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Proton multiple-quantum NMR for the study of chain dynamicsand structural constraints in polymeric soft materials

Kay Saalwachter

Institut fur Physik, Martin-Luther-Universitat Halle-Wittenberg, Friedemann-Bach-Platz 6, D-06108 Halle, Germany

Received 9 August 2006Available online 12 January 2007

Keywords: Residual dipolar couplings; Transverse relaxation; Double-quantum NMR; Polymer dynamics; Elastomers; Networks; Rubber; Swelling;Poly(isoprene); Poly(butadiene); Poly(dimethylsiloxane)

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1. Multiscale chain dynamics and residual dipolar couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. The static 1H multiple-quantum experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1. Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2. Conceptual details and data treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2.3. Advanced approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3. Limitations of transverse relaxometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4. Comparison of DQ excitation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3. Applications to elastomers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1. Chain order distributions and heterogeneities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2. Quantitative interpretation of residual couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3. Chain dynamics in elastomers: failure of the slow-motion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.4. Network swelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5. Strained and oriented networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4. Entangled melt dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245. Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1. Grafted chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2. Confined chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.3. Study of gelation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6. Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1. Introduction

This article reviews the principles and applications of oneof the currently most powerful NMR approaches for thecharacterization of chain motion in elastomers, entangled

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doi:10.1016/j.pnmrs.2007.01.001

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Progress in Nuclear Magnetic Resonance Spectroscopy 51 (2007) 1–35

polymer melts and related mobile polymeric systems wellabove the glass transition. Considering the ubiquitousapplications of rubbers, as well as the industrial relevanceof melt processing of polymers, the rheological behaviorof soft polymeric systems is of central (technological) con-cern, yet the establishment of quantitative relationshipsbetween macroscopic properties and structure and dynam-ics at the molecular level remains an open challenge.

Melt-like polymer chains exhibit a complex hierarchyof dynamic processes, starting with very fast and localconformational rearrangements on the ps scale, andextending into the range of seconds for slow, diffusiveand cooperative motions [1]. Fast local motions are thedomain of NMR methods based on longitudinal spinrelaxation of suitable nuclei such as protons or carbon-

13, where the study of the relaxation dispersion over alarge frequency range for the former via field-cyclingmethods [2–5] or the chemical resolution of the latter[6,7] are the basis of the most powerful strategies, anddo not even require isotopic labelling. This class of meth-ods is not covered here and the reader is referred to thecited literature.

Motions that involve a larger number of segments up tothe level of whole chains dominate the mechanical proper-ties of soft polymeric systems, emphasizing the need forNMR methods that are sensitive in the corresponding fre-quency range. Traditionally, transverse relaxation phenom-ena of protons or deuterons, or, equivalently, line shapeanalysis in the frequency domain, have been used for thatpurpose [8–20], and provided the starting point for more

Nomenclature

List of symbols, abbreviations and acronyms

ACF autocorrelation functiona(w) duty-cycle dependent scaling factorAW Andersen–WeissBR butadiene rubberC1 Flory’s characteristic ratioC(t) orientation autocorrelation functionCPMG Carr–Purcell–Meiboom–GillDCC dipolar coupling constantDCE dipolar correlation effectDELM dipolar-encoded longitudinal magnetizationDLS dynamic light scatteringDSC differential scanning calorimetryDQ double quantumDeff effective DCC (=Dstat/k)DG residual DCC, Gaussian distribution averageDres residual DCCDstat static-limit DCCDi inter-pulse spacingFID free induction decay�HDQ average (DQ) HamiltonianHJ (J coupling) HamiltonianI. . . spectroscopic intensityI ðiÞþ=� raising/lowering spin operatorJ scalar couplingk scaling factor for the intra-segmental DCCj power-law exponentLM longitudinal magnetizationM2eff effective static-limit second momentM2res residual second momentMc network chain molecular weightMe entanglement molecular weightMte molecular weight between trapped entangle-

mentsMAS magic-angle spinningMQ multiple quantum

N number of statistical (Kuhn) segmentsnc number of pulse sequence cyclesnDQ point-by-point normalized DQ intensityNR natural rubber (poly(cis-1,4-isoprene))P(. . .) probability densityP2(. . .) second Legendre polynomialPB poly(butadiene)PDMS poly(dimethylsiloxane)phr per hundred rubberPS poly(styrene)PS-b-PB PS-block-PB copolymerP(S-co-AMS) poly(styrene-co-aminomethylstyrene)U. . . pulse phase/. . . integrated dipolar evolution phaseQ degree of volume swelling V/V0

RDC residual dipolar couplingREDOR rotational-echo double-resonanceSBR styrene–butadiene rubberSb dynamic order parameter of the polymer back-

bonePMQ multiple-quantum sum intensity

rG standard deviation of Gaussian distributionSEDOR spin–echo double resonanceT �2 apparent transverse relaxation timeTg glass transition temperaturetp 90� pulse lengthtc pulse sequence cycle timesDQ DQ evolution timesd disengagement or terminal relaxation timese entanglement timesf/s fast/slow correlation timesR (longest entanglement-constrained) Rouse timesz z-filter timeWLF Williams–Landel–FerryZQ zero quantum

2 K. Saalwachter / Progress in Nuclear Magnetic Resonance Spectroscopy 51 (2007) 1–35

specific developments that ultimately led to the introduc-tion of multiple-quantum (MQ) spectroscopy.

An important feature of the dynamics of cross-linked,tethered or entangled chains is the semi-local anisotropy.Fast motions such as Rouse modes are ultimately hinderedby the presence of topological restrictions, and long-livedorientation correlations are induced. Their magnitude isdirectly linked to microscopic parameters such as thecross-link density, the entanglement length, or the ‘‘tubediameter’’, and their lifetime is related to parameters suchas the terminal relaxation time or the disengagement timeof the tube/reptation model. Consequently, knowledgeabout local order is highly relevant for the understandingof the material’s rheological properties. Already more than30 years ago, Cohen-Addad [8,16] and some time later yetindependently Gotlib [9] had realized that the degree oflocal order is directly reflected in the transverse evolutionof magnetization subject to residual motion-averaged ten-sorial NMR interactions.

The most straightforward observable is certainly theproton residual dipolar coupling, which renders transverserelaxation curves of long-chain polymers non-exponentialsince they are dominated by (coherently refocussable) dipo-lar dephasing. A multitude of often related theories hasbeen presented over the years that address the central prob-lem of describing the shape of relaxation curves and reso-nance lines, and separating the coherent dipolar effectfrom true spin relaxation induced by random thermalmotion [11,14,17,20–25]. As will be shown below, MQspectroscopy can help to test the validity of some impor-tant model assumptions. Conversely, transverse relaxationdecays are notoriously featureless and influenced by a vari-ety of factors, such that the use of multi-parameter func-tions based on strong model assumptions cannot beexpected to provide proof of the validity of a model andmay even lead to fitting artifacts.

The existence of ‘‘pseudo-solid’’ spin echoes representsone of the most direct proofs of residual dipolar couplings[26], yet their analysis until now remains mostly qualita-tive [27]. Improvements directed at a better separation oftrue transverse relaxation and residual couplings wereachieved by clever combinations of Hahn and solid echoes[28–30] or by approaches based on stimulated echoes[31,32]. The former yields a build-up signal that isdescribed by a sine–sine correlation function, also referredto as the ‘‘b’’ function. This function provides a more reli-able basis for fitting, separation and extraction of mean-ingful parameters, as the coherent dipolar effect causesan intensity build-up, while dynamic effects cause the com-peting decay.

The essential advantage of MQ spectroscopy is now thatit not only yields a build-up function that is essentiallyidentical to the above mentioned sine–sine correlation,i.e., a build-up curve that is dominated by spin-pair dou-ble-quantum (DQ) coherences [33], but also that the sameexperiment provides access to a fully dipolar-refocussedmultiple-quantum decay function which can be used to

independently analyze the effect of dynamics on the mea-sured data. In networks, where large-scale chain dynamicsis mostly absent, the shape of the MQ decay is almost iden-tical to the relaxation part of the DQ build-up signal, suchthat a temperature-independent normalized build-up func-tion can be obtained that solely depends on the networkstructure. In this way, reliable information on the distribu-

tion of residual couplings, and thus on semi-local dynamicheterogeneity, becomes accessible.

This review is concerned with the foundations andrecent applications of proton MQ spectroscopy to a varietyof systems such as elastomers of different types, swollengels and gelling systems, as well as free and confined poly-mer melts and chains tethered to copolymer blocks andsurfaces. It should be emphasized, however, that the cen-tral concepts and advantages of MQ spectroscopy, in par-ticular the distribution analysis in networks and theseparation of structural and dynamic information, isdirectly and without any change in the experimental strat-egy applicable to deuterium. The absence of complicationsdue to multiple coupled spins and the possibility of selec-tive deuteration means that deuterium NMR studies willhelp to further extend our understanding of polymerdynamics.

2. Basic principles

2.1. Multiscale chain dynamics and residual dipolar

couplings

The phenomenological starting point for the under-standing of the relationship between polymer chaindynamics and NMR-detected local order is the orientationdependence of the dipolar (or quadrupolar) coupling,which is given by the second Legendre polynomialP2(cosh). A priori, the angle h is the orientation of theinternuclear axis with respect to the magnetic field, whichfluctuates rapidly and thus mirrors the segmental dynam-ics. In order to simplify the treatment, one can subsumelocal and very fast conformational rearrangements onthe ps scale into a pre-averaged dipolar tensor. The jointnumber of monomer units that take part in this pre-aver-aging can be referred to as an ‘‘NMR submolecule’’[21,22,24,34]; the associated tensorial interaction is there-fore a semi-local quantity. It may be related to the statis-tical or Kuhn chain segment [35], and the analysis ofmotions within such a Kuhn segment is the domain ofT1 relaxometry and rotational isomeric states models.(Note that this submolecule definition resembles that ofBrereton [24,34], while in Cohen-Addad’s original work[21,22] it comprises a length scale up to the entanglementlength.)

The pre-averaging embodies a rescaling of the static-limit dipolar coupling constant (DCC), Dstat, by a con-stant k, for which a model needs to be adopted (videinfra). Importantly, h then takes on the meaning of theorientation of the local symmetry axis of motion rather

K. Saalwachter / Progress in Nuclear Magnetic Resonance Spectroscopy 51 (2007) 1–35 3

than an individual bond or internuclear vector orienta-tion, and this symmetry axis can safely be assumed tobe along the polymer backbone. The time dependence,h(t), therefore monitors orientation fluctuations of thepolymer backbone. Its dynamics should be describablein terms of the classical theories of polymer dynamics[1], and its spatio-temporal distribution can be quantifiedin terms of a uniaxial order parameter. The characteristicsof the uniaxial dynamics of h(t) is most convenientlydescribed by the autocorrelation function (ACF) of thesecond Legendre polynomial,

Cðjta � tbjÞ ¼ hP 2ðcos htaÞP 2ðcos htbÞi: ð1Þ

C(t) basically gives the probability of finding a chain seg-ment, that has been observed in a particular orientationhta at time ta, again in the same orientation after the timet = jta � tbj has passed. Most theoretical approachesdescribing NMR relaxation rely on a knowledge of thisfunction.

Fig. 1 gives a schematic representation of the orientationACF for the case of long-chain polymer melts and net-works. The first decay is associated with Rouse-type chainmotions. This type of dynamics is ultimately constrainedby entanglements or permanent cross-links, whereby amore or less developed plateau arises. In permanentlycross-linked networks, the plateau is only weakly affectedby slower, cooperative modes, while for entangled linearor end-tethered chains, reptation or arm retraction, respec-tively, provide effective mechanisms for further loss of cor-relation. The exact shape of the ACF depends on thecomplex hierarchy of free, constrained, and cooperativemotions. Different models and approximations for theACF, many of them based on the Rouse/reptation or tubemodels [1] have been discussed in the context of differentNMR studies, yet no consensus has currently been reachedon its exact shape.

The height of this plateau is given by the square of whatis here defined as the order parameter of the polymerbackbone,

Sb ¼ kDres

Dstat

¼ 3

5

r2

N; ð2Þ

where Dres is the residual dipolar coupling (RDC) that re-sults as a time average over the fluctuations of the dipolartensor covering the time until the plateau region is reached.The constant k describes the averaging due to very fast in-tra-segmental motions. As indicated by the right hand sideof Eq. (2), Sb is related to the ratio r of the end-to-end vec-tor r of the chain segment separating the constraints to itsaverage, unperturbed melt-state value r2

0ðr2 ¼ r2=r20Þ, and to

N, the number of statistical (Kuhn) segments between theconstraints [38]. The latter provides the link to entangle-ment theories or theories of rubber elasticity, swelling[39], and stress-optical properties [40] of elastomers. The1/N-relationship is very well supported by a variety ofNMR experiments, to be addressed below.

Due to the lack of timescale separation, Sb or, equiva-lently, Dres is not easily accessible in polymer melts or lowlycross-linked systems, while a model-free determination ofDres is feasible for the case of elastomers [36]. In this case,it is possible to fully remove the effect of dynamics from theexperimental observables, whereby reliable informationeven on the distribution of Dres is attainable. The fittedquantity always represents an average over multiple inter-and intra-segmental dipolar couplings, which means thatonly an apparent spin-pair Dres is obtained. It is often givenin terms of the related (van Vleck) second moment M2res,

M2res ¼9

20D2

res: ð3Þ

Thus, S2b ¼ M2res=M2eff , when we include the pre-averaging

by local conformational rearrangements in an effectiverather than static rigid-lattice second moment,

M2eff ¼9

20

Dstat

k

� �2

: ð4Þ

This, as well as Deff = Dstat/k, is of course a model-depen-dent quantity. Note that the occasional neglect of k is asubject of some confusion in the literature.

The effect of distributions of Dres on a simple, purelydipolar free-induction decay as well as the correspondingspectra is demonstrated in Fig. 2. Generally, distributionsof Dres lead to a disappearance of characteristic oscilla-tions, but it is important to realize that intermediate-time-scale motions and the resulting intensity relaxation havethe same qualitative effect, such that a reliable differentia-tion between the two is not straightforwardly possible. Ina variety of works [17,22,23,41–45] proton or deuteriumrelaxation decays or lineshapes of elastomers have beenanalyzed in terms of P(r), a Gaussian distribution of end-to-end distances between cross-links. In terms of the nor-malized r2 ¼ r2=r2

0, P(r) is given by

PðrÞ ¼ 4ffiffiffipp 3

2

� �3=2

r2 exp � 3

2r2

� �: ð5Þ

Fig. 1. Schematic representation of the orientation autocorrelationfunction, C(t), for entangled melt and elastomer chain segments far aboveTg. Adapted from Ref. [36].


