Análisis de resultados

Solid-state NMR is not only a powerful local structural technique, but also a sensitive probe of local ion dynamics, because the various timescales relevant to NMR spectroscopy span an

extremely wide temporal range (Figure 2.7).

directly with T1 and T2 relaxation processes, although often this is only relevant for

macroscopic processes such as flow in solution slightly faster spectral timescale (

superionic conductors, can lead to characteristic disturbances in the lineshapes of NMR resonances. Finally, the Larmor timescale is roughly the inverse of the Larmor frequency and thus probes very fast dynamics

explicitly distinguished from the so motional dependence of both motional correlation times 𝜏

magnetic fields can occur that effectively promote spin relaxation. temperature relaxometry data

MHz.168

Figure 2.7: Comparison of the various

dynamic processes on both the macroscopic and molecular levels. Figure reproduced from Levitt.

2.1.5.1

Lineshape changes in multi

Chemical exchange occurring in nuclear spin systems on the order of the spectral timescale, i.e., the FID acquisition time

lineshapes of the spectral resonances. Consider the simple case of two equally populated sites range (Figure 2.7). Very slow motional dynamics may interact relaxation processes, although often this is only relevant for macroscopic processes such as flow in solution-state NMR. Motional processes on the slightly faster spectral timescale (10-6 s), which is the same rate as "slow" ionic motion in lead to characteristic disturbances in the lineshapes of NMR Finally, the Larmor timescale is roughly the inverse of the Larmor frequency and thus probes very fast dynamics (10-9 s). While the Larmor regime (~10-9

explicitly distinguished from the so-called relaxation timescale (10-3 to 10 motional dependence of both T1 and T2 relaxation times frequently becomes

are on the order of the Larmor frequency 𝜔 occur that effectively promote spin relaxation.

temperature relaxometry data often reveal very fast ionic motion on the order of 10

various timescales probed by NMR spectroscopy and how processes on both the macroscopic and molecular levels. Figure reproduced from Levitt.

Lineshape changes in multi-site exchange

exchange occurring in nuclear spin systems on the order of the spectral timescale, (μs to ms), can lead to characteristic changes in the observed lineshapes of the spectral resonances. Consider the simple case of two equally populated sites Very slow motional dynamics may interact relaxation processes, although often this is only relevant for Motional processes on the , which is the same rate as "slow" ionic motion in lead to characteristic disturbances in the lineshapes of NMR Finally, the Larmor timescale is roughly the inverse of the Larmor frequency and

9 _{s) in Figure 2.7 is}

to 103 s), in fact the frequently becomes important when , because transient Fits to variable- motion on the order of 10–100

timescales probed by NMR spectroscopy and how they relate to typical processes on both the macroscopic and molecular levels. Figure reproduced from Levitt.169

exchange occurring in nuclear spin systems on the order of the spectral timescale, , can lead to characteristic changes in the observed lineshapes of the spectral resonances. Consider the simple case of two equally populated sites

A and B that undergo exchange characterized by by 𝑘 , and assume that the

separation Δ𝜐. In the case that

resonances can be clearly distinguished in the NMR spectrum (

frequency components arising from the two sites persist long enough to contribute to the FID (Figure 2.8b and c, top).

Figure 2.8: a) Simulations of spectra influenced by a two separation between the two sites is

for 𝑘 = 200, 4440, and 15000 Hz, and using the simulation code of Forse et al.

As the exchange rate increases, the two resonances begin to broaden and eventually undergo coalescence into a single broad feature at

phenomenon of coalescence can be rationaliz

single spin. While uninterrupted precession for a time period of at least

required to distinguish the two frequency components, this is rendered impossible by the random hops occurring on this t

exchange characterized by random "hops" with an exchange rate given ssume that the resonances arising from sites A and B

In the case that 𝑘 ≪ Δ𝜐, which is called the "slow motion" regime, the two resonances can be clearly distinguished in the NMR spectrum (Figure 2.8a, top), bec

arising from the two sites persist long enough to contribute b and c, top).

