Diffusive Motions of Molecular Hydrogen Confined in THF Clathrate

Jul 19, 2012 - of hydrogen molecules confined in the small cages of the deuterated THF clathrate hydrate have been investigated at equilibrium by mean...
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Diffusive Motions of Molecular Hydrogen Confined in THF Clathrate Hydrate E. Pefoute,†,‡ E. Kemner,‡ J. C. Soetens,† M. Russina,‡ and A. Desmedt*,† †

Institut des Sciences Moléculaires, UMR5255 CNRS - Université de Bordeaux 1, Groupe Spectroscopie Moléculaire, 351 cours de la Libération, F-33405 Talence, France ‡ Helmholtz Zentrum Berlin, BENSC, Glienickerstr. 100, D-14109 Berlin, Germany ABSTRACT: Clathrates hydrates are nanoporous crystalline materials made of water cages encapsulating guest molecules. By inserting H2 molecules with the help of a promoter (tetrahydrofuran, noted THF), systems relevant for hydrogen storage application are formed by using relatively soft pressure (of the order of 100 bar) near room temperature. Dynamic properties of hydrogen molecules confined in the small cages of the deuterated THF clathrate hydrate have been investigated at equilibrium by means of incoherent quasi-elastic neutron scattering (QENS). These QENS investigations provide direct experimental evidence about the fundamental aspect of translational diffusive motions of the hydrogen molecules. A comprehensive study of the hydrogen molecules dynamics above 100 K has been achieved through a quantitative analysis of the structure factors (i.e., the spatial extend of the H2 diffusive motion) as well as of the QENS broadening (i.e., the characteristic time of the diffusive motion). On the probed time scale, the H2 molecular translations occur within localized spherical area in the cage with low activation energy of 1.59 ± 0.06 kJ mol−1. The dynamical diameter of H2 molecules varies from 2.08 Å at 250 K to 1.64 Å at 100 K, and the diffusion constant ranges from 0.16 ± 0.03 rad ps−1 at 100 K to 0.49 ± 0.03 rad ps−1 at 250 K. These results indicate that no diffusion between the cages is observed in the picosecond time scale.

1. INTRODUCTION Clathrates hydrates are nanoporous crystalline materials made of water cages stabilized by the presence of foreign (generally hydrophobic) molecules. 1 The cages are formed from polygonal rings of water molecules connected at their edges by means of hydrogen bonds. Numerous arrangements of these cages exist to form a three-dimensional crystalline structure, depending on the nature of the encapsulated molecules. Under appropriated thermodynamic conditions, clathrates hydrates crystallize into three major types of structure: two cubic structures termed type I and type II (Figure 1)2,3 and one hexagonal structure termed type H.4 The natural existence of large quantities of hydrocarbon hydrates in deep oceans and permafrost is probably at the origin of numerous applications in the broad areas of energy and environmental sciences.5,6 At a fundamental level, the thermodynamics and phase equilibrium properties have been extensively studied and numerous studies are devoted to the formation, inhibition, and decomposition of gas hydrates,7 to their anomalous thermal conductivities,8 or to the properties of ionic clathrate hydrates.9,10 The immediate relevance of this paper concerns the H2 storage application. Mao et al.11 showed that pure hydrogen clathrate hydrate could be formed by applying 2000 bar hydrogen pressure on powdered ice. By forming a binary H2 clathrate hydrate with the help of a promoter (by coincluding tetrahydrofuran, noted THF), Florusse et al.12 showed that the clathrate formation pressure could be reduced to typically 100 bar. Since these discoveries, numerous experimental and theoretical studies focused on the formation mechanisms and on methods to improve the formation kinetics and the © 2012 American Chemical Society

Figure 1. Representation of the two types of cages forming the type II H2-THF clathrate hydrate. The large cage 51264 contains the THF guest molecule (represented by a black sphere), and the small cage 512 is filled with one hydrogen molecule at the most. The oxygen atoms of the water molecules are located at each corner and the H-bonds are contained at the edge.

