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On the role of experimental imperfections in constructing 1 H spin diffusion NMR plots for domain size measurements

We discuss the precision of 1D chemical-shift-based 1 H spin diffusion NMR experiments as well as straightforward experimental protocols for reducing errors. The 1 H spin diffusion NMR experiments described herein are useful for samples that contain components with significant spectral overlap in the 1 H NMR spectrum and also for samples of small mass (< 1 mg). We show that even in samples that display little spectral contrast, domain sizes can be determined to a relatively high degree of certainty if common experimental variability is accounted for and known. In particular, one should (1) measure flip angles to high precision (≈ ±1° flip angle), (2) establish a metric for phase transients to ensure their repeatability, (3) establish a reliable spectral deconvolution procedure to ascertain the deconvolved spectra of the neat components in the composite or blend spin diffusion spectrum, and (4) when possible, perform 1D chemical-shift-based 1 H spin diffusion experiments with zero total integral to partially correct for errors and uncertainties if these requirements cannot fully be implemented. We show that minimizing the degree of phase transients is not a requirement for reliable domain size measurement, but their repeatability is essential, as is knowing their contribution to the spectral offset (i.e. the J 1 coefficient). When performing experiments with zero total integral in the spin diffusion NMR spectrum with carefully measured flip angles and known phase transient effects, the largest contribution to error arises from an uncertainty in the component lineshapes which can be as high as 7 %. This uncertainty can be reduced considerably if the component lineshapes deconvolved from the composite or blend spin diffusion spectra adequately match previously acquired pure component spectra.

INTRODUCTION

A straightforward, robust method for determining domain sizes and miscibility in two component polymer blends and pharmaceutical formulations is highly desired. One such approach is based on 1 H spin diffusion NMR, which has classically been utilized for degrees of mixing in blends [ 1 , 2 , 3 , 4 , 5 , 6 ], and continues to be a vital analytical tool for polymer blends [ 7 , 8 ], copolymers [ 9 ], pharmaceuticals [ 10 ], and dispersions [ 11 ]. Due to its high sensitivity, Combined Rotation And Multiple Pulse Spectroscopy (CRAMPS) [ 12 ] based 1 H spin diffusion is a fast method for determining degrees of miscibility. In CRAMPS experiments, strongly coupled spin systems like 1 H are modulated such that the homonuclear dipolar interactions are minimized during the multiple pulse cycle while other interactions (i.e. chemical shift) are left intact, though scaled. As a result, CRAMPS can yield 1 H NMR spectra with high signal-to-noise and chemical shift resolution of approximately 1 ppm or better. In chemically simple systems such as polymers, polymer blends, and composites, this resolution is often sufficient and, when combined with 1 H spin diffusion methods, domain sizes in solid polymer blends can be determined.

In the seminal work by Clauss et al [ 1 ], it was shown that domain sizes could be accurately determined via 2D chemical-shift-based (CSB) 1 H spin diffusion NMR when utilizing MREV-8 homonuclear decoupling [ 13 ]. This 2D CSB 1 H spin diffusion NMR experiment is a straightforward method for measuring domain sizes in blend samples that are not mass limited (10 – 100 mg) and when the two components do not overlap each other in the 1 H CRAMPS spectrum [ 1 , 8 ]. In this experiment, a 2D spectrum is collected for each spin-diffusion time investigated, which makes extensions of the technique to mass-limited samples (< 1 mg) difficult. We present here the 1D variant of that original experiment which uses a fixed number of MREV-8 cycles for preparation for each spin diffusion time chosen. This experiment has been utilized for determining domain sizes and miscibility of polymer blends [ 4 , 5 , 6 ] with applications in photoresists [ 14 ] and organic electronics [ 15 ] in which samples are typically mass-limited. While the 1D experiment has the drawback that it requires additional background experiments for analysis, it has advantages in (1) shorter acquisition times and, hence, higher signal-to-noise ratios, so as to be applicable for measuring domain sizes in small samples (≈0.1 mg) such as thin films [ 16 , 17 ] and (2) samples in which the components exhibit extreme spectral overlap in the 1 H CRAMPS spectrum.

Domain sizes are extracted via spin diffusion NMR by measuring the rates of change of nuclear spin polarizations associated with the two phases in a blend or composite. When a step-function initial polarization gradient is imposed on a two-component system, the initial rate of exchange is directly proportional to the inter-component surface-area/volume ratio in the sample. From that ratio, the domain size can be extracted if the morphology is known (spheres, rods, lamellae) [ 18 ]. If the morphological character is not known, considerable error in the domain size can exist (factor of one to three) because similar curves could be predicted for different domain sizes depending on the morphological dimensionality [ 19 ]. Similarly, if the composition is not known (and cannot, for instance, be obtained via inspection of the 1 H spectrum), then additional error can exist since spin diffusion curves of blends with different compositions will differ even for similarly sized domains [ 19 ]. We will not comment on these errors in this paper because it is assumed that the sample morphology and/or composition can be ascertained via additional NMR measurements or another technique such as scattering or microscopy.

Due to the Fickian nature of spin flips, intra- and inter-monomer spin exchange occur simultaneously at early times of the spin diffusion process, which, depending on monomer size, is typically < 1 – 3 ms. At longer times, intra-monomer spin equilibration has been reached and inter-monomer spin exchange can unambiguously be monitored. In the 2D experiment, spin diffusion is measured from the build-up of cross-peaks between the two components’ resonances on a 2D contour plot [ 1 , 8 ] and due to this greater spectral separation, intra -monomer spin exchange can, in principle, be distinguished from inter -monomer spin exchange. However, if 1 H chemical shift contrast is not sufficient, no spectrally isolated “cross-peak” would be identified on the contour plot, making it difficult to distinguish between intra- and inter-monomer spin exchange. In these cases either 13 C detection can be employed [ 20 , 21 , 22 , 23 ] which can require acquisition times that are orders of magnitude longer, or the spectra can be edited via addition/subtraction with the 1 H CRAMPS spectra of the components, necessitating background experiments.

The 1D experiment measures spin diffusion by monitoring the decay of the polarization of one of the components after a gradient has been established. In cases where both components in the blend contain aromatic and aliphatic resonances, there will be extreme spectral overlap in the 1 H spectrum and, similar to the 2D version, the feasibility of the experiment hinges on there being spectral isolation of one of the components somewhere in the spectrum. An example is given in Figure 1 , which shows the high resolution 1 H CRAMPS spectrum ( Figure 1a ) of a blend of poly(3-hexylthiophene) (P3HT) and phenyl-C61-butyric acid methyl ester (PCBM) (50:50 by mass, 85:15 mol of 1 H). Also shown are spin diffusion spectra ( Figure 1(b) – (d) ). The multiple pulse sequence is MREV-8 [ 13 ]. As shown in Figure 1a , there is significant spectral overlap in the CRAMPS spectrum between P3HT (red) and PCBM (blue), but there is no appreciable spectral overlap at chemical shifts greater than 7.5 ppm. Despite severe spectral overlap from 1 ppm to 7.5 ppm, domain sizes could be ascertained from 1D 1 H CRAMPS spin diffusion by establishing, and then monitoring, the decay of the polarization gradient. The polarization gradient is realized by partially inverting the polarization of one of the components; in this case PCBM ( Figure 1b , blue) is inverted while P3HT (red) has positive polarization. (The converse is not practical due to limitations in acceptable frequency offsets). The rate of decay is measured by monitoring the polarization of one of the components, P(t) , as a function of time; linear decays in polarization when plotted vs. the square root of time can directly be correlated to the domain size because of Fick’s second law after accounting for T 1 relaxation effects.

(a) 1 H CRAMPS spectrum of a P3HT-PCBM blend (black) showing the individual components P3HT (red) and PCBM (blue), which were acquired separately. (b) 1 H spin diffusion spectrum at 2 ms mixing time (black) showing the individual components P3HT (red) and PCBM (blue), and mixing times of (c) 80 ms, and (d) 160 ms. (e) The spin diffusion plot of the P3HT-PCBM blend following different thermal annealing cycles.

The polarization is normalized via the equation:

where P(t=∞) is the polarization level at the infinite time limit, and P(t=0) is the polarization level at the beginning of the mixing time. In blends in which the components’ spectra overlap like this, one cannot know P(t=0) directly from the sample’s spin diffusion spectra since intra-monomer polarization gradients exist which can cause changes of intensity at early times. Conversely, since there can potentially be domains that are much larger than the longest spin diffusion radius ( 4 D T 1 < 200 nm , T 1 < 1 s usually ) , the P(t=∞) value is not known either. These two polarization values are found by running a separate experiment on a composite that is coarsely grained (≫ 100 nm) by mechanical mixing (i.e. a “physical mixture”) with identical chemical composition to the unknown blend sample so that the y-axis (ΔM) can be properly normalized via Equation (1). As shown in Figure 1e , when placed on a normalized spin diffusion plot the changes in domain sizes in a blend of P3HT and PCBM can be followed by monitoring the changes in the various plots. As shown from the relatively flat profile and larger intercept at longer times ( Figure 1e , red triangles), the 150 °C annealing of this particular sample caused dramatic coarsening because a portion of the PCBM was incorporated into large PCBM crystals (> 1 μm) [ 15 ].

As discussed above, the advantage of this particular experiment is that it can be successfully performed on systems that exhibit significant spectral overlap in the 1 H spectrum (i.e. both components contain aliphatic and aromatic protons) and in samples with low masses (< 1 mg). The disadvantage is that additional background experiments have to be performed on a physical mixture sample (and perhaps on the neat components, see below) for normalizing the spin diffusion plot. The focus of this paper is to identify the causes and potential limits of precision in measuring domain sizes via the spin diffusion plot as constructed from 1D 1 H spin diffusion NMR experiments. In particular, we discuss the effects of experimental limitations on the precision of establishing ΔM values which are critical for accurate and precise domain size determination. Since spin diffusion plots from 1D experiments on intimately mixed blends are normalized by comparing intensity levels to those of separate experiments on either a coarsely grained physical mixture sample or the neat components, determining the origin of experiment-to-experiment variability is critical for estimating precision and, ultimately, improving measurements. To determine the magnitude of these instrumental imperfections, we intentionally varied pulse amplitude (B 1 ), probe tuning (phase transients), sample morphology (lineshape), and frequency offset. We report the first three parameters as a function of frequency offset due to its convenience as a tunable parameter for establishing spin diffusion NMR experiments with zero total integral, which we will show below leads to higher precision.

RESULTS/DISCUSSION

The cyclic nature of the multiple pulse decoupling in CRAMPS is responsible for minimizing the homonuclear dipolar coupling, but experimental imperfections slightly perturb the doubly rotating frame of observation (i.e. the toggling frame). Imperfections such as pulse amplitude (B 1 ) variability, phase offsets, and pulse rise/decay time effects (“phase transients,” see below) all cause deviations from ideal CRAMPS behavior. The impacts of these imperfections were laid out over 40 years ago in the seminal paper on multiple pulse decoupling by Rhim and coworkers [ 24 , 25 ] and further expanded upon by Mehring [ 26 ]. Here, we utilize their formalism to demonstrate the sensitivity of these variables, which commonly occur in NMR, on 1D chemical-shift-based 1 H spin diffusion NMR experiments. Amplifier (B 1 ) drift, phase transients, and sample-to-sample variability all impact the reproducibility of the 1 H NMR chemical shift scale, which in turn, affects the frequency of oscillations in the spin diffusion preparation stage. Small deviations in frequency in the gradient preparation stage can significantly influence the polarization levels, and hence ΔM values, used for the spin diffusion plot for calculating domain sizes. Below we will show that, despite these polarization level deviations, they are not likely to lead to dramatic losses of precision in 1D 1 H spin diffusion NMR experiments if (1) polarization levels of known calibration standards (neat components and/or a physical mixture) are measured, (2) accurate values of B 1 are known, including susceptibility to drift, and (3) experiment-to-experiment variability of phase transients is minimized by way of direct measurement of tuning parameters such as the reflected voltage-to-forward voltage ratio, V R / V F . Furthermore, if these experimental conditions can be established, variations in ΔM values can be kept low (< 7 %), which ultimately will allow for domain sizes to be determined to high precision if the sample stoichiometry and morphological dimensionality are both known. Again, we quote no level of precision in domain size extracted here, only levels of precision of ΔM values, because the sensitivity of the domain size value to stoichiometry and sample morphology (i.e. spheres vs. lamellae) is so great (factor of 3 or greater).

