Microstructure and molecular dynamics of elastomers as studied by advancedlow-resolution nuclear magnetic resonance methods∗

Kay Saalwachter∗†

Institut fur Physik – NMR, Martin-Luther-Universitat Halle-Wittenberg,Betty-Heimann-Str. 7, D-06120 Halle, Germany

(Dated: February 2012)

Nuclear Magnetic Resonance (NMR) certainly belongs to the most powerful spectroscopic tools inrubber science. Yet, the often high level of experimental and in particular instrumental sophisticationrepresents a barrier for its wide-spread use. This contribution reviews recent advances in low-resolution, often low-field, proton NMR characterization methods of elastomeric materials. Chemicaldetail, as normally provided by chemical shifts in high-resolution NMR spectra, is often not neededwhen just the (average) molecular motions of the rubber components is of interest. Knowledge ofthe molecular-level dynamics enables the quantification and investigation of coexisting rigid andsoft regions, as often found in filled elastomers, and is further the basis of a detailed analysis of thelocal density of crosslinks and the content of non-elastic material, all of which sensitively affect therheological behavior. In fact, specific static proton NMR spectroscopy techniques can be thought ofas “molecular rheology,” and they open new avenues towards the investigation of inhomogeneities inelastomers, the knowledge of which is key to improving our theoretical understanding and creatingnew rational-design principles of novel elastomeric materials. In this review, the methodologicaladvances related to the possibility to study not only the crosslink density on a molecular scale butalso its distribution, and the option to quantitatively detect the fractions of polymer in differentstates of molecular mobility, and to estimate the size and arrangement of such regions, are illustratedon different examples from the rubber field. This concerns, among others, the influence of thevulcanization system and the amount and type of filler particles on the spatial (in)homogeneity ofthe crosslink density and the amount of non-elastic network defects, and the relevance of glassyregions in filled elastomers.

Keywords:

Contents

I. Introduction 1

II. Conceptual Basics and Experiments 3A. Basic principles of pulsed NMR 3B. Proton NMR and mobility 3C. Quantitative detection of immobilized

components 6D. Spin diffusion studies at low field 7E. Time-domain signals of polymers far above Tg 8F. MQ NMR measurement of the crosslink

density and spatial inhomogeneities 10

III. Recent low-field NMR applications inrubber science 11A. Are crosslinked rubbers homogeneous or

inhomogeneous? 11B. Quantative correlation of NMR and

equilibrium swelling 13C. Filler effects I: NMR-detected crosslink

density and inhomogeneities vs. macroscopicproperties 14

∗published in Rubber Chem. Technol. 85 (2012) 350–386;doi:10.5254/rct.12.87991†Electronic address: kay.saalwaechter@physik.uni-halle.de;URL: www.physik.uni-halle.de/nmr

D. Filler effects II: surface-immobilizedcomponents 16

IV. Summary and conclusions 19

Acknowledgments 20

References 20

Biographical Sketch 24

I. INTRODUCTION

This review is concerned with applications of modernlow-resolution (LR) proton nuclear magnetic resonance(NMR) techniques in rubber science. As we will see, asmall set of robust and in principle easy-to-use pulsedNMR experiments, implemented on simple and cost-efficient low-field (LF) equipment, can be combined toreveal detailed information on inhomogeneities in rubbermaterials. These comprise the quantitative assessmentof not only the average crosslink density, but its (spatial)distribution, the quantification of the amount of elasti-cally inactive, no-load-bearing polymer chains associatedwith connectivity defects or free chains, and the amountof immobile polymer contributions, as for instance foundas absorbed species on the surface of filler particles, or inthe form of giant crosslinks in thermoplastic elastomersor physical gels.

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Rubbers are soft solids, and the elastically active poly-mer component, as discussed below, features an almostliquid-like chain mobility. Nevertheless, the essentialspectroscopic information arises from a specific featureof solid-state NMR, namely the orientation dependenceof certain magnetic interactions of the spin-carrying nu-clei. The term spin interaction refers to corrections tothe Zeeman energy that is measured in NMR, such asthe chemical shift, defining the exact position of a spec-tral line, or the through-bond isotropic J coupling be-tween different nuclei, leading to small splittings of thelines. The rather broad spectroscopic lines and thus badresolution featured by rubbers and even linear polymersin solution now arise from through-space dipole-dipolecouplings, which, owing to their orientation dependence(anisotropy), are usually averaged to zero in low-molarliquids with isotropic mobility. However, residual dipo-lar couplings (RDCs) persist in rubbery materials due tothe fact that the motion of network chains is anisotropicdue to chemical crosslinks and physical entanglements [1].The RDC phenomenon is the reason for the low spectralresolution in rubbers.

Therefore, solids-specific line-narrowing techniquessuch as magic-angle spinning (MAS) are needed if rub-bers are to be investigated with chemical detail. Forprinciples and applications of high-resolution solid-stateNMR, the reader is referred to established monographs[2–4] and review articles [5–8], of which refs. [5, 6] arespecifically focussed on solid-state NMR applications inthe rubber field. High-resolution NMR will not be furtherdiscussed in this article, yet its particular use should behighlighted by the 13C MAS NMR studies of the chemicalstructure of crosslinks in rubber by J. L. Koenig and hiscoworkers [9–11], which can be considered groundbreak-ing. 1H MAS NMR, in particular when applied underthe conditions of very fast spinning, can further be usedto detect and characterize even small amounts of rigidfiller-bound rubber components with chemical resolution[12, 13].

The RDC phenomenon, which is directly related to theshort transverse relaxation times (T2) of rubbers mea-sured in spin-echo experiments [1, 15–18], is at the coreof this article, as it provides an indirect yet quantitativemeasure of the crosslink density. Similar studies havebeen carried out using deuterium (2H) NMR, for whichthe quadrupolar coupling is the relevant anisotropic in-teraction, and which are based on much the same prin-ciples [19–21]. However, such studies require isotope la-belling and the use of high-field instruments, stressingthe practical advantages of proton NMR. Apart from thequantitative study of crosslink density via proton RDCs,possibly in low-field/low-resolution instruments [22, 23]or even in highly inhomogeneous fields of surface NMRdevices [24, 25], it can be used as a contrast mechanismin NMR imaging experiments [26, 27], whereby inhomo-geneities in crosslinking down to the 10 µm range areaccessible. Below, we will deal with the detection of in-homogeneities on even smaller scales. Anisotropic spin

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Figure 1: Sketch of the inhomogeneous nanoparticle distribu-tion in a filled rubber for, e.g., tire applications (upper part),and storage modulusG′ in linear response at a shear frequencyof 100 Hz of unfilled vs. silica-filled SBR as a function oftemperature (lower part). The inset shows the reinforcementR = G′filled/G

′unfilled as a function of temperature difference

to the glass transition. Tire image by courtesy of ContinentalAG; data from ref. [14].

interactions further reflect the mechanical stress at thechain level, which can thus be visualized in imaging ex-periments on macroscopic, strained samples [28]. NMRexperiments on strained samples generally provide a sen-sitive means to explore the validity of rubber elasticitytheories [21, 29], yet the large body of related NMR workis beyond the scope of this application-oriented review.

In the following, an introduction into the underlyingprinciples of the proton NMR response of elastomers ispresented, with an emphasis on observables that helpexplaining the mechanical properties of unfilled, and inparticular filled rubbers. Various low-field NMR ap-plications will then be presented, ranging from stud-ies of model compounds that demonstrate the feasibilityof studying crosslinking inhomogeneities, comparisons oflow-field NMR results with mechanical and equilibriumswelling experiments, and finally a closer look at rubbersfilled with a variety of often nanometric particles.

Fig. 1 highlights one of the intriguing features offiller effects in rubber. While unfilled SBR exhibits theexpected positive temperature dependence as expectedfrom the entropic models of rubber elasticity in the high-temperature plateau region of G′, filled SBR has a muchhigher modulus which usually decreases significantly onheating. This is emphasized by plotting the reinforce-ment factor R(∆T ) = G′filled/G

′unfilled [30] at constant

temperature difference to the glass transition shown inthe inset, which is seen to be highest at temperaturesaround and above the glass-rubber transition. High val-ues exceeding R = 10 and the decrease at higher temper-

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atures indicate synergistic and non-trivial effects of thefiller network, where the NMR study of interfacial phe-nomena in the as-made compound is of particular rele-vance, as a “glassy layer” of immobilized polymer mate-rial on the particle surface may constitute the “sticker”between the particles, thus explaining the softening athigher temperature [30].

As we will see, low-field NMR techniques can be usedto detect quantitatively the amount of immobilized poly-mer in such filled systems, and further to study, inde-pendently and specifically, the crosslink density and po-tential inhomogeneities in the rubbery matrix of filledelastomers. In particular the latter two quantities arenot directly accessible by any other comparably simpletechnique, but are of relevance for the understanding,modelling, and optimization of the mechanical proper-ties.

II. CONCEPTUAL BASICS ANDEXPERIMENTS

This section gives an overview of the NMR con-cepts and experiments that are relevant for low-field/low-resolution (LF/LR) studies of protons (1H), emphasizingthe relevant observables, the approaches for data treat-ment, and the relation of the results to molecular dy-namics and structure in rubbers. Importantly, LF/LRNMR implies that spectra are not of concern. The oftenvaluable chemical-shift information is always obscured byeither the intrinsic signal width, or (for the case of mo-bile polymer components) by magnetic-field inhomogene-ity, as cost-efficient permanent magnets are used. Thismeans that the free-induction decay (FID), that is ei-ther directly acquired after the common single 90◦ radio-frequency (rf) pulse or after a more complicated pulsesequence, is usually not Fourier-transformed but ratheranalyzed directly in the time domain (TD). TD NMRand LF/LR NMR are thus synonyms. Since the readermay be more familiar with NMR spectra rather thantime domain signals, some phenomena below will alsobe discussed in terms of their spectral features. Basicknowledge of solution-state NMR is assumed, and thenecessary background in pulsed solid-state NMR is pre-sented. Table I provides a list of the most relevant NMRexperiments and concepts.

A. Basic principles of pulsed NMR

A LF TD NMR experiment in its simplest form justconsists of a single 90◦ rf pulse, with the rf frequencytuned to the resonance condition for 1H nuclei in thegiven magnetic field. The rf frequency corresponds tothe Larmor frequency νL = ∆E/h, where ∆E is the dif-ference in the Zeeman energy levels of the spin-up and

spin-down states of a spin-1/2 nucleus in the primarymagnetic field B0. In LF NMR, magnetic fields are typ-ically in the range of 0.2–1.5 T, corresponding to νL of10–60 MHz, which can be provided by nowadays perma-nent magnets.