Gaussian statistics is a cornerstone assumption in most the-ories of polymer dynamics, and it is important to appreci-ate its potential effect on the measured data. Using Eq. (2),the distribution of Dres is easily obtained as [37]

P ðjDresjÞ ¼2ffiffiffipp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi27jDresj

8�D3res

se�3jDresj=ð2�DresÞ: ð6Þ

This is a gamma distribution, whose positive distribution-averaged �Dres is directly related to its standard deviation,r2

c ¼ 23

�D2res. It should be kept in mind that P(jDresj) neglects

any influence of a potentially serious network chain poly-dispersity. The solid line in the inset of Fig. 2b may be re-ferred to as ‘‘super-Lorentzian’’, and its observation indeuterium spectra of elastomers was interpreted as a confir-mation of the relevance of Gaussian statistics [41,44,45].Below, evidence will be presented that suggests that thisconclusion is in fact not fully correct.

2.2. The static 1H multiple-quantum experiment

2.2.1. Basic principles

The pulse sequences of the experiments to be discussedin the following are schematically depicted in Fig. 3, anddetails are given in the cited literature. Many of the resultsreported herein have been obtained using simple low-fieldequipment, which was shown to yield data of almost thesame quality as a modern high-field spectrometer [47].The major effects on the NMR observables are dipolar innature and thus field-independent, and the loss of chemicalresolution due to the low field or the absence of MAS doesnot pose a serious restriction when single-component sys-tems or materials with large mobility contrast (such asblock-copolymers) are to be investigated. Intensities arealways obtained either by simply measuring the initialFID amplitude, or by signal integration after FT, providedthere is sufficient chemical-shift resolution at high field.

The advantages of the MQ experiment are best appreci-ated by comparison with a simple dipolar FID of a spin-

pair, as for instance detected in a Hahn (not solid!) echoexperiment. The echo intensity, Iecho, is related to /echo,the net phase factor (see Fig. 3a), by

Fig. 2. (a) Distribution functions, P(jDresj), for residual dipolar coupling constants, Dres, with a common average value �Dres=2p ¼ 100 Hz. rG denotes thestandard deviation of a Gaussian distribution. (b) Relaxation functions calculated from Eqs. (7) and (8) using the distributions in (a). The inset shows thecorresponding spectra after Fourier transformation. Data replotted from Ref. [37].

Fig. 3. (a) Schematic pulse sequence for (sometimes combined) Hahn- orsolid-echo experiments, where the net phase factor /echo is obtained as asum or difference, respectively, of phases /d associated with purely dipolartransverse spin evolution. (b) Static multiple-quantum experiment [46].The phase factors relevant for evolution under the pulse sequence blocks,/DQ, generally represent spin evolution under a dipolar DQ Hamiltonian.Coherence selection is realized via an incremented shift DU of the carrierphase U0 of the reconversion pulse sequence with appropriate receiverphase cycling to obtain the DQ-filtered (UDQ) or reference (Uref)intensities. (c) and (d) Detailed pulse sequence of the improved DQexperiment of Baum and Pines [37,46]. Details on the exact timing, theequivalence of cycle time (tc) or cycle number (nc) incrementation to realizea specific sDQ, and the duty-cycle dependence, can found in from Ref. [37].


IechoðsechoÞ ¼ hcos /echoðsechoÞi; ð7Þ

The time-dependent phase factor /echo (in radians) associ-ated with purely dipolar time evolution reads

/echoðsechoÞ ¼3

2Deff

Z secho

0

P 2ðcos htÞdt: ð8Þ

The time dependence of ht reflects the full hierarchy of mo-tions ranging from orientation fluctuations of the Kuhnsegment to large-scale cooperative modes. Assuming allmodes up to the length scale of the fixed topological con-straints in a permanent network to be fast, the integral inEq. (8) simply reduces to Sb. As sketched in the upper leftof Fig. 4, one would in this case expect to observe purelydipolar FIDs (or spectra) characterized by a well-definedresidual dipolar coupling, SbDeff. This scenario will be re-ferred to as the ‘‘quasi-static’’ or ‘‘fast-motion’’ limit.

In practice however, there will always be some contribu-tion of molecular motion on the timescale of the experiment.The explicit time dependence of h must therefore be consid-ered, and phenomenologically, a damping of the dipolarFID (or a broadened spectrum) is observed. This is sketchedat the lower left of Fig. 4. Note that in a dipolar-refocussed

solid-echo-type experiment in the absence of motion leadingto relaxation, the net phase would be the difference of twoequal terms (see Fig. 3a) and thus simply be zero. Conse-quently, the echo duration dependent signal would then beconstant in the absence of relaxation, and a homogeneouslydecaying function when motion is present [48].

While the term ‘‘multiple-quantum’’ experiment implies aconceptually complicated experiment, its theoretical treat-ment is in fact rather compact when the arguments arerestricted to simple spin pairs. In most general terms, timeevolution occurs under a pure dipolar DQ Hamiltonian [46],

�HDQ ¼ �aðwÞ

2

Xi<j

DðijÞeff P 2ðcos bijÞðIðiÞþ I ðjÞþ þ I ðiÞ� I ðjÞ� Þ: ð9Þ

Note that the duty-cycle dependent scaling factor a(w) isusually absorbed into the time axis and neglected in thefollowing for clarity [37]. Focussing on a single pair inter-action, the corresponding intensity build-up of DQ coher-ence, IDQ(sDQ), is obtained as

IDQðsDQÞ ¼ hsin /DQ1 sin /DQ2i; ð10Þ

where the phase factors /DQab ” /DQ1 and /DQ2 for differ-ent time intervals sa to sb (see Fig. 3b) are given by

/DQab ¼ Deff

Z sb

sa

P 2ðcos htÞdt: ð11Þ

Thus, the only formal differences between time evolutionin Hahn echo and MQ experiments are (i) the type oftrigonometric functions appearing in the signal functionsIecho(secho) and IDQ(sDQ) and (ii) the prefactor of 3/2that appears for dipolar transverse time evolution, butnot for evolution under the DQ Hamiltonian given byEq. (9). Note that other pulse sequences may have a dif-ferent average Hamiltonian and thus exhibit a differentprefactor.

A salient advantage of the MQ experiment is now that asimple change in the 4-step DQ filtration phase cycle of thereceiver provides access to a reference intensity,

I refðsDQÞ ¼ hcos /DQ1 cos /DQ2i; ð12Þ

which simply comprises all magnetization that has notevolved into DQ (or, more precisely, (4n + 2)-quantum)coherences. In the spin pair limit, it just comprises dipo-lar-encoded longitudinal magnetization (DELM), whichcan also be used for separate analysis [49].

Fig. 4. Comparison of the results of ‘‘conventional’’ experiments (dipolar or quadrupolar spectra and the corresponding shift-compensated FIDs) and anMQ experiment for the cases of isolated spins or spin pairs without relaxation and real elastomers. The given relations describe the form of the signalfunction for the ideal, ‘‘quasi-static’’ case without relaxation.


The sum of the two possible signal functions IRMQ is

IRMQðsDQÞ ¼ hsin /DQ1 sin /DQ2i þ hcos /DQ1 cos /DQ2i:ð13Þ

This is a fully dipolar refocussed intensity that decays onlyas a result of molecular motion that renders /DQ1 „ /DQ2

The phenomenological behavior of the signal functionsIDQ(sDQ) and IRMQ(sDQ) for the quasi-static and real casesis shown at the right side of Fig. 4.

It turns out that in networks, the relaxation contributionsto IDQ(sDQ) and IRMQ(sDQ) are nearly equal, such that theeffect of molecular motion on IDQ(sDQ) can be removed bypoint-by-point division to give InDQ, the so-called normalized

DQ intensity (not to be confused with the ‘‘trivial’’ and cus-tomary normalization of experimental intensities to, forexample, the full intensity after a 90� pulse):

InDQðsDQÞ ¼ IDQðsDQÞ=IRMQðsDQÞ: ð14Þ

This now carries only structural information, and can beanalyzed in the quasi-static limit even in terms of distribu-tions of Dres The approach is easily validated by the obser-vation that InDQ(sDQ) is independent of temperature overan interval exceeding 100 K, as is shown in Fig. 6.

In the absence of relaxation effects, InDQ(sDQ) is givenby Eq. (10) with equal phases, hsin /2

DQi. Under theassumption of static Gaussian statistics for the associatedinteraction frequency x � /DQ/sDQ, which is approxi-mately provided by the powder average (see Fig. 5), thesecond-moment approximation of the DQ build-up reads

InDQðsDQÞ ¼1

21� exp � 8

9M2ress

2DQ

� ��

¼ 1

21� exp � 2

5D2

ress2DQ

� �� : ð15Þ

A comparison of this result with the powder-averaged sig-nal hsin /2

DQi, Eq. (10), as well as experimental data isshown in Fig. 6. The strong oscillations of the powder-averaged spin-pair signal are not very apparent in experi-mental data, which is a consequence of multiple couplings,as evidenced by multi-spin simulations (see Fig. 20). In par-ticular this latter comparison shows that Eq. (15) approxi-mates the build-up function of a monomeric unit very wellup to InDQ = 0.45. Equating this value with Eq. (15) yieldsa definition of an operationally defined fitting limit,

smaxDQ ¼ 2:4=Dres; ð16Þ

that should always be enforced to minimize systematic er-rors. When more complicated functions comprising distri-butions or correlation times (see Section 3.3) are used forfitting, smax

DQ must be iteratively adjusted to the currentbest-fit value of Dres.

A distribution of residual couplings, which is in the sim-plest case related to the chemical heterogeneity (e.g., sty-rene–butadiene rubber, SBR), lead to deviations from theinverted Gaussian shape. The most general approach isto use Eq. (15) as the kernel function in a regularized inver-sion of the distribution interval, and details and limitationsare discussed in Ref. [37]. For narrower spreads of frequen-cies, one can assume the shape of the distribution functionto be Gaussian (see Fig. 2), whereby the distribution inte-gral over Eq. (15) can be evaluated analytically, giving

InDQðsDQÞ ¼1

21�

exp �25D2

Gs2

DQ

1þ45r

2G

s2DQ

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4

5r2

Gs2DQ

q0BB@

1CCA ð17Þ

DG and rG are the average apparent RDC constant and thestandard deviation, respectively. Note that fits to this func-tion make physical sense only as long as rG is substantiallysmaller than DG. For samples with clear-cut bimodality,stable fits of weighted two-component superpositions ofEqs. (15) or (17) are also possible.

For completeness, I also give the the second-momentapproximation to the Hahn echo decay function, Eq. (7),

IechoðsechoÞ ¼ exp � 9

40D2

ress2echo

� �: ð18Þ

Fig. 5. Comparison of a Pake-like (solid line) and a Gaussian (dashedline) frequency distribution. The second-moment approximation corre-sponds to approximating the former by the latter.

Fig. 6. Normalized 500 MHz proton DQ build-up curves of a PDMSnetwork demonstrating the virtual independence of the data on experi-mental conditions such as temperature and DQ sequence parameters (seeFig. 3). The long-time performance of a time-incremented 1-cycle versionis inferior, which is why for such experiments it is recommended to alwaysuse nc = 2, where the second cycle has inverted phases. This is also theversion used on the low-field instrument (minispec) [47], where addedpulse imperfections of too many appended cycles have a deleterious effect.The lines are fits to analytical DQ build-up functions, and the gray verticalbar indicates the operationally defined fitting limit given by Eq. (16). Datareplotted from Ref. [37].


2.2.2. Conceptual details and data treatment

Following the established rules of coherence order selec-tion [50], the acquisition of a reference intensity that formsthe full complement of IDQ(sDQ) is only possible for DQcoherence selection using a 4-step phase cycle {x, y, �x,�y} of the carrier phase difference DU (see Fig. 3). Then,the receiver phase cycle is {x, �x, x, �x} for IDQ(sDQ), and{x, x, x, x} for Iref(sDQ). These phase cycles in fact selectall coherences of order (4n + 2) and (4n), respectively, andmany higher coherence orders do indeed play a role, sincethe average DQ Hamiltonian, Eq. (9), excites all 2n coher-ence orders. An important consequence of long-time spindynamics spanning the full Liouville space of even quantumorders is the equal intensity partition among IDQ(sDQ) andIref(sDQ). This means that InDQ(sDQ) in networks alwaysattains a long-time intensity plateau at 50% (the same holdsfor isolated pairs, but not for finite odd-numbered clusterssuch as methyl groups). This is shown in Fig. 7. Note thatInDQ(sDQ) becomes unreliable at longer evolution times,where IDQ(sDQ) has decayed to a very low level.

Spin counting experiments [46], which are based on adiscrete smaller-step incrementation of DU and subsequentFourier transformation over this phase dimension, can beused to assess the effect of higher-quantum coherences onthe measured intensities [37]. The data plotted in Fig. 7gives convincing evidence that higher-quantum orders areindeed not important until well into the plateau region ofthe proton nDQ build-up curve of a typical rubber. Thespin-pair treatment sketched above thus appears well justi-fied, as long as the data analysis is retricted to the region ofthe initial rise. Note that transverse relaxation data, whichare commonly treated in terms of spin-pair theories, areaffected by multi-spin effects on a similar timescale, suchthat fits to the full relaxation function will inevitably sufferfrom systematic errors.

So far the simplified assumption that the investigatedmaterial consists only of elastically active network chainshas been made. Usually, additional components such asshort dangling chains, sol, or some solvent may be present.The dynamics of these components are usually isotropic onrather short timescales, such that no observable dipolarcoupling remains. These parts are therefore detected onlyin the reference experiment, and thus form more slowlydecaying long-time tails of IRMQ(sDQ). In order for the nor-malization approach to be successful and to observe a 50%intensity plateau for InDQ(sDQ), it is necessary to removethese contributions by suitable fitting and subtraction ofthe tails. Tails are also observed in T2 relaxation curves;they are schematically shown in Fig. 4.

Multicomponent fits of transverse relaxation data arealways subject to potentially uncontrolled interdependen-cies of individual fitting parameters, in particular whenthe exact functional forms of the components areunknown. The MQ experiment has some features that pro-vide a more reliable separation of different components inthe case of permanent elastomers [51].

Singly exponential tails, as observed in our work on dryand swollen PDMS networks [37,52] are easily identified bystraight lines in a semi-logarithmic plot of IRMQ(sDQ), seeFig. 8a. Notably, the amount of the mobile contributionschanges with temperature, which indicates its origin to befrom partially hindered dangling chains: larger and largerfractions of such chains appear isotropically mobile whenthe entanglement effect on their ends is reduced by armretraction processes. Fitting and subtracting them, andusing Eq. (14), leads to InDQ(sDQ) curves such as the onesin Fig. 6 that are temperature independent over the wholeinvestigated range.

In the case of styrene–butadiene rubber, and sometimesalso butadiene rubber and natural rubber, a second, morerapidly decaying component of IRMQ(sDQ) (or Iref(sDQ))appears. Since these polymers are less mobile (=closer toTg at common experimental temperatures), this componentmay be associated with dangling chains (B), while the longtime tail is attributed to low molecular weight sol compo-nents (C). The associated value of T �2C is usually longenough to be reliably fitted and subtracted.

