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1 Solid State Nuclear Magnetic Resonance 34 (2008) Contents lists available at ScienceDirect Solid State Nuclear Magnetic Resonance journal homepage: Spin-diffusion NMR at low field for the study of multiphase solids M. Mauri a, Y. Thomann b, H. Schneider a, K. Saalwächter a, a Institut für Physik, Martin-Luther Universität Halle-Wittenberg, Friedemann-Bach-Platz 6, D Halle (Saale), Germany b Freiburg Materials Research Center, Albert-Ludwigs-University, Stefan-Meier-Straße 21, D Freiburg, Germany article info abstract Article history: Received 30 June 2008 Available online 3 July 2008 Dedicated to Hans-Heinrich Limbach on the occasion of his 65th birthday Keywords: NMR relaxation Spin diffusion Dipolar filter Double-quantum filter Block copolymers SBS Computer simulation The use of spin-diffusion NMR for the measurement of domain sizes in multiphase materials is becoming increasingly popular, in particular for the study of heterogeneous polymers. Under conditions where T 1 relaxation can be neglected, which is mostly the case at high field, analytical and approximate solutions to the evolution of spin diffusion are available. In order to extend the technique to more general conditions, we performed a comprehensive study of the diffusion of magnetization in a model copolymer at low field, where T 1 tends to be of the same order of magnitude as the typical spindiffusion time. In order to study the effects of T 1 and to delineate the optimal T 1 values for back correction prior to applying the initial-rate approximation, we developed a numerical simulation based on the diffusion equation and including longitudinal relaxation. We present and discuss the limits of simple correction strategies for initial-slope analysis based on apparent relaxation times from saturation-recovery experiments or the spin-diffusion experiments themselves. Our best strategy faithfully reproduces domain sizes obtained by both TEM investigations and full simultaneous fitting of spin-diffusion and saturation-recovery curves. Full fitting of such independent data sets not only yields correct domain sizes, but also the true longitudinal relaxation times, as well as spin-diffusion coefficients. Effects of interphases with distinct mobility on spin-diffusion curves, as well as practical hints concerning the reliable component decomposition of the detected low-resolution FID signal by help of different magnetization filters are also discussed in detail. & 2008 Elsevier Inc. All rights reserved. 1. Introduction 1.1. Low-field NMR Recent applications of NMR spectroscopy are mostly performed at high field, and the trend towards ever increasing magnetic fields has provided powerful tools for the study of many problems in chemistry, physics and bioscience. On the other hand, low-field NMR is a growing and increasingly independent field of research, where the inherent limitations, such as the inability to detect field-dependent effects (chemical shift), imply a concentration on field-independent effects, such as dipolar couplings. Moreover, due to low cost, low maintenance, short recycle times and ease of application, this technology has a vast potential for use in industrial and technical applications. The primary application is relaxometry, focussed on the quantitative detection of differently relaxing (i.e. solid or liquid) fractions in samples ranging from polymers [1] to foods [2], and there are strong efforts focussed on further developing the pool of possible Corresponding author. address: kay.saalwaechter@physik.uni-halle.de (K. Saalwächter). experiments. More advanced experiments have been successfully applied at low field, including multiple-quantum techniques for the study of chain dynamics in elastomers [3] and linear polymers [4]. This work is focussed on the specific limits and possibilities of performing spin-diffusion measurements at low field Spin diffusion at low field The spatial arrangement and the dimensions of different phases in solid state can be investigated using spin diffusion if they can be resolved with respect to their NMR properties. The technique has an operational range extending up to the order of tens of nanometers, so it has become a useful tool in polymer science [5 7]. Polymer blends and copolymers often have macroscopic properties depending on microscopic structure on the nanometer scale [8]. Generally speaking, spin-diffusion NMR experiments are made up of three steps: selection of magnetization from a single phase, variable diffusion time and phase-resolved detection. Selecting magnetization at high field is most commonly done by exploiting differences in chemical shift or transverse relaxation. Since chemical shifts are not readily accessible at low field due to an insufficient shim of typical magnets at around 20 MHz proton /$ - see front matter & 2008 Elsevier Inc. All rights reserved. doi: /j.ssnmr