hydrogen content.9,13 At a fundamental level, hydrogen clathrate hydrates represent a model system for exploring the confinement effect on the dynamics of H2 molecules. The dynamical properties of hydrogen molecules confined in clathrate hydrates has been investigated at equilibrium by means of Raman spectroscopic measurements12,14 and inelastic neutron scattering.15−18 Both H2 roton and vibron bands are at Received: January 26, 2012 Revised: July 19, 2012 Published: July 19, 2012 16823

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hydrogen molecular diffusion through the cages of the THF-H2 clathrate hydrate may be observed as suggested in a previous QENS investigation.18 Unfortunatly, no analysis of the QENS spectra as a function of the momentum transfer has been performed in these QENS experiments. In the present paper, a comprehensive study of the hydrogen molecules dynamics above 100 K is provided through a quantitative analysis of the structure factors (i.e., the spatial extent of the diffusive motion) as well as of the QENS broadening (i.e., the characteristic time of the diffusive motions), indicating that the H2 molecular translations occur within the cage and not between the cages on a picosecond time scale.

lower frequencies in the cage structure than in the free gaseous phase, representative of the attractive interaction existing between the H2 molecule and the aqueous cage in the frame of the Buckingham model.19,20 Broadening of the H2 modes is also observed in the clathrate phase compared to the gas phase. In this issue, high-resolution inelastic neutron scattering experiments15−17 have led to a detailed analysis of the free rotor at ca. 14 meV and rattling modes at ca. 10 meV. Both modes split into triplet due to the anisotropy of the potential energy experienced by the H2 molecules. All these studies address the fundamental interest due to the inherent quantum mechanical behavior of H2 molecules “isolated” within cages. One important issue about the dynamical processes of hydrogen molecules confined in clathrates hydrates deals with their diffusion from cage to cage. Indeed, such information is crucial not only for the understanding of the clathrate at equilibrium but also for the H2 (de)insertion mechanism. The method used to form the THFH2 clathrate hydrate is to apply a pressure of the order of 100 bar on a grounded THF clathrate hydrate. The model for the formation mechanisms assumes that the H2 loading within THF clathrates hydrates is done in two steps: hydrogen adsorption onto the clathrate particle surface, followed by subsequent diffusion of hydrogen into the clathrate hydrate particle.21 The limiting step in the storage kinetic (of the order of days) is associated with the diffusion step, characterized by a high activation energy of 78.7 kJ/mol.21 This high activation energy should be related to recent electronic structure calculations.22 The potential energy barrier associated with H2 guest molecules migration between cages of the type II clathrate hydrate has been estimated to be 23 kJ/mol for H2 migration through a hexagonal face of the large cage and to range from 104.7 to 121.4 kJ/mol (depending on the orientation of the H2 molecule) for migration through the smaller pentagonal faces of the small cages. Finally, the hydrogen diffusion coefficient value gradually decreases with increasing pressure or increasing hydrogen concentration in the hydrate.23 This is because the transport of hydrogen molecules between the cages may proceed only when there is a vacant guest site in the neighboring cage, and diffusion is slower when a larger fraction of the guest sites is already filled. The fundamental aspect of H2 diffusion between cages plays a key role in the understanding of the system. This issue has been addressed by means of pulse-field-gradient H1 NMR,24 in situ neutron diffraction,25 volumetric measurements,21 or theoretical calculations.22 However, contradictory results have been obtained: the determined H2 diffusion coefficient ranges from ca. 10−8 to ca. 10−11 cm2 s−1 in a similar pressure− temperature domain. A detailed description of the spatial and time characteristics of this diffusive mechanism is missing at a microscopic level. In order to shed more light on this issue, incoherent quasi-elastic neutron scattering (QENS) is particularly well adapted since this technique allows, in both space and time, the individual displacements of hydrogen nuclei to be probed. Recent QENS investigations report that no significant quasi-elastic broadening due to the hydrogen molecules was observed at temperature below 150 K while on the basis of the analysis of the Debye−Waller factor, onset of this motion may be observed between 100 and 150 K.17 Above ca. 100 K, Choi et al. report QENS spectra clearly showing quasielastic broadening.18 These observations indicate that no diffusive motion is observed on picoseconds time scale of the performed QENS experiments below ca. 100 K. Above this temperature,