In Figure 2 , we have plotted a typical time domain profile of a 1D 1 H spin diffusion NMR experiment. Our example is P3HT:PCBM. In the first stage (the preparation stage), the polarization gradient is created by allowing the magnetization to evolve under a fixed number of multiple pulse cycles, in this case ten. As shown in the blue circle, the intensity of the first data point, I 0 (the y-axis is scaled by I 0 for clarity), is identical to the total integrated intensity of the thermal equilibrium Fourier transformed CRAMPS spectrum ( Figure 1a ) and is proportional to the proton magnetization, which is dictated by the Boltzmann distribution of spins. The intensity of the second data point (red dotted circle) is reflective of the magnetization after a fixed number of CRAMPS preparation cycles (i.e., ten). The magnetization then evolves during the mixing time, after which it is then detected via CRAMPS. The intensity of the first point in the detection period (third circled data point from the left, green circle, Figure 2 ), which we denote I , is equivalent to the total integral of the Fourier transformed spin diffusion spectrum ( Figure 1b ). For a neat component, the ratio of these integrals, I/I 0 , is equivalent to the polarization relative to the thermal equilibrium polarization (Equation 2a). For a composite/blend of components A and B this ratio is a sum of the relative polarization values (Equation 2b) weighted by their proton mole fractions, ρ A and ρ B , respectively.

Time domain profile of a 1D 1 H spin diffusion NMR experiment. After a fixed number of multiple pulse cycles in the preparation, some of the spins have inverted (PCBM, blue arrow) and some have not (P3HT, red arrow), also see Figure 1b . The intensity in the red dotted circle, I , relative to the blue dotted circle, I 0 , is the weighted sum of both initial polarizations (see Equation 2). Spin diffusion between components will cause each polarization to decay during the mixing time (red “P3HT” line and blue “PCBM” line), but, in the absence of T 1 effects, the total intensity will not change (black “Sum” line). The polarizations are detected via CRAMPS after the mixing time and then Fourier transformed to obtain the “spin diffusion spectrum” (also see Figure 1b-d ).

In Figure 2 , the black line marked “Sum” is the weighted sum of the P3HT (red line) and PCBM (blue line). Furthermore, assuming negligible intensity loss due to longitudinal (i.e. T 1 ) proton relaxation, intensities of the second and third data points (in the red and green dotted circles) are essentially the same. Using equations 2(a) and 2(b), one can use the following procedure for finding the polarization values in a blend needed for calculating ΔM, the y-axis of the spin diffusion plot (Equation (1)). P(t) is determined using the blend sample and P(t=0) and P (t=∞) are obtained from the physical mixture. Since measurements of at least two samples, the “unknown” blend and the “standard” physical mixture, are needed in determining ΔM, it is critical to minimize variability in these measurements. If one seeks accurate ΔM values, how close to matching the experimental conditions does one have to be to ensure an identical polarization gradient is prepared in separate experiments?

What experimental variables affect the polarization gradient, and to what degree? Since a component’s polarization is equal to the time domain intensity I (of that given component) relative to I 0 , the polarization gradient is clearly sensitive to the observed frequency under CRAMPS decoupling, which we denote by the oscillation in the preparation stage in Figure 2 . If the observed frequencies change as experimental parameters are altered, there will be a concomitant change in the polarization gradient. Fortunately, the sensitivity of MREV-8 decoupling to pulse width/amplitude offset, phase transients, phase errors, frequency offsets, rf inhomogeneity, and power droop have all been beautifully laid out [ 24 , 25 , 26 ]. In those previous works, the various contributions of errors to the rf Hamiltonian were separated and quantified by assuming the total rf Hamiltonian is the sum of an ideal, cyclic component and a non-ideal, non-cyclic component. The error factors of MREV-8 are given in Table 1 .

Contributions of various experimental errors to the Hamiltonian during MREV-8 decoupling. The variables are as follows: δ i , is pulse width/height misadjustment with phase i , I i is the spin quantum number in the i th direction, J 1 is phase angle accumulated during a phase transient, t c is the total cycle time (in this case 39.6 μs), π i , is the phase offset of the pulse with phase i , Δω is the angular frequency offset, ω 0 is the Larmor frequency, σ zzi is the zz th component of the chemical shift tensor of the i th spin, a is the scaling factor of MREV-8 ( = 3 t w t c [ 4 π − 1 ] ) , ω s is the steady-state nutation frequency after power droop, b is the decay time of the power droop, and B ij is the dipolar coupling constant between spins i and j . Higher order effects and cross-terms are not included.

Effect of pulse amplitude on I/I 0 uncertainty

We measured I/I 0 on individual samples of P3HT and PCBM as well as their physical mixture. In the case of neat materials (P3HT and PCBM), the polarization level, which is constrained to be within the range from −1 to 1, is calculated via the integrated intensity, I , in a spin diffusion experiment relative to the CRAMPS spectrum intensity, I 0 . We note again that this ratio is identical to the intensity of the (n + 1) st data point in the preparation period ( Figure 2 , red circle) relative to the first data point ( Figure 2 , blue circle), where n is the number of preparation cycles. In Figure 3 , we plot I/I 0 as a function of offset frequency for (a) P3HT, (b) PCBM, and (c) their physical mixture (50:50 by mass). The mixing time is fixed at 3 ms, and the range of acceptable carrier frequency offsets (3500 – 5000 Hz) is limited by the effectiveness of CRAMPS (offset < 5000 Hz) and avoided overlap with the spectrum (offset > 3500 Hz). As shown in the figure, even for small drifts in pulse amplitude of approximately 1.4 kHz, which correspond to ±0.6 ° flip angle offsets, δ x , (see Table 1 , second row), significant polarization variability can be observed of up to ± 0.04. In the case of a 50:50 blend of P3HT with a typical PCBM polarization P PCBM = −0.44, this would correspond to an uncertainty spread in ΔM of 18% (= 0.08/0.44), clouding any meaningful analysis.

Integral of the spin diffusion spectrum ( I ) relative to the CRAMPS spectrum ( I 0 ) recorded after 3 ms mixing time for samples of (a) P3HT, (b) PCBM, and (c) their physical mixture for slight increases (blue) and decreases (red) in B 1 as a function of frequency offset. The thin dotted lines between points in (a) and (b) are a guide for the eye. The thick lines in (c) are calculated from Equation 2(b) with ρ PCBM = 0.15 and ρ P3HT = 0.85.

Clearly the potential exists for greater precision to be garnered from a 1 H spin diffusion plot. To do so, the same I / I 0 value must be maintained in both the physical mixture and the blend sample; this is done by intentionally changing the frequency offset until I/I 0 = 0. When looking at the plot of the physical mixture ( Figure 3c ), one observes that (like Figures 3a ,b) the I/I 0 values change with frequency offset and that different curves are observed for different B 1 values (black, red, and blue dots). The black dots correspond to I/I 0 values in which the B 1 field was precisely known to be 166.6 kHz; the blue and red dots correspond to B 1 values that were intentionally mis-set by 1.2 kHz too high and too low, respectively. If the experiment is conducted while keeping I/I 0 = 0 (see the dotted black circle in Figure 3c ), then drifts in the B 1 value by ± 1.2 kHz would result in a range or band of frequency offset values that would satisfy I/I 0 = 0 ( Figure 3 , gray box). This range of frequency values lies within the purple (frequency offset = 4810 Hz) and green (frequency offset = 4880 Hz) vertical dotted lines in Figure 3 and these positions represent the extremes of error in CRAMPS frequency due to B 1 drift for the spectrometer in our laboratory. The polarization values for the two components that would be observed with these CRAMPS frequencies can be known by following the vertical dotted lines (purple and black) up into Figures 3a ,b and monitoring at what I/I 0 values the lines intersect. The polarization values (y-axes, I/I 0 ) at these conditions (horizontal dotted lines) represent the extrema of the uncertainty ranges, which upon inspection of Figure 3a,b , are relatively narrow (< 1%), translating in a spread of ΔM values of < 2%. Hence, one sees that although a spread in B 1 values might cause a spread in CRAMPS frequencies, the impact on uncertainty or spread in the polarization levels for spin diffusion experiments performed when maintaining I/I 0 = 0 is low.

To further test the reliability and repeatability, we have synthesized what the physical mixture I/I 0 values should be (solid lines through the data, Figure 3c ) based on the known polarization levels of the neat components ( Figures 3a ,b) and the proton fractions (ρ P3HT = 0.85, ρ PCBM =0.15) using Equations (2a,b). As one can see, the agreement with experiment and prediction is quite good. The two caveats are that (1) agreement exists between the deconvolved physical mixture lineshapes and neat component lineshapes, and (2) the rf probe tuning is consistent. We next explore the impact of uncertainty in those two parameters.

Effect of phase transients on polarization level uncertainty

Each NMR probe suffers from ring down effects because of its inherent RLC time constant or quality factor (Q factor). CRAMPS experiments use cycles of phase sensitive pulses for decoupling, and the rise/decay times of these pulses create non-ideal fields which are non-linear and out-of-phase with the peak of the pulse; these perturbations are called “phase transients” [ 27 , 28 ] and schemes for reducing their effects have been outlined [ 29 ]. Phase transients will cause a frequency shift in the observed CRAMPS spectrum, but will not result in spectral broadening [ 24 , 25 , 26 ]. Tuning the probe effects the Q factor directly, and consequently any variability in tuning will result in a frequency shifts under multiple pulse decoupling.

In Figure 4 we have plotted the magnitude of the observed CRAMPS resonance frequency offset for the narrow resonance of the methyl protons of poly(di-methyl-sulfoxide) (PDMS) as a function of carrier frequency offset for different tuning levels. A clear shift in the observed frequency occurs upon changing the tuning. This is expected since tuning the probe would alter the rise/fall times of the pulses and, hence, the J 1 coefficient ( Table 1 , third row). The frequency offset and phase transient correction terms ( Table 1 , rows 1 and 3, respectively) both contain I z terms, and should add linearly. Hence, when plotting the observed frequency offset vs. carrier frequency offset as in Figure 4 , the curves should be linear with the slope equal to the scaling factor and with an ordinate equal to 2 J 1 2 π t c . However, no curves pass through the origin; the minimum CRAMPS frequency is at ≈65 Hz, demonstrating that phase transient effects persist for all tuning configurations, in violation of the first order effects mentioned here. We attribute the persistence of phase transient effects and the curves not passing through the origin to higher order and/or cross-term effects as discussed in Ref. 24. Importantly, as we show below, the persistence of phase transients does little to effect the precision with which ΔM values can be calculated if the sensitivity of the frequency change to the tuning is predictable (see Figure 4b below).

(a) The absolute value of the observed resonance frequency offset of PDMS (non-quadrature, cosine detection) as a function of the carrier frequency offset for different reflected voltage-to-forward voltage ratios (V R /V F ) by changing the tuning capacitor clockwise (cw) or counterclockwise (ccw). Changes in the matching capacitor showed similar behavior as long as the V R remained constant (see the blue dotted line as an example). (b) The observed frequency offset of PDMS with a 1000 Hz carrier frequency offset as a function of V R /V F difference from the minimum value; V R min = (40 ± 10) mV and V F = 9.8 V. For convention, we denote negative V R values as clockwise turns of the tuning capacitor and positive as counterclockwise. The values of the best fit line (red curve) to y = m*x +b are m = (2141 ± 37) Hz, b = (432 ± 2) Hz.

For a given carrier frequency (in this case 1 kHz), the observed CRAMPS frequency will change linearly with the reflected voltage level via the empirical equation:

Furthermore, when the receiver is in phase with the signal, the polarization is a cosine function and any changes in polarization go as:

since P is simply a cosine function assuming negligible damping. Here, ν obs is the observed CRAMPS frequency offset ( Figure 4a , y-axis), which is a function of carrier frequency offset and tuning (see Figure 4a , x-axis); t is the total preparation time, which in these experiments is 0.396 ms. The maximum expected rate of change in polarization is the coefficient −2 πt , which is ‐0.0029 in units of polarization/Hz. Combining this value with Equation (3) yields a maximum change in polarization upon changes in tuning to be 6.2 (units of polarization per unit of V R /V F ). Typical (normalized) variations in tuning are less than 0.02 (see below), translating into changes of polarization of less than 0.1.

