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Multiscale Water Dynamics within Dense Clay Sediments Probed by 2 H Multiquantum NMR Relaxometry and Two-Time Stimulated Echo NMR Spectroscopy Patrice Porion,* Anne Marie Faugère, and Alfred Delville* Centre de Recherche sur la Matière Divisée, CNRS-Université d’Orléans, FRE3520, 1b rue de la Férollerie, 45071 Orléans Cedex 02, France

ABSTRACT: 2H NMR spectroscopy, relaxometry, and two-time correlation measurements are used to investigate the structural and dynamical properties of water molecules confined within the multiscale porous network of dense clay sediment. The residual quadrupolar splitting detected by 2H NMR spectroscopy is the fingerprint of the specific orientation of the water molecules pertaining to the first hydration shell of clay lamellae. Multiquantum 2H NMR relaxation measurements are used to quantify the distribution of the clay platelet orientations within the dense sediment. The average residence time of the water molecules confined within the clay interlamellar space is determined by exploiting 2H multiquantum NMR relaxation measurements under spin-locking conditions. Finally, long-time scale diffusion of the water molecules within the multiscale porous network of the clay sediment is quantified by measuring the attenuation of the 2H NMR two-time stimulated echo.

I. INTRODUCTION For the last decades, confined fluids have been the subject of numerous theoretical1−3 and experimental4−14 studies because of the strong modifications of their thermodynamical properties in comparison with those of bulk liquids. In that context, clay− water solid−liquid interfaces were frequently investigated for two main reasons: first, these lamellar materials are perfectly characterized and atomically smooth, leading to ideal models of solid−liquid interfaces, and second, they exhibit a large variety of physical-chemical properties (swelling, gelling, ionic exchange capacity, and water and polar solvent adsorption) exploited in numerous industrial applications like drilling, paint and cosmetic industries, or waste storing.15 For various applications, including heterogeneous catalysis and waste management, it appears crucial to carefully monitor the retention capacity of the clay lamellae by measuring the mobility of the confined probes over a broad range of diffusing time. For that purpose, the short-time mobility (time < 100 ns) of confined liquids was frequently investigated by neutron scattering experiments.16−30 In addition, the long-time mobility of the fluids may be investigated either by using field-cycling NMR relaxometry31−36 (10 ns < time < 100 μs) or pulsed gradient spin−echo NMR attenuation37−42 measurements (time > 1 ms). Unfortunately, these two © 2013 American Chemical Society

experimental procedures become useless for a large variety of natural clay platelets because the NMR relaxation rate of the confined probes is largely enhanced by the presence of paramagnetic impurities within the clay atomic network. In that framework, we developed alternate NMR experimental procedures to extract dynamical information on the mobility of water molecules and neutralizing counterions confined within natural or synthetic clay platelets. First, multiquantum NMR relaxation measurements under spin-locking conditions leads to dynamical information over roughly three time decades (between microsecond and millisecond). In addition, two-time stimulated echo NMR spectroscopy was shown to adequately probe the water mobility within a broader dynamical range (between 10 μs and 0.1 s),43−46 leading to useful information on the multiscale organization of the porous network probed by the diffusing probe. As displayed in Figure 1, dense clay sediments exhibit complex and multiscale structures: (i) at short distances, the clay sediment is composed of individual hydrophilic and highly anisotropic Received: September 18, 2013 Revised: October 28, 2013 Published: November 18, 2013 26119

dx.doi.org/10.1021/jp4093354 | J. Phys. Chem. C 2013, 117, 26119−26134

The Journal of Physical Chemistry C

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H multiquantum NMR relaxation measurements are performed under spin-locking conditions to determine the mobility44,46 of the confined water molecules and characterize their average residence time within the clay interlamellar space. Finally, twotime stimulated echo 2H NMR spectroscopy64−67 is used to probe the long-distance mobility of the water molecules within the multiscale structure of the clay sediment. The Appendix at the end of this article is added to provide the theoretical basis necessary to understand the elaboration and analysis of our multiquantum NMR measurements.

Figure 1. Schematic view of the multiscale organization of the clay sediment resulting from the coexistence of clay aggregates with various orientations of the platelet directors.