After the 90◦ pulse has rotated the sample magneti-zation from ||z (along B0) to the transverse plane, themagnetization vector starts to precess around the B0 fieldwith the same νL, thus inducing an AC voltage with thecorresponding frequency in the same coil that was usedfor the pulse. Using clever electronics, the very weak nu-clear induction signal can be separated from the strongpulse (subject to a so-called dead time problem, see be-low), and then amplified and digitized. This so-calledFID signal then persists for a time of a few tens of µsto many ms, as limited by the so-called apparent T2 re-laxation time. Importantly, before digitization the signaloscillating with n×10 MHz is mixed down by subtractionof the νL carrier frequency (“rotating frame”), essentiallyremoving any oscillation if exactly the right frequency issubtracted (“on-resonance FID”).

In more complicated experiments, so-called pulse se-quences consisting of one or many rf pulses of variableduration and phase are applied either before or after theactual 90◦ pulse providing the detected signal. The mostsimple one of such experiments is certainly the popu-lar Hahn-echo experiment, where a 180◦ pulse is appliedafter duration τ after the 90◦ pulse. Signal acquisitionthen starts another period τ later on the top of an echo inwhich effects of time-evolution due to magnetic-field in-homogeneity are removed. In this way, true T2 relaxationtimes can be measured by following the signal intensityin the time domain as a function of τ . Below, a numberof more complicated experiments based upon the sameprinciple (time variation in a complex pulse sequence,monitoring changes in signal intensity) will be discussed.

B. Proton NMR and mobility

The only NMR “fine structure” effect relevant for1H TD NMR is the distance-dependent through-spacehomonuclear dipole-dipole coupling (DDC). In ordinaryorganic solids, the coupling constant between neighbor-ing protons is on the order of 20–30 kHz and thus dom-inates the spectra even at the highest fields, leading tobroad spectra with Gaussian shape. Since its distance de-pendence follows ∼ r−3, it is dominated by next-neighborinteractions, but due to the high abundance of protons,it is nevertheless intrinsically a many-spin phenomenon.Since the Fourier transform of a Gaussian function isagain a Gaussian (with inverse width), the correspond-ing time domain signal is a half-Gaussian function witha decay time constant of typically 15–20 µs.

The origin of the DDC interaction can be easily under-stood on the basis of two interacting bar magnets repre-senting magnetic dipoles with fixed orientation of theirprincipal axes (corresponding to the B0 field direction

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Table I: Explanations of common abbreviations and acronyms of various NMR techniques.

meaning short explanationHR NMR high-resolution NMR NMR at high (n×100 MHz) Larmor frequencies in very homogeneous fields pro-

vided by superconducting magnets, requiring specific line-narrowing techniqueswhen solids are investigated

LF NMR low-field NMR NMR at Larmor frequencies of typically 50 MHz or less, often provided by per-manent magnets with high field inhomogeneity, leading to low resolution

LR NMR low-resolution NMR NMR in inhomogeneous, often low magnetic fields; spectral features are blurred,chemical-shift resolution is lost

TD NMR time-domain NMR NMR without Fourier transformation when it is not necessary due to low reso-lution; analysis of time-dependent NMR intensities rather than spectra, eitherdirectly on the detected free-induction decay (FID) or as a function of a specialtiming parameter in a pulse sequence, mainly to determine relaxation times

MAS magic-angle spinning alternative to “static” NMR: fast (n kHz) rotation of the sample in a ceramic tubeinclined by 54.7◦ w/r/t the magnetic field, leading to high-resolution solid-statespectra

1H NMR proton NMR NMR with protons as the most sensitive and most abundant nucleus; in thesolid state, the high abundance leads to strong dipole-dipole couplings and badspectral resolution

13C NMR carbon NMR low natural isotopic abundance of 1% and low sensitivity leads to low signal ofthis and other heteronuclei, requiring signal enhancement techniques such as CP

CP cross polarization pulse technique used in solid-state NMR of 13C or other lowly abundant, insen-sitive nuclei to enhance their low polarization and thus increase the signal

MSE (mixed) magic-sand-wich echo

special spin echo pulse sequence leading to a time-reversal of all relevant spininteractions in static 1H NMR; used to overcome the instrumental dead timeand, with long echo durations, as a “dipolar filter” in samples with regions ofdifferent molecular mobility

MQ NMR multiple-quantumNMR

NMR based upon pulse sequences creating special coherently superposed statesof many spins, mostly used in TD experiments for the study of dipole-dipolecouplings; can be used to select and probe polymer regions of different mobilityon the basis of the different dipole-dipole couplings

DQ NMR double-quantum NMR special case of MQ NMR when only two-spin pair interactions are relevant

acting as quantization axis) and fixed distance of theircenters. Obviously, the potential energy of one magnetin the field of the other is changing its sign when the othermagnet is turned around by 180◦. The magnets of courserepresent two spins, and the dipolar potential energy is asmall correction to the Zeeman energy. Since the spin-upand spin-down states are in an ensemble almost equallypopulated, the resulting NMR spectrum shows a doubletfor each nucleus (or only one doublet if the two nucleiare magnetically equivalent), offset from the bare Lar-mor resonance frequency by plus or minus the couplingconstant.

The bar magnet analogy goes even further: even withfixed distance and fixed magnet orientations, differentconfigurations can be realized by rotating the inter-magnet connection vector relative to the fixed individ-ual magnets’ orientations (identified with the externalB0 field direction). Such an overall rotation of thepair connection vector relative to B0, henceforth asso-ciated with the angle θ, changes the potential energyas well. The limiting cases for bar magnets are well-known, i.e, two bar magnets on top of each other withthe same N/S orientation attract each other, but theyrepel each other in side-by-side (θ = 90◦) configuration.The dipolar splitting observed for a single pair of equiv-alent nuclei is thus a function of orientation. Specifi-

cally, it follows the second Legendre polynomial of cos θ:P2(cos θ) = 1

2(3 cos2 θ − 1). The splitting for θ = 90◦

(P2(cos 90◦) = −0.5) is thus half as large as for θ = 0◦

(P2(cos 0◦) = 1); note that the sign change cannot bedetected. Since most NMR studies are performed onpolycrystalline powders or amorphous substances, it isclear that all possible orientations contribute to a givenspectrum. For just spin pairs, this leads to the charac-teristic double-horned “Pake” spectrum [31]. Since pro-tons are abundant, each proton couples with many others(splitting of a splitting of a splitting...), which leads toa smearing-out of the characteristic features and to theGaussian spectral shape mentioned above.

In the time domain, the quick decay can be understoodas an interference effect: the signal consists of many dif-ferently oscillating components, each with its individualfrequency due to the different orientation-dependent cou-plings, which loose their phase relation. This process istermed “dipolar dephasing”, and it is distinguished bythe characteristic Gaussian shape as opposed to the com-mon exponential decay arising from motion-induced truerelaxation effects. The difference between coherent de-phasing and true relaxation is that the former can betime-reversed in a suitable spin echo experiment (see be-low).

The DDC discussed so far is thus an important mea-

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Figure 2: (a) FIDs of poly(styrene), PS, at different temper-atures. The dashed lines are fits using a suitable theoreticaldescription. The decrease of the initial intensity with tem-perature reflects the Curie effect. Due to the finite receiverdead time (rdt.) the initial 12 µs cannot be detected. (b)FIDs of another glassy polymer, poly(ethyl acrylate), PEA,in an equivalent temperature range, detected after a magic-sandwich echo which solves the dead time problem. The solidlines are again fits to theory. (c) Initial FID intensities takenfrom (b) and multiplied by T/(400 K) vs. temperature. Theintensity minimum identifies the temperature at which thecorrelation time of motion is in the µs range. Data in (b) and(c) are taken from ref. [32].

sure of structure, as it reflects distance and orientationof a pair of nuclei. More intriguing features come intoplay when the spins do not to stay in place for the

timescale of FID acquisition, which is on the order ofthe inverse width of the dipolar spectrum. If molecu-lar dynamics changes the spins’ orientation during thistime, the appearance of the spectrum (and the TD signal)changes. In the fast-motion limit, the spin pair changesits orientation many times on this timescale. If all pos-sible orientations are sampled, the static DDC frequencyis replaced by its time average and since the average〈P2(cos θ(t))〉t = 0 for dynamics sampling an isotropicorientation distribution, dipolar broadening effects van-ish in isotropic liquids. This is the phenomenon of mo-tional narrowing in the spectral domain, and is high-lighted in the time domain by FIDs of poly(styrene)shown in Fig. 2a. It is seen that the action of DDCs,which are responsible for the quick decay at low temper-atures, vanishes upon heating above the glass transition,when the monomer units start rotating almost isotropi-cally (α process).

Motional narrowing of a spectrum is thus equivalentto an increase of the apparent T2 relaxation (decay) timeof the FIDs. The biggest changes occur around 419 K,which is about 35 K above the glass transition temper-ature. This temperature may be tentatively called the“NMR Tg”. It is defined as the temperature at which thecorrelation time of motion is on the order of the inverseDDC constant, i.e., in the µs range. NMR thus measuresa “high-frequency Tg” in a similar sense as in dielectricor mechanical spectroscopy. The width of a proton spec-trum, or equivalently, the FID decay time is thus a sensi-tive means to distinguish phase-separated polymer com-ponents by their mobility, ranging from the rigid solid(tens of kHz linewidth, FID decays within ∼50 µs) tothe isotropic liquid limit (<1 kHz linewidth, not resolveddue to the field inhomogeneity, FID decays on the manyms timescale).

Fig. 2a also highlights two other experimental issues.First, the intensity of an NMR signal follows the Curielaw (∼ 1/T ), resulting from the decreasing thermal pop-ulation difference of the two spin states in the high-temperature approximation. Provided that experimentsare conducted with a sufficiently long relaxation delay(recycle delay � T1), the Curie law is followed exactlyand a correction can easily be applied if needed. Sec-ond, the initial part of the FID is undetectable becauseof the instrumental dead time, which is typically longerthan 10 µs on LF instruments. During this time, thestrong rf pulse power is dissipated in the circuit, onlythen enabling the acquisition of the much weak NMR sig-nal induced in the rf coil. Since theoretical predictionsof the signal shape are not generally possible or feasible,one cannot easily extrapolate the FID signal to zero ac-quisition time (t = 0), which is an obstacle to a precisequantification of the NMR signal (the signal at t = 0 isproportional to the total proton number in the sample).

