Collective dynamics and self-motions in the van der Waals liquid tetrahydrofuran from meso- to inter-molecular scales disentangled by neutron spectroscopy with polarization analysis

Arantxa Arbe, Gøran J. Nilsen, Mark Devonport, Bela Farago, Fernando Alvarez, José A. Martínez González, Juan Colmenero

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Arantxa Arbe, Gøran J. Nilsen, Mark Devonport, Bela Farago, Fernando Alvarez, et al.. Collective dynamics and self-motions in the van der Waals liquid tetrahydrofuran from meso- to inter-molecular scales disentangled by neutron spectroscopy with polarization analysis. The Journal of Chemical Physics, 2023, 158 (18), pp.184502. 10.1063/5.0147427. hal-05330186

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# Collective dynamics and self-motions in the van der Waals liquid tetrahydrofuran from meso- to inter-molecular scales disentangled by neutron spectroscopy with polarization analysis

Arantxa Arbe 🐶 ; Gøran J. Nilsen 🐶 ; Mark Devonport; Bela Farago 🐶 ; Fernando Alvarez 🐶 ;
José A. Martínez González 🐶 ; Juan Colmenero 🐶

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J. Chem. Phys. 158, 184502 (2023)

https://doi.org/10.1063/5.0147427

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# Collective dynamics and self-motions in the van der Waals liquid tetrahydrofuran from meso- to inter-molecular scales disentangled by neutron spectroscopy with polarization analysis EP

Cite as: J. Chem. Phys. 158, 184502 (2023); doi: 10.1063/5.0147427

Submitted: 22 February 2023 • Accepted: 18 April 2023 •

Published Online: 8 May 2023

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Arantxa Arbe,<fnref n="1" />,ª ID Gøran J. Nilsen,<fnref n="2" /> ID Mark Devonport,<fnref n="2" /> Bela Farago,<fnref n="3" /> ID Fernando Alvarez,<fnref n="1" />,<fnref n="4" /> ID
José A. Martínez González,<fnref n="5" />,ᵇ ID and Juan Colmenero<fnref n="1" />,<fnref n="4" />,<fnref n="5" /> ID

## AFFILIATIONS

## ABSTRACT

By using time-of-flight neutron spectroscopy with polarization analysis, we have separated coherent and incoherent contributions to the scattering of deuterated tetrahydrofuran in a wide scattering vector (Q)-range from meso- to inter-molecular length scales. The results are compared with those recently reported for water to address the influence of the nature of inter-molecular interactions (van der Waals vs hydrogen bond) on the dynamics. The phenomenology found is qualitatively similar in both systems. Both collective and self-scattering functions are satisfactorily described in terms of a convolution model that considers vibrations, diffusion, and a Q-independent mode. We observe a crossover in the structural relaxation from being dominated by the Q-independent mode at the mesoscale to being dominated by diffusion at inter-molecular length scales. The characteristic time of the Q-independent mode is the same for collective and self-motions and, contrary to water, faster and with a lower activation energy (≈1.4 Kcal/mol) than the structural relaxation time at inter-molecular length scales. This follows the macroscopic viscosity behavior. The collective diffusive time is well described by the de Gennes narrowing relation proposed for simple monoatomic liquids in a wide Q-range entering the intermediate length scales, in contraposition to the case of water.

© 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0147427

<footnote n="1">¹Centro de Física de Materiales (CFM) (CSIC-UPV/EHU) – Materials Physics Center (MPC), Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain</footnote>

<footnote n="2">²ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot OX11 0QX, United Kingdom</footnote>

<footnote n="3">³Institut Laue-Langevin, 71 avenue des Martyrs, Grenoble Cedex 9, 38042, France</footnote>

<footnote n="4">⁴Departamento de Polímeros y Materiales Avanzados: Física, Química y Tecnología (UPV/EHU), Apartado 1072, E-20080 San Sebastián, Spain</footnote>

<footnote n="5">⁵Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain</footnote>

<footnote>ªAuthor to whom correspondence should be addressed: a.arbe@csic.es</footnote>

<footnote>ᵇCurrent address: School of Pharmacy, Universidad San Pablo-CEU, CEU Universities, Urbanización Montepríncipe, 28660 Boadilla del Monte, Spain.</footnote>
# I. INTRODUCTION

By now, efforts over several decades have been devoted to the understanding of the dynamics of the so-called α-relaxation in glass-forming liquids of different nature. Probably influenced by the mode coupling theory,¹⁻⁴ the term “structural relaxation” has been used since the 1990s to refer to the mechanism leading to the decay of the coherent scattering function $S(Q, t)$—Q being the scattering vector— just at the first structure factor peak ($Q_{max}$), revealing thereby the time dependence of inter-molecular correlations. In this context, neutron scattering (mainly neutron spin echo, NSE) experiments on fully deuterated materials conducted at $Q_{max}$ have provided a direct microscopic observation of utmost value.⁵ However, the concept behind “structural relaxation” is much broader and should be applied to the diverse mechanisms leading to the decay of density fluctuations at different length scales. In fact, nowadays, one of the most crucial open questions in the field of liquids and glass-forming systems is to understand the collective dynamics in the region of intermediate length scales (ILS). These correspond to distances longer than the inter-molecular ones but not yet in the hydrodynamic regime. The lack of a definite theoretical framework is one of the main reasons for our ignorance of this problem. From an experimental point of view, accessing $S(Q, t)$ in the ILS region is a real challenge due to very low scattering intensities. More crucially, the incoherent contribution in this Q-regime—even using completely deuterated samples—is not negligible, in general, and has to be carefully taken into account. Therefore, to date, there are not many works dealing with this subject. We can cite the pioneering NSE study on the glass-forming polymer polyisobutylene⁶ where the incoherent signal was measured on a protonated sample and properly subtracted from the NSE signal. However, a qualitative step forward has been given in this direction with the availability of recently developed wide-angle time-of-flight neutron spectroscopy with polarization analysis. In particular, the instrument PLET—LET spectrometer with the polarization (P) analysis option—at the ISIS Neutron and Muon Source, Oxfordshire, UK, offers the possibility of separating coherent and incoherent contributions to the scattering with a sub-meV resolution.⁷ In a recent study on deuterated water, PLET was applied to the investigation of (deuterated) water dynamics.⁸ In that study, it was shown that the structural relaxation of water at the mesoscale is dominated by a constant-Q mode. When approaching inter-molecular distances, it crosses over to being mainly driven by diffusion. The experimental results were nicely described by a model that—in addition to collective excitations—assumes a simultaneous occurrence (convolution) of both processes,⁸⁻¹¹ the Q-independent mode and diffusion. The Q-independent mode has been interpreted as localized motions involving breaking and reforming of H-bonds (HBs) at the origin of the relaxation of the HB network.

It is clear that the HB network in water could be crucial to determining the relaxation of density fluctuations at mesoscales; conversely, in a polymer such as polyisobutylene, the intramolecular connectivity is obviously an inherent and essential ingredient that complicates the problem. The question we address in this work is as follows: How does the collective dynamics behave, especially at ILS, in a system with a “simpler” kind of inter-molecular interaction, namely a van der Waals molecular liquid? In particular, does the low-Q mode manifest also in the absence of a HB-network? And,

if yes, how? Also, to what extent does a “simple liquid” follow the de Gennes narrowing prediction relating collective and self-diffusion?

With these ideas in mind, we have again applied the novel PLET capabilities to disentangle the coherent and incoherent contributions to the neutron intensity scattered by the van der Waals liquid tetrahydrofuran (THF). To enhance the coherent contribution, a deuterated sample has been investigated. We have explored the temperature dependence of the dynamics in this liquid from temperatures slightly above the melting point up to slightly below the boiling point. To complete the information on the self-motions at low temperatures, we have also applied the recently developed NSE instrument WASP at the Institut Laue-Langevin (ILL) in Grenoble, France, to a protonated sample. This instrument provides an excellent resolution and extension toward much lower Q-values. This combined experimental approach has allowed us to fully characterize the self- and collective motions of THF over an unprecedented Q- and temperature range.

We have first treated the experimental results with a phenomenological approach in the susceptibility representation. This has allowed identifying, in an unbiased way, the main features of the dominant processes contributing to the incoherent and coherent dynamic structure factors. The phenomenology found for THF is qualitatively similar to that observed for water, thus justifying the application of the model used for water dynamics also in the case of THF. The application of the model to both incoherent and coherent scattering is successful and allows a quantitative characterization of the processes involved in each case, which can be compared with the water results. We conclude that, although exhibiting qualitatively similar dynamic features, these two liquids show clear differences in the behavior of the Q-independent mode and in the manifestation of collectivity in the diffusive component approaching the intermediate length scales region.

