., cross-polarization (CP) and magic-angle spinning (MAS) methods, measurements of relaxation times and some ¹³C and ²H wide-line NMR, as applied to a variety of different polymers.⁷  In turn, the complementary monograph of Schmidt-Rohr and Spiess unfolds the wealth of multidimensional solid-state NMR spectroscopy,⁸  with an emphasis on dedicated experiments such as exchange NMR, to study details of complex motions and orientation/order phenomena. Finally, the book edited by Ando and Asakura⁹  is a rich source of information not only on solid-state NMR studies of a very broad range of polymer materials, but also on the use of less common nuclei such as ¹⁵N, ¹⁷O and ¹⁹F. In the present book, Chapters 9 (Asano) and 17 (Fu, Chen and Yao) give perspectives on ¹⁹F NMR, and the latter also includes ⁷Li NMR, which offers high potential for the characterization of lithium ion conductors in battery applications. Chapter 11 of Sun and co-authors presents some applications of ²H NMR, and Chapter 20 of Yamada gives an account of the use of less common ³³S NMR as mostly applied to vulcanized rubbers.

This brings us back to the classification scheme in Figure 1.1, where part (c) illustrates the introduction of crosslinks into chains in bulk or solution, resulting in an elastomer/rubber (when far above T_g) or a gel, respectively. The chains in such a system still move rather quickly, leading to large and in some regards liquid-like angular excursions of the monomers and thus significant averaging of anisotropic interactions such as dipole–dipole coupling (DC) and chemical-shift anisotropy (CSA). However, the chain-end connectivity imparts some anisotropy on the segmental motion, meaning that a small but significant anisotropy remains. The corresponding small degree of time-persistent local ordering can be quantified by a dynamic order parameter S, which is simply calculated as the ratio of the residual magnitude (after averaging) of a given tensorial interaction, e.g., DC constant D, to its static counterpart:

The constant k takes into account deviations of the main principal axis direction of the respective tensor (e.g., the internuclear vector) from the polymer chain direction, so as to define S relative to the backbone orientation. The time and ensemble average 〈⋯〉_t,ens over the time-dependent backbone orientation θ[t] covers a time up to the relevant probed evolution/acquisition time in the experiment. This should be longer than the relaxation time of the slowest chain mode, after which the crosslinks prevent further averaging. Thus, D_res reaches a plateau beyond a certain temperature and, otherwise, the measurement does not yet reflect the long-time limit. Given the internal timescale of typical inverse ¹H–¹H DC constants, this is commonly the case about 50 K above T_g. At lower temperatures, and in polymer melts at any temperature (segments eventually move isotropically), the apparent (measured) D_res is temperature-dependent. For more details, see Section 1.2.

The last part of eqn (1.2), first established in a famous paper concerned with the phenomenon of strain birefringence,²²  makes the link to structure in terms of the number of statistical (Kuhn) segments of the network chain, N_K. In addition, the dependence on the actual end-to-end distance R of the long chain relative to its Gaussian average R₀ imparts a dependence on a microscopic deformation (note that the distributed, instantaneous R is always subject to a time average, at least within the Phantom model). Therefore, S and with it D_res are directly proportional to the number density of network chains (or, equivalently, the crosslink density). Thus, D_res is proportional to the elasticity modulus of a rubber.²³  With N_K of the order of a hundred, S is in the percent range, and typical ¹H residual dipolar coupling constants (RDCCs) are thus around 100–200 Hz. The resulting degree of line broadening is intrinsic to static spectra of such materials, and is an obstacle to high-resolution spectroscopy. I stress the high similarity of this phenomenology to NMR studies of thermo- and lyotropic liquid crystals (LC).²⁴  The main difference is of course that typical S values for nematic LC and alkyl chains of membrane lipids are about one order of magnitude larger than in elastomers and gels, and that LC can easily form macroscopically ordered phases, sometimes simply by orientation along the B₀ field of the NMR magnet. This may lead to a simple spectral splitting or shift arising from the significant residual tensorial interaction. In contrast, isotropic polymer samples always feature powder spectra (unless a bias is applied by, e.g., external deformation).