This problem can be circumvented by applying a spinecho, which means that the FID decay is time-reversed,such that data acquisition can start at the top of an echoat effectively t = 0. At this point, it is important to stress

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that the well-known Hahn echo (90◦−τ/2−180◦−τ/2−acq.) cannot be used for this purpose, as it does not af-fect spin evolution due to homonuclear DDC. A solid echo(using a 90◦ instead of a 180◦ pulse) is a possible solu-tion that has been frequently applied in the filled-rubbercontext [33]. However, for quantum-mechanical reasonsit is not very efficient, and additional signal decay hasto be corrected for by back-extrapolation over a series ofecho delays τ . We have previously suggested the use ofthe more efficient so-called “mixed magic sandwich echo”[34, 35], MSE, which can also be easily implemented onLF equipment [36, 37]. The MSE is a complex pulse se-quence of duration 6τ , which serves to refocus the timeevolution due to both, field inhomogeneities (as does theHahn echo) and multi-spin homonuclear DDC. The in-terval τ is the time between the last pulse in the se-quence and the echo maximum, which must cover thedead time. Fig. 2b demonstrates the efficiency of thisapproach. Such MSE-FIDs are now amenable to quanti-tative multi-component fitting, see the next section.

A last point to comment in this context is that theMSE is also not fully quantitative in reconstructing thetotal magnetization, see Fig. 2c. At very low temper-atures (glassy range), there is a ∼20% signal loss as-sociated with experimental limitations (finite minimumpossible pulse length), which can be easily corrected for.The more important effect is the intensity minimum ob-served for temperatures around the “NMR Tg.” This isnot only the temperature at which the sensitivity of theTD signal to temperature changes is highest, but also thetemperature at which the true T2 relaxation time duringthe MSE is shortest. Since the MSE is at least 60 µs longto overcome the instrument dead time, the minimum trueT2, which is on the order of 100–200 µs, leads to another∼30% signal loss at Tg + 30–50 K. The condition for theT2 minimum is the same as the one given for the “NMRTg: echo formation is impeded if the correlation time ofmotion is on the order of the inverse NMR interaction(DDC) frequency. This is referred to as the “interme-diate motional regime,” and its effect on the MSE can,with some limitations, be used to study the timescale ofthe α process [32, 38].

C. Quantitative detection of immobilizedcomponents

The sensitivity of the FID signal to motion can be uti-lized not only to study the molecular dynamics of ho-mopolymers at different temperatures, but more impor-tantly, to study coexisting rigid and mobile components.A case specifically relevant to the rubber field is the pos-sible immobilization of the rubber phase by adsorptionto the surface of filler particles. This phenomenon will bediscussed in more detail in the application section below.Our data for a model-filled nanocomposite rubber systemconsisting of a crosslinked PEA matrix and monodisperse

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Figure 3: MSE-FIDs of crosslinked PEA, pure and filled with20 vol-% of silica particles 27 nm in diameter, modified toform chemical grafts to the rubber phase. The signal of thefilled PEA can be fitted (solid line) and thus decomposed into3 components of different mobility (broken lines). Data takenfrom ref. [32].

∼30 nm silica particles shown in Fig. 3 shall serve as anexample.

TD signals of pure phases can in almost all relevantcases be fitted very satisfactorily with a modified expo-nential function, ∼ exp{−(t/T app

2 )β}. For each compo-nent in an inhomogeneous sample, one thus has 3 inde-pendent fit parameters: a fraction f , the apparent (!)relaxation time T app

2 , and an exponent β. Compressedexponentials, also termed Weibullian functions, cover therange from Gaussian (β = 2) to exponential (β = 1) de-cay, corresponding to temperatures below and above the“NMR Tg”, see Fig. 2a,b. Stretched exponentials (β < 1)indicate rather mobile phases, combined with a distribu-tion of decay times (inhomogeneous superposition of in-distinguishable but dynamically different components).Note that FID data should never be fitted for acquisitiontimes longer than 200–300 µs! First, the modified expo-nential function, as many other theoretical functions likethe ones calculated on the basis of the Anderson-Weissapproximation [39], is only an approximation that worksfor sufficiently short times. Second, and much more prob-lematically, the shape of the FID on an LF instrumentis generally ill-defined at long times due to the unknownB0 field inhomogeneity.

The FID of the nanocomposite in Fig. 3 clearly showsfeatures of a rigid solid (signal decay within ∼50 µs) ontop of the signal of the pure matrix. In fact, a two-component fit is not sufficient, thus at least 3 componentshave to be used to obtain a satisfactory representation ofthe data. These components can be associated with therubber matrix, a phase of intermediate mobility closerto the particles, and the rigid adsorption layer. In ourwork, we could show that this is only a minimal model,in fact, the data is fully consistent with a region witha gradient in glass transition temperatures, arising fromthe strong immobilization of the monomers in immediate

7

contact with the silica surface [32, 40].This of course raises the question whether such de-

compositions are unique, considering the many indepen-dent fit parameters. It must be stressed that fits with3 or more components are only stable and make senseif some of the fitting parameters can be determined in-dependently and thus be fixed during the fit. For ex-ample, the FID of the pure PEA matrix can, with somelimitations, taken to represent the mobile phase in thenanocomposite. Suitable magnetization filters based onpulse sequences that are sensitive to the different mobil-ity (such as the MSE itself) can further be used to isolatethe signal of only the mobile or only the rigid phase. Suchstrategies are discussed in detail in refs. [32, 37]. Anotherpossible approach, applicable to systems with a smoothgradient in mobility, is to find a suitable model that an-alytically describes this gradient and the correspondingsignal functions with only a few parameters, and deter-mine these from the fits [40].

Similar strategies can of course be applied to studyquantitatively the phase coexistence in semicrystallinepolymers [36, 41, 42] or thermoplastic block copolymersconsisting of hard and soft blocks, such as SBS [37, 43].

D. Spin diffusion studies at low field

The different NMR properties of phases with very dif-ferent mobility can not only be utilized for their quan-tification, but also for the study of domain sizes, pro-vided they are in the range of about 1–100 nm. For thispurpose, a quantum-mechanical process termed spin dif-fusion can be employed. Spin diffusion arises from theexchange of the magnetization state of two neighboringspins (“flip-flop process”) mediated by the DDC betweenthem [31], and becomes visible if regions in the sample,such as different phases in an inhomogeneous system, canbe selectively polarized.

In the ensemble average, a sequence of many such spinflip-flops leads to a time-dependent magnetization profilethat can be modelled as a diffusion process. Note that noactual material transport is involved, the magnetizationcan “diffuse” among spatially fixed spins. Since estimatesand calibration procedures exist for the spin diffusion co-efficient of many different glassy and mobile polymers[4, 45, 46], spatial dimensions can be determined from thetime-dependence of the overall magnetization in the dif-ferent phases. The course of spin diffusion experiments isschematically shown in Fig. 4 for the cases of phase selec-tion and distinction by chemical shift (spectral, high-fieldonly) and by mobility. The latter approach is relevant forLF TD NMR.

The basis of such experiments are the very samemobility-based magnetization filters mentioned above. Aspin diffusion experiment is nothing more than such afiltering pulse sequence, followed by a spin diffusion pe-riod (mixing time, tm) during which the magnetization is

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Figure 4: (a) Schematic illustration of the spin diffusionprocess after applying a magnetization selection in a nanos-tructured, two-component inhomogeneous polymer sample,adapted from ref. [44]. (b) Spectroscopic observation ofthe spin diffusion process after selection of one phase forcases where the two components have distinguishable chem-ical shift. (c) Time-domain observation of the spin diffusionprocess after selection of the mobile phase for phases withlargely different mobility. Data taken from ref. [37].

“stored” along the z axis, and then “read out” by another90◦ pulse, possibly using an MSE to detect a dead-time-free FID. Typical data is shown in Fig. 4c, where it isseen that rigid-phase signal is gradually re-appearing ina time range of many ms after it has been suppressed bya suitable filter.

The essential problem of such TD experiments at lowfield, as compared to high-field high-resolution versions[4, 44], is that T1 relaxation times of mobile componentsare particularly short at low field, down to tens of ms. Insuch cases, T1 relaxation competes with the spin diffusionprocess, which is represented by the decay of the initialintensities upon increasing tm in Fig. 4c. While compo-nent analysis is still straightforward, the analysis of thetm dependence of the different signal fractions becomesrather involved if the extracted domain sizes should bemore that just rough estimates. This ultimately requiresinvolved numerical simulation of the spin diffusion data,which is the actual subject of ref. [37]. Even though we

8

have tried to formulate some simple rules for approximatedata analysis along the lines of the established initial-slope analysis [4], currently limited to lamellar morpholo-gies, the truly quantitative evaluation of low-field spindiffusion data involving components with short T1 can-not be considered routine at the moment.

E. Time-domain signals of polymers far above Tg

Let us now take a closer look at highly mobile com-ponents, defined by the condition that the experimentaltemperature classifies as “far above Tg,” referring to therange in which the motional narrowing effects on the FIDare essentially complete and effects of the static DDCshould be gone. We thus deal with components whose ap-parent T2 relaxation time is on the order of or longer thanthe short 200–300 µs range of the detected and fitted FID.In order to properly characterize the molecular dynamicsof such components, the B0 inhomogeneity effects haveto be overcome, which is traditionally done by taking aHahn-echo decay curve, which is acquired by increment-ing the echo delay, and simply evaluating the intensityof the FID signal acquired at the echo top. The Hahnecho time-reverses and thus removes field-inhomogeneityeffects, and the resulting decay curve reflects pure relax-ation and – in fact – still some significant dipolar dephas-ing effects.

Fig. 5a shows that the rotational dynamics of seg-ments and thus the spin pairs within the monomerunits is anisotropic if the ends of a given network chainare fixed at the crosslink junctions. This means thateven when the motion is very fast, the average Sb =〈P2(cos θ(t))〉t 6= 0. Sb is the so-called dynamic orderparameter of the fluctuating chain, and it is directly pro-portional to a finite and measurable residual dipolar cou-pling (RDC), characterized by a RDC constant Dres inunits of rad/s (D/2π is in Hz). Specifically,

Sb =Dres

(Dstat/k)=

3

5N. (1)

Dstat is the average static-limit DDC constant, and k isa correction factor < 1 accounting for the spin arrange-ment and motions within a statistical segment. N is thenumber of statistical segments of the network chain, or,more precisely, the number of segments between topolog-ical constraints. The last part of this relation was firstcalculated by Kuhn and Grun in the context of strainbirefringence [47], and the first NMR observations andtheoretical explanations in polymer melts and elastomersare due to due to Cohen-Addad [1] and Gotlib [15], re-spectively. In the former case, entanglements take therole of the crosslinks, and the associated reptation mo-tion of long linear chains complicates the matter, andwill not be discussed further, see refs. [48, 49] for details.Without entanglements, the first appearance of a mea-

solvent,oil

loops

dangling ends

sol

(t)

b(t)|n|~Sb~Dres

B0(a)

(b)

Figure 5: (a) The different lines signify snapshots, i.e., possi-ble conformational states of a given fluctuating network chain.The orientational dynamics of a segmental vector b(t) (thatcan be associated with individual spin pair orientations) isanisotropic in the long-time limit, characterized by a finite dy-namic chain order parameter Sb. The associated order tensor↔n is oriented along the connecting line of the crosslinks and isapproximately the same for all monomers of a given chain. Itis of course isotropically distributed in an unstretched sampleconsisting of many chains. (b) Sketch of typical elasticallyinactive, isotropically mobile components in networks.

surable Dres as a function crosslink formation betweenmobile chains is directly related to the gel point [50].