# II. EXPERIMENTAL

Neutron scattering experiments with uniaxial polarization analysis were performed at the LET direct geometry time-of-flight spectrometer at the ISIS Neutron and Muon Source, Oxfordshire, UK.⁷,¹²⁻¹⁴ Deuterated tetrahydrofuran (dTHF, $C_4D_8O$, Eurisotop D149F) was filling a hollow cylindrical sample holder (1 mm space between concentric cylinders). Incident energies $E_i$ of 3.70, 1.77, and 1.03 meV were selected by the chopper system running in an intermediate resolution setting, yielding full-width half-maximum (FWHM) resolutions of $\Delta\hbar\omega = 110$ µeV (3.0%$E_i$), $\Delta\hbar\omega = 39$ µeV (2.2%$E_i$), and $\Delta\hbar\omega = 18$ µeV (1.7%$E_i$), respectively, at the elastic line. Polarization, flipping, and analysis of the beam were achieved using a supermirror polarizer, precession coil flipper, and hyperpolarized ³He spin filter analyzer, respectively. The data collected with the flipper on and off were corrected for the finite polarizing efficiency of the instrument using a procedure detailed in Ref. 8, and the results were combined to yield the coherent and incoherent dynamical structure factors. Background correction was performed by using the measurement on the empty cell. Three temperatures, 175, 220, and 300 K, were investigated.

Neutron spin echo experiments¹⁵ were performed using the WASP instrument¹⁶ at the ILL. The WASP detector angle coverage in one position is 90°. It can be positioned to cover, for example,
5°–95° or 45°–135°. The incoming wavelength $\lambda$ can be chosen between 3 and 10 Å. In these experiments, $\lambda = 4$ Å corresponding to an incident energy of 5.11 meV was used to study a protonated sample (hTHF, C₄H₈O, Scharlab TE02252500) at 175 K. The detector was positioned at 5°–95°. The thickness of the sample between the walls of the hollow cylinder was 0.1 mm. The data were deconvolved by division using a resolution measurement on a 1 mm TiZr null alloy, which gives no Bragg peaks but only incoherent scattering. It was in the same hollow cylindrical shape as the sample.

Complementary x-ray experiments were performed with a RIGAKU SAXS instrument at the CFM in San Sebastian, Spain, in the temperature range between 170 and 340 K for both dTHF and hTHF samples.

### III. MAGNITUDES MEASURED BY NEUTRON SCATTERING EXPERIMENTS

#### A. General considerations

Neutron scattering results from the interactions of neutrons with atomic nuclei.17,18 These interactions are characterized by the scattering length $b_\alpha$, which can be positive, negative, or complex and depends on the isotope $\alpha$ considered ($\alpha = \text{H, D, }^{12}\text{C, }^{16}\text{O, etc.}$) as well as on the relative orientation of the neutron–nuclear spin pairs. The double differential scattering cross section $\partial^2\sigma/\partial\Omega\partial\hbar\omega$ measured in a “standard” neutron scattering experiment is the number of neutrons scattered into a solid angle comprised between $\Omega$ and $\Omega + d\Omega$ and which have experienced a change in energy $\hbar\omega$, with respect to the total number of incident neutrons.17 The difference between the wavevectors of the scattered ($\vec{k}$) and incident ($\vec{k}_0$) neutron determines the momentum transfer $\vec{Q}$, whose modulus, in elastic conditions, is given by $Q = 4\pi \sin(\theta/2)/\lambda$ ($\theta$: scattering angle; $\lambda = 2\pi/k_0$: incident wavelength). $\partial^2\sigma/\partial\Omega\partial\hbar\omega$ has a coherent and an incoherent contribution,

$$ \frac{\partial^2 \sigma}{\partial \Omega \partial \hbar \omega} = \frac{1}{4\pi\hbar} \frac{k}{k_0} \left[ \sum_{\alpha, \beta} \bar{b}_\alpha \bar{b}_\beta S_{\alpha\beta}^{\text{coh}}(\vec{Q}, \omega) + \sum_{\alpha} \overline{\Delta b_\alpha^2} S_{\alpha}^{\text{inc}}(\vec{Q}, \omega) \right], \quad (1) $$

where the indices $\alpha$ and $\beta$ run over all the different kinds of atoms in the sample ($\alpha, \beta \in \{\text{H, D, C, O, . . .}\}$). For simplicity of nomenclature and language, in the following, we will denote the double differential cross section as “I” (“intensity”) and assume an isotropic sample. Equation (1) can then be written as

$$ I_{\text{tot}}(Q, \omega) = I_{\text{coh}}(Q, \omega) + I_{\text{inc}}(Q, \omega), \quad (2) $$

where

$$ I_{\text{coh}}(Q, \omega) = \frac{1}{4\pi\hbar} \frac{k}{k_0} \sum_{\alpha, \beta} \bar{b}_{\alpha} \bar{b}_{\beta} S_{\alpha\beta}^{\text{coh}}(Q, \omega), \quad (3) $$

$$ I_{\text{inc}}(Q, \omega) = \frac{1}{4\pi\hbar} \frac{k}{k_0} \sum_{\alpha} \overline{\Delta b_{\alpha}^2} S_{\alpha}^{\text{inc}}(Q, \omega). \quad (4) $$

The Q and $\omega$-dependencies of the coherent and incoherent contributions are determined by the scattering functions. $S_{\alpha\beta}^{\text{coh}}(Q, \omega)$ is the coherent scattering function involving pairs of atoms of kind $\alpha$ and $\beta$ and $S_{\alpha}^{\text{inc}}(Q, \omega)$ is the incoherent scattering function of $\alpha$-nuclei. These are related, via Fourier transformation to the time domain, with the intermediate scattering functions $[S_{\alpha\beta}^{\text{coh}}(Q, t)$

and $S_{\alpha}^{\text{inc}}(Q, t)]$. The van Hove correlation functions $[G_{\alpha\beta}(r, t)$ and $G_{\alpha}^{\text{self}}(r, t)$, respectively] are obtained by Fourier transformation from the reciprocal to the real space. Although experimentally not accessible, these functions have an easy interpretation: in the classical limit, $G_{\alpha\beta}(r, t)$ can be written as

$$ G_{\alpha\beta}(r, t) = \left\{ \frac{1}{N} \sum_{i\alpha, j\beta}^{N_{\alpha}, N_{\beta}} \delta\{r - |\vec{r}_{i\alpha}(t) - \vec{r}_{j\beta}(0)|\} \right\}, \quad (5) $$

where $\vec{r}_{i\alpha}(t) [\vec{r}_{j\beta}(0)]$ is the position vector of the $i$th atom of kind $\alpha$ [jth atom of kind $\beta$] at time $t = t$ [time = 0] and the sum runs over all the different atoms of kinds $\alpha$ and $\beta$ [$N_\alpha$ ($N_\beta$): total number of atoms of kind $\alpha$ ($\beta$); $N = \sum_\alpha N_\alpha$]. Thus, $G_{\alpha\beta}(r, t)dr$ is the probability that, given a particle of kind $\beta$ at the origin at time $t = 0$, any particle of kind $\alpha$ is within a distance between $r$ and $r + dr$ at time $t$. On the other hand, the self-part of the van Hove correlation function $G_{\alpha}^{\text{self}}(r, t)$ is obtained by restricting the correlations considered in Eq. (5) to those relating the positions of a single particle of kind $\alpha$ at different times,

$$ G_{\alpha}^{\text{self}}(r, t) = \left\{ \frac{1}{N} \sum_{i\alpha}^{N_{\alpha}} \delta\{r - |\vec{r}_{i\alpha}(t) - \vec{r}_{i\alpha}(0)|\} \right\}. \quad (6) $$

$G_{\alpha}^{\text{self}}(r, t)$ is the Fourier transform of $S_{\alpha}^{\text{inc}}(Q, t)$ in space: incoherent scattering relates to single particle motions.

As shown in Eqs. (3) and (4), the weights of the coherent and incoherent contributions to the scattered intensity are determined by the scattering lengths of the isotopes involved. Table I shows the mean values of this parameter for the isotopes commonly present in organic materials and common liquids, e.g., water or THF, and for deuterium. In the table, we can also find other useful parameters, including the coherent and incoherent scattering cross sections of these isotopes, defined as $\sigma_{\text{coh}}^{\alpha} = 4\pi\bar{b}_{\alpha}^2$ and $\sigma_{\text{inc}}^{\alpha} = 4\pi\Delta b_{\alpha}^2$.

Most spectrometers work in the frequency domain, addressing the total scattered intensity [Eq. (2)]. We also note that the measured magnitudes are affected by the instrumental resolution function $R(Q, \omega)$ through convolution. $R(Q, \omega)$ can be determined from the intensity elastically scattered by a “frozen” sample.

The differential scattering cross sections $(\partial\sigma/\partial\Omega)_{\text{coh}} = I_{\text{coh}}(Q)$ and $(\partial\sigma/\partial\Omega)_{\text{inc}} = I_{\text{inc}}(Q) (= I_{\text{inc}})$ are obtained by integration of the corresponding double differential scattering cross sections [“intensities,” Eqs. (3) and (4)] over all possible energy transfers.