In LC as well as in elastomers/gels, already rather moderate MAS can tackle the problem of line broadening and provide high resolution (“HR-MAS”), as already shown in 1974 by Doskočilová and coworkers.²⁵  Notably, this was before Cohen-Addad²⁶  and Fedotov et al.²⁷  independently explained and exploited the appearance of RDCs in polymer melts and elastomers for the first time. Not until the 1980s did applications of the simple yet powerful HR-MAS approach appear, and it even took until 1996 for the technique to become widely recognized under the given name.²⁸  Early examples include ¹³C MAS NMR on thermoplastic elastomers²⁹  and vulcanized rubbers,³⁰  solvent-swollen poly(styrene) resins³¹  and superabsorbing gels.³²  Thanks to the rather moderate T₁ relaxation time, arising from the fast segmental motion, quantitative ¹³C spectroscopy by direct polarization (DP) rather than CP is easily possible. The polymer industry has recently started showing increased interest in this latter aspect, because ¹³C melt-state MAS NMR represents so far the only feasible method for the quantification of long-chain branching in polyolefins.³³  In this book, applications of ¹H/¹³C HR-MAS NMR to swollen gels and bulk rubbers are the subject of Chapters 7 and 8 by Liu and Feng and Kawahara et al., respectively.

In many applications, in particular those of rubbers made of simple homopolymers or statistical copolymers, the chemical resolution provided by HR-MAS is not of primary relevance. Rather, D_res as a measure of crosslink density is of interest. On a qualitative level, D_res is governing the transverse (T₂) relaxation measured in a Hahn-echo or related experiment, with non-refocused dipolar dephasing being the dominant source of signal loss, rendering ¹H T₂ experiments a good choice to obtain the desired insights, as illustrated in Chapter 9 by Asano. However, the most reliable measurement of D_res itself is certainly achieved by static multiple-quantum NMR,³⁴  which will be addressed in Section 1.2.

Turning to Figure 1.1, parts (d) and (e), we enter the domain of thermoplastic polymers used in the majority of packaging and engineering applications, namely block copolymers and semicrystalline polymers, respectively. In such systems, the dynamic heterogeneity present in single-phase systems (in particular close to T_g) is further amplified and coupled to morphology, i.e., the existence of phase-separated domains on the nanometre scale. Spectroscopically, there are now different complementary approaches to tackle such complexity.

Following our caveat, the most unbiased approach is clearly the detection of a fully relaxed ¹H or (sufficiently short T₁ permitting) even ¹³C signal under MAS or even static conditions after a single 90° pulse. The static ¹H NMR option is described in Section 1.2 (see also Figure 1.3). In short, fitting of a distortionless on-resonance ¹H low-field free-induction decay (FID), possibly combined with suitable filtering experiments and Hahn-echo decay data covering longer evolution times, can provide us in favourable cases with absolute values for the fractions of distinct components reflecting (nano)phases with different overall mobility.³⁵  Such an approach works well when the (nano)phases have distinctly different mobility, i.e., one being glassy or crystalline, and the other being far above T_g. This is the case for many phase-separated polymer blends, semicrystalline polymers and block copolymers made of glassy as well as rubbery components (for instance, different kinds of poly(styrene-co-butadiene)s such as HIPS, SBS etc.). Another separate class of materials with similar NMR signature are nanocomposites of a rubbery polymer and, e.g., silica spheres. In such cases, adsorptive interactions lead to an immobilized polymer layer of variable thickness in the few-nm range that can conveniently be quantified by ¹H low-field NMR.³⁶ 

Filtering experiments which make use of controlled signal loss (or generation) arising from either strong ¹H DCs, or intermediate mobility, are in such cases instrumental for meaningful fits, and they can be used to generate suitable initial conditions for spin diffusion experiments to estimate domain sizes in the nm range.^1,8,35  Experiments along these lines as applied to various polymer systems including nanostructured ones and nanocomposites are described in Chapter 9 by Asano, Chapter 10 by Pickering and White, Chapter 11 by Sun and co-authors and Chapter 12 by Wang. Finally, Chapter 13 by deAzevedo and co-authors describes how such filtering techniques in static low-field ¹H NMR can be applied to quantify segmental mobility in terms of correlation times, activation energies and dynamic heterogeneity.

