Multiple-quantum NMR studies of polymer chain dynamics∗

Kay Saalwachter∗1, †

1Institut fur Physik – NMR, Martin-Luther-Universitat Halle-Wittenberg, D-06120 Halle, Germany(Dated: September 1, 2016)

Multiple-quantum (MQ), specifically the analysis of double-quantum build-up curves, pro-vides quantitative information on proton–proton dipolar couplings, or alternatively on deuteriumquadrupolar couplings. The study of these interactions, in particular when they are partially av-eraged by molecular motion, provides valuable information on amplitude and timescale of rotionalmotions of the related molecular segments. MQ NMR has in recent years evolved as one of the mostpowerful methods for the study of polymer chain motion in a variety of system, also owing to thefact that the proton-detected variant is readily applicable on cost-efficient low-field instrumentation.This chapter provides an overview of the most relevant technical aspects, recent improvements andapplications to different polymer materials, with a focus on work published during the last decade.Particular emphasis is devoted to studies of elastomers and hydrogels, for which the method providesunique structural information. Confined chain motion, found in entangled melts, block-copolymersor grafted systems, is also addressed.

Keywords: order parameter, polymer networks, hydrogels, rubber, polymer melts, entanglements,supramolecular polymers, confinement effects, nanocomposites, filled elastomers

INTRODUCTION

Anisotropic interactions in solid-state NMR providevaluable measures of orientation-dependent phenomena,in particular on rotational dynamics of specific molecu-lar units. In soft materials such as lipids, liquid crys-tals or in particular polymers, fast anisotropic dynamicsof substantial amplitude is common, leading to signifi-cant motional averaging and correspondingly low resid-ual interactions. Disorder and inhomogeneities, result-ing in interaction distributions, provide another chal-lenge for a quantitative assessment. In this context,multiple-quantum (MQ) NMR as applied to protons ordeuterons, providing a robust and quantitative accessto residual dipolar [1–3] or quadrupolar [4–6] couplings(RDCs or RQCs, respectively), their distribution, andtime-dependent phenomena (“intermediate” dynamics),has evolved as the probably most robust spectroscopictool [7, 8].

The main methodological features of MQ NMR, specif-ically of double-quantum (DQ) NMR build-up curveanalysis, were recently reviewed [8]. The first applicationto polymer melts is due to Graf et al., who used magic-angle spinning (MAS) recoupling DQ NMR to study site-resolved RDCs [1]. MAS DQ NMR in fact provides dif-ferent modes of extracting the (R)DCs in one- or two-dimenstional spectra [8, 9], but the study of DQ build-upcurves in static samples is the most robust and quanti-tative approach [7], provided that site-resolution is notof concern. This is the case for many chemically sim-ple homo- or copolymers. The application of static low-resolution MQ NMR to polymeric soft materials was al-

∗to be submitted to Modern Magnetic Resonance, 2nd ed.

ready the subject of a review article in 2007 [7].

Here, we focus on more recent methodological develop-ments as well as new applications to different polymer-based materials comprising crosslinked systems (elas-tomers, gels) and constrained (entangled, end-fixed andnano-confined) chains. This review chapter is exclusivelyconcerned with experiments on rather dynamic systems,but it should be noted that also in rigid polymers DQNMR also provides unparalleled quantitative access toweak pairwise [10] or multi-spin [11] 13C–13C dipolar in-teractions. These can be used to probe molecular prox-imities and thus conformations and packing, respectively.For instance, recent work of Miyoshi and coworkers hasdemonstrated that previously unaccessible informationon the chain folding structure in semi-crystalline poly-mers can be gleaned in this way [11, 12].

POLYMER CHAIN MOTION AND NMROBSERVABLES

The orientation dependence of 1H–1H dipolar as wellas 2H quadrupolar couplings follows in both cases thesecond Legendre polynomial P2[cos θ]. For mobile poly-mer chains far above the glass transition temperatureTg, the main difference is that in the former case mul-tiple pair couplings combined with fast intra-segmentalmotions lead to the appearance of a single effective RDC(Deff) that is the same for all protons in the monomerunit provided that they do not reside in motionally de-coupled side-chains. This is ultimately the consequenceof the dipolar truncation effect [8]. The multi-spin situ-ation further justifies the use of the second-moment ap-proximation in theoretical treatments [13, 14]. In the2H case, RQCs remain distinguishable for the differentlabeled positions, which can provide additional informa-

2

(a)

cooperative chain modes

s...s

segmentalmodes ns...s

C(t)

Sb2

1

0.1

0.01

constraint

(b)DQ excitation acq.

time

network

entangled melt

b

DQ reconversion

0.001

0.0001

0.6

0.8

1.0

tens

ity

(c)IDQIref and tail fit (14% defects)IMQ = IDQ IrefInDQ = IDQ/(IMQtail) and fit

DQ DQ

0 2 4 6 8 10 12 14 20 40 60 800.0

0.2

0.4

rel.

int

DQ evolution time / ms

A-l vs. Gauss kernel

DQ evolution time / ms

Figure 1: (a) Orientation fluctuations of a fluctuating polymerchain can be described by a orientation autocorrelation func-tion C(t) of the second Legendre polynomial. In case of long-time stable topological constraints (e.g. chemical crosslinks),this function attains a plateau described by a semilocal dy-namic order parameter Sb ∼ 〈P2[cosβ]〉, which is the norm of

the time-averaged segmental ordering tensor |↔Sb |. Its princi-

pal axis is parallel to the end-to-end vector ~R. (b) Schematicpulse scheme of an MQ NMR experiment. (c) MQ NMR sig-nal functions of a typical rubber sample as a function of pulsesequence time τDQ: the reference intensity Iref , from which aslowly relaxing defect contribution can be obtained, the DQintensity build-up IDQ, and the two relevant derived quan-tities, the MQ sum intensity IΣMQ and the normalized DQbuild-up InDQ.

tion or complicate the data analysis. Further, theoret-ical signal functions are easily calculated, as we have asingle-coupling situation. In the following I only referto RDCs, but imply that the same or at least similarprinciples apply for RQCs. Unless noted otherwise, alldiscussed experimental data is from 1H MQ NMR.