Eq. (1) is the basis of the NMR determination of thecrosslink density of rubbers, since N ∝Mc, the molecularwight between crosslinks. Since N is roughly of the order100, Sb ≈ 0.01, which means that Dres is in the percentrange of Dstat. Note that entanglement effects also con-tribute to Dres at moderate crosslink density, since moreprecisely Dres ∝ (1/Mc + 1/Me), where Me is the entan-glement molecular weight. There are numerous examplesin the literature addressing quantitative relationships be-tween NMR observables reflecting RDC effects and thecrosslink density [16, 18, 51, 52], many of them usingLF/LR NMR.

The most important consequence of the presence of asmall but finite DDC is that the transverse relaxation(T2) decay is non-exponential. For homogeneous net-works at high enough temperature, the decay is in factGaussian, in perfect analogy with the rigid-solid case dis-cussed in Section II B, but of course on a roughly 100times longer timescale (some ms instead of tens of µs):

Iecho = exp{ 9

40D2

resτ2} (2)

9

There is considerable disagreement in the literature as towhether eq. (2) is correct, or should be modified in termsof models taking into account, for instance, a quasi-staticGaussian distribution of end-to-end distances [51] or ef-fects of intermediate motions [17, 53]. As also discussedfurther below, our work has evidenced that these effectnever play an essential role at sufficiently high tempera-ture, and that deviations of actual data from eq. (2) aremost often due to sample inhomogeneity [54–56].

In order to reach the regime where the above func-tion is applicable, the experimental temperature has tobe high enough to avoid a large additional influence ofincoherent (true) T2 intermediate-regime relaxation pro-cesses, which lead to a faster decay and lend some expo-nential character to the Gaussian dipolar dephasing. Inorder to check whether the regime of dominant RDC ef-fects is reached, the temperature should be raised until a“T2 plateau” is observed. An alternative to temperaturechange is partial swelling [52, 57]; the solvent then acts asplasticizer and speeds up the molecular dynamics. It isoften argued that working at moderate swelling degreescan help to remove entanglement effects from Dres andthus quantify their relative contribution, however, onemust keep in mind that Dres changes rather non-triviallyin swollen samples, see below and refs. [58–60]

As mentioned, there are many reasons why a fit us-ing eq. (2) can be difficult. As is apparent from thedata in Fig. 6a, there are often other, sometimes signif-icant signal contributions from components with longerand almost exponential relaxation behavior. These canbe associated with network defects, solvent, or significantamounts of extender oil, which all move isotropically andthus do not exhibit RDC effects, but may differ in theirT2 relaxation times. See Fig. 5b for example of such elas-tically inactive components. The use of a two- or three-component fitting function is thus advised, which canlead to ambiguous fitting results, in particular when theinitial decay due to the network component is not per-fectly Gaussian. Apart from the mentioned exponentialrelaxation contributions, network inhomogeneities, lead-ing to a distribution in Dres and a non-Gaussian signaldecay, are a major reason for deviations from eq. (2).See refs. [54, 62] for discussions of possible artefacts inT2 relaxation studies of rubbers.

This is why the method of choice for a precision mea-surement of Dres ∼ 1/Mc is multiple-quantum (MQ)NMR, which can be straightforwardly implemented onLF equipment. The technique has its predecessors in lessrobust combinations of solid and Hahn-echo signal func-tions [63–65], and was first applied to the study of orderphenomena in linear polymer melts [66] and networks[67] under high-resolution (MAS) conditions. Care hasto be taken under such conditions, as for instance theT2 relaxation times are strongly affected by MAS. Es-sentially, the RDC effects are averaged out by MAS andthe T2 is not an easily interpretable quantity any more,requiring the use of so-called recoupling pulse sequencesto re-introduce the RDC and make it measurable. This

rel.

inte

nsity

DQ evolution time / ms

0.01

0.1

1

0 10 20 30

0 5 10 150.01

0.1

1

Hahn echo delay / ms

0 5 10 150.0

0.2

0.4

0.6

0.8

1.0IrefIDQIMQ = Iref + IDQ – defectsInDQ = IDQ / IMQ

rel.

inte

nsity

DQ evolution time / ms

rel.

inte

nsity

(a)

(b)

(c)Irefmobile defects (C)

constrained defects (B)

Iref - C

IDQ

Iref – IDQ – C

non-exponential!

Figure 6: Time-domain signals for crosslinked SBR with 80phr silica measured at 80◦C, taken from refs. [54, 61]. (a)Hahn-echo decay curves in semi-log representation, with linesindicating two exponentially relaxing defect fractions. (b)As-measured (red up-triangles) and processed (black down-triangles) data from a multiple-quantum experiment. (c) Pro-cessing involves a possible stepwise subtraction of defect com-ponents from Iref in semi-log representation, see text.

being not trivial, it was later realized that the static ver-sion of MQ NMR, first applied to rubbers in ref. [68], is infact much more robust and can without any compromisein data quality be carried out on LF equipment [22]. Adetailed account of the technique and many applications

10

in soft-matter science are reviewed in ref. [23]. The bigadvantage of this method is that a largely temperature-independent response function (InDQ) can be generatedwhich is free of relaxation effects and which can be ana-lyzed solely in terms of Dres (and distributions thereof).Sample data is shown in Fig. 6b, and without any re-course to the somewhat involved theoretical background,the discussion will just focus on some phenomenologicalaspects of data acquisition and treatment.

The raw data comprise not one but two signal func-tions that are measured as a function of pulse sequenceduration τDQ. The DQ build-up function IDQ and the ref-erence decay function Iref are acquired with one and thesame pulse sequence with slightly different internal set-tings (the phase cycle differs). IDQ(τDQ) reflects Dres inits (inverted-Gaussian) initial rise, but is also subject toincoherent relaxation effects at longer times. These canbe removed by point-by-point division of a “relaxation-only” function, the sum MQ decay IΣMQ(τDQ), leadingthe mentioned temperature-independent normalized DQbuild-up InDQ(τDQ). IΣMQ is obtained from the sum ofthe two experimental raw signal functions IDQ and Iref .It is in principle equivalent to an echo signal function inwhich all coherent effects, DDC and shift/field inhomo-geneity, are refocussed.

Calculating IΣMQ for the network component only isnot straightforward, since Iref contains, in the same wayas Iecho shown in Fig. 6a, also the defect components,which need to be subtracted before normalization. Thissubtraction procedure can be performed in a step-wisefashion, possibly taking recourse to the artificially con-structed dataset Iref−IDQ, in which even signal tails canbe identified whose apparent T2 relaxation is almost thesame as the one of the network component itself. SeeFig. 6c for sample data and refs. [23, 61] for details.

MQ NMR is thus a more robust method than T2 relax-ometry for the identification of signal components and forthe precise determination of Dres. The normalized InDQ

build-up function has a built-in quality control of thedefect-fraction determination, as it must always reachan intensity plateau of 0.5 in the long-time limit. Anexception to such a favorable behavior are inhomoge-neous gels with components exhibiting largely differentRDC and thus component-dependent and strong decayof IΣMQ; see ref. [69] for a discussion of possible dataanalysis strategies.

F. MQ NMR measurement of the crosslink densityand spatial inhomogeneities

Fitting functions published in refs. [23, 70, 71] can beused for the analysis of InDQ, obtaining a reliable aver-age value for Dres. For a very homogeneous network (noDres distribution), the generic fitting function applicableto any simple polymer such as natural rubber (NR), bu-tadiene rubber (BR), poly(dimethyl siloxane) (PDMS),

100% = sulfur contentin standard recipe

0.22 0.23 0.24 0.25 0.26 0.27 0.282.2

2.3

2.4

2.5

2.6

2.7

2.8 linear fit through (0,0)

90%

100%

110%120%

cros

slin

k de

nsity

from

she

ar e

xp.

/ 10

26m

-3

residual dipolar coupling Dres/2 / kHz

Figure 7: Linear NMR-elasticity relation for sulfur-crosslinked SBR (21% styrene, 63% vinyl). The crosslinkdensity ν is calculcated from the plateau modulus in shearaccording to G′ = ν k T . Data taken from ref. [14].

and the like, reads [71]

InDQ(τDQ, Dres) = 0.5(1− exp

{−(0.378DresτDQ)1.5

}× cos(0.583DresτDQ)) . (3)

For inhomogeneous networks and rubbers based oncopolymer chains or more complex monomers (with dy-namically decoupled side groups such as in alkyl acry-lates), Dres distributions must be taken into account,calling for multi-component versions of eq. (3). Conver-sion factors turning Dres into crosslink density (∼ 1/Mc)depend on the type of polymer and on the microstruc-ture. Examples were published and gauged against re-sults from swelling experiments for the cases of NR, cis-BR and PDMS [70, 72, 73]. An alternative is to use thelinear relation between Dres and the shear modulus [74],using G′ ≈ ρRT/Mc = ν k T for calibration, as done inref. [14] and shown in Fig. 7.

As noted, a key point of the MQ method is that theInDQ(τDQ) signal function can be reliably analyzed interms of distributions of Dres. This can be done by suit-able fitting functions based upon, e.g., a Gaussian dis-tribution of RDCs [23]. One then has the distributionwidth σ (square-root of the variance) as a second parame-ter characterizing network inhomogeneity. Alternatively,one can use a numerical fitting procedure based upon in-version of the distribution integral, which is related to theinverse Laplace transform often used for the analysis ofT2 relaxation data. Note however that the latter is boundto fail in soft materials, since we have seen that T2 re-laxation is intrinsically non-exponential. Inverse Laplacetransforms assume by definition a superposition of ex-ponential components, and using it on intrinsically non-exponentially relaxing components can give meaninglessresults. Our recent approach published in ref. [71] in-volves the use of a reliable and generic Kernel functionin an algorithm based upon Tikhonov regularization [75].

11

Dres/2 / kHz

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

3.5

0%10%20%30%50%70%90%100%

nDQ

inte

nsity

short-chaincontent:

DQ evolution time / ms

0 0.4 0.8 1.2 1.60

.2

.4

.6

.8

rel.

ampl

itude

/ a

.u.

0%

10%

20%

30%

50%

70%

90%

100%

(a)

(b)

Figure 8: (a) Normalized DQ build-up curves for a series ofbimodal end-linked PDMS model networks (Mc = 41 and 0.8kg/mol), with the weight fraction indicated. The dashed linesare not fits but merely linear combinations of interpolatedpure long- und short-chain network data in the known pro-portion. (b) Dres distributions obtained by numerical analysisof the nDQ build-up curves in (a). Data taken from ref. [76].