II. MATERIALS AND METHODS II.i. Sample Preparation. Hectorite (from Hector, CA) purchased from Ward’s Natural Science is a natural clay with the general formula Si8Al0.22Fe0.05Mg4.93Li0.8(OH)3.6F0.4O20Na+0.6. That swelling clay47 results from the sandwiching of one layer of octahedral magnesium oxide between two layers of tetrahedral silica. Solvated interlamellar sodium cations neutralize the negative charge of the clay network resulting from the substitution of some octahedral MgII by LiI. Prior to use, the natural clay sample was purified according to classical procedures,68 and the cations were exchanged,68,69 leading predominantly to monoionic clay samples. The clay particles are further selected according to their size by using centrifugation.68 As shown previously by TEM,46 individual hectorite clay particles appear as lath, with average size of (700 ± 300) nm and width of (70 ± 20) nm. Because of the juxtaposition of 7 ± 2 individual hectorite laths, the average size of the clay particle detected by TEM is 500 × 800 nm2. A self-supporting film (0.5 mm thick) is obtained from dilute aqueous clay dispersion (12 g/L) by ultrafiltration under nitrogen (3−5 atm) using a membrane with an average pore size of 0.1 μm (Osmonics, Inc.). The clay film was dried under nitrogen flux before equilibration with a reservoir of heavy water at a fixed water chemical potential (P/P0 = 0.33) using a saturated salt solution (MgCl2). The water partial pressure (P/P0 = 0.33) was selected because it corresponds mainly to an interlayer space28,49,50 with a period of roughly 12 Å. This interlayer space is large enough to accommodate one layer of confined water molecules. A lamella (30 × 5.5 mm2) is cut into the film and inserted into a glass cylinder which fits the gap inside the solenoid coil used for the NMR measurements.

(thickness ≈ 7 Å, diameter < 1 μm) charged platelets neutralized by interlamellar counterions (see Figure 1); (ii) at intermediate distances, microscopic domains are formed by the stacking of numerous (10−100) parallel clay platelets47,48 (Figure 1) containing highly confined water molecules; and (iii) at the largest distances, clay sediment is formed by the juxtaposition of microdomains characterized by slightly different orientations (Figure 1). This work focuses on a natural clay sample (hectorite) with low water content, i.e., in conditions similar to those used for waste storage. As shown experimentally by X-ray49,50 and neutron49,50 diffraction measurements in addition to numerical modeling,50−57 the interlamellar space of such swelling clays evolves as a function of the water partial pressure and accommodates in order to contain one or two hydration layers. In that study, a self-supporting clay film is prepared by ultrafiltration leading to partially oriented clay platelets. The hydration state of the film is controlled by exposure to vapor of heavy water under controlled temperature and chemical potential, leading to a wellcharacterized water monolayer.50 First, 2H NMR spectroscopy is performed to identify the specific orientation of these confined water molecules.58−61 Second, multiquantum 2H NMR relaxation measurements are performed to quantify46,62 the strength of the intrinsic quadrupular relaxation mechanism of the confined heavy water molecules in addition to the heteronuclear dipolar coupling induced by the paramagnetic impurities of the natural clay network. This study also quantifies the distribution of the clay platelets orientation within the clay sediment.63 Third,

Figure 2. (a) 2H NMR spectra recorded as a function of the film orientation βLF into the static magnetic field B0; (b) variation of the residual quadrupolar coupling νQ extracted from the 2H NMR spectra and the Hahn spin−echo attenuation (see text) as a function of the film orientation βLF into the static magnetic field B0. 26120



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relaxation rates is required to separately quantify46,62 the contributions from the quadrupolar and the heteronuclear dipolar relaxation mechanisms, respectively. The spin-locking relaxation rates72 were further measured to determine the mobility of the water molecules confined in the interlamellar space between two elementary clay particles by extracting the dispersion curves of the spectral densities quantifying the quadrupolar and heteronuclear dipolar relaxation mechanisms. For that purpose, three different pulse sequences were used (Figure 4) to determine the time evolution of the