a) Simulations of spectra influenced by a two-site exchange process of rate 𝑘

separation between the two sites is Δ𝜐 = 2000 Hz. b) Depiction of individual contributions of spins to the FID, = 200, 4440, and 15000 Hz, and c) the summed FID of all the simulated spins. Figure

et al.170,171

As the exchange rate increases, the two resonances begin to broaden and eventually undergo coalescence into a single broad feature at 𝑘 =

√ Δ𝜐 ≈ 2.22Δ𝜐 (Figure 2.8

phenomenon of coalescence can be rationalized by considering precession at the level of a single spin. While uninterrupted precession for a time period of at least

required to distinguish the two frequency components, this is rendered impossible by the random hops occurring on this timescale (Figure 2.8b, middle). Moreover, the individual random "hops" with an exchange rate given have a frequency which is called the "slow motion" regime, the two a, top), because the arising from the two sites persist long enough to contribute separately

, where the frequency tions of spins to the FID, Figure based on Keeler137

As the exchange rate increases, the two resonances begin to broaden and eventually undergo Figure 2.8a, middle). The ed by considering precession at the level of a single spin. While uninterrupted precession for a time period of at least 1/Δ𝜐 would be required to distinguish the two frequency components, this is rendered impossible by the b, middle). Moreover, the individual

spins also rapidly dephase from each other, leading to a FID with a short persistence length (Figure 2.8c, middle) corresponding to a maximally broadened resonance.

For fast exchange rates 𝑘 ≫ Δ𝜐, in the limiting case of the "fast motion" regime, a single resonance is observed which further narrows with increasing exchange rate (Figure 2.8a, bottom). In this scenario, the individual spins hop many times within a single precession period and the discrete frequency components are averaged (Figure 2.8b, bottom); as the exchange rate increases, less dephasing occurs, the FID persists for a longer time (Figure 2.8c, bottom), and the resonance narrows further.

Importantly, in the case of two unequally populated sites, in the fast motion regime, the coalesced resonance appears at the population-weighted average of the shifts of the two resonances. Otherwise, the observed spectral behavior, including the characteristic exchange rate at coalescence, is analogous to the situation of equally populated sites.172 Spectra resulting from symmetric as well as asymmetric exchange of several sites can be simulated with software such as the McMaster Exchange Code (MEXICO) developed by Bain.173

Figure 2.9: Simulations of motional narrowing with a uniform distribution of frequencies spanning a range of 6 kHz (i.e., between -3 kHz and 3 kHz). Figure based on Keeler137 using the simulation code of Forse et al.170,171

Many real spin systems comprise a broad distribution of frequency components arising from effects such as local disorder, so that in contrast to the two-site scenario previously described, the individual resonances cannot be distinguished. In this case, exchange between the resonances can also lead to motional narrowing (Figure 2.9). However, motional narrowing is only experimentally observed when the exchange rate is on the order of the inherent

frequency distribution among the exchanging sites; in the slow- and fast-exchange limits, static linewidth values are observed.

Indris et al. have extracted motional correlation times for dynamic processes that give rise to motional narrowing, using the implicit expression:

(Δ𝜐) = (Δ𝜐 ) + (Δ𝜐 ) ⋅2

𝜋⋅ tan (𝛼 ⋅ Δ𝜐 ⋅ 𝜏 ) (2.19)

where Δ𝜐 is the experimental linewidth at a given temperature, Δ𝜐 and Δ𝜐 are the limiting values of the linewidth in the high-temperature and low-temperature plateaus, respectively, and 𝛼 is a lineshape-dependent constant on the order of unity.174,175 Dunstan et al. later reported an approximation to Equation (2.19) that was combined with the Arrhenius law to give:

𝑑 ln(Δ𝜐) 𝑑(1/𝑇) = −

𝐸

𝑅 (2.20)

where T is the experimental temperature, 𝐸 is the activation energy, and R is the molar gas constant.132 In this analysis, only the linear region of ln(Δ𝜐) vs. 1/𝑇 is used to extract the activation energy, i.e., the values of Δ𝜐 in the limiting high- and low-temperature regions are not included. Equation (2.20) is used in the present work to extract activation energies for ion dynamics causing motional narrowing; this motion can be interpreted physically as corresponding to site-specific dynamical processes (e.g., rattling or polyanion rotation) that do not lead to large changes in the local chemical environment.