<table><caption>TABLE I. Values of the average scattering lengths $\bar{b}_\alpha$ (average over the possible relative orientation of the neutron–nuclear spin pairs), their squares $\bar{b}_\alpha^2$, their deviations $\overline{\Delta b_\alpha^2}$, and the coherent and incoherent scattering cross sections for different isotopes $\alpha$.</caption><thead><tr><th>Isotope α</th><th>bar<sub>α</sub>/fm</th><th>bar<sub>α</sub><sup>2</sup>/fm<sup>2</sup></th><th>bar<sub>α</sub><sup>2</sup>/fm<sup>2</sup></th><th>σ<sub>coh</sub><sup>α</sup>/barns</th><th>σ<sub>inc</sub><sup>α</sup>/barns</th></tr></thead><tbody><tr><td><sup>1</sup>H</td><td>-3.7406</td><td>13.992</td><td>638.78</td><td>1.758</td><td>80.27</td></tr><tr><td><sup>2</sup>H (D)</td><td>6.6710</td><td>44.502</td><td>16.322</td><td>5.592</td><td>2.05</td></tr><tr><td><sup>12</sup>C</td><td>6.6511</td><td>44.237</td><td>0</td><td>5.559</td><td>0</td></tr><tr><td><sup>16</sup>O</td><td>5.8030</td><td>33.675</td><td>0</td><td>4.232</td><td>0</td></tr></tbody></table>
They correspond to the $t = 0$ value of the $I_{coh}(Q, t)$ and $I_{inc}(Q, t)$ functions.

From Table I, it is clear that due to the large value of $\overline{\Delta b}_H^2$, in H-containing systems, the total signal measured is dominated by the incoherent scattering from hydrogens, revealing their self-motions; substituting H by D, this incoherent contribution is drastically reduced. The intensity scattered by fully deuterated samples has an important coherent contribution and is sensitive to the collective dynamic structure factor. However, the intensity measured on a deuterated sample in a “regular” neutron scattering experiment might be strongly contaminated by incoherent contributions, mainly at low Q-values.

## B. Polarization analysis

Contrary to coherent scattering, which does not induce a neutron spin flip, incoherent scattering of a sample constituted of aleatory oriented spins has a 2/3 probability to spin flip the scattered neutrons. Hence, the separation of incoherent and coherent nuclear scattering processes can be achieved using a polarized incident neutron beam and counting separately the neutrons scattered with and without spin-flip with regard to the incident beam polarization, obtaining two different Q-dependent intensities: the spin flip intensity $I_{SF}(Q, \omega)$ and the non-spin flip intensity $I_{NSF}(Q, \omega)$,

$$I_{SF}(Q, \omega) = \frac{2}{3} I_{inc}(Q, \omega), \qquad (7)$$

$$I_{NSF}(Q, \omega) = I_{coh}(Q, \omega) + \frac{1}{3} I_{inc}(Q, \omega). \qquad (8)$$

From these equations, $I_{coh}(Q, \omega)$ and $I_{inc}(Q, \omega)$ can be deduced.

## C. The neutron spin echo signal

Polarized neutrons are also involved in the experiments performed by the neutron spin echo (NSE) technique.¹⁹ NSE instruments are very particular since they access the scattering functions directly in the time domain, being the signal measured

$$I_{NSE}(Q, t) = \frac{I_{coh}(Q, t) - \frac{1}{3}I_{inc}(Q, t)}{I_{coh}(Q, t=0) - \frac{1}{3}I_{inc}(Q, t=0)}, \qquad (9)$$

where $I_{coh}(Q, t)$ and $I_{inc}(Q, t)$ are, respectively, the Fourier transforms of $I_{coh}(Q, \omega)$ and $I_{inc}(Q, \omega)$ given by Eqs. (3) and (4) (i.e., the scattered intensities, proportional to the scattering functions weighted by the corresponding scattering lengths). NSE presents several advantages with respect to “regular” instruments: first, directly accessing the time domain allows easy deconvolution of the instrumental function by simple division. Second, NSE provides the best energy resolution of all neutron scattering instruments. However, we remark that it does not allow the separation of coherent and incoherent dynamic contributions but delivers the mixed signal expressed by Eq. (9).

## D. Scattering by THF

Since $\overline{\Delta b}_C^2 = \overline{\Delta b}_O^2 = 0$, the incoherent intensity scattered by THF uniquely arises from its hydrogens, i.e., from the protiums in the

protonated sample hTHF or from its deuterons in the deuterated sample dTHF. Thus, $I_{inc}^{THF}(Q, \omega) \propto S_H^{inc}(Q, \omega)$. The coherent scattering involves pair correlations of all different nuclei weighted by the products of the corresponding (non-zero) averaged scattering lengths, $I_{coh}(Q, \omega) \propto \sum_{\alpha, \beta} \overline{b}_\alpha \overline{b}_\beta S_{\alpha\beta}^{coh}(Q, \omega)$. Since $\overline{b}_D$ and $\overline{b}_C$ are very similar and $\overline{b}_O$ is not very different from them (see Table I), for dTHF, we can approximate $I_{coh}^{dTHF}(Q, \omega) \propto \sigma_{coh} S_{coh}^{coh}(Q, \omega)$. Here, we have introduced the coherent scattering cross section of the sample $\sigma_{coh} = 4\pi \sum_\alpha \overline{b}_\alpha^2 = \sum_\alpha \sigma_{coh}^\alpha$, where the sum runs over all the nuclei $\alpha$ of the molecule, and the dynamic structure factor of the sample $S_{coh}^{coh}(Q, \omega)$, where all the atomic pair correlations are equally weighted. In the static case, this is the total static structure factor $S(Q)$.

It is common to express some measured magnitudes in terms of the scattering cross sections of the sample $\sigma_{coh}$ above introduced and $\sigma_{inc} = \sum_\alpha \sigma_{inc}^\alpha$ (for dTHF, $\sigma_{inc} = 16.41 \text{ b/mol}$ and $\sigma_{coh} = 71.21 \text{ b/mol}$). In particular, $I_{inc}(Q) = I_{inc} = \sigma_{inc}/(4\pi v)$ (Q-independent) and $I_{coh}(Q \to \infty) = \sigma_{coh}/(4\pi v)$, where $v$ is the molecular volume, which can be assumed to be independent of isotopic labeling to a good approximation. We can deduce $v = 1.36 \times 10^{-22} \text{ cm}^3$ at 300 K from the density of hTHF (0.880 g/cm³, Ref. 20). Thus, at this temperature, $I_{inc}^{dTHF} = 0.0096 \text{ cm}^{-1}$ and $I_{coh}^{dTHF}(Q \to \infty) = 0.0417 \text{ cm}^{-1}$. Another useful magnitude to define is the scattering length density $\rho = \sum_\alpha \overline{b}_\alpha^2/v$, which for dTHF at 300 K is $6.312 \times 10^{10} \text{ cm}^{-2}$.

## IV. RESULTS

Figure 1 shows the ratio between coherent and incoherent differential cross sections $I_{coh}(Q)/I_{inc}$ deduced from the PLET experiments on dTHF at the three temperatures investigated. They have been calculated from the total amplitudes of the model functions employed to describe the spectra in order to avoid incomplete integration over the limited accessible dynamic window. As argued above, $I_{coh}(Q) \propto S(Q)$ is a good approximation since all pair correlations are nearly equally weighted in this sample (see Table I). The structure factor shows a main peak centered at

<figure><img src="image_12.png" /><figcaption>FIG. 1. Ratio between coherent and incoherent differential scattering cross sections obtained for dTHF from PLET at the three temperatures investigated. The arrows indicate the expected asymptotic ($Q \to 0$ at 300 K and $Q \to \infty$) values.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>FIG. 2. Temperature dependence of (a) the average inter-molecular distance as determined from the first peak position of S(Q) and (b) the compressibility with respect to its value at 300 K. The results are obtained for dTHF from PLET (filled circles) and from SAXS (empty circles) and for hTHF from SAXS (squares). The lines are linear (a) and parabolic (b) fits.</figcaption></figure>