The segment-based effective RDC tensor can safely beassumed to be parallel to the polymer backbone, whichmeans that the additional averaging (lowering) of theRDC reports on chain modes, as illustrated in Fig. 1a.The dipolar interaction frequency relevant for a segment(monomer) that is part of a chain with fixed ends, as for

instance found in an elastomer, is

ωD(θ) =3

2Deff Sb P2[cos θ] , (1)

where θ is the orientation of the end-to-end vector withrespect to B0 subject to powder averaging, and

Sb =1

2

[(3〈cos2(β)〉 − 1)

]=

3

5

(l R

R20

)2

. (2)

is the segmental dynamic order parameter of the back-bone. Here, l is the segmental length, β the segmentalorientation with respect to the end-to-end vector ~R, and〈. . .〉 and [. . .] indicate thermal conformational and struc-tural (segmental) ensemble averages, respectively. Ef-fects of chain deformation (due to macroscopic strain orswelling) enter through the dependence on the end-to-end vector R2/R4

0, R0 being the unperturbed dimension[15, 16]. For bulk samples, we have R4

0/R2 = N l2, lead-

ing to a simpler expression for the RDC,

Dres = Deff Sb = Deff3

5N. (3)

This quantity is thus a measure of the length of thenetwork chains in terms of the number of segments N ,thus, it is proportional to the density (inverse molecularweigth) of network chains 1/Mc in units of mol/kg. Thevalue for Deff is polymer-specific, and values have beenreported for a variety of systems [17, 18]. The argumentsabove imply a pre-averaging due to fast chain modes andthe virtual absence of slower processes. This situation isonly realized in elastomers at rather high temperaturesabove Tg. Otherwise, ωD(θ) is a random function in time,meaning that only

C(t) = 5〈P2[cos θ(t+ τ ]P2[cos θ(τ)]〉τ,ens. , (4)

the orientation autocorrelation function (OACF) ofP2[cos θ], is a well-defined quantity (θ is now the segmen-tal orientation with respect to B0). This general case isrelevant for, e.g., linear entangled chains or chains closerto Tg. See again Fig. 1a for sketches of C(t) for differentcases. If C(t) does not exhibit a well-defined plateau onthe µs to ms timescale (corresponding to the inverse oftypical (R)DC values), the apparent (!) RDCs obtainedfrom the data analysis will be a function of temperature,and can be used to conclude on the shape of C(t). A morequantitative theory-based analysis of the signal functionsis of course the best approach in this case, see below.

Turning to the MQ NMR experiment, Fig. 1b showsthe schematic pulse sequence consisting of suitably phase-cycled DQ excitation and reconversion periods of equallength τDQ followed by a z filter and detection after a 90

pulse. See refs. [7, 8, 19] for information on technicaldetails and suitable pulse sequences. A pulse sequencewith a DQ Hamiltonian not only creates DQ but alsozero-quantum and higher-order coherences (thus “MQ

3

NMR”), but the analysis can safely be restricted to thedominating DQ coherences [3]. The most relevant techni-cal point is that two phase-cycled controlled signal func-tions are recorded, namely the DQ build-up IDQ(τDQ)and reference decay intensity Iref(τDQ). The former con-tains the desired information on the RDC, while the lat-ter represents an intensity complement subject to thesame relaxation effects. Note that given high-resolutionconditions (high field, MAS), such data can be obtainedfor any peak in the spectrum [8], but for simplicity, we re-strict this chapter to integral 1H detection of chemicallysimple polymers. See Fig. 1c for sample data taken onan elastomer, and information on further data processingsteps.

Summation provides a “relaxation-only” full-echofunction IΣMQ(τDQ) = IDQ(τDQ) + Iref(τDQ), which, to-gether with IDQ(τDQ) is the basis of further data anal-yses. Note that Iref(τDQ) also contains all signals fromisotropically mobile moieties (Dres = 0), related to net-work defects (dangling chains, sol) and solvent. Theseare characterized by a long effective T2 relaxation time,leading to a long-time tail that can be fitted and sub-tracted [20]. After tail subtraction, the sum function canbe used to relaxation-correct (point-by-point normalize)the DQ build-up,

InDQ(τDQ) = IDQ(τDQ)/(IΣMQ(τDQ)− tail) . (5)

As illustrated in Fig. 1c, this function reaches a long-timeplateau of 50% of the total intensity, which indicates, sim-ply speaking, the equal partitioning of the signal betweenmodulated longitudinal magnetization (the main contri-bution to Iref) and DQ coherences. This is only the casefor network chains or other permanently anisotropic sit-uations (e.g. liquid crystals), for which C(t) exhibits along-time plateau. In this most simple case, the nDQbuild-up curve is calculated as

InDQ(τDQ) = 〈sin2(k ωD(θ) τDQ)〉θ,ens. , (6)

where k is a scaling factor that depends on the pulse se-quence. For the case of the static Baum-Pines sequence[21], which is the best choice for sufficiently mobile sys-tems with small RDCs [3], k = 2/3, which will be im-plied in all further equations below. See also below forsuitable approximate solutions of eq. (6). Data analy-sis proceeds differently for the more general situation ofchains that exhibit orientation relaxation on the relevantµs–ms timescale. The following sections thus treat, first,the case of networks, and then the more general and ul-timately more complicated cases of supramolecular net-works, entangled polymer melts and confined chains.

POLYMER NETWORKS: ELASTOMERS ANDSWOLLEN GELS

InDQ data measured on an elastomer as shown inFig. 1c cannot be fitted well to a simple numerical pow-der average of eq. (6), as the latter exhibits decisive long-time oscillations. Their absence in experimental data isprimarily due to multiple RDCs among many protons,which can be accounted for by way of a second-momentapproximation [7]. The resulting inverted-Gaussian func-tion reads

InDQ(τDQ, Dres) =1

2

(1− exp0.4D2

res τ2DQ

). (7)

This result also cannot describe actual data on homoge-neous rubbers perfectly well, due to a small overshootrelated to multiple-spin quantum dynamics [3].

Another factor to consider is that the overall shapeof the build-up function is affected by a distribution ofRDC values p(Dres) in an inhomogeneous sample, mean-ing that

InDQ(τDQ) =

∞∫0

InDQ(τDQ, Dres) p(Dres, 〈Dres〉, σ) dDres .

(8)The distribution parameters 〈Dres〉 and σ (width in termsof standard deviation) can be assessed either by (i) solv-ing the integral analytically based upon an assumed dis-tribution function (e.g. Gaussian), (ii) by numerical in-tegration as part of the fitting procedure, or (iii) by nu-merical integral inversion procedures using eq. (7) as theKernel function. In any case, errors due to small misfitsat longer τDQ due to eq. (7) being an approximation canbe minimized by focusing on the initial-rise region [3, 22].

In order to be able to fit the whole range of reliableInDQ(τDQ) data, we have more recently used data takenon elastomers that were known to be very homogeneous(negligible σ) to derive an empirical build-up functionthat is suitable for any elastomer with protons restrictedto chemical groups close to the main chain,

InDQ(τDQ, Dres) = 0.5(1− exp

−(0.378DresτDQ)1.5

× cos(0.583DresτDQ)) . (9)

Owing to its similarity to the function describing anon-resonance 1H free-induction decay (FID) of a rigiddipolar-coupled solid as popularized by Abragam [23],it is referred to as “Abragam-like” (A-l) function. Thisfunction can now be used as Kernel build-up function inthe different implementations of eq. (8). The data inFig. 1c demonstrate that the use of this function leads toa near-perfect fit.