Due to the similarity of its apparent T �2B to the long-timedecay of actual multiple-quantum coherences in IDQ(sDQ)and Iref(sDQ), it is often not possible to simply subtract asecond exponential tail. The trick to solve this problem isbased on the notion that IDQ(sDQ) = Iref(sDQ) in the long-time limit. As shown in Fig. 8b, the B fraction is more reli-ably identified and fitted in a semi-logarithmic plot ofI refðsDQÞ � IDQðsDQÞ � Ce�2sDQ=T �

2C vs. sDQ, where the inter-val over which the B component decays linearly is substan-tially prolonged. With B determined, the final formula forIRMQ(sDQ) is thus

IRMQðsDQÞ¼ IDQðsDQÞþ I refðsDQÞ�Be�2sDQ=T �2B�Ce�2sDQ=T �

2C ;

ð19Þ

Fig. 7. Experimental 500 MHz MQ intensities for protons of a PDMSnetwork separated by quantum order. Results obtained using the regular4-step (4n + 2)-quantum selection phase cycle (small open squares) arecompared with intensities from spin counting experiments that differen-tiate up to 16 quantum orders. All data is point-by-point-normalized forrelaxation effects using the full sum intensity over all quantum orders. Thelines merely guide the eye. Data replotted from Ref. [37].


where B and C are the fraction of dangling and sol chains,respectively. The fraction A of network chains is easily ob-tained as A = 1 � B � C when Iref(sDQ) is normalized to 1for sDQ = 0.

It should be mentioned that the amplitudes, and to a lar-ger extent, the apparent relaxation times of the contribu-tions B and C, are affected by the long-time performanceof the pulse sequence, which in turn depends on the rfhomogeneity, the quality of the spectrometer setup, andother factors. This is probably the reason why solventand dangling-chain contributions cannot be separated inthe case of PDMS, where the apparent T2 is long and theseeffects dominate. This is also why quantitative interpreta-tions of long T2 values should be avoided. Given all ambi-guities, however, the initial rise of InDQ(sDQ) is only weaklyaffected even when the mobile contributions are not mod-elled and subtracted precisely [51].

2.2.3. Advanced approachesWhen the network chain content is known already or

not of central interest, the isotropically mobile contribu-

tions can also removed by appending a second fixed DQexcitation/reconversion sequence block before the actualincremented DQ pulse sequence [37]. The nested phasecycle is constructed such that DQ coherences (that ariseonly from the network fraction) are selected by the firstblock, while the second block is used to probe eitherIDQ(sDQ) or Iref(sDQ). The success of this procedure isshown in Fig. 9a.

Of crucial importance is the period sz between the twoparts of the experiment, during which spin exchange mayoccur. The DQ pre-selection of course excites segmentsassociated with different orientations of the RDC tensorwith respect to B0 with different efficiency, which meansthat the isotropic powder average, needed for an unambig-uous analysis of the final build-up curve, is broken. Thiseffect is demonstrated in Fig. 9b for the case of sz = 0. Inorder to avoid such complications one can just use a valueof sz on the order of 100 ms to ensure complete re-equili-bration. Note that for experiments with short sz it is impor-tant to construct the phase cycle such that only coherencesof order 0 (mostly longitudinal magnetization and someZQ coherences that result from two consecutive DQ tran-sitions) are retained during this interval.

A powerful experimental strategy, yet to be used inactual applications, is to monitor the recovery of the isotro-pic powder distribution by incrementing sz (see Fig. 9c).Flip–flop-mediated spin diffusion or slow reorientationsof the RDC tensor are possible processes explaining thebehavior, and temperature-dependent studies can be usedto differentiate between the two scenarios. Considering thatslow reorientations are probably absent in permanent elas-tomers (vide infra), spin diffusion is the most likely candi-date. Thus, using spin diffusion coefficients that can beestimated from T2 experiments [54], one could gain accessto the length scale of correlated RDC tensor orientations,or the size of regions of different cross-link density in het-erogeneous rubbers.

2.3. Limitations of transverse relaxometry

As mentioned, Hahn echo experiments are a popularalternative to MQ experiments in rubber applications. Inthe framework of the Andersen–Weiss theory [55], the echointensity Iecho(secho) can be evaluated from Eqs. (7) and (8)on the basis of a slow-motion model for the RDC tensorreorientation process. That is, only an exponential long-time decay of the correlation function in Fig. 1, with adecay time of ss, is explicitly considered [11]. This yieldsa three-parameter fitting function for the network compo-nent (A):

IechoðsechoÞ ¼ exp �secho=T 2Af g

� exp � 9

20D2

ress2s e�secho=ss þ secho

ss

� 1

� �� :

ð20Þ

Fig. 8. IRMQ(sDQ) for protons at 500 MHz of dry PDMS rubber (a) andSBR (b), both plotted on a semi-logarithmic scale. In (b), the strategy for areliable stepwise subtraction of different tails associated with differentmobile parts of the sample is sketched. Data replotted from Refs. [37] and[51], respectively.


T2A should phenomenologically parametrize the influenceof the initial fast decay of the correlation function, and isoften neglected. As will be unambiguously proven in Sec-tion 3.3, the slow-motion model is incorrect, yet since it

is still rather popular [13,15,18,56], the serious limitationsinherent in the use of Eq. (20) must be emphasized [53].

Using proton Hahn-echo experiments at 400 MHz pro-ton frequency in combination with Eq. (20), Luo et al.[56] have studied residual dipolar couplings in a series ofstyrene–butadiene rubber filled with different amounts ofcarbon black and silica. In agreement with earlier work inthe field, their data was interpreted as indicating a substan-tial increase of the effective cross-link density with increas-ing filler content (Fig. 10a). This, however, is an artifactrelated to fitting ambiguities resulting from parameter inter-dependencies, which is directly proven by the observation ofa contradictory trend when the same samples are investi-gated at 20 MHz and analyzed in the same way(Fig. 10b). Therefore, slight field-dependent changes in theshape of the relaxation curves, which are of course not cov-ered by the model underlying Eq. (20), bias the fitting resultsinto different directions. The origin of the (weak) fielddependence is as yet unclear, yet molecular motion in com-bination with susceptibility contrast around nanoscopicvoid spaces appears to be a possible candidate [57].

In fact, the rubber matrix turned out to be virtuallyunaffected by the presence of filler, as clearly corroboratedby proton MQ experiments conducted on the same sampleseries at 500 MHz [51], see Fig. 10c. In this graph, both theaverage RDC from a regularization analysis, and resultsfrom a single-parameter fit using Eq. (15) are given, andas expected for distributed quantities, the values differdue to the way the average is taken. Notably, as shownin Fig. 10d, fitting the 20 MHz data with the static-limit fit-ting function Eq. (18) yields the same results. The corre-sponding fits are of course only poor representations ofthe actual data (since no distribution is explicitly accountedfor), but they do provide a stable average over the existingRDC distribution. In conclusion, Eq. (20) should not beused for the analysis of Hahn echo experiments onelastomers.

Another, even more subtle artifact in RDC determina-tions arises when elastomers with different chemical func-tionalities are investigated by proton Hahn-echorelaxometry at high magnetic field [58]. At Larmor fre-quencies of a few hundred MHz for protons, a limitedchemical shift resolution makes it possible to detect RDCsspecifically for the resolved resonances even in static spec-tra. Along these lines, Steren et al. [59] have claimedweaker RDCs associated with the olefinic proton methineresonance in natural rubber when compared to the RDCsdetected at the (overlapped) CH2/CH3 signal positions.The effect can be inspected for the resolved Hahn echo dataplotted in Fig. 11. These observations are again in markedcontrast to the results of MQ experiments [60], which indi-cate virtually identical Dres (see Section 3.2).

The explanation is as follows: the methine resonancecomprises a single proton that is primarily coupled toneighboring methene and methyl protons rather than itsown kind one monomer unit away (for a proof see the2D DQ correlation spectrum in Ref. [58]). Apart from

Fig. 9. DQ pre-filtering (see coherence pathways in the inset) for theremoval of isotropic signal components and exchange studies. (a)InDQ(sDQ) build-up curves for protons at 500 MHz in a slightly swollenPDMS network (tc = 0.2 and 0.8 ms, small and large symbols, respec-tively) showing the equivalence of a �10% tail subtraction and pre-filtering for a long z-filter period sz = 300 ms. (b) Build-up curves fordifferent DQ pre-selection times and sz = 0. The arrows indicate therespective pre-selection times on the regular build-up curve. (c) Build-upcurves using a fixed preselection time of 2.2 ms and different sz.Magnetization or orientation exchange during sz lead to an approach ofthe regular DQ build-up (dashed line). Data replotted from Ref. [37].


the dipolar couplings between CH and CH2/CH3, thechemical shift separation of 3.5 ppm amounts to 1750 Hzat 500 MHz, which clearly exceeds the average dipolar cou-pling of a few hundred Hz experienced by the CH group.Since the homonuclear dipolar and the chemical shift dif-ference Hamiltonians do not commute, the homonucleardipolar coupling is effectively averaged to the weak cou-pling limit by the chemical shift difference [61]. Thisamounts to a factor of 2/3 for the apparent dipolar cou-pling, in agreement with the observation of the Hahn echodecay at high field.

When the chemical shift difference is effectively removedby either performing experiments at low Larmor frequencyor by using a repetitive pulse sequence that provides rapidshift refocussing, the strong coupling limit applies and thefactor 2/3 disappears. The Baum/Pines MQ experiment,which we apply in a repetitive fashion (nc incrementation)at high field for exactly this reason, does provide this rapidrefocussing, as chemical shifts are compensated over a sin-gle cycle. Alternatively, CPMG experiments can be per-

formed, where of course always the last echo of anincremented train needs to be Fourier-transformed toobtain a spectrum. The results in Fig. 11 nicely show thatthe CPMG-detected initial decay of CH and CH2/CH3

magnetization is indeed almost identical. Note that thisresult is not much dependent on the pulse spacing in theCPMG trains; slight differences arise only for the apparentT2 values of the sol contributions detected at the differentresonances (this may well be a genuine effect related tothe fact that the fast segmental averaging process that ulti-mately leads to similar RDCs and causes this T2 decay hasdifferent local geometries).

In summary, MQ spectroscopy has been found to be themost reliable tool for the analysis of elastomer microstruc-ture, as it intrinsically avoids artifacts related to non-dipo-lar effects on the analyzed build-up functions. Further, as agenuine advantage, it provides access to the RDC distribu-tion, which in the case of Hahn echoes is always masked byeffects of intermediate motions.

2.4. Comparison of DQ excitation schemes

As an alternative to the lengthy Baum/Pines sequencepresented above, one may simply use a two-pulse segmentðp

2Þx � sDQ � ðp2 Þx for MQ excitation and reconversion.

Transverse magnetization created by the first pulse thenundergoes free dipolar evolution (as in a Hahn echo exper-iment), and after sDQ, the resulting two- and higher-spinantiphase coherences are converted into various MQcoherences by the second pulse. Free dipolar evolution dur-ing sDQ features the common prefactor of 3/2, renderingthe build-up quicker, yet less efficient as to the excitationof higher-order coherences than the Baum-Pines sequence.When just spin pairs are considered, the theory discussed inSection 2.2.1 is fully valid apart from this multiplicativecorrection. In order to improve the long-time performanceof the two-pulse sequence, a refocussing p pulse needs to beadded in the center, and the version with equal phases

Fig. 10. Proton residual dipolar couplings measured in SBR as a function of filler loading, given in per hundred rubber (phr). (a) and (b) Hahn-echorelaxometry at 400 and 20 MHz, both using Eq. (20) for data analysis, (c) static DQ spectroscopy at 500 MHz, (d) single-parameter fits to 20 MHz datausing Eq. (18). Sol and dangling chain contributions were always subtracted before fitting. Data replotted from Ref. [53].

Fig. 11. Proton transverse relaxation data of the resonances of NR thatare resolved at 500 MHz. For the CPMG experiments, the p pulse spacingDp is given in the legend. Data replotted from Ref. [58].


ðp2Þx � sDQ=2� ðpÞx � sDQ=2� ðp

2Þx has shown the best per-

formance in our experiments.The short two-pulse segment has the decisive advantage

that it provides access to strong, even rigid-limit dipolarcouplings on the order of 10 kHz. In this regime, the inten-sity decays almost completely during a single cycle of theBaum/Pines sequence, whose minimum duration is around50 ls under favorable conditions (short pulses, short phaseswitching times). The two-pulse segment has been usedextensively in many applications of MQ spectroscopy bythe Aachen group. They are reviewed in Ref. [62], andRefs. [49,63,64] are the most detailed accounts of applica-tions to elastomers. However, these papers do not go intotoo much detail as to the actual data treatment, the fittingprocedure and its limitations (they are particularly vagueabout the essential prefactor in the functions used for fit-ting). In this section, I therefore present some as yet unpub-lished results concerning a detailed comparison of the twoalternative experiments, theoretically as well asexperimentally.

The most important difference between the two exper-iments is that dipolar time-reversal is not possible withthe two-pulse segment. The sign of the average Hamilto-nian of the Baum/Pines sequence, Eq. (9), is easilyinverted by a 90� shift of the carrier phase. This providesthe possibility of assembling a fully dipolar refocussedsum intensity IRMQ(sDQ), to be used for point-by-pointnormalization and removal of relaxation effects. Thisstrategy may not be straightforwardly applicable to thetwo-pulse segment, where an additional intensity decayoccurs due to the homogeneous nature of free multi-spinhomonuclear dipolar evolution. The same effect is alsoresponsible for the well-known inability of the solid echoto refocus multiple dipolar couplings. The Baum/Pinesexperiment in turn can in that sense be compared to amagic-sandwich echo [65], which does provide full dipolartime reversal.

Fig. 12 shows build-up data based on 6-spin simula-tions of a part of a poly(butadiene) chain fluctuating rap-idly (fast, quasi-static limit) with an order parameter of0.01 (see [60] for details). The lines are for the Baum/Pines MQ experiment, where it is seen that the DQ (moreprecisely: 2Q + 6Q) build-up curve reaches the expectedintensity plateau at 50% of the full magnetization. Inthe absence of motion IDQ(sDQ) therefore equalsInDQ(sDQ), and IRMQ(sDQ) = Iref(sDQ) + IDQ(sDQ) isalways unity (full dipolar ‘‘echo’’). To the contrary,IDQ(sDQ) excited by the two-pulse segment does not reachthe 50% limit, and Iref(sDQ) + IDQ(sDQ) is subject to nota-ble homogeneous dephasing. A normalization may never-theless be attempted, and it is seen that, apart fromoscillations arising from the limited number of simulatedspins, InDQ(sDQ) also approaches 0.5. As expected fromthe higher prefactor of free dipolar evolution, this curvesrises faster than IDQ(sDQ) of the Baum/Pines sequence,and their initial parts coincide once the time axis is scaledby 3/2.

These observations are quantitatively reproduced in thecorresponding experiments, see Fig. 13. Here, the decay ofIRMQ(sDQ) from the Baum/Pines experiment is mainlycaused by molecular dynamics, while for the two-pulse seg-ment, a strong additional dephasing is apparent. Thismeans that information about dynamic timescales cannotstraightforwardly be extracted from the latter experiment.Consequently, Demco and coworkers have introduced theanalysis of the fully shift- and dipolar-refocussed decayof a mixed magic-sandwich echo [66] as the appropriatemeans to assess dynamic timescales in elastomers [48].Their theory provides a basis for the more reliable jointanalysis of intensity build-up and decay in the Baum/PinesMQ experiment discussed in Section 3.3.