The fitting program and instructions are included in ourpublication [71] as supporting information.

An experimental verification of the possibility to quan-titatively characterize not only the average crosslink den-sity but also its local variations was earlier realized withbimodal model networks [22, 76]. Such networks aremade by end-linking of mixtures of long and very shorttelechelic chains with four-functional crosslinkers. Theyare known to be phase-separated systems composed ofnm-sized clusters of short chains interconnected by longchains, as a result of mere statistics, since the short chainsalways contribute the major fraction of cross-linkableends [77]. They represent an ideal test case for the abilityof the MQ technique to detect an inhomogeneous distri-bution of crosslink density in an elastomer.

Results from ref [76] are summarized in Fig. 8. Thebi-component character of the build-up curves in (a) isclearly evidenced by the fact that the curves for the bi-modal networks can be modelled by mere superpositionof the experimental curves for the pure-component net-works, weighted by their respective fractions. Actual fit-ting of such curves can be performed by using a build-upfunction consisting of two components with different av-erage Dres and variable fraction [71, 76]. Alternatively,our Tikhonov regularization procedure yields an estimatefor the actual distribution function without assumptionson its shape. The corresponding results in Figure 8bnicely demonstrate the two-component nature of thesedistributions for bimodal networks. Notably, the maxi-

mum of the more weakly ordered (less crosslinked) long-chain component does not change appreciably by addi-tion of short chains, indicating that these chains are nothindered by the presence of the short ones. Such a dy-namic decoupling of highly and lowly crosslinked regionsrequires a spatial separation in the range of several nm.As explained in more detail in the next section, directlyconnected short and long network chains (as present instatistically crosslinked, i.e. vulcanized) rubbers show anaveraged response and are not a priori distinguishable bythe technique.

III. RECENT LOW-FIELD NMRAPPLICATIONS IN RUBBER SCIENCE

This section gives an account of recent TD NMR ap-plications that are of direct relevance for technical elas-tomers and rubbers, based on the principles and us-ing the techniques presented in the previous section.First, typical results for the NMR-determined crosslinkdensity and its potential inhomogeneity are presentedfor various elastomer types, also including the complexchanges occurring when swollen rubbers are investigatedby NMR. A particularly useful approach is the correla-tion of the crosslink density determined by NMR in drysamples with analogous results from the popular equi-librium swelling (Flory-Rehner) experiments [78]. Thesegive valuable insights into the quantitative character ofthe latter, and also provide an interesting new means tostudy interactions between the rubber matrix and fillerparticles. Results concerning filled elastomers also com-prise relations of the NMR results and the mechanicalproperties, and the relevance of surface-immobilized rub-ber (“glassy layers”), for which very recent data and chal-lenges are presented and discussed.

A. Are crosslinked rubbers homogeneous orinhomogeneous?

NMR results for NR crosslinked with different cure sys-tems [79] are shown in Fig. 9a and b. In (a) the frac-tion of elastically inactive defects (see Fig. 5b) is plot-ted vs. the NMR-detected crosslink density. For sulfur-based vulcanization, it is seen that the defect fraction de-creases strongly with increasing crosslink density, as ex-pected from a random crosslinking process based on longprecursor chains, for which the defect fraction mainlyconsists of the progressively shorter chain ends. In con-trast, peroxide-crosslinked NR always contains between20 and 25% defects, indicating an important contributionof chain scission reactions. Similarly high defect fractionsare more commonly observed only in networks based onshorter precursor chains, as recently studied in detail forthe case of PDMS [73]. In this work, we found that the

12

0.0 0.2 0.4 0.6 0.8 1.00

0.1

0.2

0.3

0.4

0.5

0.6

rel.

ampl

itude

0.1 0.2 0.3 0.4 0.5 0.60

5

10

15

20

25

30

non-

elas

tic n

etw

ork

defe

cts

/ %

residual dipolar coupling Dres/2 / kHz

residual dipolar coupling Dres/2 / kHz

NR-peroxide (P)NR-sulfur efficient (E)NR-sulfur conventional (C)

NR-PNR-CNR-E

(a)

(b)

0 1 2 3 4 50

2

4

6

8

10

12

14

concentration of vulc. agent x104 / mol/g

NR-E(1:5.5, x=1.4)

BR-C(1:1.1, x=7.3)

NR-C(1 : 1.3, x=6.1)

BR-P(1:10)

BR-E(1:6.9, x=1.2)

NR-P(1:1.6)

C–Sx–C

cros

slin

k de

nsity

N

MRx1

04/

mol

/g(c)

Figure 9: (a) Fraction of non-elastic network defects in natu-ral rubber samples vulcanized with two different sulfur-basedcure systems and dicumyl peroxide, measured at 80◦C andplotted vs. the average NMR crosslink density (in terms ofDres). (b) Representative crosslink density distributions forone sample from each series. The peroxide-based system israther inhomogeneous. (c) Efficiency of vulcanization as ob-tained by correlation of the crosslink density from NMR withconcentration of the vulcanizing agent. For sulfur-based sys-tems, the slopes can be converted into the average length xof the sulfur bridges. Data taken from ref. [79].

statistical treatment of Miller and Macosko [80, 81] pro-vides a precise quantitative prediction of the inelastic de-fects.

The Dres distributions in Fig. 9b show that sulfur-vulcanized rubbers are highly homogeneous, whileperoxide-based crosslinking leads to substantial spatialinhomogeneity in the crosslink density. The distributioncomponent at higher Dres, related to highly crosslinkedregions, may arise from a secondary polymerization ofdouble bonds in the unsaturated backbone, leading tolarge multifunctional crosslinks. An interesting correla-tion is presented in Fig. 9c, where the actual crosslink

density (derived from the average Dres) is plotted vs. theconcentration of the respective vulcanizing agent (sulfuror peroxide). The slope represents the efficiency of thecrosslinking process, and in the case of sulfur, the lengthof the –Sx– bridges can be estimated. It is seen that the“efficient” vulcanization system (based on a larger con-centration of an amine accelerator) leads to significantlyshorter bridges for both NR and BR.

It is further seen that peroxide-based crosslinking ismuch more efficient for BR than for NR. This goes alongwith the observation that peroxide-crosslinked BR doesnot contain a large fraction of defects, indicating thatthe mentioned chain-scission reactions are specific forNR, possibly related to radical reactions that are in-fluenced by its methyl group. However, it should bementioned that the peroxide-crosslinked rubber matrix issimilarly inhomogeneous for BR as it is for NR. A similardifference in matrix (in)homogeneity comparing sulfur-and peroxide-based crosslinking was also evidenced forthe case of EPDM-based rubber [82]. Another recentMQ-NMR study has evidenced a significant increase ofthe defect fraction, a decrease in the overall crosslinkdensity and the appearance of network inhomogeneitiesupon thermal aging of nanoparticle-filled EPDM [83].Maxwell and colleagues have early on observed similarthermal and radiation-induced aging effects in differenttypes of silocone elastomers (mainly PDMS) by T2 relax-ation [84] and later by MQ NMR [84–87]. It was generallyfound that aging leads to more defects and more inho-mogeneities, but in this case to higher average crosslinkdensities.

Some basic considerations involving the in fact surpris-ing finding of apparently very homogeneous rubbers aresummarized in Fig. 10. Taking up eq. (1), according towhich Dres ∝ 1/Mc, we realize that Mc is in fact a dis-tributed quantity, for instance, in a randomly crosslinkedrubber based on long precursor chains, a most-probablemolecular weight distribution (Mw/Mn = 2) is expected.In addition, the fixed-junction model also predicts a pro-portionality between Dres and the squared end-to-endseparation 〈r2〉 of the network chain, which again is a(Gaussian-)distributed quantity. The thus theoreticallyexpected broad distributions of Dres are compared to atypical experimental result in Fig. 10a, and the questionarises why effects of the undoubtedly present distribu-tions are not reflected in the data [56]. It is importantto note that the phenomenon of narrow Dres distribu-tions is also found in computer simulations of realistic,disordered networks [88].

This issue, comparing and contrasting this surprisingfinding with earlier experimental and theoretical results,is the subject of ref. [56], where we have proposed somepreliminary explanations. The efforts to completely un-derstand this phenomenon are still ongoing, and with ourmore recent work [60, 73], the picture is getting clearer.More detailed analytical theory suggests that the NMR-detected local order is proportional to the square of theforce acting on the chain ends [60], and this force is of

13

backbone order parameter Sb Dres

0.00 0.05 0.10 0.15

Gaussian ree distribution

+ polydispersity

rel.

wei

ght

/ a.

u.

experimental (NR)

Q = 1 ( Sb 0.06)

1.241.59 1.882.552.92

0.00 0.10 0.20 0.30 0.40

rel.

wei

ght

/ a.

u.

backbone order parameter Sb Dres

1.0 1.5 2.0 2.5 3.0 3.5 4.00.24

0.28

0.32

0.36

0.40

0.44

affin

e/G

auss

ian

res.

dip

. cpl

. Dre

s/2

/ kH

z

degree of swelling Q = V/V0

solventgood solvent

effect ofexcludedvolume!

(a)

(b)

(c)

Figure 10: (a) Apparent NMR crosslink density distribution(in terms of the backbone order parameter Sb, see eq. 1),for a sulfur-vulcanized NR sample, as compared to theorypredictions assuming a Gaussian distribution of end-to-enddistances of the network chains, and additionally chain lengthpolydispersity. (b) Order parameter distributions of a PDMSend-linked model network (linear precursor of 5.2 kDa) as afunction of swelling degree Q = V/V0 in octane below andat equilibrium. (c) Change of the average RDC ∝ Sb as afunction of swelling for the same type of sample comparinggood and θ solvent. Data taken from refs. [56], [59], and [60],respectively.

course balanced when short and long network chains,or chains with very different instantaneous end-to-endseparation, are connected. Such a force balance is onlypossible if the crosslinks are allowed to move, which isthe essence of the phantom model of rubber elasticity[89, 90]. The results presented below also corroboratethat the phantom model provides a more quantitativebasis for the analysis of equilibrium swelling experimentsthan the affine fixed-junction model [73, 79]. In sum-mary, local force balances are probably responsible forthe observation of the uniform response of homogeneous(but still disordered) networks. An apparent exceptionfrom this phenomenon are defect structures in swollen yethighly homogeneous model hydrogels based upon four-

functional star precursors made from poly(ethylene ox-ide), PEO, where well-defined defect structures such asdouble-stranded links between two crosslinks can be re-solved from the normal network chains [69].