II.ii. 2H NMR Measurements. 2H NMR spectra of heavy water were recorded on a DSX360 Bruker spectrometer operating at a field of 8.465 T. On this spectrometer, the pulse duration for the total inversion of the longitudinal magnetization is equal to 23 μs. Spectra were recorded using a fast acquisition mode with a sampling time of 0.25 μs, corresponding to a spectral width of 4 MHz. The spectra and relaxation rates were recorded for different orientations βLFof the film director nF,L ⃗ with reference to the static magnetic field B0 (see Figure 2a) using a homemade sample holder and detection coil.46 Figure 2a displays some 2H NMR spectra as a function of the orientation of the clay director within the static magnetic field. As previously shown by Monte Carlo simulations,46,49,50,62,63 the specific orientation of the confined water molecules is responsible for the splitting of the deuterium resonance lines displayed in Figure 2a. Such specific orientation of confined water molecules was already detected within partially oriented sediment of two-layer Nahectorite hydrates.13,46 Furthermore, the reduced asymmetry of the water resonance line is induced by the second-order relaxation mechanism resulting from the cross-correlation61 between the quadrupolar and heteronuclear dipolar couplings felt by the confined water molecules. A complete basis set, with eight independent operators, is required to describe the time evolution of spin I = 1 nuclei during any pulse sequence. A possible basis set is defined by the irreIR IR IR IR ducible tensor operator70,71 TIR 10, T11(a,s), T20, T21(a,s), T22(a,s). The first three operators correspond to the longitudinal (TIR 10) and transverse (TIR (a,s)) components of the spin magnetization. 11 The five residual operators describe the five components of the quadrupolar Hamiltonian. The relaxation rate constant of the TIR 10 coherence, also called longitudinal relaxation rate (R1), is noted here R10. It is measured by the classical inversion−recovery pulse sequence.72 The relaxation rate constant of the TIR 11(a,s) coherences, also called transverse relaxation rate (R2), is noted here R11. It is measured by the Hahn echo pulse sequence.73 The IR relaxation rates of the TIR 20 and T22(a,s) coherences are measured by adequate pulse sequences46 (Figure 3) and noted R20 and R22, respectively. This set of measurements of different 2H NMR

Figure 4. Pulse sequences and coherence transfer pathways used to measure the multiquantum spin-locking relaxation rates of the (a) IR IR IR TIR 11ρ(s), (b) T21ρ(a) and T21ρ(s), and (c) T22ρ(a) coherences, noted T11ρ(s), T21ρ(a), T21ρ(s), and T22ρ(a), respectively (see text). The delay ε is set equal to 10 μs. IR IR TIR 11(s), T21(a,s), and T22(a) coherences under irradiation, noted T11ρ(s), T21ρ(a,s), and T22ρ(a), respectively. As displayed in Figure 4, the transition between the zero-order initial coherence IR (TIR 10) and the required second-order coherences (T21(a,s) and 46 IR T22(a)) is performed by a single pulse, noted ψ10→21 and ψ10→22, respectively, exploiting the residual static quadrupolar coupling ωQ. Figure 5 exhibits the optimal transfer between these coherences as predicted by the simplified master equation which describes their time evolution under irradiation by neglecting their relaxation (see eq A3). Seven different irradiation powers are used, corresponding to the angular velocities ω1 equal to 1.36 × 105, 7.7 × 104, 4.3 × 104, 2.2 × 104, 1.0 × 104, 5 × 103, and 3 × 103 rad s−1. The measurement of two-time correlation function is performed using the pulse sequence illustrated in Figure 6a.

Figure 3. Pulse sequences and coherence transfer pathways used to IR measure the NMR relaxation rate of the (a) TIR 20 and (b) T22 (a,s) coherences. The delay δopt is selected to optimize46 the coherence transfer, and ε is set equal to 10 μs. 26121



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confined water molecules diffuse within the clay sediment and exchange between clay aggregates with different orientations. After that mixing period, these labile water molecules sample another residual coupling (ωQ(τM)) during the second evolution period (te). As a consequence, the detected magnetization varies according to74,75 I(te , τM) ∝ exp(− R 20τM − 2R11te)⟨cos(ωQ (0)te) × cos(ωQ (τM)te)⟩

(1)

The interferences between these various local environments lead to a net reduction of the measured intensity67,74,75 if the time scale quantifying their exchange is longer than the transverse relaxation time, corresponding to the so-called slow exchange regime. To avoid artifacts resulting from the recovery of the TIR 10 coherence during the mixing period (τM), a double-quantum filtering is applied in order to select only the contribution from the TIR 20 coherence. For that purpose, the duration of the fourth pulse of the sequence, noted ψ (Figure 6a), is carefully set to 33 μs (Figure 6b) to simultaneously optimize67 the transfer from IR the TIR 20 coherence to the T22 coherence and minimize the transfer IR from the T10 coherence using the set of eqs A4a and A4e of the Appendix by exploiting the set of angular velocities (k1, k2, k3) previously determined by spin-locking relaxation measurements (Figure 12e−h).