Fig. 12. Comparison of simulated MQ data using the Baum/Pinessequence or the two-pulse segment for excitation and reconversion. Thesimulations were performed for two methylene protons, including theirfour most strongly coupled neighbors as passive coupling partners (6-spinsimulation), in a uniaxially rotating cis-poly(butadiene) chain withSb = 0.01 (see [60] and Section 3.2).

Fig. 13. Proton MQ data for cis-butadiene rubber (1 phr sulfur) at 80 �Cmeasured at 20 MHz on a Bruker minispec mq20 using the Baum/Pinessequence and the two-pulse segment. For the latter, the x-axis is scaled by3/2 in order to account for the higher prefactor in free dipolar evolutionand thus render the data directly comparable.


The build-up of InDQ(sDQ) of the two-pulse segment isdelayed at intermediate times, in accordance with dephasingeffects that cannot be normalized away completely. Thismeans that single-parameter fits using Eq. (15) with the cor-rect prefactor yield lower values of RDCs. Fits may well berestricted to the very initial rise, yet a bias towards large val-ues must be expected for networks with broader distribu-tions of residual couplings. As is however apparent fromthe observation of an intensity plateau at InDQ = 0.5, thearguments concerning an equal intensity distribution among4n + 2 and 4n higher-order coherences over IDQ(sDQ) andIref(sDQ) remain valid, whereby the analysis ofIref(sDQ) � IDQ(sDQ) proposed in Section 2.2.2 should alsobe a good way of performing the important extraction ofthe mobile fraction. This aspect is crucial, because whenIDQ(sDQ) is to be fitted without point-by-point normaliza-tion, its intensity scale must be adjusted to network compo-nents only. Otherwise, an uncontrolled average overnetwork chains and non-coupled sol is obtained.

Initial build-up data from the two-pulse segment is com-monly fitted to the first parabolic term � D2

ress2DQ in the

Taylor expansions of Eqs. (10) or (15), corrected by 3/2and damped by an exponential function to account forrelaxation and dephasing effects. However, no informationabout the fitting limit is given in Refs. [49,63,64]. Anincreased validity range can be expected from a dampedversion of Eq. (15),

IDQðsDQÞ¼1

2ð1� expf�2M2ress

2DQgÞexp �2sDQ

T �2

� �

¼1

21� exp � 9

10D2

ress2DQ

� �� exp �2sDQ

T �2

� �; ð21Þ

where the first term on the right hand side then properlydescribes the intensity plateau. Note that the analogousEq. (13) in Ref. [64] is incorrect by a prefactor of 4 for M2.

Results of fits to data from the two experiments are pre-sented in Fig. 14. Contrary to the expectation from Fig. 13,the residual couplings from the two-pulse segment surpassthe values from the Baum/Pines experiment by 10–20% (fit-ting InDQ(sDQ) indeed yields decreased values of 141 and154 Hz for Dres/2p at the two temperatures). This systematicerror is due to the fact that the intensity decay is forced to beexponential, while Fig. 13 clearly shows a non-exponentialdecay of the sum intensity. One can of course always decreasethe fitting limit for a better approximation, but then again,systematic errors arise for networks with broader distribu-tions. In summary, the two-pulse segment is the sequenceof choice for investigations of strongly coupled systems, rigidor close to the glass transition, but more quantitative data isprovided by the Baum/Pines sequence with its well-definedpure dipolar DQ average Hamiltonian.

3. Applications to elastomers

This section summarizes results that were mostlyobtained using the Baum/Pines version of the proton MQ

experiment, and many of them were measured on simplelow-field spectrometers. MQ NMR is shown to provideinformation on heterogeneous microstructure and swellingbehavior, the cross-link density, and the geometry andtimescales of chain dynamics of common elastomers.

3.1. Chain order distributions and heterogeneities

Bimodal end-linked poly(dimethylsiloxane) networkswere investigated as a test case for the quantitative assess-ment of heterogeneity in the local cross-link density [37].Bimodal networks with a large difference in constituentchain length are known to be phase-separated on the nano-meter scale [67,68]. This is basically a statistical conse-quence of the fact that the short chains provide the vastmajority of reactive chain ends and thus mainly undergocross-linking with themselves, even at rather lowconcentration.

The increasingly steep build-up curves in Fig. 15 give adirect indication of the growing average cross-link densityupon increasing the short-chain content. Notably, thebuild-up curves of the mixed systems show distinct differ-ences in their shape. In order to provide unambiguousproof that the shape of the curves reflect microscopic het-erogeneity, one can simply compare these curves with lin-ear combinations of the pure-component responses usingthe correct stoichiometry. The success of this approachspeaks for itself; the results thus indicate that normalizedDQ build-up curves obtained using the Baum/Pinessequence can with good confidence be analyzed in termsof distributions of residual couplings. Generally, it shouldbe pointed out that this applies only to permanent net-works, for which the point-by-point normalization is suc-cessful in rendering the results temperature-independent.

Fig. 14. Fits to proton DQ build-up data of protons at 20 MHz for cis-butadiene rubber at two different temperatures. In agreement withcommon practice in the respective literature, normalized InDQ(sDQ) datausing the Baum/Pines sequence were fitted to Eq. (15), while the fits tonon-normalized IDQ(sDQ) excited by the two-pulse segment contain anadditional exponential damping term, Eq. (21). Note that the latter datawas divided by (1-%sol) to take into account that DQ intensities arise onlyfrom the network fraction. The fitting limits are indicated by vertical barsand are based on the validity limit of the spin-pair approximation.


Apart from fitting build-up curves to the quasi-staticapproximation formula, Eq. (15), or multi-componentsuperpositions thereof, one can assume the coupling distri-bution to be Gaussian (see Fig. 2a), and use Eq. (17) toobtain an averaged value of DG and the standard deviationrG. The most general approach is to use Eq. (15) as theKernel function in a regularized inversion procedure toobtain an estimate of the actual distribution function [37].

It must be mentioned that systematic errors on the 20%level cannot be avoided, as Eq. (15) does not describe theslight intensity maximum observed experimentally,whereby the more strongly coupled components are alwaysslightly overestimated. Another limitation arises for verywide distributions with components that have very differentIRMQ(sDQ) decay times. Then, the more weakly coupledcomponents are progressively overestimated, as the fullIRMQ(sDQ) used for point-by-point normalization lacksthe more strongly coupled contributions and changes therelative scale of InDQ(sDQ). Generally, the distributionassessment is reliable in this respect as long as IRMQ doesnot decay substantially until InDQ(sDQ) has reached 0.45.Raising the temperature, it is usually possible to reachthe safe regime in which the result is temperatureindependent.

A comparison of Dres distributions measured on differ-ent types of elastomers is presented in Fig. 16. The bimodalnetworks yield nicely bimodal distribution functions(Fig. 16a). Notably, the maximum associated with themore weakly coupled longer chains does not shift apprecia-bly, indicating that the long-chain dynamics is not substan-tially influenced by the presence of the short-chain clusters.

For the case of styrene–butadiene rubber (Fig. 16b),rather wide distributions are obtained, which, in viewof the much narrower distributions obtained for chemi-cally uniform systems, is interpreted in terms of differentRDCs associated with the different monomeric units. Asit is also apparent from this set of data, the presence offiller particles, carbon black as well as silanized silica,does not appreciably change the cross-link density ofthe rubber matrix. This finding is in stark contrast toresults from T2 relaxometry, and the results shown herelead the way to the identification of the serious fittingartifacts associated with the latter approach, as discussedin Section 2.3.

Fig. 15. Normalized 500 MHz proton DQ build-up curves for a series ofbimodal poly(dimethylsiloxane) networks consisting of co-endlinked longand very short chains (Mn � 47 and 0.8 kg/mol, respectively). The dashedline is a fit to Eq. (15), and the solid lines are linear combinations ofexperimental curves of the pure short- and long-chain networks, weightedby the corresponding stoichiometries. Data replotted from Ref. [37].

Fig. 16. Distribution functions of residual couplings Dres between protonsin various types of elastomers. (a) Bimodal end-linked poly(dimethylsilox-ane) networks [37], (b) pure and filled styrene–butadiene rubber [51], and(c) sulfur-vulcanized natural rubber [60]. The dashed line in (c) is thatexpected for a Gaussian distribution of network chain end-to-enddistances, Eq. (6), scaled to the same average Dres of the sample with10 phr sulfur.


The most important results from the point of view ofpolymer physics are the distributions of Dres measuredfor natural rubber (Fig. 16c). Apart from weak contribu-tions of more strongly coupled chains, associated withpreparation-specific heterogeneities, these distributionsare extremely narrow over the whole range of cross-linkdensities investigated, and a number of important conclu-sions can be drawn.

The virtual single-component nature of the result is indecisive contrast to what is expected from (i) the Gaussiandistribution of end-to-end distances of network chains,which should lead to a gamma distribution of couplings,Eq. (6), and (ii) the polydispersity of network chain lengths,which should be described by an exponential distribution[69]. The non-observation of any type of distribution effectmeans that chain order, as perceived by NMR, is measuredas an average over a region in space spanning several net-

work chain dimensions. Even more consequentially, thismeans that network chain entropy, as determined by thenumber of accessible conformational microstates, cannot

be described in terms of single-chain concepts; it is domi-nated by cooperativity.

In this regard, it is interesting to consider the result of arecent transverse relaxation study of Cohen-Addad [70], inwhich the influence of chain ends on orientationcorrelations in entangled melts was studied by means ofcomparing fully hydrogenous chains with selectively end-deuterated ones. The central result was that the long tailof the relaxation curves, that is commonly attributed tothe isotropically mobile unentangled end parts, is stillobserved when extended end sections are deuterated. Thiswas interpreted in terms of mobilization of the inner partsof chains that are in proximity to the ends of other chains,confirming a picture where the detected chain order is aver-aged over a certain region in space. This interpretation nowprovides a rationale for the non-observation of substantialdistribution effects on the distribution of residual couplingsbetween protons measured by MQ spectroscopy, and suchobservations may provide a crucial test criterion for refinedtheories of chain dynamics and rubber elasticity.

These findings challenge many (NMR-) works that,directly or indirectly, rest upon the assumption of Gaussianchain statistics. Proton relaxation functions [17,22,23] aswell as 2H lineshapes [41–43] have been described in termsof such statistics. A question to be addressed is of coursewhy experimental 2H lineshapes [41,44,45] sometimes doresemble the ‘‘super-Lorentzian’’ calculated on the basisof the gamma distribution (Fig. 2c). We explain this firstby the presence of sol- and dangling-chain contributions,which always lead to a sharp spectral center, and by theadditional effect of intermediate motions, that do affect2H lineshapes, but can be normalized away in an MQexperiment.

The importance of cooperativity is also supported by theobservation of a splitting in 2H spectra of strained elasto-mers [41,71,72], which is unexplainable in terms of thecommon single-chain model [42,43]. An orientational mean

field has been invoked as an explanation [42,73], and ourresults indicate that the associated cooperative packingeffect, potentially attributable to entanglement effects anddescribable in terms of tube models [1,74,75], might alsobe active in unstrained elastomers. Note also that deuteronsignals from probe molecules or oligomers that perform a(restricted) diffusive average over a certain region in spacealways exhibit a well-defined splitting in strained elasto-mers [76–78], and sol and dangling chains fall in the samecategory. Therefore, the observation of a splitting in thesharp spectral center is always expected.

This section is concluded with an actual application ofdistribution analysis in c-irradiated, silicone elastomersby Maxwell and coworkers [79,80]. The subject of thisstudy was a commercially available silica-filled peroxide-cured system that consists mainly of poly(dimethylsilox-ane), and the aim was to investigate the resistance of thishighly robust material to radiation-induced damage. Typi-cal proton nDQ build-up curves are shown in Fig. 17. Theirshape is similar to those of the bimodal networks ofFig. 15, and indeed, it was possible to reliably fit the datato a bimodal model based on Eq. (15), yielding two residualcouplings and a relative weight.

The bimodality was attributed to the highly abundant(�30 wt%) silica filler, with more highly cross-linkeddomains in the vicinity of the particle surfaces, that mighteither contribute additional chemical or physical links, oraffect the cross-linking reaction. The results shown inFig. 18 indicate a slight increase of the cross-link densityin both domains upon c-irradiation. Notably, the amountof more highly cross-linked material is also increased sub-stantially, which means that the irradiation effect is nothomogeneous over different domains.

Fig. 17. Normalized 400 MHz proton DQ build-up curves for filledsilicone elastomers exposed to different doses of c-irradiation given inkGray. For the sample with the highest dose, a single-component fit to ananalogue of Eq. (15), and the two components of a bimodal fit are shown.ÆXdæ is equivalent to Dres. Figure reprinted with permission fromRef. [79].

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3.2. Quantitative interpretation of residual couplings

In order to convert the NMR-determined RDCs into across-link density (or the average molecular weight of anetwork chain, Mc), a model needs to be adopted [60]. Thiscomprises two major steps. Following Eq. (2), a properlydefined ‘‘rigid-limit’’ coupling Dstat/k is first to be deter-mined in order to obtain a backbone order parameter Sb.This reference value should comprise an average over allfast conformational fluctuations that occur within a singlestatistical (Kuhn) segment.

Second, under the assumption of Gaussian statistics, Sb

yields the number of statistical segments N, which in turncan be converted into Mc using the known characteristicratio C1 [81]. As is obvious from the preceding section,the assumption of Gaussian statistics, and therefore the pre-factor 3/5 in Eq. (2), is critical. Our computer simulations[82] show that the value of the prefactor in fact depends ondetails of the fluctuation statistics (as a consequence of thetensorial nature of the averaged quantity), and that particu-larly serious deviations from the value of 3/5 can occur in theswollen state, where solvent quality (size of the excluded vol-ume interaction) plays a significant role (see Section 3.4 fordetails). In the following, these complications are neglected,and a comparison of results for different network systemswill show that this is not the only source of ambiguity.

Coming back to the first step, our approach to define areference Dstat/k is based on spin dynamics simulations of asmall segment of the polymer backbone comprising each ofthe different types of protons and its most strongly coupledneighbors (4–7 ones need to be taken into account toobtain build-up curves whose initial parts are almost inde-pendent of the spin system size). The chain within theKuhn segment is supposed to be extended (a commonbut not unquestioned assumption [83]) and best approxi-mated by the lowest-energy conformation adopted in thecrystal structure. Intra-segmental motions are mimickedby a simple rotation, where the only free parameter ofthe model is the orientation of the local rotation axis withrespect to the backbone orientation.