We have plotted the I/I 0 values of P3HT (a), PCBM (b), and their physical mixture (c) as a function of frequency offset in Figure 5 for three tuning levels. The black dots, which correspond to the lowest V R /V F value (0.04/4.9 = 0.008) and hence “best” tuning scenario, are shown for P3HT ( Figure 5a ), PCBM ( Figure 5b ), and the physical mixture ( Figure 5c ). The blue and red dots in Figure 5 are polarizations measured upon detuning the tuning capacitor clockwise and counterclockwise, respectively, to attain V R /V F = 0.024. In the case of P3HT ( Figure 5a ), the polarization levels change significantly ( ca . ± 0.1) depending on the tuning capacitor direction; this value is as predicted from via Equation (4) for ( V r − V r m i n V f ) = 0.016 , t = 0.396 ms , and ν obs ≈ 2.1 kHz (at 4800 Hz). The I/I 0 changes for PCBM ( Figure 5b ) are less dramatic because the higher fraction of aromatic protons (0.36:0.64 aromatic:aliphatic protons) moves the observed offset frequency to lower ν obs values (≈ 1520 Hz at a carrier frequency offset of 4800 Hz), decreasing the d P d ν quantity. Note that at a frequency offset of ca. 4700 Hz, there is essentially no change in the PCBM polarization level with detuning; this is due to the fact that 2 π o b s t  ≈ 0.

Integral, I , of the spin diffusion spectrum (10-cycle CRAMPS preparation) relative to the equilibrium CRAMPS spectrum ( I 0 ) as a function of frequency offset for P3HT (a), PCBM (b), and their physical mixture (c) for different V R /V F values as observed upon changing the tuning capacitor cw (red dots) and ccw (blue dots). The forward voltage was measured to be 4.1 V. Spectra were recorded after 3 ms mixing time. (c) The solid lines are calculated from the values in (a) and (b) using Equation 2(b).

The precision of determining ΔM values on the spin diffusion plot ultimately depends on the variability of the polarization levels at I/I 0 = 0 (black dotted circle, Figure 5c ). If the ( V r − V r m i n V f ) values changed on the order of ± 0.024 from one experiment to the next, then in order to maintain I/I 0 = 0, a spread of possible frequency offsets would be required ( Figure 5 , gray box) in order to maintain I/I 0 = 0 that ranged from 4700 Hz ( Figure 5 , purple dotted line) to 4920 Hz ( Figure 5 , green dotted line). We use these tuning conditions as extremes of drift; one can typically maintain V R values to within a factor of 0.2, not 3. At these extremes, the I/I 0 values for PCBM range from – 0.43 to – 0.48 ( Figure 5b , horizontal purple/green lines); those of P3HT range from 0.01 to 0.02 ( Figure 5a , horizontal purple/green lines). When using P PCBM to calculate ΔM from Equation (1) (because of its greater spectral isolation at chemical shifts > 7.5 ppm, see Figure 1a ), then the observed spread in polarization levels (±0.025) translates into a spread of ΔM values of ±0.028 if the probe tuning (as measured via the V R /V F ) is not known to within a factor of three. The V R /V F value is typically known to with a factor of 0.2 (not 3) and will allow for greater predictability. For instance, the solid lines in Figure 5c are I/I 0 values as predicted from a linear combination of the neat component I/I 0 values ( Figures 5a ,b) using Equation (2a,b); the agreement is good.

Sample-to-sample variability

When performing experiments with I/I 0 = 0 in the physical mixture or blend, a third source of precision loss comes from variations in the spectra. Changes in the resonance positions can occur as a result of variations in magnetic susceptibility anisotropy, molecular packing, and molecular conformations. We tested this variability by making ordered and disordered samples of P3HT and PCBM. We assume similar spin diffusion coefficients between ordered and disordered samples because we have generally observed similar T 2 values amongst sample sets and T 2 times are used to calculate spin diffusion coefficients [ 1 ]. We have plotted in Figure 6 the CRAMPS spectra for P3HT (a), PCBM (b), and the physical mixture (c) for both ordered and glassy samples. There are subtle differences in linewidths between the various samples depending on order, in particular the PCBM samples, in which the glassier sample exhibits essentially no fine structure ( Figure 6b , red spectrum), but the ordered PCBM does ( Figure 6b , black spectrum).

CRAMPS spectra of (a) P3HT, (b) PCBM, and (c) physical mixtures of P3HT and PCBM. Samples were either ordered (black) or disordered (red). (d)-(f) Plots of polarization vs. frequency offset for P3HT (d), PCBM (e), and physical mixture of P3HT and PCBM (f) for either ordered (black) or glassy (red) samples.

These spectral changes translate into changes for the spin diffusion experimental conditions. As shown in Figure 6(d)-(f) , the I/I 0 vs. frequency offset curves depend on whether the sample was ordered (black dots) or glassy (red dots) and result in a band of possible offset frequency values for which I/I 0 = 0 in the blend ( Figure 6c-e , gray box). For the ordered physical mixture, I/I 0 = 0 at 4900 Hz ( Figure 6 , vertical green dotted line); for the glassy sample I/I 0 = 0 at 4780 Hz ( Figure 6e , vertical purple dotted line). This spread of conditions translates into an uncertainty of the P3HT I/I 0 values of 0.07 to 0.1 ( Figure 6d , purple and green horizontal lines, respectively). For PCBM, the spread of I/I 0 values range from −0.42 to −0.39 ( Figure 6e , horizontal purple and green dotted lines) depending on order. If the PCBM polarization values are used to construct the spin diffusion plot using Equation (1), then, based on the spread in those values, one would expect a spread of ΔM values of approximately 0.07.

With spectral deconvolution, error can potentially be reduced if 1) one can reliably synthesize (via spectral differences) artifact-free, adequately phased spectra of the neat components from the composite CRAMPS spectrum and 2) there are negligible differences between the deconvolved spectra and those of previously acquired spectra of the neat components. In our lab, we have generally found this to be feasible, particularly if spin diffusion spectra are acquired together with CRAMPS spectra via block averaging to minimize drift on short time scales (< minutes). In practice, one could qualitatively check the extent to which the spectral features resemble either previously run samples, and use the most appropriate I/I 0 value (i.e. “ordered” or “disordered”) to calculate ΔM. This procedure will result in a much more precise ΔM. For instance, since we knew the lineshapes of the ordered and glassy samples and their polarizations, we were able to easily predict the I/I 0 values in the physical mixture ( Figure 6e , solid lines) with a high degree of certainty as witnessed by the agreement of the experimental data (black/red dots, Figure 6e ). However, if the experimental spectrum does not resemble either synthesized single component spectrum, an estimate of the applicable I/I 0 value could be obtained (but as of yet untested in our lab) by the inverse Fourier transform of the synthesized spectrum to obtain the time-domain data, in which the ratio of the (n+1) st data point to the first data point would yield the proper I/I 0 value. One potential issue preventing a precise I/I 0 estimate using this method would be from artifacts arising from baseline and phasing issues, which could perturb the intensity of the first time domain point.

CONCLUSIONS

Despite significant spectral overlap in the 1 H CRAMPS spectrum, with straightforward experimental protocols, errors can be significantly reduced to < 1 % in 1 H CRAMPS-based spin diffusion NMR measurements depending on the sample. Domain sizes can be determined to relatively high degree of certainty if a morphology dimensionality is known, flip angles are known to high precision (≈ ±1°), the reflected voltage-to-forward voltage is known and reproducible, the blend spectrum can be deconvolved into the neat-component spectra, and experiments are performed close to zero-total integral where corresponding spectra are adjusted, using the equilibrium lineshape, to the zero-integral condition before analysis. Criteria b through d relate mainly to the establishment of the proper scaling for the ΔM plots, or, equivalently, to the correct determination of the initial average-polarization gradient between the two components. If the component spectra that are deconvolved in the blend sample do not adequately match those of previously acquired neat component spectra, the largest contribution to error can be as high as ≈ 7%.

EXPERIMENTAL

P3HT (Plexcore 2100, Plextronics Inc., Pittsburgh, PA) and PCBM (99.5 %, NanoC Inc., Westwood, MA) were used as received. According to the manufacturer the P3HT is ultra-high purity (< 25 ppm trace metals) and highly regioregular (> 98% head-to-tail), with a number averaged molar mass of 64500 g/mol and polydispersity index (PDI) of ≤ 2.5. Ordered films were prepared by drop casting solutions (15 mg/mL) in chlorobenzene into a Teflon-well plate; films formed in approximately (4 to 6) h. Glassy films were prepared by drop casting solutions (15 mg/mL) in chloroform (> 99.8 %) onto a 70 °C heated Teflon substrate; films formed in < 5 s. The samples were ≈5 mg. The chloroform (> 99.8 %) and chlorobenzene (> 99.8 %) were used as received. P3HT films were sliced into approximately fifty fine flakes of ≈0.1 mm dimension and (lightly) pressed into disks to ensure isotropy and homogeneity in the NMR experiments. PCBM powder was collected with a non-magnetic stainless steel spatula. The P3HT/PCBM blend sample ( Figure 1 ) was drop cast into a chlorobenzene Teflon-well plate, removed with a non-magnetic stainless steel spatula and, if applicable, thermally annealed on a hot plate at 150 °C in a glovebox for 30 minutes. Additional characterization of the blend samples are given in [ 15 ].

NMR characterization

CRAMPS NMR experiments were performed on Bruker DMX300 spectrometer (7.05 T) at 300.13 MHz, a 5 mm Doty CRAMPS probe, and with the MREV-8 pulse sequence [3 with the following parameters: eight 1.5 ⍰s ⍰/2 pulses, cycle time 39.6 μs, 400 data points, 8-32 scans, 8 s recycle delay and 65136 zero filling points. Silicon nitride rotors with Kel-F caps and spacers were used for magic angle spinning.

ACKNOWLEDGMENT

M. P. Donohue, V. M. Prabhu, D. L VanderHart, and C. L. Soles are acknowledged for their thoughtful comments and edits of this manuscript.

This work was carried out by the National Institute of Standards and Technology (NIST), an agency of the U. S. government, and by statute is not subject to copyright in the United States. Certain commercial equipment, instruments, materials, services, or companies are identified in this paper in order to specify adequately the experimental procedure. This in no way implies endorsement or recommendation by NIST.

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I. INTRODUCTION

Ii. methods, a. computational methods, b. experimental solid-state nmr, c. experimental processing, iii. theory, a. low-order correlations in liouville space theory, b. basis set selection, iv. results and discussion, a. effect of resonance offset on spin diffusion, b. relation to perturbation approaches, c. effect of dynamics on spin diffusion, d. spin diffusion experiments, v. conclusions and outlook, supplementary material, acknowledgments, author declarations, conflict of interest, author contributions, data availability, nuclear spin diffusion under fast magic-angle spinning in solid-state nmr.

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Ben P. Tatman , W. Trent Franks , Steven P. Brown , Józef R. Lewandowski; Nuclear spin diffusion under fast magic-angle spinning in solid-state NMR. J. Chem. Phys. 14 May 2023; 158 (18): 184201. https://doi.org/10.1063/5.0142201

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Solid-state nuclear spin diffusion is the coherent and reversible process through which spin order is transferred via dipolar couplings. With the recent increases in magic-angle spinning (MAS) frequencies and magnetic fields becoming routinely applied in solid-state nuclear magnetic resonance, understanding how the increased 1 H resolution obtained affects spin diffusion is necessary for interpretation of several common experiments. To investigate the coherent contributions to spin diffusion with fast MAS, we have developed a low-order correlation in Liouville space model based on the work of Dumez et al. (J. Chem. Phys. 33 , 224501, 2010). Specifically, we introduce a new method for basis set selection, which accounts for the resonance-offset dependence at fast MAS. Furthermore, we consider the necessity of including chemical shift, both isotropic and anisotropic, in the modeling of spin diffusion. Using this model, we explore how different experimental factors change the nature of spin diffusion. Then, we show case studies to exemplify the issues that arise in using spin diffusion techniques at fast spinning. We show that the efficiency of polarization transfer via spin diffusion occurring within a deuterated and 100% back-exchanged protein sample at 60 kHz MAS is almost entirely dependent on resonance offset. We additionally identify temperature-dependent magnetization transfer in beta-aspartyl L-alanine, which could be explained by the influence of an incoherent relaxation-based nuclear Overhauser effect.