2 126 M. Mauri et al. / Solid State Nuclear Magnetic Resonance 34 (2008) frequency, phase resolution is limited to exploiting T 2 differences between the phases: polymer phases with higher mobility display longer transverse relaxation times. These differences depend on dipolar coupling, so they are field-independent. In this work, we compare results using both the mobile and the rigid phase as magnetization sources for the spin-diffusion process. For the slowly relaxing mobile phase, we use a filter based on a magicsandwich echo (MSE) [9 11], which avoids artefacts arising for the simple two-pulse Goldman Shen filter [12] at short spin-diffusion times [13]. A double-quantum filter, based on the field-independent excitation of double-quantum coherences in the strongly dipolar-coupled rigid phase, is used for selecting magnetization in the rigid phase [14,15]. It must be stressed that both filters are ultimately based on the local mobility of the polymer, rather than chemical composition or crystalline structure, and as such they reveal areas of reduced or high mobility, without implying any long-range order, or necessarily separating polymers of different composition. Detection is of course limited by the same principle, i.e. detecting the different phases by virtue of their mobility difference, which is often favorably reflected in the FID. It should be noted that spin-diffusion studies at low field, based on mobility contrast as selection and detection principle, are advisable only in systems with known and well-defined larger-scale (dynamic) heterogeneity. More subtle differences in mobility, which can also be employed in spin-diffusion experiments using e.g. the 12-pulse dipolar filter (DF) with different, rather long filter times [16], may well be of a very local, molecularscale origin, such as a mobile side chain. The experiment then characterizes a local magnetization exchange process, for instance caused by the nuclear Overhauser effect or chemical exchange [17]. Avoiding such phenomena, and separating flip-flop-mediated spin diffusion from other magnetization exchange mechanisms inevitably requires additional high-field experiment employing a site-resolved detection using for instance cross-polarization and magic-angle spinning, as demonstrated in an earlier paper by Hans-Heinrich Limbach [18]. Spin-diffusion NMR data at high field can often be treated analytically: the diffusion equation has actually been solved for a number of different geometries. Such solutions [15,19] are sometimes very convoluted, and invariably require longitudinal relaxation to be neglected, which is not generally correct, especially at low field. With the B 0 field strength and inhomogeneity inherent to low-field technology, T 1 is short enough for relaxation and spin diffusion to occur on the same timescale. This is a well-known limitation, and is only partially mitigated by different modified versions of the Goldman Shen sequence [20]. When different regions have different relaxation times, these correction schemes do not work properly. Suitable analytical solutions are not available when diffusion and relaxation occur simultaneously, and one has to resort to numerical approaches [19,21 23]. A much faster way, providing satisfactory results for small values of lamellar size, is the initial-rate approximation [5 7]. To verify the validity of approximate results at low field, we have numerically solved the (spin) diffusion equation including the effect of relaxation by means of a computer simulation program with innovative features. We are able to perform a fitting of any evolution of magnetization from a known starting point to equilibrium, taking account of both diffusion and T 1 relaxation. Similar to the aforementioned spin-diffusion sequences, simple phase-resolved saturation-recovery (SR) experiments also contain information on diffusion, because the inhomogeneous relaxation creates a magnetization gradient that in turn drives spin diffusion. Consequently, simultaneous fitting of specifically filtered spin diffusion and SR data sets is more stable and exploits more information on the real system. Since many of the parameters including the diffusion coefficients of mobile and rigid phase are known or can be calculated by independent measurement, the fitting of many curves with many parameters is still manageable. This work is mainly concerned with relaxation issues arising in low field, as is the case with polymeric solids. It can be conceived that similar issues can be present even at high field if, for any reason, one or more of the phases have rather small relaxation times, or if the domains are large and the spin-diffusion times are long. 2. Experimental 2.1. Sample A sample of polystyrene-polybutadiene-polystyrene (SBS) star block copolymer, referred as K-resin (K-Resin s KR03, Chevron Phillips Chemical Company LLC) was investigated in this work [24]. The different homopolymers do not mix, so the PB cores tend to segregate and produce disordered lamellar phases with rather regular width of about 1071 nm, as proven by TEM (Fig. 1). The relative lack of mixing can be experimentally verified over a large range of temperature by high-resolution MAS experiments: the volume fraction of PB units in the PB-dominated phase is around 95% at 298 K and 91% at 336 K, with the mobilized PS mainly located at the interface (details of the corresponding study are the subject of a separate publication, Ref. [24]). Moreover, the glass transition temperature (T g ) of the two polymers is very different: 388 K for the polystyrene, and less than 200 K for polybutadiene. In the range between 300 and 360 K PS is glassy while PB is rubbery, so the two phases can be resolved at low field using their mobility, while the sample as a whole does not melt or suffer structural rearrangements. The PB content is around 25% in weight and the proton signal fraction associated with the mobile phase can be estimated to more than 32% of the total proton signal. About 120 mg of sample in the form of granules was used in the low-field NMR experiments. With mass densities of 1.05 g/cm 3 for PS and 0.89 g/cm 3 for PB, the corresponding proton (spin) densities are 81 and g/cm 3, respectively NMR parameters All low-field experiments were performed on a Bruker minispec mq20 at 20 MHz proton resonance frequency, equipped with a BVT3000 heater working with nitrogen gas. The minispec K-Resin KR03 Fig. 1. Approximate structure of the PS-PB star block copolymer (left) and lamellar structure as probed by TEM (right, scalebar: 100 nm).