With the current level of methodological development,MQ NMR is ideally suited to address relevant scientificquestions in the field of elastomers and gels. By way

4

of the analysis of the relevant signal functions, the con-tent of non-elastic defects, the average crosslink densityin terms of 〈Dres〉 (the angular brackets are often, andwill henceforth be, omitted) and the width or shape ofits distribution (the latter quantified by σ) are straight-forwardly accessible. The method thus provides infor-mation on network micorstructure and topology that isoften physically more relevant than the analogous butmore qualitative insights from traditional Hahn-echo 1HT2 relaxometry or the purely chemical information (e.g.on crosslink structure) obtained from 13C MAS NMR[24]. To illustrate this point, Fig. 2a shows Dres distribu-tions of three different types of elastomers. It is immedi-ately seen that the curing system (sulfur- vs. peroxide-based) affects the microstructure significantly [25], andthat copolymer-based rubbers such as EPDM may beunexpectedly inhomogeneous [26], up to a level at whichthe applicability of the classical theories of rubber elas-ticity may be questioned [26, 27].

Swollen elastomers always exhibit rather broad Dres

distributions arising from inhomogeneous swelling andthe resulting variations in local chain stretching [28].Such inhomogeneities can only be avoided in rather spe-cial hydrogel model systems synthesized in the swollenstate [29, 30]. In previous related studies, it was shownthat in the absence of spatial inhomogeneities it is evenpossible to distinguish between regular network chainsand different types of connectivity defects, see Fig. 2b.These are identified by largely different associated Dres

values of the monomers in these respective structures.The wide range of Dres values in this system (vary-

ing over 2 decades) poses a challenge to the normaliza-tion procedure discussed above. This should be elabo-rated upon due to its exemplary character: Instead ofincreasing step-wise to the value of 0.5 (the heights ofthe steps reflecting the relative amplitudes), the nDQbuild-up curve shows distinct maxima and a long-timedecay. This is due to the fact that DQ intensity normal-ization, see eq. (5), assumes that all components of aninhomogeneous system have the same relaxation decayas described by IΣMQ. This condition is often fulfilledwell enough to obtain an analyzable nDQ signal func-tion (exhibiting the expected long-time plateau) if theoverall decay of IΣMQ occurs on a comparably long τDQ

timescale. But if IΣMQ differs among the componentsand is significant already during the DQ intensity build-up, normalization will fail. Then, one needs to resort toa simultaneous multi-component fit of the constitutingsignal functions [29]:

IDQ (τDQ) =

n∑i=1

aiInDQ(τDQ, D(i)res) exp

[− (τDQ/τi)

βi

](10)

IΣMQ (τDQ) =

3∑i=1

ai exp[− (τDQ/τi)

βi

]. (11)

2.5

3.0

/ a

.u. NR (peroxide x-linking)

EPDM (peroxide x-linking)

NR (sulfur vulcanization)(a)

0.5

1.0

1.5

2.0

prob

abili

ty d

ensi

ty /

0.0 0.2 0.4 0.6 0.8 1.00

0.5p

Dres/2 / kHz

1.0 regular links (~65%) IMQ

IDQ

(b)

0.2

0.4

0.6

0.8

norm

. int

ensi

ty

double links (~25%) higherdefects (~10%)

IDQ

DQ

InDQIMQ

0 10 20 30 40 50 60 70 80 90 1000.0

DQ evolution time / ms

Figure 2: (a) Dres distributions characterizing the extentof spatial crosslinking inhomogeneities in different types ofrubber. The homogeneous case of sulfur-vulcanized naturalrubber (NR) is compared to the less well-defined cases ofperoxide-based radical crosslinking of NR [25] and ethylenepropylene diene terpolymer (EPDM) rubber [26]. (b) MQNMR data and multi-component fits revealing “quantized”defect structures in hydrogels based upon end-linking of starprecursors. The structural complexity requires in this case ajoint fit to the raw signal functions [29].

Here, ai are the amplitudes of the different components,τi and βi their transverse relaxation parameters (assum-ing a stretched or compressed exponential shape), and

D(i)res the corresponding RDCs. Such a multi-parameter

fit must be carefully checked for stability, using e.g. asimulated-annealing algorithm, and is probably not fea-sible for systems with more than n = 3 distinct com-ponents. The blue solid lines in Fig. 2b represent anexample for a 3-component fit.

Microstructure studies of bulk elastomers

In addition to the exemplary data discussed above(Fig. 2), many more applications related to the mi-crostructure characterization of elastomers by low-field1H MQ NMR have appeared in the last years. Chinn,Maxwell and colleagues have used it extensively to studyfor instance thermal degradation [31], radiation-induced

5

crosslinking [32], and the relation between experimentallyaccessible and computer-simulated microstructure [33] ofsilicone-based elastomers.

Valentın et al. focused on open questions concern-ing natural rubber (NR), including the study of pre-vulcanization of dispersed latex [34] and a comparisonof NMR and dielectric spectroscopy results highlight-ing the microstructural differences between sulfur- andperoxide-based curing and their relation to the segmen-tal dynamics [35]. The latter work is in fact a follow-upof the earlier results shown in Fig. 2a, which stressed thatperoxide-based crosslinking leads, in particular in case ofNR, to a high level of non-elastic defects and pronouncedcrosslinking inhomogeneities [25]. Finally, Simonutti etal. demonstrated the sensitivity of MQ NMR to smallstructural changes during the late stage of rubber vul-canization, comparing different bulk and filled diene elas-tomers [36]. This work relied on our previous work onthe linear correlation between crosslink densities of bulkrubbers determined by NMR and by the Flory-Rehnerswelling method, which in fact allowed for a critical evalu-ation of data analysis procedures involved in the swellingtechnique [37]. The effect of filler particles [38] deservescloser attention, and is deferred to the section dealingwith composites.

Swollen elastomers and gels

Pioneering work concerning NMR studies of networksswollen in solvent is due to Cohen-Addad, who has veryearly on realized the decisive role of local chain deforma-tion upon swelling, the relevance of topological interac-tions (entanglements) and the influence of solvent quality[41–44]. As is obvious from eq. (2), Dres ∝ Sb ∝ R2 issensitive to the state of chain stretching upon swelling. Inour earlier work, relying on the unique sensitivity of MQNMR to Dres distributions, we could demonstrate thefirst direct NMR observation of swelling inhomogeneities,a phenomenon so far characterized only by scattering ex-periments [28].

