# Probing Small-Angle Molecular Motions with EPR Spectroscopy: Dynamical Transition and Molecular Packing in Disordered Solids

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

_{0}, we can write:

_{0}, U

_{0}= U(q

_{0}).

_{Tg}). These fluctuations provide a source of stochasticity of motion, so librations may become stochastic. Stochasticity may appear also for random transitions between vibrational levels in an anharmonic potential, for which the mean amplitude, <s>, is a non-zero fluctuating value (Figure 1a, the small solid arrow). Another source of stochasticity could be random jumps between two or more closely spaced shallow wells (Figure 1a, the curved arrow). Small-angle stochastic reorientations of the molecules are here called stochastic (molecular) librations. Furthermore, these motions may be called wobbling motion [20], or quasi-librations [21,22].

_{Tg}these intermolecular potentials are destroyed.

_{Tg}, molecular solids transform into supercooled liquids, for which also unusual phenomena are known: secondary Johari–Goldstein β-relaxation seen in dielectric relaxation [23,24,25,26], dynamical transition in neutron scattering and Mössbauer absorption [19,27,28,29], cooperativity of motions and nanoscale heterogeneity [30,31], and others. The phenomenon of secondary β-relaxation was also addressed by NMR [32,33]. Some theoretical aspects of supercooled liquids may be described with idealized “mode-coupling theory” [34].

## 2. Methodology

#### 2.1. CW EPR Spectra of Nitroxides in Molecular Glasses

^{−7}s and 10

^{−10}s, for studying anisotropy of the motion, for elucidating its heterogeneity. In solids, unrestricted rotations, however, are suppressed and only small-angle motions (librations, wobbling, etc.) exist.

**B**

_{ext}the external magnetic field,

**g**is the g-tensor, g

_{e}is the g-factor of free electron,

**A**is the hyperfine interaction (hfi) tensor for interaction with a nitrogen nucleus, expressed in the magnetic field units. Nuclear Zeeman and quadrupole interactions are here neglected because of their smallness. The laboratory framework typically used suggest that the z axis is directed along the magnetic field

**B**

_{ext}, when an alternative magnetic field with the microwave frequency ω is also applied to the system, the solution of Hamiltonian (2) results in the EPR line positions:

^{14}N m acquires three values, −1, 0, 1.

^{8}rad/s. This is a measure of spectral anisotropy that is an essential value for analyzing motions detected by CW and pulsed EPR.

_{ext}is not equal to B

_{m}(θ,φ) that is given by Equation (3), and in polyoriented media the EPR lineshape $g({B}_{ext})$ is determined by averaging over all angles θ and φ:

^{12}s

^{−1}[43]).

#### 2.2. CW EPR: Dynamical Librations

_{XX}, A

_{YY}, and A

_{ZZ}as the principal values of the hfi tensor. Let librational motion occur for simplicity via rotations around the X molecular axis, and let α(t) be the instant small deviation angle (α

^{2}(t) << 1) for the motion from the equilibrium position (see Figure 3). Then the new motion-averaged principal values are [42]:

_{Tg}), all of them depended linearly on temperature T. This linearity may be ascribed to harmonic oscillations for which the relation is expected:

_{XX}> will be determined by the Bose factor $n(\mathsf{\Omega})=1/(\mathrm{exp}(\hslash \mathsf{\Omega}/kT)-1)$:

_{XX}> linearly depends here on a temperature that is in line with Equation (8).

_{XX}> principle values were measured at cryogenic temperatures by a pulsed electron-nuclear double resonance (ENDOR) technique for

^{15}N-substituted nitroxides. The results are given in Figure 4 for glycerol and o-terphenyl glasses, along with the <A

_{ZZ}> principle values found from the splitting between the two outmost spectral components (cf. Figure 2). The fitting of <A

_{XX}> temperature dependence allowed to assess the ${\mathsf{\Omega}}_{libr}^{}/2\pi $ value as 60 cm

^{−1}for glycerol and 90 cm

^{−1}for o-terphenyl. This result coincided fairly well with the Raman scattering data also obtained for these glasses [43], which showed that dynamical librations of guest nitroxide molecules is determined by the vibrations of the host molecules. The similar results were also obtained for glassy liquid crystals [46].

_{XX}> temperature dependences in Figure 4 manifest a kink with a slope slightly smaller at higher temperatures. The reason for this kink is not clear. It is likely a result of influence of anharmonicity of the motion, which may become essential at higher temperatures—up to the overcoming of one of the barriers in the potential well (see Figure 1a).

