Intracrystalline Jump Motion in Poly(ethylene oxide) - ACS Publications

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Intracrystalline Jump Motion in Poly(ethylene oxide) Lamellae of Variable Thickness: A Comparison of NMR Methods Ricardo Kurz,† Anja Achilles,† Wei Chen,§ Mareen Schaf̈ er,† Anne Seidlitz,† Yury Golitsyn,† Jörg Kressler,‡ Wolfgang Paul,† Günter Hempel,† Toshikazu Miyoshi,§ Thomas Thurn-Albrecht,† and Kay Saalwac̈ hter*,† †

Institut für Physik and ‡Institut für Chemie, Martin-Luther-Universität Halle-Wittenberg, D-06099 Halle, Germany Department of Polymer Science, The University of Akron, Akron, Ohio 44325-3909, United States

§

ABSTRACT: Helical jumps in poly(ethylene oxide), which are the molecular processes underlying the intracrystalline chain diffusion, are studied on the microseconds to milliseconds time scale by means of NMR. Using a simple proton time-domain technique, a wide range of melt-crystallized morphologies is investigated ranging from extendedchain crystals of short chains to crystals with disordered fold surfaces of longer chains up to 190 kg/mol. From variable-temperature data we directly determine the Arrhenius activation parameters and find that the activation energy is always around 65 kJ/mol. At a given temperature, average correlation times vary from sample to sample over about 1 decade and increase approximately linearly with the lamellar thickness. The observed linear relation is reproduced by a generic Monte Carlo simulation model implementing a mechanism of diffusing defects. The experimental results are compared to 1D carbon-13 MAS exchange NMR (CODEX) and proton rotating-frame relaxation (R1ρ) data, for which we highlight the challenges and significant bias effects arising from the significant distribution of correlation times. Effective spin-diffusion averaging of the proton R1ρ demonstrates that monomers with different jump dynamics are spatially close; i.e., they coexist in neighboring stems.

I. INTRODUCTION Semicrystalline polymers are industrially applied in many fields. Depending on chemical structure and processing, their macroscopic mechanical properties can differ substantially. When crystallized from the melt, the morphology will be semicrystalline (SC), consisting of crystalline lamellae separated by remaining amorphous layers with higher segmental mobility.1 Although there is a high degree of order in crystalline lamellae, there exists a chain diffusion process in the crystalline regions of many SC polymers, the so-called αc-relaxation,2 the presence of which dictates the macroscopic characteristics such as flexibility, creep, brittleness3,4 and drawability.5 Polyethylene (PE) is the polymer that has been studied most thoroughly with regards to the αc-relaxation,2 mostly using mechanical and dielectric spectroscopy. Early studies mostly concentrated on its influence on macroscopic properties, focusing on single crystals prepared from the dilute solution as model systems.6,7 Ashcraft and Boyd showed that the αc-type relaxation is indeed associated with the crystalline region of SC polymers.8 In this work, we build upon and extend the even earlier work of Hikichi and Furuichi9 as well as Olf and Peterlin,10−12 who used simple proton wide-line NMR observables to detect intracrystalline motions referred to as “oscillations” or “rotations” in different PE10,12 and poly(ethylene oxide) (PEO) samples9,11at that time without time scale information. Only much later (see below) this was unambiguously explained by symmetry-conserving jumps of the monomers and © XXXX American Chemical Society

thus of whole stems. Already in these early days it was assumed that the morphology, i.e., mainly the crystallinity and the lamellar thickness dc, as varied by changing the isothermal crystallization temperature Tc,1 may play an important role.13 However, due to the qualitative and indirect nature of the early NMR and mechanical as well as dielectric results, respectively (the latter relying on motions of oxidative defects such as CO groups), no clear picture was obtained. Boyd later introduced a classification into low-, medium-, and high-crystallinity polymers, wherein the high-crystallinity class was suggested to be associated with the presence of an αcprocess.13 In their important more recent work collecting direct NMR3 and indirect (mechanical and dielectric spectroscopy) evidence for different polymers, Hu and Schmidt-Rohr demonstrated that the presence of an αc-relaxation is directly related to macroscopic ultradrawability.5 The former is in turn associated with helical jumps, upon which the chain undergoes simultaneous rotation and translation through the crystal.3 Finally, a classification of SC polymers into αc-mobile and crystal-fixed polymers was proposed, supporting Boyd’s earlier suggestion. Up to the present day an exact understanding of the αcrelaxation on the molecular scale is still under discussion. Early on, Reneker associated the αc-process to certain defect species Received: April 24, 2017

A

DOI: 10.1021/acs.macromol.7b00843 Macromolecules XXXX, XXX, XXX−XXX

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Detailed NMR investigations of other αc-mobile polymers are much rarer and mostly limited to small numbers of samples.3,5,27−32 Among the 13C NMR methods used, the centerband-only detection of the exchange (CODEX) technique,27 probing reorientations of the chemical-shift anisotropy (CSA) tensor and relying on high-resolution by magic-angle spinning (MAS), stands out as the most efficient (but still comparably time-consuming) method in current use.33 Here, we report on a detailed study of the SC polymer poly(ethylene oxide), PEO. It has long been known to exhibit an αc-relaxation that is about as fast as in PE at comparable undercooling below Tm9,11,34 and jumps along its 72 helix have been identified as the molecular origin of the αc-mobility.3,31 We rely on the molecular weight and Tc dependence of the lamellar thickness dc,1 and use even simpler 1H NMR timedomain data to characterize the helical-jump motion in PEO samples with widely variable dc, spanning the range between extended-chain crystals of short-chain and switchboard morphologies of long-chain samples. We compare our results with alternative NMR results using 13C CODEX27 and 1H R1ρ35 and explain the origin of the significant deviations between those and our results, which arise from ensemble-bias and specific averaging effects, respectively, inherent to these other approaches, in all cases caused by the significant correlation time distribution.

that diffuse through the crystal.14,15 Mansfield and Boyd employed conformational-energy calculations for the crystalline regions of paraffins and PE and developed a model based upon a localized 180° twist propagating through the crystal.2 Ewen, Strobl, and Richter discussed the jump process in crystals of long alkanes as defect-driven mechanism, which finally leads to a translation of an extended all-trans chain.16 Therefore, a defect has to originate at one side of the crystal and has to propagate through it to let one stem perform translational motions. Early molecular dynamics (MD) simulations suggested that chain diffusion over large scales is activated by different sorts of defects, i.e., a combination of localized defects as well as longer-range soft twists.17 In this work, the process was also found to be a local phenomenon independent of crystal thickness,17 which remains unproven, since observations probing the αc-relaxation macroscopically (and thus indirectly) do reveal a dependence.13 More recently, Mowry and Rutledge studied the defect migration and chain translocation in PE crystals by atomistic MD simulation, with the result that a specific combination of different conformational defect structures is involved in the effective defect migration from one unit cell to the next.18 Nowadays, NMR has emerged as the most suitable method to investigate the intracrystalline jump motion in SC polymers,3 since it provides direct access to the molecular process. In addition to the very early 1H line shape investigations of PE,10,12 the use of rotating-frame relaxation rates (R1ρ) to estimate the time scale of the molecular process has also first been described in the 1960s.19 Only much later, a rather sophisticated and time-consuming dipolar 13C exchange NMR method was used to directly and unambiguously prove the presence of 180° helical jumps in PE crystallites.20 Less sophisticated but still time-consuming 13C 1D and 2D exchange NMR methods3 probing the large-scale exchange of monomers between the crystalline and the amorphous regions of PE were extensively applied to probe the time scale of the chain diffusivity and the associated activation energy.21−24 In particular, Yao et al.23,24 revealed a significant dependence on sample morphology, comparing predominantly adjacentreentry crystals obtained as reactor powders or by solution crystallization with melt-crystallized samples featuring a disordered switchboard-type fold surface. Specifically, the process was found to be faster and feature a lower activation energy in the former case, which was rationalized in terms of a reduced entropic barrier. Furthermore, they introduced the notion of an effective jump rate, explaining deviations in activation energies between the actual local chain flips and the effective process leading to chain diffusion into and out of the crystal. Only more recently, our group revived the possibility to use rather simple and time-efficient 1H NMR methods, which probe directly the jump process rather than the larger-scale chain diffusivity in chemically sufficiently simple samples, such as linear homopolymers without hydrogen-containing conformationally flexible side groups. Owing to the rather small spectral changes associated with the 180° jump motion in PE, we resorted to measuring the transverse (T2) relaxation behavior under a dipolar-refocusing magic-sandwich echo pulse sequence.25 A comparison of different adjacent-reentry and switchboard-type crystal morphologies26 confirmed the earlier observation of Yao et al. now on a local scale, with the fastest chain jumps occurring in rather finite-size nanocrystals featuring adjacent reentry.