Later, benefiting from a more quantitative data anal-ysis on the basis of eq. (9) and improved experimentaldata [15, 40], we were able to complement the earlierwork of Cohen-Addad in two main directions. Our start-ing point was the insight that the unperturbed dimensionR0 of the network chain, relevant for eq. (2), is modifiedin the swollen (=“semi-dilute”) state when using a goodsolvent. Under these conditions, the chains are subjectto additional expansion due to excluded-volume interac-tions, and the point to consider is that the size scalingbehavior is different below and above the so-called cor-relation length ζ corresponding to the “mesh size.” Anin-depth treatment of Sb measured in the equilibrium-swollen state using scaling theory [45] provides a uni-versal power-law dependence on the degree of volume

(a)

R2

0.1

NR peroxide-curedNR sulfur curedpa

ram

eter

Seq

~

1.480.07

1.480.03

0.15

0.01

1042 86

NR sulfur-curedPDMS end-linkedPDMS randomly x-linked

equilibrium swelling degree Qeq

orde

r p

0.07

(b)

0.05

0.10

0 5

Dre

s/2

/ kH

z Dres(Q=1)

Dres,PRN = Dres,swQ-2/3 (affine range) phantom reference

Dres,sw(Qeq)Dres

desinter-

( )

1 2 3 4 5 6 70.00

D

swelling degree Q

phantom reference network state

desinter-spersion

Figure 3: (a) Segmental order parameters derived from av-erage Dres values measured in different equilibrium-swollennetworks (good solvent case) as a function of the equilibriumdegree of volume swelling Qeq = Veq/V0 [15]. The universalscaling law is in agreement with theoretical predictions andemphasizes the role of thermodynamic effects on the NMR ob-servable [15, 16, 39]. (b) Dres values as a function of partialbelow-equilibrium swelling. The late-stage data proves affinedeformation of the network chains, and allows for the defini-tion of a “phantom reference state” value Dres,PRN via back-extrapolation, which quantifies the actual chemical crosslinkswithout entanglement effects [40].

swelling Q = V/V0 in good solvent, see Fig. 3a [15].The same result has been found earlier by Cohen-Addad,but was mistaken as a proof of de Gennes’ influential c∗-theorem [43]. We found that also a suitably modifiedclassic Flory-Rehner treatment can also explain the re-sult, providing new insights into the phenomenon of gelswelling. In a follow-up study, we extended our experi-ments to the poor solvent range, establishing the sensi-tivity of the NMR observables to thermodynamic effects[16, 39].

A second, even more relevant finding related to themicrostructural basis of rubber elasticity was the insightthat the dependence of Dres(Q) below swelling equilib-rium is in agreement with simple affine deformation ofthe chains, Dres(Q) ∼ Q2/3 according to eq. (2), once aninitial desinterspersion stage is overcome (Fig. 3b). Back-extrapolation of the affine region to Q = 1 provides theso-called “phantom reference network” value Dres,PRN,which quantifies the actual chemical crosslinks without

6

entanglement effects. In this way, by comparison withtheoretical models describing entangled rubber elasticity[40], we could establish for the first time a clear relationbetween microstructure (connectivity vs. topology), andmacroscopic behavior [27].

An in-depth comparison of crosslink densities obtainedby NMR vs. equilibrium swelling, including the quanti-tative consideration of non-elastic defects in experimentand theory, allowed us to clarify the role of latter andand prove the validity of simple statistical models of thecrosslinking process [46]. Finally, MQ NMR proved alsouseful to characterize the crosslink density and structuralinhomogeneities in swollen superabsorber gels [47], whichare often inhomogeneous-by-design. The microstructuralcharacterization provided by MQ NMR can thus aid inimproving such materials.

Uniaxial deformation

The concepts that apply for elastomer swelling are eas-ily transferred to the study of mechanically deformedsamples, providing another platform to put elasticity the-ories to a test. Also in this case we rely on the depen-dence of Dres ∝ Sb ∝ R2 on the distance between con-straints, see eq. (2). The main difference is that while freeswelling corresponds to isotropic dilation, stretched rub-ber samples are anisotropic, introducing a dependenceof the measured data on the orientation Ω of the mainstretching direction with respect to B0 (for the uniaxialcase). This offers new opportunities for data fitting, butalso poses some new challenges.

Early insights into stretched elastomers were gleandedfrom static 2H NMR spectra of either deuterated low-molecular probe molecules [48] or deuterated networkchains [49]. In the former case, only diffusion-averaged“mean-field” information is obtained, while in the lattercase full microscopic information on local deformationis in principle within reach. The early lineshape anal-yses relied on partially critical model assumptions, andbecause of this, the application of MQ NMR withoutthe need for isotope labeling appeared promising [50, 51].Our first application was concerned with quantifying theso-called overstrain of the network matrix for the caseof filled elastomers [50], followed by a quantitative as-sessment of microscopic deformation in bulk elastomers[51].

In Fig. 4a InDQ build-up curves are compared for anundeformed vs. a uniaxially strained unfilled rubber sam-ple. In the latter case we observe significantly increasedDres values and a clear orientation dependence. At thattime it was not possible to model the strained-networkbuild-up data, so we introduced an “artificial powder av-erage” by superposition of Ω-dependent data [50, 51]. Inthis way, eq. (9), which relies on an isotropic sample, canbe used to extract Dres distributions and averages. The

(a)0.6

ty 10

.u.

0.4

0.2

nDQ

inte

nsit

= 1 = 3 = 0° = 54° = 90°

0

5

ob. d

ensi

ty /

a.

affine phantom i t b3(b)

0.03210

DQ evolution time / ms

00

Dres/Dres,=1

pro

1 2 3

>1phantom iso. tube3

2

DQ

Inte

nsity

1

9060300sample orientation / °

= 3rel.

D

0

sample orientation / (c)

0 6

0.8

sity

1.0 eq.(12) 0°eq.(12) 90°eq.(9) A-leq.(6) sin2

0.2

0.4

0.6

nDQ

inte

n

0 1 2 3 4 5 6 7 80.0

DQ evolution time / ms

Figure 4: (a) InDQ build-up curves for a uniaxially deformedNR sample (λ = 3), comparing three orientations Ω (stretch-ing direction vs. B0, symbols) and the “artifical powderaverage” (lines) constructed by weighted superposition, alsocompared to the unstretched case. The right panel showsthe derived powder-averaged Dres distributions. (b) RelativenDQ intensity at fixed τDQ = 0.3 ms (λ = 3) as a func-tion of Ω along with network model predictions. (c) Newangle-dependent generic nDQ build-up function (two orien-tations) whose powder average corresponds to eq. (9), to beused in future modeling of nDQ build-up curves measured onanisotropic samples, compared with a powder-averaged sin2-build-up, eq. (6). Data adapted from refs. [51, 52].

inset of Fig. 4a demonstrates that deformation not onlyleads to an increase, but also to a considerable broaden-ing of the Dres distribution, reflecting an inhomogeneousensemble of chains in different stretching states.

Such a distribution is in fact expected from simplemodeling, see the inset of Fig. 4b. Depending on the ori-entations of the different network chains with respect tothe strain (represented by a “spokeswheel”), chains areeither stretched or even compressed upon deformation.A comparison with model predictions demonstrated thatthe classical models of rubber elasticity (affine, phan-tom) cannot describe the observations, thus, more elab-orate “tube” models have to be employed, which takeinto account effects of topological entanglements. The

7

mentioned comparison was possible for Dres distributions(Fig. 4a) and for nDQ intensities measured at short τDQ

as a function of Ω (Fig. 4b), and did not yet provide afully consistent picture, as different tube models are ableto capture certain features differently well [51].