^{2}.

#### 2.3. ESE: Stochastic Librations

_{c}<< τ), and occurs via random walks within a restricted frequency interval, Equation (10) results in an exponential decay [50,51]:

_{c}<< τ of fast motion may be alternatively presented as [50,51]

^{2}(t) << 1 we may write [52,53]:

_{N}is of the order or less than 10

^{5}s

^{−1}, which is much smaller than the rate of echo decay in organic solids that is typically larger than 10

^{6}s

^{−1}(cf. Figure 5). Therefore, nitrogen nuclear relaxation in nitroxides normally may be neglected then analyzing the spin relaxation induced by stochastic molecular librations, and m in Equation (14) may be considered as a constant.

_{ext}= 3400 G (0.34 T), ${A}_{\parallel}=35\text{}\mathrm{G},\text{}{A}_{\perp}=5\text{}\mathrm{G}$, and an individual Gaussian line broadening was of 3 G. The insert in Figure 6 shows the ln(E

_{2}(2τ)/E

_{1}(2τ)) ratio at different delays 2τ. One can see that this ratio depends almost linearly on 2τ:

^{6}s

^{−1}. Comparing Equations (15) and (18), we can write

_{12}= 0.9 × 10

^{17}rad

^{−2}s

^{−2}.

#### 2.4. Stimulated ESE: Slow Motions

_{SE–}90°–τ–echo detection, where t

_{SE}is an additional time delay. Stimulated echo is sensitive to motions in the microsecond time scale [20,75] that is determined by the spin-lattice relaxation time, T

_{1}. Reorientations by the angles larger than ~2° can be studied by pulse excitation near the canonical orientations of the nitroxide spin probe, at which the spectral anisotropy is small [20]. For pulse excitation at the spectral positions between canonical orientations [75], the anisotropy is high, ~5 × 10

^{8}rad×s

^{−1}at X-band, which for t

_{SE}~10

^{−6}s results in the acquired phase around unity for the angle of ~0.1°. The detection of such small-amplitude molecular reorientations is a unique property of the ESE technique.

_{SE}delay.

_{SE}, instead of Equation (24), is given by the expression:

_{SE}, which is in agreement with the experimental observation presented by Equation (25).

_{SE}($\tau $ is also small, typically $\tau <<{t}_{SE}$), Equation (26) may be expanded in terms of the parameter $|{R}_{m}(\theta ,\varphi )|{\mathsf{\Omega}}_{rot}({\tau}^{2}+\tau {t}_{SE})$. Assuming that the distribution $g({\mathsf{\Omega}}_{rot})$ is normalized, $\int g({\mathsf{\Omega}}_{rot})}d{\mathsf{\Omega}}_{rot}=1$, the first-order Taylor expansion can be formally presented as

^{−7}÷ 10

^{−6}s).

_{12}= 0.9 10

^{17}rad

^{−2}s

^{−2}, one can obtain the angular velocity ${\mathsf{\Omega}}_{0}$. From data in Figure 7 it follows that near, say 260 K, ${\mathsf{\Omega}}_{0}$ is of the order of 10

^{3}rad/s. Then, for t

_{SE}~10

^{−5}s, we obtain that the reorientation angle is of the order of 1°.

_{c}becomes larger than 10

^{−7}s. (The τ

_{c}may increase with temperature because of increase of cooperativity of the motion).

_{SE}is observed, this supports the model of fast stochastic librations, with correlation times τ

_{c}less than 10

^{−7}s [75].

## 3. General Features of Small-Angle Motions in Solids

#### 3.1. Dynamical Librations and Transition in Molecular Glasses

_{d}. [27,28,29,83]; a similar effect was observed in a Mössbauer absorption experiment [27,84]. Both these techniques probe mean square displacement of atoms (MSD), <x

^{2}>, where x is the individual displacement. Neutron scattering probes the motion of hydrogen atoms. It is sensitive to the motions developing in the picosecond-nanosecond time scale. Mössbauer absorption is sensitive to motions of ferrous atoms naturally presenting in some proteins, or artificially inserted into molecular glass formers. It is sensitive to motions faster than 10

^{−7}s. For harmonic motions, <x

^{2}> is expected to linearly depend on the temperature, which is indeed observed below T

_{d}. Above T

_{d}, a drastic enhancement of <x

^{2}> is observed, which is attributed to the onset of anharmonic or diffusive motions. Typically, <x

^{2}> attains a value of ~0.1 Å

^{2}.