II. EXPERIMENTAL DETAILS Samples. PEO samples from different sources with various weightaveraged molecular weights and polydispersities below 1.2, crystallized isothermally from the melt at different temperatures Tc, were investigated (sample names PEOn-m with n being Mw in kg/mol and m being Tc in °C). PEO187 and PEO53 were purchased from Polymer Standards Service. PEO6 and PEO1 were provided by SigmaAldrich. The Mws of PEO187 and of PEO53 are well above the critical molecular weight Mc = 5.9 kg/mol,36 while PEO6 and PEO1 are not entangled. All samples except PEO5-25s were crystallized under ambient conditions at a given Tc for several days after annealing in the melt at 90° for about 1 h prior to cooling to Tc at about 1 K/s. They were then cooled down to −50 °C at about 3 K/s and afterward reheated to the individual measurement temperatures at about 10 K/s. At each given temperature the samples were annealed for 30 min prior to the experimental duration of another 30 min. To check for effects of ambient humidity and oxygen, a series of experiments using a sample sealed in an evacuated glass tube did not afford any differences. PEO525s was crystallized from dilute acetonitrile solution (0.01 wt %) and consists of cubic single crystals of 10 nm thickness. Long periods L were measured by small-angle X-ray scattering (SAXS) using a Kratky compact camera equipped with a focusing Xray optics and a temperature-controlled sample holder (see refs 37 and 38 for details). The melting temperatures were determined by differential scanning calorimetry (DSC). Crystallinities fcr were obtained by NMR (see below) and by analysis of the SAXS interface distribution function,38,39 assuming a perfect lamellar stack, with overall good agreement. The reported lamellar thickness dc was derived from the NMR-based weight crystallinity fcr: dc = Lfcr,NMR (we neglect a small systematic error arising from the different densities of the crystalline and amorphous phases). The dc values vary significantly with Mw and Tc, while the melting points (maxima positions in DSC) primarily depend on Mw. Relevant sample parameters are summarized in Table 1; the crystallinity results are discussed below. All samples except PEO1-30 and PEO5-25s form switchboard-like semicrystalline structures,1 with highly ordered crystalline lamellae, disordered fold surfaces, and amorphous regions exhibiting fast segmental motions at temperatures above its Tg located at about −60 °C. PEO5-25s is a powder of single crystals with nominally twicefolded chains, while PEO1-30 features extended-chain crystals with B


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for the individual experimental temperatures Texp (given in K). In this way, the full signal at t = 0 is always defined, overcoming ambiguities related to back-extrapolating the quickly decaying crystalline signal. The FIDs were decomposed into three fractions following the strategies published by Hansen et al.44 and Schäler et al.,42,43 who did similar analyses for other SC polymers. The strongly homonuclear dipole−dipole coupled fraction fcr represents the highly ordered chains in the crystalline lamellae, while an intermediately mobile interphase and a mobile fraction, f i and fam, respectively, make up the amorphous phase. The MAPE-filtered FIDs reflect the most mobile amorphous fraction quantitatively, provided that the filter time is chosen appropriately.42 Since the helical-jump motion within the crystalline phase is strongly anisotropic, proton homonuclear dipole−dipole couplings always remain strong, leading to comparably fast dipolar dephasing of the associated signal. This enables a clear distinction from the amorphous fraction, the MAPE-filtered more mobile part of which can be fitted by a stretched exponential function of the form fam(t) = fam exp{−(t/T*2am)νa}. After subtraction of this component (see Figure 1), the two remaining fractions can be fitted by

Table 1. Sample Parameters

a

sample

Tm (°C)

La (nm)

dc (nm)

fcr,NMRa,b

PEO187-60 PEO187-54 PEO187-30 PEO53-60 PEO53-54 PEO53-30 PEO6-30 PEO5-25s PEO1-30

63 63 63 61 61 61 54 60 39

60.9 40.5 22.7 41.6 32.8 19.9 19.9 10c 6.9d

52.4 32.2 16.9 37.2 27.6 15.3 17.5 9.7 5.9

0.86 0.85 0.75 0.89 0.91 0.76 0.89 0.97 0.94

Experimental error ca. 5%. bAt 25−30 °C. cDetermined by AFM. Average of two populations.

d

two long-period populations (6.3 and 7.5 nm) due to fractionated crystallization even within the rather low polydispersity of 1.03.40 NMR Spectroscopy. 1H NMR. These measurements were performed on a 200 MHz Bruker Avance III spectrometer equipped with a static 5 mm Bruker probe featuring a short dead time (2.5 μs). The temperature accuracy is estimated to about ±1 K with a gradient of 0.5 K over the sample. 90° excitation pulses of 2 μs length and recycle delays (d1) of 10−30 s were applied, the latter fulfilling d1 ≫ 5T1. Heating and cooling was realized with a flow of dried air, where for lower temperatures an AirJet XRII851 cooling device was used. Free-induction decay (FID) signals were acquired exactly onresonance in pure-absorption mode, analyzing only the decaying real part of the signal (the imaginary part is always near zero). Furthermore, the magic and polarization echo (MAPE) sequence with an overall filter duration of 0.6 ms was used as a filter to selectively suppress the crystalline signal as an aid in component decomposition.41,42 All FIDs were only analyzed up to t = 200 μs acquisition time, since at longer acquisition times field-inhomogeneity and sample susceptibility effects distort the signal and preclude proper fitting.42,43 13 C 1D MAS Exchange NMR (CODEX). These experiments27 were performed on 300 as well as 400 MHz Bruker Avance systems (13C Larmor frequencies of 75.5 and 100.6 MHz) using Bruker 4 mm MAS double-resonance probes. The rotor-synchronized CODEX experiments were performed at a spinning rate of 4000 ± 2 Hz. The 90° pulse lengths for 1H and 13C were both set to 4 μs. The crosspolarization (CP) and recycle delays (d1) were 1 ms and 2−8 s, respectively. On the 1H channel, high-power proton decoupling (CW or TPPM) was used during 13C evolution and signal acquisition periods. On the 13C channel, the recoupled evolution time Ntr was set to 2.0 ms, with tr the MAS period (250 μs) and N an even integer multiple. The 13C CSA was recoupled by rotor-synchronized 180° pulses during two evolution periods, which were separated by a mixing time, tmix. The exchange-reduced intensity S(tmix,Ntr) is thus obtained. Besides slow motions on the seconds−milliseconds scale, other factors, such as T1 and T2 relaxation arising from faster dynamic modes and instability of the NMR spectrometer, could also lead to a signal decay. In order to compensate for the additional decay, a reference spectrum S0 was obtained by switching tmix with a final z-filter time, tz,27 where a tz of one MAS rotor period was chosen. In addition, a correction for exchange by spin diffusion was applied for sample PEO5-25s as described in ref 28. Component Analysis and Crystalline Signals. For all PEO samples regular and MAPE-filtered FID signals were measured in a temperature range from −50 °C up to the melting point. FIDs in the molten state at Tmelt were collected, and the t = 0 extrapolated signals reflecting the full sample signal were used to apply a Curie correction of all data taken at lower temperatures, following