Spin diffusion is a reversible and coherent process through which spin order may be transferred via dipolar couplings in the solid state. In 1 H solid-state nuclear magnetic resonance (NMR) at slow magic-angle spinning (MAS) frequencies, spin diffusion occurs in a manner analogous to macroscopic diffusion owing to the nucleus’s low chemical shift dispersion and strong dipolar couplings. The spatial diffusional nature of this transfer has led to it being applied to the study of systems from materials 1 and biomaterials, 2,3 to small molecules 4–6 and proteins. 7,8 Additionally, spin diffusion plays an important role in dynamic nuclear polarization NMR (DNP-NMR). 9–15 Recently, experimental methods relying on selective pulses to exploit the increased resolution at faster MAS and higher magnetic fields have been introduced. For example, selective pulses have found use in reducing t 1 noise, 16 in increasing the rate of experimental acquisition, 17 and in selectively investigating pharmaceuticals in the presence of excipients. 18 In addition, low power pulses with narrow bandwidth were used for implementing chemical exchange saturation transfer (CEST) in the solid state, 19,20 where spin diffusion may be an alternative mechanism to chemical exchange that needs to be considered. 21 Modeling spin diffusion transfer is of key importance to understanding the results of such experiments.

The spin dynamics at slow spinning frequencies have been shown to be adequately reconstructed using diffusion-based perturbation theory simulation approaches. 4,6,22,23 In these approaches, perturbation theory is used to derive rate expressions that are then used to model the system as a diffusive process. 24,25 It has even been shown that such models are able to solve crystal structures from known unit cell parameters to excellent precision. 22 However, with the increase in resolution obtained using higher MAS frequencies and higher magnetic fields, the assumption that spin diffusion may be treated in an entirely spatial manner begins to break down. 24–26 As an energy conserving process, it follows that spin diffusion between spins with dissimilar energy level separations (i.e., different chemical shifts) is only possible if interacting with a spin energy bath, such as that provided by a dense dipolar coupled proton spin network. The decrease in spectral overlap with higher spinning frequencies arises because these dipolar coupling networks are more effectively averaged, and this combined with the larger energy level separations at higher magnetic fields means that spin diffusion becomes strongly dependent on the resonance offset between two spins. The importance of this resonance offset dependence was recently demonstrated in the work of Agarwal, 27 where it was shown that, in proton spin diffusion spectra of L-Histidine · HCl · H 2 O, negative cross peaks may be observed arising due to the interaction of four spins simultaneously, where the difference in differences between pairs of spins lead to a n = 0 rotational resonance transfer with inverse sign.

This four spin interaction effect would not arise in perturbation theory-/diffusion-based approaches, with the exception of qualitative models intended explicitly to study the effect. 28 Indeed, the majority of such models published to date either include the resonance offset through an exponential or Gaussian approximation of a zero-quantum line shape 23 or exclude it entirely. 4,22 Computational calculations in which the spin evolution of the density matrix is simulated under the spin Hamiltonian would, in theory, accurately reconstruct the coherent spin dynamics. Unfortunately, owing to their exponential scaling (∝2 n , where n is the number of spins), such simulations are typically restricted to systems with fewer than 12 spins. 23,29 As a result, they are unable to accurately model spin diffusion for which interactions with many more spins must be considered.

One approach that has been used to remedy this scaling problem is the use of restricted basis sets. 30–33 In such approaches, the number of basis states for which the evolution must be considered are drastically reduced by omitting those that can be assumed to contribute negligibly to the evolution of the spin system. Restricted basis set methods have been shown to enable accurate simulation of spin systems containing thousands of interacting spins. 11 In the work of Dumez et al. , 31 the low-order correlation in Liouville space (LCL) method was introduced, where only zero-quantum operators are considered, and product states are limited to those containing at most q interacting spins. Such an algorithm scales polynomially as n q and allows for the number of spins in simulations to be drastically increased. The LCL method was further developed by Perras and Pruski, 32 who introduced local restriction (LR-LCL), where only the N closest spins to each spin were considered to be interacting, resulting in a linear scaling algorithm (∝ n × N q −1 ). Such an approach has been applied to modeling DNP in systems containing thousands of atoms. 11,34

LR-LCL simulations are, however, considered to be accurate up to only ∼40 kHz MAS frequencies. 11 This limitation arises due to the aforementioned increasing dependence on resonance offset and the chemical anisotropy. Though resonance offset was included in the LR-LCL model introduced in the work of Perras and Pruski, 32 to our understanding, it was included solely as an isotropic chemical shift.

We introduce a new method for basis set selection and further develop the LCL method to include both isotropic and anisotropic chemical shift (chemical shift anisotropy, CSA). We then consider the effects of the full chemical shift, MAS frequency, magnetic field, and dynamics on the evolution of the spin system. We show experimental results for the dipeptide β -aspartyl L-alanine ( β -AspAla), for which we find agreement with the simulated trends, but additionally observe temperature-dependent behavior, which may be indicative of an incoherent 1 H– 1 H homonuclear nuclear Overhauser effect. Finally, we show that the efficiency of polarization transfer via 1 H spin diffusion in a deuterated fully back-exchanged protein at 60 kHz MAS is dominated by resonance offset.

The crystal structure of β -AspAla (CCDC: FUMTEM) 35 was geometry optimized by density functional theory (DFT) using CASTEP 16.1. 36–38 CASTEP implements DFT using a plane-wave basis set. The default CASTEP 16.1 ultrasoft pseudopotentials were used. The Perdew–Burke–Ernzerhof (PBE) implementation of the generalized gradient approximation was used as the exchange–correlation functional. 39 Plane waves up to 700 eV were used. The same cutoff energies were then used to determine magnetic resonance parameters using the Gauge-Including Projector Augmented Wave (GIPAW) method 40–44 under the same DFT conditions to determine the CSA tensors, which were then extracted using Magresview. 45  

2. Spin diffusion simulations

We develop upon the excellent Tourbillon model introduced in the work of Dumez et al. 31 Specifically, we have further extended this model to use locally restricted basis sets and to implement the use of an unordered map for storing the density states, as introduced by Perras and Pruski. 32 Additionally, we implemented isotropic and anisotropic chemical shift evolution, the ability to output zero-quantum line shapes, and an implementation of our basis set selection method ( see below ). The code may be found at https://www.github.com/ThatPerson/Tourbillon_fastMAS .

Spin diffusion simulations were run using a complete unit cell of β -AspAla (4 molecules in the unit cell and 12 1 H per molecule, i.e., n = 4 × 12 = 48 spins) using periodic boundary conditions. Unless otherwise stated, all simulations began with the inversion of all carboxylic acid protons (the site with the most separated 1 H NMR resonance), for which the interatomic distances and resonance offsets are shown in Table I . This system is illustrated in Fig. 1 , where the spins of particular interest are color coded: Asp HA (orange), the spatially closest proton to COOH; Ala NH (green), the closest in chemical shift to COOH; and Ala CH 3 (lilac), for which there is particularly interesting spin evolution. In the case of the alanine CH 3 , we only considered one of the protons (labeled 12 in Table I ) when plotting the trajectories as the evolution differs slightly between nonsymmetrically equivalent sites. The spins are numbered 1–48, where the protons are numbered sequentially for each individual β -AspAla molecule, i.e., molecule 1 is numbered 1–12, molecule 2 is numbered 13–24, etc. Experimental isotropic chemical shifts were used, 46 but the CSA tensors were calculated as described in a prior section. We note that these experimental isotropic chemical shifts differ from those given at lower spinning frequencies likely due to sample rotation heating. 47 The rotational motion of the NH 3 and CH 3 groups was considered by assuming averaging of the chemical shift tensors in the molecular frame prior to conversion into the interaction frame for both of these sites, though no explicit averaging of dipolar couplings was considered unless indicated explicitly; the effect of dynamics on spin diffusion will be considered in Sec. IV C .

Nearest neighbor distances and resonance offsets between the carboxylic acid proton (COO H ) and the other sites in β -AspAla. Spins of particular interest are in bold.

β -aspartyl L-alanine ( β -AspAla), the model system used here. The spins of interest are highlighted in their respective colors. (a) A representation of the DFT (CASTEP) geometry-optimized structure, centered on one of the carboxylic acid protons. (b) A 1 H one-pulse MAS NMR spectrum acquired at a spinning frequency of 55 kHz and a 1 H Larmor frequency of 600 MHz. These spins were chosen to best illustrate various principles of this system: The Ala COOH (gray) is the most isolated resonance in the spectrum and so the easiest to selectively invert/saturate experimentally without interfering with other sites; the Ala NH (green) site is the closest in chemical shift to Ala COOH; the Asp HA (orange) is the closest in space to Ala COOH; the Ala CH 3 (lilac) experiences “inverse sign spin diffusion” (see discussion in Sec. IV A ).

Simulations were performed for t mix = 100 ms [see Fig. 2(a) ] using a REPULSION-48 set of crystallite orientations 48 and a time step of 0.2 µ s. Simulations were run using the University of Warwick Scientific Computing Research Technology Platform (SCRTP) High Performance Computing clusters, on nodes consisting of two Intel Xeon 24 core processors giving 48 cores per node. Parallelism was implemented with each crystallite running in an individual thread using OpenMP. 192 GB of RAM was present per node; however in the case of some larger models, high memory nodes were used with up to 1.5 TB of RAM. We additionally ran simulations using a REPULSION-128 set of crystallite orientations using the HPC Midlands Tier 2 High Performance Computing cluster Sulis, on nodes containing two AMD EPYC 7742 (Rome) 2.25 GHz 64 core processors, giving 128 cores and 512 GB of RAM per node. There was no additional benefit to using more crystallites (see Fig. S1).

(a) Effective pulse sequence used for simulations. (b) The pulse sequence used experimentally to probe spin diffusion. (c) Pulse program used for experiments on protein samples (see Sec. IV D 2 ), where the additional 15 N dimension was necessary for resolution of different sites. The saturation pulse was applied at the 1 H frequency of interest (see main text). In (b), an echo was used for background suppression. The phase cycle for (b) was as follows: Gaussian pulse {y}, 90° {x −x}, 180° {x −x y −y}, detect {x −x −x x}.

β -AspAla was purchased from Bachem (Switzerland) and packed as received into both a 1.3 mm zirconia rotor and a 1.6 mm zirconia rotor. The 1.6 mm rotor then had a plug of KBr inserted prior to cap insertion. Perdeuterated ( 2 H, 13 C, 15 N) GB1 with 100% back-exchange was prepared as described in Ref. 49 and packed into a 1.3 mm zirconia rotor, sealed with silicon glue.

2. Spectrometer and MAS probe

Experiments were performed at three fields: a 1 H Larmor frequency of 599.5 MHz with a Bruker Avance Neo console using a Bruker 1.3 mm HXY probe; a 1 H Larmor frequency of 850 MHz with a Bruker Avance Neo console using a Phoenix 1.6 mm HXY probe; and a 1 H Larmor frequency of 1 GHz with a Bruker Avance Neo console using a Bruker 1.3 mm HCN probe.