The orientation of the rotation axis is optimized by com-paring the results of numerous simulations to experimentalconstraints, namely (i) group-specific build-up curvesdetected for the different types of protons that are resolvedat high field, and (ii) the ratio of intra-group DQ coherenceintensities to coherences involving two different types ofprotons. The latter information can be obtained from 2DDQ correlation spectra such as the one shown in Fig. 19,where the former appear on the diagonal, and the latteras off-diagonal intensities. This general approach was firstintroduced by Graf et al. [33], who used highly resolvedmagic-angle spinning (MAS) spectra and a DQ recouplingsequence to detect the site/pair-specific build-up in linearpoly(butadiene) (see also Fig. 32a). Their conclusions,based on spin-pair considerations, may be somewhat chal-lenged by the complication that the intensity scale for eachof the build-up curves is not easy to define. It must comprise

Fig. 19. Two-dimensional proton DQ correlation spectrum of cis-buta-diene rubber (1 phr sulfur) acquired at 500 MHz with sDQ = 1.3 ms. Sucha spectrum is easily obtained by introduction of an incremented t1

evolution period between DQ excitation and reconversion in a DQ-filteredMQ experiment (see Fig. 3b). The projections are taken over the shadedareas. Spectrum replotted from Ref. [60].

Fig. 18. Relative population of the weakly cross-linked contribution (a)and average residual couplings (b) derived from two-component fits to400 MHz proton nDQ build-up curves of filled silicone elastomers as afunction of c irradation dose. ÆXdæ is equivalent to Dres. Figure reprintedwith permission from Ref. [79].

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information on the number of couplings partners, and thefitting result depends on this choice. Our approach wastherefore to always represent intensities relative to the fullsum magnetization detected at a specific spectral position.

Natural rubber and cis-butadiene rubber share almostthe same conformations in their crystal structures, suchthat the same set of simulation geometries can be used toobtain a calibration for both systems, just replacing oneof the olefinic protons by a methyl group. Fig. 20a showsresults for BR, where an inclination of the local rotationaxis with respect to the backbone by 12� was found tomatch the experimental constraints best. Note that this‘‘best-fit’’ orientation should not be given too much physi-cal significance. It merely parameterizes the intra-segmen-tal dynamics in a way that is consistent with theobservations, under the assumption that the actual degreeof averaging afforded by the true local conformationaljumps is similar to that of the simplified rotation. A sur-prising result from our simulations that also highlightsthe dangers inherent to simplified models was that thedominant coupling for the CH2 protons is not the intra-group coupling but the coupling to one of the CH2 protonslocated on the other side of the central cis double bond(protons 3 and 5 in Fig. 20a). The reason is the geometryof local motion (‘‘rotation’’ around the backbone) as wellas the close spatial proximity afforded by the favored skew

conformations of the H2C–CH bonds.

A weighted average of the simulated site-specific build-up curves can finally be fitted to Eq. (15) to obtain the ref-erence couplings,

Dðcis-BRÞstat =k ¼ 2p� 8:1 kHz; ð22Þ

DðNRÞstat =k ¼ 2p� 6:3 kHz: ð23Þ

The results of Graf et al. [33] indicate that the residual cou-pling associated with trans-BR should be about twice aslarge as for cis-BR, which means that either the associateddynamic order parameter is higher or that the intra-seg-mental averaging is less efficient (higher Dstat/k). It is worthmentioning that the variation of the final Dstat/k for bothNR and BR was never more than 25% over the wholerange of orientations studied.

When the model cannot be constrained appropriately, asis the case for PDMS with its single type of protons, oneneeds to adopt an assumption of the local orientation ofthe rotation axis. For PDMS, we simply used an extendedall-trans segment, with an angle of 90� between the Si–Cbond and the backbone (Fig. 20b). The reference couplingis then [60]

DðPDMSÞstat =k ¼ 2p� 7:58 kHz: ð24Þ

Note that in our first paper on PDMS rubber [37], amethyl-group specific build-up function was used, fits towhich yield different values for Dres than Eq. (15). Since

Fig. 20. Multi-spin simulations of proton DQ build-up in rigidly rotating chain segments. (a) Individual and averaged build-up curves for the differentprotons in cis-butadiene rubber, based on 8-spin simulations using the indicated protons (Sb = 0.01), and fits to the averaged simulation result [60]. (b)Simulated build-up curves for a single CH3 and a poly(dimethylsiloxane) monomer unit (Sb = 0.02, dotted and dashed lines, respectively), compared to afit to Eq. (15) and actual experimental data [37].


the lower InDQ(sDQ) plateau of 0.33 that is specific for thepecularities of MQ dynamics in isolated methyl groups[84,85] is never observed due to multiple extra-group cou-plings (see Fig. 20b), I recommend to always use Eq.(15), which yields an apparent Dres that is properly definedby its relationship to the dipolar second moment, Eq. (3).

Finally, using the definitions of Flory’s characteristicratio and the Kuhn segment length (and assuming anextended chain conformation within the Kuhn segment),one obtains relationships between experimental RDCsand Mc:

M ðcis-BRÞc ¼ 656 Hz

D res=2pkg=mol; ð25Þ

M ðNRÞc ¼ 617 Hz

Dres=2pkg=mol; ð26Þ

M ðPDMSÞc ¼ 1266 Hz

Dres=2pkg=mol: ð27Þ

Results for NR and PDMS for various cross-link densi-ties are collected in Fig. 21, where they are compared withresults from established Flory–Rehner swelling experi-ments. The most prominent feature is the nice linear rela-tionship between the NMR observable and themacroscopic measure of network structure, and such linearrelationships have been reported for a large number of dif-ferent elastomer systems [17,27,49,86–92]. There are, how-ever, a few exceptions in the literature [31,32,48,93], wherespecific deviations from Gaussian chain statistics are dis-cussed as possible causes. These works have in common thatthe data analysis is based on a slow-motion model, the inap-plicability of which is discussed in Section 3.3. Judging fromthe overwhelming evidence for a linear relationship, it ap-pears that the latter results are model-specific artifacts.

Using the model described above, we can now attempt amore quantitative assessment of the linear relationship.There are several implications that follow from our com-parison in Fig. 21, and details can be found in Ref. [60].

The salient points are: (i) NMR and swelling results dis-agree by a factor of 2, which is fully acceptable consideringthe many model assumptions. It is, however, unclear whyNR and PDMS show deviations in different directions.(ii) The intercept with the y-axis depends on both theentanglement density (�1/Me), that contributes to theNMR observable but not necessarily to the swelling result,and on the density of trapped entanglements (1/Mte) thatdo affect the swelling The results for NR are roughly con-sistent with what is known from the literature after a cor-rection factor of 2 is considered, while the entanglementdensity (or contribution from ‘‘local chain packing’’) forPDMS is largely overestimated. This is specifically surpris-ing because the slope is underestimated, thus agreementcannot be reached by introducing a correction factor.

At this point, we conclude that a consistent quantitativeinterpretation of residual couplings in terms of an averageorder parameter of the polymer backbone is not possible inthe framework of the simple Kuhn chain model. While atleast some papers in the literature do report good agree-ment between NMR-derived cross-link densities and valuesobtained from other techniques, it would appear that itresults from a cancellation of errors arising at variousstages of the modelling. This certainly does not invalidatesuch approaches; it merely shows that the final conversionfactor of an RDC to 1/Mc should be considered a material-dependent calibration quantity. Most importantly, comingback to our observation of narrow RDC distributions, therole of end-to-end distance distributions and the networkchain polydispersity is fully unresolved.

3.3. Chain dynamics in elastomers: failure of the slow-motion

model

As opposed to the special case of InDQ(sDQ) curves mea-sured for protons in permanent elastomers, the directlyobtained build-up and decay functions of the MQ experi-ment, Eqs. (10) and (13), always exhibit a marked temper-ature dependence. This is demonstrated in Fig. 22. In Ref.[94], it is shown that instrumental factors (duty cycle, pulseimperfections) play a minor role, such that IDQ(sDQ) andIRMQ(sDQ) can be analyzed in terms of chain dynamics.

Using the Andersen–Weiss (AW) approximation [55] incombination with a suitable model for the loss of orienta-tion correlation as sketched in Fig. 1, analytical fittingfunctions can be obtained for different scenarios [94]. Inshort, the first step comprises the rearrangement of, e.g.,Eq. (10) under the assumption of a Gaussian distributionof interaction frequencies:

IDQðsDQÞ ¼ sinhðh/DQ1/DQ2iÞ expð�h/2DQ1iÞ: ð28Þ

The second step consists in evaluating the time and ensem-ble averages of the phase factors in this equation in termsof the orientation autocorrelation function, C(t), of theeffective dipolar tensor, Eq. (1). For example, for the mixedproduct of DQ phases, the average becomes

Fig. 21. NMR-determined cross-link densities (in terms of 1/Mc) ascompared to results of Flory–Rehner swelling experiments. Data replottedfrom Ref. [60].


h/DQ1/DQ2i ¼4

9M2eff

Z sDQ

0

Z 2sDQ

sDQ

Cðjta � tbjÞdta dtb; ð29Þ

where the powder average simply leads to a factor of 1/5that is absorbed into the definition of M2eff, Eq. (4). Start-ing from here, and using some rules for the evaluation ofthe double integral [3], specific scenarios for C(t) can beimplemented.

The detailed account of the theory presented in Ref. [94]builds upon and extends the treatment of the b functionpublished by Ball, Callaghan and Samulski [30], which isa direct analogue of IDQ(sDQ) constructed by a clever com-bination of Hahn- and solid echoes [28]. Demco andcoworkers presented how the Andersen–Weiss model canbe generalized for the evaluation under an actual pulsesequence that consist of a succession of evolution intervalsrather than a single dipolar phase [48], and using these con-cepts, it turns out that relaxation under an MQ experimentcan be treated in terms of evolution under an averageHamiltonian, Eq. (9), with only minor effects of finitepulses and the finite cycle time [36,94].

Up until now, the majority of models dealing with trans-verse relaxation phenomena in mobile polymer systems restupon the assumption that the segmental modes that affordthe primary averaging down to Dres do not exert a domi-nant influence on the decay [28–32,48]. The slow-motionmodel, in particular the one that assumes an exponentialloss of correlation,

CðtÞ ¼ S2b expf�t=ssg; ð30Þ

goes back to Fedotov and coworkers, who applied it toproton transverse relaxation experiments of elastomers[11]. The dangers inherent to using the resulting signalfunction, Eq. (20), for fitting were highlighted inSection 2.3.

Fits of analogous analytical expressions [36] forIDQ(sDQ) and IRMQ(sDQ) to proton MQ experiments onvulcanized NR are shown in Fig. 22. The backbone orderparameters calculated from the fitted RDCs are plottedin Fig. 23. These results, along with the correlation times

ss discussed below, show that the slow-motion model isindeed not applicable to elastomers. Strikingly, fits toIDQ(sDQ) and IRMQ(sDQ) yield very different Sb values,and results from the latter show large changes with temper-ature. It is also not possible to perform a simultaneous fitto both functions using shared parameters, which shouldbe possible if the model is correct.

Other ACFs considered are a single exponential thatmodels only the fast segmental process,

CðtÞ ¼ ð1� S2bÞ expf�t=sfg þ S2

b; ð31Þ

a (seemingly more realistic) model that combines two expo-nential processes,

CðtÞ ¼ ð1� S2bÞ expf�t=sfg þ S2

b expf�t=ssg; ð32Þ

where sf and ss are the fast and slow correlation times,respectively, and finally a power-law ACF for the fast ini-tial decay,

CðtÞ ¼1 for jtj < s0

ð1� S2bÞðs0=jtjÞj þ S2

b for jtjP s0

�; ð33Þ

with s0 as the onset time and j as the characteristic expo-nent. Kimmich and coworkers have proposed the use ofEq. (33) for a more realistic account of polymer dynamicsmainly in the context of T1 relaxometry [5]. It is probablythe most realistic model, as superpositions of processeson different timescales (e.g., Rouse modes) must be ex-pected. Note that in their work on the dipolar correlationeffect in rubbers [31], they have used such a function tomodel the alleged slow process, and our corresponding testsshowed that this leads to the same unphysical temperaturedependence that we have observed for the exponentialslow-motion ACF.

As an example, only the explicit signal functions derivedfor the combined model, Eq. (32), are given:

Fig. 23. Backbone order parameters Sb for natural rubber (3 phr sulfur)as derived from fits to the different MQ signal functions based on the slow-motion model. Data replotted from Ref. [36].

Fig. 22. 20 MHz proton MQ intensity build-up and decay data for anatural rubber sample (3 phr sulfur) measured at different temperatures.The lines are fits to the slow-motion model. Data replotted from Ref. [36].


IDQðsDQÞ¼ exp �8

9ð1�S2

bÞM2effs2f

�

� e�

sDQsf þ sDQ

sf

�1

� ��8

9S2

bM2effs2s

� e�sDQss þ sDQ

ss

�1

� ��

� sinh4

9ð1�S2

bÞM2effs2f

�

� e�2sDQ

sf �2e�sDQsf þ1

� �þ4

9S2

bM2effs2s

� e�2sDQ

ss �2e�sDQss þ1

o; ð34Þ

IRMQðsDQÞ ¼ exp � 4

9ð1� S2

bÞM2effs2f

�

� 4e�

sDQsf � e

�2sDQ

sf þ 2sDQ

sf

� 3

� �

� 4

9S2

bM2effs2s

� 4e�sDQss � e�

2sDQss þ 2sDQ

ss

� 3

� ��: ð35Þ

The slow- and the fast-motion-only expressions are pub-lished in Ref. [36]. They are easily obtained as limitingcases letting the correlation times sf fi 0 and ss fi1,respectively.

Fits of the temperature-dependent IDQ(sDQ) andIRMQ(sDQ) to expressions based upon these models arecompared in Fig. 24. In all cases, simultaneous fitting withshared parameters was possible, and the quality of all fits inthe validity region given by Eq. (16) is similar. Best agree-ment is found for the power-law model, that also gives thebest long-time prediction. Note that experimental data arealways expected to fall below the model predictions atlonger times, as multi-spin correlations that lead to moreeffective relaxation are not included in the theory.

The fast correlation time sf obtained from fits to thefast-motion and the combined model are very similar,

and are in Fig. 25 compared to ss obtained from the (nec-essarily independent) slow-motion fits to IDQ(sDQ) andIRMQ(sDQ). Strikingly, ss derived from the latter two signalfunctions not only differ by up to two orders of magnitude,but they also exhibit a completely unphysical temperaturedependence. The same is true for the ss derived from thecombined model (for data see Ref. [36]). From this we con-clude that there is no sustainable evidence for a detectableinfluence of a slow cooperative process in networks. Thesame conclusion was drawn earlier from 2H spin-alignmentexperiments on PDMS networks [95].

The sf, however, do exhibit a reasonable temperaturedependence, and a fit to a Williams–Landel–Ferry functionyields an extrapolated glass transition that matches theDSC value within 1 K. This provides an unambiguousproof that the overall intensity decay is indeed dominatedby fast segmental modes that are coupled to the segmentala relaxation. The results from the power-law model arequalitatively similar, yet at the moment we lack the theoret-ical understanding to attach physical significance to theonset time s0 and the exponent j. The work of Kimmichand Fatkullin suggests that the true ACF might in fact fol-low different power laws over different time regimes [5], andwe hope that future theoretical work will reveal the trueshape of network chain autocorrelation functions.