2. EXPERIMENTAL DETAILS 2.1. Sample Preparation. The samples were prepared by mixing deuterated water (99.8% deuterated) and deuterated tetrahydrofuran (noted TDF, 99.5% deuterated) in stoichiometric proportion (17:1 mol). The prepared solution has been stirred in a thermal bath at a constant temperature of 275 K (the melting point of the hydrogenated THF clathrate hydrate is 277 K) until crystallization occur (in about 1 day). The resulting TDF clathrate hydrate has been ground to form a fine powder, and about 5 g of TDF clathrate hydrate has been loaded into a precooled cylindrical aluminum cell. To avoid sample decomposition and water condensation, all these manipulations have been done under a cold nitrogen atmosphere. The cell, cold-transferred into a cryostat at 270 K, has been slowly purged with hydrogen gas (three times) and then pressurized up to the desired value (265 bar in the present experiment). The hydrogen loading lasts 1 week, and over 7 days, the temperature of the bath cycled between 270 and 277 K. This method is similar to that described by Stern et al.26 for converting ice particles to methane hydrate. The H2-loaded sample and the storing materials (i.e., the TDF clathrate hydrate) has been ground to form a fine powder and filled into flat aluminum containers sealed with indium wire (under a cold nitrogen atmosphere) to perform the QENS experiments. 2.2. QENS Experiment. Neutron scattering experiments were carried out on the reference TDF clathrate hydrate sample and on the hydrogen-charged sample, TDF-H2 clathrate hydrate. The QENS experiment was performed on the cold time-of-flight (ToF) spectrometer NEAT27 at the Berlin Neutron Scattering Centre of the Helmotz Zentrum Berlin (Berlin, Germany). The scattering angles covered by this instrument are in the range 13°−136°, and the energy resolution was ΔE ≈ 150 μeV for an incident wavelength λ0 = 4.8 Å. The samples have been cold-transferred into the cryostat of the spectrometer, and the slab orientation was 135° to the beam. The sample thickness was 0.2 mm so that the multiple scattering can be neglected (the transmission coefficient is more than 0.9). QENS spectra of the H2 filled clathrate hydrate have been recorded every 50 K between 100 and 250 K. In order to take into account the contribution of the storing matrix, QENS spectra of the H2-unloaded TDF-17D2O sample have been recorded in the same experimental conditions. Empty cell spectra have been recorded for background correction, and a vanadium spectrum has been used to correct for detector efficiency and to determine the experimental resolution function. Two sets of data have been extracted from the neutron scattering experiments. For the structural analysis, only elastic scattering has been extracted from the experimental data at each scattering angle. For QENS analysis, several detectors were grouped together to improve 16824

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about concerted or correlated motions. These two contributions are weighted by the incoherent and coherent scattering cross sections of the sample. The present QENS experiments aim at studying the molecular hydrogen dynamics, i.e., at collecting the incoherent signal arising from the hydrogen molecules. In this issue, the incoherent scattering due to the cages has been minimized by deuterating the storing materials, i.e., the water and THF molecules (the incoherent cross section of deuterium is 40 times less than the one of hydrogen). Hence, the coherent contributions can be neglected, since Bragg peaks were carefully removed from the experimental data and coherent inelastic scattering is well separated from the quasielastic region of the spectra. Thus, in the quasi-elastic region (i.e., energy transfer window of ±3 meV), the experimental scattering law Sexp(Q,ω) is related to the incoherent scattering law S(Q,ω) by (see refs 36 and 37 for details about the relationship between the measured spectra and the scattering function)

the statistical accuracy, and Bragg peaks were carefully removed from experimental data. These data reduction was carried out with the INX software,28 and QENS data analysis has been done by using the NEMO program.29 2.3. Characterization of H2-TDF Clathrate Hydrate. The powder diffractograms recorded at 250 K on the ToF spectrometer NEAT are shown in Figure 2 for TDF and TDF-