3 M. Mauri et al. / Solid State Nuclear Magnetic Resonance 34 (2008) features typical 901 pulse lengths of down to 2 ms (125 khz nutation frequency), reliable phase cycling in 901 steps with switching times in the 2 ms range, and a minimum dwell time of 400 ms, making it well adapt to rather advanced solid-state pulse sequences. The only serious problem, to be tackled experimentally (see below), is the rather long dead time of 10 ms or more. For every sample temperature, one or two series of spin diffusion and one series of SR experiments were performed, each series comprising data points. Each point usually required 1024 scans with 1.5 s recycle time, because low noise and full recovery of the magnetization were necessary for quantitative point-by-point evaluation of the ratio between the two phases. A few high-field SR experiments were performed at 200 MHz on a Tecmag Apollo console, requiring 128 scans per point Pulse sequences inv. x y±x Selection Diffusion Detection MAPE inv. x ±x φ 1 2τ 2τ 2τ τ 2τ 2τ τ t = 0 DQ filter x -x φ DQ -φ DQ τ DQ 2μs τ DQ φ 1 τ 2τ 2τ τ t = 0 Generally, fully complex NMR time-domain signals were acquired on resonance with the receiver phase tuned to full absorption mode and analysis of the real part only, applying a two-step inversion phase cycle on the 901 magnetization read pulse for imperfection compensation (full CYCLOPS is thus not necessary). SR measurements were performed with a train of dephasing pulses followed by variable recovery time before detection. Spin-diffusion sequences are more complex, insofar as they require a magnetic phase selection, or filter, that prepares the sample in a state where one of the two phases (either rigid or mobile/rubbery) is completely depleted, while the other has as much magnetization as possible. We implemented sequences to study spin diffusion in two directions: from the mobile to the rigid phase using a magic-and-polarization echo (MAPE) DF sequence [11] and from the rigid to the mobile by means of a short doublequantum (DQ) filter [14]. We have tested several different DF approaches for mobilephase selection, and place a few comments here. First, at short spin-diffusion times, the two-pulse Goldman Shen filter [12] generally leads to the appearance of a short negative (dipolar) oscillation in the initial part of the FID, which essentially reduced the starting intensity and rendered fitting results for the growing rigid-phase signal unreliable. This artifact is well-known and is usually attributed to multiple-quantum coherences in the rigid phase. However, since trying to remove it using the pulsed-fieldgradient feature of our minispec during the diffusing time remained unsuccessful, we rather attribute it to multi-spin dipolar-ordered states (created from multi-spin antiphase coherences by the second 901 pulse), which also explains that it also cannot be removed by phase cycling. Thus, the creation of dipolar-mediated coherences in the rigid phase during a transverse dephasing/selection delay must be avoided, which calls for selection approaches based upon z-magnetization and dipolar time-reversed sequences, with a filter action related to the breakdown of the zero dipolar average Hamiltonian in strongly coupled spin systems. Schmidt-Rohr s 12-pulse DF [16], as well as the filters based on a (mixed) MSE [11], fulfill the requirement. The MSE can be applied to z-magnetization (just as the 12-pulse filter), where it forms a magic and polarization echo (MAPE DF [11]) or between two 901 pulses, where it refocuses transverse magnetization rather than polarization (MSE filter). Note that the latter was referred to as MAPE filter in the paper of Demco and coworkers [11,15], yet we suggest that our terminology is a bit more descriptive. We tested all variants, and found that all filters worked similarly well in suppressing the rigid-phase signal at comparable filter times, with small differences arising in the unwanted (see below) suppression of mobile signal. In this regard, the MAPE filter as shown in Fig. 2(a), with a 500 ms filter time ( ¼ 6t), worked best. Generally, φ 2 φ 3 φ 3 φ 3 φ 3 -φ 3 -φ 3 -φ 3 -φ 3 φ 2 τ φ 2τ φ the mobile-phase filter sequences (MAPE in Fig. 2(c)) were phasecycled by inverting all component pulses once, keeping all other phases constant. The DQ filter works by selectively exciting DQ coherences in the rigid phase. The sequence illustrated in Fig. 2(b) excites and reconverts DQ coherences in dependence of t DQ, and selection (filtering) of DQ signals is realized by a 4-step phase cycle over one of the two pulse pairs flanking t DQ. We measured the corresponding excitation behavior in separate experiments, shown in Fig. 3, and then consistently used the t DQ value providing the maximum DQ-filtered signal (20 ms) in the selection part of the DQ spin-diffusion sequence. For the diffusion process, it is important that magnetization is stored alternately along the z and z directions. The difference of the two signals is finally detected to limit spurious effects from the recovery of magnetization during the diffusion time. The alternate storage is realized by phase-cycle controlled inversion of the 1 H magnetization either after or prior to the filters, where for the case of the DQ filter, we use a composite inversion/flip-back pulse 90x 90y 907x that is compensated for finite pulse effects in combination with large dipolar couplings [25] Detection 2τ φ 2τ φ τ φ τ φ n MSE The two phases described above both contribute to the detected FID, and their relative intensities depend on the final magnetization state after any of our sequences. Since the two 2τ φ 2τ φ 2τ φ τ φ n MSE Fig. 2. Two spin-diffusion experiments with different selection filter, following the 3-step concept explained in the text. Case (a) displays a dipolar (MAPE) filter, while (b) shows a DQ filter. t DQ is the double-quantum excitation time. In both cases, a phase-cycle controlled magnetization inversion is applied, and the detection is preceded by a mixed magic-sandwich echo (MSE) with variable number of cycles (c) for an almost complete refocusing of the rigid-phase signal. On the minispec, n MSE ¼ 1 was sufficient to overcome the 10 ms dead time. The MSE is also part of the MAPE filter (with f 2 on the last pulse), where longer filter times are realized by increasing t f.

