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Heterogeneous Rotational and Translational Dynamics in Glasses and Other Disordered Materials Studied by NMR

  • Roland BöhmerEmail author
  • Michael Storek
  • Michael Vogel
Living reference work entry

Abstract

Disordered materials such as glass formers and amorphous solid electrolytes are characterized by ubiquitous nonexponential molecular and/or ionic dynamics. This chapter focuses mostly on their investigation via two-time stimulated-echo-based correlation functions. Underlying concepts are briefly reviewed and recent experimental examples from 2H, 7Li, 17O, 23Na, and 31P NMR are presented which encompass nonselective and selective central-transition excitation and a variety of relevant spin quantum numbers. Several recent methodological developments render also four-time stimulated-echo techniques applicable to a large array of probe nuclei. The higher-order correlation functions thus accessible enable quantitative insights into the origins of the nonexponentiality of atomic, ionic, or molecular motions. Provided that heterogeneous dynamics prevails, these experiments elucidate the temporal evolution of fast and slow subensembles, in particular by monitoring exchange processes among them. Together with corresponding frequency-domain techniques that are also touched upon, NMR methods to unravel the nature of nonexponentiality in a host of materials have become available for almost any probe nuclei.

Keywords

Dynamic heterogeneity Glasses Hexagonal ice Microcoils Multicomponent systems Multidimensional NMR Nonexponential dynamics Oxygen NMR Quadrupolar nuclei Second-order quadrupolar interactions Stimulated echoes 

Introduction

In disordered materials, transport and reorientation governed by a unique time scale at a given temperature and pressure is the exception rather than the rule: In condensed matter, the molecular, atomic, or ionic motions are typically characterized by nonexponential correlation functions [1]. This statement concerns the dynamics in all glasses , in most viscous liquids, and in many crystals even if they exhibit a well-defined center-of-mass lattice [2]. To rationalize the ubiquitously observed dispersive characteristics, fundamentally different scenarios have been suggested [3, 4, 5], see Fig. 1.
Fig. 1

Schematic illustration of (a) the homogeneous and (b) the heterogeneous scenario. The macroscopic, ensemble-averaged correlation function ϕ displays the same nonexponential, i.e., dispersive, shape in both cases. However, for scenario (a), identical microscopic correlation functions are averaged, while in (b) the overall response involves processes evolving on distinguishable time scales. This simplified representation ignores any dynamic exchange processes, i.e., the possibility that fast relaxations can become slow and vice versa (Adapted from [4, 5] and reproduced courtesy of R. Richert)

An appropriate modeling of the underlying motional mechanisms requires a detailed understanding of the origin of the nonexponential dynamics which depends on the local environment of the relaxing units, on their mutual interactions, and on the coupling among various degrees of freedom. A number of different experimental methods and computer simulations were devised [4, 6, 7, 8, 9] to address these questions in terms of dynamic heterogeneity . Among them is a class of approaches that are sensitive to motional trajectories on the single-molecule scale. Another family of experiments is based on the selection or manipulation of dynamically distinct subensembles with various electrical, mechanical, optical, and magnetic resonance methods that are available in the experimenter’s toolbox. This chapter summarizes how, starting from now classic work [10, 11], particularly the NMR methods have turned out as enormously versatile.

In the previous edition of this series [12], the status of relevant multidimensional NMR approaches was reviewed mostly from the perspective of stimulated-echo-based pulse sequences. At that time, just a few nuclei, basically the 2H, 13C, and 109Ag nuclear probes, were exploited for the detection of dynamical heterogeneities. In other words, the approach was restricted to spin-1/2 and spin-1 isotopes which are subjected to anisotropic chemical shift or first-order quadrupolar interactions. The materials then under scrutiny were mostly polymers and liquids composed of (small) organic molecules.

However, about two-thirds of the NMR-active isotopes are half-integer quadrupolar nuclei with typical quadrupolar couplings ranging from a few kHz to many MHz. Owing to recent methodological developments, such probes have now become accessible no matter whether their NMR spectrum can be excited nonselectively (usually this requires spectral widths not exceeding a few 100 kHz) or whether using conventional methods the central transition can be irradiated only selectively because the satellite spectrum is close to 1 MHz wide or even broader.

To gain an overview also including spin-1/2 and spin-1 nuclei, the next section entitled “Probing Nonexponential Dynamics Using Stimulated Echoes,” outlines the theoretical framework and some practical details regarding measurements of two-time correlation functions which allow one to quantify the degree of the nonexponentiality of motional processes. Then, the Section “NMR Measurements of Slow Atomic, Molecular, and Ionic Motions in Glasses and Crystals” presents recent examples of how organic and inorganic materials featuring spin-1/2 and spin-1 nuclei, as well as half-integer quadrupolar probes such as 7Li, 23Na, and 17O, can be explored using stimulated-echo techniques. Finally, the Sections “Homogeneous Versus Heterogeneous Transport and Reorientation: Concepts” and “Homogeneous Versus Heterogeneous Transport and Reorientation: Time-Domain/Frequency Domain Techniques” survey the state of the art regarding multidimensional NMR studies of dynamic heterogeneity. Several additional NMR experiments useful in the present context are mentioned as well. This contribution focuses mostly on experiments on stationary (nonrotating) samples.