FIDnorm (t , Texp) =

FIDexp(t , Texp) melt

FID

norm FIDcr/ i (t ) = fcr (t ) + fi (t )

= fcr exp{−0.5(at )2 } sin(bt )/bt + fi exp{− (t /T2*i)νi }

where fcr(t) is represented by the so-called Abragam function describing the FID of strongly dipolar-coupled spin systems.45 From the parameters a and b the dipolar second moment of the corresponding spectral (frequency-domain) line shape can be calculated, i.e., M2 = a2 + b2/3. Alternatively, neglecting the slight oscillation, M2 can also be obtained by a fit assuming a Gaussian shape * )2}, then M2 = 2/T2cr *2. It is related to the apparent static∼exp{−(t/T2cr limit dipolar coupling constant via M2 = (9/20)Dstat2. The apparent * and the larger T2i* are typically of the transverse relaxation times T2cr order of ten to several tens of microseconds. By way of this decomposition, the experimental pure crystalline component subject to further analyses can be isolated by another subtraction of the fitted interphase signal (the fitted shape of fcr according to eq 2 is only relevant for the component decomposition). We can of course not exclude that a more immobilized part of the interphase fraction remains to be part of the apparent fcr component and exerts some systematic error. On the basis of the rather high overall crystallinities, we consider this error to be small if not negligible. The right part of Figure 1 demonstrates the significant temperature dependence of the pure crystalline signal arising from the helical-jump motion. Note that Abragam or Gaussian fits to this data are necessarily approximate, as in the dispersion region where the signal changes markedly with temperature, the functional form is qualitatively different (see below). However, enforcing a Gaussian shape, the decay of the fitted M2 is a suitable qualitative indicator of the presence of intracrystalline dynamics;10,12,25 see ref 40 for a sample application to PEO. It is noted that the time-domain decomposition procedure is more reliable than spectral integration/devonvolution procedures, which are subject to baseline problems and uncertainties with regards to the used spectral line shape functions, the analytical form of which is often not known.

III. THEORY 1 H FID Analysis. As is apparent from Figure 1, the intracrystalline jump motion has a large effect on the associated 1 H FID signal, fcr(t), as the helical jumps in PEO lead to a significant reorientation of the 1H−1H dipole−dipole coupling (DDC) tensor associated with the CH2 groups. This enables a direct analysis, in some contrast to our previous study on PE,25,26 where the effect of motional averaging is significantly weaker due to the symmetry conserving 180° jumps (this restricts their effect to the modulation of secondary interchain contributions). While the previous work has focused on the

Texp

(t = 0, Tmelt) Tmelt

(2)

(1) C


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fast-limit dephasing due to the residual anisotropy, and (1 − S2) M2 = ΔMdyn is the magnitude of the fluctuating part of the 2 DDC interaction. Details of the fitting strategy are deferred to the Results and Discussion section. We assess the relationship between τc in eq 3 and the residence time in the helical-jump process by comparison with computer simulations of eq 4, as provided by a first-principles integration of the stochastic Liouville−von Neumann equation for a dipolar spin pair using a home-written code.51 We compare the 72 helical jump in PEO with constrained 3-site (next neighbor back or forth) jumps along the same helix. The latter mimics a potentially constrained translation, e.g., caused by restrictions within the fold surface. The two processes are described by exchange matrices3

Figure 1. Decomposition of a 1H FID into different signal fractions on the example of PEO187-60 measured at 0 °C. (left) A lowtemperature FID (black open squares) is compared with a 0.6 ms MAPE-filtered FID (red solid circles), the latter representing the mobile-amorphous fraction. Subtraction and fitting reveals the intermediately mobile amorphous (green dashed line) and, after its subtraction, the crystalline fraction (blue solid line). (right) Renormalized crystalline FIDs for different temperatures. The black dotted line indicates the fitting limit.

⎛− 2 ⎜ ⎜1 Π 72 = k ⎜ 0 ⎜ ⎜⎜⋮ ⎝1

analysis of magic-sandwich echo decays, which is experimentally more challenging and subject to specific ambiguities, we can here fit the fcr(t) data directly, however, based upon the same theoretical approach. The basis of motional averaging, resulting in a prolongation of the FID signal (i.e., T*2 ) and a narrowing in the spectral domain, is the orientation dependence of the 1H−1H DDC frequency, ωD (θ) = (3/2)D stat P2 [cos θ], θ being the instantaneous principal-axis orientation of the DDC tensor with respect to the B0 field. Related observables measured on samples featuring an overall isotropic ensemble of orientations can be modeled on the basis of Anderson−Weiss (AW) theory,46−49 relying on an assumed exponential shape of the orientation autocorrelation function of the second Legendre polynomial

∫t

t+t

0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟⎟ 0 0 1 −2 ⎠

0 0 1 ⋮

0 0 0 ⋮

0 0 0 ⋮

1 0 0 ⋮

(6)

(7)

respectively. As to the initial conditions, in the former case all sites are evenly populated, while in the latter case only the central site is populated. The reorientation angles around the zaxis of the 72 helix are multiples of 102.86°. Thus, the sites are not evenly distributed around the z-axis in the case of 3-site back-or-forth jumps, as the jump angle is not equal to 120°. The angle is, however, close, which is why the constrained process resembles a symmetric 3-site jump. The comparison of the two scenarios thus enables an assessment of the effect of the number of sites M accessible to the process (7 and 3, respectively). It is noted that the residence time for random jumps connecting M communicating sites (a situation also realized by any 3-site jump process) is τ = (Mk)−1.3

(3)

′

ω[θ(τ )] dτ ]⟩ens, t ′ ′ ⎤ ⎡ 1 ≈ exp⎢ − S2M 2t 2 ⎥ exp[− (1 − S2)M 2τc 2 ⎦ ⎣ 2

(4)

× (e−t / τc + t /τc − 1)]

fnorm cr (t)

0

⎛− 1 1 0 ⎞ ⎜ ⎟ Πc3 = k ⎜1 −2 1 ⎟ ⎝ 0 1 −1⎠

which describes the effect of orientation fluctuations with the correlation time of reorientation τc.46−50 This ansatz correctly describes uniaxial rotational diffusion (diffusion on a cone) as well as helical jumps as a discrete representation. S2 is a dynamic order parameter quantifying the degree of anisotropy of the motion. For a static dipolar spin pair, the FID reads ⟨cos ωD(θ)t⟩ens, while under the given assumptions the effect of random thermal motions can be calculated as47−49 f crnorm (t ) = ⟨cos[

0 1 −2 ⋮

and

C2(t ) = 5⟨P2[cos θ(t ′)]P2[cos θ(t + t ′)]⟩ens, t ′ = S2 + (1 − S2) exp[−t /τc]

1 −2 1 ⋮

(5) Figure 2. Simulated 1H−1H spin-pair FIDs comparing 72 and constrained 3-site helical jumps with the given rate constants k (solid and dashed lines, respectively) and selected fits (solid and open symbols, respectively) based upon the second-moment approximation. The parameters Dstat/2π = 30 kHz (M2 = 16 000 ms−2) and S2 = 0.09 for 72 jumps, the latter arising from β = 45° (tilt of the tensor away from z around the y-axis), were chosen so as to roughly match the experimental results.

