Solid State Nuclear Magnetic Resonance 72 (2015) 50–63

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Solid State Nuclear Magnetic Resonance

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journal homepage: www.elsevier.com/locate/ssnmr

Basic principles of static proton low-resolution spin diffusion NMR innanophase-separated materials with mobility contrast$

Kerstin Schäler, Matthias Roos, Peter Micke 1, Yury Golitsyn, Anne Seidlitz,Thomas Thurn-Albrecht, Horst Schneider, Günter Hempel, Kay Saalwächter n

Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle, Germany

a r t i c l e i n f o

Article history:Received 13 July 2015Received in revised form28 August 2015Accepted 1 September 2015Available online 3 September 2015

keywords:Low-field NMRSemicrystalline polymersPolymer crystallizationPolymer dynamicsNanophase separationPoly(ϵ-caprolactone)Block copolymers

x.doi.org/10.1016/j.ssnmr.2015.09.00140/& 2015 Elsevier Inc. All rights reserved.

ial issue 40 yrs CPMAS & 25 yrs REDOR honoesponding author.ail address: kay.saalwaechter@physik.uni-hall: http://www.physik.uni-halle.de/nmr (K. Saalesent address: Max-Planck-Institut für Kerneidelberg, Germany.

a b s t r a c t

We review basic principles of low-resolution proton NMR spin diffusion experiments, relying on mobilitydifferences in nm-sized phases of inhomogeneous organic materials such as block-co- or semicrystallinepolymers. They are of use for estimates of domain sizes and insights into nanometric dynamic in-homogeneities. Experimental procedures and limitations of mobility-based signal decomposition/filter-ing prior to spin diffusion are addressed on the example of as yet unpublished data on semicrystallinepoly(ϵ-caprolactone), PCL. Specifically, we discuss technical aspects of the quantitative, dead-time freedetection of rigid-domain signals by aid of the magic-sandwich echo (MSE), and magic-and-polarization-echo (MAPE) and double-quantum (DQ) magnetization filters to select rigid and mobile components,respectively. Such filters are of general use in reliable fitting approaches for phase composition de-terminations. Spin diffusion studies at low field using benchtop instruments are challenged by rathershort 1H T1 relaxation times, which calls for simulation-based analyses. Applying these, in combinationwith domain sizes as determined by small-angle X-ray scattering, we have determined spin diffusioncoefficients D for PCL (0.34, 0.19 and 0.032 nm2/ms for crystalline, interphase and amorphous parts,respectively). We further address thermal-history effects related to secondary crystallization. Finally, thestate of knowledge concerning the connection between D values determined locally at the atomic level,using 13C detection and CP- or REDOR-based “1H hole burning” procedures, and those obtained by ca-libration experiments, is summarized. Specifically, the non-trivial dependence of D on the magic-anglespinning (MAS) frequency, with a minimum under static and a local maximum under moderate-MASconditions, is highlighted.

& 2015 Elsevier Inc. All rights reserved.

1. Introduction

Proton spin diffusion experiments have been used widely inestimations of nanometer scale domain sizes in organic materials,in particular in polymers [1–16]. Such experiments are often per-formed under magic-angle spinning (MAS) conditions involvingcross-polarization (CP) transfer for protons to 13C nuclei [17] inorder to benefit from the high site resolution. Yet, their muchsimpler single-channel implementation under static low-resolu-tion conditions is often sufficient in those cases where the do-mains differ appreciably in their molecular dynamics.

One of the first applications of static proton-only time-domain

ring Jack Schaefer.

e.de (K. Saalwächter).wächter).physik, Saupfercheckweg 1,

(TD) spin diffusion NMR based upon mobility filtering using theclassic Goldman–Shen filter [18] is due to Assink [1]. The principleof such experiments is highlighted in Fig. 1a, where the re-equi-libration after filtering for mobile-phase material is sketched, asdetected by TD free-induction decay (FID) signals. Different mag-netization filters have been developed to select either mobile-phase [3,6,18] or rigid-phase [2,9,12] signals. In favorable cases,the phase-selective magnetization data as a function of spin dif-fusion time diffτ can then be analyzed on the basis of a simpleinitial-slope analysis [4,5,10].

The proton FID and its corresponding Fourier-transformedspectrum is fully dominated by effects of dipole–dipole couplings,which are field-independent. Therefore, mobility-selective ex-periments can be performed on cost-efficient and robust low-fieldequipment rather than relying on high-field instruments withsuperconducting magnets. Low-field NMR is very popular in manyareas of applied research [19], such as for instance food science[20], polymer or pharmaceutical technology [21]. However,systematic applications of spin diffusion NMR for domain-size

rd I( diff)

selection detection

MSEspin diffusion

difffilter

Mz Mz Mz Mz Mz

Mx Mx

t t

Mx

t

Mx

t

Mx

t

selection spin diffusion…

τ τ

Fig. 1. Principle of 1H spin diffusion experiments based upon mobility filtering.(a) Magnetization profiles, adapted from Ref. [4], and corresponding FID signals;(b) schematic pulse sequence (rd: recycle delay, MSE: magic-sandwich echo).

Fig. 2. (a) Typical proton spectra of an isotropically mobile (liquid, dotted line) anda rigid solid material. The latter is represented by a Pake spectrum for a proton pair(r¼1.8 Å, solid line) and the Fourier transformation of the Abragam function, Eq.(1). (b) Time-domain signals of poly(ethyl acrylate) at different temperatures,showing the typical features of dipolar averaging upon crossing the “NMR glasstransition” at around T 40 Kg + . Solid lines are fits to Eq. (3) with resulting apparentcorrelation times of the α process given in the legend, and thin dotted lines are fitsto Eq. (2) with exponent ν varying between about 2 (rigid limit) and 1. Data in(b) are taken from Ref. [24].

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–63 51

estimation performed on low-field instruments [13–16] are sur-prisingly rare. The reason is probably that T1 relaxation times canbe quite short at low field (sometimes only a few tens of ms at0.5 T corresponding to 20 MHz 1H Larmor frequency), whichmeans that the spin diffusion process competes with T1 relaxation.An analytical treatment of spin diffusion [5,12] is then not possibleany more, and one has to resort to numerical, simulation-basedfitting approaches [2,8,11,14,22]. Hedesiu et al. [13] were appar-ently the first to compare results from high- and low-field in-struments, and found very good agreement despite neglecting T1issues, which is likely due to the small domain size in the in-vestigated system. For larger domain sizes, T1 effects may in factalready become non-negligible even on 200 MHz instruments[8,11,22].

In this contribution, we review and elaborate on fundamentaland technical aspects of low-field mobility discrimination and spindiffusion experiments on the example of as yet unpublished datafor poly(ϵ-caprolactone) (PCL), a semicrystalline engineeringpolymer with favorable properties such as medium crystallinity,well-defined lamellar morphology and convenient melting pointof around 60 °C. We first address the fundamental connectionbetween molecular mobility and proton TD signals, and describepulse sequences appropriate for low-field instruments.