_{d}lies in the interval from 170 to 230 K. The importance of dynamical transition for proteins is determined by the experimental fact that it can correlate with the onset of protein function [27,28,29,83].

^{2}implies that for reorientation angle α

^{2}~ <x

^{2}>/l

^{2}~ 0.1 rad

^{2}, which is in agreement with that found from CW EPR spectra, see the assessments in Section 2.2 and in [103]. Therefore, one may assume that CW EPR should provide results consistent with the neutron scattering and Mössbauer absorption spectroscopies.

^{2}> and <α

^{2}>, respectively, as a function of temperature. One can see a rather good agreement between the two data sets. This can be easily understood by assuming that the tempone and orthoterphenyl molecules reorient by the same angle α, so that the relation $x=l\alpha $ indeed holds, where l ≈ 1 Å.

^{15}N-ENDOR at 112 K (see Figure 4). This coincidence is in favor of the above suggestion that near 112 K onset of anharmonicity of the motion takes place, and the deviation from the linear dependence taking place at 243 K (that coincides with

_{Tg}) just implies the dynamical transition (see above), i.e., T

_{d}= 243 K.

_{d}. And it is interesting to note that above 243 K the $<{\alpha}^{2}(t)>{\tau}_{c}$ motional parameter was found to increase with temperature remarkably faster than the <x

^{2}> parameter determined from neutron scattering

#### 3.2. Cooperativity of Stochastic Librations, Influence of Hydration for Biological Systems

#### 3.3. Individual Stochastic Librations on an Inorganic Surface

_{2}surface. The anisotropic relaxation rate ΔW was found to demonstrate a saturating behavior with the temperature increase, with the maximum ΔW

_{max}~ 1 μs

^{−1}attained near 250 K.

^{2}>τ

_{c}

^{2}<< 1, when also $\tau \gg {\tau}_{c}$, Equation (30) is reduced to the exponential dependence, $E(2\tau )=\mathrm{exp}(-2\tau \mathsf{\Delta}{\omega}^{2}{\tau}_{c})$, that is in full agreement with Equation (11). The advantage of this simple model is its applicability to any $\mathsf{\Delta}{\omega}^{2}{\tau}_{c}^{2}$ value, i.e., this model is not restricted by the condition of fast motion that is used in the Redfield’s theory of spin relaxation (see above). The saturation behavior for the ΔW temperature dependence then appears as a consequence of violation of the <Δω

^{2}>τ

_{c}

^{2}<< 1 condition in the Redfield’s theory.

_{c}attains a value of several tens of nanoseconds, while the angle α is around 0.02 rad [111]. From comparison ΔW for lipid bilayers, it was concluded that the saturating behavior is an exclusive feature of the individual molecular motions. The ΔW

_{max}close to the value of 1 μs

^{−1}was found in the experiment to be close for very different molecules—a small highly polar nitroxide radical and a large spin-labeled peptide—so the effect seems to be independent on the type of the molecule. Then, for any molecular system, the excess of ΔW > 1 μs

^{−1}may be ascribed to the effects of cooperative motions, and this excess implies that motion involves independent reorientations around several different axes.

_{max}observed in ESE decays is enhanced for highly flexible molecules, such as stearic acid, attaining the value of 2 μs

^{−1}. This effect was interpreted as a result of the two-axial (or planar) motion appearing instead of the uniaxial motion for more rigid molecules.

#### 3.4. Stochastic Librations and Softness/Rigidity of Molecular Packing

## 4. Applications

#### 4.1. Dynamical Transition in Membranes and Proteins

_{g}was reported to be near 200 K [121,122]. Therefore, one may expect that EPR-detected dynamical transition, as it was found for simple molecular glass-formers (see Section 3.1), takes place for biological media as well.

_{d}indicated. (In the cases when T

_{d}was not explicitly indicated by authors, it was assessed here from the presented <α

^{2}> temperature dependences).

_{d}for all of the investigated membranes and proteins lies between 190 and 240 K. Second, the dynamical transition is a property of only hydrated biosystems, for lyophilized samples it disappears. And finally, for membranes it is possible to indicate the relationship between the quantitative T

_{d}value and the obvious qualitative stiffness/cohesion characteristic of the molecular packing: both characteristics are larger for the label positioned closer to the membrane surface (5-PCSL as compared to 16-PCSL, e.g.,), for the membranes with more ordered lipid conformation (DPPC as compared to POPC), for the case of interdigitated lipid chains (DHPC as compared in [128] with DPPC).