2

Here, = fcr(t)/fcr, and M2 = (9/20)Dstat is the second moment of the spectral line shape, which in a real system subsumes also secondary couplings that ultimately lead to Gaussian rather than Pake-type features in the time and spectral domains. The fact that the frequency distribution is close to Gaussian improves the validity range of AW theory, which includes a second-moment approximation. S2M2 describes the D


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Macromolecules Corresponding results are presented in Figure 2, where simulated results are compared with fits to eq 5. It is noted that the initial decay of spin-pair response provided by the simulations is very well fitted, while the long-time oscillations cannot be captured (they are of course much reduced in actual data due to secondary couplings, see Figure 1, thus improving the applicability of the Anderson−Weiss approximation). This is why the fits are restricted to the initial 40% of the decay. At the limits of the range of simulated rate constants k, eq 5 with τc → ∞ and τc → 0 reduces to a Gaussian providing M2 and S2M2, respectively. Only in the dispersion region (k between about 5 × 104/s and 106/s) is the full fit sensitive enough to provide a reliable value for τc. The circles in Figure 2 correspond to such a fit, and it is seen that 72 and 3-site jumps provide different fitted values of S2. The expected input value is uniaxially averaged not reproduced by constrained 3-site jumps, since the 3 sites are not arranged symmetrically, precluding full uniaxial averaging. This emphasizes the need to have data at sufficiently high temperature available to reliably determine S2. Conveniently, however, the value of τc is in both cases close to the calculated residence time for 3-site jumps (1/(3k) = 3.33 μs). We can thus conclude that the fitted τc can be associated with the average residence time in a given raster of the helix, give and take a systematic error of the order of 10% on the relevant logarithmic scale. An experimental test of the outlined 1H FID analysis approach, applied to a model substance in which methyl groups undergo a well-defined 3-site jump, is described in Appendix A. 13 C CODEX. The CODEX technique is sensitive to slow changes in the orientation of the 13C chemical shift anisotropy (CSA) tensor, as probed under MAS conditions by recoupling through rotor-synchronized 180° pulses during two evolution periods separated by tmix. The phase cycle ensures that the detected signal corresponds to an echo in which all evolution is refocused. If, however, molecular motion occurs during tmix, refocusing will not occur during the second evolution time, leading to a decay of the exchange intensity S. The reference spectrum S0, obtained in the limit of short tmix, serves to compensate for possible relaxation effects (T1, T2) and decay due to other imperfections (see the Experimental Details section). Specific limitations of CODEX were previously studied by spin-dynamics simulations51 as well as experimentally.51,52 Here, we extend the simulation work to the PEO case, using the same software as above,51 assuming δ-pulses. Exchange decays for the two scenarios, i.e., 72 helical vs constrained 3-site jumps using eq 6 or 7 as simulation input, are plotted in Figure 3. Even long mixing times can be efficiently simulated with our code51 by simply suspending the quantum-mechanical propagation during tmix, restricting the time propagation to a mixing between the components of the Liouville state vector (i.e., the density matrices representing the different sites). Again, we are interested in the relationship between the τc, now fitted directly to S/S0 using S(tmix ) = p + (1 − p) exp[− (tmix /τc)β ] S0(tmix = tr)

Figure 3. Simulated 13C CODEX decay data (symbols) comparing 72 and constrained 3-site helical jumps using the experimental parameters and CSA tensor parameters from the literature.53 The S0 intensity was obtained as simulated intensity with tmix,min = tr in analogy to the experiment. It is noted that for k = 500/s S0 is reduced to about 15% of the starting intensity as a result of intermediate-motional (T2) effects.51,52 The lines are fits to eq 8.

specific nonlinear nature of the correlation function probed by CODEX,27,54 being related to the correlation function probed by stimulated-echo and 2H spin-alignment experiments,55,56 it is not equivalent to C2(t) (eq 3). The latter can only be probed in the limit of short evolution (recoupling) times.3 For long recoupling times in the plateau region of the Ntr dependence,27,54 the τc fitted by eq 8 is equivalent to the residence time in a certain jump position, which for jumps among M sites is given by τ = (Mk)−1, and the plateau value p is 1/M for M symmetry-distinguishable sites.3,54−56 The results shown in Figure 3 for the slow (k = 10/s) 3-site jumps are in full support of these arguments, while the decay for the 72 helical jumps is multiexponential. The initial decay for the two cases is nearly identical and is dominated by nextneighbor jumps among three sites, while consecutive jumps further away along the helix, with ultimately diffusive character, contribute longer decorrelation times. An analytical solution is provided in Appendix B and fits the simulated data perfectly. However, for the sake of generality we suggest to use the common KWW fit (β = 0.76) and employ a correction factor for the fitted τc, which is a factor of 1.55 longer than the actual next-neighbor jump correlation (= residence) time. The data shown in Figure 3 for the faster decay (k = 500/s) address intermediate-motional regime effects. These were the subject of previous publications51,52 and are here highlighted for completeness. In this case, the jumps occur already on the time scale of the rotor period; i.e., they are relevant already during recoupling. This means that a part of the tensorial decorrelation occurs during Ntr, reducing the decorrelation, i.e., the effective difference tensor probed during tmix.54 This is demonstrated to affect the apparent plateau value p, which is seen to be increased significantly. The fitted τc does not deviate from the expectation τ = 1.55 × (3k)−1, while the stretching exponent β is reduced. An important effect not visible for the normalized quantity S/S0 is that under the simulated conditions the S0 reference intensity is reduced during Ntr by about 85% as a result of transverse relaxation arising from the intermediate mobility.51,52 This is the source of the most relevant systematic errors, to be discussed below.

(8)

This Kohlrausch−Williams−Watts (KWW) or stretchedexponential function can account for a correlation time distribution and other intrinsic sources of nonexponentiality (e.g., due to a diffusive process; see below). FBecause of the E


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Figure 4. Apparent crystallinities of the melt-crystallized samples. Black and red symbols indicate NMR data, while smaller blue and violet symbols represent corresponding SAXS results. Vertical dotted lines indicate the three different Tc values. All values are subject to an estimated experimental error of ±2%.

IV. RESULTS AND DISCUSSION Sample Crystallinity. The results of the decomposition procedure presented above in terms of crystallinities ( fcr) (see Figure 1) are presented in Figure 4. While the thickness of crystalline lamellae increases strongly with Tc, the crystallinity shows a similar but naturally weaker trend, as the crystallinity detected at and above −10 °C is mostly of the order of ∼80% or even higher (see Table 1). The thickness of the amorphous region was generally found to be only weakly dependent on crystallization conditions. Note that only above −40 °C (Tg + 20 K with Tg = −60 °C) the amorphous phase is sufficiently mobile to be identified in terms of f i, while above −25 °C a mobile-amorphous fraction fam becomes detectable. Within the systematic deviations expected for the different methods, the NMR data compare rather well with the SAXS data. Significant deviations are only apparent above 50 °C. The onset of melting probably modifies the regular lamellar stack structure assumed in the SAXS analysis, leading to systematic errors. The data collected in Table 1 demonstrate that depending on Tc and Mw, different morphologies could be created covering a wide range of structure parameters (dc, L). In combination with samples PEO5-25s (twice-folded single crystals) and PEO1-30 (chain-extended crystal), we also cover a variety of fold surface structures. Intracrystalline Dynamics Probed by 1H FID Data. The renormalized crystalline FID data fcr(t)/fcr shown in Figure 5a are the subject of further analyses. They feature a very short decay time scale due to dipolar dephasing, as discussed above. Because of the approximate nature of eq 5 which cannot describe the features at longer evolution times, we restrict the analyses to the first 40% of the decay. At the lowest temperatures, the data in this range is indistinguishable from a Gaussian decay ∼exp(−(1/2)M2t2), reflecting the static-limit M2 that is representative of strong intra-CH2 primary coupling and the proton density in the crystal. Observations on poly(ϵcaprolactone), a crystal-fixed polymer, have demonstrated a weak, nearly linear temperature dependence up to the melting point43 due to thermal expansion and the excitation of oscillations on the picosecond time scale.57,58 In order to account for this effect, and neglecting a small potential contribution from an inseparable contribution of the immobilized amorphous fraction, M2(T) was fitted in the temperature interval between −55 and −25 °C as M2(T) = [13500(±700) − T × (20(±1)/°C)] kHz2 and extrapolated to