So far, the analysis of whole InDQ build-up curveswas precluded by the lack of a suitable single-orientationbuild-up function representative of a protonated poly-mer segment. For 2H NMR, the situation is simple inthat eq. (6) can be used. For 1H NMR, a single-anglebuild-up function taking into account the multiple dipo-lar couplings was not known. In principle, spin dynamicssimulations can be used for that purpose [17], but thenumber of spins necessary to obtain a converged resultsis still unfeasibly large. In a recent effort, we have thusresorted to simple reverse fitting of an empirical functionwith limited number of fitting parameters, formally sim-ilar and optimized to fit eq. (9) upon powder averaging.The result is

InDQ(τDQ, Dres, θ) =

0.5(1− exp

−(0.205DresP2[cos θ]τDQ)2

× cos(0.839DresP2[cos θ]τDQ)) . (12)

Fig. 4c shows this nDQ build-up for two canonical ori-entations along with its powder average, eq. (9). Weare currently exploring the use of eq. (12) in analysesof stretched-network data, and expect further uses inother cases where polymer chains move in macroscopi-cally anisotropic environments.

Supramolecular and phase-separated networks

In supramolecular networks chemical crosslinks are re-placed by physically interacting entities, e.g. hydrogen-bonded units, phase-separated regions of chemically dis-tinct and matrix-incompatible moieties, or ionic clusters.Consequently, the operating temperature plays a veryprominent role in that thermal motion can render suchphysical links labile. Such materials can thus flow – oreven self-heal [53] – at elevated temperatures, but arepermanently elastic at lower temperatures. In terms ofthe segmental OACF C(t) (eq. (4), see also Fig. 1a),we can thus expect a decay at longer times beyond therubbery plateau, leading to an apparently temperaturedependent RDC, Dres(T ). While a quantitative under-standing for such a situation has been achieved for en-tangled polymer melts (see below), we here summarizethe results of some studies based upon this qualitativeobservation.

A first, recent study of a supramolecular self-healingrubber material consisting of low-molecular hydrogen-bonded components, combining MQ NMR with ther-mal and mechanical studies, demonstrated the use ofMQ NMR to reveal aging phenomena in the investi-

gated system. Under aging conditions actual chemi-cally linked network chains are formed in addition topurely supramolecular interactions [53]. For a phase-separated poly(urethane) elastomer with chemically re-versible links based upon the Diels-Alder reaction, InDQ

build-up curves, which are found to be temperature-dependent only above a certain critical temperature,could directly reveal the presence of thermally labile links[54].

Ionomeric rubbers, obtained for instance by ioniza-tion of carboxylic acid groups pending from a olefinicrubber main chain, are chemically rather simple, butreveal a surprisingly complex microstructure and mor-phology, related to the formation of ionic cluster struc-tures. A significant amount of main-chain segmentsis entrapped and immobilized in such structures [19].Consequently, such samples exhibit wide Dres distribu-tions, requiring the combination of different DQ pulsesequences (specifically, the three-pulse segment and theBaum-Pines sequence) to capture the full coupling range[19, 54]. The high-Dres component reflects the cluster-immobilized segments, while the low-Dres component isrelated to entropically elastic chains. Increasing the tem-peratures mainly reduces the fraction associated with theformer, corresponding to the softening of the linking re-gions, and decreases the absolute value of the latter, re-flecting the loss of elasticity and the onset of long-timeflow [19]. A similar combination of MQ NMR methodshas also been used to monitor the reverse situation ofcuring an epoxy resin, where increasing temperature in-duces cross-linking and the appearance of high-Dres com-ponents [55].

ENTANGLED POLYMER DYNAMICS

Understanding the dynamics of entangled linear chainswith high molecular weight (MW) in the melt is a keyquest in polymer physics, and a rather mature theoreti-cal understanding has been reached [45, 56, 57]. Chain-based theories can represent the complex macroscopicmechancial behavior rather well, but many features ofthese theories are yet to be confirmed on a microscopiclevel. The main feature is that long chains can relax ter-minally only by reptation motion, i.e., one-dimensional(1D) diffusion along their own contour [45, 58], which ex-plains the extraordinarily high viscosity of high polymermelts. Later refinements include the modeling of the con-straints posed by neighboring chains by a confining tubewith certain properties [56]. De Gennes in his famouspaper [58] has in fact already foreseen the possibility topredict the signature of reptation by NMR measurementof the dipolar OACF, C(t) (eq. (4), see also Fig. 1a),in terms of power-law regimes with time exponents −1/4and −1/2 for tube-constrained local fluctuations (regimeII) and larger-scale reptation (regime III), respectively

8

[59]. Based upon groundwork of Ball, Callaghan andSamulski [59], the first experimental observation of thesescaling exponents is due to Graf, Heuer and Spiess [1],who used DQ MAS NMR to directly probe C(t). Atthat time, only a single point in time was accessible dueto pulse sequence restrictions, meaning that the actualC(t) had to be constructed by time-temperature super-position [57].

Our more recent work represents an elaboration andquantification of this early work, providing full quanti-tative information on tube-confined chain motion [6, 60–62]. Static MQ NMR is perfectly suited, as one can inprinciple access C(t) over a continuum of times repre-sented by τDQ, and thus directly probe the shape of C(t).MQ NMR thus complements traditional relaxation timemeasurements, lineshape analyses or real-time exchangeNMR [63, 64] by providing direct time domain access toC(t) in the range of about 10 µs and a few ms, i.e., therange of typical τDQ.

A non-constant C(t) means that the signal functionsare to be calculated as follows [14, 59, 62]:

IDQ(τDQ) = 〈sinφ1 sinφ2〉 (13)

≈ sinh〈φ1φ2〉 exp−〈φ21〉, (14)

IΣMQ(τDQ) = 〈sinφ1 sinφ2〉+ 〈cosφ1 cosφ2〉 (15)

≈ exp〈φ1φ2〉 exp−〈φ21〉. (16)

The brackets denote an ensemble average. The keyphysics is encoded in the phases φi, obtained as

φi(ta, tb) = Deff

∫ tb

ta

P2[cos θ(t)] dt, (17)

with the interval tb− ta = τDQ. With this, φ(0, τDQ) andφ(τDQ, 2τDQ) are identified with φ1 and φ2, represent-ing the excitation and reconversion periods, respectively.This treatment neglects pulse sequency details by assum-ing the continuous action of a well-defined average Hamil-tonian. Discrete-interval effects have previously been ad-dressed and turned out to be minor [14]. Importantly,P2[cos θ(t)] is a random function in time. The (withinthe spin pair limit) exact results eqs. (13,15) can for in-stance be evaluated on the basis of dynamic simulationmodels supplying trajectories θ(t) over sufficiently longtimes, evaluating eq. (17) via a finite-interval summa-tion. This approach has been followed in previous work,relying on a simple 1D random-walk model of reptationmotion along a tube, which in itself is represented by arandom walk structure in 3D [65].