$Q_{max} \approx 1.45 \text{ Å}^{-1}$, in good agreement with the results previously reported.²¹ This position slightly shifts with $T$, as expected for a peak arising from correlations of inter-molecular origin.²² As shown in Fig. 2(a), the deduced inter-molecular distance in the Bragg approximation $d = 2\pi/Q_{max}$ increases linearly with $T$ in good agreement with the counterpart results obtained from the x-ray experiments. On the other hand, the neutron results shown in Fig. 1 might

be compatible with the expected theoretical asymptotic value of $I_{coh}(Q \to \infty)/I_{inc} = \sigma_{coh}/\sigma_{inc} = 4.34$ indicated by the arrow. Below $Q_{max}$, the coherent intensity drastically decreases with decreasing $Q$, and below $Q \approx 1 \text{ Å}^{-1}$, it becomes approximately constant. In liquids, the coherent intensity in the low-Q limit is determined by the isothermal compressibility $\beta_T$, as $I_{coh}(Q \to 0) = \rho\beta_T k_B T$ with $k_B$ the Boltzmann constant. The reported values for $\beta_T$ in hTHF around RT are $1.0 \times 10^{-9} \text{ Pa}^{-1}$, Ref. 20, and $8.09 \times 10^{-10} \text{ Pa}^{-1}$, Ref. 23. This value is not expected to significantly depend on isotopic labeling (see, e.g., Ref. 24 for the case of water). Using the above obtained value for the scattering length density $\rho$, for dTHF, we deduce $I_{coh}(Q \to 0)/I_{inc} \approx 1.4-1.7$ around 300 K, in excellent agreement with our results at this temperature (see Fig. 1). The decrease in compressibility with decreasing temperature leads to coherent scattering becoming similar to—or even lower than—incoherent scattering at the lower temperatures investigated. As shown in Fig. 2(b), there is a good agreement between the PLET and the SAXS results obtained in this study regarding the $T$-dependence of the compressibility deduced from the $Q \to 0$ intensities. Nearly indistinguishable SAXS results on deuterated and protonated THF indicate no appreciable influence of isotopic labeling on this dependence, which can be well described by $\beta_T(T)/\beta_T(300 \text{ K}) = 1.04 - 0.00655T + 2.16 \times 10^{-5}T^2$ for the deuterated sample.

From the above results, we can easily appreciate the difficulties in accessing coherent scattering of dTHF in the low-Q range of the ILS, especially at low temperatures. However, separating coherent and incoherent dynamic contributions is possible with PLET, even in such a challenging situation. Figure 3 shows PLET results for the intermediate temperature investigated at two representative Q-values: in the ILS and at $Q_{max}$. Since no scaling factors have been applied, here, one can appreciate which contribution dominates the different parts of the total spectra at these Q-values. The broadening of the scattering function in the frequency domain (characterized, e.g., by the half-width at half-maximum) is related to the inverse of the characteristic time of the underlying dynamic process.

<figure><img src="image_14.png" /><figcaption>FIG. 3. PLET results on dTHF at 220 K for $Q = 0.55 \text{ Å}^{-1}$ (a), i.e., in the intermediate length scales region, and for $Q = 1.45 \text{ Å}^{-1}$ (b), i.e., close to $Q_{max}$ (see Fig. 1). The blue filled circles correspond to coherent scattering, and the red empty circles correspond to incoherent scattering. The black lines are fits by the proposed model, convoluted with the instrumental resolution (dashed lines).</figcaption></figure>
A first sight on the spectra reveals very different average characteristic times for self- and collective dynamics in both cases: at ILS, coherent scattering shows much faster dynamics than incoherent, and in the central part, the total signal (coherent plus incoherent) is completely dominated by the slow incoherent contribution. We recall that in a liquid, the quasielastic broadening at low-Q-values is mainly reflecting the self-diffusion. On the contrary, at $Q_{max}$, the collective dynamics—determining the central part of the total spectrum—is clearly slower than the self-dynamics. Although spectral mode contributions can, in principle, be disentangled by mode distribution analysis, as, e.g., that proposed in Ref. 25, a univocal identification of these contributions is impossible without PA.

The susceptibility representation $\chi''(Q, \omega)$ can be very useful to identify the presence of different dynamical contributions to the spectra. In this representation, a dynamic process appears as a peak, whose characteristic time can be identified as the inverse of the frequency of the maximum. The imaginary part of the coherent susceptibility is obtained as $\chi''_{coh}(Q, \omega) = S^{coh}(Q, \omega)/[1 + n(\omega)]$, where $n(\omega)$ is the Bose factor $n(\omega) = (e^{\hbar\omega/k_B T} - 1)^{-1}$. Some representative examples obtained from the system energy-loss side of the spectra are shown in Fig. 4. In the accessed dynamic window, we can identify at least two main processes, which we shall refer to as relaxational (dominating at lower frequencies, corresponding to slower characteristic times) and vibrational (at higher frequencies, faster times) contributions. We have thus used the first approach to describe the results,

$$\chi''_{coh}(Q, \omega) = S(Q)\{[1 - C(Q)]\chi''_{V}(Q, \omega) + C(Q)\chi''_{R}(Q, \omega)\}, \quad (10)$$

with $\chi''_{V}(Q, \omega)$ the vibrational contribution—weighted by the amplitude $[1 - C(Q)]$—and $\chi''_{R}(Q, \omega)$ the relaxational contribution—weighted by $C(Q)$. For the latter, we have assumed the simplest process, namely a Lorentzian with characteristic time $\tau$,

<figure><img src="image_15.png" /><figcaption>FIG. 4. Imaginary part of the susceptibility for coherent scattering obtained from PLET results with $E_i = 1.77$ meV on dTHF at 220 K and the Q-values indicated. The error bars are shown for two of the Q-values. The dashed lines are fits considering a vibrational and one Lorentzian function for the relaxational contribution; the solid lines are the fits by the proposed model. The dashed area shows the energy range below the resolution.</figcaption></figure>

$\chi''_{R}(Q, \omega) = \omega\tau/[1 + (\omega\tau)^2]$ (corresponding to a single exponential in the time domain), whereas a damped resonance term has been used for the vibrational contribution,

$$\chi''_{V}(Q, \omega) = \frac{k_{0}\omega_{0}^{2}\omega}{[(\omega_{0}^{2} - \omega^{2})^{2} + (k_{0}\omega)^{2}]} \quad (11)$$

The parameters $\omega_0$ and $k_0$ are, respectively, the frequency and damping of the resonance. The obtained fits are shown in Fig. 4 as dashed lines. We used $\omega_0/(2\pi) = 1.52$ THz and $k_0/(2\pi) = 19.6$ THz, and the characteristic relaxation times are plotted in Fig. 5. A strong modulation by $S(Q)$ can be observed for these relaxation times, tending to an approximately constant value at low $Q_s$. We note that in this preliminary analysis, no convolution with the experimental resolution has been considered. This is not a problem when the processes have characteristic times shorter than those corresponding to the instrumental resolution. However, when $\tau$ approaches the resolution, this kind of analysis leads to apparent values of the times shorter than the actual ones, and that depend on the incident energy of the neutrons. This is the situation of the results obtained at 175 K around $Q_{max}$ (see Fig. 5). On the other hand, from Fig. 4, we can appreciate that the description is reasonable for low Q-values; however, approaching $Q_{max}$, the spectra demand for a more complex model function. The same kind of approach was applied to the incoherent contribution. The resulting times for the intermediate temperature are included in Fig. 5 as empty symbols. Their Q-dependence is slightly more pronounced than a $\tau \propto Q^{-2}$-law corresponding to simple diffusion below $Q_{max}$. They tend to become Q-independent above it.

From the above first approach to the phenomenology involved, we can deduce that (i) at ILS, coherent and incoherent scattering functions decay with very different time scales; the characteristic time for incoherent scattering displays the Q-dependence expected

<figure><img src="image_16.png" /><figcaption>FIG. 5. Characteristic times obtained from the fit of the relaxational part of the coherent susceptibility (dynamic structure factor) of dTHF to a single Lorentzian (single exponential) as a first approach (filled symbols). Different colors correspond to the different temperatures investigated. For 220 K, the times obtained for the incoherent counterpart are included with empty symbols. The dotted line is a $\tau \propto Q^{-2}$-law. Different symbols correspond to different incident energies (circles: 1.03 meV; squares: 1.77 meV; and diamonds: 3.70 meV).</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>FIG. 6. Characteristic times obtained for D₂O at 300 K from PLET results⁹ from a similar analysis as that leading to the times for dTHF shown in Fig. 5. Different symbols correspond to different incident energies of the neutrons (circles: 1.05 meV; squares: 1.81 meV; and diamonds: 3.84 meV).</figcaption></figure>

for a diffusive process. However, coherent scattering presents nearly Q-independent values for the characteristic time. Thereby, the relaxation time of the self-motions is markedly slower and this difference is amplified toward lower Q-values; (ii) within the uncertainties, at ILS, besides a vibrational contribution, the susceptibility corresponding to the relaxational part of the coherent spectrum can be rather well described by a single Lorentzian (Fig. 4) (equivalently, a single exponential in the time domain); (iii) this kind of description ceases to be satisfactory at higher Q-values, suggesting an additional mechanism of relaxation; and (iv) the steeper Q-dependence of the incoherent characteristic time with respect to the Q⁻²-law in the range 0.6 Å⁻¹ ≲ Q ≲ Qmax also points to a contribution superimposed to the simple diffusion for the relaxational part of the spectrum.