Generally, the RDC phenomenon has to be handledwith care in swollen systems, as we could experimen-tally show that networks deform highly non-affinely, seeFig. 10b,c and refs. [59, 88]. These observations areagain based upon the fact that Dres ∝ Sb ∝ 〈r2〉/Nunder ideal condistions (θ solvent, affine fixed-junctionbehavior). Experimental data are obviously at variance.In addition, the phenomenon of swelling heterogeneities,well known from many scattering studies to occur in the100 nm range and above, is directly evidenced in theNMR data in Fig. 10b. An important complication ishighlighted by the data in Fig. 10c, where we not onlysee strongly sub-affine behavior, but also that solventquality (excluded-volume effect) rather than geometriceffects (chain stretching) alone plays a significant role[60]. Therefore, MQ NMR experiments (and the lessquantitative but feasible T2 studies) should be performedon bulk samples, or at small concentrations in case somesolvent is needed to speed up the chain dynamics andmake the sample amenable to NMR study.

B. Quantative correlation of NMR and equilibriumswelling

The quantitative relationship between the NMR-detected average Dres and the network structure has al-ready been discussed in the context of Fig. 7, wherethe linear relation with the crosslink density derivedfrom the plateau modulus determined by rheology ordynamic-mechanical analysis was used for calibration.In Fig. 11, analogous correlations between the NMR-determined crosslink density and results from Flory-Rehner swelling experiments [78] are presented, as pub-lished in refs. [72, 73]. Again, we observe near-perfectlinear dependencies. This time, Dres was converted toactual crosslink density based on models of the spin dy-namics within the statistical segments of NR, cis-BR, andPDMS [70].

We studied in detail the dependence of the resultsfrom swelling experiments on the way the experimentsare conducted and evaluated [72], addressing for instancethe issue of defining proper volume fractions in systemswith fillers and significant amounts of other non-swellablecomponents such as ZnO. In particular, we addressed thedifferences arising from using Flory’s fixed-junction affinevs. James’ and Guth’s phantom model for the elasticcontribution in the Flory-Rehner treatment [78], whereFig. 11a suggests a better mutual agreement with the lat-ter. The consequences of phantom-like behavior, whichwe have seen in the previous section to be physically rea-sonable also from a chain dynamics point of view (forcebalance), have been addressed in another more recent

14

0.0 0.1 0.2 0.30.0

0.1

0.2

0.3

0.4

0.5

affine modelphantom model

slope=1.26

slope=1.98

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(a)

1/M

c (N

MR

) /

mol

/kg

1/Mc (swelling) / mol/kg

1/Merheo

PDMS 11k, 4% vinylPDMS 13k, 8% vinylslope = 2.06

1/M

c (N

MR

) /

mol

/kg

(b)

NR 80°C

PDMS 40°C

Figure 11: (a) Crosslink density ν ∝ 1/Mc of sulfur-vulcanized NR samples from NMR as compared to resultsfrom equilibrium swelling based on Flory-Rehner theory [78]using the affine and the phantom model. (b) The same forrandomly vinyl-functionalized PDMS networks vulcanized bya bifunctional linker, based on the phantom model and usingeq. (4), taking account of the effective weight-averaged func-tionality of crosslinks. Data taken from refs. [72] and [73],respectively.

work focussing on PDMS networks [73]. We in fact foundthat phantom behavior introduces a dependence on thefunctionality of the crosslinks not only into the final rela-tion for the evaluation of swelling experiments, but alsointo the calibration relating crosslink density and Dres,here given for the example of PDMS:

1

MPDMSc

=Dres/2π

1266 Hz

f

f − 2mol/kg . (4)

This correction ultimately arises from the fact that thecrosslink fluctuations can be treated in terms of vir-tual chains, which effectively lengthen the actual networkchains, leading to the factor of f/(f − 2) [73, 90]. Thisaffects both, the evaluation of swelling and NMR exper-iments. The correction was not yet part of the NMR-based data shown in Fig. 11a [72], which means thatthe validity of the phantom model actually cannot be in-ferred from this data, as the associated slope would resultto be 2.52 instead of 1.26. However in our recent work[73], we studied a large series of networks prepared un-der very different conditions and used high-resolution 1HMAS NMR in combination with calculations based uponthe Miller-Macosko theory of random crosslinking [80]to obtain reliable absolute-value results for the crosslinkdensity as a gauge. In less highly crosslinked samples,

the average functionality of the crosslinks was found tobe significantly less than 4, and based on this variation inf (which does not appear as a parameter in affine fixed-junction models) it is possible to experimentally provethat the phantom description is qualitatively correct. Asto absolute values, the NMR results (Fig. 11b, ordinate)are roughly a factor of 1.4 overestimated, while equilib-rium swelling results (abscissa, based on reasonable liter-ature values for χ) are underestimated by about the samefactor. These deviations are well within the model depen-dencies underlying the NMR [70] as well as the swellingexperiments.

Deficiencies of the thermodynamic contribution toswelling equilibrium are in fact a significant source ofuncertainty. For instance in our work, we referred thethermodynamic interaction parameter χ from the liter-ature, where it is reported to be a function of solventcontent, and further to be qualitatively different for so-lutions of linear chains and swollen networks [72]. Thebasis of the need for such ad-hoc adjustments is certainlythe simplistic nature of the Flory-Huggins treatment [78],and possibly to some degree one of the basic assump-tions of Flory-Rehner theory, according to which elas-tic and thermodynamic contributions should be strictlyseparable. While the use of more realistic equation-of-state approaches is certainly advised, the Flory-Hugginstreatment is at present still the method of choice due tothe large body of data in the literature for χ of manypolymer-solvent pairs.

We finally comment on the apparent ordinate inter-cept that is visible in Fig. 11, in particular in part (a).This intercept is due to entanglement effects and does notappear in NMR-vs-rheology correlations (Fig. 7), as en-tanglements contribute equally to Dres and to the plateaumodulus. In swelling, however, entanglements contributemuch less; their effect is reduced to topologically active(“trapped”) links. This is why the intercept is found tobe close to the inverse entanglement molecular weightMe derived from the plateau modulus of an equivalentlinear-chain melt. The effect is much less pronounced forPDMS (Fig. 11b) due to its higherMe. It should be notedthat the additivity concept (Dres ∝ 1/Mc + 1/Me) onlyholds for small Mc. For weakly crosslinked networks, adominance of entanglement effects with Dres scaling asM−0.5

e is theoretically expected [91], but cannot be ob-served because such long network chains have very longrelaxation times, violating the necessary prerequisite offast-limit averaging on the timescale of the experimentin the ms range.

C. Filler effects I: NMR-detected crosslink densityand inhomogeneities vs. macroscopic properties

Filler effects on various molecular parameters of therubber matrix in which they are embedded are frequentlydiscussed, and are of relevance for an in-depth under-

15

0.0 0.5 1.00

.1

.2

.3

.4

.5

.6

.7

.8NR1 pureNR1 33 wt% silicaNR2 pureNR2 31 wt% carbon black

rel.

ampl

itude

residual dipolar coupling Dres/2 / kHz

0 10 20 30 40 50 60 70 800.00

0.05

0.10

0.15

0.20

0.0

0.2

0.4

0.6

0.8

1.0

silica content / phr

res.

dip

. cpl

. Dre

s/2

/ kH

z

non-elastic defect fraction (oil etc.)

(a)

(b)

Figure 12: (a) NMR crosslink density distributions of pureand filled sulfur-vulcanized NR samples. (b) Average NMRcrosslink density (in terms of Dres), left ordinate, and defectcontent, right ordinate, of sulfur-vulcanized SBR samples con-taining different amounts of silica. Data taken from refs. [92]and [14], respectively.

standing of the synergistic effects of fillers on the mate-rial performance (see Fig. 1). Often, such information isobtained indirectly, either by performing solvent extrac-tion experiments yielding the so-called “bound rubber”fraction, as critically discussed in the next section, or byfitting theoretical models to rheological data, which usu-ally offer no option for an independent test of whetherthe resulting apparent changes in the matrix crosslinkdensity upon filling are true or not. This calls for a local,spectroscopic approach that selectively probes the rub-ber phase, and TD NMR is here advocated as the moststraightforward choice.

Typical results from our work are presented in Fig. 12.In part (a), it is evidenced that filling with both silica andcarbon black (CB) has virtually no effect on the rubbermatrix. Only a slight shift of the crosslink density tolower values is seen, which is explained by inactivationof parts of the vulcanization system by adsorption to thehigh-surface filler, leading to somewhat lower crosslinkdensities. This effect can be significant, as demonstratedin Fig. 12b on the example of oil-extended SBR (thuscontaining 20% NMR-detected non-elastic components)filled with increasing amounts of silica [14]. Along withsimilar observations on a variety of other conventionalrubber-filler systems [92], we can conclude that filler ef-fects on the matrix are usually rather minor, which isat variance with a number of works based on indirect

0.2 0.3 0.4 0.5 0.6

3

4

5

6

7

shea

r m

odul

usG

' / d

Nm

1/Mc (NMR) / mol/kg

0.0 0.5 1.0

NR pure

NR-clay

NR-organoclay

NR-amine

rel.

ampl

itude

/ a

.u.

residual dipolar coupling Dres/2 / kHz

NR-organoclay:exfoliated!

(a)

(b)

Figure 13: (a) NMR crosslink density distributions (interms of Dres) of pure NR and samples filled with 10 phr(organo)clay, also compared to a sample containing the sameamount of amine used as organo-modifier. (b) Correlationof the NMR crosslink density of the same samples with theplateau modulus. Data taken from ref. [92].

methods. It must be emphasized that the NMR result isobjective and model-free.

Fig. 13 shows results for NR filled with clay minerals[92], which are a very promising new class of nano-sizedfiller materials, as it is possible to obtain nanocompos-ites characterized by almost perfect dispersions of molec-ularly thin (alumo)silicate sheets, which significantly en-hance the mechanical and barrier properties of rubberand many other commodity polymers already at verylow content [93, 94]. A good dispersion of the sheetsis only possible if the clay is modified beforehand by anorganic surfactant, often an alkyl amine, which pre-swellsthe galleries between the silicate sheets and enables ef-ficient exfoliation upon processing. In Fig. 13a, signif-icant changes of the NR matrix upon adding clay areevidenced, in particular, the exfoliated organoclay sam-ple exhibits a significantly increased crosslink density ascompared to pure NR.