3. 1D 1 H selective spin diffusion experiments on β -AspAla

Spin diffusion experiments were performed using the 1.3 mm HXY probe operating in double resonance mode spinning at 55 kHz at a 1 H Larmor frequency of 599.5 MHz. 1D proton spin diffusion (PSD) experiments 50 were performed in a manner analogous to saturation transfer difference methods in the solution state, 51,52 using variable length “trains” of Gaussian inversion (180°) pulses for the saturation of the highest ppm resonance, the carboxylic acid resonance, at 13.1 ppm. Gaussian inversion pulses were optimized for a pulse length of 1.1 ms. Hard pulses were applied to 1 H with a nutation frequency of 100 kHz, corresponding to a 90° pulse length of 2.5 µ s. The pulse sequence for this is shown in Fig. 2(b) , where it is compared with the effective pulse sequence for the simulations [ Fig. 2(a) ]. Note that simulations were performed using a single inversion [ Fig. 2(a) ] as the Gaussian pulse trains used experimentally have the potential to generate nonzero-quantum coherence, which cannot be represented using our model. On the other hand, experimentally saturation was used as this ensures a greater degree of magnetization transfer. The number of inversion pulses applied was varied linearly from 0 to 100, with eight coadded transients acquired per increment. Except where stated, a recycle delay of 2 s was used. The resulting data were then analyzed by taking the peak intensities for each peak. Spectra were referenced according to the resonance offset for the methyl group used for simulations at 1.4 ppm. 46 Temperatures were calibrated using the chemical shift dependence on temperature of the 79 Br resonance in KBr. 53 Further experiments were performed: 1D PSD experiments were performed using a 1.6 mm HXY probe at 40 kHz at a 1 H Larmor frequency of 850 MHz over a wider range of temperatures; these results are shown in the supplementary material (Figs. S7 and S8). A 2D PSD experiment was performed using a 1.3 mm HCN probe at 60 kHz at a 1 H Larmor frequency of 1 GHz.

4. Spin diffusion in GB1 probed via a 2D 1 H– 15 N Experiment

Experiments were performed using the 1.3 mm HXY probe operating in triple-resonance HCN mode spinning at 60 kHz. Cooling was applied to achieve a sample temperature of ∼300 K as calibrated using the difference in shift between 2,2-dimethyl-2-silapentane-5-sulfonate sodium salt (DSS) and H 2 O. 54 2D spin diffusion experiments were performed in a manner analogous to saturation transfer difference methods in the solution-state 51,52 or CEST experiments. 19 Low-powered (∼100 Hz) bandwidth saturation was applied for 500 ms, both on resonance with various N–H sites and off resonance. The pulse sequence for this is shown in Fig. 2(c) . A total of 32 coadded transients were acquired for each of 128 t 1 free-induction decays (FIDs), using States-TPPI in F 1 for sign discrimination and a 2 s recycle delay. Except for the saturation pulse, the 1 H transmitter was placed at 2.46 ppm with a spectral width of 39.7 ppm. The 15 N transmitter was placed at 120 ppm, with a spectral width of 54.9 ppm. Hard pulses were applied to 1 H with a nutation frequency of 100 kHz, corresponding to a 90° pulse length of 2.5 µ s. On 15 N, hard pulses had a nutation frequency of 50 kHz, corresponding to a 90° pulse length of 5 µ s. Cross-polarization (CP) was applied with a 70:100% linear ramp 55 on 1 H, meeting the Hartmann–Hahn condition with 45 kHz on 1 H and 15 kHz on 15 N. 56–58 This was applied for 800 µ s for the 1 H → 15 N CP and for 600 µ s for the 15 N → 1 H CP. 100 ms MISSISSIPPI water saturation using 10 kHz 1 H irradiation was used. 59 Spectra were referenced according to DSS at 0 ppm, with 15 N referenced indirectly to liquid NH 3 at 0 ppm using the IUPAC recommended frequency ratio. 54,60 Low-powered ( ν 1 = 10 kHz) WALTZ-16 heteronuclear decoupling was applied to the 1 H channel during 15 N evolution and again to the 15 N channel during 1 H acquisition. 61  

β-aspartyl L-alanine experiments : Recorded experimental spectra were integrated using a python script using both nmrglue 62 and lmfit 63 to fit a polynomial background and Lorentzian line shapes to each peak (code can be found at https://github.com/ThatPerson/PeakInt1D ). Following optimization of the offset on the first slice, the offsets were fixed and only the peak height and width were allowed to vary. Fitting of the peak width was performed to account for noise, noting that we would not expect variation in peak width with the duration of the saturation pulse.

Perdeuterated GB1 experiments : Peaks in 2D experimental spectra were integrated using CARA. 64,65 Assignments were taken from Ref. 65 .

A brief introduction to the LCL approach is given here, based on that presented in the work of Brüschweiler and Ernst, 66 Dumez et al. , 31 and Perras and Pruski. 32 For clarity, a full list of symbols and associated descriptions is given in Table II .

List of symbols used in the text.

The low-order correlation in Liouville (LCL) method relies on basis set reduction by excluding certain terms. 30,31,67,68 For the simulation of spin diffusion, the most obvious terms to omit are those that are not zero-quantum since spin diffusion is fundamentally a zero-quantum process. 24,25 Further basis set reduction can be performed as considered below in Sec. III B below.

Ideally, we would like to consider a basis set in which we consider all spin interactions. Such a simulation, akin to a full density matrix treatment as implemented by SIMPSON 70,72,73 and SpinEvolution, 74 would scale exponentially (∝ 2 3 n assuming Hilbert space simulations). 73 To reduce the complexity of the problem, the simulation can be approximated using a reduced basis set. In the work of Edwards et al. , 75 three basis sets were introduced for the solution state with the aim to reduce the number of terms that must be included: (1) IK-0, in which only product operator states consisting of a given number of spins are considered; (2) IK-2, in which only product operator states containing spins that are close in proximity are included; and (2) IK-1, where both criteria are applied. In the solid state, the analogous restricted basis sets used in the work of Dumez et al. 31 correspond to an IK-0 basis set. The local restriction introduced by Perras and Pruski 32 further develops this into an IK-1 basis set.

Simulations of the pulse sequence in Fig. 2(a) applied to the β -AspAla unit cell ( n = 48) at ν r = 60 kHz, ν 0 ( 1 H) = 600 MHz, with a variable basis set size ( N avg , approximate number of neighbors per spin). (a) Evolution of the z -magnetization, M z , of an Asp NH spin after inversion of the carboxylic acid proton. The spread about the lines indicates twice the standard deviation for all symmetry equivalents. (b) The computational time required to run a model of each size, relative to the model with N avg = 2. The simulated z -magnetization evolution rapidly converges, with a basis set consisting of on average 16 spin neighbors per spin approximating well a system containing an average of 24 spin neighbors. Only up to a spin system containing 24 spin neighbors per spin was considered as above this was not computationally feasible.

In our modeling, we limited ourselves to product operator states containing at most q = 4 spins, an upper limit that has been shown to allow for accurate simulation when compared to exact simulation at slower spinning. 68 While a larger value of q max may model the spin evolution better, 30,31 such a larger basis set is infeasible. The faster spinning frequencies modeled here mean that the rotor cycle time is shorter and, therefore, more steps are required to sample the rotation adequately for any given time, and additionally the slower rate of spin diffusion means that longer times must be simulated. A significantly larger absolute number of steps, P , therefore needs to be calculated. As such, given the exponential scaling nature of q , 30,31 [see Eq (16) ] it is not feasible to simulate with larger product operators included.

A method for selecting which spin pairs contribute significantly is necessary to perform a basis set reduction in spin space. It has been shown that applying spatial restrictions (i.e., selecting the N max closest spins) to the choice of spin states to include is valid 30–32,68,75 when the evolution of the spin dynamics is determined predominantly by through-bond J-couplings (as in solution-state NMR) or dipolar couplings in the presence of a strongly coupled spin bath (as at slow spinning frequencies). However, this assumption begins to fail for solid-state NMR in the fast spinning regime. 26,27,46 At attainable fast spinning frequencies, the dipolar couplings are not averaged to zero as in the solution state but are averaged sufficiently to make the transfer of magnetization strongly truncated by resonance offset.

Spin-pair interaction scores as calculated using Eq. (17) over a representative simulation of the pulse sequence in Fig. 2(a) including all spin pairs at ν r = 60 kHz, ν 0 ( 1 H) = 600 MHz, beginning with inversion of the carboxylic acid proton. LCL simulations without restriction are run for 1000 steps (0.2 ms), with the weighted population of two, three, and four spin states plotted above. Spin pairs involving the COO H and the Ala N H , Asp H A, and Ala C H 3 in β -AspAla are shown in green (solid), orange (dashed), and lilac (dotted), respectively (see Fig. 1 and Table I ). Beyond ∼0.1 ms, the contribution of the 2, 3, or 4 spin states no longer significantly changes such that the ordering of which spin interactions to include is likely fixed at this point.

The spin pairs that contribute the most are not necessarily those closest in proximity; Fig. 5 shows the magnitude of the score in Eq. (17) as a function of distance and resonance offset for a methyl group proton (spin-10 in Table I ), where there is no clear dependence on either parameter. Figure S2 depicts the spatial arrangement of the spins included as interacting for this methyl group proton at ν r = 20 kHz and ν 0 ( 1 H) = 600 MHz.

Basis-set selection according to the method outlined in Sec. III B for the pulse sequence in Fig. 2(a) starting with COOH resonance inversion at the indicated spinning frequencies (ν 0 ( 1 H) = 600 MHz) as a function of their resonance offset and distance from spin-10, a CH 3 proton in molecule 1. The size of each point represents the score as in Eq. (17) , i.e., the sum of the weighted 2, 3, and 4 spin terms. Spins that were included as “neighbors” to the spin of interest are shown highlighted in blue by our method of restricting the average number of spin neighbors per spin, N avg , to 16, while spins that were not included as neighbors are shown in black. The number of “neighbors” included per spin are shown in Fig. 6 . The spatial arrangement of the spin pairs included for the ν r = 20 kHz simulation are shown in Fig. S2 in the supplementary material .

Number of neighboring spin pairs selected for N avg = 16 as a function of the Spin ID of interest for three different spinning frequencies, for simulations of the pulse sequence in Fig. 2(a) at ν 0 ( 1 H) = 600 MHz beginning with inversion of the carboxylic acid proton resonance (gray), with chemical shift anisotropy included and no dipolar averaging. Colors are as in Fig. 1 and sites with asterisks are those used in the following figures. The spin IDs are as given in Table I , noting that the unit cell contains four symmetry equivalent molecules (1–12, 13–24, 25–36, 37–48).

Note here that we use the average number of spin neighbors per spin ( N avg ) as opposed to a fixed number of spin neighbors per spin ( N max ). This is because when ordering the spin pairs by score, we find that some spins are generally more important to the evolution of other spins; for example, the carboxylic acid protons are inverted at the start of the simulations and as a result have the greatest magnetization gradient with their neighbors and therefore tend to have a much larger contribution to many other spins. This is seen in Fig. 6 , where generally the carboxylic acid protons (gray) are found to contribute more to the spin evolution of far more other spins than is the case for the NH 3 protons (lilac). This is also evident in Fig. 5 where the biggest circles (highest scores) are at the greatest resonance offset. At faster magic-angle spinning frequencies, the relative spin pair scores change as shown in Fig. 5 , leading to a different basis set selection, thereby making different spin pairs contribute more or less as in Fig. 6 . For example, two spins that are spatially proximate but well separated in resonance frequency will contribute more at lower spinning frequencies, but, with the truncation of resonance offset at faster spinning frequencies, they will consequently contribute less and therefore be less likely to be included. This has important consequences for the evolution of the spin system as will be discussed in the following sections.

The chemical shift of a spin relates to how the local environment changes the effective magnetic field experienced by that spin and therefore the change in the difference in energy between the spin-up and spin-down states. Spin diffusion, being an energy conserving diffusion of spin order between sites, is therefore truncated by the offset between two resonances corresponding to two chemically distinct sites in the solid-state structure. Dipolar couplings to the large proton bath provide an external reservoir of energy to allow for spin diffusion to occur as apparent by the broad overlapping peaks in 1 H solid-state NMR spectra. In the limit of low magnetic fields and low spinning frequencies, the contribution from this spin bath is sufficient to ensure that the rapid magnetization transfer between different spins occurs in an approximately spatial manner (i.e., that the efficiency of the polarization transfer is approximately related to the distance between spins). 22,23

With faster magic-angle spinning and higher magnetic fields, however, the spatial nature of this transfer begins to break down. Figure 7 shows the effect of including both isotropic and anisotropic chemical shift in our spin diffusion model. It is notable that, in the absence of isotropic or anisotropic chemical shift (blue dotted lines), the reduction in the rate of spin diffusion is negligible as higher spinning frequencies are attained, as seen by comparing with the red dashed lines. This suggests that it is the truncation by resonance offset that dominates the reduction in spin diffusion rather than this being due to the direct averaging of dipolar couplings. Note that, in the long-time limit, the curves converge at 83.3%. This occurs because, in the absence of any relaxation, the diffusion will eventually cause the system to equilibrate at the average of the initial magnetization. Noting the inversion of some of the spins, namely, the carboxylic acid protons, this average is then { 44 × + 1 + 4 × − 1 / 48 = 0.833 ⁠ .