Probing Nonexponential Dynamics Using Stimulated Echoes

In NMR spectroscopy, motiona l processes can be tracked by the time-dependent precession frequencies ω(t) of suitable spin ensembles, so-called isochromats. Depending on the relevant nuclear spin interactions, ω can encode the molecular orientation or other aspects regarding the local environment of the considered spin probes. When monitoring motional processes, the interaction tensors which govern ω can change actively and in a single-particle fashion, for instance if the probe nucleus jumps from site to site or if the molecule containing the nucleus changes its orientation. Alternatively, its precession frequency can be altered passively if particles (in general several of them) in its vicinity change its tensorial properties. In many circumstances, these two limiting scenarios are not strictly separable, e.g., if rotational-translational coupling is relevant so that molecular orientation and local environment change simultaneously.

Three-pulse stimulated-echo sequences, cf. the dashed boxes in Fig. 2, provide powerful basic building blocks (i) to acquire two-time correlation functions suited to access ultraslow motions, defined to take place on typical time scales of milliseconds and longer, and (ii) to record four-time correlation functions suitable to address the issue illustrated in Fig. 1 [13, 14]. Stimulated-echo sequences are useful for application in the frequency as well as in the time domain; this chapter mainly focuses on the latter.
Fig. 2

Pulse sequence useful to measure four-time stimulated echoes. Four distinct evolution intervals t p with precession frequencies ω 1 to ω 4 are separated by three mixing times t m1, t m2, and t m3. The first three pulses of this sequence generate a two-time stimulated echo, F 2(t m1 = t m ), and can be regarded as a low-pass filter to select a slow subensemble. The phase information regarding this subensemble is then stored by the fourth pulse. The sequence represented by the last three pulses eventually acts as an analyzer for the properties of the selected subensemble usually either in the limit t m2 → 0 in terms of G 4(t m3) or for t m1 = t m3 in terms of F 4(t m2)

First, nonselectively excitable probe nuclei with I = 1/2, 1, and 3/2 are treated and afterward selective excitation of the central (\( -\frac{1}{2}\leftrightarrow +\frac{1}{2} \)) transition of nuclei with I > 3/2 will be a concern. Assuming that the spin evolution is dominated by first-order quadrupolar interactions or anisotropic chemical shifts, properly phase-cycled sequences comprising three pulses (with flip angles φ i for i = 1,2,3), and in many instances combined with proton decoupling [15, 16, 17], can produce signals of the form [18, 19, 20, 21]:
$$ {F}_{2,I}^{\mathrm{cos}}\left({t}_1,{t}_{\mathrm{m}},{t}_2\right)={\overline{s}}_I^{\mathrm{cos}}\left\langle \cos \left({\omega}_I(0){t}_1\right)\cos \left({\omega}_I\left({t}_{\mathrm{m}}\right){t}_2\right)\right\rangle $$
(1a)
and
$$ {F}_{2,I}^{\mathrm{sin}}\left({t}_1,{t}_{\mathrm{m}},{t}_2\right)={\overline{s}}_I^{\mathrm{sin}}\left\langle \sin \left({\omega}_I(0){t}_1\right)\sin \left({\omega}_I\left({t}_{\mathrm{m}}\right){t}_2\right)\right\rangle . $$
(1b)
These equations involve the overall prefactors \( {\overline{s}}_I^{\alpha }= \) \( {s}_I^{\alpha }{p}_I^{\alpha}\left({\varphi}_1,{\varphi}_2,{\varphi}_3\right) \) with α = {cos,sin}. Here, the coefficients \( {s}_I^{\alpha } \) are numerical factors and the coefficients \( {p}_I^{\alpha } \) describe the signal reduction caused by nonideal flip angles. Explicit expressions for these coefficients are compiled in Table 1.
Table 1

Coefficients entering the stimulated-echo functions given in Eq. (1). The flip-angle-dependent amplitudes \( {p}_I^{\sin, \cos}\left({\varphi}_1,{\varphi}_2,{\varphi}_3\right) \) are abbreviated as p(1,n,n) = sin(φ 1) sin( 2) sin( 3). The amplitude prefactors \( {s}_I^{\sin, \cos } \) and the carrier states (in terms of the irreducible spherical tensor operator T 0 of rank and order 0), as they are relevant for nonselective cos–cos and sin–sin experiments, are also summarized

Spin I

1/2

1

3/2

\( {p}_I^{\mathrm{cos}} \)

p(1,1,1)

p(1,1,1)

Eqs. 6 and 7 in [22]

