^{1,a)}, J. R. Tozoni

^{1}, K. Schmidt-Rohr

^{2}and T. J. Bonagamba

^{3}

### Abstract

One-dimensional (1D) exchange NMR experiments can elucidate the geometry, time scale, memory, and heterogeneity of slow molecular motions (1 ms–1 s) in solids. The one-dimensional version of pure-exchange (PUREX) solid-state exchange NMR, which is applied to static samples and uses the chemical shift anisotropy as a probe for molecular motion, is particularly promising and convenient in applications where site resolution is not a problem, i.e., in systems with few chemical sites. In this work, some important aspects of the 1D PUREX experiment applied to systems with complex molecular motions are analyzed. The influence of intermediate-regime (10 μs–1 ms) motions and of the distribution of reorientation angles on the pure-exchange intensity are discussed, together with a simple method for estimating the activation energy of motions occurring with a single correlation time. In addition, it is demonstrated that detailed information on the motional geometry can be obtained from 1D PUREX spectral line shapes. Experiments on a molecular crystal, dimethyl sulfone, confirm the analysis quantitatively. In two amorphous polymers, atactic polypropylene (aPP) and polyisobutylene (PIB), which differ only by one methyl group in the repeat unit, the height of the normalized exchange intensity clearly reveals a striking difference in the width of the distribution of correlation times slightly above the glass transition. The aPP shows the broad distribution and Williams–Landel–Ferry temperature dependence of correlation times typical of polymers and other “fragile” glass formers. In contrast, the dynamics in PIB occur essentially with a single correlation time and exhibits Arrhenius behavior, which is more typical of “strong” glass formers; this is somewhat surprising given the weak intermolecular forces in PIB.

The Brazilian agencies FAPESP and CNPq supported this research.

I. INTRODUCTION

II. EXPERIMENT

III. THEORETICAL BACKGROUND

A. Review of the 1D PUREX method

B. 1D PUREX intensity in the slow-motion regime: Discrete jumps

C. Slow-motion regime: Diffusive motions

D. 1D PUREX line shapes and geometry of motion

E. Intermediate motional regime: Single correlation time

F. Intermediate motional regime: Distribution of correlation times

G. Exchange intensity as a function of temperature

H. for a single correlation time

I. for a distribution of correlation times

IV. EXPERIMENTAL RESULTS

A. Demonstrations on dimethyl sulfone

B. Dynamics near in amorphous polymers

V. CONCLUSIONS

### Key Topics

- Rotational correlation time
- 111.0
- Nuclear magnetic resonance
- 16.0
- Polymers
- 14.0
- Activation energies
- 13.0
- Diffusion
- 7.0

## Figures

Pulse sequence used in the 1D PUREX experiments with cross polarization (CP), chemical-shift evolution and refocusing times , mixing time , and -storage period . Standard dipolar decoupling of protons (DD) is applied during evolution and detection of transverse magnetization. pulses are shown in black. In the absence of exchange, the phase cycle yields a pure stimulated echo after the second period; the signals obtained in the absence of exchange (full stimulated echo) are outlined in the figure. After acquiring the signal with exchange, and are interchanged and the reference signal without exchange is recorded. The normalized pure-exchange intensity is obtained as .

Pulse sequence used in the 1D PUREX experiments with cross polarization (CP), chemical-shift evolution and refocusing times , mixing time , and -storage period . Standard dipolar decoupling of protons (DD) is applied during evolution and detection of transverse magnetization. pulses are shown in black. In the absence of exchange, the phase cycle yields a pure stimulated echo after the second period; the signals obtained in the absence of exchange (full stimulated echo) are outlined in the figure. After acquiring the signal with exchange, and are interchanged and the reference signal without exchange is recorded. The normalized pure-exchange intensity is obtained as .

Reorientation-angle distributions for isotropic rotational diffusion and the corresponding doubly normalized 1D PUREX intensities as a function of the mixing time, calculated in the slow-motion regime. (a) Reorientation-angle distribution at selected mixing times; (b) vs at selected evolution times.

Reorientation-angle distributions for isotropic rotational diffusion and the corresponding doubly normalized 1D PUREX intensities as a function of the mixing time, calculated in the slow-motion regime. (a) Reorientation-angle distribution at selected mixing times; (b) vs at selected evolution times.