Figure 5. Schematic view of the optimization of the first pulse duration, denoted as Ψ10→21 and Ψ10→22 in the sequences illustrated in Figures 4b and 4c, respectively; used to perform the multiquantum spin-locking relaxation measurements T21ρ(a), T21ρ(s), and T22ρ(a) (see text). For this example, βLF = 0° and ω1 = 1.36 × 105 rad s−1.

III. RESULTS AND DISCUSSION III.i. 2H NMR Spectra. The residual quadrupolar splittings displayed in Figure 2a are the fingerprint of the specific orientation of the confined water molecules and are in complete agreement62,63 with predictions obtained by molecular modeling of the clay−water interfaces. As previously shown,62 the apparent residual quadrupolar splitting, noted ωobs Q , can also be extracted from the time evolution of the TIR 11(a,s) coherences (Figure 7) using the simple relationship IR IR T11 (τ ) = T11 (0) cos(ωQobsτ ) exp(−τR11)

(2)

Figure 2b displays the angular variation of the apparent residual splitting ωobs Q as a function of the orientation of the lamella βLF within the static magnetic field B0. If the single-spin population exists, the maximum value of the detected quadrupolar coupling results from the specific orientation13,46,49,50,62,63 of the water molecules confined between two clay platelets. As a consequence, Figure 2b should exhibit perfect agreement between the observed residual splitting ωobs Q and the theoretical relationship ωQobs(β LF) = ωQmax |P2 cos(β LF )| = ωQmax Figure 6. (a) Pulse sequence and coherence pathway used to measure the attenuation of the two-time 2H NMR echo I(te,τM) as a function of the evolution period te and the mixing time τM; (b) schematic view of the optimization of the fourth pulse duration, noted Ψ20→22, used to measure the two-time stimulated echo attenuation; for this example, βLF = 0° and ω1 = 1.36 × 105 rad s−1 (see text).

3 cos2(β LF) − 1 2 (3)

where P2 is the second-order Legendre polynomial and ωmax Q is the maximum value of the residual quadrupolar coupling detected for a perfect alignment between the static magnetic field B0 and the film director n⃗F,L. However, the data displayed in Figure 2b do not perfectly match the prediction of eq 3. Two conditions are required to explain such a discrepancy: first, the coexistence of various water environments, with different intrinsic residual quadrupolar coupling; second, the time scale (τexch) characterizing the exchange of the water molecules between these various spin environments must be longer than their transverse relaxation time R−1 11 . As a consequence, the 2H NMR spectra of the confined water molecules (Figure 2a) are characteristic of powder spectra of a

That experiment exploits the heterogeneities of the residual quadrupolar coupling67,74 felt by the water molecules confined within clay lamellae pertaining to different aggregates with various orientations into the static magnetic field B0. During the first evolution period (te), the first-order coherence TIR 11 oscillates according to the initial value of the residual quadrupolar coupling (ωQ(0)). During an appropriate mixing period (τM), the 26122



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Figure 8. Variations as a function of the film orientation βLF into the static magnetic field B0 (a) of the apparent multiquantum relaxation IR IR IR rates of the TIR 10, T11(a,s), T20, and T22(a,s) coherences, noted R10, R11, R20, and R22, respectively, and (b) of the apparent spectral densities JQ0 (0), JD0 (0), UQ, and UD extracted from these Rij values (see eqs 4a−4e).

the focus is placed on the leading contributions, it is possible46,62 to simplify the set of eqs A7 and A10:

Figure 7. Raw data and fits of the Hahn spin−echo attenuation obtained from a complete simulation of the time evolution of the TIR 11(a,s) coherences during the NMR pulse sequence for few clay film orientations βLF into the static magnetic field B0: (a) βLF = 0°, (b) βLF = 30°, (c) βLF = 50°, (d) βLF = 70°, and (e) βLF = 90°.

