**3. PFG HR-MAS NMR to measure diffusion in materials characterization**

Modern HR-MAS probes include a gradient coil that can produce a magnetic gradient along the long axis of the MAS rotor which is set at the magic angle (θ = 54.7o). Standard MAS probes have also been combined with micro-imaging gradient systems in which gradient coils wrapped around the stator were not employed, but instead rely on imaging gradients that were external to the probe. [82] This is not the common configuration, and has been replaced by significant development efforts from the instrumental vendors involving the integration of gradients directly into the HR-MAS probes. With the gradient coil along the magic angle pulsed field gradient (PFG) experiments can be performed under MAS conditions. Fortuitously, enhanced *T2* times are generally observed under MAS allowing PFG experiments with longer diffusion times to be implemented than would have been accessible with static conditions.

During the PFG diffusion experiments, the application of a gradient "tags" a spin with a phase that is related to its spatial position. Figure 8A provides a pictorial representation of the dephasing of spins around the magic angle caused by the magic angle gradient. If the position of the spin does not change during the diffusion period (Δ), this dephasing is refocused and the original signal intensity (*S0*) is recovered. If on the other hand the spin changes spatial position (diffuses) during Δ, the dephasing for that spin is not refocused, and the signal intensity decreases. The loss in signal intensity with increasing gradient strength is related to the self-diffusion rate with the classic Stejskal-Tanner equation:

$$\frac{S}{S\_0} = e^{-\nu^2 g^2 \delta^2 \left(\Delta - \frac{\delta}{x}\right) D} \tag{1}$$

where *S* is the experimental amplitude of the signal, is the gyro magnetic ratio, *g* is the gradient strength, is the gradient pulse length, Δ is the diffusion time, and the *D* is the diffusion constant.[83] By fitting this decay the diffusion constant can be determined. Figure 8B shows an example of the signal intensity loss observed during diffusion experiments for the different two water environments present in swollen AEM with increasing gradient strength.

PFG diffusion HR-MAS NMR experiments can also be used to obtain diffusion-filtered NMR spectra through the separation of different motional regimes present in complex mixtures. This filtering is accomplished by selecting Δ times where the signal intensity (Eqn. 1) for the fast diffusing components has been highly attenuated, while the slow diffusing components have significant signal intensity remaining. Practical aspects of diffusion measurements using HR-MAS have been previously discussed by Viel *et al.* [84] A significant finding from this study was that sample volume played a role in the reliability of diffusion rates measured. It was shown that small volumes (~12μL) exhibit reproducible diffusion rates, while larger volumes (~50μL, a full 4 mm rotor) produced inconsistent or unreliable data.

294 Advanced Aspects of Spectroscopy

humic materials.[81]

accessible with static conditions.

strength.

complexes.[76, 80] These types of studies have been extended to three-dimensional (3D) HMQC-TOCSY to further increase the resolution of the highly overlapping spectra from

**3. PFG HR-MAS NMR to measure diffusion in materials characterization** 

Modern HR-MAS probes include a gradient coil that can produce a magnetic gradient along the long axis of the MAS rotor which is set at the magic angle (θ = 54.7o). Standard MAS probes have also been combined with micro-imaging gradient systems in which gradient coils wrapped around the stator were not employed, but instead rely on imaging gradients that were external to the probe. [82] This is not the common configuration, and has been replaced by significant development efforts from the instrumental vendors involving the integration of gradients directly into the HR-MAS probes. With the gradient coil along the magic angle pulsed field gradient (PFG) experiments can be performed under MAS conditions. Fortuitously, enhanced *T2* times are generally observed under MAS allowing PFG experiments with longer diffusion times to be implemented than would have been

During the PFG diffusion experiments, the application of a gradient "tags" a spin with a phase that is related to its spatial position. Figure 8A provides a pictorial representation of the dephasing of spins around the magic angle caused by the magic angle gradient. If the position of the spin does not change during the diffusion period (Δ), this dephasing is refocused and the original signal intensity (*S0*) is recovered. If on the other hand the spin changes spatial position (diffuses) during Δ, the dephasing for that spin is not refocused, and the signal intensity decreases. The loss in signal intensity with increasing gradient

strength is related to the self-diffusion rate with the classic Stejskal-Tanner equation:

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where *S* is the experimental amplitude of the signal, is the gyro magnetic ratio, *g* is the gradient strength, is the gradient pulse length, Δ is the diffusion time, and the *D* is the diffusion constant.[83] By fitting this decay the diffusion constant can be determined. Figure 8B shows an example of the signal intensity loss observed during diffusion experiments for the different two water environments present in swollen AEM with increasing gradient

PFG diffusion HR-MAS NMR experiments can also be used to obtain diffusion-filtered NMR spectra through the separation of different motional regimes present in complex mixtures. This filtering is accomplished by selecting Δ times where the signal intensity (Eqn. 1) for the fast diffusing components has been highly attenuated, while the slow diffusing components have significant signal intensity remaining. Practical aspects of diffusion measurements using HR-MAS have been previously discussed by Viel *et al.* [84] A significant finding from this study was that sample volume played a role in the reliability of diffusion rates measured. It was shown that small volumes (~12μL) exhibit reproducible

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**Figure 8.** A) Pictorial representation of the gradient produced along the magic angle of the rotor. B) The decay of two different water signals found in a 1N methanol solution of an AEM membrane with increasing gradient strength. Gradient strength values (G/cm) are shown above the stack plot.

Three commonly used PFG diffusion pulse sequences are shown in Figure 9. The basic spinecho diffusion sequences is depicted in Figure 9A, but is limited by loss of signal intensity due to spin-spin *T2* relaxation during the diffusion period Δ. Two variations of the stimulated echo (STE) sequence are shown in Figure 9B and 9C, and in this case spin-lattice *T1* relaxation is occurring during the diffusion Δ period. For most materials *T1* values are longer than *T1* making these STE sequences the preferred choice for material analysis. It should be noted that all of the gradient pulses in these sequences are trapezoidal shaped to compensate for the inability of instrumentation to generate perfect rectangular gradients. Shaped pulses, like sine or trapezoidal shapes, are used to produce experimentally reproducible gradient pulses. The PFG stimulated echo with dipolar gradients and spoil gradient, depicted in Figure 9B, is beneficial for the use with many HR-MAS samples which exhibit differences in magnetic susceptibility across the sample.[85, 86] The PFG stimulated echo in Figure 9C has an additional delay to the PFG stimulated echo in Figure 9B that is utilized to address eddy currents within the sample.[87]

**Figure 9.** Diffusion pulse sequences. Pulse Field Gradient (PFG) A) Spin-Echo, B) PFG Stimulated Echo with dipolar gradients and spoil gradient based on Cotts *el al.* 13-interval sequence[85], and C) PFG Stimulated Echo with dipolar gradients and spoil gradient with an additional eddy current delay. GMAS indicates that the gradient is applied along the magic angle.