We find a strong analogy with the water results obtained by applying a similar phenomenological approach (see Fig. 6). We recall that a third component, in addition to vibrations and diffusion, was considered in a model proposed to describe water motions in our previous studies.⁸,¹⁰,¹¹ Its bases were first laid for describing the merging of the α and β-relaxations in polymers.⁹ This model was also used by other authors on water dynamics.²⁶ We will see whether it is also applicable to the case of THF. In the following, we briefly sketch this model.

## V. CONVOLUTION MODEL

The proposed model considers both a vibrational and a relaxational contribution. The relaxational part is assumed to consist of two processes—a diffusive and a Q-independent one—taking place simultaneously. Accordingly, the corresponding scattering functions in the frequency domain are combined with a convolution. The formulation of the model is simpler in the time domain, where the convolution becomes a product. In this domain, the function F(Q, t)—representing either the intermediate incoherent

scattering function S<sup>inc</sup>(Q, t) or the normalized dynamic structure factor S<sup>coh</sup>(Q, t)/S(Q)—is expressed as

$$F(Q, t) = [1 - C(Q)]F_V(Q, t) + C(Q)F_R(Q, t), \quad (12)$$

with F<sub>V</sub>(Q, t) the vibrational contribution—weighted by the amplitude [1 − C(Q)]—and F<sub>R</sub>(Q, t) the relaxational contribution—weighted by C(Q). F<sub>R</sub>(Q, t) is assumed to be the combination of two independent processes,

$$F_R(Q, t) = F_d(Q, t)F_c(Q, t), \quad (13)$$

namely a diffusive contribution F<sub>d</sub>(Q, t), given by

$$F_d(Q, t) = e^{-\frac{t}{\tau_d}}, \quad (14)$$

with τ<sub>d</sub>(Q) a diffusive time, and a Q-independent mode contribution F<sub>c</sub>(Q, t). This is parameterized as

$$F_c(Q, t) = A(Q) + [1 - A(Q)]e^{-\frac{t}{\tau_c}}, \quad (15)$$

where [1 − A(Q)] is the amplitude of the Q-independent mode [A(Q) is the elastic incoherent structure factor (EISF) in the incoherent case] and τ<sub>c</sub><sup>o</sup> is a Q-independent relaxation time. Then,

$$F_R(Q, t) = [1 - A(Q)]e^{-\frac{t}{\tau_c}} + A(Q)e^{-\frac{t}{\tau_d}}, \quad (16)$$

where we have defined the effective time τ<sub>c</sub> as

$$\tau_c \equiv \left( \frac{1}{\tau_c^o} + \frac{1}{\tau_d} \right)^{-1}. \quad (17)$$

The average relaxation time of the relaxational contribution is

$$\langle \tau \rangle = \int_{0}^{\infty} F_{R}(Q, t) dt = [1 - A(Q)] \tau_{c} + A(Q) \tau_{d}. \quad (18)$$

Introducing Eq. (16) in Eq. (12), we obtain

$$F(Q, t) = [1 - C(Q)]F_V(Q, t) + C(Q)[1 - A(Q)]e^{-\frac{t}{\tau_c}} \\ + C(Q)A(Q)e^{-\frac{t}{\tau_d}}. \quad (19)$$

The corresponding scattering function in the frequency domain, assuming again a damped resonance term for the vibrational contribution, is

$$F(Q, \omega) = [1 - C(Q)] \frac{k_o \omega_o^2}{\pi [(\omega_o^2 - \omega^2)^2 + (\omega k_o)^2]} \\ + C(Q)[1 - A(Q)] \frac{\tau_c}{\pi [1 + (\omega \tau_c)^2]} \\ + C(Q)A(Q) \frac{\tau_d}{\pi [1 + (\omega \tau_d)^2]}, \quad (20)$$

where F(Q, ω)—as F(Q, t)—is a normalized function. When fitting this function to the experimental data, it is affected by a global intensity prefactor: I<sub>o</sub>I<sub>inc</sub> for incoherent scattering and I<sub>o</sub>I<sub>coh</sub>(Q) for coherent scattering, where I<sub>o</sub> is a constant related to the instrumental conditions. In the following, we will denote with a sub- or superscript “inc” the parameters for incoherent scattering; no specification means the coherent case. It is worth recalling that in the
<figure><img src="image_18.png" /><figcaption>FIG. 7. Coherent scattering function of dTHF at 220 K and the different Q-values indicated. The incident energy of the neutrons used in each case is indicated. The solid lines are fits with the proposed model, and the components are shown as dashed (diffusional), dashed-dotted (effective), and dotted (vibrational) lines.</figcaption></figure>

framework of this model, the Q-independent mode is contained
in the actually observed Lorentzian (or exponential, respectively)
contribution with characteristic time $\tau_c$, which *does* depend on
Q as dictated by Eq. (17). This means the effective time $\tau_c$ is
that experimentally observed for the contribution arising from the
Q-independent mode, modified by diffusion.

A. Model fit for coherent scattering of dTHF

We first note that for Q ≲ 1 Å⁻¹, where S(Q) is flat, besides a
vibrational contribution, the relaxational part of the spectra can be
reasonably well described by a single Lorentzian (Fig. 4). Since in
this Q-range the diffusive time is very slow—much longer than the
observed collective time—in the framework of the proposed model
the characteristic time of this single Lorentzian should be τ_c, which,
through Eq. (17), is just the characteristic time of the Q-independent

mode $\tau_c^o$. Thus, we fixed the values of $\tau_c^o$ as those deduced from
the previous single Lorentzian fits of the susceptibility in the low
Q-range (see Fig. 5). These values are $\tau_c^o \approx 7, 2.8$, and 1.3 ps for 175,
220, and 300 K, respectively. Using the same vibrational contribu-
tion as before, we fitted the full expression [Eq. (20)] convoluted with
the instrumental resolution to the $I_{coh}(Q, \omega)$ results. The descrip-
tions obtained are very good, as shown in Figs. 3 and 7 for several
Q-values at 220 K, and in Fig. 8 for a representative Q-value in
the ILS region and the three temperatures investigated. From these
fits, we obtained the Q-dependent values of the rest of the para-
meters involved, $\tau_d$, A, and C, and the global intensity proportional
to $I_{coh}(Q)$. The upper panels in Fig. 9 show the results correspond-
ing to the characteristic times $\tau_d$. The value of the Q-independent
mode characteristic time $\tau_c^o$ used in the fits and the effective time
$\tau_c$ obtained through Eq. (17) are also shown for comparison in
these panels. Relevant magnitudes are also the relative amplitudes
of the three contributions to the coherent scattering function. They

<figure><img src="image_19.png" /><figcaption>FIG. 8. Coherent scattering function of dTHF at Q = 0.85 Å⁻¹, representative of the ILS region, and the three temperatures investigated: 175 K (a); 220 K (b); and 300 K (c). The incident energy of the neutrons used is 1.77 meV. The lines are fits with the proposed model.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>FIG. 9. Upper panels: characteristic times $\tau_d$ (filled symbols) and $\tau_c$ (empty symbols) obtained from the application of the model to the coherent scattering function of dTHF. The circles correspond to $E_i = 1.03$ meV, the squares correspond to 1.77 meV, and the diamonds correspond to 3.7 meV. The horizontal lines show the value of $\tau_c$. The dashed lines are the self-diffusion characteristic time for deuterons $D^{-1}Q^{-2}$ deduced from the analysis of $I_{inc}(Q, \omega)$ (or $I_{inc}(Q, t)$), and the dotted lines display the de Gennes narrowing prediction $D^{-1}Q^{-2}S(Q)$. Lower panels: Q-dependence of the relative contributions of the three processes to the coherent structure factor: vibrational (lower yellow area), Q-independent mode (middle green area), and diffusive (upper red area) contributions. These areas are delimited by (1 – C) (up-triangles) and (1–AC) (down-triangles); the values represented are the average of the results obtained from the different values of $E_i$ employed. The results correspond to 175 K [(a) and (d)]; 220 K [(b) and (e)]; and 300 K [(c) and (f)].</figcaption></figure>

are represented in the lower panels of Fig. 9. We remind you that they are given by AC for the diffusive component, $C(1-A)$ for the Q-independent mode, and $1-C$ for the vibrational contribution [Eqs. (19) and (20)].

## B. Model fit for incoherent scattering of dTHF and hTHF

To evaluate the incoherent scattering, we used a different strategy. Due to the long diffusion times observed at low Q-values, we exploited the excellent resolution and wide Q-range accessed by the NSE technique to properly determine the $\tau_d$-values and, from them, the rest of the parameters involved in the model. We focused on the lowest temperature, 175 K, for which the signal is well centered in the WASP dynamic window. Since NSE does not separate coherent and incoherent intensities, we addressed the incoherent scattering function of hydrogen atoms in THF on a protonated sample. As explained above, in this case, the incoherent scattering has exactly the same information—besides weak isotopic effects—as in dTHF [$I_{inc}^{THF}(Q, \omega) \propto S_H^{inc}(Q, \omega)$, equivalently $I_{inc}^{THF}(Q, t) \propto S_H^{inc}(Q, t)$], but the incoherent contribution is overwhelming with respect to the

coherent one ($\sigma_{coh}/\sigma_{inc} = 0.063$ for hTHF). The NSE signal [Eq. (9)] is clearly dominated by the incoherent contribution.