In Part 1, we then present a detailed account on mobility-basedsignal decomposition and filtering, focusing on the magic-sand-wich echo pulse sequence (MSE) [23] in its use to overcome thedead time issue and to select mobile-phase magnetization [6], andthe double-quantum (DQ) filter [9,12] to select rigid-phase mag-netization. Special emphasis is put on the inevitable dynamic in-terphase and robust multi-parameter fitting, addressing aniso-tropic bias effects in powder samples. In Part 2, we describe low-field spin diffusion experiments on PCL in combination with si-mulation-based fitting approaches to obtain spin diffusion coeffi-cients D, relying on domain sizes determined by small-angle X-rayscattering (SAXS) experiments. We also address effects of thermalhistory related to secondary crystallization, and stress the ne-cessity to consider a more sophisticated, island-like structure ofthe interphase to account for the experimental data. Part 3 reviews13C-based 1H magnetization hole burning experiments for thedetermination of D values at a very local scale, and their applica-tion to assess differences in D values under static vs. MASconditions.

2. Principles and experimental implementations

2.1. Proton time-domain signals of bulk polymers

The typical rigid-limit Pake spectrum of a dipolar-coupledproton pair is shown in Fig. 2a as the solid line. Its width, ascharacterized by the separation of the “horns”, is 1.5 times thedipole–dipole coupling constant D /2HH π (the unit of DHH is rad/s).In multi-proton system, this feature is of course washed out, but inmaterials with abundant methylene (CH2) groups, some of the paircharacter prevails. The TD (FID) signal can then be well describedby the so-called Abragam function,

I t e b t b tsin / , 1a tAbr

/22( ) = ( ) ( ) ( )−( )

whose Fourier transform is shown in Fig. 1a as the dashed line. Thesecond moment of this spectrum depends upon the two shapeparameters a and b: M a b /32

2 2= + [25]. As will be shown below,the major part of the TD decay is also well represented by a simpleGaussian. In contrast, mobile organic matter such as isotropic

Fig. 3. Pulse sequences: (a) FID detection, (b) full FID detection by MSE refocusing, (c) DQ filter for rigid signal components including a phase-supercycled composite-pulse zinversion, (d) MAPE filter for mobile signal components including a phase-supercycled z inversion. Solid bars are 90° pulses, the dashed boxes are optional 180° pulses. Thephase cycle for the MSE is x x1φ = { }, y y y y x x x x2φ = { }, x x x x y y y y3φ = { }, x x x x x x x xrecφ = { }. The DQ selection phase supercycle is x y x yDQφ = { }, inverting thereceiver for every other scan.

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–6352

liquids or also polymers far above the glass transition temperatureTg( ), display slowly, near-exponentially decaying FIDs, corre-sponding to a Lorentzian-type spectrum. A generic function thatfits the TD signal of almost any homogeneous material is thestretched exponential, often referred to as Kohlrausch–Williams–Watts (KWW) function,

I t e . 2t TKWW

/ 2( ) = ( )−( )β

Here, T2 is an apparent relaxation (decay) time. The exponent βcan vary between values significantly below 1 (in which case itindicates an inhomogeneous distribution of exponential decays),to 2. Note that for β values between 1 and the rigid-solid Gaussianlimit value of 2, the function is often referred to as a compressedexponential or Weibullian, but for simplicity, we will always referto it as KWW.

Typical TD signals of a polymer undergoing a glass transition attemperatures spanning the rigid to mobile limits are shown inFig. 2b. They reflect a “motional narrowing” phenomenon (refer-ring to spectra rather than TD signals), arising from an increasinglycomplete, near isotropic average of the dipolar coupling tensor.Single KWW functions shown as thin dotted lines, with exponentsvarying from 2 to around 1, are able to represent the data ratherwell over the whole range, which emphasizes its use as a

component function in fits to data of dynamically inhomogeneoussystems.

For a semi-quantitative analysis, one can use the theory ofAnderson and Weiss [26] to obtain an analytical (second-moment)approximation of a dipolar FID for a spin pair:

⎡⎣⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥

I t t t

M et

cos d

exp 13

t

t

AW

2 c2 /

c

c

∫ ω θ

ττ

( ) = ( ( ′)) ′

≈ − + −( )

τ

τ

τ

τ

+

−

The dipolar frequency tω θ( ( )) is proportional to D P tcosHH 2 θ× ( ( )),and the orientation angle tθ ( ) of the internuclear vector is a ran-dom function in time. For the simple case of isotropic rotationaldiffusion, for which the autocorrelation function of P tcos2 θ( ( )) is asimple exponential texp / cτ∼ [ − ], standard procedures [27] can beused to arrive at the above analytical result, where the secondmoment M D9/202 HH

2≈ ( ) is related to the largest dipole–dipole paircouplings DHH in the system. Note that for cτ → ∞ (rigid limit), Eq.(3) reduces to I t M texp /2Gauss 2

2( ) = [ − ]. Eq. (3) also provides verysatisfactory fits to data of homogeneous organic solids (neglectinghere the issue of significant dynamic heterogeneities close to Tg).

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–63 53

Note that the most significant changes occur at about T 40 Kg + ,which is because the α relaxation time of the glass process needsto reach values of around D1/ HH, which of the order of 10 μs ratherthan the typical 100 s near Tg.

As can be seen in Fig. 2b, Eq. (3) also describes the data ratherwell. It captures the very initial part somewhat better than theKWW function, and is rather similar to it over most of the interval.Therefore, TD signals of organic solids with intrinsic dynamic in-homogeneities can always be analyzed with a suitable (minimal)number of components, f I ti

ni1 i∑ ( )= , where KWW, or, if indicated

by the data, an Abragam function for the most rigid component,should be used as component functions. As a very importanttechnical remark, we stress that TD data, in particular those takenat low-field instruments with significant magnetic field (B0) in-homogeneity, should never be fitted for times exceeding about200 μs, as shim limitations (or chemical-shift effects at higherfield) render the fits invalid. For longer times, Hahn-echo decaycurves are preferably analyzed.

2.2. Pulse sequences

Pulse sequences suitable for low-field instruments are shown inFig. 3. See also our previous Ref. [14] for a detailed account. Thesimple FID signal shown in Fig. 3a typically suffers from problemsdue to the significant receiver dead time recτ( ), which is of the orderof 10 15 s… μ on most low-field instruments. This covers much of arigid-component decay. The problem can be overcome by a suitablespin echo, where for dipolar couplings (rather than shift or in-homogeneity effects) the solid echo acq90 90x yτ τ( ° − − ° − − )± is themost simple, but in multi-spin systems not very effective solution(vide infra). In our hands, the magic-sandwich echo (MSE) [23]shown in Fig. 3b represents the best choice. Its most efficient im-plementation should nominally be the one employing two con-tiguous phase-inverted n�360° burst pulses of length 2τ each [23].However, the inevitable phase-switching delay τφ, which is of theorder of 1–2 μs on most low-field instruments, renders a pulsedversion with explicitly programmed τφ more efficient and robust[14]. Higher values of τφ exceeding 1–2 μs challenge the efficiency ofthe MSE (vide infra). Note that the phase of its last pulse includes acomposite 180° inversion, termed mixed MSE [28].