Simulated effect of adding both isotropic and anisotropic chemical shift to the employed LCL model, as a function of spinning frequency, at ν 0 ( 1 H) = 600 MHz, for a unit cell of β -AspAla (n = 48) with periodic boundary conditions. The evolution of the magnetization ratio, M z , is shown for the Ala NH (spin 5 in Table I ), after inversion of the COOH proton resonance [see pulse sequence in Fig. 2(a) ].

Figure 7 further exemplifies the importance of including not only the isotropic component of the chemical shift tensor but also the anisotropic component. In the absence of anisotropic chemical shift, it appears that the rate of spin diffusion is further reduced as the anisotropic component of the chemical shift tensor does not commute with the homonuclear dipolar interactions and so aids the spin diffusion.

The increasing dependence of spin diffusion on resonance offset between sites may also be seen in Fig. 8 . Here, the transfer of magnetization for the Ala NH (4.6 ppm offset from COOH), Asp HA (8.6 ppm offset), and an Ala CH 3 proton (11.7 ppm offset) is shown as a function of spinning frequency (10–150 kHz) and magnetic field ( 1 H Larmor frequencies of 100 MHz, 600 MHz, and 1 GHz). At 100 MHz, it is observed that there is little truncation due to offset even for high spinning frequencies (≥60 kHz), indicating little isotropic chemical shift resolution even at these higher spinning frequencies. This is exemplified by comparison between 20 kHz MAS at 1 GHz and 60 kHz MAS at 100 MHz, where the rate of transfer appears qualitatively similar. At 1 GHz, spinning frequencies greater than 100 kHz would be expected to severely truncate coherent spin diffusion.

(a) Visualization of the system with spins of interest highlighted with their respective colors. (b)–(r) Simulated effect of the spinning frequency and applied magnetic field (given as 1 H Larmor frequency) on the rate of spin diffusion after inversion of the COOH proton [see pulse sequence in Fig. 2(a) ]. Simulations were run for a full unit cell of β -AspAla ( n = 48), with periodic boundary conditions, in the absence of dipolar averaging. Colors are matched with Fig. 1 , i.e., solid green (Ala NH), dashed orange (Asp HA), dotted lilac (Ala CH 3 ) (see also Table I ). Left to right: ν 0 ( 1 H) = 100, 600 MHz, 1 GHz. Top to bottom: ν r = 10, 20, 40, 60, 100, 150 kHz. The spread around each line corresponds to twice the standard deviation for all four equivalent spins in the system. This error increases at higher spinning frequencies likely due to faster MAS giving shorter rotor cycle times and therefore sparser sampling of the evolution of anisotropic chemical shift and dipolar couplings under MAS (noting that we used a timestep of 0.2 µ s for all simulations).

Coherent inverse sign spin diffusion, as observed in L-Histidine · HCl · H 2 O at 60 kHz at 500 and 700 MHz, 27 is also seen. In these simulations, such spin diffusion is manifested by the magnetization associated with a site increasing, e.g., M z > 1, as is evident for the inset axes on Fig. 8 (l, n, o, q, and r). This is expected to occur when four interacting spins, I, S, R, and P, meet a resonance condition of ω I − ω S − ω R − ω P = 0 ⁠ . In this system, we envisage that it is the Ala COOH (13.1 ppm), Ala NH (8.5 ppm), Ala HA (5.4 ppm), and Ala CH 3 (1.4 ppm) protons that approximately meet this four spin resonance condition, as 8.5 − 1.4 = 7.1 ≈ 13.1 − 5.4 = 7.7. It is found to be particularly relevant in this system for the methyl protons (lilac) at spinning frequencies >60 kHz at 1 GHz, and >100 kHz at 600 MHz; a more thorough analysis will be provided in Sec. IV C .

While the restricted basis set low-order correlation in the Liouville space model used here is orders of magnitude faster than simulating the complete evolution of the full density matrix, it is still far slower than perturbation theory diffusion equation-based approaches. As such, relating the resonance-offset dependence here to these methods is of interest to the broader applicability of such models at faster spinning frequencies.

In such models, the dependence of spin diffusion on both spinning frequency and resonance offset arises via this line shape function. As such, understanding the nature of this function has been of interest. 77,78 Typically, however, this has been done using convolutions of single-quantum line shapes 78 or through master equation-based modeling. 77 By comparison, here, we present simulations in Fig. 9 at ν 0 = 600 MHz, where we consider the evolution of the I + S − state for different spinning frequencies during the normal evolution of the system after inversion of the carboxylic acid proton resonance. In Sec. S2 of the supplementary material (see Figs. S4–S6), we present the resulting ZQ line shapes arising from higher-order terms that are zero-quantum in these same spins (e.g., I + S − R z ). We find that these higher-order zero-quantum operators show the same trend upon increasing MAS frequency, however, with increased broadening.

Simulated zero-quantum line shapes at the first spinning sideband as obtained from evolution of the I + S − ZQT coherence after inversion of the carboxylic acid proton resonance. Simulations were run with ν 0 ( 1 H) = 600 MHz and variable ν r , for a full unit cell of β -AspAla ( n = 48) with periodic boundary conditions. The interactions between spin-2 and spin-5 are shown where spin-2 and spin-5 are the NH 3 + and the Ala NH proton of molecule 1, respectively (see Table I ).

Specifically, the zero-quantum line shape of the first spinning sideband between an Asp NH 3 + proton and Ala NH protons of molecule 1 extracted from our model are shown in Fig. 9 for a range of spinning frequencies. At spinning frequencies ≤15 kHz, the line shape is broadened such that it is not possible to identify a single peak, which suggests that spin diffusion in this regime will have little dependence on the exact form of this function. As the spinning frequency increases, however, the line shapes narrow significantly. It is also noted that the intensity of the spinning sidebands is significantly reduced with higher MAS frequencies, reflecting the reduction in spin diffusion rate with spinning frequency.

A result of this zero-quantum line narrowing is that we would expect spin diffusion to occur only within a well-defined range of resonance offsets, with relayed transfer slightly broadening this by allowing magnetization to transfer between rather more separated spins. Modeling this line narrowing in the manner utilized here may enable the development of perturbation theory approaches at faster MAS frequencies by accounting more directly for the change in ZQT line shape, rather than approximating this effect with a simple decaying exponential as performed in earlier work. 23  

Many of the systems of interest for characterization with spin diffusion-based techniques exhibit dynamics. 2,79–81 Dynamic motions affect the coherent evolution in spin space by leading to averaging of dipolar couplings and CSA tensors. 82,83 Even in crystalline solid samples, it is known that significant motions may be present. 83–88 Of particular importance are rotational motions of axially symmetric groups, such as methyl groups, primary amines, and phenyl groups, which are typically fast on the NMR timescale and with limited energetic barriers. 89,90

Motions occurring at frequencies coincident with various spin interactions can give rise to relaxation. In the case of interacting proton spins, motion could lead to incoherent cross relaxation and the nuclear Overhauser effect. Such an effect would interfere with the coherent spin evolution. A relaxation superoperator-based treatment 91 of such effects is beyond the scope of this paper, where we are solely interested in the coherent evolution of the system.

In our model, we introduce dynamics through two parameters. Overall motion is reflected by an order parameter S , which applies a linear scaling to all dipolar couplings between spins. C 3 rotation of the NH 3 and CH 3 groups was modeled by averaging the dipolar couplings between protons within the same group by P 2 (cos( θ )) ( ⁠ = 1 2 3 ⁡ cos 2 θ − 1 ⁠ ), where θ is the angle between the rotational axis and the interaction of interest. The angle is usually taken to be 90° so that the averaging factor is 0.5. As stated previously, we also averaged the CSA tensors in the molecular frame between all protons within a given CH 3 or NH 3 group prior to rotation into the interaction frame. 77  

The spin evolution as shown in Fig. 10 (60 kHz MAS, 600 MHz) appears relatively insensitive to the overall motion order parameter. Under CH 3 or NH 3 rotation, however, the dynamics significantly change. Given that six of the 12 protons in each molecule of β -AspAla are within these axially symmetric groups, this is unsurprising.

The simulated variation in spin diffusion evolution is shown for the Ala NH, Asp HA, and Ala CH 3 sites after inversion of the carboxylic acid resonance (see pulse sequence in Fig. 2(a) under different models of dynamical averaging. All dipolar couplings within each system were scaled by the parameter S to reflect overall dynamical motion. In (b), (d), (f), all dipolar interactions within a CH 3 or NH 3 group were additionally scaled by 0.5 to represent their axial rotation. Simulations were run with ν 0 ( 1 H) = 600 MHz and ν r = 60 kHz, for a full unit cell of β -AspAla ( n = 48) with periodic boundary conditions.

Of important note is the behavior of the inverse sign spin diffusion (M z > 1) apparent for the methyl (lilac) group. As was noted by Agarwal, 27 this inverse transfer is aided by scaling the couplings involved. For this effect to occur, the four spin third-order Hamiltonian contribution must dominate over the two and three spin Hamiltonians. While at such fast spinning frequencies, the two spin terms are effectively completely averaged, relayed transfer may still come to dominate over the four spin interactions at long simulation times, hence giving the short lifetime of the effect observed in Fig. 10(e) in the absence of methyl rotation. When these relay interactions are further averaged by the methyl rotation, however, the four spin term can dominate for far longer as seen in Fig. 10(f) .

1. Example 1: Spin diffusion in β -aspartyl L-alanine

Experimental spin diffusion results obtained at 55 kHz MAS and 600 MHz are presented in Figs. 11(a) – 11(f) , with the six plots corresponding to the resolved 1 H resonances [see Fig. 1(b) and Table I ]. In these experiments, variable length trains of selective Gaussian inversion pulses were applied selectively to the carboxylic acid proton peak [see Fig. 2(b) ] and the corresponding intensity of all other sites recorded in a manner analogous to saturation transfer difference experiments conducted in the solution state. 51,92 Under these conditions, it is expected that the transfer of spin order will occur via spin diffusion with the saturated site acting as a magnetization sink. We chose to apply a saturation type experiment because, experimentally, we would experience influence from incoherent relaxation processes and so would expect a fast recovery of the COOH spin making analysis of the behavior more complex. On the other hand, under saturation, the system would evolve toward a “steady state” and, thus, the rate at which this occurs may be more readily quantified. Additionally, mis-set inversion pulses may give rise to transverse components that cannot be simulated in our model, and may not completely invert the site, meaning that the initial state would not be completely along −z as simulated. On the other hand, saturation would cause the magnetization of this site to approach 0.

(a)–(f) Experimental spin diffusion curves under saturation of the COO H proton resonance for β -AspAla at 55 kHz MAS and a 1 H Larmor frequency of 600 MHz [recorded using the pulse sequence in Fig. 2(b) ], at temperatures of 279–320 K (calibrated using KBr). There is a significant difference in the spin diffusion observed under each temperature condition. Errors are shown as twice the standard error from voigt line shape fitting with lmfit. Simulated spin diffusion curves under the same conditions are shown in red. (g) A 2D 1 H– 1 H spin diffusion spectrum recorded at 60 kHz MAS and a 1 H Larmor frequency of 1 GHz with a mixing time of 50 ms.

While we additionally present a comparison with simulation in Figs. 11(a) – 11(f) , we note that the presence of incoherent auto- and cross relaxation means that full agreement cannot be expected: Experimentally, the spin diffusion is a combination of incoherent and coherent processes, while our modeling here is purely coherent.