One-dimensional PUREX spectra as a function of the reorientation angle for . (a) . Full lines: Non-normalized spectra for reorientation angles varying from 10° to 90°. Dashed lines: Point-by-point normalized spectra for reorientation angles varying from 10° to 90°. (b) Same as (a) for .

One-dimensional PUREX spectra as a function of the reorientation angle for . (a) . Full lines: Non-normalized spectra for reorientation angles varying from 10° to 90°. Dashed lines: Point-by-point normalized spectra for reorientation angles varying from 10° to 90°. (b) Same as (a) for .

Calculated 1D PUREX spectra as a function of the rotation axis for a fixed rotation angle of . The CSA principal values used in the calculations are typical for the carbonyl carbons in the backbone of peptides and proteins, . (a) Full powder spectrum for reference. (b) Rotation of around the principal axis of the chemical-shift tensor. (c) Rotation of around the axis. (e) Rotation of around the C–C backbone bond in a glycine residue of a poly-Gly -helix peptide (Ref. 31). The Euler angles relating the CSA principal axes with a molecular frame where the axis is parallel to the C–C bond are . (f) Rotation of around C–N bond in a glycine residue of a poly-Gly -helix peptide (Ref. 31). The Euler angles relating the CSA principal axes with a molecular frame where the axis is parallel to the C–N bond are .

Calculated 1D PUREX spectra as a function of the rotation axis for a fixed rotation angle of . The CSA principal values used in the calculations are typical for the carbonyl carbons in the backbone of peptides and proteins, . (a) Full powder spectrum for reference. (b) Rotation of around the principal axis of the chemical-shift tensor. (c) Rotation of around the axis. (e) Rotation of around the C–C backbone bond in a glycine residue of a poly-Gly -helix peptide (Ref. 31). The Euler angles relating the CSA principal axes with a molecular frame where the axis is parallel to the C–C bond are . (f) Rotation of around C–N bond in a glycine residue of a poly-Gly -helix peptide (Ref. 31). The Euler angles relating the CSA principal axes with a molecular frame where the axis is parallel to the C–N bond are .

Normalized 1D PUREX intensities plotted as a function of and in the intermediate motional regime for two-site jumps, with CSA parameters of and . (a) vs for various values of the correlation time , with and ; (b) vs for values ranging from 0.2 to 50 ms, with and ; (c) vs for values between 0.2 and 10 ms, with and a reorientation angle . (d) vs with and for several values, considering a two-site jump with a log-Gaussian distribution of correlation times (Refs. 9 and 15). (e) Curves of as a function of , plotted for different KWW distributions of correlation times with of 1, 0.4, and 0.3 and the corresponding fitting using KWW (stretched-exponential) correlation functions. (f) Simulations of vs for different mean correlation times and values. Corresponding fits using KWW correlation functions and the mean correlation times extracted from these fits are also shown.

Normalized 1D PUREX intensities plotted as a function of and in the intermediate motional regime for two-site jumps, with CSA parameters of and . (a) vs for various values of the correlation time , with and ; (b) vs for values ranging from 0.2 to 50 ms, with and ; (c) vs for values between 0.2 and 10 ms, with and a reorientation angle . (d) vs with and for several values, considering a two-site jump with a log-Gaussian distribution of correlation times (Refs. 9 and 15). (e) Curves of as a function of , plotted for different KWW distributions of correlation times with of 1, 0.4, and 0.3 and the corresponding fitting using KWW (stretched-exponential) correlation functions. (f) Simulations of vs for different mean correlation times and values. Corresponding fits using KWW correlation functions and the mean correlation times extracted from these fits are also shown.

One-dimensional PUREX intensity curves as a function of temperature, calculated for a two-site jump with a single correlation time following the Arrhenius behavior. (a) Normalized exchange intensity calculated for different activation energies based on the slow-exchange approximation according to Eq. (9), with and ; (b) same as (a) but with and different ; (c) linearized plot for independently determining the activation energy, and ; (d) full curves calculated according to Eq. (10) for several activation energies with . (e) for varying mixing times with fixed and . (f) Exchange intensity (without normalization by ) for several mixing times and . The CSA parameters used in all the calculations were and .