Looking closer, however, one realizes that the largestincrease is found for a sample prepared without clay butwith the same amount of amine used to pre-swell theorganoclay (NR-amine). Thus, the increase is simply dueto the amine acting as accelerator, rendering the vulcan-ization process more efficient. There is no “nano” effecton the crosslink density, rather, each clay-filled samplehas a lower crosslink density than its appropriate coun-

16

with surfaceattachment

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

no surfaceattachment

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

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0.3

0.4

0.5

0.6

0.7

NR-4% FGS

NR-1.5% FGS

NR-16%CB

NR-1% FGS

NR-40%CB

unfilled NRNR-silica and NR-CBunfilled NRNR-CB, NR-FGS

NR-40%silica

NR-1.5% low-surface FGS

1/M

c (N

MR

) /

mol

/kg

1/Mc (swelling) / mol/kg

NR pureNR-amineNR-clayNR-organoclay

(a)

(b)

+ solvent

1/M

c (N

MR

) /

mol

/kg

Figure 14: (a) Correlation of crosslink density ν ∝ 1/Mc ofsulfur-vulcanized pure and clay-filled NR samples (same sam-ple series as in Fig. 13) determined by NMR and equilibriumswelling based on the phantom model. (b) Same type of corre-lation for samples containing different types of fillers (surface-modified silica, carbon black, and functionalized graphenesheets). The master curves (solid lines) are the ones fromFig. 11a. Deviations from them prove strong (chemical andabsorptive) links exist between the filler and the matrix, thusrestricting the swelling degree. Data taken from ref. [92].

terpart. The correlation of the average crosslink densityfrom NMR with the plateau modulus in Fig. 13b is the ex-pected linear one, with the exception of NR-organoclay,which is obviously the only sample with significant re-inforcement related to a filler network. NR-clay is notexfoliated, and the low level of filling (10 phr) does inthis case not visibly enhance the mechanical properties.This once again demonstrates the advantage of NMR asa local technique, allowing for a separation of the effectsthat do or do not contribute to reinforcement. Weakeryet similar effects were also found for the ternary sys-tem NR/clay/poly(ethylene glycol) [95], where the latteraides in the exfoliation of the nanosheets.

In evaluating the possible contributions to reinforce-ment, the adhesion of the rubber matrix to the filler sur-face is another important issue. For this purpose, correla-tions of the NMR-detected crosslink density with resultsfrom equilibrium swelling turn out to be extremely useful[92], see Fig. 14. In part (a) samples from the NR-clayseries are shown to all follow the same master line, prov-ing that the presence of filler does not affect the swelling

process in any way. In such a case, the network sim-ply swells away from the filler, forming a solvent-filled“bubble” around the non-swellable filler. The presenceof this bubble can independently be proven by studyingthe freezing point depression of the swelling solvent byDSC [92].

In significant contrast, fillers with active surface modi-fication, forming either chemical bonds to the rubber ma-trix, or at least offering adsorption sites that are stableagainst the competing solvent, lead to lower degrees ofswelling and thus to apparently enhanced crosslink den-sity, which is evidenced by positive a shift along the ab-scissa (Fig. 14b). In such systems, the degree of swellingcan be expected to be inhomogeneous in the vicinityof the particles, which thus act as giant nano- or evenmicron-sized crosslinks [96]. The most dramatic effecthas been observed for samples filled with functionalizedgraphene sheets (FGS), also referred to thermally ex-panded graphite oxide [97, 98]. This is another new, ex-tremely promising filler system, leading to dramatic mod-ification of the material’s properties at extremely low lev-els of filling. In our case, this is evidenced by the largestshift of the apparent crosslink density from swelling for asample containing only 4% of FGS. Note that the NMR-detected crosslink density as well as its distribution isagain not significantly affected, which proves that the av-erage spatial distance between the stable adsorption sitesof different NR chains on the FGS must be larger than orat least on the order of the distance between crosslinks.In other words, the NR adsorbed to FGS does not form adense brush. In the following section, it is demonstratedthat filler-induced changes in the mobile rubber matrixcan indeed occur under such special circumstances.

D. Filler effects II: surface-immobilizedcomponents

Fig. 15 shows NMR-determined crosslink density dis-tributions for an interesting class of model-filled elas-tomers, based upon an almost perfect dispersion ofmonodisperse silica spheres with diameters in the 20–50 nm range in a matrix formed by poly(ethyl acrylate),PEA [30, 32, 99]. The system is special in that the dis-persion quality can be controlled by playing with the in-terparticle interactions in a colloidal suspension of theparticles in the monomer before crosslinking. The systemcan thus be tuned between the limits of perfect disper-sion and a high level of aggregation, where filler networkeffects only arise in the latter case. See ref. [100] for a re-view of the work related to these model nanocomposites,which constitute a perfect model system for the study ofrubber reinforcement effects.

Two types of fillers are compared in Fig. 15, namelypartially silanized silica spheres exhibiting just absorp-tion sites, and spheres that are densely modified witha reactive linker forming covalent bonds to the network

17

0.01 0.1 1 100.00.10.20.30.40.50.60.7

25% 42 nm silica44% 42 nm silica25% 26 nm silica37% 26 nm silica

PEA 0.3% x-linker

rel.

cont

ent

/ a.

u.

0.01 0.1 1 100.00.10.20.30.40.50.60.7

18% 42 nm silica30% 42 nm silica20% 26 nm silica30% 26 nm silica

PEA 0.3% x-linker

rel.

cont

ent

/ a.

u.

residual dipolar coupling / kHz

non-grafted

grafted

Figure 15: NMR crosslink density distributions (in terms ofDres) of model-filled poly(ethyl acrylate) networks, comparingsamples filled with silica spheres just presenting absorptionsites vs. silica spheres with a high density of chemical graftsto the network matrix. Data taken from ref. [32].

matrix. It is seen that in the latter case, the matrix isindeed significantly modified: the overall crosslink den-sity is increased and the apparent distribution is broader,with indications for a bimodal behavior at high filler load-ings. This phenomenon is explained by the very highdensity of effective grafting sites [32]. When the distancebetween such constraints is much smaller than the av-erage crosslink separation, the local order and thus themolecular-level elastic response of the shorter networkchains close to the particles can be expected to be muchincreased.

It seems that the grafting densities that can be reachedusing CB or silica as fillers in conventional rubbers aregenerally much lower, explaining the absence of compa-rable effects in any of the conventional rubbers systemsthat we have investigated so far (e.g. Fig. 12). Notethat silica fillers are usually used in combination withsilanization agents that should provide chemical bondsbetween the rubber and the filler, however, the graft-ing densities reached in this way are apparently not veryhigh. Another issue is the absence of filler aggregationin the presented system: aggregation significantly lowersthe internal surface area and thus the relative amount ofsurface-modified polymer species.

The final issue to be covered in this review concernspolymer species in immediate contact with the filler.While the phenomenon discussed in the last paragraphsaddresses topology effects on still mobile (rubber elas-tic) chains, we now turn to species whose mobility issignificantly reduced as a result of strong adsorption in-teractions. The NMR response of such material, oftenreferred to as “glassy layer,” is solid-like, and its quan-tification was discussed in the context of Fig. 3. Resultsof such component analyses for the case of the PEA-silicamodel nanocomposites introduced above are compiled in

[Tg(PEA) 250 K]

300 320 340 360 380 4000.0

0.2

0.4

0.6

0.8

1.0sample: 30% 42 nm silica, grafted

mobile networkintermediate glassy fractionfr

actio

n

temperature / K

0.00 0.01 0.02 0.03 0.04 0.050.0

0.2

0.4

0.6

0.8

1.0

glas

sy +

inte

rmed

iate

frac

tion

Si surface-to-volume ratio / nm1

T = 353 Ksilica surface treatment:

graftedabsorptive

(a)

(b)

mobile network

glassy and intermediate layers

SiO2

Figure 16: Results from FID component analysis (see Fig. 3)in model-filled PEA networks, where significant amounts ofimmobilized polymer are evidenced. (a) Signal fractions as afunction of temperatures. (b) Total amount of immobilizedmaterial as a function of inner silica surface for the two typesof samples. The thickness of the layer at 350 K is estimatedto around 5–6 nm from the known filler size and total volumein the grafted samples. Data taken from ref. [32].

Fig. 16, and are the subject of ref. [32].The NMR signal (essentially simple FIDs) can be de-

composed into 3 distinguishable species, whose proton(=volume) fraction is plotted as a function of tempera-ture in Fig. 16a. At temperatures of about 60 K abovethe Tg of the PEA matrix , around 50% of all poly-mer in the given (grafted) sample is found to exhibit ahigher Tg. The interpretation in terms of an effectivelyincreased Tg is based upon the finding that the immobi-lized polymer fractions decrease upon heating. The high-temperature plateau value for this fraction correspondsto species with apparently infinite Tg, i.e., species that donot desorb below the sample decomposition temperature.More detailed fitting of the NMR data, based upon thetemperature-dependent response of the pure matrix poly-mer (Fig. 2b) in combination with DSC results showedthat the phenomenon is fully consistent with a smooth

18

Tg gradient [40], for which the 3-component fit is simplythe minimal model that allows for reliable fitting.

The total immmobilized polymer fraction for the givenseries of samples at a fixed temperature (100 K above thematrix Tg) is plotted against the known specific innersurface in Fig. 16b. It is seen that very large (� 20%)immobilized fractions are only found for the grafted sam-ples, where the chemical bonds between network and sil-ica contribute to raising the effective Tg in the vicinity ofthe particles. Nevertheless, the fraction is still substan-tial in the samples where the PEA just adsorbs to thepartially hydrophilic silica, most probably by hydrogenbonds. From the known geometry (perfectly dispersedspherical filler particles of known size), one can estimatethe thickness of the immobilized layer to around 5–6 nmfor the grafted samples at 350 K. The high-temperatureplateau representing strongly adsorbed material in thenon-grafted case is thus estimated to around 1–2 nm.

Complete immobilization leading to solid-like behaviormediated by hydrogen bonds is a well-documented phe-nomenon that was first studied in detail for the systemPDMS/silica [101–104]. Even though PDMS is generallya hydrophobic polymer, its Si–O–Si backbone is locallypolar and can form strong hydrogen bonds with a hy-drophilic silica surface. Quantitative analyses of the pro-ton NMR response of systems with good dispersion andknown particle size suggested a rigid layer thickness of1–2 nm [103, 105], and the same is true for the systemPEO/silica [106].

If linear PMDS of sufficiently high molecular weight isblended with silica, one in fact obtains permanent andeven swellable elastomers [107–109], in which the strongsurface adsorption withstands the competition with anunpolar solvent. Using MQ NMR, we have studied thestructure and dynamic composition of such physical net-works in which the the silica particles thus form giantphysical crosslinks [110]. Similar phenomena are againobserved for linear PEO/silica model nanocomposites[106]. Related phenomena are also responsible for theformation of physical gels made of aqueous solutions oflinear poly(vinyl alcohol) subjected to freeze-thaw cycles.In this special case, PVA crystallites form (the amount ofwhich can be quantified by proton FID decomposition),and act as giant solid-like crosslinks connecting mobile,rubber-elastic chains embedded in water as solvent [111].

Immobilized surface-associated species are, however,an elusive phenomenon in conventional filled rubber com-pounds. One of the earliest reports of small amounts ofimmobilized polymer species in cis-BR/CB compounds,as identified by proton FID and T2 measurements, is dueto McBrierty and coworkers [112]. Fractions in the rangeof 10% or more in as-made composites have so far onlybeen measured in the mentioned model systems, whereasconventional filled rubber compounds show immobilizedfractions of less than 5% of the total rubber [113]. In fact,our own work has so far not indicated fractions in excessof 1–2% in NR or SBR systems at typical loadings, irre-spective of the filler type. Often, the immobilized fraction

is virtually undetectable.