4 128 M. Mauri et al. / Solid State Nuclear Magnetic Resonance 34 (2008) Intensity (Arb. Units) MSE Solid Echo Fig. 3. DQ-filtered intensity build-up in the rigid phase at 315 K, plotted relative to the full (rigid+mobile) magnetization. Note that at longer times, the mobile (rubbery) phase signal rises via its residual dipolar couplings. phases have very different mobility and T 2, the separation of these contributions is based on direct FID analysis, as described in Section 2.5. The (apparent) relaxation time for the rigid (PS) phase is in the range of tens of microseconds, while for the softer PB phase it is in the range of hundreds of microseconds. In fact, the relaxation of the rigid-phase signal during the dead time (10 ms on our minispec) is not negligible, which leads to a possible underestimation of its relative contribution. We addressed this problem by applying a mixed MSE refocusing block [26,27] at the end of each sequence ( mixed refers to equal phases of the two sandwich 901 pulses, which includes a hidden composite 1801 pulse that leads to chemical shift refocusing for the flanking t periods). This block, shown in Fig. 2(c), is also phase-cycled once by inverting all component pulses. The MSE is theoretically capable of fully refocusing magnetization in presence of multispin dipolar coupling. From the decay of the initial intensity (first point of the FID), and the (linearly extrapolated) signal that remains from the amorphous phase, both plotted in Fig. 4, it is readily apparent that the rigid phase has a much faster decay. To evaluate the signal loss from the shortest possible MSE echo sequence (one cycle), we extrapolate the intensity of the first point of the FID to zero echo time, using a Gaussian function. It must be stressed that this extrapolation is much more reliable than the extrapolation of analogous solid-echo data (also shown in Fig. 4), which decay much faster, requiring a larger backextrapolation. It is apparent that the refocusing is much more effective for the shortest possible MSE block (60 ms) than for the shortest possible solid echo of 20 ms (both are dictated by the need to accommodate a 10 ms dead time). From a Gaussian back-extrapolation, we can reliably estimate a loss of only 10% of the rigid-phase signal during the refocusing block, and have correspondingly up-corrected all our data. The loss in the mobile part is negligible. Generally speaking, not only the shortest possible MSE performs better than the shortest solid echo, but the back-extrapolation of the MSE is also higher than its solid-echo equivalent. This suggests that the very popular solidecho approach, used systematically in spin-diffusion literature [28], underestimates the rigid fraction even after correction. As a final remark, we also stress that the performance of the MSE (as well as a solid echo) is severely degraded when the rigid parts to be refocused exhibit intermediate (khz MHz) mobility. This is, for instance, the case in polyethylene or poly(ethylene Echo time (us) Fig. 4. Comparison between solid-echo and MSE intensities at low field. The intensity of the first FID point is plotted against total echo time. There are less MSE points due to the integer cycle increment used here (see Fig. 2). For the MSE, the intensity of the linearly extrapolated mobile part is also represented and fitted with a line (see also Fig. 5). oxide) crystallites or in polymers close to T g in such cases, the use of a probe with a short dead time is the only option for quantitative rigid-phase detection Quantitative phase determination The rigid fraction f r can be quantified by extrapolating the linear part of the FID to obtain a measure of the mobile part or, more precisely, by fitting the initial part of any fully relaxed FID with the following equation [27]: FIDðtÞ FIDð0Þ ¼ f rð1 f 0 n ri Þe ðt=t 2r Þ2 þ f r f 0 ri e ðt=tn 2ri Þv ri þð1 f r Þe ðt=tn 2m Þnm. (1) The T n 2x are apparent transverse relaxation times for the rigid (r), rigid-interfacial (ri), and mobile (m) phases, respectively, and v x are coefficients that parameterize a deviation of the more slowly relaxing signals from purely exponential or Gaussian shape. In most cases, v x took on values between somewhat below 1 (stretched exponential) and 1.5 (Weibullian). Fitting the initial part only, in our case from 0 to 200 ms, is essential, as the shape of the FID at longer times may be complex, depending on the field inhomogeneity of the low-field instrument or joint effects of residual dipolar couplings, motion, and inhomogeneity of the mobile phase. The last term in Eq. (1) works very well in fitting the initial part of almost any mobile-phase relaxation curve (or FID), but if the exact decay function is not known (which, considering the many unknowns, is in fact always the case!), imposing a specific shape or introducing more sub-components is most often hardly physical. Introducing a rigid-interfacial contribution (f 0 ri, defined as relative to the rigid part only) proved necessary, but resolving the potential ambiguities arising from many fit parameters is very important. First of all, the Gaussian shape (and T n 2r ) of the core of the rigid phase can independently be confirmed by fitting a Gaussian to the initial signal in a DQ-filtered spin-diffusion experiment. Note that in this experiment, the FID shape is subject to small changes when spin-diffusion times shorter than 100 ms are used, which is due to a re-equilibration over different powder orientations (biased by the DQ filter) by spin diffusion within the rigid phase.