Parts (c) and (d) of Fig. 3 show pulse sequences for magneti-zation selection for rigid or mobile parts using a double-quantum(DQ) [9,12] or a magic-and-polarization-echo (MAPE) filter [6,14],respectively. These precede the MSE-FID sequence. For spin dif-fusion experiments they are separated by a variable spin diffusiontime diffτ . Both filters include a phase-cycle controlled magneti-zation inversion, which compensates for T1 relaxation artifactsduring diffτ and further removes unwanted coherences. The DQfilter relies on the fact that for short excitation times DQτ , DQ co-herences can only be excited (and later reconverted to observablemagnetization) in the presence of strong and time-stable paircouplings DHH. We note that the refocusing 180° pulses are onlynecessary if DQ evolution times DQτ beyond about 100 μs are ofinterest.

The MAPE filter is in fact nothing else than an MSE applied to zmagnetization. Its performance is very similar to Schmidt-Rohr's12-pulse sequence [3], and both avoid some artifacts related tomultiple-quantum coherences inherent to the classical Goldman–Shen two-pulse filter [18]. The main difference between the MSEand MAPE sequences is that the former should employ as-short-as-possible τφ, with a possible length incrementation by using oneor two loops (see Fig. 3b) for maximum efficiency, while the MAPErelies on the breakdown of the refocusing efficiency for strongdipolar couplings, which is achieved by incrementing τφ. See belowfor an experimental demonstration.

2.3. Experimental details

Most experiments were performed on different Bruker minispecmq20 instruments ( B 0.5 T0 ≈ , 20 MHz proton Larmor frequency)equipped with wide temperature range static probes operating witha flow of heated (or cooled) air. Note that the rf coil is woundaround the sample dewar, meaning that temperature effects on thedetection system are minimized. This enables precise Curie cor-rection by multiplication with T T/ ref of spectral intensities, providedthat fully relaxed signals are acquired. This was always ensured byvariation of the recycle delay. The instruments feature 90° pulselengths of around 2 μs, dead times between 11 and 14 μs, and adwell time of 0.4 μs. All time-domain NMR signals were recordedon-resonant and in full-absorption mode receiver setting, analyzingonly the real part of the complex time-domain signal. The sampleswere placed in evacuated and flame-sealed NMR tubes. Im-portantly, the sample height should never exceed about 8 mm inorder to avoid problems with B1 inhomogeneity. Only the data inFig. 5b have been recorded on a 200 MHz Bruker Avance III in-strument with a low-dead time (∼3 μs) static probe, with similarother parameters as above.

Small-angle X-ray scattering (SAXS) investigations of the do-main sizes (amorphous thickness da, crystalline thickness dc andlong period L d da c= + ) were performed following previouslypublished procedures. See e.g. our previous work [16,29]. The re-sulting “linear” volume crystallinity is f d L/c c= .

Experiments were performed at different temperatures aboveroom temperature (mostly 30 °C or 40 °C) on an industrial sampleof poly(ϵ-caprolactone) (PCL) of M 42.5 kg/moln = and PDE1.5(Sigma Aldrich), crystallized isothermally at 45 °C. Following ourprevious work [14,30], the comparison of 1H NMR signal fractionsand actual dimensions from SAXS requires a correction by therelative proton densities of the different phases. For instance, at27 °C the densities of PCL amount to 1.137 and 1.075 g/cm3 in thecrystalline and amorphous domains, respectively [31]. The weightfraction of protons in PCL is 0.088. As an example to illustrate theeffect of temperature, the relative proton density of crystalline vs.amorphous domains increases from 1.057 at 27 °C to 1.065 at45 °C, reflecting the larger thermal expansion coefficient of theamorphous phase.

3. Part 1: Mobility-based signal decomposition and filtering

Free component fitting in PCL: Semicrystalline PCL, with its pair-dominated dipolar response due to the predominance of CH2

groups and the use of the Abragam function for the crystallinepart, represents a favorable showcase for FID component decom-position. We will first demonstrate that, in this special case (whichincludes similar CH2-based polymers such as polyethylene), it ispossible to achieve reliable component decomposition evenwithout the aid of filter experiments. The latter are, however, in-evitable for cases involving e.g. glassy PS, which exhibits a Gaus-sian FID signal. Then, multi-parameter fitting becomes unstable,such that the shape parameters of one or two components need tobe pre-determined. This procedure, and some of its specificitiesand limitations, will also be discussed on the example of PCL aswell as a PS-based diblock copolymer.

Fig. 4a shows FID signals of semicrystalline and molten PCL at30 and 90 °C, respectively. The latter can be used to obtain a re-liable measure of the total sample magnetization by linear back-extrapolation of the FID despite the receiver dead time recτ , ofcourse using Curie correction to render the FID intensities com-parable. In this way, the first point of the FID at 30 °C is fixed, and a3-component fit using a linear combination of the Abragamfunction, Eq. (1), and two KWW functions, Eq. (2), is stable even

0.00 0.05 0.10 0.15 0.20

0.0

0.2

0.4

0.6

0.8

1.0

amorphous-phase signal (fa = 41.0 %)

interphase signal (fi = 10.1 %)

norm

. int

ensi

ty

acquisition time / ms

measured FID:(normalized/Curie-corrected)

melt (T = 90°C)semi-crystalline (T = 30°C)

crystalline signal (fc = 48.9 %)

240 280 3200.0

0.2

0.4

0.6

0.8

1.0

sign

al fr

actio

n f

T / K

TgDSC

TgNMR Tf

cryst. (fc)rigid-am. int. (fi)mobile-am. (fa)

240 280 3200.0

0.1

0.2

0.3

0.4

0.5

T / K

T* 2a

/

ms

Fig. 4. (a) FID signals of molten (grey) and crystalline (black) PCL with fitted signalcomponents, the crystalline fraction being represented by the Abragam function,Eq. (1). (b) Signal fractions of PCL and (c) apparent relaxation time of the mobileamorphous fraction as a function of temperature.

Fig. 5. (a) Comparison of PCL signal functions, demonstrating the effects of MSErefocusing and magnetization filters. (b) Decay of echo intensities, comparing thecommon solid echo with different time- and cycle-incremented variants of theMSE. The solid lines are power-law fits ( c a t− α). (c) Effect of MAPE filtering ofincreasing duration on the FID (note the dead time). The inset shows a fit of thefiltered mobile amorphous component to Eq. (2).

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–6354

though a significant part of the initial FID is not detected. Includingthe crystalline, interphase and amorphous fractions fc/i/a(amounting to two independent fit parameters, as they sum up to1), the resulting free 8-parameter fit is possible because theparameters a and b in Eq. (1) are robustly determined from theoscillatory feature.

Stable fitting is even possible upon cooling the sample towardsTg of the amorphous phase, as demonstrated by the data in Fig. 4band c. The crystalline fraction (Fig. 4b) apparently increases onlysomewhat below the “NMR Tg ” (at which 10 sτ ≈ μα ), while theapparent T2a

⁎ relaxation time of the mobile phase drops to values ofaround 50 μs (Fig. 4c). At the same time, the associated fa de-creases apparently, while the fi increases. This is due to the lack ofdynamic contrast between these two phases, so it cannot be ex-pected that they can still be distinguished. Nevertheless, as long assome dynamics on the dipolar-coupling timescale exists in theother phases to render them distinguishable, the crystalline part isfaithfully fitted thanks to the oscillatory feature due to the rigidCH2 groups.