One-dimensional PUREX intensity curves as a function of temperature, calculated for a two-site jump with a single correlation time following the Arrhenius behavior. (a) Normalized exchange intensity calculated for different activation energies based on the slow-exchange approximation according to Eq. (9), with and ; (b) same as (a) but with and different ; (c) linearized plot for independently determining the activation energy, and ; (d) full curves calculated according to Eq. (10) for several activation energies with . (e) for varying mixing times with fixed and . (f) Exchange intensity (without normalization by ) for several mixing times and . The CSA parameters used in all the calculations were and .

Distributions of correlation times and corresponding normalized exchange intensities : (a) Log-Gaussian distributions of correlation times; (b) curves for two-site jumps calculated for the distributions shown in (a) with ; (c) KWW distributions of correlation times; (d) curves for two-site jumps calculated for the distributions shown in (c) with . (e) curves for random jumps, , and log-Gaussian distributions of correlation times. (f) curves for random jumps, , and KWW distributions of correlation times. The Arrhenius parameters used to perform the linear conversion from to temperature are calculated with and .

Distributions of correlation times and corresponding normalized exchange intensities : (a) Log-Gaussian distributions of correlation times; (b) curves for two-site jumps calculated for the distributions shown in (a) with ; (c) KWW distributions of correlation times; (d) curves for two-site jumps calculated for the distributions shown in (c) with . (e) curves for random jumps, , and log-Gaussian distributions of correlation times. (f) curves for random jumps, , and KWW distributions of correlation times. The Arrhenius parameters used to perform the linear conversion from to temperature are calculated with and .

One-dimensional PUREX spectra for DMS at 20 °C with a mixing time of 50 ms. Left: reference spectrum. Right: Point-by-point normalized spectrum. Simulations with a reorientation angle of are represented by dotted lines. The measuring time of each pair of spectra was about 1 h.

One-dimensional PUREX spectra for DMS at 20 °C with a mixing time of 50 ms. Left: reference spectrum. Right: Point-by-point normalized spectrum. Simulations with a reorientation angle of are represented by dotted lines. The measuring time of each pair of spectra was about 1 h.

Experimental 1D PUREX intensities for DMS and corresponding fit curves, in the intermediate motional regime. (a) dependence of the normalized exchange intensity at , 308, 318, and 325 K; (b) dependence of at , 308, 318, 325, and 333 K; (c) Arrhenius plot of the correlation times determined in (a) and (b). The measuring time per spectrum was 20 min.

Experimental 1D PUREX intensities for DMS and corresponding fit curves, in the intermediate motional regime. (a) dependence of the normalized exchange intensity at , 308, 318, and 325 K; (b) dependence of at , 308, 318, 325, and 333 K; (c) Arrhenius plot of the correlation times determined in (a) and (b). The measuring time per spectrum was 20 min.

Experimental 1D PUREX intensity vs temperature for DMS. (a) Normalized exchange intensity for and . Calculated curves using the slow-motion approximation and the general formula are shown as full and dashed curves, respectively; (b) plot of vs on the low-temperature side of . The linear fit based on Eq. (10) provides the Arrhenius parameters.

Experimental 1D PUREX intensity vs temperature for DMS. (a) Normalized exchange intensity for and . Calculated curves using the slow-motion approximation and the general formula are shown as full and dashed curves, respectively; (b) plot of vs on the low-temperature side of . The linear fit based on Eq. (10) provides the Arrhenius parameters.

Experimental 1D PUREX intensities for amorphous atactic polypropylene (aPP) and polyisobutylene (PIB), measured with and . Simulated curves with isotropic rotational diffusion model are also shown. The fits were obtained using KWW distributions of correlation times with for PIB and for aPP. In both cases, the temperature dependencies of the correlation times were assumed to be WLF functions with the parameters given in the text. The experimental data were corrected for spin-diffusion effect by subtracting a constant background measured at low temperatures.

Experimental 1D PUREX intensities for amorphous atactic polypropylene (aPP) and polyisobutylene (PIB), measured with and . Simulated curves with isotropic rotational diffusion model are also shown. The fits were obtained using KWW distributions of correlation times with for PIB and for aPP. In both cases, the temperature dependencies of the correlation times were assumed to be WLF functions with the parameters given in the text. The experimental data were corrected for spin-diffusion effect by subtracting a constant background measured at low temperatures.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content