We observe that, under saturation, there is a strong resonance offset dependence with regard to the magnetization change under saturation. Figures 11(a) – 11(f) are ordered with the site highest in chemical shift (closest in resonance offset to the carboxylic acid) at the top, with the chemical shift progressively getting lower (with greater resonance offset) on moving down the figure. Interestingly, we note that the Ala CH 3 group experiences significant inverse sign spin diffusion similar to that which has been seen in other systems, 27,28,93 with an enhancement of ∼20%. This is significantly greater than any inverse sign spin diffusion we observed within our simulation [the maximum seen being 1.82%, in Fig. 10(f) with S = 0.94]. Moreover, we find that this effect exhibits a significant temperature dependence, with an increase in the amount of inverse sign spin diffusion at higher temperatures. Section S3 of the supplementary material investigates this temperature dependence in more detail. This observation is in contrast to the similar effect observed for L-Histidine by Agarwal 27 and suggests that this effect may be an incoherent cross relaxation effect.

Calculated nOe [see Eqs. (19) – (21) ] as a function of temperature at a range of magnetic field strengths. This was calculated assuming an Arrhenius dependence of timescale on temperature, with an activation energy of 17.4 kJ/mol, and a correlation time of 200 ps, though these are approximate. 89,95 Temperatures of 279 and 320 K are shown using vertical dashed lines.

That this is an incoherent effect is additionally supported by the 2D 1 H– 1 H spin diffusion MAS NMR spectrum of β -AspAla [ Fig. 11(g) ], where we observe negative cross peaks to other sites (specifically, Ala NH, Ala HA), which would not arise in the case of the inverse sign coherent spin diffusion mechanism.

While both homonuclear 96 ( 11 B) and heteronuclear nOes 97–100 ( 1 H– 13 C, 1 H– 15 N) have been observed before in the solid state, we believe that this is one of the first observations of a 1 H– 1 H solid state nOe. Indeed, the only example we were able to find following an extensive literature review is the suggestion that a 1 H– 1 H homonuclear nOe is responsible for the difference between 1 H– 1 H SD and 1 H– 1 H fp-RFDR spectra occurring in a study of bone at 110 kHz MAS. 101 However, in light of the results of this study, we would suggest that what was observed in those spectra was not actually a nuclear Overhauser effect, rather it was the result of resonance offset truncation in spin diffusion. While this paper was under review, a study of hydrogen–π interactions with trapped water molecules was published showing a similar nOe effect involving a methyl group. 102  

The presence of this temperature-dependent incoherent effect may suggest that at fast spinning frequencies and high magnetic fields, the spin terms responsible for coherent spin evolution are sufficiently truncated by resonance offset such that there may be a competing influence of incoherent effects.

2. Example 2: Spin diffusion in 100% back-exchanged perdeuterated GB1

Since we have observed strong resonance offset dependence for spin diffusion at fast MAS in a protonated sample, we also wanted to investigate the extent of this phenomenon in a system with a more dilute proton network. For perdeuterated GB1 with 100% back-exchange, the combination of perdeuteration along with fast magic-angle spinning (at 60 kHz) leads to excellent resolution with proton linewidths smaller than 50 Hz. Selective saturation (∼100 Hz bandwidth, 500 ms) was applied to the LYS 10 N–H site, and the ratio of the intensities of all other resolved resonances under saturation on resonance and off resonance was recorded as a function of the resonance offset from the saturation transmitter. We observe a strong dependence of magnetization change on resonance offset [ Fig. 13(a) ], yet negligible to no dependence on distance [ Fig. 13(b) ]. Thus, under these conditions, considering the intensity of peaks as even a very rough indicator of spatial proximity may lead to wrong conclusions without careful consideration of the chemical shift. Additional discussion of this effect, including saturation of other sites within perdeuterated GB1, is given in Sec. S4 of the supplementary material .

Experimental spin diffusion in 100% back-exchanged [U– 2 H, 13 C, 15 N]-GB1 at ν r = 60 kHz, ν 0 ( 1 H) = 600 MHz. Saturation at a 1 H nutation frequency of 100 Hz was applied selectively on resonance to a sidechain proton of LYS 10 for 500 ms. The ratio of peak intensity under saturation to a reference unsaturated spectrum was then calculated, as a representation of the z-magnetization. (a) Variation of saturation ratio as a function of the resonance offset from the transmitter. (b) Variation in saturation ratio as a function of the distance from the LYS 10 N–H site in GB1, considering both intramolecular and intermolecular nearest neighbors.

As faster magic-angle spinning and higher magnetic fields become routine for accessing higher resolution in 1 H solid-state NMR, understanding the behavior of spin diffusion becomes more important for designing experimental methods in which spin diffusion is utilized or where it is a competing effect to, e.g., chemical exchange like in CEST. Specifically, as 1 H sites become better resolved, the transfer of spin order between them via spin diffusion becomes increasingly dependent on their resonance offset. We showed that the inclusion of chemical shift evolution, and specifically chemical shift anisotropy, is of key importance for simulating how these systems evolve. We also investigated the effect of dynamics and found it to play an important role in the coherent evolution of the system. Finally, we observed that experimentally the situation is likely further complicated by the presence of incoherent cross relaxation effects.

While here we have only considered direct proton to proton spin diffusion, we note that the consequences and features observed likely affect other experiments. For example, CHHC 23,103 experiments as commonly applied at slower spinning rates will suffer the same resonance offset dependence under fast MAS as the proton spin bath relied upon for transfer becomes increasingly more resolved. This analysis highlights the importance of application of active recoupling under fast magic-angle spinning conditions both to collect spatial structural constraints 104–110 and in order to overwhelm the presence of incoherent cross relaxation-based nOe phenomena. On the other hand, better understanding of proton spin diffusion at fast MAS has implications for our ability to measure site-specific 1 H relaxation and quantification of 1 H CEST in the solid state.

See the supplementary material for additional discussion and data relating to the experimental phenomena observed here, as well as discussion of other considerations with regard to the model presented here, such as why the evolution of the spin system is marginally different for crystallographically equivalent sites, and on the number of crystallites that must be included.

We thank Zainab Rehman for useful discussions. The UK High-Field Solid-State NMR Facility used in this research was funded by EPSRC and BBSRC (Grant No. EP/T015063/1 as well as Grant No. EP/R029946/1). Collaborative assistance from the Facility Manager Team (Dinu Iuga, University of Warwick) is acknowledged. BT thanks Bruker and the University of Warwick for Ph.D. funding as part of the Warwick Centre for Doctoral Training in Analytical Science. Computing facilities were provided by the Scientific Computing Research Technology Platform of the University of Warwick. Some calculations were performed using the Sulis Tier 2 HPC platform hosted by the Scientific Computing Research Technology Platform at the University of Warwick. Sulis is funded by EPSRC Grant No. EP/T022108/1 and the HPC Midlands + consortium. The renewal of the 600 MHz solid-state NMR console (Avance NEO) was funded by BBSRC (Grant No. BB/T018119/1), EPSRC and University of Warwick. J.R.L. acknowledges funding from BBSRC (Grant No. BB/W003171/1). We thank Dr. Andrew P. Howes and Patrick Ruddy for supporting the operation of the Millburn House Magnetic Resonance Laboratory.

The authors have no conflicts to disclose.

Ben P. Tatman : Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). W. Trent Franks : Conceptualization (equal); Investigation (equal); Writing – review & editing (equal). Steven P. Brown : Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Józef R. Lewandowski : Conceptualization (lead); Data curation (lead); Funding acquisition (equal); Methodology (supporting); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are openly available in ZENODO at https://doi.org/10.5281/zenodo.7521397 .

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• Published: 14 September 2008

Time-resolved dynamics of the spin Hall effect

• N. P. Stern 1 ,
• D. W. Steuerman 1 ,
• S. Mack 1 ,
• A. C. Gossard 1 &
• D. D. Awschalom 1  

Nature Physics volume  4 ,  pages 843–846 ( 2008 ) Cite this article

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The generation and manipulation of carrier spin polarization in semiconductors solely by electric fields has garnered significant attention as both an interesting manifestation of spin–orbit physics as well as a valuable capability for potential spintronics devices 1 , 2 , 3 , 4 . One realization of these spin–orbit phenomena, the spin Hall effect 5 , 6 , has been studied as a means of all-electrical spin-current generation and spin separation in both semiconductor and metallic systems. Previous measurements of the spin Hall effect 7 , 8 , 9 , 10 , 11 have focused on steady-state generation and time-averaged detection, without directly addressing the accumulation dynamics on the timescale of the spin-coherence time. Here, we demonstrate time-resolved measurement of the dynamics of spin accumulation generated by the extrinsic spin Hall effect in a doped GaAs semiconductor channel. Using electrically pumped time-resolved Kerr rotation, we image the accumulation, precession and decay dynamics near the channel boundary with spatial and temporal resolution and identify multiple evolution time constants. We model these processes with time-dependent diffusion analysis using both exact and numerical solution techniques and find that the underlying physical spin-coherence time differs from the dynamical rates of spin accumulation and decay observed near the sample edges.

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Theories have predicted 5 , 6 , 12 , 13 , and experiments confirmed 7 , 9 , that an electric current in a crystal with spin–orbit coupling gives rise to a transverse spin current through the spin Hall effect (SHE). Spin-dependent scattering of carriers by charged impurities (the extrinsic SHE) 5 , 6 , 14 or the direct effect of spin–orbit coupling on the band structure (intrinsic SHE) 12 , 13 causes spin-dependent splitting in momentum space and a resulting pure spin current. Although not locally observable, the presence of this bulk spin current can be inferred from the existence of non-equilibrium spin accumulation near sample boundaries. Whereas extrinsic spin Hall currents generated by impurity scattering evolve on momentum scattering timescales (<1 ps), spin Hall accumulation is expected to develop on the much slower spin-coherence timescale τ ( ∼ 1 ns). As this timescale is of the same order as that desired for fast electrical manipulation of spin polarization in spintronics devices, understanding dynamics on this timescale is critical for both physical and practical insights into the extrinsic SHE processes.

Steady-state observations of electrically generated spin accumulation 7 , 15 , 16 are effective for inferring τ , but they cannot directly access the dynamical processes on the nanosecond timescale. In contrast, time-resolved spin dynamics with picosecond resolution are routinely measured using ultrafast optical pump–probe techniques 17 , 18 . Time resolution of bulk current-induced spin polarization was achieved using a photoconductive switch 15 , but only precessional dynamics were observed owing to the short duration of the ultrafast current pulse. Furthermore, in contrast to the boundary accumulation from the SHE, current-induced spin polarization is a bulk phenomenon and consequently neither the steady-state 4 , 15 nor the time-resolved 15 measurements investigated spatial dynamics near the sample edge. Here, we combine the spatial resolution afforded by scanning Kerr microscopy 19 with an optical probe pulse delayed relative to the electrical pump pulse to achieve both temporal and spatial resolution of spin polarization generated electrically by the extrinsic SHE in an n-doped GaAs channel. Details of the experimental technique are shown in Fig. 1 and are discussed in the Methods section.

a , Schematic diagram of time-resolved measurement of the SHE (EOM, electro-optic modulator; BS, beam splitter; PD, photodiode; FG, function generator; RF, radiofrequency). b , Optical microscope image of the sample with coordinate system defined (units in micrometres) showing the origin (black circle) and the location for most of the measurements (red circle). The bright yellow regions are the gold contacts and the grey region is the GaAs channel. c , Illustration of the a.c. pulse scheme for lock-in detection. The pure a.c. components of the switch output (red) are added to a square wave at f V to create a triggered pulse train with amplitude modulation at f V ≈1 kHz.

Figure 2 b–d shows the Kerr rotation θ K ( t ) as a function of delay time t in a fixed magnetic field B applied along the y axis. The spin polarization is generated by the SHE from a voltage pulse V ( t ) with length t p =6 ns and amplitude V 0 =2 V ( Fig. 2 a). The laser is positioned at y =126 μm so as to be close to the boundary spin generation with minimal clipping of the spot by the edge. During the pulse (0< t < t p ), spin polarization builds up owing to the SHE. After the pulse has passed ( t > t p ), the accumulated spin polarization undergoes decay and precession ( Fig. 2 b–d). We fit θ K ( t ) for t > t p to an exponentially decaying cosine to extract an inhomogeneous depolarization time τ * and Larmor precession frequency ν L = g μ B B / h , where g is the Lande g -factor, μ B is the Bohr magneton and h is Planck’s constant. Linear fits to ν L ( B ) give | g |=0.346±0.002 ( Fig. 2 e), which is consistent with g expected for this doping level 20 . The depolarization time τ * =2.8±0.1 ns measured at a specific spatial location should differ from the physical spin-decoherence time τ . Because spin polarization can diffuse owing to accumulation gradients, there is a second pathway for spin depolarization beyond decoherence that depends strongly on the measurement location. We reconcile the dynamically measured τ * with the intrinsic decoherence time τ later in our discussion.

a , Voltage pulse profile V ( t ) measured by an oscilloscope (red line) and by reflectivity modulation Δ R / R (black diamonds). The arbitrary units of Δ R / R are scaled to match the scope voltage. The vertical dashed lines mark the time region 0< t < t p . b – d , Representative scans of Kerr rotation θ ( t ) for t p =6 ns and y =126 μm. The blue lines are calculations from equation (1) with τ =4.2 ns, L s =3.9 μm and y =126 μm. e ,  ν L extracted from cosinusoidal fits to θ K ( t ). f – h , Kerr rotation θ K ( B ) as a function of magnetic field at three representative times. The blue lines are calculations from equation (1).