MSE-based signal refocusing: The above component decom-position can be verified by use of the different refocusing and filterexperiments, the results of which are shown in Fig. 5a. From thecomparison of the FID of PCL with its MSE-detected counterpart,overcoming recτ , it is seen that the short MSE is near-quantitative,with only some minor signal loss in the initial part amounting toabout 5–10% of the total signal. The quality of MSE refocusing is

demonstrated in Fig. 5b, for which a high-field instrument withshorter recτ and shorter phase switching delays has been used toalso reveal the short-time behavior. It is seen that the popularsolid echo refocusing is only acceptable for echo delays of theorder of 2 6 srecτ ≈ μ . The two MSE versions with length variationby nMSE loop incrementation (with or without the dashed bracketsin Fig. 3b) perform rather similarly, and the decay, relating mostlyto the crystalline fraction, has a near-Gaussian shape t2( ∼ − ).This suggests that imperfection-related corrections to the MSE

average Hamiltonian (H H 0MSE dip^ = = as leading term) are of sec-

ond order. The main imperfection is here the finite sequence cycletime, which is of the order of but not much shorter than D1/ HH

[23].On the minispec, with a phase switching delay τφ and 90° pulse

lengths both around 2 μs, the minimum overall MSE duration 6 τis around 80 μs, sufficient to overcome recτ τ= = of 13 μs. This ex-plains the small but perceptible intensity loss. It should be notedthat the echo position (zero acquisition time) of the MSE on theminispec is not exactly at the theoretically expected position (seeFig. 3b), but is shifted to earlier times by about 2 μs per cycle(nMSE). This is likely due to other imperfection-related corrections

to HMSE^ related to imperfect pulses, and must be accounted for to

Fig. 6. DQ-filtered MSE-detected signal functions for PCL (a) for increasing DQexcitation time, the inset showing the evolution of maximum intensity, (b) for afixed DQτ of 16 μs with two different fits using an Abragam function, Eq. (1), andone or two additional KWW minority components, Eq. (2). The inset shows thecorresponding spectra, including the decomposed signals.

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–63 55

compare data on a common time axis. It is most easily quantifiedby locating the minimum of the dipolar oscillation feature, ifpresent, or by applying a Gaussian fit with variable time shift

M t texp /222( ∼ [ − ( − Δ ) ]) to the initial decay.

A decay of the MSE on top of the above imperfection-relatedeffects (which are proportional to M2) can further be due to in-termediate-regime dynamic processes changing the orientation ofthe dipolar tensor during the MSE duration. A theoretical accountof this T2 effect was published by Demco and coworkers [32], againon the basis of Anderson–Weiss theory [26]. As a consequence, thenormalized MSE-refocused FID data in Fig. 2b in fact go through aminimum of the absolute intensity (thus of T2) when crossing the“NMR Tg”, see Ref. [24]. We have recently used a more generalversion of the theory to detect chain flips (“helical jumps”) inpolyethylene crystallites [33,34]. Note that PCL does not exhibitsignificant intra-crystalline dynamics; its crystalline part thusqualifies as a reference for a rigid solid [30].

MAPE dipolar filter for mobile components: For the crystalline(rigid) regions, both a solid echo as well as the τφ-incrementedMSE decay much more quickly than nMSE-incremented MSEs, withapproximate t3− time scaling (see Fig. 5b). This indicates the re-levance of dephasing due to higher-order correction terms. Thisbreakdown of refocusing efficiency, resulting from the many non-commuting dipolar pair Hamiltonians in a multi-proton system,constitutes the function of the Goldman–Shen [18], Schmidt-Rohr12-pulse [3], and MAPE [6,14] dipolar filters. Applying the MSE inthe form of a MAPE (Fig. 3c) prior to the 90° excitation/detectionpulse [14], we can study its efficiency as a function of total length,see Fig. 5c. For PCL, it was found that a τφ of 37 μs (a total MAPEduration of 6 600 sMτ = μ ) results in a virtually complete, artefact-free suppression of the crystalline component, while affecting thelong-time intensity (FID at 200 μs) only rather little.

We generally recommend stabilizing multi-component fits at agiven temperature by fixing the shape parameters of the so-ob-tained mobile component (T2a, aβ ). For PCL this affords similar butnot exactly the same result as the free fit (Fig. 4). This is becausethe mobile-component selection is a priori operational in a sensethat there is some ambiguity in choosing the optimal filter times(see Fig. 5c). It is clear that even a three-component model is asimplification for a system inwhich the interphase is certainly bestcharacterized by a smooth mobility gradient [35]. Modeling theinterphase plus mobile signal with two distinct components ismerely a minimal model. In other words, the dipolar filter timeand the resulting choice of mobile shape parameters will have anindirect influence on the best-fit shape parameters for the inter-phase. This must be kept in mind in interpretations.

DQ filter for rigid components: Fig. 6a demonstrates the effect ofthe DQ filter shown in Fig. 3c for different evolution times DQτ . Thebuild-up curve of the initial intensities in the inset reveals amaximum at 16 sDQτ ≈ μ , which is expected for strongly dipolar-coupled 1H pairs of the crystallites. The shape of the correspondingTD signals is seen to vary significantly for different DQτ . This is avery important bias effect that has two origins.

First and more importantly, very short DQτ favor 1H pairs or-iented along the B0 field. This biases the normally isotropic powderaverage, and the fitted shape parameters can thus not be expectedto work well for a signal detected after a 90° pulse. Second, someinterphase and even mobile-phase signal may also be excited withof course rather small efficiency due to the much weaker dipolarcouplings in these phases. This is demonstrated by the componentdecomposition results for the DQ-filtered signal in Fig. 6b. For-tunately, choosing 13 sDQτ = μ reduces the additional componentsto below 10% while yielding Abragamian shape parameters thatcorrespond well to the isotropically excited signal. The latter isdemonstrated in the following.

Fig. 7 shows data for PCL as compared to a data for a lamellarpoly(styrene)–poly(butadiene) (PS–PB) block copolymer, in whichthe PS domain is glassy amorphous (thus rigid) at room tem-perature. The DQ-filtered MSE-detected FIDs in (a) show differentand in both cases characteristic changes as a function of a shortspin diffusion delay diffτ after the filter. For PCL, we mainly observea slight growth of interphase signal while retaining the initialdecay rate (that dominates the shape parameters a and b of theAbragamian and thus M2). For PS, which is characterized by aGaussian rather than an Abragamian rigid-phase signal, we alsosee a growth of interphase, but in addition the initial decay rate(thus M2) decreases notably.

This is quantified by the M2 data in Fig. 7b as a function of diffτfor different DQ filter times. For PCL the data do depend on DQτ (asa consequence of the powder bias effect), but do not vary uponincreasing diffτ . For PS, the powder-biased initial M2 values allconverge to the isotropic powder value within about 0.5 ms of spindiffusion. This is because different pair orientations are close toeach other in an amorphous glass (see inset). For PCL, this wouldrequire magnetization exchange beyond a given crystallite, which

Fig. 7. Effect of DQ filtration time (powder subensemble selection) on the shape of the filtered signal and effect of a small spin diffusion delay for the examples of PCL (left)vs. the rigid (PS) fraction of a lamellar amorphous PS–PB block copolymer (right). (a) MSE-refocused DQ-filtered FIDs for a fixed DQτ and variable diffτ , (b) shape of the FIDs ascharacterized by the second moment fitted from the initial decay for different DQτ as a function of diffτ . The qualitative difference between the semicrystalline andamorphous cases can be explained by the different morphologies as shown in the insets, where for the semicrystalline case different crystallites have different FID signals,corresponding to their orientation, that cannot be equilibrated by a short spin diffusion delay. The latter is easily possible for amorphous PS.