5 M. Mauri et al. / Solid State Nuclear Magnetic Resonance 34 (2008) Second, the shape ðt n 2m ; n mþ of the mobile signal can independently be fixed by fitting the signal after the MAPE DF. Finally, the two shape parameters for the rigid-interfacial fraction as well as the two weights (f r, f 0 ri) are obtained from a fit to a fully relaxed MSE-refocused FID with very good S/N (1000 scans). Experimental data and fits from this procedure are shown in Fig. 5(a). It is important to realize that the shape and the fraction of the rigidinterfacial component do depend somewhat on the function used for the mobile phase, i.e., slightly different parameters are obtained if a longer DF is used. In other words, the experimental condition defines the rigid-interfacial part. This is not surprising, as realistically, there should always be a dynamic interface with a gradient in mobility, a part of which survives the DF for the mobile phase. It should generally be noted that T n 2ri was always around ms (and thus only slightly longer than T n 2r 18 ms), which suggest that this fraction can be analyzed as part of the rigid phase (using the same spin-diffusion coefficient), as it is done in the following. In fact, our attempts to fit spin-diffusion data (see Fig. 8) for all three phases did not give very reliable results (as we also need to fit the longitudinal relaxation times) we only note that the width of the rigid interface was always close to what is expected from the intensity ratio (10 25% of the overall rigid part in the investigated temperature range) assuming a 1D layered system. Since we know from other experiments that even the mobile phase is PS-rich close to the interface, it is also clear that this fraction mainly consists of only slightly mobilized PS. In fitting the FIDs recorded in the course of spin-diffusion experiments, the only free parameters are the weights f r and f 0 ri, yet in particular in MAPE-filtered spin-diffusion experiments, the early rising intensities of the rigid-interfacial and the rigid phases are rather weak, such that f 0 ri cannot be fitted with good accuracy in this range (see full symbols in Fig. 5(b)). However, in writing Eq. (1), this ambiguity does not affect the result for f r at all. As expected, in the MAPE-filtered experiment the rigid-interfacial fraction decreases with spin-diffusion time (as it is polarized first), and reaches a final value of about 10 25%. In particular for DQ-filtered spin-diffusion experiments, where the short-time data is rather accurate, we observed that the results for f 0 ri reached the equilibrium value quicker than expected from simulations (empty symbols and dashed lines in Fig. 5(b)). We suggest to explain this by the fact that this fraction does not form a uniform lamellar interface, but partially exhibits an island-like distribution within the rigid phase (see Fig. 11) otherwise, its very quick polarization within the first ms cannot be explained. This interpretation is also corroborated by the weaker interface signatures in the DQ-filtered spin-diffusion experiments (see Fig. 13). Again, this suggests treating this fraction as part of the rigid phase. Note that for data taken during a spin-diffusion experiment, the overall intensity scale is set by FID(0) at the shortest spindiffusion delay, i.e., if the FIDs taken at later t diff and analyzed by Eq. (1), the obtained f r,m are multiplied by FID(0,t diff )/ FID(0,t diff ¼ 0) in order not to normalize the relaxation effect away (yet, this simple option is also tested below). We remind again that the fitting result f r is always multiplied by 1.1 to correct for the 10% loss during the final MSE, and the fractions are re-normalized accordingly. Results for the rigid fraction at three different temperatures are shown in Fig. 6. A sample fitting omitting the rigid interface (f 0 ri ¼ 0 in Eq. (1)) and letting the remaining parameters vary (including now a varying exponent n r for the rigid fraction), is also shown. In this case, the rigid fraction is underestimated by about 10%, which may be acceptable. A much simpler and practical linear back-extrapolation of the remaining amorphous part from about ms gives even better results, namely f r ¼ 62% instead of 65%. This procedure is obviously less biased by the existence of the rigid-interfacial component, which in the two-component fit as partially attributed to the mobile phase. In this respect, an even more reliable procedure recommendable for high-throughput processing is to just read off the FID intensity at a time where the rigid and rigid-interfacial parts have practically decayed to zero (70 ms) and back-extrapolate this intensity (i.e., multiplying it by a constant factor) using the mobile-phase relaxation parameters (see Eq. (1)). Finally, note that since we perform a direct analysis of the FID, the measured T n 2x values are apparent relaxation times, not corresponding to the real T 2. Notice also that the fraction f r is in terms of number of protons associated to the rigid phase. Since the two components of the copolymer have different chemical nature and density, it does not correspond to the weight fraction of the PS component. Only after correction for proton density, this defines the ratio between the phase dimensions (characterized by d mob and d rig when purely lamellar geometry is assumed). Normalized intensity full (refocussed) FID; f r = 7, f ri ' = 0.18 ~10% mobile loss due to 500 μs MAPE filter DQ-filtered rigid: MAPE-filtered mobile: Gaussian,T r = 18.3 μs T m = 0.73 ms, ν m = 1.24 rigid-interfacial: T ri = 25.5 μs, ν ri = Acquisition time (ms) Rigid-interfacial fraction f ri ' K MAPE 300 K MAPE 345 K DQ 300 K DQ sqrt (Spin diffusion time / ms) Fig. 5. FID decomposition at 315 K (a) and rigid-interfacial fraction during spin-diffusion experiments (b). In (a), thin lines are experimental data and the thick background lines are the fits. The full and MAPE-filtered FIDs in are to scale, so that the loss of mobile-phase signal during the filter can be quantified. The DQ-filtered FID data and the rigid-interfacial fit are scaled to match the full FID. In (b), solid und open symbols represent the rigid-interfacial fraction detected during MAPE- and DQ-filtered experiments, respectively. The lines indicate the qualitative trends for f 0 ri expected from corresponding simulations assuming an even interfacial layer.