Figure 5. (a) Simultaneous fits of 1H FID data taken at different temperatures using eqs 5 and 10 for two samples as indicated. (b) Normalized mean-square deviation of fits and data for PEO53-30 (left) and fitted S2 value (right) as a function of fixed activation energy. The results for Ea in Table 2 correspond to the median of the interval indicated by the hatched box.

higher temperatures for further analyses. The slope corresponds well to the previous observations.43 Following our previous work on PE,25,26 we assume a lognormal distribution of correlation times 1 p(ln τc , μ , σ ) = exp[− (ln τc − μ)2 /(2σ 2)] (9) 2π σ where μ = ⟨ln τc ⟩. For each temperature, a 50-step approximation of the distribution integral over eq 5 f crdistr (t ) =

3σ

∫−3σ p(ln τc , μ, σ )f crnorm (t , M2 , S2 , τc) d ln τc (10)

was carried out during fitting, using equally spaced intervals on the ln τc scale. The physically relevant average correlation time is calculated from the median of the distribution on the logarithmic scale τc = exp[⟨ln τc⟩] = exp[μ] = τ0 exp[Ea /kT ]

(11)

and is taken to follow an Arrhenius temperature dependence. In this way, data sets for a given sample at all available temperatures were fitted simultaneously, including also temperatures above Tc but before the onset of the final melting. Sample fits are shown in Figure 5a. Note that the observation of virtually no change in the FIDs below −25 °C is rather relevant. Even though the separation of then immobilized amorphous and interphase contributions (which comprise only 20% or less of the signal) is then challenged, we can in this way exclude wide correlation time distributions (large σ). This is actually why the low-T data were included in the simultaneous fit. The free parameters left for fit are the anisotropy parameter S2, the activation energy Ea, the front factor τ0, the standard deviation σ (which roughly corresponds to the fwhm of the F


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Macromolecules Table 2. Activation Parameters, Distribution Widths, and Correlation Times at 303 K for All Samples sample

Eaa (kJ/mol)

τ0 (10−17 s)

σ

τc̅ (30 °C)b (μs)

PEO187-60 PEO187-54 PEO187-30 PEO53-60 PEO53-54 PEO53-30 PEO6-30 PEO5-25s PEO1-30 PEO5-25sc PEO1-30c

66.4 66.7 66.3 58.9 64.9 58.7 68.7 62.2 57.7 72.5 60.1

2.5 2.4 1.0 51.2 3.7 19.4 0.9 10.9 26.2 0.079 7.8

1.0 1.0 0.8 0.8 1.0 1.0 1.1 1.0 2.3 1.0 2.3

13.5 7.8 2.6 7.9 5.8 2.6 6.5 6.3 2.1 2.2 1.8

Figure 6. (a) Correlation times recalculated from the fitted activation parameters (see Table 2); the symbols merely indicate the available experimental temperatures. The gray shaded region indicates the range in which the 1H FIDs are most dependent on τc (see Figure 2). (b) Correlation of τc̅ at 30 °C with the lamellar thickness dc. The red symbols and lines (= linear fits) are predictions from a Monte Carlo model with variable defect generation probability p per defect migration step (see section “Defect Diffusion Model”).

Estimated error ±5 kJ/mol. bEstimated error ±30%. cUsing S2 = 0.13.

a

distribution on a log τ scale, i.e., in units of decades), and an amplitude factor fnorm cr (0) for each temperature. The latter were always very close to 1, thus confirming the accuracy of the intensity normalization inherent to the fit by eq 2. S2 could not be measured directly because the fast limit is not reached before the final melting. It can be assumed that S2 does not depend on the sample (it just reflects the structure of the PEO helix), and this assumption was confirmed by analysis of the normalized mean-square deviation χ2 between fit and data. We have fixed the value of Ea at different values over a given interval and optimized S2, σ, and τ0. For each sample, S2 was about 0.09 ± 0.01 in the best-fit region (see Figure 5b). In order to stabilize the fits, we have therefore fixed S2 = 0.09 in the final fitting run and optimized only σ and τ0 for each Ea value along the search grid. The best-fit Ea and its uncertainty (always about ±5 kJ/mol) were determined from the location and width, respectively, of the region of minimal χ2 up to a 20% increase above the minimum (see also Figure 5b). All obtained parameters are collected in Table 2. For the chain-extended (PEO1-30) and the twice-folded single-crystal (PEO5-25s) samples it was not possible to confirm the value of S2 = 0.09, as these samples could not be measured above 30 °C due to an earlier onset of melting and crystal reorganization, respectively. Following the results of Figure 2, and hypothesizing that the tight register in these samples may only allow for restricted 3-site back and forth jumps, we also performed fits using the value of S2 = 0.13 expected in this case. The corresponding results in Table 2 are not too much different from the general trend. This means that jumps clearly occur also in these samples, but we cannot exclude them to be restricted. The chain-extended sample stands out in that it features a significantly wider τc distribution covering more than 2 decades. Figure 6 illustrates the results in the form of a recalculated Arrhenius plot. We realize that all samples feature a similar activation energy of the order of 65 kJ/mol, but differ in the front factor. The values for τ0 given in Table 2 vary significantly and display an anticorrelation with Ea. This is expected, as the actual correlation times τc, being of the order of several microseconds in the dispersion region around 30 °C (see the gray shaded region and the dashed vertical line), are most sensitively encoded in the data (see Figure 2). Even with small ambiguities in Ea the extrapolation to zero intercept, 1/T → 0, is obviously subject to large ambiguities on the log τc scale. Note that the unrealistically small absolute values of τ0 suggest

a cooperative nature of the jump process, possibly being related to glassy dynamics in the fold surface (see below). In the following we thus focus on analyzing τc̅ (30 °C). These values vary within less than a decade, and we attempted to correlate them with various morphological parameters (L, fcr, Tc). We found the most significant correlation with dc (see Figure 6b). The scatter does not allow us to empirically identify a specific functional dependence; e.g., both a linear dependence of τc̅ or log τc̅ on dc could be reconciled with the data. Since dc is directly related to the crystallization temperature Tc, the correlation of τc̅ with the latter is also visible but less significant. Comparing to earlier work, the conclusion made previously for only two samples of isotactic poly(propylene) crystallized at different Tc, where it was found that samples crystallized at higher Tc feature up to 40 times slower intracrystalline dynamics and higher Ea,30 can thus be supported partially. In that case, the rather large difference probably arises partially from small variations in the helix structures within the unit cells. Our finding of a nearly morphology-independent Ea stands in some contrast to our previous study of different morphologies of PE, where a range of values of 60−114 kJ/mol relevant for the local jump dynamics was covered,25,26 with the most weakly activated samples featuring the fastest jumps being the ones with adjacent-reentry fold surfaces.23,24 Of course, the given series does not comprise samples with many adjacent re-entries per chain, but the fact that the highly crystalline chain-extended and twice-folded morphologies of PEO do not display a significantly different trend is remarkable. Defect Diffusion Model. In trying to rationalize the findings, it should be kept in mind that the defect diffusion mechanism established for PE16,18 implies that in that case the defects are created in the amorphous region, as the introduction of gauche conformers into the all-trans 21 helix requires that additional monomers move into the crystal (“compressional defect”). The residence time of the helical jump process is thus determined by the amount of defects passing through a given part of the chains. Thus, it is determined not only by the activation energy required to move the defect but also by the defect concentration. The question whether the measured Ea is dominated by the local defect barrier or by the barrier involved in creating defects G