\( {s}_I^{\mathrm{cos}} \)

1

1

\( \sqrt{2}/5\approx 0.283 \)

Carrier

T 10

T 10

T 10 & T 30

\( {p}_I^{\mathrm{sin}} \)

p(1,1,1)

p(1,2,2)

p(1,2,2)

\( {s}_I^{\mathrm{sin}} \)

1

3/4

9/20

Carrier

T 10

T 20

T 20

Note that for I = 3/2, the cos-cos experiment constitutes a special case, because in addition to the expression in Eq. (1a), an ω-independent pedestal occurs. Furthermore, the sequence works best if the third flip angle is adjusted to a specific value, e.g., to φ 3=\( \arccos \sqrt{5/9}\approx {41.8}^{{}^{\circ}} \) [22]. Additionally, setting φ 1 = 90° and φ 2 = 45°, the amplitude factor is \( \sqrt{2}/5 \) (see Table 1) and the pedestal is two-thirds of this value.

In order to circumvent receiver overload arising after the last pulse, full transverse magnetization refocusing can be achieved by augmenting the sequence by a fourth pulse that generates a Hahn echo for I = 1/2 or a solid echo for I = 1 [13]. Experiments with I = 3/2 nuclei require more involved procedures [22]. Information about phase cycles is available [23, 24, 25, 26].

To implement the t p → 0 limit (required, e.g., in two-dimensional exchange spectroscopy), a zeroth pulse can be added for I = 1/2 and 1 [23]. However, the t p → 0 limit is interesting in other contexts as well, e.g., when considering nonselectively excitable spins with I ≥ 3/2. The 133Cs nucleus, with I = 7/2, represents probably the most relevant example. As one may work out, e.g., from [27], for t p → 0 and various I ≥ 1 and with the abbreviation p(1,n,n) = sin(φ 1)sin( 2)sin( 3) for \( {p}_I^{\sin, \cos}\left({\varphi}_1,{\varphi}_2,{\varphi}_3\right) \), the stimulated-echo signals can be written as:
$$ {F}_{2,I}^{\mathrm{sin}}\left({t}_p\to 0,{t}_{\mathrm{m}},{t}_p\to 0\right)={s}_I0.5em {t}_p^20.45em p\left(1,2,2\right)0.24em \left\langle {\omega}_I(0){\omega}_I\left({t}_m\right)\right\rangle $$
(2)
with \( {s}_{I,\mathrm{halfinteger}}=\frac{3}{20}\left[I\left(I+1\right)-\frac{3}{4}\right] \). This coefficient gives 9/20, 6/5, 9/4 [2], and 18/5 for I = 3/2, 5/2, 7/2, and 9/2, respectively. Incidentally, Eq. (2) is also applicable for integer spins I ≥ 1, but then \( {s}_{I,\mathrm{integer}}=\frac{3}{5}\left[I\left(I+1\right)-\frac{3}{4}\right] \). The special situation with the nonselectively excited I > 3/2 isotopes is that the relevant density matrices in general involve sums of different trigonometric functions of precession frequencies. These sums become irrelevant only in the t p → 0 limit and the simple form of Eq. (2) results.

However, for most of the half-integer species with I > 1, the quadrupolar coupling constants are so large that only a selective irradiation of the central transition can be achieved unless special measures are taken. For example, one can exploit microcoils to excite spectra well in excess of 1 MHz [28].

Stimulated-echo and related sequences can also be applied for half-integer quadrupolar spin systems subjected to dominant second-order quadrupolar interactions [29]. Also for this case, two-time correlation functions of the form given by Eq. (1) are relevant. For purely selective pulses exciting the central transition of half-integer I ≥ 3/2 spins, one finds that the amplitude factors are s I,select = 3/[4I(I + 1)]. This expression applies to sin–sin and to cos–cos experiments and yields 1/5 for I = 3/2, 3/35 for I = 5/2, 1/21 for I = 7/2, and 1/33 for I = 9/2. The attenuation due to nonideal flip angles \( {p}_I^{\sin, \cos}\left({\varphi}_1,{\varphi}_2,{\varphi}_3\right) \) is here given by \( {\Pi}_{i=1}^3\sin \left[\left(I+\frac{1}{2}\right){\varphi}_i\right] \) [29]. Relevant phase cycles, essentially corresponding to those suitable for spin-1/2 systems, have been summarized [23, 26]. For selectively excited half-integer quadrupolar spins, the longitudinal carrier states which store the encoding during the mixing times in terms of irreducible spherical tensor operators are summarized in [Table 4 of 29].