Sexp(Q , ω) = F(Q )e−ℏω / kBT S(Q , ω) ⊗ R(Q , ω) + B(Q ) (1)

where F(Q) is the scaling factor for which the Q dependence is 2 2 due to the Debye−Waller factor (e−Q ⟨u ⟩ where ⟨u2⟩ is the mean-square displacement), T is the temperature, and kB is the Boltzmann constant. B(Q) is the background term reproducing the inelastic contributions in the quasi-elastic region. The function R(Q,ω) represents the experimental energy resolution, which acts as a time filter on the scattering law. Indeed, the width of the energy resolution function (noted ΔE hereafter) gives the time scale on which a given motion will be observed on the experimental spectra: it allows the definition of an observation time (of the order of picoseconds in the present case). In QENS experiment, any dynamical process, occurring on a time scale significantly longer than the observation time, will give rise to a QENS broadening narrower than the instrumental resolution function, and the experimental scattering law reduces to a purely elastic component. 3.1. Specifications of Models. The scattering law, S(Q,ω), contains the key information about the dynamical processes occurring in the clathrate. In the present case, this function may be decomposed into two terms associated with the chemical species constituting the materials:

Figure 2. Powder difractograms of the TDF and TDF-H2 clathrates hydrates recorded at 250 K with the ToF spectrometer NEAT (λ0 = 4.8 Å). The Bragg peaks are indexed in the type II clathrate structure (i.e., space group Fd3m with a = 17.27 Å for the TDF clathrate and with a = 17.26 Å for the TDF-H2 clathrate hydrate). The peaks marked with asterisks are due to hexagonal ice Ih.

H2 clathrates hydrates. The integrity of the clathrate structure after the hydrogen uptake is maintained since both diffractograms are indexed in the type II clathrate structure with space group Fd3m̅ and cubic cell parameter of ca. 17.3 Å. On both diffractograms, a small amount of ice is observed. Such a contamination has a negligible impact on the analysis of the QENS spectra as it has been shown in the case of methyl iodide clathrate hydrate for instance.30 In order to estimate the cage occupancy of the present sample, 0.5 g of the sample was transferred to a small can (of known volume) under a cold nitrogen atmosphere (to prevent ice forming and clathrate hydrate decomposition). Gas-release thermodynamic measurements lead to hydrogen uptake of ca. 0.7 wt % (assuming that only molecular hydrogen is involved in the pressure release). Such value corresponds to a cage occupancy of 74% by molecular hydrogen, i.e., not more than one hydrogen molecule per small cage. While initially a multiple occupancy of the cages by the H2 molecules was considered to be possible,11,31,32 subsequent reports indicate that only one H 2 molecule is hosted in the small cages.23,25,33−35 The value obtained for the present sample is thus in agreement with these latter results.

S(Q , ω) = cHSH(Q , ω) + csSs(Q , u′)

with cH + cs = 1 (2)

where the scattering laws SH(Q,ω) and Ss(Q,ω) are due to the molecular hydrogen and to the storing matrix (i.e., the deuterated THF and water molecules), respectively. The fractions cH and cs of each chemical species are weighted by their respective incoherent scattering cross sections, taking into account for the imperfect deuteration of the TDF and water molecules and for the incoherent cross section of the deuterium. To quantify the incoherent contribution of each chemical species, one has to consider a chemical formula of the present clathrate hydrate written as [gH2, dTDF, (1 − d)THF]/17[dD2O, (1 − d)H2O]. The H2 cage occupancy of 0.74 (see previous section) leading to g = 1.48 and assuming an effective molar deuteration d of 0.99, one obtains that cH = 0.67 and cs = 0.33. Thus, because of the large incoherent scattering cross section of the proton, QENS experiments allow a relatively simple access to the dynamic of hydrogen molecules inserted within the clathrate.