Typical optical studies of electrically generated spin accumulation measure the time-averaged projection of spin precession as a function of applied transverse magnetic field B . In analogy with the Hanle effect of luminescence depolarization, the z -axis spin polarization s z should depolarize for increasing B when 2π ν L ∼ τ −1 (ref.  21 ). Therefore, coherence times in steady-state experiments are typically extracted from the linewidths of s z ( B ). Near the sample edge, s z ( B ) is a Lorentzian line shape analogous to the Hanle effect 7 , whereas it becomes more complicated away from the edges owing to the interplay of spin precession and diffusion 11 , 21 . In the current experiment, measurement of θ K ( B ) does not represent a time-averaged steady-state accumulation, but rather a snapshot at a fixed time t of the dynamic behaviour of an electrically generated spin ensemble in a magnetic field.

Representative scans of θ K ( B ) at y =126 μm are shown for t =2, 5 and 9 ns in Fig. 2 f–h with the magnetic field applied along the y axis. For small t , θ K ( B ) grows in a broad peak that narrows as t increases ( Fig. 2 f). Only for t ∼ t p > τ does θ K ( B ) approach the Lorentzian line shape expected from a conventional Hanle analysis ( Fig. 2 g). For t > t p , θ K ( B ) is primarily governed by spin precession, exhibiting characteristic periodic lobes of decreasing amplitude away from B =0 ( Fig. 2 h).

In Fig. 3 a, we use a longer pulse t p =15 ns to investigate accumulation dynamics with the current flowing for various V 0 . We fit θ K ( B =0) to an exponential saturation with a time constant τ acc . For each V 0 , τ acc is around 40% of the τ * measured from decay of the spin polarization ( Fig. 3 b). Both τ * and τ acc decrease weakly with V 0 , which is expected owing to electron heating 11 , 22 .

a , Kerr rotation θ K ( t ) (upper panel), and inverse field width B 1/2 −1 (lower panel) with t p =15 ns at y =126 μm for V 0 =2.0 V (black), 1.5 V (red), 1.0 V (blue) and 0.5 V (green). The solid lines are fits from our model. b , The accumulation time τ acc (green triangles), Hanle time τ 1/2 (blue circles), decay time τ * (black squares) and coherence time τ (red diamonds) as a function of voltage. The error bars represent one standard error of the parameter obtained from a least-squares fitting routine. c ,  θ K ( t ) for y =126 μm (black), 124 μm (red), 122 μm (blue) and 120 μm (green). The solid lines are calculations with amplitude fixed by a fit at y =126 μm. d , Spatially resolved Kerr rotation θ K ( y ) near the edge for t =5 ns (black), 7 ns (red), 9 ns (blue) and 11 ns (green). The inset shows the decay of j y z ∝ ∂ s z ( y )/ ∂ y extracted at y =126 μm after the voltage pulse is complete ( t p =6 ns).

We characterize the magnetic-field line shapes by their inverse half-width B 1/2 −1 , which increases with t before quickly saturating ( Fig. 3 a). We can understand this evolution of B 1/2 in a simple physical picture 21 . Soon after the pulse turns on at t =0, spins are all recently generated at the sample edge and have had little time for spin precession about B . For later times, spins have a larger spread in generation times (up to t p ) and have correspondingly more time for precession; hence, there is more depolarization for a given B and the Hanle curve narrows ( Fig. 2 g). For t ≫ τ , the average precession time is governed by τ rather than t and the Hanle width becomes constant as in a steady-state measurement. The coherence times τ 1/2 calculated from the saturation of B 1/2 near t ∼ t p agree with decay times τ * and are consequently also longer than the accumulation times τ acc ( Fig. 3 b). Diffusion analysis of the SHE accumulation is necessary to reconcile the observed differences between the timescales τ acc , τ * and τ 1/2 .

The weak spin–orbit coupling of the GaAs system enables spin conserving hard-wall boundary conditions normal to the edges at y =± w /2 ( j y i =− D ∂ y s i − σ SH E δ i z =0). Boundary conditions accounting for spin–orbit effects at the sample edge (such as in the intrinsic SHE) would require modification for effects on the scale of the mean free path 21 , but these effects can be ignored in the extrinsic case studied here.

We first develop an intuitive picture of the time-dependent spin Hall processes using the exact solution to equation (1) under the simplest conditions. For B =0, the components of s are uncoupled and equation (1) can be solved using a Green’s function to obtain an infinite series solution for s z ( y , t ). The diffusion equation for s z can be written as:

F ( y , t ) is a source function that contains all SHE terms. The homogeneous F =0 Green’s function for equation (2) is:

where m is an integer, Θ ( t ) is the Heaviside step function, k m =(2  m +1)π/ w and λ m =1/ τ + k m 2 L s 2 / τ . For time-independent E , integrating equation (3) yields a series representation of the one-dimensional steady-state solution to the spin Hall diffusion equation 7 . To obtain a time-dependent solution, we assume an ideal square electric-field pulse of width t p and amplitude E 0 , E = E 0 [ Θ ( y + w /2)− Θ ( y − w /2)][ Θ ( t )− Θ ( t − t p )]. The corresponding source function is F ( y , t )= σ SH E 0 [ δ ( y + w /2)− δ ( y − w /2)][ Θ ( t )− Θ ( t − t p )] and the solution is found by integrating:

The three regimes of the term-by-term time-dependence function T m ( t ) in equation (4) can each be observed in Fig. 2 b. The lines in Fig. 3 a represent fits of θ K ( t ) to equation (4) keeping the first 200 terms in equation (4) and convoluting the solution with the Gaussian profile of the laser spot. As L s =3.9±0.2 μm was found independently from steady-state spatial measurements, the only fit parameters are τ and an overall amplitude scaling. The best fit values for the parameter τ are plotted in Fig. 3 b and are significantly longer than the experimentally measured timescales τ acc , τ * and  τ 1/2 .

Figure 3 c shows θ K ( t ) and calculations from equation (4) convoluted with the laser profile for y =126, 124, 122 and 120 μm for V 0 =2 V using τ =4.2 ns obtained from the earlier fits. We fix the amplitude of the calculation from a fit to y =126 μm, and the remaining curves have no free parameters. For y away from the edge, θ K ( t ) does not grow exponentially in t and we cannot define the time constant τ acc as in Fig. 3 a. Comparison of calculations from equation (4) with and without the spot size averaging reveals that the apparent asymmetry between the growth and decay times τ acc and τ * near the edge is primarily due to spatial averaging of these diffusion profiles over the Gaussian laser spot. The difference between τ * and τ is real, however; dynamically measured spin polarization near the sample edge evolves with a faster time constant than the underlying spin-coherence time.

We can understand the fast evolution of spin polarization from the interplay of diffusion and spin decoherence. As polarization gradients cause spins to diffuse away from the sample boundary, spin depolarization must occur faster than decoherence of the electrically generated spins. These dynamics are captured in the diffusion analysis of equation (4) by the fast decay rate λ m of terms with large m in equation (4). Higher m terms are primarily responsible for the discrepancy between the best fit value for τ and the faster timescales τ acc and τ * observed for spin accumulation and decay in Fig. 3 b, but they contribute significantly only to s z near y = w /2 where all terms are in phase. In this boundary region, timescales should differ most from the coherence time τ .

We numerically calculate ∂ s z / ∂ y at y =126 μm from spatial scans in Fig. 3 d. The spatial derivative of s z ( y ) is proportional to the diffusive spin current and is non-zero at the sample edge in the presence of a compensating spin Hall current for 0< t < t p . The spin Hall current itself tracks the pulse V ( t ) as fast as the momentum scattering timescales, but diffusive spin accumulation responds slower. After the spin Hall current disappears at t = t p , the measured ∂ s z / ∂ y relaxes with time constant τ j =1±0.1 ns to satisfy the diffusive j y z =0 boundary condition ( Fig. 3 d, inset). Averaging of our diffusion solution over the finite laser spot size, we calculate the value τ j =1.15 ns for the evolution time of ∂ s z / ∂ y at y =126 μm, consistent with our observed value.

a , Kerr rotation θ K ( B , t ) at y =126 μm for t p =6 ns. b , Calculations of s z ( B , t ) from equation (1) with τ =4.2 and L s =3.9 μm at y =126 μm.

The agreement between the experiments and calculations demonstrates that a single homogeneous decoherence time τ captures the various timescales observed in time-resolved measurements of the accumulation, decay and diffusive dynamics of boundary spin polarization due to the extrinsic spin Hall effect. Although diffusive timescales are set by the spin-coherence time, evolution near sample boundaries can be limited by the faster response of the spin current. This spatial dependence of timescales could prove helpful for using electrically generated spin polarization in high-frequency semiconductor devices.

Channels of width w and length l are processed from a 2- μm-thick silicon-doped GaAs epilayer on 200 nm of undoped Al 0.4 Ga 0.6 As grown on a semi-insulating (001) GaAs substrate by molecular beam epitaxy ( Fig. 1 b). The n-GaAs has doping density n =1×10 17  cm −3 and mobility μ =3,800 cm 2  V −1  s −1 at T =30 K. The sample is mounted in a helium flow cryostat so that the channel ( x direction) is perpendicular to the externally applied in-plane magnetic field B ( y direction). All measurements are at temperature T =30 K. A voltage V ( t ) applied across annealed Ni/Ge/Au/Ni/Au ohmic contacts creates an in-plane electric field E ( t )= V ( t )/ l along x . We desire an impedance of ∼ 50 Ω to deliver the maximum broadband electrical power to the device; choosing w =256 μm and l =130 μm yields a device with d.c. resistance R =48 Ω at  T =30 K.

Time resolution of SHE accumulation is achieved by electrically pumped Kerr rotation microscopy using a mode-locked Ti:sapphire laser tuned to 1.51 eV that emits a 76 MHz train of ∼ 150 fs pulses. The pulse repetition rate is reduced to 38 MHz by pulse picking with an electro-optic modulator. Each laser pulse is divided into a trigger and a linearly polarized probe pulse. Spin polarization is generated at the sample edges by the SHE due to the current from a square electrical pulse of width t p , amplitude V 0 and 0.8 ns rise time applied to the sample from a pulse pattern generator triggered by the optical pump pulse. The linearly polarized probe beam is focused through a microscope objective to a 1 μm spot on the surface of the sample that can be scanned with submicrometre resolution. A balanced photodiode bridge measures the Kerr rotation of the linear polarization axis θ K of the reflected beam which is proportional to the spin polarization along the z axis s z . The leading edge of the electric pulse profile V ( t ) arrives at an electronically programmable delay time t before the arrival of the optical pulse. All reported measurements are taken at the centre of the length of the channel ( x =0).

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Acknowledgements

We thank NSF and ONR for financial support. N.P.S. acknowledges the support of the Fannie and John Hertz Foundation and S.M. acknowledges support through the NDSEG Fellowship Program.

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N. P. Stern, D. W. Steuerman, S. Mack, A. C. Gossard & D. D. Awschalom

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Stern, N., Steuerman, D., Mack, S. et al. Time-resolved dynamics of the spin Hall effect. Nature Phys 4 , 843–846 (2008). https://doi.org/10.1038/nphys1076

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Received : 20 May 2008

Accepted : 15 August 2008

Published : 14 September 2008

Issue Date : November 2008

DOI : https://doi.org/10.1038/nphys1076

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