Many of these experiments can be performed also under conditions of magic-angle spinning (MAS). For example, CODEX is a stimulated-echo MAS experiment which relies on a recoupling of the chemical shift anisotropy to measure rotational correlation functions of specific structural units [30]. Moreover, it should be mentioned that other NMR techniques are also well suited to characterize the degree of the nonexponentiality of molecular dynamics. For instance, field-cycling relaxometry allows one to map out spectral densities over wide ranges of frequency. While approaches using I = 1/2 nuclei have been successfully employed to investigate motional processes since decades [31, 32], the capabilities of 7Li field-cycling studies were demonstrated only recently [33, 34]. Furthermore, highly dispersive molecular dynamics manifest themselves also in so-called “two-phase” NMR line shapes. These line shapes can be described as a superposition of a rigid-lattice and a motionally narrowed spectrum resulting from the fast and the slow particles, respectively, of a broad distribution of correlation times [35].

NMR Measurements of Slow Atomic, Molecular, and Ionic Motions in Glasses and Crystals

Stimulated-echo studies proved particularly useful to investigate multicomponent systems . Then, the isotope selectivity can be exploited to characterize the molecular dynamics of specific constituents separately. This capability was demonstrated, e.g., for a mixture of tripropyl phosphate (TPP) and deuterated polystyrene (PS) [36, 37]. Recording 2H and 31P stimulated-echo decays it was found that the components of such mixtures exhibit mutually decoupled rotational motions which are faster and more nonexponential for TPP than they are for PS. For example, see Fig. 3. Likewise, in work on solid electrolytes featuring several mobile ion species, such approaches allow one to selectively investigate the hopping motion of either one of the species. This was shown, for instance, in 6Li and 109Ag stimulated-echo studies on Li x Ag1–x PO3 glasses [38].
Fig. 3

Reorientational correlation functions, F 2(t m ), measured by the stimulated-echo technique for a 50% mixture of tripropyl phosphate and deuterated polystyrene (all curves corrected for spin-lattice relaxation): top 2H NMR, bottom 31P NMR (the latter data are additionally corrected for spin diffusion) (Adapted from [36] and reproduced courtesy of E. A. Rössler with permission from AIP Publishing LLC)

Concerning isotopes with relatively large quadrupolar couplings, the I = 5/2 probe 17O is a very useful and important nucleus; for a review, see [39]. Figure 4 presents 17O data for the example of hexagonal ice . Here, the quadrupolar coupling constant C Q of the nuclear probe is 6.66 MHz [40] so that in powder samples only the central transition is selectively excited. In ice the quadrupolar coupling constant of 2H is about 40 times smaller (0.215 MHz), thus allowing for nonselective excitation. Figure 4a shows measurements employing the 17O and the 2H isotopes [29]. These experiments are sensitive to the reorientational motion of the H2O molecules taking place in their local environments that are characterized by a well-defined tetrahedral symmetry.
Fig. 4

(a) Two-time cos–cos correlation functions of hexagonal ice measured using 2H and 17O NMR at the given evolution times t p and Larmor frequencies ν L . The inset sketches the environment of an H2O molecule in ice. The larger spheres refer to O atoms and the smaller ones to H atoms. The broken and full arrows indicate principal axes of the deuteron and oxygen EFG tensors, respectively. Upon reorientation, the 2H tensors perform tetrahedral jumps, whereas the 17O tensors perform 90° jumps. Adapted from [29]. (b) Normalized mixing-time-dependent decays of the 23Na sin–sin echo amplitude recorded for t p = 20 μs for a disordered crystal. About 1500 scans were coadded for each data point. The inset depicts the temperature-dependent time scales recorded for this crystal with the line representing an Arrhenius law (Adapted from [28])

One should be aware that although studies using the two isotopes reveal essentially the same motional time scale, 2H and 17O are sensitive to different aspects of water motion: deuteron measurements probe the reorientation of the OD bonds and oxygen NMR monitors how the two-fold axis of the H2O molecule reorients, cf. the inset of Fig. 4a. At mixing times longer than those covered in Fig. 4a, the F 2(t m ) functions display another step stemming from rotationally coupled translational jumps [41, 42] before spin-lattice relaxation eventually sets in. At 183.5 K, the 17O–T 1 time is 2.5 s and the 2H–T 1 time is 430 s demonstrating that in studies of ice dynamics the 17O probe has the potential to reduce the measuring time significantly.

In addition to exploiting the conventional selective central-transition excitation of broad spectra with a satellite width of several MHz, it is useful to employ microcoils to enable their nonselective excitation [43]. Recently, microcoils were exploited to detect 23Na (I = 3/2) sin–sin correlation functions for (Na2SO4)0.1(Na3PO4)0.9 [28]. According to Witschas and Eckert [44], this disordered crystal is characterized by a quadrupolar product C Q (1 + η 2/3)1/2 of 2.5 ± 0.2 MHz. Using a 13-turn solenoid coil with an inner diameter of 0.8 mm and a π/4 pulse length of 150 ns, spectral intensity covering a range from roughly −1 MHz to +1 MHz could be excited at a power level of 1300 W [28]. The results in Fig. 4b show that it is possible to record spin-alignment echo curves under these challenging conditions. Mapping out the thermally activated 23Na ion motion this way, an activation energy barrier is obtained, cf. the inset of Fig. 4b, which is in agreement with that reported on the basis of spin-lattice relaxometry at much higher temperatures [44].