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–6356

is clearly unfeasible. Fortunately, it is also seen that 13 sDQτ = μgives almost the same M2 as the isotropic powder value.

Coming back to the comparison of full and filtered signals inFig. 5a, it must be stressed that the DQ-filtered signal in that plot isthe only one that is not to scale. DQ filtering is not quantitative in asense that a maximum of around 50% of the signal of a dipolar-coupled system can be filtered through DQ coherences, and that itis additionally subject to relaxation even at moderate DQτ [36]. TheDQ-filtered signal in Fig. 5c has therefore been scaled to its am-plitude fitted from the full PCL signal.

In summary, it is demonstrated that the choice of DQτ is veryrelevant in determining the right shape parameters (a and b of anAbragamian or M2 of a Gaussian) for the rigid phase. This is ofparticular importance for cases such as PS, which do not benefitfrom the stable full-signal fitting that is possible with an Abraga-mian. In such cases, the multi-component fit requires fixed shapeparameters for the rigid and mobile components [14].

Filtering in side-chain polymers: As a final remark, it is notedthat component selection is also possible rather locally for, e.g.,polymers with rather rigid main chains and dynamically de-coupled, more mobile side chains, such as poly(n-alkyl metha-crylates). In such a case, magnetization filters can be appliedsuccessfully, and spin diffusion experiments can be performed, butthese will not necessarily reflect the length scale of a possiblyphase-separated side-chain domain [37], but be governed bymagnetization transfer via the nuclear Overhauser effect (NOE). In

this case, the experiment rather reports on the timescale of side-chain dynamics [37].

4. Part 2: Spin diffusion experiments and calibration of spindiffusion coefficients

We will now turn to applying the above discussed magnetiza-tion filters in actual spin diffusion experiments, specifically, usingthese to obtain spin diffusion coefficients D for PCL. These back upthe values that we have reported recently [16], but for which wehave not published the details of the underlying systematic study.Following the experimental scheme in Fig. 1b, the filters are fol-lowed by a variable diffτ ranging between about 100 μs and a few100 ms. The ultimate limitation is here the shortest T1 relaxationtime in the system. Generally, MSE-detected FID data as a functionof diffτ are fitted with all shape parameters fixed, as determinedfrom filtered (rigid and mobile) and unfiltered signals. The latterprovide the parameters for the interphase. Thus, the diffτ -depen-dent magnetization component fractions fx diffτ( ) are the only freefit parameters in such an experiment.

A few remarks on the principles of the analysis procedure areon order. The time-dependent phase fractions fx diffτ( ) can gen-erally be analyzed on the basis of solutions to the classical diffu-sion equation [5,38]. Apart from the usually straightforward initialand boundary conditions, it is crucial to use an appropriate model

A

BC

DE

7

8

9

0

20

40

14

16

18

thic

knes

s / n

m crystalline dc

amorphous da

E

DC

BA

T / °

Clo

ng p

erio

d / n

m

0.00.20.40.60.8

0.4

0.5

0.6

fract

ion

amorphous phase: SAXS / NMRinterphase: NMR

SAXSNMR

crys

t. fra

ctio

n

0 2 4 6 8 10 12time / days

Fig. 8. Results of a combined SAXS and NMR study of domain sizes and fractions inPCL as a function of time after initiating isothermal crystallization at 45 °C.(a) Temperature profile showing the measurement intervals (see text), starting theNMR experiments 4 days after initiation; SAXS results for (b) the long period and(c) crystalline and amorphous thickness; comparison of the (d) crystalline fractions(SAXS: linear crystallinity) and (e) amorphous and interphase fractions. The latter isonly available from NMR.

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–63 57

for the nanoscale morphology. In particular, the diffusion di-mensionality (1D for lamellar systems, 2D for rod-like domains, 3Dfor insular/spherical domains) plays a central role [5]. Since thedata is often featureless, independent knowledge of the mor-phology of the sample at hand is important. Once the appropriatemodel is identified, analytical solutions for two-domain modelsare available [5], and even for the lamellar three-domain modelincluding an interphase appropriate fitting functions for thefx diffτ( ) have been published [12].

A simple, feasible and popular approach is the initial-slope (orinitial-rate) analysis [4,5,10] for phase fractions plotted as a func-tion of diffτ . A simple linear fit to the initial behavior of either thesource or the sink phase in combination with a suitable formula, aswell as known effective D value and diffusion dimensionality, di-rectly yields the respective domain size. However, this approach,as well as the above-mentioned use of analytical fits to the wholedata range, is significantly challenged by signal decay due to T1during the spin diffusion time. This effect is particularly serious atlow field, and may be non-negligible even on 200 MHz instru-ments when the domain sizes are large [8,11,22]. The short T1 atlow field significantly affects already the early-time data, chal-lenging simple initial-rise analyses. In our previous work, we haveassessed possible, empirical and semi-quantitative correction ap-proaches [14], and just stress here that even a T1 measurement(e.g. by saturation recovery) provides non-trivial results that re-flect spin diffusion effects rather than clean T1 values. An analysisof the spin diffusion data on the basis of finite-difference lattice-based numerical simulations including T1 effects is thus preferred[2,8,11,22], as presented in much detail in our previous Ref. [14].

Lamellar dimensions and temperature effects: In order to obtain afull picture of the potential morphological complexity of PCL, wehave conducted a long-time annealing experiment following iso-thermal crystallization at T 45 Cc = ° , see Fig. 8a for the temperatureprotocol. Two independent samples were investigated in parallelby SAXS, obtaining the lamellar long period L and the individualthicknesses dc and da, and by NMR, obtaining phase fractions withhigher time resolution and performing rigid- and mobile-phasefiltered spin diffusion measurements at 5 different points A…E intime as follows:

: measurements at T 45 Cc = ° after 4 days to ensure the existenceof well-ordered, stable crystallites of preferably uniformstructure,

: repetition of the measurements within 4 more days at Tc,: cooling the sample to 27 °C and measurements after tempera-ture equilibration (less than one hour),

: repetition of the measurements within 4 more days at 27 °C,: heating back to Tc and measurements after temperature equili-bration.

Note that at 45 °C the crystallization half time is about 30 min, andthe primary crystallization is complete within 2–3 h (followed by avery slow, small secondary increase in fc by a few %).

This protocol was devised in order to check whether significantchanges occur in samples that were crystallized at elevated tem-perature but stored and/or measured at lower (room) tempera-ture. The results shown in Fig. 8 show the variation of the SAXSdimensions (b, c) and a comparison of SAXS- and NMR-basedcomponent fractions (d, e). We observe a small but detectabledecrease in L and dc when cooling to 27 °C, the effect being ratherfast and reversible. Note that the changes in dimensions amount to0.5 nm at most, which is less than the length of a monomer unit(0.8 nm).

At the same time, fc from NMR reversibly increases by about 4%on cooling. The comparison of all fractions demonstrates that bothtechniques yield rather comparable results; SAXS overestimates

the crystallinity only somewhat. We attribute this to its sensitivityto periodic electron density changes and its lack of ability to ac-count for an interphase. Only a small, denser part of what NMRdetects as interphase is in SAXS apparently detected as crystalline.The small discrepancy in fc apparently vanishes at the lowertemperature.