6 130 M. Mauri et al. / Solid State Nuclear Magnetic Resonance 34 (2008) Normalized intensity 1.0 T / K f r K 68.8% 315K 66.7% K 64.8% (lin. 62%) K 62.5% 360K 60.3% Acquisition time / ms Fig. 6. FIDs acquired at different temperatures, normalized to emphasize the changes in the line shape due to changes in the mobile fraction. The solid line is a fit to Eq. (1) with f 0 ri ¼ 0 (only two components), yielding f r ¼ 0, and the dashed line is a linear back-extrapolation of the mobile fraction from 55 to 90 ms, yielding f r ¼ 2. A temperature dependence of f r and thus of the ratio between the different domain sizes is also apparent from Fig. 6, with the rigid fraction decreasing from 69% at 300 K to 60% at 360 K. The effect is explained by progressive mobilization of PS located at the interfacial regions, as will be shown in an upcoming communication. A combined study of this phenomenon, using also high-field MAS NMR, will be reported elsewhere [24]. 3. Mathematical approach to the study of spin diffusion 3.1. Analytical/numerical solution including relaxation The diffusion equation has been solved for different geometries in the case of material, heat and electricity diffusion problems [29], and conceptually similar solutions exist for fitting the spin-diffusion curves. For applications in heterogeneous polymer systems with lamellar or similar structure, analytical solutions were given for two [19] and three [30,31] regions in one or in two and three [32] dimensions. These results fit very well the magnetization behavior for measurements in high fields, where the influence of the longitudinal relaxation can be neglected. However, in cost-efficient low-field NMR instruments T 1 tends to be in the tens of milliseconds range, and the effect of T 1 relaxation during the spin-diffusion time becomes nonnegligible. In the case of diffusion out of a region of length d with diffusion coefficient D, the process will have a characteristic diffusion time d 2 /D, and spin diffusion will dominate the evolution of magnetizationonlyinthecased 2 /D5T 1 [20]. Thisconditionisnot fulfilled at low field for typical lamellar sizes, and methods for reducing the T 1 influence (e.g., modified Goldman Shen techniques [33,34] or correction by back-multiplication with a raising exponential function [5,6]) were tested by different authors. These techniques work well only when T 1 is the same in all regions; however, in general there are also different intrinsic values of T 1 for the different domains. For the correct consideration of T 1 in the diffusion process, the diffusion equation for the magnetization M as a function of space r and time t should include a relaxation term, and thus becomes: " # qmðr; tþ ðk 1Þ qmðr; tþ ¼ D þ q2 Mðr; tþ qt r qr qr 2 þ 1 T 1 ½M 0 Mðr; tþš, (2) where M is the magnetization, with equilibrium value M 0, D is the diffusion coefficient and T 1 is the longitudinal relaxation time. Analytical solutions are very complicated, but for some special cases, numerical solutions were treated in a number of papers [19 22,34,35]. With constant diffusion coefficient D and longitudinal relaxation time T 1 within each region, Eq. (2) can be solved for each region with boundary conditions corresponding to lamellar systems (k ¼ 1), hexagonally packed cylinders (k ¼ 2) and spheres on a hexagonally packed lattice (k ¼ 3). Analytical and numerical approaches to these solutions take advantage of the boundary conditions dictated by the geometry. The case of three cyclically arranged lamellar domains (e.g. crystalline, amorphous and an intermediate region) is exemplified in Fig. 7. For symmetry reasons, the condition qmðr; tþ ¼ 0 (3) qr must be fulfilled at the symmetry planes, while at the boundary planes (here shown for A/B), the magnetization must be continuous across the boundary, M A ðr AB ; tþ ¼M B ðr AB ; tþ (4) and the boundaries cannot act as sources or sinks of magnetization, r A D A qm A ðr AB ; tþ qr qm ¼ r B ðr AB ; tþ B D B. (5) qr Here, the spin densities r A/B need to be taken into account. For calculation of the total magnetization of each domain, which is the NMR observable, one needs to integrate the magnetization over the specific region, multiplied by the corresponding spin density (which corresponds to summing up over the discrete space elements of our numerical simulation). For the main purpose of this paper, the 3-domain lamellar geometry is chosen to represent the material, and the mathematical constraints were included in a generalized program for the numerical solution of the spin-diffusion equation with a large variability of parameters and applications. Importantly, the program can simulate relaxation or spin-diffusion data (depending on initial and final magnetization states), as well as fit simulation results to experimental data. In addition, the influence of polydispersity of the lamellar sizes can also be explored by simulations. The features of the program and the numerical implementation are explained in Appendix A, where we also highlight applications to diffusion in higher dimensions (cylindrical and spherical domains). Note that the fitting not only comprises domain sizes (the primary target quantities), but also the T 1 relaxation times of the different regions and even the Fig. 7. Representation of three cyclically arranged lamellar domains A, B, C with different values for D, T 1, d, M(0), and spin density r. Dashed lines indicate symmetry planes, solid lines boundary planes.