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associated with the mean time between defect passages at a given raster site, obtained as ensemble average over many MC trajectories. Since the number of sites along a stem is defined by dc, the only relevant free parameter in the simulation is the probability p to form a defect on the first or last raster position per MC step. A defect jumping away from the stem end is considered annihilated. The results of such simulations are compared to the experimental data in Figure 6b, where the mapping of MC steps onto actual time (scaling along y) is of course arbitrary. The simulation data, subject to weak statistical scatter, are very well represented by linear relationships τc̅ (dc). This is also supported theoretically.59,60 All simulation results were first scaled to have the same intercept and then scaled by a common factor so as to match the fitted experimental slope with the p = 0.01% data. In this way the common intercept was found to be compatible with the fitted experimental value of 0.77 ± 0.56 μs. Lower p values leading to fewer overall defects are thus found to lead to a steeper dc dependence. Unfortunately the large scatter of the experimental results, in particular the large error of the extrapolated intercept τc̅ (dc → 0), does not allow for further conclusions on a specific value for p. For p of the order of 0.01% the simulations generate an average defect concentration of less than one per stem over the given dc range. Yet, the picture does not change qualitatively upon increasing the defect generation probability. For larger values and thus higher defect concentrations, further assumptions on their interaction are required. We assumed “phantom” defects moving independently of each other. However, introducing the possibility of pairwise annihilation was found to not change the qualitative observation of a linear dependence. It is noted that the assumption of defects “reflecting” each other is unrealistic, as this scenario (singlefile diffusion) leads to defect accumulation in the center of the stems. Taken together, the near-linear behavior with an intercept governed by the defect generation rate predicted by the model is robust and compatible with the experimental data. We take this a a strong hint toward the validity of the defect diffusion model. Intracrystalline Dynamics Probed by 13C CODEX NMR. It is noted that the obtained activation energy of the order of 65 kJ/mol is significantly larger than the value found in an earlier study based upon the analysis of 1H R1ρ.35 Therefore, in this and the next section we now turn to an in-depth comparison of different NMR methods, thus highlighting the important limitations inherent to these other techniques. The CODEX method has been widely used to investigate the molecular dynamics in semicrystalline polymers.27,29,30,32,33 It is thus important to evaluate potential shortcomings of this convenient technique, as already addressed in previous publications.51,52 For a comparison with the 1H FID data, CODEX experiments have been performed on three of our samples; see Figure 7 for a compilation of the results. In Figure 7a, the S/S0 exchange decays of a melt-crystallized sample are seen to almost reach the expected plateau value of 1/7. A higher plateau value can be expected in the case of strong intermediate-motional effects (see the Theory section) and/or a contribution from immobile or fast-limit mobile chains,27 while spin diffusion effects even in natural abundance lead to lower values and long-time decay. In the right panel of Figure 7a it is also demonstrated that under the given conditions, T2 effects, again related to

Figure 7. (a) Exemplary CODEX tmix decay curves of the methylene resonance of PEO187-30 (Ntr = 2 ms) at two different temperatures (left) along with reference (S0) decay curves as a function of recoupling time (right). The first point for Ntr = 0 was obtained from an equivalent 13C CP MAS spectrum without recoupling. (b) Comparison of the fitted and corrected correlation times τc from CODEX (see the Theory section) and corresponding data from 1H FID analyses in an Arrhenius representation. On the right ordinate a schematic τc distribution for T = −20 °C is shown, indicating as shaded regions the ensemble members that are retained after T2 filtering during Ntr.

in the crystal−amorphous interphase cannot be answered. Notably, the range 17−63 kJ/mol of local barrier heights estimated by atomistic simulation18 is not compatible with the range of experimental data cited above, indicating that defect creation may be the rate-limiting step in the case of PE. Given a constant rate of defect creation in a given fold surface, combined with their diffusive nature (finite return and annihilation probability without chain translation), one would expect on average fewer local jumps for thicker lamellae, explaining the trend toward slower jump dynamics upon increasing dc observed here and in previous works on other polymers. For the given case of PEO, the 72 helical chain features trans and gauche conformations, which means that both compressional and extensional defects may play a role. Therefore, it is not immediately clear whether the arguments for PE can be applied. Given the current lack of molecular insights and the clearly observed dc dependence, we anyway adopt the hypothesis of defect generation in the fold surface. We have implemented a generic Monte Carlo (MC) simulation model combining single random back or forth jumps of a defect along a stem of given length as represented by a simple linear arrangement of raster sites (i.e., a 1D diffusive process) with a fixed defect generation probability p at either end of the stem. In this model, the “jump time” τc̅ as probed by NMR (identified with the residence time of a monomer in a given position) is H


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refrain from actually applying a correction, since the fitted β partially arise from the mentioned intrinsic nonexponentiality. We conclude this section by stressing that the 1H FID-based method is not only much more time-efficient than CODEX (experimental times of minutes rather than dozens of hours), but in the given case also more reliable due to the complete absence of T2-related ensemble bias effects in a dynamically inhomogeneous sample. The techniques are of course complementary in that their dynamic ranges do not overlap. Also, CODEX can provide (albeit biased) τc values at a single given temperature and is applicable to more complex, possibly polymorphic samples. The 1H FID analysis is in turn restricted to simple systems and morphologies, and data taken at different temperatures need to be analyzed simultaneously for stable fitting. Intracrystalline Dynamics Probed by 1H R1ρ. In earlier work, Johansson and Tegenfeldt investigated chain dynamics in the crystallites and the amorphous regions in PEO via measurements of 1H rotating-frame relaxation rates R1ρ,35 obtaining a significantly lower activation energy of 32 kJ/mol. Since this technique, relying on recording the magnetization decay under a spin-lock condition during radio frequency (rf) irradiation, is simple and popular for the study of dynamics on the time scale of the inverse rf frequency (ω1), we set out to reproduce the previous results. In ref 35, R1ρ was measured during multiple-pulse spin locking62 and analyzed according to standard theory47,63

intermediate motions,51 lead to a significant decay of the S0 reference intensity down to a level below 30%, meaning that the majority of signal is lost in this case. Losses due to imperfect heteronuclear decoupling in the strongly coupled CH2 groups are also expected, but our previous work has shown that these effects cannot explain the large signal loss in PEO.52 The comparison of the two temperatures in Figure 7a (right panel) demonstrates that at −33° at least 50% of the signal is lost due to intermediate motions. Strong bias effects can thus be expected. The observed temperature trend corresponds to the slow-motion branch of the “recoupled T2”, which has a minimum when the motions occur on the time scale of tr.51 In Figure 7b, the CODEX and 1H FID-based results for τc, the former analyzed by fits to eq 8, are compared in an Arrhenius plot. We also include in the background all data from Figure 6. An average upward shift of the CODEX results by almost an order of magnitude is apparent, while the associated Ea (60 ± 10 kJ/mol) is, if at all, somewhat lower. At the lowest temperatures, the extrapolated 1H-based results and the CODEX results appear to merge. Because of the longer time scales probed, a different apparent Ea may not be unexpected, but it is noted that a cooperative non-Arrhenius process, e.g., a Vogel−Fulcher law, would usually exhibit an Ea that increases, rather than decreases, upon cooling. The observations can all be explained by the substantial signal loss during recoupling, combined with the significant correlation time distribution at T = −20 °C of about 1 decade width. A sample distribution is shown on the right ordinate of Figure 7b. At the highest temperatures at which CODEX is feasible, the T2 decay of S0 (and thus also S) will remove those chain segments from the detected ensemble that have correlation times of the order of milliseconds, i.e., that move significantly during tr. Mainly signal from the slowest-moving units is retained, which consequently biases the τc result toward too large values. In addition, if the distribution is sufficiently broad, even the fastest ensemble members may contribute detectable signal, as they could be located on the fast side of the mentioned T2 minimum.51 These fast-limit ensemble members feature an effective time-averaged CSA tensor that does not cause a decay during tmix. This will further increase the observed intensity plateau p27 and thus render its interpretation critical. We note in passing that the CODEX decays of the PEO5-25s sample (not shown) only reach plateau values of the order of 0.5, which may arise from stronger intermediate-motional effects (possibly due to a different τc distribution shape) or from more restricted back-and-forth jumps covering only 3 sites. The latter possibility is reinforced by the 1H-based results for this sample (see Table 2). Similar complexities are also expected for the chain-extended PEO1-30 sample. Detailed CODEX studies of these and related samples will be published elsewhere. One should keep in mind that the average τc̅ = exp[⟨ln τc⟩] defined by eq 11 is not identical to the τc obtained from the KWW fit. Apart from the correction factor of 1.55 related to the nonexponential nature of the CODEX helical-jump correlation function (see the Theory section), which is considered in Figure 7b, one should also take into account a correction related to the specific averaging over the τc distribution. According to ref 61, the KWW-fitted τc is larger than τc̅ by factors 1.16 or 1.78 for β exponents of 0.8 or 0.5, respectively. Taking this effect into account would alleviate the discrepancy between 13C CODEX- and 1H FID-based somewhat. We