We interpret the observed changes in terms of the so-calledinsertion mode [39], and the (rather small!) discrepancies be-tween the two methods by their different sensitivity to an orderedperiodic lamellar arrangement and molecular dynamics, respec-tively. Upon cooling, secondary lamellae grow within the existingamorphous phase. This is manifested in SAXS mainly as a slightdecrease of all three average dimensions, while NMR appears tofaithfully reflect the amount of the additional crystallites.

Spin diffusion experiments and analysis: We have performedspin diffusion experiments at the indicated times, and have ana-lyzed them on the basis of known SAXS-based dimensions. Forthis, we have subdivided L from SAXS according to fa, fi and fc

fixed (SAXS)da = 6.69 nmdi = 0.89 nmdc = 7.63 nm

fit A:Da = 0.036 nm2/msD i = 0.26 nm2/msDc = 0.40 nm2/msT1a = 86 msT1i = 262 msT1c = 554 ms

fit B:Da = 0.031 nm2/msDi = 0.16 nm2/msDc = 0.34 nm2/msT1a = 89 msT1i = 240 msT1c = 467 ms

cryst. / int. / am./ / exp. datafit A fit B

0 100 200 300 400 5000.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

0 50 10 0 1500.00

0.05

0.10

0.15

0 50 100 1500.0

0.1

0.2

MAPE-filtered

sign

al fr

actio

n

diff / ms

sign

al fr

actio

n

DQ-filtered

anomaly!“interphase

effect”

τ

Fig. 9. Results of two different simulation-based fitting runs, applied simulta-neously to two sets of signal fractions from three-component fits to spin diffusiondata for PCL after (a) rigid-phase and (b) mobile-phase filtering using a DQ andMAPE filter, respectively. The fit employed fixed dimensions based upon SAXS re-sults, and the results of the two different runs (D and T1 values), reflecting thestability limits of the fits, are shown on the right.

Fig. 10. Spin diffusion coefficients for the three phases in PCL obtained from re-peated runs of simulation-based fits as a function of annealing time (see Fig. 8a).The dashed lines indicate average values that do not vary significantly in the stu-died temperature range.

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–6358

from NMR (considering the different proton densities) to obtainthe used da, di and dc, respectively [16]. Fig. 9 shows one set ofsignal fractions after (a) rigid-phase and (b) mobile-phase filteringas a function of diffτ . In each case, the source phase monotonicallylooses polarization, while the two distinguishable sink phases at-tain polarization on different timescales, but then go through amaximum and finally decay towards 0. As mentioned above, this isdue to rather short T1 relaxation times at low field, in particular ofthe mobile amorphous phase, which ultimately acts as magneti-zation sink. Thus, iterative fits based upon simulations taking intoaccount the T1 effect were performed simultaneously to DQ- andMAPE-filtered spin diffusion data. Since component fractions andlamellar dimensions are known, we have three D values and threeT1 relaxation times as free parameters, the results for which arealso shown in Fig. 9. Individual fitting runs with different initialparameter values in fact converge to slightly different final results,which prompted us to perform multiple fittings for each data set.

Spin diffusion coefficients: The relevant D values are shown inFig. 10. Together with the different measurements taken at timesA…E we obtain satisfactory statistics. Considering that the di-mensions and fractions change only slightly during the annealingrun, we simply report average values over the whole series, as alsoshown in Fig. 10. We did not observe any significant dependenceon temperature. Only the Da values, which feature rather smallscatter, appear to increase somewhat on cooling, which is ex-pected due to a concomitant increase in residual dipolar couplingsby slower dynamics in the amorphous fraction.

The D value of 0.34 nm2/ms for the crystalline phase is in goodagreement with theoretical estimations [6,13] based upon M2 ofpolyethylene, which has a crystal structure rather similar to and asecond moment M2 only somewhat higher than that of PCL [30].

The value is, however, significantly smaller than the often used“universal” value of 0.8 nm2/ms reported for PS and poly(methylmethacrylate) [4,5]. The discrepancy is in line with our own pre-vious, SAXS-based calibration of D for PS, which gave 0.38 nm2/ms.The reason will be addressed in more detail in Part 3 below; it islikely due to the use of static vs. slow-MAS conditions in the citedreferences.

Non-trivial interphase structure: In the inset of Fig. 9b, a per-sistent discrepancy between fits and actual MAPE-filtered spindiffusion data at short τdiff is highlighted. Experimentally, it ap-pears that the interphase signal does not rise first (as is the casefor DQ-filtered data, Fig. 9a), but in parallel with the crystallinephase. This is counterintuitive for a layer-like arrangement, inwhich magnetization diffusing from the rigid to the mobile regionor vice versa always has to first cross the interphase [35]. As si-mulations based upon a stacked layer arrangement (effectively a1D diffusion scenario) were not able to account for this feature, wehave extended our simulation to 2D, as reported previously [16].

Relevant results from this study [16] are shown in Fig. 11. Fol-lowing earlier, related work of VanderHart [2], we have turnedaway from the common assumption of an interphase that merelyrepresents the region of an extended mobility gradient [35], andhave tested the assumption of an “island-like” interphase that maybe located at the interface between the rigid crystallites and themobile regions. This allows for significant direct contact betweenthe latter two, which turned out to explain the anomaly qualita-tively. This is not an unphysical assumption, considering thatmobility transitions can be rather abrupt. For example, in an ear-lier NMR T2 relaxation study of poly(dimethylsiloxane) chains at-tached to silica [40], domains with distinct mobility were found tocomprise only a few monomers, with a layer thickness in the 1 nmrange. Notably, the rigidly adsorbed and the adjacent intermediatefraction in this study featured a clear T2 contrast of one order ofmagnitude, much higher than the difference we observe.

Fig. 11 shows our “best-fit” data in a sense that over a range ofsimulations testing different geometries, we found best agreementwith the data for scenarios inwhich the interphase was in fact buriedcompletely within the crystalline phase. This is a novel and relevantinsight worthy of closer inspection in other semicrystalline polymers.We found a similar phenomenon of an island-like interphase, nowlocated directly at the rigid-mobile interface, in the lamellar PS–PBblock copolymer [16]. In an earlier study employing spin diffusion

x= 15 nmq= 0.6

x= 15 nmq= 0.7

x= 15 nmq= 0.8

q0 0

0.1

0.2

0 0

0.5

1.0

sim. 1Dsim. 2D

fra

ctio

n

interphasemobile phaserigid phase

sourcesinks

0.0 0.0

0 6 12 18 240.0

0.1

0.2

0 6 12 18 240.0

0.5

1.0

fra

ctio

n

q

diff0.5 / ms0.5

0 6 12 18 24 0 6 12 18 24

diff0.5 / ms0.5τ τ

Fig. 11. (a) DQ-filtered and (b) MAPE-filtered spin diffusion data for PCL as com-pared to the 1D (lamellar stack morphology) best-fit vs. 2D model calculations. Thesketches show 2D cross sections of the simulated “elementary cells” (using periodicboundary conditions). The parameter q quantifies the relative direct rigid-to-mo-bile contact area. Data replotted from Ref. [16].