For a more theory-based evaluation, the Anderson-Weiss (second-moment) approximation [66, 67] can beemployed to arrive at eqs. (14,16), where an ensembleand time average is now to be taken over products of theindividual phases, which is a rather straightforward ex-ercise [59, 67]. The final signal functions are calculated

by simple time integrals over C(t), eq. (4), namely

〈φ21〉 =

4

9M2eff × 2

∫ τ

0

(τ − t′)C(t′) dt′ (18)

and

〈φ1φ2〉 =4

9M2eff

[∫ τ

0

t′ C(t′) dt′

+

∫ 2τ

τ

(2τ − t′)C(t′) dt′]. (19)

The dipolar second moment follows the common defini-tion M2eff = 9

20D2eff ; note that the two equations include

a correction factor (2/3)2 = 4/9 taking into account thespecific scaling factor of the most-used Baum-Pines DQpulse sequence.

A simple way to link the experimental observables withthe target quantity C(t) is a direct time-domain anal-ysis of IDQ or actually better, InDQ. Using the lattermainly corrects for transverse relaxation effects arisingfrom dynamics on timescales t < τDQ and improves thevalidity range of the inherent approximations. InDQ datafor a very high MW polymer (with >1000 entanglementsper chain) are shown in Fig. 5a. Temperature variationdemonstrates that an apparent backbone order parame-ter

Sb(T ) = Dres(T )/Dref , (20)

as encoded in the initial rise, decreases with temper-ature. In addition, relaxation at long times unac-counted for by normalization prevails, and both phe-nomena demonstrate that C(t=τDQ) is not constant intime. An approximate numerical value for the apparentRDC, Dres(T ), can be obtained by fitting InDQ(τDQ) upto τmax

DQ (where it reaches its highest value) to eq. (7),combined with an empirical exponential relaxation termexp−2τDQ/T2,app, see the dashed line in Fig. 5a. Thisis only a good strategy if InDQ reaches values exceeding0.3. Otherwise, one can use a fit to a short-time expan-sion of eq. (7), InDQ(τDQ) ≈ 0.2Dres(T )2 τ2

DQ withoutthe use of a relaxation term. In this way, one can quan-tify C(t) at a time corresponding the center of the fittedinterval, C(0.5τmax

DQ , T ) ≈ Sb(T )2.For a more detailed analysis, we rely on a short-time

approximation first published by Graf et al. [1, 69]: Ex-panding InDQ(τDQ) = 〈sinφ1 sinφ2〉 for small φ1,2 andassuming C(t) to be constant for t < τDQ, one can showthat

InDQ(τDQ) ≈ k τ2DQ C(τDQ) , (21)

which we found to be a good approximation for eitherInDQ ≤ 0.1 or up to half the τDQ it takes to reach theInDQ maximum if it is significantly lower than 0.1. Seeref. [61] on the relation between the numerical constant kin eq. (21) and Sb(T ) obtained from eq. (20). Typically,

9

(a) 0 4(a)

0.2

0.3

0.4

243 K303 K

363 K

DQ

inte

nsity

absolute-value fitDQC(t) ~ DQ

2 1.71

(b)100

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.80.0

0.1

PB2000k (N/Ne = 1100)

norm

.

validity limit of eq. (21)

DQ evolution time / ms

10-4

10-3

10-2

10-1

100

t 0.85

0

Se2 ~

1/Ne2

C(t

/e)

II

III IV

IPB2000k

t 0.29

0

10-7

10-6

10-5

s eC

CC

10-4 10-3 10-2 10-1 100 101 102 103 104 105

t/e

II

PB23kPB35k

PB87k

(c)1.0

PB87k (N/N = 45)

0 2

0.4

0.6

0.8

273 K

MQ DQ

norm

lized

inte

nsity

353 K

PB87k (N/Ne = 45)

full C(t)eqs. (23,24)

0 2 4 6 8 100.0

0.2n

DQ evolution time / ms

Figure 5: (a) InDQ build-up curves of highly entangledpolybutadiene (PB; MW given in in the sample name) at dif-ferent temperatures along with fits as described in the text.(b) C(t) for a wide range of PB samples covering all regimesof the tube model [56], combining data from MQ NMR (solidsymbols) covering the high-MW range [60] and T1 field-cyclingrelaxometry (open symbols) covering the low-MW range [68].(c) IΣMQ and IDQ signal functions for PB at two different tem-peratures [61], along with a comparison to theory [62]. Thedotted lines mark the upper end of the fitting interval to eqs.(23,24). N in the captions is the degree of polymerization,and Ne the number of monomers per entangled stand.

a power law C(t) ∝ τ−εDQ represents a good approxima-tion over a sufficiently short time. The data shown inFig. 5a representing the highly entangled limit is com-patible with ε ≈ 0.29 for all temperatures. This valueis close to de Gennes’ prediction of −1/4 for the tubemodel regime II (“local reptation”). For shorter chains,a cross-over to tube model regime III (actual reptation)and, beyond the terminal time τd, into the free-diffusionregime IV is expected. This could be confirmed quantita-

tively for several polymers over large MW ranges by con-structing C(t=τDQ) ∝ InDQ(τDQ)/τ2

DQ, measured piece-wise over short time intervals at different temperatures,and combining the data to master curves as a function ofa master variable t/τe(T ) [61]. The reference τe(T ) [56]is the MW-independent “entanglement time” that sepa-rates regime II from the Rouse regime I (locally uncon-strained chain modes), and its temperature dependenceis known.

The data in Fig. 5b presents on overview of resultsobtained, combining data from MQ NMR [60] with dataderived from T1 field-cycling relaxometry (FCR) coveringthe range of shorter times [68], i.e. the “glassy” regime0 (segmental fluctuations) and the Rouse regime I. Typi-cal polymer timescales, prominently the disentanglementtime τd(M) marking the transition between the reptationregime III and the free-diffusion regime IV, could be ex-tracted from the MQ data and were shown to be in verygood agreement with expectations from tube theory andother experiments [61]. It was in fact possible to highlightsmall but relevant deviations from the tube model pre-dictions, in particular of the exponent ε mentioned above[60, 61], providing new evidence for additional chain re-laxation mechanisms unaccounted for in the simple tubemodel [56].

Note that recent work has shown that the apparentlyvery good fit of results from MQ NMR and FCR is par-tially coincidental, as the two methods exhibit differentsensitivities to inter-chain dipole-dipole couplings [6, 70].Notably, MQ NMR results provide a direct time-domainmeasure of C(t), with comparable results for 1H and 2HNMR, the latter probing purely intra-chain dynamics,which are further in good qualitative agreement with dif-ferent polymer simulation studies [6, 71].