Overall, recent methodological advances regarding the stimulated-echo technique now enable efficient measurements of motional time scales, no matter whether the spectra are typically only a few kHz broad (such as for 6Li [38]), whether they are in the “convenient” 30…150 kHz range (like, e.g., for many spin-1/2 nuclei and for deuterons), or whether they are much broader as for the examples shown in Fig. 4. Equally important in the present context is that the ability to perform two-time stimulated-echo measurements also paves the way the application of its four-time variants. The corresponding experiments are useful to tell homogeneous from heterogeneous motional processes.

Homogeneous Versus Heterogeneous Transport and Reorientation: Concepts

It is a relatively straightforward matter to distinguish the two limiting scenarios sketched in Fig. 1 based on whether or not one can suppress the contribution of some, e.g., the faster ones of these motional processes. If such a suppression or filtering is possible because a distribution of effective relaxation rates exists, see Fig. 5a, one deals with heterogeneous dynamics. In this case, the remaining response will be shifted toward the slower end and, as sketched in Fig. 5b, it will be based on a narrower distribution and thus exhibit a more exponential shape than without low-pass filtering. If however, a homogeneous scenario prevails, subensemble selection does not induce a shape change but an amplitude reduction of the filtered response function, see also Fig. 5b.
Fig. 5

(a) Effective distribution of motional rates Γ that may give rise to a nonexponential decay of an F 2(t m ) function. (b) Selection of small motional rates will narrow the detected distribution and shift its mean to longer times (area in dark green) if the heterogeneous scenario applies. Conversely, as the light green area in (b) illustrates, in the homogeneous limit only the amplitude and not the shape of the detected distribution is altered. The short- and the long-time limits of the F 4(t m2) function that is recorded during the dynamic exchange experiment are illustrated in (c) and (d), respectively (Adapted from [48])

A basic pulse sequence suitable for potential filtering is depicted in Fig. 2. For a first intuitive understanding of this sequence, one may say that only the relaxing units that are slow on the scale set by the mixing time t m1 contribute to the echo after this period, more precisely, that a molecular or ionic unit has not changed its orientation, site, etc. during the time scale t m1 set by the experiment.

The amplitude of F 2(t m1) is thus a measure for the fraction of slow units and FE ≡ 1 – F 2(t m1) a measure for the (low-pass) filter efficiency [45, 46]. The last three pulses of the sequence in Fig. 2 can be used to analyze the dynamics of the selected subensemble by varying the mixing time t m3. If G 4 and F 2 display the same dependence on the variable mixing time, the response is said to be homogenous. An echo function G 4(t m3) that, however, decays slower and more exponential than the ensemble average, i.e., than F 2(t m ), is an indicator of heterogeneous dynamics. Note that the notion of simple filtering and thus the latter statement may need to be augmented by more elaborate analyses if the nuclear probes in the materials under study feature only a finite number of distinguishable sites [52, 54]. It should also be mentioned that apart from low-pass filtering, other selection schemes can be implemented [47] and that some of the literature denotes G 4(t m3) as F 3(t m3).

So far, the second mixing time of the pulse sequence shown in Fig. 2 was tacitly assumed to vanish. By varying t m2 and keeping the other mixing times constant, one can check whether a slow subensemble stays slow or whether it becomes faster on a prolonged t m2 scale. The simplest way to test for slow↔fast exchange among the relaxing units is to set t m1 = t m3 so that the same low-pass filter has to be passed twice. If the slow units remain slow, e.g., if t m2→0 is chosen, the F 4(t m2) signal will be maximum; if some of the selected slow units will become fast during a sufficiently long t m2 period, these units will get stuck in the second filter, resulting in a reduced F 4(t m2) amplitude [11]. The limits of such a t m2 dependence are illustrated in Fig. 5c, d. The time scale τ ex characterizing the F 4 decay thus quantifies the lifetime of the dynamic heterogeneity.

The pulse sequence applied in studies of dynamic exchange (using F 4) is the same as that employed for the low-pass filtering (G 4) experiment. The various four-time functions differ only with respect to the choice of the variable mixing time.