Fig. 12. (a,b) Pulse sequences for 1H magnetization hole burning based upon de-phasing by a nearby 13C rare spin using (a) single and (b) phase-inverted doubleLee–Goldburg CP [50,51] followed by 1H spin diffusion and site-resolved 13C de-tection. (c) Principle of rare-spin-based 1H hole burning and re-equilibration. SeeRef. [46] for details.

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–63 59

experiments under MAS conditions, we could also demonstrate thatthe interphase mainly consists of mobilized PS units [41]. The na-nometric size scale of these domains suggests that the phenomenonmay be related to the size of dynamic heterogeneities that are typicalfor materials close to Tg [42].

5. Part 3: Macroscopic vs. local spin diffusion coefficients andtheir behavior under MAS

As already alluded to above, absolute values for spin diffusioncoefficients may be obtained by theoretical estimates based upondipolar second moments or T2 values of the respective phases[6,13], or by calibration experiments on a given polymer usingsamples of known dimensions as determined by SAXS or electronmicroscopy [4,5,10]. Very recent work has also demonstrated afirst-principles calculation of spin diffusion coefficients based oncrystal structure data [43].

Alternatively, one can make use of the known proton densityand rely on experiments that probe the spin diffusion directly andrather locally [44–47]. We here review such experiments in orderto highlight non-trivial effects of MAS onto D values, which wehave discovered recently [47]. 13C-based detection of 1H spin dif-fusion under MAS is rather popular in that it allows for chemical(site) resolution in more complex systems. Commonly and ex-pectedly, the faster the MAS frequency MASν( ) the slower the spindiffusion, as corroborated by many studies [43-45,48,49]. How-ever, in expanding the MAS range to rather slow spinning andcomparing to the static case, we found a pronounced maximum ofthe D value at moderate MAS [47], to be demonstrated and ex-plained below.

After a first demonstration on the special case of ferrocene byErnst and coworkers [44], Chen and Schmidt-Rohr [45] were thefirst to report on a systematic determination of D values in dif-ferent polymers by help of such localized experiments. They reliedon the well-defined dephasing action of the famous rotational-echo double-resonance (REDOR) pulse sequence invented byGullion and Schaefer [52], using it in a natural-abundance 13C–1HCP-MAS [17] experiment to create 1H “magnetization holes” ofvariable and defined size and shape. Such holes re-equilibrate bylocal spin diffusion (see Fig. 12c), and the hole-filling process canbe followed locally and in real time using 13C detection at the locusof the initial hole after a short CP.

We have performed similar experiments [46,47], restrictingourselves to rather localized hole burning using Lee–Goldburg CP[50,51], see Fig. 12a and b. This strategy is similar to the one ofErnst and coworkers [44], and is advantageous in that it simplifiesthe data analysis (see below). It also allows for a second phase-inverted LGCP step that can be used to double the experimentalcontrast by nearly inverting rather than just nulling the 1H holemagnetization (Fig. 12b).

Illustrative results of such experiments are collected in Fig. 13.Panel (a) shows the behavior of a 1H hole located at a central,symmetric aromatic 13C–H position in an effectively linear spinarrangement in an oriented rigid-rod type liquid crystal molecule[46]. Note that no powder average is active here, and that fastuniaxial rotations of the bent molecule render the dipolar couplinggeometry linear. This represents a nice model case of 1D spindiffusion within a finite 52-proton system. Due to well-definedand equal dipolar couplings to its only two equivalent neighbors,the re-equilibration process at the detected site is primarily

Fig. 13. Local 1H hole re-equilibration under static conditions in (a) a B0-orientedsmectic-A “bent-core” liquid crystal detected at the central aromatic 13C7–H posi-tion and (b) the crystalline region of polyethylene, in both cases with (circles) andwithout (squares) 13C dipolar decoupling during spin diffusion. (c) Principle of thedata analysis procedure represented by Eq. (4). The fit results are shown as dashedlines in (a) and (b). Data re-plotted from Refs. [46,47].

0.0

0.1

0.2

0 4 8 12 16 20

PE

PS

0 4 8 12 16 200.0

0.1

0.2

MAS / kHz

D/

nm2 /m

sD

/ nm

2 /ms

ν

Fig. 14. MAS-dependent local spin diffusion coefficients for (a) amorphous, glassyPS and (b) the crystalline regions of PE. Data re-plotted from Ref. [47].

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–6360

oscillatory, but on a baseline contribution that follows a diffusivebehavior. This example is a nice illustration of the intrinsicallycoherent nature of the spin flip–flops as the base process of spindiffusion, which is in fact time-reversible [23,53].

In a system of less well localized, multiple 1H pair couplings, asrepresented by most organic substances including polymers, os-cillations cannot be observed, rather, local hole re-equilibrationappears to be purely diffusive. This is illustrated on the example ofthe crystalline signal of polyethylene (PE), see Fig. 13b [47]. Thedata were fitted [46] to the following simple solution of the 3Ddiffusion equation describing the broadening of a point source(this being mathematically equivalent to a point hole) in terms ofits intensity detected at the locus of the source,

⎡⎣ ⎤⎦I I A D1 4 , 4diff eq H1

diff3/2( )τ ρ π τ τ( ) = − ( + Δ ) ( )

− −

where Ieq is the equilibrium magnetization, Hρ the 1H spin density,and A a numerical factor depending on the experiment (1 forFig. 12a, 2 for Fig. 12b). An important parameter is τΔ , which is avirtual time shift parameter that makes up for the finite initialwidth d D4 ln 2 0.3 nm0 τ= Δ ≈ of the magnetization hole. In thisway, the initial hole shape is approximated to be Gaussian, d0being its full width at half maximum, which is well justified forlocalized hole burning [46]. See Fig. 13c for a visualization of theanalysis concept.

Note that in both the above cases, 13C dipolar decoupling has arather strong positive effect in that active additional heteronuclearcouplings effectively quench the local-scale spin diffusion. Theorigin of this effect is explained below. Such “bottlenecks” are ofcourse rare given the large spatial separations determined by 13Cat natural abundance, so the main effect observed here relates tothe 13C nucleus at which the hole was created and later detected.On larger length scales, the effect is expected to be rather small.

So-obtained spin diffusion coefficients for PS and PE, re-presenting rigid amorphous vs. crystalline polymers, are plotted inFig. 14 as a function of MASν . Note that the crystalline structure ofPE is rather similar to the above-discussed case of PCL. The static-limit values of 0.15 and 0.1 nm2/ms, respectively, are significantlylower than values calibrated on the basis of larger-scale spin dif-fusion. An increase of the order of 50% is observed until MASfrequencies of the order of 2–3 kHz are reached, only above whichthe values drop. The latter trend is well understood by averaging ofdipolar couplings by MAS, but the increase is less trivial and ex-plained below. We are aware of only one case reported by Eicheleand Grimmer [54], where a MAS-related minimum in bulk 31P T1value in a material with sparse paragnetic centers (acting as re-laxation sinks) was an indirect indication of a similar effect.