3. DYNAMICS OF MOLECULAR HYDROGEN In a neutron scattering experiment, the probability to find a neutron leaving the sample within a solid angle element in a given direction with a given energy is measured. This double differential scattering cross section can be reduced to an experimental scattering law Sexp(Q,ω) recorded as a function of momentum transfer ℏQ and energy transfer ℏω. This law consists into the superimposition of two terms: the incoherent scattering (related to self-correlation) providing information about the motion of individual atoms and the coherent scattering (related to pair correlation) providing information 16825

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broadening at all studied temperature. In order to proceed with a quantitative analysis, the following phenomenological scattering law has been considered:

Subtracting the TDF-17D2O contribution by using eq 2, the scattering law of molecular hydrogen, SH(Q,ω), has been extracted from the QENS spectra of the TDF-H2/17D2O sample. As shown in Figure 3, these spectra exhibit a QENS

SH(Q , ω) = A 0(Q )δ(ω) + [1 − A 0(Q )]L(ΔH )

(3)

where A0(Q) is the elastic incoherent structure factor (EISF) giving the amplitude of the elastic term represented by a Dirac function δ(ω) and provides information about the geometry of the H2 diffusive motion. The Lorentzian function, L(ΔH), represents the quasielastic contributions for which the halfwidth at half-maxima (denoted HWHM), ΔH, provides information about the characteristic time of the diffusive motions. This phenomenological approach leads to a reasonable agreement as shown in Figure 4. The measured EISF and HWHM are showed in Figures 5 and 6, respectively. Let state a few preliminary assessments concerning the hydrogen dynamics (rotational and translational motions) at the origin of the observed QENS signal: (i) H2 molecules behave as free rotor, with the first rotational transition observed at ca. 14 meV.15−17 In other words, the rotational dynamics of H2 molecules gives rise to inelastic peak outside the quasielastic region (i.e., ±3 meV). Thus, the observed quasi-elastic broadening is not due to rotational motion but to translational motion of H2 molecules occurring on picoseconds time scale.

Figure 3. QENS spectra of the H2 molecules confined in the TDF clathrates hydrates recorded at various temperatures with the ToF spectrometer NEAT (λ0 = 4.8 Å). The dashed line corresponds to the spectrometer resolution function with ΔE = 152 μeV (determined from the vanadium QENS spectrum). All spectra are normalized.

Figure 4. Fitted (lines) and experimental (points) QENS spectra at selected temperatures (ToF spectrometer NEAT; λ0 = 4.8 Å and ΔE = 152 μeV). The continuous lines correspond to the fitted scattering laws, the dashed line to the fitted H2 QENS components; and the dotted lines to the fitted backgrounds. 16826

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Model B. One can assume that the H2 molecule may explore the whole volume of the small cage. Such a model is equivalent to diffusion within a sphere with impermeable walls. The EISF writes36 A 0B(Q )

(5)

where j1(x) is the spherical Bessel function of first order and r is the sphere radius. In both cases, two limiting cases may be considered for the sphere radius by considering the H−H bond length of 0.74 Å. The first case consists in considering a radius corresponding to one-half of the difference between the small cage diameter (ca. 7.8 Å) and the H−H bond length, i.e., r1 = 0.97 Å (considering atomic van der Waals radiuses of 1.35 and 1.2 Å for oxygen and for hydrogen, respectively). The second case, based on radial distribution functions obtained by means of MD simulations39 indicating an averaged water−hydrogen molecules distance of ca. 3 Å, leads to a sphere radius r2 = 0.9 Å. 3.2. Model Discussion. For the model A, the calculated EISF with the help of the two radiuses values (represented by means of dotted lines for both radiuses in Figure 5) is not in agreement with the experimental EISF. In the case of model B, the two limit values of the sphere, r1 and r2, have been considered, and both cases are shown as dashed lines in Figure 5. While considering a radius of 0.97 Å could be clearly ruled out in view of the disagreement, the EISF calculated for a radius of 0.9 Å may reproduce the experimental EISF at 250 K. However, at lower temperatures, the model B with r2 = 0.9 Å does not reproduce the experimental EISF. In view of the high value of the experimental EISF, one may consider both models with sphere radius smaller than r2. Thus, the experimental data have been fitted by means of models A and B, given by expressions 4 and 5, in which the sphere radius may vary. Both models lead to identical EISF in agreement with experimental EISF (continuous lines in Figure 5), but to different fitted sphere radiuses (Table 1). In the accessible Q range of the

Figure 5. Experimental EISF of molecular hydrogen inserted within the small cages of the TDF-17D2O clathrate hydrate at various temperatures (filled circles: 250 K; open circles: 200 K; squares: 150 K; triangles: 100 K). The dotted line corresponds to the model A (diffusion on a sphere) and the dashed line (diffusion within a sphere) to the model B for sphere radiuses indicated on the figure. The continuous lines are fits of models A and B with sphere radiuses given in Table 1 (see text for details).