_{d}in Table 1 are in general agreement with those found by neutron scattering in purple membranes (T

_{d}= 230 K [130] and T

_{d}= 260 K [131]), in 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) bilayer (T

_{d}= 250 K) [132], in DPPC bilayer (T

_{d}= 230 K) [133], in model membranes by Raman spectroscopy [118] and by differential scanning calorimetry [122].

_{d}derived from the temperature dependence of CW EPR spectra may serve as a quantitative measure of the stiffness/cohesion of the molecular packing. The comparison of T

_{d}data for proteins lysozyme and casein [108] show that for the latter protein, this characteristic is larger. (This is discussed below in Section 4.8 in more detail).

_{d}[104,134], which may be related to dynamical transition observed in neutron scattering [106]. In biological membranes, however, this onset occurs above 130 K or even above 100 K (see Figure 9). On the other hand, one can see in Figure 9 that above 200 K, the ΔW value becomes larger than 2 μs

^{−1}. In Section 3.3, it was indicated that ΔW ≈ 2 μs

^{−1}corresponds to the maximal rate achievable for individual motions of flexible biological molecules. Therefore, it would be reasonable to suggest that dynamical transition in the membrane corresponds to appearing of cooperative lipid motions. However, the validity of this suggestion needs further investigation.

#### 4.2. Lipid Packing in Biological Membranes

#### 4.3. Stochastic Librations and Slow Rotations near T_{d} in Membranes

_{c}< τ, which in turn implies that τ

_{c}is smaller than 10

^{−7}s. This result supports validity of the librational model of molecular motion for this system.

_{c}at elevated temperatures becomes larger than 10

^{−7}s.

#### 4.4. Proteins and Antimicrobial Peptides in Membranes

^{n}] with n = 1, 8, or 16 in DMPC membrane were studied at a concentration of 1 mol %. The amino acid sequences of the three TOAC

^{n}-Alm derivatives were:

- Ac-
**TOAC**-Pro-Aib-Ala-Aib-Ala-Glu(OMe)-Aib-Val-Aib-Gly-Leu-Aib-Pro-Val-Aib-Aib-Glu(OMe)-Glu(OMe)-Phol [TOAC^{1}] - Ac-Aib-Pro-Aib-Ala-Aib-Ala-Glu(OMe)-
**TOAC**-Val-Aib-Gly-Leu-Aib-Pro-Val-Aib-Aib-Glu(OMe)-Glu(OMe)-Phol [TOAC^{8}] - Ac-Aib-Pro-Aib-Ala-Aib-Ala-Glu(OMe)-Aib-Val-Aib-Gly-Leu-Aib-Pro-Val-
**TOAC**-Aib-Glu(OMe)-Glu(OMe)-Phol [TOAC^{16}].

^{8}sample), which confirmed the used small-angle approximation. The T

_{d}values derived from CW EPR (see Table 1) were 220 K for TOAC

^{1}and TOAC

^{8}samples, and 160 K for TOAC

^{16}. It was suggested [124] that associated with the stochastic librations the fluctuations in polar fields from the peptide could facilitate ion permeation through the membrane.

- nOct-Aib1-Gly-Leu-Aib-Gly-Gly-Leu-Aib8-Gly-Ile-Lol (native trichogin GA IV)
- Fmoc-
**TOAC**1-Gly-Leu-Aib4-Gly-Gly-Leu-Aib8-Gly-Ile-Leu-OMe (FTOAC1) - Fmoc-Aib1-Gly-Leu-Aib4-Gly-Gly-Leu-
**TOAC**8-Gly-Ile-Leu-OMe (FTOAC8).

#### 4.5. Lipid Bilayers Interacting with Cryoprotectants

#### 4.6. Supercooled Ionic Liquids

_{Tg}of the anisotropic relaxation rate ΔW (in [166,167,168,169,170,171,172,173] another denotation was used which is however coincides almost quantitatively with the ΔW rate used in this review). A typical example of the obtained temperature dependence is shown in Figure 14. The most intriguing result seen here is the drastic suppression of molecular mobility observed with temperature increase near

_{Tg}. This suppression implies that local density around the nanoscale spin probe solute grows with temperature, which is highly uncommon, because substances typically become less dense upon a temperature increase.

_{Tg}.

_{2}surface, which also demonstrates temperature dependence with maximum. These data were explained by large reorientation angles resulting in a violation of Redfield’s theory. However, this explanation cannot be employed for IL in Figure 14. First, for ILs in Figure 14 the dependence starts at 60 K and attains its maximum at 150 K while for adsorbed molecules it starts at 100 or at 130 K and attains its maximum at 240 K, which can be easily understood as a consequence of the fact that molecules on the surface possess much more freedom of motion, which indeed results in large reorientation angles. Secondly, for ILs, the maximum in temperature dependence in Figure 14 is followed by the minimum and subsequent further increase. This effect certainly cannot be explained by large reorientation angles.