Brereton has repeatedly criticized the use of the AWapproximation that is central to our and earlier approaches[23,24,34], and this potentially serious objection deserves afew comments. His treatment of transverse relaxation ofmobile polymer chains is based on a scale-invariant ‘‘sub-molecule’’ model [21,22,24,34], and it is argued that thecartesian components of the submolecule end-to-end vec-tors are statistically independent and that Gaussian statis-tics should apply. This specifically implies a Gaussiandistribution of end-to-end separations that is either sam-pled rapidly in time in the case of short melt chains, orleads to a quasi-static response function (‘‘frozen-bond’’limit) that then consists of a superposition of frequencycontributions given by Eq. (6). In Section 3.1, it was how-ever shown that Gaussian chain statistics and the resulting

Fig. 24. Simultaneous fits to the two MQ signal functions for differentmodels of correlation loss. Data replotted from Ref. [36]. The grey barindicates the fitting limit given by Eq. (16).

Fig. 25. Correlation times for natural rubber (3 phr sulfur) as derivedfrom fits to the two MQ signal functions based on the slow- and the fast-motion model. The dashed line is a Williams–Landel–Ferry (WLF) fit tothe results for the fast-motion model. Data replotted from Ref. [36].


gamma distribution of RDCs do not describe the data wellin this limit; a rather narrow distribution of couplings isfound experimentally. Therefore, the AW approximationwith its single residual second moment describes experi-mental data better, and Gaussian single-chain models willhave to be improved by introduction of a degree of coop-erativity of chain motion that screens such localdistributions.

In Ref. [36], we have also performed Monte-Carlo sim-ulations of a rotational diffusion process, that is character-ized by exponential loss of correlation and should modellocal conformational jumps, to show that the AW modelrepresents a good approximation all the way from the slowto the fast limit. In addition, the second-moment approxi-mation that is part of the AW model (i.e., that a Pake-likequasi-static dipolar coupling distribution is reasonably wellapproximated by a Gaussian) can be tested experimentallyas it allows the derivation of a conversion formula of MQto dipolar-dephasing data, which matches the experimentalHahn-echo decay very satisfactorily.

Finally, the shape of the IRMQ(sDQ) relaxation functionsshould be discussed. Close inspection of the fits in Fig. 24shows that the decay of IRMQ(sDQ) for the fast-limit modelis mono-exponential, and for small sf and ss fi1, anapparent T2 can be derived from Eq. (35):

T 2MQ ¼8

9ð1� S2

bÞM2effsf

� ��1

� 2:5Dstat

k

� ��2

s�1f : ð36Þ

A relaxation term of similar form was predicted by Brer-eton for the case of transverse relaxation of Rouse chainsin melts and networks [23,34]. In the latter case,ð1� S2

bÞM2eff is associated with the coupling within a sub-molecule, similar to when using a pre-averaged M2eff �(Dstat/k)2 for the Kuhn segment, and sf is associated withthe fastest Rouse mode.

In contrast, experimental IRMQ(sDQ) decays can be deci-sively non-exponential (see Fig. 22). Since a true slow-motion contribution that would introduce some convexityin the decay curve appears unlikely, one could of courseattribute the convexity to a more complicated (e.g.,power-law) ACF that extends into the experimental time-scale. Yet, another effect that deserves careful discussionis the influence of isotropic J couplings. Fig. 26 shows someunpublished simulation data featuring a realistic set ofdipolar and J couplings. Clearly, IRMQ(sDQ) is significantlyaffected by J couplings that are about an order of magni-tude weaker than the average RDC. This somewhat coun-terintuitive behavior can be explained superficially by thefact that dipolar couplings to a third spin renders twootherwise equivalent spins magnetically inequivalent in thesame way as a chemical shift difference; the formal deriva-tion involves the non-zero commutator ½ �HDQ; H J � [96].Note that this must also be taken into account for rubberswith chemically equivalent protons such as PDMS.

Importantly, this higher-order effect causes a downwardcurvature of any IRMQ(sDQ) decay. Even though it appears

to be rather weak in relevant elastomer cases, the effect maybecome more pronounced in weakly coupled entangledpolymers. More importantly, such deviations of true exper-imental behavior from what is provided by the fittingmodel may always lead to severe fitting instabilities oncemany parameters are involved, as exemplified already inSection 2.3. This stresses the importance of simultaneousfitting of IRMQ(sDQ) and IDQ(sDQ), which stabilizes theresult even when the experimental trend cannot be fullyreproduced by the theoretical expression. A factor that alsodeserves closer attention in the future is the necessarily lim-ited validity range of spin-pair based theories for any typeof experiment; enforcing a fitting limit as given by Eq. (16)might help to stabilize fits and improve the results.

3.4. Network swelling

A subject area of pronounced current controversy is themicroscopic picture of network swelling. In short, the affinetheory of Flory and Rehner [35] that stipulates additivity ofthe entropy-elastic and osmotic free-energy contributionsupon solvent take-up and volume increase has long beenchallenged by thermodynamic investigations that suggestan increase in chain entropy rather than a decrease at inter-mediate degrees of swelling [97]. The role of complex topo-logical rearrangements has been stressed more recently;these ultimately lead to the appearance of pronounced het-erogeneities that can be detected by a variety of experimen-tal methods, and are also found in computer simulations,see Ref. [52] and references therein.

The first systematic NMR investigations on the swellingdependence of orientation correlations in permanent net-works were made by Cohen-Addad et al. [27,98–101],who found that at least for systems cross-linked in solution,an NMR quantity that is related to the RDC first decreasesand then increases upon swelling [98,99]. Such behavior hasbeen interpreted in terms of a chain ‘‘desinterspersion’’,whereby chains first increase their conformational space

Fig. 26. Simulations of proton IRMQ(sDQ), IDQ(sDQ), and InDQ(sDQ)curves acquired with the Baum/Pines sequence, with and without theinclusion of typical geminal and vicinal J couplings. See caption of Fig. 12for details of the simulations.


and their entropy as packing-induced orientation correla-tions are relieved, while a pronounced loss in conforma-tional entropy is observable only at later stages. Eventhough the general idea was published almost 30 yearsago [102], the attractively simple picture of a nematic-likeorientational mean field contribution that can explain ther-modynamic as well as NMR observations [103], has not yetfound its way into current theories of rubber elasticity.

While the earlier NMR work of course discussed only asingle averaged NMR observable, our contribution to thisfield has been focussed on the quantification of changes inthe chain order parameter distribution, that may of coursebe related to the heterogeneities found in many scatteringstudies [52,82]. Before discussing the experimental findings,it is instructive to first consider the behavior that isexpected from the simple Flory–Rehner model. A homoge-neous affine and isotropic volume increase of a networkwill lead to an affine stretching of each network chain,which means that any chain order distribution function(see Fig. 16) will just be stretched (scaled) along the x-axis,according the change of r2 ¼ r2=r2

0 in Eq. (2).This is not what is observed experimentally [52,82], as

well as in large-scale computer simulations [82]. Fig. 27shows that the majority of chains, even at swelling equilib-rium, remain unstretched or even relaxed, while only asmaller fraction experience a pronounced increase of Sb,thus contributing to balancing the osmotic driving force.Most notably, the maximum indicating the most probableorder parameter does not shift towards higher values, andin all cases, the width of the distribution increases uponswelling. The almost semi-quantitative agreement of exper-iments and simulations is encouraging in that additionalinsight into, e.g., the internal structure of the swollen net-works can now be extracted from the computer simula-tions, that are thus shown to model essential features ofreality rather well.

An example of such an approach is provided in Fig. 28,where large concentration fluctuations are apparent. Thephenomenon is not surprising as such, as they also appearin polymer solutions (and are responsible for light, neutron,

and X-ray scattering phenomena). The important finding isthat the regions are perfectly stable in time, i.e., the concen-tration fluctuations are static. The regions of high and lowswelling have a length scale of a few radii of gyration of sin-gle network chains, which is thus in agreement with what isknown from many scattering studies [104].

An important result of our extensive computer simula-tions concerning the quantitative interpretation of Sb interms of network chain length is summarized in Fig. 29.It concerns the validity of Eq. (2), the variation of Sb alongthe chain, and the changes observed upon swelling. In a drynetwork, the distribution of Sb along the chain is not verypronounced (±20%), and probably even less significant forlonger chains. From our data, we concluded that the non-observation of large distribution effects, in particular of dif-ferent end-to-end separations, is simply because chains thathave rather short end-to-end separations are very likely to

Fig. 27. Chain order parameter distributions of swollen end-linked model networks at different degrees of volume swelling Q = V/V0. The experimentaldata in (a) is from PDMS swollen in octane as a good solvent, and the simulated data in (b) is from large-scale bond-fluctuation Monte-Carlo simulations.Data replotted from Ref. [52,82].

Fig. 28. Time-averaged 3D density fluctuations of a swollen end-linkedmodel network. Color (green to red) codes polymer chain concentrationthat varies by as much as 50% over the different regions. The picture isbased on the simulations described in Ref. [82], and was kindly providedby Jens-Uwe Sommer. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this paper.)


participate in topological links (‘‘trapped entanglements’’)and thus cannot reorient freely as required by the model.One could of course also invoke nematic-like orientationcorrelations or a ‘‘tube’’ model to describe the phenomena[74,75]. Importantly, the NMR results are not in disagree-ment with neutron scattering studies that indicate that ran-dom networks indeed exhibit static Gaussian statistics overwhole, long network strands.

Most significant is the deviation between vector and ten-sor orientation correlation, which should just differ by afactor of 3/5 (see Eq. (2)). For the dry network, the devia-tion is of course expected, since we have already seen thatthe Gaussian end-to-end separation distribution isscreened, which changes the average. For the swollen net-work, however, the difference is much increased, which isa consequence of a change in the fluctuation statistics andthe tensorial nature of Sb (=(3/5)sG from the simulations).It was further found that the plateau value of sG dependsstrongly on solvent quality. With excluded volume (goodsolvent), the average sG is lower, which is rather counterin-tuitive. Detailed studies along these lines are currentlyunderway, and a closer understanding of the role of thefluctuation statistics is indispensable for a quantitativeinterpretation of the relationship between NMR-detectedchain order, chain entropy, and the actual length of net-work chains.

3.5. Strained and oriented networks

The effects of macroscopic strain on the observables hasbeen an important aspect from the earliest days of NMRinvestigations of elastomers. 2H NMR was the method ofchoice, and the importance of induced nematic-likeorientation effects was first recognized by observation ofwell-defined splittings in 2H spectra of deuterated probemolecules dissolved in stretched elastomers [76]. Later

work on lineshapes of stretched elastomers with deuteratedchains stressed the importance of non-affine local stretch-ing and distribution effects [72], and the use of deuteratedoligomers or free probe chains allowed the quantificationof the degree of orientational coupling between chains ina dense system [73,78,105]. Theoretical models employingan orientational mean field, as caused by excluded-volumeinteractions, have been successfully used to describe thephenomena [41,42,45], and the preparation conditions,i.e., the presence of trapped entanglements, have beenshown to exert an appreciable influence on induced orien-tation [106]. More recently, biaxial deformations have beenused to show that the orientational mean field follows thepredictions of the phantom model [107].

Proton NMR investigations are less frequent and wereinitially restricted to conventional transverse relaxationstudies [108–110]. Recently however, the Aachen grouphas demonstrated the use of proton DQ NMR using thetwo-pulse segment to directly detect the apparent RDCchanges upon elongation as well as employing RDCs as acontrast mechanism in DQ-filtered NMR images [111].As a proof of principle, Fechete et al. showed that angle-dependent DQ build-up (Fig. 30a) and dipolar-encodedlongitudinal magnetization (DELM) decay curves can beused to study the degree of chain orientation and stretching[112]. The change in the maxima positions already gives aqualitative impression of the phenomenology, and morequantitatively, apparent values for Dres can be extractedfrom the initial slopes. As expected, the angular variationof the apparent, normalized RDCs shown in Fig. 30b fol-lows the second Legendre polynomial jP2(cos h)j, and theanalysis of the absolute values could be the basis of a quan-titative assessment.

It should be noted that the fitted values of Dres areapproximate because the fitting function is based on an iso-tropic distribution of residual tensor orientations, but thisshould be replaced by a uniaxially biased distribution ina stretched sample. An appropriate analysis was presentedfor the case of RDCs detected for water molecules thatexhibit strongly anisotropic mobility in tendon [113,114],where the angular distribution function of collagen fibrilswas determined. Consistent results were obtained fromthe analysis of orientation-dependent DQ build-up, DELMdecay, and splittings in DQ-filtered spectra [114]. Theapplication of proton and deuteron MQ NMR to the char-acterization of order in biological tissues was pioneered byEliav and Navon [115,116], and it should be pointed outthat this is a particularly favorable application, whererelaxation effects due to polymer chain dynamics are lar-gely absent, as rapidly exchanging/diffusing H2O isdetected. In addition, well-resolved sDQ-independent split-tings are observed in DQ-filtered spectra as a result ofthe strong uniaxiality in such tissues.

For direct detection of elastomer chain signal, it isexpected that the analysis of normalized DQ build-upcurves using the Baum/Pines sequence offers an advantagewith respect to an exact quantification. The data in Fig. 9b

Fig. 29. Simulated chain order parameters as a function of position alonga chain in a dry and an equilibrium-swollen end-linked model network.Q = V/V0 indicates the degree of volume swelling, m is the plateau value ofthe bond vector orientation correlation function <b (0) b (t) >/<b2> andsG ¼ 5

3Sb � 1=N is the rescaled tensorial order parameter measured by

NMR. Gaussian network theory predicts the dashed line for bothquantities. Data replotted from Ref. [82].


shows that such build-up curves are rather sensitive tochanges in the powder distribution, suggesting that itshould ultimately be possible to extract similar detailedinformation as obtained from 2H NMR lineshapes.

As a final remark, it should be stressed that the richinsights on induced orientation and lateral chain packingthat were gained from experiments on stretched elastomershave not yet found their counterpart in the description ofthe undeformed state. The qualitatively new insights intoactual RDC distributions presented in Section 3.1, whichshow that single-chain models are clearly insufficient todescribe the narrow distributions found, may ultimatelybe explained by establishing direct correlations of thestretched and unstretched states. Work along these linesis in progress.

4. Entangled melt dynamics

The decisive difference between chain dynamics in net-works and entangled melts is of course the possibility forlarge-scale chain motion on longer timescales in the latter,

i.e., the occurrence of reptation [1]. Depending on themolecular weight of the chains and thus the number ofentanglements per chain, the timescale of correlation lossassociated with this process is more or less separated fromthe more localized Rouse modes (see Figs. 1 and 31). Thetimescale for reptation (between the longest constrainedRouse mode sR and the tube disengagement time sd) inhighly entangled melts ranges from about 0.1 ms up tothe range of seconds [35]. This is outside the typical timewindow of inelastic neutron scattering and longitudinalNMR relaxation, yet NMR methods based on RDCs andtransverse relaxation phenomena are ideally suited todirectly probe this type of dynamics.