7 M. Mauri et al. / Solid State Nuclear Magnetic Resonance 34 (2008) diffusion coefficients. However, the latter can only be fitted when at least one thickness or one other diffusion coefficient is known (since diffusion processes are scale-invariant) The initial-rate approximation (IRA) An alternative to numerical treatment is the simple initialslope approximation [5 7], which is particularly feasible for samples with an (a priori unknown) distribution of domain sizes, where the model on which the numerical solution is based may be pffiffiffiffi hard to verify or falsify. In principle, the intercept t 0 of the extrapolation of the initial slope of the normalized intensity of the source region against the square root of the mixing time with pffiffiffiffi the root-of-time axis can be used, as well as the intercept t 0 of the extrapolation of the initial slope of the normalized intensity of the sink region against the square root of the mixing time with the parallel to the root-of-time axis across the initial value of the source curve [7]. This is sketched in Fig. 8, on the example of data that exhibits strong relaxation effects. In a perfectly lamellar system and in the absence of relaxation, both intercepts should be the same and give a consistent value for the size of the domain that acts as the source, while the other size results from the knowledge of the phase composition. In a mobile-phase selected experiment, with the dimensionality k (k ¼ 1 for a lamellar system, 2 for cylinders, 3 for spheres or cubes), the width d mob of the mobile phase directly follows from pffiffiffiffi for the source as well as for the sink curve [5]: t 0 d mob ¼ p 4k ffiffiffi p r mob pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r rig D mob D rig p ffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi t 0. (6) D mob þ r rig D rig Note that in the equivalent equations in Refs. [6] and [7], the second fraction is subsumed in an effective spin-diffusion coefficient, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D eff ¼ D mob D rig pffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi þ =2 Normalized intensity D mob D rig total magnetization = FID (0,t diff )/FID(0,0) mobile - source rigid - sink rigid-interfacial t 0,r 5 10 t 0,m sqrt (Spin diffusion time / ms) Fig. 8. Typical results of a mobile-phase filtered spin-diffusion experiment performed at low field (345 K), where T 1 effects are strong. Different ways of performing the initial-rate approximation, using the source or the sink phase are shown. Note that the two intercepts should be equal (as they are in the absence of relaxation), thus T 1 effects must be corrected for. The dashed line represents the theoretical plateau at 63% rigid phase. Open circles are the overall rigid-interfacial fraction, f 0 ri f r FID(0, t diff )/FID(0,0), that we always treat as part of the rigid phase. The delayed initial decay and rise are due to an invisible mobileinterfacial fraction that is addressed in Section 5.4. However in these references, the spin densities were neglected. An equivalent expression holds for d rig from a rigid-phase selected experiment, where, however, the spin density r mob must be used in the numerator. With d mob determined by TEM and SAXS experiments on different model block copolymers, and D rig from other measurements, the authors of ref. [7] were able to determine a calibration of the diffusion coefficient of the mobile phase as a function of T 2 measured in Hahn-echo or CPMG experiments, of course, for negligible T 1 influence only. We have used this calibration below. Now, for the inclusion of the influence of the longitudinal relaxation, a suitable T 1 back correction must be incorporated in the initial-rate approximation method. A priori, one might assume that for the source curve in a mobile-phase selected experiment, the selective T 1 of the mobile phase (that is usually shorter) should to be chosen, whereas T 1 values of both domains influence the initial part of the sink curve. Unfortunately, the correct selective T 1 values can be determined only by a numerical solution of the diffusion equation. Therefore, by comparison with such solutions, a recommendation for the (more advantageous general) use of the sink curve and for a suitable choice of the effective T 1 time will be discussed in Section 4.2, allowing practical applications of the simpler initial-rate approximation technique. 4. Simulation results for combined spin diffusion and relaxation In this section, we discuss the interrelation of diffusion and relaxation using numerical solutions of the diffusion equation. We have simulated many scenarios, and derived conclusions on the basis of the analysis of such exact data by simpler means, e.g., we performed fits based on the neglect or simple phenomenological correction of the T 1 effects. First conclusions on the feasibility, limitations and errors of simple fitting of experimental spin-diffusion data are presented, and distribution effects of lamellar sizes and effects of spin diffusion of higher dimensionality are addressed Discussion of errors due to the interplay T 1 and spin diffusion The statement in Section 3.1, that a, the ratio between the longitudinal relaxation time T 1 and the time d 2 /D, characteristic for the spin diffusion (a ¼ T 1 D/d 2 ), is crucial for deciding if the relaxation must be taken into account for the investigation of the spin-diffusion process, will now be substantiated more quantitatively. In this section, we merely summarize observations made from various simulations using different parameters. For simplicity, these discussions are restricted to two domains, for which the same ratio a was used. As a guiding line, the following limiting cases can be defined: On the one hand, in a MAPE or DQ-filtered experiment, the long time behavior of the magnetization for both regions converges in the relevant time region to a constant nonzero value only if there is no influence of relaxation. On the other hand, without diffusion the signal for the initially non-polarized region remains zero for all times, and the relaxation time for the other component can of course be found by usual fitting. We first address the possible determination of T 1 from a onephase-filtered spin-diffusion experiment by simply analyzing the initial decays (the first 10% signal loss) of the source or the sum signals (both plotted in Fig. 8). Not unsurprisingly, from the diffusion-dominated source signal, even for the case of rather slow diffusion (a ¼ 0.1), the simulations show that T 1 of the source component can be estimated only with an error of about 20 25%, if the diffusion is not considered (in the case of SR, the error can be even larger). To the contrary, the sum signal of a mobile-phase MAPE-filtered experiment indeed provides a very good estimate of