Figure 6. Experimental HWHM of molecular hydrogen inserted within the small cages of the TDF-17D2O clathrate hydrate at various temperatures (filled circles: 250 K; open circles: 200 K; squares: 150 K; triangles: 100 K). The lines are fits (see text for details).

Table 1. Diffusion Constants and Radii of the Spherical Volume Explored by the Center of Mass of Hydrogen Molecule Confined in the Small Cage of the TDF-H2 Clathrate Hydrate in the Case of Model A (ron and α = 2) and in the Case of Model B (rin and α = 4.332 96)

(ii) The spatial extent of the observed translational diffusive motion is probed through the EISF. The observation of large EISF indicates that the probed translational motion occur on short spatial range: the characteristic distance of the observed translation is shorter than π/Qmax ∼ 1.8 Å. Such a distance being significantly shorter than the cage−cage distance, no H2 diffusion through cages is observed on picoseconds time scale.38 On the basis of these observations, the quasi-elastic signal can be attributed to localized translational diffusive motion of the whole H2 molecule within the cage. To reproduce this dynamical process, various models may be considered and summarized as follows: Model A. Let assume that the H2 molecule explores the surface of the small cage. Such a dynamical process can be reproduced by means of a model of diffusion on a sphere for which the EISF writes36 A 0A (Q ) = [j0 (Qr )]2

⎡ 3j (Qr ) ⎤2 ⎥ =⎢ 1 ⎣ Qr ⎦

T [K] 100 150 200 250

ron [Å] 0.45 0.53 0.61 0.67

± ± ± ±

0.05 0.05 0.05 0.05

rin [Å] 0.58 0.68 0.79 0.86

± ± ± ±

0.05 0.05 0.05 0.05

αDH [rad ps−1] 0.32 0.56 0.80 0.98

± ± ± ±

0.06 0.06 0.06 0.06

present experiment, the model of diffusion inside a sphere is equivalent to the one of diffusion on a sphere, except for the fitted values of sphere radius. To disentangle the two considered models, a larger momentum transfer range (not accessible in the present QENS experiment) would be required. In the case of model B, the effects of the sphere walls would become negligible for momentum transfer significantly greater than π/r (corresponding to Q ∼ 3.7 Å−1 at 250 K in the present case), so that the Fick law behavior (leading to EISF closed to zero) would be retrieved for Q ≫ π/r while in the case of model A, the EISF would not be equal to zero for Q ≫ π/r.36

(4)

where j0(x) is the spherical Bessel function of zero order and r is the sphere radius. 16827

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the H−H bond of 0.74 Å, one obtains that the dynamical diameter of hydrogen molecules is about 2.08 Å for model A and about 2.46 Å for model B. Recent neutron diffraction experiment and DFT dynamics25 show that the H2 molecules explore a volume with a diameter of the order of 2 Å. Such a value is in favor of diffusion on a sphere. Assuming the model of diffusion on a sphere, the dynamical diameter of H2 molecules decreases with the temperature from 2.08 Å at 250 K to 1.64 Å at 100 K. In view of the thermal expansion of the type II structure,41 one would have expected a less important shrinkage of the H2 dynamical diameter. Such thermal evolution suggests that the molecular hydrogen tends to be localized in the cage by lowering the temperature. One arising question concerns the fact that this localized area of the cage is centered or off-centered. Indeed, an off-centered position of H2 molecules would involve the existence of specific H2−H2O interactions and so the existence of adsorption sites at the cage surface. The understanding of the nature of such adsorption sites would then open opportunities for multiple H2 occupancy of small cages and thus the improvements of H2 storage capacities.