The resolution of finer details of course requires the use of MAS to achieve high resolution and, in the case of ¹³C, an additional signal enhancement by use of CP is common and reasonable. The resulting signal is well known not to be quantitative in a sense of spectral integrals reflecting the correct stoichiometry. But at least for protonated ¹³C, CP build-up is often complete within a few hundred μs, while signal loss due to rotating-frame relaxation (T_1ρ) is not yet significant, resulting in at least semi-quantitative intensities. Nevertheless, in order to avoid conclusions on sub-ensembles, one should always vary the CP time, compare the obtained signal integrals with what is expected for a sample of the given size and certainly also compare with DP excitation. In this regard, for the (possibly simpler) case of lipid membranes, Topgaard and coworkers³⁷  have suggested that the comparison of three ¹³C excitation schemes, CP, DP and INEPT, can even provide semi-quantitative information on the timescale of motion and a level of local transient anisotropy (as represented by an order parameter S). INEPT makes use of J couplings to transfer the polarization during a rather long coherent evolution delay, which limits its efficiency to rather liquid-like components.

In ¹H MAS spectroscopy at moderate (up to ∼10 kHz) spinning it may be difficult to resolve and correctly integrate a broad, tens of kHz wide background of a potential rigid component that may be hidden below the baseline. Also in such a case an absolute-intensity calibration (e.g., by adding a known amount of material with a sharp resonance, such as PDMS rubber) is essential to quantify the dynamic fractions in the sample. Chapter 10 by Pickering and White presents an example of such a “spin counting” procedure as applied to the quantification of comonomer partitioning in tapered block copolymers.

In order to derive site-specific information on structure (proximities or even internuclear distances), timescales and amplitudes of motion (mostly quantified by an order parameter S, see eqn (1.2)), one needs to resort to multidimensional approaches. This means that the intensities of the spectral peaks are modulated with a relaxation process or a coherent spin evolution under the interaction to be isolated and measured (e.g., DC). Under MAS, the latter is achieved by specific recoupling pulse sequences. A specific example can be found in Chapter 14 by Miyoshi and co-authors, where the application of double-quantum (DQ) dipolar recoupling is described as a means to measure DCCs and thus the spatial relations (distances) of multiple ¹³C-labelled sites in polymer crystallites in order to conclude on chain-folding motifs.

A now rather popular recoupling experiment is the centreband-only detection of exchange (CODEX) technique by Schmidt-Rohr and coworkers,³⁸  which includes recoupling of the CSA and enables the modulation of (most frequently) ¹³C peak intensities by slow orientational jumps in the ms to s range, enabling the study of site-resolved slow dynamics. This experiment can serve as another example of sub-ensemble bias effects that may arise in the presence of correlation time distributions and related T₂ losses during the recoupling periods of the experiment, along the same lines as outlined for the ²H NMR example in Figure 1.2. We have studied such effects to some extent in my group theoretically as well as experimentally,^39,40  and the consequences will be illustrated in Section 1.2 by comparison with a less biased ¹H-based technique. I have also discussed similar effects in DQ recoupling experiments for DC measurements in a recent review.⁴¹ 