2.1.5.2

Motional effects on nuclear relaxation rates

2.1.5.2.1

Low-frequency ion dynamics and T

relaxation

In the case where high-frequency processes such as electron relaxation do not contribute to T2

relaxation, one can assume that low-frequency motion (i.e., on the order of the spectral timescale between μs and ms) is the predominant T2 relaxation mechanism.176–178 This results

in a T2 which is much shorter than the T1, as high-frequency processes tend to contribute to

conditions, one first assumes that the motion is characterized by an exchange rate 𝑘 leading to random hops between sites, where the resonances arising from these sites have a maximal frequency separation Δ𝜐. (This situation may manifest either as a two-site exchange where the Δ𝜐 is well-defined, or as a single spectral feature comprising a distribution of resonances where the Δ𝜐 is roughly on the order of the linewidth.) Then, the motion has characteristic effects on the T2 relaxation time in the slow and fast motion regimes (i.e., for the cases

𝑘 ≪ Δ𝜐, and 𝑘 ≫ Δ𝜐, respectively).

In the slow motion regime, the T2 is inversely proportional to 𝑘 , that is, 𝑇 = 𝑐/𝑘 . As the

motional rate increases, the T2 relaxation time becomes shorter. The corresponding physical

picture is that the increasingly faster motion shortens the effective coherence lifetime of the magnetization, which lowers the value of T2. In the fast motion regime, the T2 is directly

proportional to 𝑘 , that is, 𝑇 = 𝑐′ ⋅ 𝑘 . At this point, the T2 relaxation time begins to

increase with the motional rate; the T2 relaxation becomes progressively less sensitive to this

motional process, which no longer interacts with the spin system on the timescale of the frequency separation, Δ𝜐. The boundary between the two regimes is given by the coalescence condition 𝑘 ≈ 2.22Δ𝜐, where the T2 value is minimal.176

Thus by measuring T2 as a function of temperature in this type of system, the values of 𝑘

can be extracted, and assuming Arrhenius behavior the activation energy can be measured. In the case where the T2 values cannot be measured directly because of experimental time

constraints, the influence of T2 relaxation on the spectral intensity can be used as a proxy for

the T2 value, which allows for determination of the activation energy of the relevant motional

process. Further details on the derivation of this method and its specific application to the present work are given in Appendix A.

2.1.5.2.2

Probing fast motion on the order of the Larmor frequency via

T

relaxation

In diamagnetic nuclear spin systems containing quadrupolar nuclei, T1 relaxation is

predominantly driven by local variations in quadrupolar coupling. Because the quadrupolar interaction is typically (one of) the largest nuclear spin interaction(s), motion-induced

fluctuations in the EFG at the rate of the Larmor frequency give rise to transient fields that are very effective at promoting relaxation between spin states of quadrupolar nuclei. In this situation, the following equation holds:

1 𝑇 = 2𝜋 25 ⋅ 𝐶 1 + 1 3𝜂 ⋅ 𝜏 1 + 𝜔 𝜏 + 4𝜏 1 + 4𝜔 𝜏 (2.21)

where 𝐶 and 𝜂 are the quadrupolar coupling constant and quadrupolar asymmetry parameter, respectively, 𝜔 is the Larmor frequency, and 𝜏 is the motional correlation time.174,175,179 Provided that the values of 𝐶 and 𝜂 are known, by measuring the T1 of a

given resonance as a function of temperature in a system with a dominant quadrupolar relaxation mechanism, values of 𝜏 can be extracted which are related to the local motion surrounding each site with a spectrally distinct resonance. The magnitude of the motional rate 𝜏 is typically on the order of the Larmor frequency, e.g., tens of MHz. Assuming an Arrhenius relationship between 𝜏 and the temperature T, the activation energy 𝐸 can then be determined. These rapid motional processes have previously been ascribed to softening of lattice vibrations or motion of defects in systems ranging from glasses to ionic conductors.168,180 In this thesis, the method is applied to characterize the motion of oxygen vacancies in SOFC electrolyte materials (Chapter 5).