Figure 10 shows the WASP results on hTHF at 175 K. As to a good approximation $I_{NSE}^{hTHF}(Q, t) = S_H^{inc}(Q, t)$, we fitted Eq. (19) to the WASP results on hTHF. The vibrational contribution is very fast for the NSE window, i.e., $F_V(Q, t)$ has already decayed to 0 at shorter times than those experimentally accessed. Therefore, the first term of the RHS of Eq. (19) is 0—vibrations are only “visible” through the amplitude factor $C_{inc}$. The excellent quality of WASP data on hTHF in the range $Q \lesssim 1.2 \AA^{-1}$ allowed determining the parameters involved in the model for self-motions at 175 K ($C_{inc}$, $A_{inc}$, and $\tau_d^{inc}$) with high accuracy. We obtained very good fits (see Fig. 10) by fixing $\tau_d^{inc} = D^{-1}Q^{-2}$ with $D = 0.023 \AA^2/\text{ps}$ and a value for $\tau_c^o = 7$ ps. Notably, this value agrees very well with that obtained for the collective counterpart $\tau_c^o$ from the first analysis of $\chi''_{coh}(Q, \omega)$ at ILS (see Fig. 5), and that has been used to describe with the model the coherent scattering function measured on dTHF by PLET. The amplitude parameters $A_{inc}$ and $C_{inc}$ are represented in Fig. 11(a).

For Q-values higher than $1.2 \AA^{-1}$, the incoherent scattering function of hydrogens decays too fast to be analyzed in the NSE
<figure><img src="image_21.png" /><figcaption>FIG. 10. NSE results on hTHF at 175 K and the Q-values indicated. The solid lines are fits with the proposed model.</figcaption></figure>

window; there are also other problems related to coherent contribu-
tions and a shadow of the instrument that prevent extracting further
information from the WASP results. We, thus, applied the model
to describe the incoherent scattering of dTHF at 175 K, fitting the
theoretical scattering function [Eq. (20)] convolved with the instru-
mental resolution to the PLET results, and fixing the values of the
diffusion coefficient and τ<sup>o</sup><sub>c,inc</sub> as obtained from the WASP analy-
sis. For the vibrational contribution, we fixed that previously used
to describe the susceptibility data. Thus, only the amplitude para-
meters were allowed to vary in the fits. As shown in Fig. 12, the
description obtained—now extending the Q-range investigated up
to much higher Q-values—is excellent, taking into account that most
of the parameters were fixed and the scatter of the experimental data.
We also find an impressive agreement of the amplitude parameters
with the WASP results (see Fig. 11) in the overlapping Q-range. This
confirms that the isotopic effect is very weak, proves the robustness
of the analysis procedure, and supports the validity of the proposed
model in a large Q-range, at least at this low temperature.

The model was also applied to the incoherent scattering func-
tion of deuterons at higher temperatures. In order to reduce the
number of free parameters, we considered the same functional form
for the vibrational contribution in all cases. The fittings with free
τ<sup>o</sup><sub>c,inc</sub> yielded values scattered around those found for the collective
relaxation time at ILS τ<sup>o</sup><sub>c</sub> (see Fig. 5). Given the coincidence of these
times for 175 K—where τ<sup>o</sup><sub>c,inc</sub> could be very accurately determined
from the hTHF results—we decided to fix the asymptotic Q → 0
limit of the collective time (2.8 ps at 220 K and 1.3 ps at 300 K) as
the value of τ<sup>o</sup><sub>c,inc</sub> in the fit of the incoherent scattering function. In
this way, we obtained good descriptions of the experimental results
(see Fig. 12) with diffusion coefficients of 0.06 and 0.25 Å<sup>2</sup>/ps for 220
and 300 K, respectively. The values obtained for the amplitudes are
shown in Fig. 11.

VI. DISCUSSION

A. Self-motions

In the framework of the model proposed, Hs undergo
simultaneous vibrations and diffusion and participate in a local
motion. Regarding the vibrational component, for all temperatures,
the amplitude C<sub>inc</sub> can be reasonably described in the whole
Q-range accessed by a Lamb–Mössbauer (LM) factor-like behavior:
C<sub>inc</sub> = exp[-⟨u<sup>2</sup>⟩Q<sup>2</sup>/3]. u is the vibrational displacement of the scat-
tering center in each of the three dimensions. The values obtained
for √⟨u<sup>2</sup>⟩ are 0.8, 0.9, and 1.1 Å for 175, 220, and 300 K, respec-
tively. These displacements are rather large, but we have to take into
account that, at these temperatures, the sample is in the liquid state.

Concerning diffusion, the obtained diffusion coefficients are
represented in Fig. 13 as a function of the inverse temperature.
The value obtained at 175 K for hTHF, together with the litera-
ture results<sup>27,28</sup> also on protonated samples, shows an Arrhenius
dependence D = D<sub>∞</sub> exp(-E<sub>a</sub>/kT) with D<sub>∞</sub> = 10 Å<sup>2</sup>/ps and an acti-
vation energy of E<sub>a</sub> = 91 meV (2.1 Kcal/mol). The isotopic effect
on the diffusion coefficient of THF was investigated in Ref. 27. The
results obtained in that study on the deuterated sample are also dis-
played in Fig. 13. The values show a small influence of deuteration

<figure><img src="image_22.png" /><figcaption>FIG. 11. Amplitude parameters *A*<sub>inc</sub> (circles) and *C*<sub>inc</sub> (squares) for the incoherent scattering at 175 K (a), 220 K (b), and 300 K (c). The solid symbols correspond to the analysis of the PLET results on dTHF, and the empty symbols correspond to the analysis of the WASP results on hTHF. The solid lines are fits by a LM factor, and the dashed lines are descriptions of *A*<sub>inc</sub> by Eq. (21) with *R* = 1.9 Å.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>FIG. 12. Incoherent scattering function of dTHF obtained from PLET at 175 K (a)–(c), 220 K (d)–(f), and 300 K (g)–(i) for Q = 0.85 Å⁻¹ with Eᵢ = 1.77 meV (a), (d), and (g), Q = 1.05 Å⁻¹ with Eᵢ = 1.03 meV (b), (e), and (h), and Q = 1.65 Å⁻¹ with Eᵢ = 3.70 meV (c), (f), and (i). The solid lines are fits with the model.</figcaption></figure>

(6%–9% smaller values of *D* for dTHF than for hTHF). The value obtained in this study from the incoherent scattering function of the deuterated sample at 300 K agrees well with the reported ones. For 220 K, a somewhat more pronounced isotopic effect would be deduced from our result, but, in view of the uncertainties involved in its determination, the agreement with the rest of the results is reasonable.

The third dynamic process is the local one. Its characteris-
tic time has been identified, within the uncertainties, with that of
its coherent counterpart τ_c°. This time follows an Arrhenius-like
temperature dependence (see Fig. 14): τ_c° = τ_∞ exp (E_a/kT) with

τ∞ = 0.13 ps and E_a = 59 meV (1.4 Kcal/mol). Regarding the rela-
tive amplitude of this process for incoherent scattering, we recall that
A_inc is the EISF characterizing the geometry of a localized motion.
Unfortunately, in the present case, the statistics of PLET results on
incoherent scattering prevent the determination of the values of A_inc
with high accuracy at high Qs (where the highest energy data are
used). In fact, for 175 K, we could only infer that they are smaller
than about 0.3 above Qmax. In the first approach, the LM expres-
sion might also be used to describe A_inc in the low Q-region (see
Fig. 11). The values deduced for √⟨u²⟩ from A_inc are of about 2 Å,
pointing to a rather large amplitude for the underlying localized
<figure><img src="image_24.png" /><figcaption>FIG. 13. Inverse temperature dependence of the self-diffusion coefficients obtained from WASP (filled square) and reported in the literature on hTHF (squares$^{27}$; crossed squares$^{28}$). The whole set of data is described by the Arrhenius solid line. The circles show the results obtained on dTHF in this work (solid circles) and in Ref. 27 (empty circles).</figcaption></figure>

motion. Its geometry can, in principle, be deduced from the values of $A_{inc}$ (equivalent to the EISF). At 220 K—the temperature at which the values of the parameters could be determined with better accuracy—the EISF corresponding to jumps among four equivalent sites in a circle of radius R,

$$EISF = \frac{1}{4} \left[ 1 + 2 \frac{\sin(\sqrt{2QR})}{\sqrt{2QR}} + \frac{\sin(2QR)}{2QR} \right], \quad (21)$$

can reasonably give an account for the behavior, with a value of R = 1.9 Å (Fig. 11). This geometry could be sensible since the diameter of this circle (3.8 Å) is very close to the average distance between hydrogen atoms bonded to two carbons that are not