⎤ ⎡3 5 R1ρ = M 2(1 − S2)⎢ J(2ω1) + J(ωL) + J(2ωL)⎥ ⎦ ⎣2 2

(12)

with the Larmor frequency ωL and the spectral density J(ω) = τc/(1 + ω2τc2). J(ω) and the factor (1 − S2) result from the Fourier transform of eq 3. Our results for R1ρ were obtained from exponential fits to the crystal-related component of the detected FID intensity data measured as a function of the duration of a contiguous spinlock pulse tSL applied after a 90° excitation pulse, fnorm cr (tSL) = exp[−tSL/R1ρ]. Sample data are shown in Figure 8. A fit of the resulting R1ρ values taken at different temperatures to eq 12 in combination with an Arrhenius law reveals Ea ≈ 30 kJ/mol (see the dashed line in Figure 8b). This confirms the earlier result but is in strong disagreement with the present results discussed above. It can be suspected that the deviation is again due to the substantial correlation time distribution, which was not considered in ref 35. We assumed this to be combined with an averaging of the observed R1ρ decay by spin diffusion. In other words, magnetization exchange mediated by dipolar flipflops may be the cause of the nearly monoexponential behavior shown in Figure 8a, whereas an R1ρ distribution and thus multiexponential decay could be expected. In order to test this assumption and quench spin diffusion during the spin lock, we performed R1ρ measurements under off-resonance spin-lock satisfying the Lee−Goldburg (LG) condition, where the effective spin-lock field is oriented along the magic angle βM = 54.7° with respect to B0.64 The effective field is then ω1e = (ω12 + ωoff2)1/2 with the resonance offset ωoff = ω1/tan βM. Experimentally, it is of course necessary to adjust the excitation pulse to a tilt angle of βM; otherwise, residual precession around the spin lock field challenges the measurement. Signal decays for on- and off-resonance LG spin lock are compared in Figure 8a. The decay under LG spin-lock is noticeably nonexponential, and KWW fits provide an exponent I


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Macromolecules β of around 0.78 at all investigated temperatures, in agreement with the expectation. We performed model calculations in order to assess the effect of a τc distribution on the average R1ρ along the lines of classical early work on spin−lattice relaxation.65 Under off-resonance conditions, the term (3/2)J(2ω1) in eq 12 is replaced by (3/2)(sin2 β cos2 βJ(ω1e) + sin4 βJ(2ω1e)), where β is the tilt angle of the effective field ω1e.63,66 Neglecting the ωLdependent terms, whose contribution is negligible when ωLτc ≫1, the main off-resonance effect on R1ρ is thus, roughly, a reduction by sin2 βM = 0.667. We have calculated averaged R1ρ values for different temperatures using the activation parameters known for the given sample, at each temperature based upon τc values covering a log-normal distribution with σ = 0, 0.8, and 1.2. The resulting data are compared with experimental results in Figure 8b. The presence of the distribution leads to a substantial flattening of the R1ρ maximum and to a reduction of the slope in the low-temperature region next to the maximum, in agreement with the experimental data. Note that at very low temperatures in the linear region of the Arrhenius representation the slope equals Ea/R, since R1ρ ∝ τc−1 when ω1(e)τc ≫ 1. Taking a closer look at the regular spin-lock decay data, we noticed that these are also weakly stretched (see Figure 8a), with β of the KWW fit starting above 0.9 but reaching values of around 0.8 or even below at temperatures at or above 290 K (20 °C). From Figure 8b (see the dotted line) we take that R1ρ becomes larger than 2/ms in this range. We can therefore estimate the relevant magnetization exchange time to about 0.5 ms. If R1ρ is longer than that, then the decay becomes closer to exponential. Note that the apparent R1ρ then reflects the arithmetic average of the individual values, which is biased toward the more quickly relaxing ensemble members.67 An estimate of the length scale related to the spatial separation of monomers with different τc and thus different R1ρ, which is averaged on the relaxation time scale by spin diffusion, requires knowledge of the spin diffusion coefficient D. Under a spin lock D is lower than the static-limit value because the dipolar Hamiltonian is scaled by 1/2,41 and it is further subject to interference with intermediate chain mobility. On the basis of previous assessments,68,69 we take D ≈ 0.1 nm2/ms as a necessarily crude but realistic estimate of its lower bound. Using t = 0.5 ms ≈ 1/R1ρ, the lower bound for the spatial separation of dynamically different chain segments is thus ⟨Δr2⟩1/2 = (6Dt)1/2 ≈ 0.5 nm. This is an atomic length scale, compatible only with adjacent stems. Note that monomeric exchange (identity exchange by helical jumps within the unit cell rather than magnetization exchange) cannot explain the R1ρ averaging because it would not be sensitive to the apparent “switching off” of the exchange by use of the LG condition. It is thus clear that the observed averaging phenomenon cannot be due to dynamics variations along a given stem, but only between adjacent stems.

Figure 8. 1H R1ρ data for sample PEO53-30 and comparison with fits and predictions. (a) Crystal-related relaxation decays taken at −10 °C, comparing regular and Lee−Goldburg spin locks with the same effective field ω1(e)/2π = 125 kHz (ω1/2π = sin βMωe/2π = 102 kHz for LG spin lock), along with exponential and KWW fits. (b) Arrhenius plot of R1ρ values as compared with results from model calculations based upon results from 1H FID analyses assuming different logarithmic widths σ of the τc distribution. The dashed line is a fit to eq 12 assuming an Arrhenius law for τc(T).

analytical data analysis approach based upon the Anderson− Weiss approximation was validated by comparison to simulation data and by experiments on a model substance. As the central result, experiments on a variety of samples, including disordered fold surfaces of bulk-crystallized long chains, twice-folded single crystals and extended-chain crystals of shorter chains, revealed that the dc dependence of the jump residence time can, within the given limited accuracy, be approximated by a linear relation, which in turn could be reproduced by a generic Monte Carlo simulation model based upon diffusing defects that are generated in the fold surface. Combining this dependence with the average fitted activation parameters, one obtains an approximate relation ⎛ dc ⎞ τc(T , dc) = 8 × 10−18 s × ⎜ + 0.77⎟ × e(64.5 kJ/mol)/ RT ⎝ 4.6 nm ⎠

Note that the experimental dc only varies in a range between about 5 and significantly below 100 nm, rendering the dc dependence weak on a logarithmic scale. In our model, the jump time τc,0 for dc → 0 is governed by the defect generation rate. Whether this model picture, as derived from the abundant literature data on PE with its all-trans conformation, really applies to the conformationally more flexible PEO helix (with a possibility of defect generation along the stem) remains to be checked by atomistic simulation. For two special morphologies, namely samples featuring chain-extended and twice-folded single crystals, the above conclusions hold equally well, but we could not exclude the presence of restricted 3-site as opposed to 7-site jumps along

V. SUMMARY AND CONCLUSIONS We have used a combination of NMR methods to probe the molecular process underlying the αc relaxation in the semicrystalline polymer PEO. In the given case of a simple homopolymer without crystal polymorphism, the analysis of simple 1H FIDs allows for (i) an estimation of the different phase components, i.e., the crystallinity, and (ii) the extraction of the residence times of the jump process on the microseconds to milliseconds time scale over a range of temperatures. The J


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relation between the helical jump dynamics and the crystal growth.