After this brief outline of the conceptual rationale for the utility of various four-time functions, implementations for specific spin quantum numbers I will briefly be summarized in the following. For this purpose, it is useful to define the abbreviations:
$$ {c}_i=\cos \left({\omega}_i{t}_p\right)\ \mathrm{and}\ {s}_i=\sin \left({\omega}_i{t}_p\right)\ \mathrm{with}\ {\omega}_i{t}_p\equiv {\omega}_I\left({t}_{mi}\right){t}_p $$
(3)
for each mixing time i = 1, 2, and 3. Then, for nonselectively excited spin I = 1/2 and spin-1 nuclei, one arrives at [23, 25]:
$$ {E}_{4,I}^{\mathrm{cccc}}={s}_I^{\mathrm{cos}}{s}_I^{\mathrm{cos}}\left\langle {c}_1{c}_2{c}_3{c}_4\right\rangle $$
(4a)
$$ {E}_{4,I}^{\mathrm{ccss}}={s}_I^{\mathrm{cos}}{s}_I^{\mathrm{sin}}\left\langle {c}_1{c}_2{s}_3{s}_4\right\rangle $$
(4b)
$$ {E}_{4,I}^{\mathrm{sscc}}={s}_I^{\mathrm{sin}}{s}_I^{\mathrm{cos}}\left\langle {s}_1{s}_2{c}_3{c}_4\right\rangle $$
(4c)
$$ {E}_{4,I}^{\mathrm{ssss}}={s}_I^{\mathrm{sin}}{s}_I^{\mathrm{sin}}\left\langle {s}_1{s}_2{s}_3{s}_4\right\rangle \operatorname{} $$
(4d)
if signal maximizing flip angles are assumed: For the pulses 1, 2, and 3, these angles are identical to those in Table 1; the pulses 2, 3, 6, and 7 should be equally long and the flip angles of the pulses 4 and 5 should be 90°.
For I = 3/2 spins, only the four-time function
$$ {E}_{4,I=3/2}^{\mathrm{ssss}}=\frac{81}{320}\left\langle {s}_1{s}_2{s}_3{s}_4\right\rangle $$
(5)
is experimentally accessible so far [48]. This expression is written here for optimum pulse lengths which do not differ from those required for \( {E}_{4,I=1}^{\mathrm{ssss}} \).

Under conditions of selective central-transition excitation, all four functions appearing in Eq. (4) can be generated for I > 1. However, assuming optimum flip angles, each \( {s}_I^{\alpha }{s}_I^{\alpha^{\prime }} \) factor is to be replaced by s I,select = 3/[4I(I + 1)]. This is because, like for the two-time experiments, all selective pulses act only on the central-transition subspace. The amplitude reduction for nonideal flip angles is described by the factor \( {\Pi}_{i=1}^7\sin \left[\left(I+\frac{1}{2}\right){\varphi}_i\right] \) [49] which shows that all seven angles should be chosen equally long and with care to avoid unwanted attenuation of the echo signal.

Based on Eq. (4), linear combinations such as
$$ {\displaystyle \begin{array}{l}{E}_4\propto \left\langle {c}_1{c}_2{c}_3{c}_4+{c}_1{c}_2{s}_3{s}_4+{s}_1{s}_2{c}_3{c}_4+{s}_1{s}_2{s}_3{s}_4\right\rangle \\ {}\phantom{\rule{0ex}{1.25em}}1.92em =\left\langle \cos \left[\left({\omega}_1-{\omega}_2\right){t}_p\right]\cos \left[\left({\omega}_3-{\omega}_4\right){t}_p\right]\right\rangle \operatorname{}\end{array}} $$
(6)
can be formed. From Eq. (6), the working principle of a low-pass filter which does not reduce the signal amplitude if ω 1 = ω 2 during t m1 is appreciated directly. But, based on arguments outlined in [48] and in accord with experimental findings, e.g., [14], the four-time echo amplitudes represented by Eq. (4a) and Eq. (4d) or Eq. (5) provide essentially the same information as that of the linear combination in Eq. (6).
For the execution of the four-time experiments, there is a subtlety to consider concerning the choice of the evolution time t p . Usually, t p should be chosen sufficiently long so that any molecular, ionic, etc. jump leads to a decay of F 2 [2]. The t p →0 limit, however, can be useful as well and has interesting consequences. In the context of four-time correlation functions, this limit, cf. Eq. (2), was first introduced in [50] and can be written as:
$$ {E}_{4,I,{t}_p\to 0}^{\mathrm{ssss}}\propto {t}_p^4\left\langle {\omega}_1{\omega}_2{\omega}_3{\omega}_4\right\rangle . $$
(7)
Normalization and suitable choice of various time intervals leads to the definition of what is called:
$$ {L}_{4,I}\left({t}_{m2}\right)=\frac{E_{4,I}^{\mathrm{ssss}}\left({t}_{\mathrm{p}}\to 0,{t}_{\mathrm{m}1},{t}_{\mathrm{m}2},{t}_{\mathrm{m}3}={t}_{m1}\right)}{E_{4,I}^{\mathrm{ssss}}\left({t}_{\mathrm{p}}\to 0,{t}_{\mathrm{m}1},{t}_{\mathrm{m}2}\to 0,{t}_{\mathrm{m}3}={t}_{m1}\right)}. $$
(8)

Some applications of this function were discussed previously [12, 51].