8 132 M. Mauri et al. / Solid State Nuclear Magnetic Resonance 34 (2008) the shorter T 1, as the diffusion effect arises only indirectly in that the influence of the longer T 1 of the rigid phase contributes at later times, once it grows in significantly. In a rigid-phase filtered experiment, the value is much less reliable, as the spin-diffusion mediated contribution of the shorter mobile T 1 grows in earlier on the timescale of the longer rigid T 1. In all cases, an apparent nonexponentiality of the decay (appearance of a second component) represents a good criterion as to the decreasing reliability of the T 1 determination. The more interesting case is when diffusion more or less dominates the process (a41), i.e. the question, for which conditions the T 1 influence can be neglected in actual spindiffusion experiments. Simulations for several parameter sets showed that the root-mean-squared (rms) deviation of both source and sink curves (in all cases relative to the common maximum initial signal, which we usually normalize to 1, see Fig. 8) without relaxation (T 1 -N) from those with the same parameters (except for T 1, a from 2 to 20) are approximately 12.5/ a%. A better fit of these curves with other parameter sets, but without relaxation, could not be achieved, as the fitting procedure with this model without relaxation shows always very different curve shapes (this can be well understood, because the relaxation decay of the curves for longer times cannot be explained by the diffusion process alone). The above-mentioned rms error estimate can help to identify the limitations for using a solution without consideration of relaxation. e.g., with T 1 being ten times longer than the characteristic diffusion time (i.e. a ¼ 10), the tolerable mean deviation is about 1.3%. However, unlike the fitting of artificial (simulated) data, fitting of experimental data may be subject to other uncertainties. The dependence on noise can be checked by simple decreasing the signal-to-noise ratio, while the delineation of the systematic influence of, e.g., domain polydispersity, requires other sources of information (e.g., TEM) on at least a similar sample. Further, considerable ambiguities are expected for a values that are not much larger than 1. Deviations that follow the prediction can in such cases result in rather different parameter sets (e.g. for a ¼ 0.5, another fit with a 25% deviation was found with about 50% changes in the diffusion coefficients and up to 20% in the relaxation times). As mentioned, an approach to correct the T 1 influence is the multiplication of the measured curves with an exponential function with positive exponent t/t 1,whereT 1 has been determined by an independent experiment, and this treatment is of course correct only for homogeneous T 1 relaxation. Simulations showed once more, that in the case of equal relaxation times in the source and the sink regions, the results of the T 1 correction are in full agreement with the calculations including relaxation. However, if there is a difference in the relaxation behavior of the two regions (which is usually the case), the correction method fails already for a difference in the relaxation times of about 5%! In fact, one can find an apparent T 1 correction value (somewhere between the two relaxation times), for which both curves reach a plateau value for long times, but the curve shapes differ strongly from the correct ones, and even the plateau values are incorrect and do not represent the phase composition. Yet, the initial part of both curves is always in good agreement with the correct results, and the best coincidence is found when the correct T 1 of the source region is used for correction. However this value, as pointed out above, is not in all cases experimentally accessible. We will show in the following that the use of the sink signal can in fact be more advantageous Comparison with the initial-rate approximation In order to formulate suggestions for a simple data treatment, the simulation can also be used for checking the applicability of several variants of the initial-rate approximation. As a start, we could confirm a well-known fact for the case without relaxation: Even though it seems a bit surprising that the initial slope is independent of an interchange of D mob and D rig, and of changes in d rig (see Eq. (6)), the comparisons of d mob values calculated by this equation with those of the exact numerical simulations showed only differences between less than 1 and about 5%, where the largest deviations were found in the cases when the diffusion coefficients and the volume of the regions are very different. Since an exponential curve, representing the T 1 effect, has an initial Gaussian-like behavior in a representation over the square root of the time, the very initial part of the source diffusion curve is in fact not exactly linear in this representation. This is clearly seen in simulations, and it is in principle possible to use this fact as a diagnostic criterion, however we note that this really pertains only to the first few percent of decay, which is usually masked by other uncertainties (noise or the inability to measure very short diffusion times, where unwanted and spurious coherences can contribute, and most importantly, interphases). On the other hand, as stated in the last section, the initial source decay can faithfully be analyzed if the correct value for T 1 of the source region is used. This T 1 value has to be measured selectively in a separate experiment, in which the effect of spin diffusion must be eliminated, which is not trivial. As pointed out above, it is only possible to extract a reliable value for the shorter of the two T 1 values from the initial sum decay of a mobile-phase selected spin-diffusion experiment. More generally, we found that the sink (build-up) curve is in fact the better candidate for initial-rate approximation. The sink results turned out to be much less dependent on the variation of (i) the actual value of the apparent T 1 correction time, (ii) the data interval over which the straight line is approximated to the measured curve, and (iii) the experimental uncertainties related to deviations from ideal behavior. In order to correct for the relaxation effect, however, both relaxation times, for the source and the sink region, must be considered in this case, because both affect the magnetization behavior already at the very beginning. In Table 1, we compile results of fits to simulated data covering a typical range of sizes and relaxation times. From simulations of spin-diffusion data, initial relaxation times T ini 1 were obtained from single exponential fits to the first 10% of overall sum (source+sink) decay. From SR data, apparent relaxation times were obtained by mono-exponential fits to the individual phaseresolved intensities, and to the total rise. We also tabulate the arithmetic mean, T app 1avg ¼ðTapp 1tot þ Tapp Þ=2. The last columns give 1mob the IRA results for d, using T ini 1, Tapp 1mob, Tapp 1tot, and Tapp 1avg in an exponential up-correction of source (T ini 1 ) and sink-curve (3 different T app 1 ) spin-diffusion data. The latter three options for sink-curve analysis were in all cases superior to using the much longer T app 1rig, which indicates that the mobile phase (the one with the shorter T 1 ) contributes the dominant relaxation effect of the sink signal in all cases. The region over which the curves were fitted to a straight line was chosen to cover a region up to 20% overall intensity loss (up to 6ms 0.5 ), sometimes less if the linear region was visibly shorter. Inspection of the tabulated data immediately reveals the overall best strategy: IRA analysis is best applied to sink (build-up) data from either mobile-phase (MAPE) or rigid-phase (DQ) selected experiments, both corrected by the arithmetic average of the shorter (mobile) and the total apparent T 1 s from a phase-resolved SR experiment, yielding the mobile or rigid domain sizes, respectively. It should however be noted that still within 10% accuracy, the mobile-domain thickness can also be obtained by IRA analysis of the source decay, corrected by T ini 1mob, that (in contrast to Tini 1rig from a DQ-filtered experiment) is reliably obtained from the sum signal