In order to better be able to study such species, it hasbecome customary to subject unvulcanized compositesamples to a solvent extraction procedure. A gravimet-ric analysis of such samples gives the so-called “bound-rubber” fraction, which represents a qualitative measureof the adsorption capacity, related to the inner surfaceof the filler. As evidenced by proton NMR on suchextracted samples, the bound-rubber fraction is in it-self dynamically inhomogeneous and consists of immobi-lized and still mobile network-like dangling and tie chains[33, 113–116].

The early work of Nomura and coworkers provideda first detailed picture of the changes of the differentbound-rubber fractions and their T2 relaxation behav-ior as a function of processing time of NR/CB com-posites in a Brabender mixer [114]. Legrand and col-leagues [115] pointed out the similarities of CB- andsilica-filled rubbers, where surface deactivation by graft-ing of alkyl chains was also studied and shown to lead tolower amounts of highly immobilized species, in particu-lar for silica. The very detailed study of McBrierty andcoworkers [113] has revealed that solvent extraction ofNR/CB composites of increasing severity can yield com-pounds in which up to 35% of the polymer is immobilized.Analyses of the signal components in EPDM/CB samplesin combination with an in-depth filler characterization,providing its inner surface area, have resulted in an esti-mated layer thickness of again 1 nm [33]. Based on thequantitative analysis of the T2 of the mobile network-likecomponent, Gronski and coworkers found for SBR/CBa typical mobile-layer thickness of 2–3 nm on top of theimmobilized layer [116]. For all of the cited studies con-cerned with solvent-extracted samples it must be empha-sized that the extraction procedure destroys the integrityof the filler network and may further change the interfa-cial interactions substantially, putting into question therelevance of the so-obtained results for explaining the re-inforcement effects in actual rubber compounds.

Detecting and quantifying potential glassy-layer ma-terial in as-used elastomers, and correlating it with theNMR-detected (true) crosslink density, the filler disper-sion as obtained from scattering experiments, and thelinear and in particular non-linear mechanical properties,is a promising route towards a better understanding ofreinforcement mechanisms. The presented model systembased on PEA/silica is particularly suited for this pur-pose, and significant progress has already been achieved[30, 99, 100]. According to our current understanding,the glassy layer acts in this case as a “glue” holdingtogether the filler network. Upon increasing the tem-perature, glassy bridges soften and the reinforcement isconsequently decreasing, see Fig. 1. Long and colleagueshave formulated a mesoscale model based on the perco-lation of glassy bridges as one of the salient features ofrubber reinforcement, which in fact correctly reproducesthe Payne and Mullins effects [117].

This survey of current NMR results should be con-

19

glassy-phase filtered

network-phase filtered

?

0.0

0.2

0.4

norm

aliz

ed in

tens

ity mobile networkintermediate part

0.0

0.1 glassy partintermediate part

spin diffusion time tm0.5 / ms0.5

0 5 10 15 20 25

Figure 17: Results from TD spin diffusion experiments on oneof the model-filled PEA samples, see also Fig. 4. The resultsfor the glassy-layer-selected experiment exhibit the right orderof magnetization flow, first into the intermediate componentand then into the network. The results for the mobile-phaseselected experiment are at variance, and may be explainedby a non-trivial, disordered arrangement of glassy and inter-mediately mobile material, as sketched at the bottom right.Unpublished data, by courtesy of Kerstin Schaler.

cluded with an open question concerning the exact ge-ometry of the glassy layer and the associated Tg gradi-ent. As mentioned, the thickness of the combined in-termediate and glassy component layer in model com-posites of PEA with grafted silica is around 5–6 nm at350 K. This figure can be obtained from a simple volumeconsideration, as mentioned above, and was also con-firmed experimentally using spin diffusion experiments,see Fig. 4c for an example. The analysis of such data isso far based upon rather approximate linear extrapola-tion procedures, treating the system as consisting of twophases [37], combined immobilized material vs. mobileelastomer matrix.

Data from spin diffusion experiments on a PEA/silicamodel composite are presented in Fig. 17. In the twoplots we see results from spin diffusion experiments, forwhich either the glassy layer or the mobile matrix wasused as a source of magnetization, using specific filters.In each plot, only the time dependence of the sink phasesis shown, this time considering 3-component fits to thesignal functions. For the first case, it is seen that themagnetization first diffuses into the region with interme-diate mobility, and then into the network matrix. Theintensity decay at long times as due to rather rapid T1

relaxation at low field. For the second case, the result iscounterintuitive in that the magnetization that starts inthe network phase diffuses into the intermediate and intothe glassy layer with almost the same apparent rate.

This asymmetry calls into question the hypotheticalpicture of a smooth mobility gradient. As mentioned, theanalysis of such spin diffusion data at low field is so farnot routine and requires numerical modelling procedures[37]. We are currently adapting our numerical proce-dures to other possible geometries, and one scenario thatcould potentially explain the observed results is depictedat the lower right of Fig. 17. Probably, the assumptionof smooth layering on a smooth surface is simply unre-alistic. The glassy layer may in itself be inhomogeneous,with changes in mobility not only in the normal direc-tion but also laterally. Also, a silica nanoparticle is notatomically smooth, and once the surface roughness is onthe order of the layer thickness, the immobilization of ad-sorbed chains may also be influenced by the local curva-ture and the nm-scale confinement geometry. We believethis observation to be quite general, as it is reproduced inother systems with similar (alleged) mobility gradients,such as on the lamellar crystal surface in semicrystallinepolymers or at the domain interface in block copolymersconsisting of hard and soft domains.

IV. SUMMARY AND CONCLUSIONS

The aim of this review was to familiarize the readerwith the most important and current concepts of protonlow-field NMR as applied to the characterization of poly-mer materials, with specific focus on rubber science. Itwas demonstrated how the fitting of simple time-domainfree-induction decay signals, refocused by a suitable pulsesequence to overcome the dead-time problem, can givereliable and quantitative information on the composi-tion of samples consisting of dynamically distinguishedphases. Further, the spin diffusion effect can be utilizedto estimate the size of the associated domains. The cen-tral phenomenon concerning mobile chains far above theglass transition was the residual dipolar coupling of mo-bile chains, which reflects the anisotropy of chain motionas arising from constraints such as crosslinks. As themost quantitative approach to study this phenomenon,multiple-quantum NMR was discussed, which yields in-formation on the crosslink density and its spatial varia-tions (i.e., crosslinking inhomogeneities).

Such molecular-level information on the microstruc-ture of rubbers was shown to be particularly useful whencorrelated with macroscopic material properties, suchas the plateau modulus or the crosslink density as de-rived from equilibrium swelling experiments. The spec-troscopic information obtained from NMR is local andthus unbiased, which means that effects of nanoscopicfiller materials on the rubber matrix can be studied objec-tively. One central result was that most filler materials donot affect the crosslink density and the (in)homogeneityof the rubber matrix very much, with the general excep-tion of a slight decrease in crosslink density afforded bypartial deactivation of the vulcanization system by ad-

20

sorption to the particle surfaces. An interesting new per-spective was opened by the correlation of crosslink den-sities determined by NMR and by equilibrium swelling,through which active rubber-filler bonds are directly ev-idenced.

Using the sensitivity of proton NMR to mobility, it wasshown to also be possible and worthwhile to determinethe amount of polymer that is immobilized by adsorp-tion to filler particles in as-made filled samples, wherewe put our focus on a special class of tunable nanocom-posites made from poly(ethyl acrylate) and monodispersesilica spheres. These materials show a pronounced glassyshell, whose fraction depends on temperature in a well-defined way, suggesting this class of materials as theperfect model to validate models explaining rubber re-inforcement in terms of glassy bridges between filler par-ticles.

While many of the cited NMR studies have revealedincreasingly rich insights into structural and dynamic as-pects in particular in filled elastomers, it is fair to statethat a full understanding of rubber reinforcement is stillopen. For instance, considering the rather low and some-times virtually absent signal contributions related to im-mobilized rubber, it is by no means clear if glassy bridgesare really a necessary prerequisite being present in anyrubber compound for tire applications, or are only onepart of the full and even more complex picture. Low-

field NMR studies will certainly continue to be useful inaddressing these intriguing questions.

Acknowledgments

Funding of the work of the author presented in this re-view was mainly provided through different programs ofthe Deutsche Forschungsgemeinschaft (DFG): SFB 418,SFB 428, SA 982/1, SA 982/3, SA 982/6, and cur-rently SFB/TRR 102. I also acknowledge funding fromthe Land Sachsen-Anhalt, the Fonds der Chemischen In-dustrie, and infrastructural support from the EuropeanUnion (ERDF programme). The work presented hereinwould not have been possible without the contributionsof many coworkers and collaboration partners over theyears, the names of which can be taken from the citedpublications. I would specifically like to mention theinvaluable contributions of Jens-Uwe Sommer (LeibnizInstitut fur Polymerforschung, Dresden) to the theoreti-cal interpretation of NMR observables in networks, andJuan Lopez Valentın (ICTP-CSIC, Madrid), who on hispost-doctoral stay funded by the Humboldt Foundationmasterminded most of the projects foussing on current is-sues in rubber science and technology that are discussedin this article.

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Biographical Sketch

Kay Saalwachter, born in 1970, obtained his diplomain chemistry in 1997 from the Institute of MacromolecularChemistry of the University of Freiburg, working on light

scattering of polymer solutions with Walter Burchard, andgot his doctorate degree in physical chemistry in 2000 workingwith Hans W. Spiess at the Max Planck Institute for Poly-mer Research in Mainz, focusing on methods of developmentin solid-state NMR. After a short period as project leader atthe same institute, he switched back to Freiburg to obtainhis habilitation in 2004 on NMR applications in the field ofmolecular dynamics in solid polymers, again at the Institute ofMacromolecular Chemistry in the group of Heino Finkelmann.In 2005, he was appointed professor of experimental physicsat the Martin-Luther-Universitat Halle-Wittenberg. Since hisfirst contact with solid-state NMR on an exchange year at theUniversity of Massachusetts (Amherst, USA) working withKlaus Schmidt-Rohr (now at Ames, Iowa), his continuing re-search interest is the development and application of NMRtechniques to the study of structure and dynamics in poly-meric, liquid crystalline, and other ’soft’ materials, which havebeen published in about 80 papers so far. He is a member ofthe editorial board of Macromolecules, J. Magn. Reson., SolidState Nucl. Magn. Reson., and Colloid Polym. Sci., and since2008, he serves the Deutsche Forschungsgemeinschaft (DFG)as a review board member in chemistry and referee for poly-mer research.

Kay Saalwachter