This book features several chapters covering a wide range of high-resolution solid-state NMR methods as applied to complex materials, often requiring MAS and dedicated recoupling experiments.⁴²  In the field of polymer electronics, the knowledge of packing relations of the chains in the amorphous and crystalline phases, and of course the crystallinity itself, is of utmost relevance to establish structure–property relations, which in this case mounts to an understanding of charge transport phenomena. Chapter 15 by Nieuwendaal and Chapter 16 by Selter and Hansen are dedicated to NMR applications in this field, including concepts of “NMR crystallography”, i.e., the use of chemical-shift information in combination with its quantum-chemical prediction to validate structural models. A related field is the one of solid polymer electrolytes for battery applications, where not only the mobility of the polymer but also that of ionic species within it are of interest. Chapter 17 by Fu, Chen and Yao presents relevant new work along these lines. The noted dynamic complexity of nanophase-separated synthetic polymers is also relevant in the biological domain, with silk being a prominent example. Chapter 18 by Guo and Yarger gives an overview of related studies. A more general perspective on solid-state NMR applications to membrane proteins can be found in Chapter 19 by Shigeta and Kawamura.

Finally, to mention the most vibrant areas of NMR method development, the recent years have witnessed a quantum leap forward concerning, first, the availability of MAS probes capable of spinning frequencies in significant excess of 100 kHz. This opens avenues for true high-resolution NMR of rigid solid samples in the ¹H dimension of various experiments with little additional effort. The contribution by Nishiyama et al. (Chapter 21) presents the new opportunities of such new technology on the examples of model samples, substances of pharmaceutical/biological relevance and also polymers, highlighting also the sensitivity enhancement of 2D NMR by inverse detection, and ¹H chemical-shift anisotropy as a less commonly utilized phenomenon.

Second, with the development and subsequent commercialization of dynamic nuclear polarization (DNP) as pioneered by Bob Griffin,⁴³  unprecedented signal enhancements, provided by coupling of NMR and ESR transitions of suitable radicals combined with polarization transfer by spin diffusion, have enabled NMR experiments deemed so far unfeasible. The vast majority of DNP-enhanced NMR is applied to aid in protein structure determination, for which a specific variant, namely photo-induced DNP, is covered in Chapter 19 by Shigeta and Kawamura. But also in solid synthetic polymers doped with appropriate radicals, DNP can be harnessed to enhance the signal of rare species such as branch points and end groups, which is the subject of Chapter 22 by Viel and co-authors. It must, however, be stressed that radical incorporation into nanophase-separated polymers is a challenge and may lead to bias effects (e.g., towards interfaces), such that further developments can be expected. In some contrast, a related solution-NMR technique called dissolution-DNP⁴⁴  enables homogeneous polarization enhancements in liquid samples. As to polymer-related applications, Hilty has prominently utilized this technique to, e.g., probe the structure and chemical kinetics of rare intermediates located at the chain end of growing polymer chains.⁴⁵ 

Third, the increasing availability of cost-efficient low- or even high-resolution NMR equipment, possibly using permanent low-field magnets, is enabling a continuing development of affordable hyphenated, i.e., combined, techniques. Of particular relevance for polymer science is the use of a high-resolution NMR spectrometer as detector for chromatography,⁴⁶  possibly using miniaturized detection schemes compatible with microfluidic devices, such as the stripline detectors advanced by Kentgens and coworkers.⁴⁷  Chapter 5 by Ute et al. highlights representative polymer-related applications involving size-exclusion chromatography (SEC). The integration of NMR and rheological experiments is also progressing. Early rheo-NMR, pioneered by the late Paul Callaghan,⁴⁸  was mainly focused on adapting simple rheological experiments for application within the wide bore of solid-state spectrometers for wide-line spectroscopy and imaging experiments. Nowadays, advances in low-field instrumentation using Halbach magnet arrays allows for NMR detection in geometries of high-sensitivity commercial rheometers,⁴⁹  enabling new classes of mostly proton-based experiments along the lines sketched above and in the following section.

As is apparent from the preceding section, in our experience the most representative and thus meaningful NMR observable, avoiding bias effects towards sub-ensembles with specific dynamics, is the detection of the signal of abundant protons in the time domain. This can be as simple as signal detection after a single 90° pulse. For experiments that are more complicated it always involves monitoring and quantifying signal decay, in order to have absolute control on the differently behaving chemical units in the sample.⁴¹  Spectroscopic resolution to achieve chemical selectivity (via MAS) can be added if required by a potentially complex material, but is often not necessary when simple homopolymers are of interest.