Conventional proton transverse relaxometry[14,20,24,25] as well as more advanced approaches suchas the dipolar correlation effect (DCE) [117] and the b echo[30,118] have been used to study reptation. Apart fromconceptual details, they all differ in the type of orientationcorrelation function used to model this process. Breretonet al. [14,20,24,25] favor a description of transverse relaxa-tion in terms of an exponential loss of correlation and suc-cessfully demonstrated the correct scaling law for themolecular-weight dependence of the reptation time derived[14]. Kimmich et al. achieved good agreement between theexperimental DCE and theory using a power-law autocor-relation function [117], yet did not link their finding toestablished theories of polymer dynamics. Considerationsbased on the mean-square displacement predicted by theDoi–Edwards tube model (Fig. 31) and approximatingC(t), the orientation ACF, as a return-to-origin probabilityfor reptation along the primitive path yields specific powerlaws for the important dynamic regimes of C(t): C(t) � t�1/4

for constrained Rouse modes up to sR, and C(t) � t�1/2 forreptation below sd [30]. Satisfactory agreement betweenthis model and experimental b echo data for long-chainpoly(dimethylsiloxane) has been be reported [118].

Fig. 31. Time and length scales for the mean square displacement,Æ(Rn(t) � Rn(0))2æ, of a segment of an entangled polymer chain accordingto the tube model. The different scaling regimes are separated by theentanglement time se, the longest contrained Rouse time sR, and the tubedisengagement time sd. The parameters a and b are the effective tubediameter (entanglement spacing) and the segment length, respectively, andN is the number of segments. Adapted from Ref. [1].

Fig. 30. (a) Proton DQ build-up curves measured at 200 MHz for naturalrubber (3 phr sulfur) at an elongation k = l/l0 = 3, varying the angle hbetween the stretching direction and the static magnetic field. (b)Normalized RDCs as a function of h. Figure reprinted with permissionfrom Ref. [112].

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The first proton DQ NMR study of reptation effects waspublished by Graf et al. [33], who used high-resolution pro-ton DQ MAS NMR to detect pair-specific DQ build-upcurves in linear poly(butadiene) of different molecularweight. While MAS offers the possibility of high site reso-lution and thus the development of specific molecular mod-els for chain order (see Section 3.2), DQ MAS recouplingpulse sequences are less flexible in the choice of the DQevolution time due to the necessary rotor synchronizationand/or longer finite pulses, leading to less points in thebuild-up curves. Another drawback can be their larger sus-ceptibility to experimental timing imperfections, such thatan equivalent function IRMQ(sDQ) must be measured andused for intensity normalization [85].

Graf et al. extracted apparent RDCs by fitting normal-ized build-up curves, that are thus corrected for decaydue to experimental imperfections and fast-motion inducedrelaxation (see below), to

InDQðsDQÞ ¼ AD2ress

2DQ exp½�sDQ=T �2� ð37Þ

at T = Tg + 50 K (see Fig. 32a). At this temperature, it wasexpected that the semi-local segmental averaging is essen-tially complete on the timescale of the experiment, suchthat the associated value of Sb should reflect segmental or-der between entanglement constraints. Varying the temper-ature, time-temperature superposition was used to recorddata over several orders of magnitude in time. For this, afixed s�DQ (=0.5 ms) was chosen according to D�1

res atT = Tg + 50 K, such that the parabolic short-time limitingbehavior applies. Then, the exponential loss term can beneglected and the measured DQ intensity is proportionalto the squared apparent RDC, with InDQðsDQÞ �D2

ress2DQ � D2

effS2bCðt=seÞs2

DQ. This relation follows fromEqs. (28) and (29), and a formal derivation can be foundin Ref. [119]. Plotting InDQðs�DQ; T Þ=I nDQðs�DQ; T gþ50 KÞthus directly yields the autocorrelation function C(t/se)

that describes correlation loss on the lengthscale of entan-glements and beyond. Fig. 32b demonstrates that scalingregimes can be observed that are in agreement with theDoi–Edwards model.

A matter of some debate, however, is the rather highstarting value of Sb � 0.2 at Tg + 50 K. An order parame-ter of about 0.03 would be expected for a chain comprisingabout 20 statistical segments that probes all conforma-tional space between entanglements. Graf et al. concludedthat either local packing constraints or a weak localstretching between entanglements plays a role. A clarifica-tion of this issue was, however, obtained from our compar-ison of T-dependent order parameters in networks andmelts [36].

Generally, networks are an attractive starting point for abetter understanding of the NMR signals of polymer melts,as they feature the same type of fast local dynamics, whilereptation is of course not possible. Fig. 33 shows MQbuild-up and decay data for the long-chain linear precursorof the previously investigated natural rubber sample series.A direct comparison of this data with Fig. 22 shows thatthe timescale of the IRMQ(sDQ) decay is very similar forthe network and the melt, indicating that the relaxationof this quantity mainly reflects fast segmental motions onthe local scale. To the contrary, IDQ(sDQ) is much reducedin the melt and therefore carries the signature of reptation.We are currently extending the theoretical framework pre-sented in Section 3.3 to reptation dynamics; it is expectedthat again, a simultaneous fitting of both sets of data willprovide a basis for reliable fits and tests of different modelsfor the autocorrelation function.

A first insight is provided by considering InDQ(sDQ) forthe linear NR (Fig. 33, right), which, as opposed to thealmost temperature-independent network data, shouldmainly reflect reptation dynamics, since the fast-motioneffect on IRMQ(sDQ) and IDQ(sDQ) should be very similar

Fig. 32. (a) Site-specific 500 MHz proton DQ build-up curves for linear poly(butadiene) used to extract an apparent D2res � S2

b at Tg + 50 K. (b)Correlation loss as a function of t/se, showing scaling regimes predicted for the reptation model. Time is varied employing time-temperature superpositionfor different temperatures above Tg + 50 K, and C(t/se) is simply obtained as a DQ intensity ratio, as discussed in the text. Note that the matched right axisfor the apparent RDC is scaled incorrectly (C(t) should scale as D2

res), as was clarified by communication with the authors. Figure reprinted withpermission from Ref. [33].

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and thus be cancelled out. Following the work of Grafet al., apparent residual couplings of the linear melt andtwo networks were determined from InDQ(sDQ) (neglectingrelaxation terms) and are compared in Fig. 34. It is seenthat the segmental averaging process down to the entangle-ment level is far from complete at T = Tg + 50 K. Muchhigher temperatures are needed to actually observe the T-independent plateau even for rubbers with only lowcross-linking. For the melt, the expected entanglement levelis only reached at about T = Tg + 140 K.

This now explains the unusually high order parameterobserved for PB melts. The Sb values for NR are still lower,but considering the potential dependence on the model aswell as the fact that trans units induce higher chain order,most of the discrepancy in the magnitude of Sb is thusexplained. Therefore, the value of Sb does not simply reflectsingle-chain order at a given temperature, assuming thevalidity of averaging timescales derived from rheologicalmodels. The reason for this might in fact be local packingconstraints that arise from overall cooperative motions,

which could also be responsible for the screening of end-to-end distribution effects discussed in Section 3.2.

Another possible explanation is the retarded self-averag-ing behavior of fluctuating tensorial rather than vectorialinteractions used to define an order parameter [82]. Thisphenomenon, which generally raises problems with respectto the proper interpretation of NMR data, is also respon-sible for the unexpected excluded-volume effects on chainorder discussed in Section 3.4. Future developments willhave to consider both possibilities, chain packing as wellas peculiarities of tensorial self-averaging, in order tofinally link up NMR observables with parameters thatdirectly correspond to those derived from the macroscopicmechanical behavior as well as results from, e.g., neutronscattering.

5. Other applications

This section finally summarizes a few miscellaneousapplications of proton MQ spectroscopy to other subjectareas, namely grafted and confined chains as well as gela-tion phenomena.

5.1. Grafted chains

Block copolymers are an attractive model system forwhich to obtain insights into the reptation dynamicsdescribed in the previous section. In lamellar poly(sty-rene)-block-poly(butadiene) (PS-b-PB) diblocks or PS-b-PB-b-PS triblocks, PB chains are grafted at one or twoends, respectively, to a rigid glassy PS phase, and are con-fined to a domain of about 15 nm size, which spans manyentanglement lengths. In the diblocks, only less efficientarm retraction processes can lead to correlation loss belowthe order parameter plateau dictated by entanglements,while for triblocks, the situation is reminiscent of a net-work, where essentially no correlation loss should beobserved.

Experimental data published by Spiess and coworkers[120] nicely confirms this picture, as shown in Fig. 35a.In the diblock (one end fixed), the t�1/2 scaling regime char-acteristic for reptation is essentially suppressed on the time-scale of its typical occurrence in the melt, and in thetriblock, hardly any decay of the order parameter isobserved. Notably, strongly increased main-chain order ismanifested in the block copolymers, indicating an influenceof the confinement posed by the grafting to the lamellarstructure.

An exciting result is the behavior of the order parameterobserved for free, marginally entangled PB chains that aredissolved in the deuterated PB phase of a diblock(Fig. 35b). While their order parameter is almost the sameas for the bulk polymer on a comparably reduced time-scale, the characteristic power law for its decay is changedconsiderably in the diblock structure and resembles that ofthe chains in the PB block. This can only mean that thematrix imposes it characteristic dynamics on the guest

Fig. 33. 20 MHz proton MQ intensity build-up and decay data for linear,uncross-linked natural rubber (NR) measured at different temperatures.The dashed lines on the right are data for vulcanized NR (see Fig. 22).Data replotted from Ref. [36].

Fig. 34. Backbone order parameters Sb derived from fits to InDQ(sDQ) forprotons in vulcanized and linear natural rubber using Eq. (15). The linesare just guides to the eye. Data replotted from Ref. [36].


chain, which again stresses the inadequacy of single-chainmodels. In other words, the time stability of the entangle-ments or the effective tube that constrain the free chain isdominated by the relaxation time of the matrix chains.

A model scenario developed to explain the above obser-vations is reproduced in Fig. 36. In agreement with thecooperativity arguments brought forth repeatedly in the

previous sections, nm-sized regions of uniform local order(describable by a nematic director) are postulated. Theseare statistically uncorrelated in free melts, and reptationleads to complete correlation loss at sufficiently long times.In the block copolymer, the local order of the domains isincreased overall and retains some correlation with the wallconfinement over the whole lamellar width, as a conse-quence of the tethering.

As a second, more application-oriented example, weturn to chain grafting as an important process for the phys-ical modification of surfaces. It is often employed to stabi-lize multicomponent polymer systems and to achievefavorable wetting conditions or good dispersion. Of partic-ular commercial relevance is the modification of inorganicfiller particles for reinforcement or viscosity control of elas-tomers and polymer melts. Silica, either fumed or precipi-tated, is commercially available with a wide range ofsurface modifications. One example are poly(dimethylsilox-ane) (PDMS) grafts, and the chain dynamics in such a sys-tem has previously been investigated in detail by Litvinovand coworkers using conventional transverse relaxometryat low field [121] (Fig. 37).

The systems were found to consist of up to three PDMScomponents with distinctly different dynamics (=apparentT2). At low grafting density, only two majority componentsare observed. A series of such samples was subjected to aproton DQ NMR study by the Aachen group [122].Fig. 38 shows DQ build-up curves for a low temperature(221K), where two distinct maxima are observed and canbe attributed to a rather rigid surface layer and anothermore mobile domain.

It is important to note that this clear signature of pro-nounced heterogeneity arises only as a result of overallintensity relaxation. Without overall decay (or after a suc-cessful normalization), a stepwise or gradually increasingbuild-up function would be expected, as is for instanceobserved in our example of bimodal networks (seeFig. 15). This poses some limitations as to the quantitativeevaluation of such multimodal build-up curves, whereutmost care is needed when the intensity scale for each ofthe maxima is to be defined. For example, when the first

Fig. 36. Postulated length scales of local order and nanodomain structure of poly(butadiene) chains in lamellar block copolymers. Figure reprinted withpermission from Ref. [120].

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Fig. 35. Correlation loss, C(t/se), of free and copolymer-graftedpoly(butadiene) (PB) chains as a function of t/se, following the approachof Fig. 32. (a) Bulk linear and copolymer-grafted PB. (b) Comparison withfree PB chains (5%) dissolved in a PB-deuterated diblock. Figure reprintedwith permission from Ref. [120].

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maximum in the DQ build-up curve is plotted on a scale ofthe total magnetization, the fit must embody the contribu-tion of the more weakly coupled components in its prefac-tor (the more mobile components do not contribute yet tothe build-up, but are part of the intensity scale). Then, theinitial build-up intensity of the second component can beaffected because the intensity relaxation of the first compo-nent occurs on a similar timescale.

As a consequence, in their initial report, Blumich andcoworkers discussed only the positions of the maxima asqualitative indicators of the actual RDCs. Both maximawere found to shift to longer times with increasing temper-ature, pointing at a speeding-up of the dynamics and agradual increase in motional amplitude. In addition, themaximum of the more mobile contributions was found tobroaden significantly, hinting at a significant distributionof underlying couplings, thus considerable dynamicheterogeneity.

5.2. Confined chains

Purely geometric confinement of small molecules andpolymers on the nanometer scale has attracted much recentattention, mainly because of the unusual and sometimescounterintuitive changes which occur in the molecular con-formation, the dynamics, and the glass transition. Under-standing surface-induced phenomena is of paramountimportance for the proper design of new materials thatare structured on the nanometer level, such as for examplepolymers filled with nano-sized fillers or thin films for sen-sor applications.

Anapore� porous alumina membranes are a suitablemodel system for studying surface-induced effects due totheir high inner surface and controlled cylindrical poresizes in the range of 10–200 nm in diameter and tens oflm in length. A first study of polymer segment orderingin such a system was due to Primak et al., who studiedpoly(dimethylsiloxane) (PDMS) in 200 nm pores by 2HNMR [123]. Up to three well-defined quadrupolar split-tings were identified as a function or pore surface coverage,indicative of discrete monolayer structures, each with itsindividual dynamic properties.

The Aachen group has demonstrated that similarresults can easily be obtained without the need for isoto-pic labelling using proton DQ NMR [124]. Fig. 39 showsproton DQ build-up curves for PDMS in Anapore withsurface coverages spanning the first few monolayers.The first monolayer is strongly absorbed, as indicatedby the sharp maximum at 37 ls. Increasing the coveragefirst leads to the appearance of shoulder-like structures,indicative of dynamic heterogeneity. At the highest con-centrations, distinct dynamics evolves for the second,third and forth monolayer, as indicated by the highermaxima.

As in the previous example of grafted chains, the exactquantification of the related dipolar couplings is not easydue to the potentially ill-defined intensity scale via theinfluence of relaxation and intermediate motions. Noteagain that the distinct maxima structure is a result of com-peting relaxation. Estimates of the RDC at least for themost strongly coupled fraction can be obtained by eitheranalyzing the first initial rise or, alternatively, by analyzingthe width of DQ-filtered (‘‘edited’’) proton spectra [49]. Inthe present example, short-sDQ-edited spectra even exhib-ited a splitting, corroborating the well-defined dynamicsin the first layer. From the initial rise of the DQ build-up, an estimate of the associated RDC of about 3.4 kHzwas obtained, indicating very restricted (probably uniaxial)dynamics. At higher coverage, the first layer is somewhatmobilized, characterized by a reduction of the couplingby about 30%.