_{n}mim]BF

_{4}(n = 0–12) only the chains with n = 3–10 demonstrate an anomaly; moreover, remarkable even-odd n dependence was found. In [172] a variety of non-IL glasses, which also contain molecules with alkyl chains, was studied. For a series of phthalates, very similar behavior to imidazolium-based ILs was clearly demonstrated, with the same length of alkyl chain.

_{Tg}was also observed here, and the amplitude of the anomaly was found to be dependent on the structure of the IL, thus showing the effects of molecular packing inside the MOF cavity.

_{4}and water were studied; it was found that water does not influence nanoclustering of IL-rich domains in which the spin probe is located. In [173], deuterated imidazolium-based ILs were compared with their protonated analogs, to assess the role of electron–nuclear spin couplings between radical probe and alkyl chains of IL; the data obtained allowed us to exclude the relaxation-induced artifacts.

#### 4.7. Supercooled Deep Eutectic Solvents

#### 4.8. Intrinsically Disordered Proteins

_{2}O-hydrated biological systems [198]; it was concluded that spin labels are located on the peptide surface, directly exposing to the water. The hydration level h used was 0.4 g of water per gram of casein (h = 0.4); for this level, proteins are known to be enveloped with a water monolayer [199]. For comparison purposes, dry proteins were investigated as well.

#### 4.9. Molecular Glasses and Other Systems

_{g}, in all the cases the linearity is broken. As it was pointed out in Section 3.1, this departure from the linearity is related to dynamical transition, which occurs at temperature T

_{d}. One can further see from Figure 17 that T

_{d}in molecular glasses is close to T

_{g}(with probably one exception for the water–glycerol mixture).

_{libr}in Equation (8). Moreover, as mentioned above, the slopes of the straight lines in Figure 17 correlate with the ‘‘fragility’’ [45] of the glass: glycerol is known to be ‘‘strong’’ glass, which is commonly attributed to the network of hydrogen bonds, and it possesses the smallest slope.

_{Tg}; however, for all other glasses, the motions were found to start well below

_{Tg}.

#### 4.10. NMR of Small-Angle Motions, Secondary Relaxation

^{2}H NMR solid-echo spectra in molecular glasses are anisotropically broadened, similarly to the CW EPR spectra. These spectra deliver information on quadrupolar coupling constant, which depends on temperature [206] analogously to the ${A}_{ZZ}^{\prime}$ temperature dependence in CW EPR (cf. Figure 17). The change of quadrupolar constant as a function of temperature was investigated for glassy toluene-d

_{5}, etanol-d

_{5}, polybutadien-d

_{6}, and some deuterated mixtures [206]. This dependence was attributed to dynamical librations, within a cone model [207] of fast librations. The <α

^{2}(t)> temperature dependence was obtained from the quadrupolar coupling constant [206] with the relation similar to Equation (7) for CW EPR.

^{2}H NMR solid-echo spectra, with the time delay between two pulses sequentially increasing [32,33,206,208,209,210]. For these spectra, similar behavior, as for the echo-detected EPR spectra, was found: for canonical orientations, the motion-induced relaxation is slowest. The data were simulated [32] by the model of stochastic motion within a cone with a full opening angle of 6°. Therefore,

^{2}H NMR is also sensitive to small-angle motions. The spectral anisotropy in

^{2}H NMR is ~5 10

^{5}rad/s that is ~200 times smaller than that in EPR of nitroxide spin probes (~10

^{8}rad/s), so the typical time delays between two echo-forming pulses, for which spectral distortions are seen [32,33,206,208,209,210], are also ~200 time larger than in Figure 5 and Figure 6 for ESE of nitroxides.

^{2}H NMR solid-echo spectra was attributed [32,33,206,208,209,210] to the manifestation of small-angule stochastic motions (analogous experiments can be also performed employing

^{31}P Hahn echo [209]). The results of these studies in molecular glassy systems were discussed in terms of Johari–Goldstein secondary β-relaxation process (see Introduction).

^{15}N-substituted nitroxide spin probes in molecular glasses and supercooled liquids [54,55] have shown that this MT-DEER experiment is most likely related to large-angle motions, and in this connection it was shown [55] that the observed MT effect is induced by secondary β-relaxation.

## 5. Concluding Remarks

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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