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APPENDIX A. VALIDATION OF 1H FID ANALYSIS In order to validate the main NMR method, we applied it to the model substance trimethylsulfoxonium iodide (TMSI). The OS(CH3)3+ cations in this molecular crystal are dynamic, with the fast-rotating methyl groups, i.e., the S−C bond vectors, performing well-defined slower 3-site jumps around the SO axis. This substance has been investigated in previous work,70 providing a database for comparison. 1H FID data are shown in Figure 9a, along with the result of a simultaneous fit to eq 5 combined with an Arrhenius temperature dependence τc(T). Here, we did not include a τc distribution but took into account the T-dependence of M2 arising from thermal expansion and small-angle motions. A linear dependence was assumed and was part of the simultaneous fit to all temperatures, resulting in M2(T) = [5474 − T × 2.33/°C)] kHz2. The result of this simultaneous fit for Ea was 69.5 ± 2 kJ/mol, to be compared with the averaged literature result of 76 ± 4 kJ/mol. The order parameter S2 was determined as 0.169. M2(T) is directly encoded at temperatures below 0 °C, where τc is too large to exert any influence on the line shape. At higher temperatures (20 °C and above), individual values τc can also be determined by fit to eq 5 only, using the predetermined M2(T) and S2 values. Corresponding results are also shown in Figure 9b, for which a separate Arrhenius fit yields Ea = 71 ± 1 kJ/mol. The quantitative agreement of these results with the literature values70 thus demonstrates the potential and the accuracy of 1H FID analysis.

Figure 9. (a) 1H FID data for TMSI taken over a wide range of temperatures. The inset shows the data as symbols and the fits to eq 5 as solid lines. (b) Obtained correlation times in an Arrhenius representation, comparing a simultaneous fit to all data (green dashed line) to fits for individual temperatures (red symbols) and a separate Arrhenius fit (red solid line). Data from the literature are shown as gray in the background.70

the 72 helix. Further studies of such samples are currently being pursued in our laboratories. One of the currently most popular NMR techniques for dynamics investigations is the 13C-based MAS exchange method termed CODEX. It provides direct access to τc of the order of milliseconds or slower with site resolution, but at the expense of long experimental times of the order of a day. Comparing results for one sample with our time-efficient 1H FID-based method, we have revealed significant bias effects related to T2 decay during the pulse sequence, which in the given case is due to significant τc distribution covering about 1 decade. This results in an overestimation of τc by up to a decade. Since 13C CODEX remains the method of choice for dynamics investigations in more complex materials, such bias effects should be carefully considered in the future. Finally, we have addressed why earlier studies relying on 1H rotating-frame relaxation provided apparently too low activation energies. This was, again, explained by the τc distribution combined with averaging by spin diffusion, leading to almost monoexponential relaxation decay functions at lower temperatures. In contrast, measurements under homodecoupling conditions using a Lee−Goldburg spin lock clearly revealed nonexponential decays and an average relaxation rate compatible with theoretical predictions considering the τc distribution. From the fact that under normal conditions the relaxation decays also become stretched at the highest temperatures, concident with high values of R1ρ (short relaxation times), a lower bound of the relevant spin exchange time could be estimated. This translates into a length scale of less than a nanometer, on which fast and slow helical jumps must coexist. In conclusion, the 1H-based FID analysis method is in the given case of a simple homopolymer the only one providing an unbiased and direct access to the helical-jump dynamics. The method is simple, robust, and reliable but features a limited sensitivity to τc, which must be compensated by simultaneous analysis of data taken over a wide range of temperatures. This can be a limiting factor for polymers with a narrow range of sufficiently fast helical jumps before the final melting. In the present case, the τc could be measured in a time and temperature range close to the crystallization temperature. On the basis of this, we are currently exploring the possible

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APPENDIX B: CODEX CORRELATION FUNCTION FOR HELICAL JUMPS The normalized CODEX signal for jump motions between M inequivalent sites is a superposition of stimulated-echo responses. With pi (i = 1...M) as the conditional jump 54 M probabilities, it reads Snorm M (tmix) = ∑i=1 pi⟨cos(Φ1 − Φi)⟩. The argument (Φ1 − Φi) comprises all possible permutations of the evolution phases related to the recoupled difference tensor, reflecting the potential orientation change of the CSA tensor upon jumping during the mixing time. For long recoupling (encoding) times Ntr, the signal contribution for a tensor that has jumped (i ≠ 1) is near zero upon powder averaging ⟨...⟩, while it is 100% if no or a return jump has occurred (i = 1). One can then neglect details related to the CSA tensor values and jump angles: Snorm M (tmix) = pi(tmix). The time evolution of the pi(t) is governed by the master equation d p = Π·p dt

where p = (p1, ..., pM)T is the population vector and Π is the exchange matrix. Solving this system of differential equations for the cases of jumps between 7 or 3 sites, using eq 6 or 7 as exchange matrices, gives S72(tmix ) = Sc3(tmix ) = K

1 (1 + 7

3

∑ 2 exp(λiktmix)) i=1

1 (1 + 2 exp(− 3ktmix )) 3 DOI: 10.1021/acs.macromol.7b00843 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules respectively. The λi are the roots of the equation x3 + 7x2 + 14x + 7 = 0; the numerical values are λ1 = −3.802, λ2 = −2.445, and λ3 = −0.753. This confirms a triexponential decay of S72(tmix), which fits the simulated data very well, with only a small 8% adjustment of the final plateau value, accounting for the small oscillations in the plateau region of the Ntr dependence.54 Note that Sc3(tmix) = p2(tmix) is single-exponential only when just the central site (2) is populated as initial condition (reflecting the intended case of back-and-return or forth-andreturn jumps along the long 72 helix). In general, one obtains (M − 1)/2 or (M − 2)/2 exponential components for jumps along an Mn helix when M is odd or even, respectively. The decay rates are determined solely by the structure of the exchange matrix. The magnitudes of the difference tensor influences only the height of the final plateau and the prefactors of the individual exponential components. Details are deferred to a future publication.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (K.S.). ORCID

Wei Chen: 0000-0001-8334-0024 Jörg Kressler: 0000-0001-8571-5985 Toshikazu Miyoshi: 0000-0001-8344-9687 Thomas Thurn-Albrecht: 0000-0002-7618-0218 Kay Saalwächter: 0000-0002-6246-4770 Present Address

W.C.: State Key Lab of Pollution Control and Resource Reuse Study, College of Environmental Science and Engineering, Tongji University, Shanghai 200092, P. R. China. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Valuable discussions with Alexey Krushelnitsky and Detlef Reichert are gratefully acknowledged. We thank Vinay Joshi (Martin-Luther-Univ., Halle) for conducting preliminary experiments. Funding was provided by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the SFB-TRR 102, project A1, and RE1025/19.

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