Similar studies along the same lines have also been per-formed for lipid films, where effects of motional heteroge-neity as a function surface coverage also becameparticularly apparent [125]. Notably, the surface layerswith sub-monolayer coverage were found to exhibit

Fig. 38. 500 MHz proton DQ build-up curves for short PDMS graftson silica, measured using the two-pulse segment. The surface coverage,thus the average length of PDMS dangling ends and loops decreasesfrom PDMS1 to PDMS3. Note that ‘‘normalized’’ here only refers toan intensity scale where unity corresponds to the full samplemagnetization. Figure reprinted with permission from Ref.[122].

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Fig. 37. Schematic model of silica grafted with poly(dimethylsiloxane)chains. The scenario shown is for short chains or low PDMS content,where mainly two components with distinctly different dynamics canbe detected. Figure reprinted with permission from Ref.[121].

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a systematically increased mobility as compared to the bulksurfactant.

5.3. Study of gelation

The applications of proton MQ spectroscopy to polymermelts and networks presented in the previous sections ofcourse suggest that significant insight can also be gainedfor systems that undergo a transformation from one limitto the other, i.e., to linear polymers undergoing cross-link-ing and gelation. Such investigations were earlier per-formed using conventional transverse relaxation methods[126–128], and in a recent study [129], we have demon-strated the advantages of MQ spectroscopy in this context.

The subjects of our study were two conceptually differ-ent gelling systems, i.e., bulk cross-linking of short linearPDMS chains, and cross-linking of poly(styrene-co-ami-nomethylstyrene) [P(S-co-AMS)] in semidilute solutionusing deuterated toluene as solvent. For the first case, fullyreacted systems were studied as a function of cross-linkerconcentration and were compared with rheological experi-

ments, while for the latter, stoichiometric compositionswere studied as a function of reaction time and are com-pared with dynamic light scattering (DLS) results [130].

In both cases, the chains are virtually unentangled so asto not exhibit any detectable DQ intensity in the initialstate. It is then expected that significant RDCs becomedetectable only when the topological gel point is reached.At or above this point, the network structure becomes infi-nite and supports long-lived orientation correlations of thechain segments. It was indeed found that DQ intensitybecame detectable only in close vicinity to the mechanicalgel point (that is defined as the state at which the loss tan-gent becomes independent of frequency).

Fig. 40 collects proton MQ data for the end-linking pro-cess of PDMS above the gel point. The analysis of the long-time tails of IRMQ straightforwardly yields the sol content,that is seen to strongly decrease with increasing cross-linkerconcentration, as expected. Interestingly, IDQ(sDQ) showstwo distinct maxima indicating a heterogeneous gel struc-ture at the early stages just above the gel point. This behav-ior is very similar to the grafted and confined-chainapplications discussed in the previous sections, yet here,the sol-corrected IRMQ(sDQ) decays slowly enough so asto provide a reliable point-by-point normalization. Thecorresponding normalized DQ curves in Fig. 40b thereforeshow the expected two-step build-up and can easily be

Fig. 39. 300 MHz proton DQ build-up curves for molecularly thin PDMSlayers in Anapore membranes, measured using the two-pulse segment. In(b), the surface concentration Cs increases in the direction of the arrow:0.25, 0.5, 0.73, 1.4, 3.6 mg/m2. Note that ‘‘normalized’’ here refers only toan intensity scale where unity corresponds to the full sample magnetiza-tion. Reprinted from Ref. [124], Copyright 2004, with permission fromElsevier.

Fig. 40. 20 MHz proton MQ build-up and decay data for the bulk end-linking gelation of PDMS for different stoichiometric ratios r of cross-linker to end functionality. (a) Raw data, the lines are exponential fits tothe tail of IRMQ(sDQ) that give the sol parts. (b) Normalized DQ build-updata indicating a heterogeneous bimodal structure in the early stages justabove the gel point. Data replotted from Ref. [129].


analyzed in terms of two distinct RDCs and a relativemolecular weight. On the whole, these curves nicely reflectthe gradual build-up of cross-link density.

From such build-up curves, and the fact thatIDQ(sDQ) > 0 only above the gel point, it is easily inferredthat the gelation process can also be monitored by a sin-gle-point measurement of the DQ intensity at a specificsDQ. The value is best chosen such that the DQ intensityhas its maximum at full conversion. The acquisition of aDQ and a reference intensity (when nDQ is to be moni-tored) with about a hundred scans each does not take morethan about a minute or two, such that real-time experi-ments to investigate the gelation kinetics of systems inrather dilute solution are feasible even at low field.

An example is presented in Fig. 41a, where the cross-linking of poly(styrene-co-aminomethylstyrene) in solutionof various concentrations with a dialdehyde as cross-linker,that readily occurs at room temperature, is monitored. Thespecific sDQ = 4 ms as well as the limiting value ofInDQ(4 ms)tfi1 were determined in separate long-timeexperiments. A fixed InDQ(4 ms)tfi1 is needed to reliablyfit the so-obtained growth curves, that were found to beexponential within experimental noise. As is obvious,

growth only starts after a lag time that corresponds tothe time at which the gel point is reached.

The gel point time as well as the gel formation time con-stant (that reflects the rate of formation of elastically activenetwork chains) derived from the fits is plotted in Fig. 41band compared with the gel point time obtained fromdynamic light scattering. This latter method is based onthe appearance of a speckle pattern arising from fixed con-centration fluctuations when the system becomes non-ergo-dic above the gel point. Since the time and length scalesintrinsic to this method are larger than for NMR, it wasnot surprising that it detects gelation somewhat earlier. Itshould, however, be noted that the investigations couldnot be performed on the very same samples; the deviationis well within the common range of reproducibility of suchgelation reactions.

Important conclusions can be drawn from the indepen-dence of the network formation rate on concentration(Fig. 41b, right ordinate), as well as from the final analysisof the proton MQ data of the fully reacted gels. It wasfound that about 50% of the polymer contributes only tothe formation of elastically inactive microgels, loops, ordangling chains. The remaining polymer is either highlymobile sol or is converted to 12–50% true network fractiondepending on concentration. Surprisingly, the averageRDC (thus the average network chain length) is also inde-pendent of concentration. All these observations are ingood agreement with or complementary to those fromdetailed mechanical and DLS studies of this system[129,130]. The concentration-independence of the gel for-mation rate and the final cross-link density indicate thatgelation is not diffusion-controlled and mainly occurs inheterogeneous clusters that in the end link up to form amacroscopic gel.

In summary, it is seen that a large spectrum of useful,partly unique insights can be gained from the applicationof MQ spectroscopy to gelling systems. Further work willhave to show how the approach has to be adapted to thecase of long, entangled pre-polymers that exhibit signifi-cant entanglement-induced residual dipolar couplings evenwhen no crosslinks are present [33,36,126]. One might envi-sion the observation of a discontinuity in InDQ(sDQ) at thegelpoint, yet, as entanglement-induced residual couplingsare a strong function of temperature (see Section 3.3), anextended protocol might necessitate a study at differenttemperatures. Cohen-Addad has discussed other possibleapproaches along these lines based on proton transverserelaxation properties [126]; the application of MQ spectros-copy to this type of system is underway.

6. Summary and conclusions

This article was intended to deliver a broad survey overthe many applications of proton MQ spectroscopy tomobile polymer systems. While the concept and the pulsesequence appear of course somewhat involved, its imple-mentation is robust, the set-up is easy (only precise 90�

Fig. 41. (a) Real-time single-point measurements of proton InDQ(4 ms) asa function of gelation time of semidilute solutions of poly(styrene-co-aminomethylstyrene), measured at 20 MHz. The lines are exponentialgrowth fits yielding a time constant sNMR and a lag time that correspondsto the gel point, using fixed long-time limiting values InDQ(4 ms)tfi1indicated by the dashed lines. (b) Comparison of NMR- and DLS-detected gel points and NMR-detected gelation time constants sNMR.Data replotted from Ref. [129].


pulses and good on-resonance conditions are to beensured), and it can be run in an automated fashion evenon cost-efficient low-field equipment. The main advantageover more traditional NMR methods such as Hahn echorelaxometry or combinations of Hahn and solid echoes isthe reliable separability of coherent dipolar effects and sig-nal loss due to motion-induced relaxation by means ofmonitoring the DQ build-up as well as a reference inten-sity. The sum of DQ and reference intensities represents afully dipolar-refocussed intensity function IRMQ(sDQ), thedecay of which is largely dominated by dipolar relaxation.

In permanently cross-linked networks, a point-by-pointdivision of the DQ build-up by IRMQ(sDQ) yields a normal-ized intensity that is temperature-independent and reflectselastomer microstructure only. It can be analyzed in termsof distributions of residual dipolar couplings, and a num-ber of applications to different types of elastomers, bimodalPDMS networks, filled SBR, NR, and filled PDMS, hasbeen presented. A particularly important finding is thatall chemically uniform single-component rubbers are foundto display surprisingly narrow, almost unimodal couplingdistributions. This has significant consequences for thevalidity of single-chain models for the explanation of theNMR response as well as the mechanical and swellingbehavior via the conformational entropy, where Gaussianstatistics is assumed for the end-to-end separation of sub-chains. This distribution should lead to a broad gammadistribution of couplings, and this is not observed in ourexperiments. Additional cooperative contributions to localchain order, for instance describable in terms of an orienta-tional mean field or a specific tube constraint, are possibleexplanations.

Using spin dynamics simulations, as well as site-resolvedDQ build-up curves and 2D spectra, we have developedmolecular models for the quantitative interpretation ofthe measured RDCs in terms of a polymer backbone orderparameter for the cases of NR, cis-BR and PDMS. A com-parison of the resulting NMR-determined cross-link densi-ties with swelling results indicates satisfactory agreementwithin a factor of 2, yet many details concerning systematicdeviations and the significance of temporary and trappedentanglements still remain unclear.

Experiments on swollen rubbers indicate a significantbroadening of the RDC distribution that was attributedto swelling heterogeneities. The behavior is strongly subaf-fine, with only a small part of the chains being stretchedsignificantly. It can be explained in terms of competingdesinterspersion and stretching processes, where the formerleads to a reduction of chain order via the relief of cooper-ative chain packing effects.

As to the chain dynamics in elastomers, the relaxation ofoverall intensity in the proton MQ experiment, thus alsothe incoherent contribution to the decay of transverse mag-netization in Hahn-echo experiments, is shown to be solelygoverned by fast segmental processes, i.e., Rouse modesthat afford the averaging of the effective static-limit refer-ence coupling down to the plateau value given by the

dynamic order parameter. An influence of slow processesthat were often adopted in earlier work can be excludedon the basis of our data.

In linear entangled melts, reptation of course takes therole of the slow process that causes a further loss of resid-ual orientation correlation. A comparison of networks anda long-chain melt showed that the overall intensity loss inthe melt is still governed by the fast modes, while reptationhas a decisive influence on the reduction of the apparentRDC derived from the DQ build-up. Importantly, temper-atures that exceed Tg by at least 100 K are needed to detectan RDC that corresponds to the entanglement level. Thisexplains earlier findings of unexpectedly high order param-eters at lower temperatures, and exemplifies the non-trivialand as yet not well understood relationship between thetrue chain fluctuation statistics, the rheologically deter-mined timescales of polymer dynamics, and the data deter-mined by NMR. More theoretical work will have to bedevoted to the development of a proper model for the ori-entation autocorrelation function for entangled chains,which should ultimately feature fitting parameters thatcan be related to classic theories of polymer dynamics.

Miscellaneous applications include the dynamic state ofchains grafted at one or two ends in block copolymers andto silica surfaces. In the first case, the confinement is foundto increase chain order and to effectively suppress reptation,confirming a picture that stresses the importance of local,cooperative order phenomena. In the latter case, the roleof heterogeneity is apparent in build-up curves with twomaxima reflecting strongly absorbed and more freely mobilechains in the outer layer. A decisive layering with increasingbut well defined mobility was also found for molecularly thinPDMS layers in high-surface porous materials. Finally, pro-ton MQ NMR has been demonstrated to provide uniqueinsights into the gelation process of polymers in the bulk orin dilute solution. The results are in good agreement withor complementary to those from rheological and light-scat-tering studies, and indicate spatially inhomogeneous gela-tion processes in both solution and bulk.

In conclusion, MQ NMR will continue to improve ourunderstanding of polymer chain dynamics, where the largevariety of new opportunities ranges from industrial screen-ing applications of elastomers to very basic questions con-cerning the theories of polymer dynamics and rubberelasticity. It should be stressed again that one salient advan-tage of MQ NMR, which is the reliable separation of coher-ent dipolar and relaxation effects by means of analyzingboth a DQ build-up as well as an MQ sum intensity decay,also applies to deuterium, where the very same pulsesequence can be applied and where the excitation of DQcoherence is largely restricted to the dominating single-spinquadrupolar effect. It is expected that many argumentsmade herein can thus be refined, as ambiguities related tomultiple coupled spins are absent. Complications may thenof course arise due to the more limited validity of the sec-ond-moment approximation that is central to the presentedtheory, and improved approaches will have to be developed.


Possible extensions of the homonuclear MQ experimentare of course methods that measure heteronuclear RDCs,and make use of the improved site resolution of, for exam-ple, carbon-13 nuclei. This would ultimately allow for thedevelopment of yet more improved models for the geome-try of local segmental motion, since better defined localtensor orientations may become accessible. The applicabil-ity of such methods for the study of elastomers [131,132] orchains in channel confinement [133] has already been exem-plified. Specifically, methods based on SEDOR (spin–echodouble resonance, for static samples [134]) and REDOR(rotational-echo double-resonance, under MAS [135]) holdmuch promise, as they also offer the possibility of intensitynormalization [136]. It will have to tested whether suchdata can also be analyzed in terms of correlation loss dueto dynamics on various timescales. This is a particular chal-lenge under MAS, where recoupling methods are to beused and where the timescale of sample rotation introducesadditional complications [137].

Acknowledgments

This work has benefitted invaluably from the fruitfulwork with many graduate students, PhD students, andcollaborators, the names of which can be taken fromthe cited publications. I thank them all for their contri-butions. Very specific thanks are due to Jens-Uwe Som-mer and Andreas Heuer, who were not only involved instarting my interest in the subject matter, but also pro-vided the theoretical input without which our researchin the area would still be in its infancy. Insightful andimportant discussions with Robert Graf, Hans-WolfgangSpiess, and Jorn Schmedt auf der Gunne are acknowl-edged. Much of the research work of my group was per-formed at the Institut fur Makromolekulare Chemie atthe University of Freiburg, where I am grateful to HeinoFinkelmann, Rolf Mulhaupt and Alfred Hasenhindl fortheir support. Funding over the years was provided bythe Deutsche Forschungsgemeinschaft (SFBs 428 and418), the Landesstiftung Baden-Wurttemberg, and theFonds der Chemischen Industrie.

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