R10 = 5UQ + R11 = R 21 =

partially oriented sample13,46,63 under the so-called slow exchange regime. III.ii. 2H NMR Multiquantum Relaxation Measurements. NMR relaxation measurement is a powerful tool for extracting dynamical information of the probes once their relaxation mechanisms are identified. In addition to their intrinsic quadrupolar relaxation mechanism, confined water molecules are also subject to heteronuclear dipolar coupling because of the paramagnetic impurities present in the atomic clay network. As shown previously, multiquantum relaxation measurements (see Appendix) may be used to separately quantify the strength of both relaxation mechanisms in the so-called slow modulation regime. As displayed in Figure 8a, the transverse relaxation rate (R11) of the confined water molecules is roughly 3 orders of magnitude larger than their longitudinal relaxation rate (R10), fully validating the above-mentioned requirement. As a consequence, the m = 0 spectral densities (i.e., JQ0 (0) and JD0 (0) in eqs A7 and A10) are much larger than the other components (JQ0 (0) > JQm (ω), m ∈ {1, 2} and JD0 (0) > JDm(ω), m ∈ {1, 2}). When

UD 3

(4a)

3 Q 5 2 1 1 J (0) + UQ + J0D(0) + UD + J1D(ωS) 2 0 2 9 2 3 (4b)

R 20 = 3UQ + UD R 22 = 3UQ +

(4c)

8 D 1 4 J (0) + UD + J1D(ωS) 9 0 3 3

(4d)

with UQ = J1Q (ω0) ≈ J2Q (2ω0) and UD = + 2J2D(ωS + ω0)

1 D J (ωS − ω0) + J1D(ω0) 3 0 (4e)

where ω0 and ωS are respectively the angular velocities (rad s−1) of deuterium and unpaired electron of iron. When the highfrequency contribution from JD1 (ωs) is neglected, Figure 8b exhibits the selected spectral densities containing all the dynamical features of these NMR relaxation measurements. The four parameters (UQ, UD, JQ0 (0), and JD0 (0)) are extracted from the measured relaxation rates (R10, R11, R20, and R22) by 26123



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solving the set of linear eqs 4a−4d. As displayed in Figure 8b, the m = 0 component of the spectral density quantifying the quadrupolar relaxation mechanism (JQ0 (0)) varies over 1 order of magnitude as a function of the orientation of the clay lamella βLF within the static magnetic field B 0 . By contrast, the corresponding spectral density quantifying the heteronuclear dipolar coupling (JD0 (0)) exhibits only a reduced variation. By using the Wigner rotation matrices,33 one can further extract from these angular variations of the apparent spectral densities JXm(ω) (with X ∈ {Q, D} and m ∈ {0, 1, 2 }) measured in the laboratory frame, the intrinsic spectral density JX,intrinsic (ω) m evaluated in the frame attached to the clay film for each relaxation mechanism76 (i.e., X ∈ {Q, D}): 1 (1 − 3 cos2 β LF)2 J0X,intrinsic (ω) 4 3 + 3cos2 β LF sin 2 β LFJ1X,intrinsic (ω) + (1 − cos2 β LF)2 J2X,intrinsic (ω) 4

J0X (β LF , ω) =

(5a) J1X (β LF , ω) =

3 cos2 β LF sin 2 β LFJ0X,intrinsic (ω) 2

1 (1 − 3 cos2 β LF + 4 cos 4 β LF)J1X,intrinsic (ω) 2 1 + (1 − cos 4 β LF)J2X,intrinsic (ω) 2 +

J2X (β LF , ω) =

(5b)

3 (1 − cos2 β LF)2 J0X,intrinsic (ω) 8

1 (1 − cos 4 β LF)J1X,intrinsic (ω) 2 1 + (1 + 6 cos2 β LF + cos 4 β LF)J2X,intrinsic (ω) 8

Figure 9. (a) Schematic view of the Euler angles characterizing the orientation of an individual clay platelet within the macroscopic film. (b) Schematic view of the film orientation within the NMR tube used to insert the clay sample into the detection coil (see text).