It is important to recall that an existence of only a finite number of sites requires more dedicated considerations [52, 53, 54, 55]. Under such circumstances, the dynamic filters become imperfect because the initial resonance frequency can be restored by a return to the initial site or, for crystals, a jump to one of the equivalent sites in the crystal lattice. Specifically, molecules which show frequency changes Δω during t m1 and, due to a jump to the initial or a periodic (= equivalent) site, −Δω during t m3 yield contributions cos2ω t p ) to the echo signal (see Eq. (6)) although they are not immobile neither during t m1 nor during t m3. Nonetheless, explicit consideration of restoration effects enables one to perform quantitative analyses of G 4 and F 4 functions even in such cases [52, 54, 55, 56, 57].

The remainder of the chapter focuses on the detection of heterogeneous dynamics based on multiple-time stimulated echoes or “two-phase” two-dimensional spectroscopy. However, it has to be mentioned that under certain circumstances (partially relaxed) one-dimensional spectra or simple spin-relaxation measurements or combination of the latter with stimulated-echo techniques can serve this purpose as well [58, 59, 60, 61, 62].

Homogeneous Versus Heterogeneous Transport and Reorientation: Time-Domain Techniques

Since the publication of the first edition of this handbook, several studies reported on four-time correlation functions for the I = 1 spins 2H [57, 63] and 6Li [64]. Most recently, experiments were devised that enable one to perform corresponding investigations using a nonselective excitation also for I = 3/2 nuclei, such as 7Li in an ion conducting glass, and for selectively excited I = 5/2 nuclei, such as 17O, in an ice-like crystal. The example presented in Fig. 6a refers to the onsite reorientation of 17O-labeled water in a clathrate hydrate containing tetrahydrofuran as a guest molecule [42, 49, 65]. To appreciate the situation in this crystal, it is useful to recall that in hexagonal ice, the H2O molecules occupy sites characterized by a well-defined local tetrahedral symmetry and that F 2 decays in an exponential fashion, cf. Fig. 4a.
Fig. 6

Two-time and four-time functions recorded (a) for tetrahydrofuran clathrate hydrate using 17O NMR in a magnetic field of 8.5 T (adapted from [49]) and (b) for a natural abundance sample of a lithium diborate glass using 7Li NMR (adapted from [48]). In both (a) and (b), one recognizes that G 4(t m3) decays slower and less stretched than F 2(t m ): This is clear evidence for dynamic heterogeneity in the presently discussed materials. All lines guide the eye

By contrast, for the clathrate hydrate, its locally distorted center-of-mass lattice leads to nonexponential water reorientation as detected using 17O NMR [42, 66], see the F 2 function in Fig. 6a. Furthermore, for long t m , it decays to values near zero although a value of 1/6 may have been expected in the presence of a six-site molecular reorientation if heteronuclear oxygen-proton dipole interactions were negligible. However, in [42], they were found to be significant, a situation that in four-time experiments, on the one hand, typically leads to inexact restoration of NMR frequencies [64]. On the other hand, this circumstance can simplify the interpretation of the results: From the slower and more exponential decay of the G 4(t m3) function with respect to the F 2(t m ) function, cf. Fig. 6a, experimental evidence for the occurrence of heterogeneous dynamics in this ice-like crystal was found [49].

Now the discussion will turn from selective to nonselective excitation. Figure 6b compares F 2(t m ) and G 4(t m3) functions recorded using 7Li NMR for a lithium diborate glass [48] in which the quadrupolar interaction is much larger than the dipolar ones [67]. Here, the F 2(t m ) function monitors the translational Li ion hopping in the disordered matrix [68]. One recognizes that the 1/e time scales characterizing the decays of the F 2(t m ) and the G 4(t m3) functions differ by a factor of about 6 for the chosen filter efficiency [48]. This finding demonstrates a large degree of low-pass filtering and thus a large degree of heterogeneity. A detailed analysis reveals that for the data shown in Fig. 6b, this ratio as well as the degree of exponentiality of G 4(t m3) is near the maximum that is theoretically expected for purely heterogeneous dynamics [48]. An accompanying study exploiting the 6Li nucleus unraveled that this probe may be less well suited to investigate dynamical heterogeneities. This is because for 6Li the multiparticle dipolar interaction is not much smaller than the single-particle quadrupolar interaction. This circumstance can invoke a mutual coupling of various subensembles, thereby impeding the ability to perform clear-cut filtering [48, 64].