<figure><img src="image_25.png" /><figcaption>FIG. 14. Inverse temperature dependence of the average relaxation time at the structure factor peak (squares) and $\tau_c^0$ (circles). The lines are Arrhenius fits, leaving free the activation energy for $\tau_c^0$ (solid line) and imposing it to that found for the self-diffusion coefficient (dashed line) and that reported for $\eta/T^{29}$ in the case of $(\tau)(Q_{max})$.</figcaption></figure>

directly connected in the THF molecule (3.6 Å).$^{21}$ We note, however, that extracting definite conclusions about the geometry of the local motions with the present data is unfortunately not possible. On the one hand, the uncertainty of $A_{inc}$ in the high-Q range is large. Measurements on a fully protonated sample would provide better statistics for its determination. On the other hand, the model assumes that diffusion and localized motions are completely independent, and in its application, we imposed that the diffusive time $\tau_d^{inc}$ follows a continuous law $\tau_d^{inc} = D^{-1}Q^{-2}$. It is expected that this law breaks down at high Qs when the elementary process underlying diffusion becomes visible (e.g., as captured by jump-diffusion models$^{30-32}$). Actually, the characteristic time for hydrogen self-motions as obtained from the phenomenological description of the susceptibility tends to flatten above $Q_{max}$ (see the case of 220 K in Fig. 5). On-going MD simulations and/or time-of-flight experiments on a protonated sample will allow us to determine the precise geometry of these local motions and the validity of the assumptions in the application of the model.

## B. Collective dynamics

In general, density fluctuations in liquids at inter-molecular distances ($Q \sim Q_{max}$) use to decay due to diffusive processes. However, in the ILS region also explored here, we have found that the amplitude of these processes is going to zero as Q decreases and the characteristic time increases as well. Then, apart from the short-time vibrational excitations (likely hydrodynamic-like modes), the only channel for the decay of density fluctuations in this regime is the local processes mentioned above. As soon as Q increases toward $Q_{max}$, one should expect a crossover regime where both local processes and diffusion contribute to the decay of density fluctuations. Thereby, in principle, it is not surprising that we have also identified three different processes for collective dynamics (density fluctuations). Let us see whether the parameter values found agree with the qualitative picture described above. First, we can consider the relative importance of the three processes as a function of Q and temperature (lower panels of Fig. 9). The relative amplitude of the collective vibrational mode shows a pronounced minimum at $Q_{max}$; in the ILS region, this contribution is the most important one at 175 K and tends to slightly decrease with increasing temperature. Above $Q_{max}$, it shows the opposite tendency. The Q-independent mode has a relative amplitude that shows a local minimum at $Q_{max}$ that becomes more pronounced with increasing temperature. At ILS, the relative amplitude of this process gains importance with increasing temperature, changing from about 0.3 at 175 K up to about 0.5 at 300 K. Finally, the diffusive contribution has a negligible amplitude at ILS at 220 and 300 K, although at 175 K, some reminiscence of the diffusive process could be inferred. As expected, this contribution has a pronounced maximum at $Q_{max}$, and in the neighborhood of the peak it increases with increasing temperature.

Thus, the collective dynamics of dTHF at ILS [when $S(Q)$ is constant] is dominated by vibrations at low temperatures, with some contribution from the Q-independent mode. This process gains importance with increasing temperature, and at 300 K, its relative amplitude becomes about 50%. When exploring shorter length scales and $S(Q)$ increases above the constant value characteristic of the ILS and dictated by the compressibility, diffusion starts to
contribute and becomes the dominant dynamic process at the struc-
ture factor peak, governing, thus, the decay of the inter-molecular
correlations. Beyond Qmax, the amplitude of the diffusive contri-
bution to the spectra is suppressed again (Fig. 9). Therefore, the
Q-independent mode—affected by diffusion—and, mainly, collec-
tive vibrations, take the dominant role. Thus, at ILS as well as at very
local length scales (corresponding to Q-values higher than Qmax), the
diffusive process shows up in the collective spectrum mainly in an
“indirect” way, namely through its influence on the Q-independent
mode, imprinting a Q-dependence in the observed characteristic
time τc [Eq. (17)], as we discuss next.

Moving to the characteristic times, in the upper panels of
Fig. 9, we have represented the time τ_c^0 corresponding to the
Q-independent mode and the effective time τ_c given by Eq. (17). We
recall that the latter is the characteristic time of the relaxational part
of the spectrum corresponding to the Q-independent mode modi-
fied by the simultaneous occurrence of diffusion. τ_c tends to τ_c^0 in
the Q → 0 limit. With increasing Q, it first decreases, reaching a min-
imum in the Q-region where S(Q) starts to increase, followed by a
weak maximum close to Qmax. At higher Qs, this is the dominant
process in the relaxational contribution (see relative amplitudes in
the lower panels of Fig. 9). As mentioned above, τ_c^0 coincides, within
the uncertainties, with the incoherent counterpart—something that
also agrees with the picture described at the beginning of this section.
Thus, the local process observed for H-motions—that is mostly visi-
ble at high Q for incoherent scattering—seems to be the main reason
for the relaxation of the coherent structure factor at ILS and high
Q values above Qmax. On the other hand, the collective diffusion
time shows a clear modulation somehow reflecting S(Q) (see the
upper panels of Fig. 9). With the information on the self-diffusion
obtained from the analysis of the incoherent scattering function, the
validity of the de Gennes narrowing,³³ τ_d = D⁻¹Q⁻²S(Q), can be
directly checked. This is done in Fig. 9. As S(Q), we used the val-
ues shown in Fig. 1, divided by σ_coh/σ_inc = 4.34. As can be seen, at
220 and 300 K, this prediction works rather well for Q ≳ 0.7 Å⁻¹;
at lower Q-values, the collective times increase with respect to this
prediction, tending to reach the self-diffusion values. For 175 K, the

de Gennes law seems to underestimate the values of τ<sub>d</sub> close to Q<sub>max</sub>,
but it appears to be extensible toward even lower Q-values than for
the higher temperatures. We recall that this region is very difficult to
explore experimentally, mainly at such low temperatures; again, MD
simulations and additional experimental information as e.g., NSE on
dTHF, could be of utmost help for a more accurate characterization
of the collective dynamics under these conditions, pushing toward
even lower Q-values.

From the model results, we can also determine the average
structural (collective) relaxation time [Eq. (18)]. Figure 15 shows
this characteristic time as a function of Q, together with its diffusive
and Q-independent mode contributions (times weighted by their
corresponding relative amplitudes). In the Q-region where S(Q) is
constant, the strong increase of τ<sub>d</sub> toward low Q seems to be com-
pensated by the continuous decrease of its amplitude. This leads to
a moderate Q-dependence, within the uncertainties, of this contri-
bution to ⟨τ⟩. We note that, in fact, both contributions are rather
similar in this region. However, as expected from the pronounced
maximum of the diffusive relative contribution around Q<sub>max</sub>, the
relaxation time at the structure factor peak ⟨τ⟩(Q<sub>max</sub>) is clearly dom-
inated by the diffusive time τ<sub>d</sub>(Q<sub>max</sub>). We recall that ⟨τ⟩(Q<sub>max</sub>)
is what has traditionally been termed “structural relaxation time.”
This time is represented as a function of the inverse temperature in
Fig. 14, together with the τ<sub>c</sub><sup>0</sup> characteristic time. As can be appreci-
ated, ⟨τ⟩(Q<sub>max</sub>) is always sensitively longer than τ<sub>c</sub><sup>0</sup>. This difference
is amplified with decreasing temperature. The temperature depen-
dence of ⟨τ⟩(Q<sub>max</sub>) can be approximated by an Arrhenius law. It
displays the same activation energy, within the uncertainties, as the
self-diffusion coefficient (see Fig. 13), and can also be well described
by the temperature dependence expected from the viscosity (η)
behavior,<sup>29</sup> ⟨τ⟩(Q<sub>max</sub>) ~ η/T.