As C(t) for entangled polymers could thus be deter-mined in a model-free way (Fig. 5b), one can use a piece-wise analytical representation of it and feed it into eqs.(18,19), to be plugged into eqs. (14,16) for a theoreticalprediction of the signal functions [62]. The results shownin Fig. 5c provide a first-time confirmation of an ana-lytical theory for MQ NMR signal functions of entangledpolymer melts. The only free parameter in the associatedfits was M2eff , which assumed reasonable values expectedfor the static limit. It is noted that the same formalismcan also be used to quantitatively describe experimentalHahn-echo transverse relaxation curves [62].

The theory using the full C(t) is ultimately straight-forward but algebraically cumbersome, calling for possi-ble simplifications. Closer scrutiny reveals that motionsoccurring at times shorter than the shortest experimen-tally accessible τDQ only lead to a multiplicative exponen-tial decay ∼ exp−2τDQ/T2f for both signal functions,where the transverse relaxation time related to fast mo-tions T2f can just as well be fitted instead of being pre-dicted. The part of C(t) relevant for the measured datais then only the real-time interval between 2τDQmin and

10

2τDQmax. A reasonable proxy is to just assume a powerlaw [72],

C(t) =

S2

b for |t| < τ0S2

b (τ0/|t|)ε for |t| ≥ τ0, (22)

with constant time-scaling exponent ε. On this basis, fitsto IDQ/ΣMQ raw signal functions should be possible aslong as this assumption holds. If the actual C(t) deviatesfrom the assumption, a forced “best-fit” of it is obtained,with ε still providing valuable qualitative insight.

Solving the AW expressions eqs. (14,16,18,19) usingeq. (22), one obtains [14]

IDQ(τDQ) = e−

2τDQ

T2f exp

−

49M2res

(ε− 2)(ε− 1)

((ε− ε2)τ2

0

+(2ε2 − 4ε)τDQτ0 + 2τ2−εDQ τ ε0

)× sinh

49M2res

2(ε− 2)(ε− 1)

((ε2 − ε)τ2

0

+(23−ε − 4)τ2−εDQ τ εo

), (23)

IΣMQ(τDQ) = e−

2τDQ

T2f exp

−

49M2res

(ε− 2)(ε− 1)

×(

32(ε− ε2)τ2

0 + (2ε2 − 4ε)τDQτ0

+ (4− 22−ε)τ2−εDQ τ ε0

). (24)

These relations are valid under the conditions ε > 0,ε 6= 1, 2, and τ0 < τDQ. Practically, τ0 is fixed at avalue just below τDQmin. The residual second momentreflecting the RDC (thus the segmental order parameter)at τ0 is M2res = S2

bM2eff = 920D2

res. A demonstration ofthese fitting functions on the example of data from awell-entangled polymer at two different temperatures isshown in Fig. 5c, and the results are seen to be virtuallyindistinguishable from the full theory prediction. Thefit results comprise ε values (0.34±0.04 and 0.8±0.04 for273 K and 353 K, resp.) that are in good agreement withvalues expected for regime II and late-stage regime III,respectively.

This fitting approach should be of use for investigationsof a variety of more complex systems, such as supramolec-ular networks with dynamic crosslinks, or branched sys-tems. Specifically, we are currently pursuing studieswhere differences between different parts of the branchedstructures, accessible by suitable 1H vs. 2H isotropic la-beling, are in the focus.

GRAFTED CHAINS, (NANO)COMPOSITES ANDCONFINEMENT EFFECTS

The dynamics of polymer chains becomes significantlyaltered in the presence of internal solid surfaces, which

( )(a)low MW high MW

high MW, high silica content

(b)

free PEOadsorbed PEOconstant layer thickness

entropy-elasticbridging chains

hybrid hydrogels

sdre

l. D

Q in

tens

ity

sdDQ evolution time / ms

r

Figure 6: (a) Model of PEO/silica nanocomposites basedupon NMR results, where free and surface-immobilized chainscan be distinguished and quantified [76]. MQ NMR pro-vided the characterization of elastic bridging chains with well-defined but low Dres, which explain the elastic properties. (b)DQ build-up curve characterizing the adsorbed polymer frac-tion in nanocomposites of PEO with laponite platelets in wa-ter. Figure reproduced from ref. [77] with permission of theAmerican Chemical Society.

are for instance present in mixtures with (nano)particles.For hydrophilic surfaces such as –OH- and H2O-coveredoxides and polymers with polar moieties, H-bonding in-teractions usually lead to strong adsorption and immo-bilization within a layer of 1 to a few nm thickness. Thepolymer in this layer is largely immobilized, thus fea-tures high dipolar couplings, and is easily quantified by1H line shape or time-domain FID analyses [73, 74]. Incontrast, the chain parts emanating from this layer arestill highly mobile but their dynamics is constrained bythe end fixation. Further, if the chains are long enoughand the nanoparticle content is high enough, bridgingchains are formed. These network-like chains provideconnectivity and are responsible for the formation of anelastic gel. MQ NMR is ideally suited to quantify therelative amount and the relative length of these con-fined chains through their RDC, as demonstrated onmixtures of poly(dimenthyl siloxane), PDMS [75], andpoly(ethylene oxide), PEO [76], with silica spheres. Acorresponding structural model is shown in Fig. 6a.

The dynamics within the highly immobilized adsorp-tion layer can be studied by 1H time-domain techniquesbased upon the magic-sandwich echo [73, 74, 78]. In an-other recent study [77], Lorthioir et al. demonstrated theuse of (non-normalized) DQ build-up curves based upon

11

the 5-pulse sequence, suited to study stronger dipolarcouplings, to characterize the rather constrained internalmobility within the adsorption layer of PEO on clay min-eral sheets (laponite), see Fig. 6b. A semi-quantitativepicture could be derived by comparison of data taken onPEO composites with PEO in a fully amorphous modelsystem not too far above its Tg. This is in agreementwith the notion that polymers in the adsorption layerexhibit an effectively increased Tg, in fact, in-depth in-vestigations confirmed the presence of Tg gradient [74].

Similar phenomena have been revealed for an unpolarrubbery polymer (EPDM) filled with carbon black (CB),using DQ-filtered 1H FID and echo signals, and spin-diffusion experiments [79]. CB exhibits favorable inter-actions in particular to polymers with double bonds, yetthe adsorption layer is usually thinner, i.e., of the orderof 0.5 nm.

Additional crosslinking of nanoparticle-filled polymersleads to high-performance gels or elastomers, which aremechanically reinforced mainly via the formation of asecond mechanical network formed by the filler particles[80]. The RDC of the network chains easily overridesthe effect of the surface bonding, as its area density isusually low [79]. Only in cases where efficient surfacegrafting agents are used can MQ NMR provide evidenceof a locally increased crosslink density close to the surface[73]. The most common observation is in fact that thecrosslink density is reduced by filler addition, due to thepartial inactivation of the vulcanization system by ad-sorption to the high-surface filler [38, 80]. Note that thisinsight can only be gleaned from MQ NMR, which pro-vides localized information, while mechanical propertiesare dominated by the filler network, precluding quanti-tative conclusions on the properties of the rubber ma-trix. Even rather large filler particles have recently beenshown to lead to enhanced aging (mostly inhomogeneouschain scission) effects, as again quantified in detail byMQ NMR [81].