To illustrate this philosophy,³⁵  Figure 1.3 shows FIDs detected for semicrystalline poly(ε-caprolactone), PCL. We compare the normal FID, subject to a significant dead-time penalty, with one detected after a mixed magic-sandwich echo (MSE).^35,50  The latter refocuses essentially all relevant (chemical-shift, B₀ inhomogeneity and dipole–dipole) interactions, and enables detection starting at t=0. Even this rather short and very efficient pulse sequence is subject to signal loss, mostly arising from imperfections such as fine finite pulse length, and effects of molecular motion (see also Figure 1.2), but the signal's shape is hardly affected, aiding in a stable fit. The quickly decaying response from the strongly dipole–dipole-coupled crystalline fraction can even be excited selectively by applying a short DQ filter, which results in a stable determination of its shape parameters. These are either the second moment M₂ of a Gaussian decay ∼exp[−M₂t²/2] or, in the given case, the parameters of the “Abragam function” that accounts for the small residual oscillation arising from the dominance of rigid CH₂ spin pairs. The long-time decay of the most mobile amorphous components can be isolated by a magic-and-polarization echo filter (MAPE, essentially a modified MSE),^35,51  or other suitable pulse sequences such as the 12-pulse dipolar filter of Schmidt-Rohr and Spiess,¹  and then fitted to a stretched/compressed exponential ∼exp[−(t/T₂)^β].

With shape parameters for the crystalline and amorphous components fixed, one can then fit the FID (that is not subject to any imperfection-related signal loss) to a three-component function, adding an interphase component that also follows a stretched or compressed exponential. In this way, we could obtain crystalline fractions f_c of semicrystalline polymers that are in almost quantitative agreement with values from other methods.³⁵  If needed, the FID decay can be amended by Hahn-echo or CPMG decay data (on the same absolute intensity scale of NMR detection) to cover exponential long-time components. Note that it is never advisable to fit the full FID, as the signal shape at long times is unknown and dominated by magnetic-field inhomogeneities (in particular at low field) and other factors. In our experience, fits to about 200 μs are always suitable. I stress that such a time-domain approach is always preferred over fitting of spectra, due to baseline issues and essentially unknown lineshape functions (note that in the frequency domain there is no equivalence to a focus on short times!). The filters described above can of course all be used for spin-diffusion NMR studies of domain sizes.^1,8,35 

As noted, such an approach works very well also for the quantification of adsorbed polymer layers on nanoparticles, where the amount of the immobilized fractions can be a strong function of temperature when the adsorption phenomenon is coupled to an increase in T_g.³⁶  Another example where molecular motion is imparting a strong temperature dependence on the ¹H FID is illustrated in Figure 1.4. Poly(ethylene oxide), PEO, is a so-called crystal-mobile polymer, meaning that chains can readily diffuse through the crystals as a consequence of a defect migration mechanism.^8,52  In a recent study, we were able to extract jump correlation times and their temperature dependence, i.e., the activation parameters, from the (unbiased) analysis of the temperature-dependent FIDs, using a suitable theoretical approach in fitting the data.⁵²  This is illustrated in Figure 1.4(a) and (b). Importantly, meaningful fits can only be obtained by assuming a distribution of correlation times, for which we have assumed a log-normal shape. As a result, we found that the full width of the distribution at half height is about a decade, depending somewhat on the specific sample.