Obviously, the MAS-related maximum partially explains thediscrepancy of the static-limit D values reported herein and theoften used “universal” value of 0.8 nm2/ms [4,5], which was in factdetermined (among other systems) for PS by 13C-detected ex-periments using moderate MAS. However, a significant deviationbetween local and mesoscale values remains. This has been ad-dressed before in some detail by Chen and Schmidt-Rohr [45], who

0 2 4 6 8 10 120.0

0.5

inte

nsity

spi

n 1

1 kHz2 kHz3 kHz

10 kHzinitial-rate fit

diff / ms

1 10

0.1

1

spin 1

spin 2

spin 3

f 11 kHz

f = 1f = 1/2f = 1/4

W/ W

0

MAS / kHz

1 kHz

ν

τ

×

Fig. 15. Results of spin dynamics simulations of a minimal 3-spin system (with spin1 being the initially depolarized one) exploring the origin of MAS-dependent localspin diffusion coefficients. (a) Intensity of spin-1 polarization as a function ofevolution time for a primary spin-1–2 coupling of 1 kHz and a spin-2–3 secondarycoupling of 11 kHz with a sample fit I Wdiff diff

2τ τ( ) ≈ ( ) , (b) build-up rates W as afunction of MAS frequency for different spin-2–3 coupling strengths scaled by afactor f. See the inset for the used geometry. Data re-plotted from Ref. [47].

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–63 61

measured local D values at variable length scales by varying the 1Hdephasing time. Their results suggest that bulk values are onlyrecovered at d0 values beyond 1 nm. The slower local-scale processis likely due to local “bottlenecks” in the coupling topology (i.e.,longer distances thus weak couplings), which are overcome onelarger scales by possible “detours”.

In Ref. [47] we have presented two alternative but ultimatelyequivalent theoretical explanations. In general, energy-conservingspin flip–flops require a matching of energy levels of the partici-pating protons. Provided such a match, flip–flops occur at a raterelated to the dipolar coupling between the spins. It is againstressed that such a “rate process” is actually coherent, i.e., it takesplace in a continuous oscillatory fashion (see Fig. 13a).

A level mismatch can arise for various reasons, which are iso-tropic shift differences, chemical-shift anisotropy (CSA) or addi-tional dipolar couplings to a third spin. The CSA is in fact thereason for the observations of Eichele and Grimmer in their 31Psystem [54], and the latter, when coupling to a 13C spin is con-sidered, is the reason for decoupling effects in Fig. 13. More im-portantly in the given context, a homonuclear coupling to a thirdproton has the same consequences. An equivalent effect is actuallywell known in terms of so-called dipolar truncation in transverseevolution under dipolar recoupling [55]. Formally, it arises whenthe Hamiltonians of the relevant interactions do not commute; theconsequence is that a stronger coupling to a third spin preventsspin polarization to be transferred to a second spin via a weakercoupling. In all cases, the rotation of the local spin system withincreasing MASν leads to increasingly frequent level crossings,realizing favorable conditions for polarization transfer transientlyand thus explaining the initial increase of D MASν( ).

We have explored this effect systematically [47] by spin dy-namics simulations, restricting ourselves to a 3-spin system as aminimal model that exhibits the mentioned properties, see Fig. 15.Part (a) shows the powder-averaged polarization build-up of theinitially polarized spin 1 weakly coupled to spins 2 and 3, whilethe latter two are more strongly coupled. The simulation of 2 kHzMAS yields the fastest transfer, and even higher MASν lead to theexpected decrease. Simulations for individual powder orientationsnicely demonstrate that the build-up is locally oscillatory with aperiod 1/ 2 MASν( ), with local slopes being steepest when the or-ientation of the strong perturbing coupling tensor crosses themagic-angle orientation, realizing a level crossing. These oscilla-tions vanish upon adding up all powder orientations.

Polarization transfer rates obtained by fits to the initial rise ofthe simulated curves are plotted in Fig. 15b as a function of MASν ,varying further the strength of the stronger perturbing coupling.The maximum transfer rate is found to range between 1 and a fewkHz, in agreement with the experiments. The used couplings ac-tually roughly correspond to what is expected for secondary inter-and intra-chain couplings in crystallites of PE, noting that theprimary strong intra-CH2 coupling is not relevant for the spindiffusion coefficient determined in our experiment. This is becauseboth protons are on average simultaneously dephased and thenrepolarized. The latter fact, relevant for all local hole burning ex-periments but not for D values measured at the mesoscale, maywell also contribute to the finding that locally determined D valuesare always somewhat too low. At the mesoscale, polarizationtransfer from one CH2 proton to the other is fast and relevant.

6. Conclusions

In summary, we have reviewed the basic principles of phasecomposition analysis and spin diffusion experiments in dynami-cally inhomogeneous organic materials such as polymers on thebasis of simple 1H time-domain NMR. The 1H FID signal of a single-

phase material is a sensitive measure of amplitude and timescaleof large-scale motions (in the μs range). In two-component ma-terials, fits using a three-component model, including a dynamicinterphase, are in most cases sufficient and stable. In favorablecases with the local coupling topology being dominated by spinpairs in CH2 groups, a solid component is best represented by theAbragam function, leading to stable fits without further measures.

In the general case, the use of MSE-refocusing to overcome thedead-time issue, and the application of mobile (MAPE) or rigid(DQ) phase filters is advised in order to determine the functionalshape of the FID components corresponding to the respectivephases. Fits to the integral FID with fixed shape parameters arethus stabilized, in particular when analyzing the results of spindiffusion experiments. We have analyzed in some detail the actionof the MAPE filter, which is based upon a breakdown of the MSEcaused by higher-order Hamiltonian contributions arising whenthe sequence cycle time is increased. For the DQ filter for rigidregions, powder average bias effects have to be carefully con-sidered, and using spin diffusion experiments, we have high-lighted qualitative differences in the response of ordered crystal-line or glassy amorphous phases.

K. Schäler et al. / Solid State Nuclear Magnetic Resonance 72 (2015) 50–6362

A simultaneous NMR and X-ray scattering long-time annealingstudy was performed on semicrystalline PCL in order to obtain acomplete picture of potential morphological changes as a functionof temperature. For various temperatures, spin diffusion data wereanalyzed on the basis of known semicrystalline dimensions inorder to determine spin diffusion coefficients D of 0.34, 0.19 and0.032 nm2/ms for the crystalline, inter- and mobile amorphousphase, respectively, in the range of room temperature up to 45 °C.We have further stressed the necessity to perform simulation-based fits in order to account for substantial T1 effects when data isacquired at low field. Such simulations were further used to assessthe non-trivial structure of the interphase, which turned out to notform a contiguous in-between layer but rather islands that aresometimes completely immersed in another phase.

Finally, we have summarized the state of knowledge concern-ing the absolute-value determination of D on the basis of local 1Hhole-burning experiments, and their comparison to results fromcalibration-based conventional spin diffusion experiments. Hole-burning experiments were recently used to elucidate a non-trivialdependence of D on the MAS frequency. A local maximum with a∼50% increase over the static case was found at moderate MAS (2–3 kHz), as explained by quenching effects resulting from dipolartruncation and MAS-related energy level crossings. This findingshould not only be relevant for the precise analyses of spin dif-fusion experiments, but also in the context of recent dynamicnuclear polarization (DNP) experiments, which rely on fast equi-libration of enhanced 1H magnetization through spin diffusion.

Acknowledgments

Funding was provided by the DFG in the framework of theSonderforschungsbereich SFB-TRR 102, project A1.

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