Less dense surface bonding to nanoparticles can, how-ever, be detected indirectly, by comparison of resultsfrom MQ NMR with equilibrium swelling experiments[38]. As already discussed above, both the RDC quan-tified by MQ NMR as well as the equilibrium degree ofswelling provide measures of the network chain density,with near-perfect linear correlation between the resultsof the different methods, termed the master line. If therubber chains exhibit permanent bonding to the fillerparticles, the equilibrium degree of swelling is reduceddue to the filler particles acting as giant crosslinks (ifno bonds are present, the rubber just swells away fromthe filler forming a solvent-filled vacuole). This leadsto apparently high network chain densities from swellingand to deviations from the master line, the extent ofwhich characterizes the rubber-filler bonds. This methodhas been established for NR containing different types offillers [38], and has in the meantime been applied to filled

(a)l t b

B0sample tube 0

polymer

AAO

slow exchange

0.10

0.15

Q(0

.5m

s)

0°90°

(b)

fast exchange

1 10 100 kg/mol 0.00

0.05I nD

MW

bulk

hs

surface-induced i t ti

sorientation

Figure 7: (a) Schematic sample set-up for orientation-dependent NMR measurements of polymers infiltrated intonanopores in stacked AAO membranes. (b) Short-time single-point nDQ intensities at τDQ=0.5 ms, reflecting the averageapparent RDC, as a function of MW for PB confined to 20 nmpores at 383K and 0 and 90 nominal orientation, and corre-sponding model explaining the two MW domains with quali-tatively different angle dependence. Figure reproduced fromref. [65].

styrene-butadiene rubber [82], and other diene elastomers[36].

Block copolymers, with phase-separated nanometer-scale soft and hard domains [83], the latter formed bye.g. glassy poly(styrene), represent another case of con-fined motion of grafted mobile chains in the soft do-main. For the case of a diblock copolymer, with onlyone end of the mobile chain fixed, one does not necessar-ily expect anisotropic chain motion. However, anisotropyarises from the ordered environment represented by the,e.g. lamellar, nanodomain. In addition, it is known thatthe chains in such systems are stretched along the inter-face normal [83]. These phenomena have recently beenstudied by 2H DQ NMR [5, 84], revealing a substantialRDC distribution for the mobile chains in a lamellar di-block copolymer. Notably, a clearly detectable fraction ofisotropically mobile parts, attributed to a location closeto the lamelar mid plane, was identified for a fully deuter-ated [5] and an end-deuterated mobile block [84]. Thesefindings are mutually consistent and emphasize that themobile chain ends are distributed throughout the soft do-main [84].

Finally, we address the influence of a neutral, non-

12

interacting wall on polymer chain dynamics. Well-dispersed nanocomposites fulfilling this requirement arenot easy to prepare due to miscibility issues, which iswhy in many cases ordered assemblies of thin-film sam-ples have been studied. Alternatively, infiltration of anunpolar polymer into an ordered porous oxide structureis particularly attractive, as samples with high filling fac-tor amenable for NMR analyses can be prepared, retain-ing a well-ordered arrangement. Our previous work hasfocused on anodic aluminum oxide (AAO) membranesfeaturing a parallel arrangement of channels with diam-eters in the 20–100 nm range filled with PB [65, 85, 86],see Fig. 7a for a sketch of an ordered sample arrange-ment allowing for orientation-dependent measurements.Our initial work on powder samples suggested a signifi-cant wall-induced enhancement of entanglement-relatedlocal segmental orientation as reflected by the averageRDC [85], complemented again by T1 FCR studies high-lighting qualitative changes in the faster (Rouse) modes[86].

Angle-dependent MQ experiments provided morequantitative insights [65]. Extending the study to un-entangled short chains, we found that in this limit theaverage RDC tensor of the polymer segments is orientedparallel to the pore axis, with an angle dependence fol-lowing the one predicted by eq. (12). This is exemplifiedby the data in Fig. 7b for MW below 10 kg/mol, and isexplained by a fast diffusive average, where the orienta-tion is merely picked up by transient wall contacts. In aseparate study, we confirmed that the diffusivity of suchshort chains is only weakly slowed down by the wall con-tacts [87], meaning that at sufficiently high temperaturethe chains diffuse a distance of many pore diameters onthe millisecond timescale of the DQ build-up.

The angle dependence of the nDQ intensity (thus ofthe apparent RDC) nearly vanishes at higher MW, whichis compatible with the notion that for the highest MW,due to slower diffusion, an isotropic bulk-like core re-gion coexists with an interphase layer, within which morelocalized segmental motions lead to a tensor orienta-tion perpendicular to the wall. In the transition region,intermediate-motional relaxation effects play a relevantrole. These features have been confirmed by a simulationmodel [65], where following eqs. (13,15,17) signal func-tions were generated on the basis of finite-interval trajec-tories of the local segmental orientaion θ(t). The latterwere generated via a simple 1D random-walk model ofreptation motion along a random 3D tube structure con-fined to a pore, assigning appropriate tensor orientationsto tube parts in the core region (where entanglementsdominate) and in the near-wall region (where surface-induced orientation prevails).

SUMMARY

We have reviewed the basic principles of multiple-quantum NMR as applied to 1H or also 2H nuclei for theelucidation of polymer chain dynamics, covering a rangeof applications on complex sample systems as well asnew methodological developments. Improved data anal-ysis concepts, including a new single-coupling double-quantum build-up function representing a protonatedmain-chain segment, and a simultaneous fit of the dif-ferent signal functions, enhanced the study of structuralinhomogeneity and defects in different kinds of networks.Data taken on partially or equilibrium swollen elastomersallow for the quantitative study of the related chain-stretching process and the magnitude of entanglementconstraints active in the bulk. Performing and ana-lyzing orientation-dependent experiments in macroscop-ically ordered samples enables insights into molecular-scale deformation processes within elastomers as wellas the motion of chain segments in anisotropic environ-ments.

The complex time and thus temperature dependenceof the NMR observables in systems exhibiting slowdynamics in the µs–ms range, such as bond-openingin supramolecular elastomers or reptation in entangledmelts, can be tacked by different kinds of approxima-tions and theoretical approaches. Anderson-Weiss the-ory was shown to provide a powerful framework forextracting information on the shape of the orientationautocorrelation function of polymer segments. Finally,it was demonstrated that the combination of differ-ent pulse sequences, namely the three-pulse segmentand the compensated Baum-Pines sequence for a focuson large and small (residual) dipolar couplings, respec-tively, should be considered to cover the wide range ofcouplings present in polymers close to and far abovetheir respective glass transition temperature, respec-tively. This includes (nano)phase-separated systems and(nano)composites featuring surface-adsorbed polymers,in which polymer segments with widely different mobilitycoexist.

† Electronic address: kay.saalwaechter@physik.

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