These findings were compared with results obtained from ¹³C CODEX experiments, which were introduced in the last section as a powerful MAS recoupling experiment providing a direct measure of slow motions by probing reorientations of the CSA tensor.³⁸  The data in Figure 1.4(c), covering a different but slightly overlapping temperature range, yielded roughly the same activation energy, but differed in absolute magnitude of τ_c by almost a decade. We have investigated and clarified the origin of this deviation in quite some detail,^39,40,52  concluding that the origin of the deviation is due to the same effect as illustrated in the context of Figure 1.2(a), i.e., in the relevant temperature range a large fraction of segments moves at an intermediate timescale, resulting in a strong transverse relaxation decay during the CODEX pulse sequence. This removes a specific fraction of segments from the signal, while leaving only those segments moving sufficiently slowly (providing a longer apparent τ_c), or even in the fast limit (distorting the long-time plateau of the CODEX intensity which informs about the number of sites of a jump process).

In the same work,⁵²  we have also checked the use of ¹H T_1ρ relaxation to study the same process. This is also a rather feasible method because its measurement is a priori not subject to a potential sub-ensemble bias, as one measures the decay of the separated crystalline signal fraction (along the lines illustrated in Figure 1.3) during a time-incremented spin-lock pulse directly following the excitation pulse. An older study along these lines gave a value for E_a that was only half as large as our average result from ¹H FIDs and ¹³C CODEX. We could show that, again, the deviation resulted from the substantial τ_c distribution. With the T-dependent data covering the region of the T_1ρ minimum, a distribution leads to a flattening of this minimum and the T dependence in its vicinity, thus fitting without considering the distribution gives a systematically too low E_a.

In a similar spirit, over the last 15 years we have further developed a more specific experiment, namely static (possibly low-field) ¹H multiple-quantum (MQ) NMR, which is the most reliable way of determining in particular residual DCCs (D_res, see eqn (1.2)) in different polymer systems.^34,41,53  The method provides quantitative access to this monomer-specific quantity, in particular, one can also extract distributions of D_res reflecting the microstructure and heterogeneity of, e.g., different network systems such as elastomers and swollen gels. This enables us, and more and more other groups, to address many fundamental questions concerning the physics of polymer networks. Many of these applications are covered in a more recent review,⁵³  so we limit ourselves to just one illustrative example; see Figure 1.5.

The main advantage of MQ NMR consists in the measurement of not just a DQ-filtered build-up curve I_DQ(τ_DQ) as a function of DQ excitation pulse sequence time τ_DQ, reflecting the magnitude of (R)DCCs, but also a fully (dipolar and chemical-shift) refocused sum intensity function, I_ΣMQ(τ_DQ). The latter is usually obtained by combining the DQ curve and a reference curve, which are mutual complements and can be measured by changing the phase cycle via switching between the selection of 2n+2 and 4n quantum orders. Alternatively, using other phase cycling options, more efficient modes of measuring the very same data exist.⁴¹  Then, a relaxation-normalized function I_nDQ(τ_DQ) is obtained by point-by-point division of the two primary signal functions, allowing for fitting approaches just considering a single D_res value – or distributions thereof.

The data shown in Figure 1.5 were taken on a special kind of model hydrogel (“tetra-PEG”),⁵⁴  consisting of two different kinds of monodisperse 4-arm PEO star molecules that can be hetero-end-linked. This synthetic strategy leads to networks with exceptional homogeneity and the expectation of a single D_res value reflecting the pre-defined chain length, see eqn (1.2). However, the actual response shows strong qualitative indications for structural inhomogeneity in terms of a D_res distribution and the presence of uncoupled, isotropically mobile defects, since the nDQ curve does not reach the theoretically predicted long-time plateau at a level of 0.5. Notably, the nDQ curve is structured, having two maxima, indicating the presence of distinct components. In fact, a simultaneous fit to the two primary signal functions assuming the presence of three well-defined (non-distributed) components with distinct D_res and T₂ relaxation times was possible, with the results for the component fractions indicated in Figure 1.5. These were in semi-quantitative agreement with analyses of tetra-PEG gels formed in a computer simulation, and could thus be identified as connectivity defects, which are in fact expected and unavoidable. Such selective information is in the given case only obtainable because the tetra-PEG gels lack other kinds of inhomogeneities, such as the normally dominating swelling heterogeneities related to fixed spatial concentration fluctuations.³⁴ 