C. THF vs water

Self- and collective dynamics in both systems, water and THF,
can be described in terms of the proposed convolution model.
Focusing on the collective dynamics at ILS, we observe the same

<figure><img src="image_26.png" /><figcaption>FIG. 15. Average characteristic time of collective relaxation obtained by Eq. (18) (solid squares) and its components (circles: diffusive and diamonds: Q-independent mode) at 175 K (a), 220 K (b), and 300 K (c).</figcaption></figure>
phenomenology for both liquids: the predominance of a Q-independent mode on the structural relaxation. There, the amplitude of the slow diffusive component tends to vanish. This observation has been corroborated by MD-simulations on the water at 300 K.<sup>34,35</sup> The analysis of the dynamic structure factor in terms of its self- and distinct-parts, possible by the simulations, showed that in the low Q-range, where the static structure factor becomes basically Q-independent, the diffusive contribution to the self- and distinct-parts cancels each other, and thereby, the decay of the density fluctuations takes place only by hydrodynamic-like ("vibrational") modes and a basically Q-independent relaxation process. This seems also to be the case for THF at 300 and 220 K, although at the lowest temperature investigated, the persistence of a small contribution of the diffusive process at mesoscales cannot be discarded from our experiments. The trend of the relative amplitude of the Q-independent mode [$C(1-A)$] and the vibrational [$1-C$] contributions with temperature are also similar for THF and water: the first one increases while the second one decreases with increasing temperature. However, in water, the Q-independent mode is relatively more important than the vibrational contribution at ILS: close to the melting point, we find $C(1-A) \approx 1-C \approx 0.5$ in water at 280 K<sup>8</sup> while $C(1-A) \approx 0.3$ and $1-C \approx 0.6$ in THF at 175 K; close to the boiling point, $C(1-A) \approx 0.7$ and $1-C \approx 0.3$ in water at 350 K<sup>8</sup> while $C(1-A) \approx 1-C \approx 0.5$ in THF at 300 K. On the other hand, the Q-dependence of the relative amplitudes of the three processes is qualitatively similar in both liquids. In both cases, the diffusive part increases when approaching the inter-molecular length scale region and shows a pronounced maximum around $Q_{max}$. Thus, in both systems, there is a crossover in the structural relaxation from being dominated by the Q-independent mode at the mesoscale to being dominated by diffusion at inter-molecular length scales. Therefore, for both systems, the temperature dependence of

the structural relaxation time $\langle\tau\rangle(Q_{max})$ is found to be the same as that observed for the macroscopic viscosity (see Fig. 14 and insert in Fig. 16), as expected from e.g., the mode coupling theory.<sup>3,4</sup>

The differences between water and THF can be found in two aspects: the first one is the Q-dependence of the diffusive characteristic time. As shown in Fig. 16 for water, $\tau_d$ shows a weak modulation close to $Q_{max}$; however, the de Gennes narrowing prediction clearly fails immediately below $Q_{max}$ and $\tau_d$ coincides with the self-diffusion time. There is no clear minimum in $\tau_d$ below $Q_{max}$. On the contrary, in THF, such a minimum is very pronounced (see the upper panels of Fig. 9). For the extreme case of the lowest temperature investigated, there is a factor of about 10 between the collective diffusive time at $Q_{max}$ (1.45 Å<sup>-1</sup>) and entering the ILS at 0.95 Å<sup>-1</sup> (i.e., when the length scale explored is about 1.5 times the average inter-molecular distance). The simple relationship between self- and collective-diffusion via $S(Q)$ (de Gennes narrowing) holds down to Q-values of about 0.6 Å<sup>-1</sup> (or even lower for 175 K), and only at very low Q-values—where the diffusive contribution is actually negligible—we observe that the collective diffusion time converges with the self-diffusion time. Thus, THF appears as a “simple liquid” in this sense.

The second difference refers to the characteristics of the Q-independent mode. In both cases, we note the coincidence, within the uncertainties, of the value of its time for collective and self-motions (although in water, this has only been checked by now for 300 K in Ref. 10). This mode could be interpreted as due to local events driving the escape from the cage imposed by the nearest neighbors. They would allow the relaxation of density fluctuations at length scales where diffusion is too slow to do it. The activation energy found for this process in THF (1.4 Kcal/mol) is much lower than that found in water (about 4 Kcal/mol). This difference can be attributed to the different origins of the interactions involved in the local relaxation mechanisms in both systems—van der Waals in THF vs hydrogen bond (HB) breaking and forming in water. In addition, we note that, while $\langle\tau\rangle(Q_{max})$ is noticeably longer than $\tau_c^0$ and displays about twice its activation energy in THF, for water, both times nearly coincide. This implies that the transport properties in water are strongly affected by the underlying HB interactions, while the local motions in THF have much less influence on diffusion and viscosity evolution.

## VII. CONCLUSIONS

Neutron spectroscopy with polarization analysis by the PLET instrument has made it possible to separate coherent and incoherent contributions to the neutron scattering of deuterated THF in a wide Q-range from mesoscale to inter-molecular length scales. This has allowed us to explore the collective dynamics in a van der Waals liquid in the almost white area of the intermediate length scales and compare the behavior with the recently reported results on water.<sup>8</sup> In addition, it has provided information on the self-motions of hydrogens (deuterons), which has been complemented with high-resolution NSE experiments by WASP on a protonated sample at low temperatures.

At ILS, coherent and incoherent scattering functions decay with very different time scales; while incoherent scattering is mainly dominated by the self-diffusion, coherent scattering presents nearly

<figure><img src="image_27.png" /><figcaption>FIG. 16. Diffusive time deduced from the application of the model to the collective dynamics in water.<sup>8</sup> for $Q \lesssim 1$ Å<sup>-1</sup>, it followed $D^{-1}Q^{-2}$ (dashed line), with $D$ obtained from the analysis of incoherent scattering (empty circles); at higher Q-values, it was fitted (filled circles). The dotted line displays the de Gennes narrowing prediction. The inset shows the inverse temperature dependence of $\tau_c^0$ (circles) and $\langle\tau\rangle(Q_{max})$ (squares). The dotted line is an Arrhenius fit with $E_a = 4$ Kcal/mol, and the solid line shows the viscosity dependence $\eta/T$.</figcaption></figure>
Q-independent values for the characteristic time of the relaxational part. Thus, the existence of the Q-independent mode reported for supercooled water and silica by MD-simulations<sup>16</sup> and for liquid water by neutron scattering<sup>8</sup> is also proved in this liquid. At higher Q-values, more than one relaxation mechanism—in addition to the vibrational contribution—has to be invoked for both coherent and incoherent scattering. This phenomenology is qualitatively identical to that found for water.<sup>8,10</sup>

The results on both collective and self-scattering functions have been satisfactorily described in terms of the convolution model proposed for water<sup>8,10</sup> that considers vibrations, diffusion, and a Q-independent mode (affected by diffusion).

At ILS, the amplitude of the slow diffusive component tends to vanish for collective dynamics, and the dynamic structure factor is dominated by vibrations and the Q-independent mode. Since the diffusional amplitude presents a pronounced maximum at the inter-molecular peak, there is a crossover in the structural relaxation from being dominated by the Q-independent mode at the mesoscale to being dominated by diffusion at intermolecular length scales. Therefore, the temperature dependence of the structural relaxation time $\langle\tau\rangle(Q_{max})$ is found to be the same as that observed for the macroscopic viscosity (as in the case of water<sup>8</sup>).

The time scale of the Q-independent mode $\tau_c^0$ is the same for collective and self-motions, within the uncertainties, faster, and with a lower activation energy than $\langle\tau\rangle(Q_{max})$; from the incoherent scattering results, it follows that the underlying self-motions have a rather large motional amplitude.

The collective diffusive time in THF follows very well and in a rather wide Q-range entering the ILS the de Gennes narrowing relation proposed for simple monoatomic liquids. This is not the case of water, probably due to the more complex inter-molecular relaxation mechanism involving HB breaking and formation. This would also be the reason for the near coincidence of $\tau_c^0$ and $\langle\tau\rangle(Q_{max})$ in water, implying that the transport properties are strongly affected by the HB-network dynamics, while the local motions leading to the relaxation of density fluctuations in THF at ILS have much less influence on macroscopic viscosity.

## ACKNOWLEDGMENTS

We acknowledge the Science and Technology Facilities Council (STFC) for access to neutron beam time at ISIS (Grant No. 10.5286/ISIS.E.RB2010388). We acknowledge the Grant No. PID2021-123438NB-I00 funded by MCIN/AEI/10.13039/5011000 11033 and by "ERDF A way of making Europe," and the financial support of Eusko Jaurlaritza, code: IT1566-22, and the IKUR Strategy under the collaboration agreement between the Ikerbasque Foundation and the Materials Physics Center on behalf of the Department of Education of the Basque Government.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## Author Contributions

**Arantxa Arbe:** Conceptualization (equal); Data curation (lead); Formal analysis (lead); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – original draft (lead); Writing – review & editing (lead). **Gøran J. Nilsen:** Data curation (equal); Investigation (equal); Methodology (equal); Writing – review & editing (supporting). **Mark Devonport:** Investigation (equal); Methodology (equal). **Bela Farago:** Data curation (equal); Investigation (equal); Methodology (equal); Writing – review & editing (supporting). **Fernando Alvarez:** Investigation (equal); Methodology (equal); Software (lead); Writing – review & editing (supporting). **José A. Martínez González:** Data curation (supporting); Investigation (equal); Software (equal); Writing – review & editing (supporting). **Juan Colmenero:** Conceptualization (equal); Formal analysis (equal); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Writing – review & editing (supporting).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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