To analyze the characteristic time associated with the probed localized diffusive process, the limit at low Q value of the HWHM is proportional to the rotational diffusion constant, noted DH, in the frame of the rotational diffusion on a sphere or inside a sphere.36 In order to extract this limit, the HWHM of the H 2 molecule have been fitted by means of the phenomenological expression ΔH (Q ) = αDH + SQ 2

(6)

where S is a phenomenological parameter reproducing the momentum transfer dependence of the QENS broadening and α is a proportional constant equal to 2 in the case of model A36 and to 4.332 96 in the case of model B.40 The results of these fits are shown as continuous lines in Figure 6. The resulting diffusion constants are given in Table 1. Their temperature dependence follows an Arrhenius behavior with a prefactor αD0H = 2.08 ± 0.2 rad ps−1 and an activation energy Ea = 1.59 ± 0.06 kJ mol−1.

4. SUMMARY AND CONCLUDING REMARKS The results presented in this paper confirm that not more than one hydrogen molecule may be loaded in the small cage of the THF clathrate hydrate as reported in various works23,25,33−35 and contrarily to initial studies.11,31,32 The QENS spectra (with an energy resolution of ca. 150 μeV corresponding to picoseconds time scale) of the hydrogen molecules confined in the small cages of the THF clathrates hydrates clearly exhibit a QENS broadening. In addition to inelastic signature of the rattling (at ca. 10 meV) and of the free rotation (at ca. 14 meV) previously observed by means of inelastic neutron scattering,15−17 this QENS component indicates the existence of translational diffusive motions of the hydrogen molecules, thermally activated (with Ea = 1.59 ± 0.06 kJ mol−1). In complement to QENS analysis previously reported,18 the present QENS investigation includes the spatial extend of the observed diffusive motion. Indeed, the analysis of the structure factor reveals that the hydrogen molecules undergo localized translational motions. No diffusion from cage to cage is observed on picoseconds time scale, as suggested by Choi et al.18 According to pulse-field-gradient H1 NMR investigations,24 the long-range diffusion coefficient of molecular hydrogen is of the order of 3 × 10−8 cm2 s−1 at 250 K (for a pressure of 10 bar). By means of in situ neutron diffraction, this diffusion coefficient has been estimated to be of the order of 10−11 cm2 s−1 at 264 K,25 in agreement with theoretical calculations.22 Taking into account this wide range of diffusion coefficient and assuming an isotropic jump diffusion model (i.e., Chudley−Elliot model36) with a mean jump distance of the order of the cage diameter, such cage-to-cage diffusion would give a QENS broadening between 10−2 and 1 μeV, which is not observable in a time-of-flight QENS experiment (for which the energy resolution is of the order of 100 μeV). Modeling the QENS signal provides information about the volume explored by the molecular hydrogen center of mass: it does not diffuse within the whole cage and explore a restricted spherical area or volume of the cage. In view of the accessible Q range of the present QENS experiment, it was not possible to distinguish the case of diffusion on a sphere (model A) from the one of diffusion inside a sphere (model B) on the basis of QENS data only. At 250 K, the diameter of the sphere visited by the H2 center of mass is 1.34 ± 0.1 Å for model A and 1.72 ± 0.1 Å for model B. Considering the free rotation15−17 with

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AUTHOR INFORMATION

Corresponding Author

*Tel ++33 (0) 5 4000 2937; Fax ++33 (0) 5 4000 8402; e-mail [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Michel Couzi for discussions on the works of the present paper. The Berlin Neutron Scattering Centre (HZB, Berlin, Germany) is thanked for the provision of beam time for QENS experiments for which the European Commission is thanked for funding under the sixth Framework Programme through the Key Action: Strengthening the European Research Area, Research Infrastructures [Contract RII3-CT-2003-505925 (NMI3)]. The CNRS and the HZB are acknowledged for funding EP’s PhD work. Finally, this paper falls in the frame of the project ANR 2011-JS08-002-01.

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REFERENCES

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The Journal of Physical Chemistry C

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dx.doi.org/10.1021/jp3008656 | J. Phys. Chem. C 2012, 116, 16823−16829