Combined with temperature variation, the ¹H MQ NMR method further enables the direct observation of polymer chain dynamics over all relevant dynamic regimes and thus a validation of polymer theories.⁵⁵  The qualitative difference between networks and linear (or also branched) melts is that the latter can flow. NMR-wise, this means that the segmental OACF C(t), see eqn (1.1), does not reach a plateau, but continues to decrease at long times following the different relaxation modes of entangled chains until its complete isotropization by terminal diffusion. As noted in Section 1.1, FC NMR relaxometry has long been the method of choice to elucidate details of C(t), which consists of a sequence of different power laws.^10,14,17,18  We could show that the same information can also be gleaned from ¹H MQ NMR.^34,53  Our latest work⁵⁵  in a longer series of papers was concerned with an extension to lower temperatures, covering also the regime of free Rouse motion. This required the combined use of two different MQ NMR experiments covering different magnitudes of RDCCs.

The data in Figure 1.6 illustrate the steps in data analysis, starting from I_nDQ(τ_DQ) build-up data taken at different temperatures. These curves are strongly temperature-dependent and do not reach the theoretical long-time plateau of 0.5, which is due to the fact that C(t) does not reach a plateau but continues to decrease with time. In other words, a D_res value fitted to the initial rise is apparent and temperature-dependent. One can in fact show that I_nDQ(τ_DQ) is proportional to C(t)τ_DQ², meaning that I_nDQ(τ_DQ)/τ_DQ² is proportional to the desired correlation function in a sufficiently narrow interval of small τ_DQ. It is then possible to use the principle of time-temperature superposition (TTS), via dividing τ_DQ by a characteristic chain relaxation time whose temperature dependence is known (e.g., from rheological experiments) to obtain a master curve of C(t) covering many decades in time. A result is shown in Figure 1.6(c), and an analysis of the power law exponents in the different regimes as well as the MW dependence of the cross-over times are in good agreement with the tube model of polymer dynamics.⁵⁵  Recently, we have also devised a new fitting approach of a larger part of the MQ NMR signal functions to extract the amplitude of C(t) and its local slope (power law exponent) for a given temperature, thus not requiring TTS and avoiding issues with low signal in the initial-rise analysis.⁵³ 

One should mention that this ¹H MQ NMR result is robust towards isotope dilution, i.e., qualitatively the same result is obtained in sparse protonated chains diluted in a deuterated background. Such strategies can be used to separate intra- and intermolecular contributions to the relevant correlation function. This is the basis of the measurement of mean-square displacements by FC NMR.¹⁸  It is so far not clear why the MQ NMR result is robust with regards to the presence or absence of intermolecular contributions, which just seem to affect the amplitude of C(t) and can thus be normalized away. The “intra-only” result from FC NMR should in principle also follow the tube-model predictions, but in practice there are significant deviations of as yet unclear origin. A more detailed presentation of the FC NMR technique by Rössler and co-authors is given in Chapter 6 of this book.

Concluding, I hope that I have shown that even rather undemanding static ¹H NMR techniques are very useful to gain rich – and in particular unbiased – insight into structural and dynamic aspects of different polymer systems. All experiments in this section can be (and have been) carried out on low-field instruments, but using them at higher field of course provides higher sensitivity for distinct components with possibly small fractions. Also, MAS variants of these experiments exist, using e.g., specific DQ recoupling pulse sequences for the case of MQ NMR,⁴¹  allowing for site-resolved experiments in more complex and chemically less uniform polymer systems.

I thank my academic advisers and my group members, many of them being co-authors of the cited publications, for their invaluable contributions in advancing the described research fields over the years. This chapter is dedicated to my predecessor Horst Schneider on the occasion of his 80th birthday. I thank him for stimulating my personal focus on low-resolution NMR methods, his continued interest in the work in the current NMR group in Halle and our long-standing cooperation on applications and analyses of ¹H spin-diffusion NMR.