Hence, it turned out advisable to investigate a possible dynamic exchange in the lithium diborate glass using the 7Li rather than the 6Li probe. Corresponding measurements of a dynamic exchange function F 4(t m2) are plotted in Fig. 7a. One recognizes that unlike the corresponding F 2(t m ) function, F 4(t m2) decays to a finite plateau as expected from the considerations visualized in Fig. 5c, d. With these illustrations in mind, it may also be anticipated that the time scale τ ex on which the F 4(t m2) functions decay will depend on the degree of filtering, provided the motional rates and the slow↔fast exchange rates are related to each other. A dependence of τ ex on the filter efficiency was indeed observed for various disordered materials and is expected on the basis of various theoretical approaches [69, 70, 71].
Fig. 7

Two-time and four-time stimulated-echo amplitudes F 4(t m2) for two solids. (a) A natural abundance sample of a lithium diborate glass measured using 7Li NMR for mixing times t m1 = t m3 = 2 ms (Adapted from [48]). (b) The orientationally disordered crystal dimethyl sulfone measured using 2H NMR. A sketch of the molecule is given (Adapted from [52]). All data are corrected for spin-lattice relaxation. The arrows indicate 1/e decay times and the solid lines guide the eye

If measurements for only a single filter efficiency are available, as for the example presented in Fig. 7a, additional scrutiny is required to extract the heterogeneity lifetime. Detailed analyses show that the heterogeneity of the Li ion hopping in this diborate glass is not long but rather short-lived [48]. This finding may contrast with the first impression from Fig. 7a. Yet, it is in harmony with most – albeit not with all – NMR studies on supercooled liquids [2, 10, 11, 64] and ion conductors [38, 52, 72] regarding this issue. A well-known exception is the longevity of the heterogeneity in the orientationally disordered crystal dimethyl sulfone [53]. Its heterogeneity presumably originates from frozen-in lattice strains and the F 4(t m2) function does not show any signs of a decay in the experimentally accessible time window, cf. Fig. 7b. Thus, the heterogeneity persists much longer than indicated by the decay of F 2(t m ).

Homogeneous Versus Heterogeneous Transport and Reorientation: Frequency-Domain Techniques

In addition to time-domain methods discussed so far, frequency-domain techniques proved well suited to investigate dynamic exchange. Actually, the first NMR approach to measure the lifetime of dynamical heterogeneities time used “reduced” 4D spectra [10, 13]. If motional inhomogeneities are sufficiently pronounced so that they manifest themselves in the regular NMR line shape, i.e., if fast and slow fractions of a rate distribution can be discriminated by motionally narrowed and rigid-lattice spectra, respectively, “two-phase” 2D spectra are sufficient to quantify rate exchange [61]. Then, cross-like spectral intensity along the frequency axes indicates slow↔fast exchange during the mixing time (Fig. 8).
Fig. 8

31P 2D exchange spectra of TPP/PS at a TPP mole fraction of 20% and T = 269.5 K for t m = 100 ms demonstrating exchange among liquid-like and solid-like TPP molecules; left: experimental; right: calculated spectra (Adapted from [37] and reproduced courtesy of E. A. Rössler with permission from Elsevier)

Specifically, particles that are fast prior to t m and slow after this period yield a rigid-lattice spectrum along one of the frequency axes, i.e., along ω 1 = 0, while those that are first slow and then fast contribute such a line shape along the second frequency axis, ω 2 = 0. Here, the terms fast and slow refer to the time scale set by the inverse width of the broad spectral component. This simple 2D method works well when the motionally narrowed 1D line can be well distinguished from the rigid-lattice spectrum. Such a situation is often found for 2H, 13C, or 31P nuclei when acting as dynamical probes in multicomponent systems, such as those formed by tripropyl phosphate and deuterated polystyrene, cf. Fig. 3. For a related mixture, the 31P 2D exchange spectrum is shown in Fig. 8. It displays well-resolved cross-like intensity which indicates exchange between liquid-like and solid-like TPP molecules in a PS matrix during t m [36, 37].

The situation is less favorable, when the overall spectral width is moderate, as found, e.g., for 6Li, or when a narrow central transition of half-integer I > 1 nuclei such as 7Li interferes with the motionally narrowed component.

Concluding Remarks

The theoretical framework and the experimental examples presented in this chapter show that NMR stimulated-echo approaches yield valuable insights into complex motional processes in crystalline and amorphous solids as well as in viscous liquids. Two-time correlation functions allow one to determine the degree of nonexponentiality of their molecular or ionic dynamics, while higher-order correlation functions enable quantitative insights into the origin of the nonexponentiality and, if dynamical heterogeneities exist, into the time scale of exchange processes between fractions of fast and slow molecules, ions, etc. Similar information can be obtained from corresponding frequency-domain techniques. In particular, also due to recent developments, these experiments have now become available for a large number of probe nuclei.

Notes

Acknowledgement

Mischa Adjei-Acheamfour and Ken R. Jeffrey are thanked for fruitful collaborations. Financial support provided by the Deutsche Forschungsgemeinschaft under Grants No. BO1301/10-1, BO1301/13-1, VO905/8-2, and VO905/12-1 is highly appreciated.

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Fakultät PhysikTechnische Universität DortmundDortmundGermany
  2. 2.Institut für FestkörperphysikTechnische Universität DarmstadtDarmstadtGermany

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