+

(5c)

The intrinsic spectral densities quantifying the heteronuclear dipolar relaxation mechanism (JD,intrinsic (0) = (2000 ± 200) s−1, 0 −1 D,intrinsic D,intrinsic J1 (0) = (1800 ± 200) s , and J2 (0) = (900 ± 100) s−1) are directly extracted from the data displayed in Figure 8b using the set of eqs 5a−5c. The determination of the three other spectral densities (JQ,intrinsic (0), JQ,intrinsic (0), and JQ,intrinsic (0)) requires further 0 1 2 analysis of the raw data displayed in Figure 8b by including the distribution of the clay directors within the sediment because of the sharp angular variation of the apparent spectral densities. We further exploit the large variation of the apparent spectral densities quantifying the quadrupolar relaxation mechanism (Figure 8b) to determine the distribution of the clay directors within the film (Figure 2a) by carefully analyzing the time evolution of the transverse relaxation rate (R11). For that purpose, the orientation of the clay platelets by reference with the film is characterized by the two Euler angles αFC (the colatitude) and γFC (the azimuth) (see Figure 9a). In the frame of the film, the normal to the clay platelet (n⃗C,F) becomes: n ⃗C,F

⎛ sin α FC cos γ FC ⎞ ⎜ ⎟ = ⎜ sin α FC sin γ FC ⎟ ⎜⎜ ⎟⎟ ⎝ ⎠ cos α FC

within the detection coil, we obtain the angle between the static magnetic field B0 and the clay director, noted θLC, defined by cos θ LC = sin β LF sin α FC sin γ FC + cos β LF(sin δ LF sin α FC cos γ FC + cos δ LF cos α FC)

Using the set of eqs A3, A7 and A10, it is possible to simulate the time evolution of the transverse magnetization during each step of the Hahn echo−pulse sequence. These simulations are performed by sampling the angle αFC to satisfy a Gaussian distribution law while the angle γFC is generated randomly in the interval [0,2π]. As a consequence, the set of eqs 5a−5c is still valid by simply replacing the cos2 βLF and cos4 βLF formulas by their average values evaluated with the Euler angle θLC: ⟨cos2 θ LC⟩ = +

2 sin 2 β LF sin 2 α FC + cos2 β LF cos2 δ LF cos2 α FC 3

1 cos2 β LF sin 2 δ LF sin 2 α FC 3

(7a)

and ⟨cos 4 θ LC⟩ =

(6a)

In the frame attached to the tube containing the clay sample (Figure 9b), we obtain n ⃗C,L

(6c)

⎛ cos δ LF sin α FC cos γ FC − sin δ LF cos α FC ⎞ ⎟ ⎜ ⎟ =⎜ sin α FC sin γ FC ⎟ ⎜ ⎜ LF FC FC LF FC ⎟ ⎝ sin δ sin α cos γ + cos δ cos α ⎠

8 sin 4 β LF sin 4 α FC 15

+ cos 4 β LF cos 4 δ LF cos 4 α FC 1 + cos 4 β LF sin 4 δ LF sin 4 α FC 5 4 + sin 2 β LF cos2 β LF sin 2 δ LF sin 4 α FC 5

(6b)

+ 4 sin 2 β LF cos2 β LF cos2 δ LF sin 2 α FC cos2 α FC

by including a possible rotation δ of the film along the axis YL (Figure 9b). Finally, after the rotation βLF of the NMR tube LF

+ 2 cos 4 β LF sin 2 δ LF cos2 δ LF sin 2 α FC cos2 α FC 26124

(7b)



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Typical fits are displayed in Figure 7, by setting to standard deviation of the Gaussian distribution of the αFC angle (σ = (7.5 ± 0.5)°) and the angle quantifying the misalignment of the film by reference with the coil (see Figure 9b: δLF = (6 ± 1)°). The fit also leads to the intrinsic spectral densities quantifying the quadrupolar relaxation mechanism (JQ,intrinsic (0) = (2 200 ± 0 200) s−1, JQ,intrinsic (0) = (30 000 ± 5000) s−1, and JQ,intrinsic (0) = 1 2 (100 ± 50) s−1). Such a large enhancement of the intrinsic spectral density (JQ,intrinsic (0)) by comparison with the two other 1 components of the quadrupolar relaxation mechanism (JQ,intrinsic (0) and JQ,intrinsic (0)) has already been reported for 0 2 various quadrupolar nuclei confined in dense clay sediments46,77 and predicted by multiscale numerical modeling.45 Because of the angular distribution of the clay directors within the film, the m = 1 spectral density of the quadrupolar coupling always contributes significantly to the transverse relaxation of the confined water molecules, whatever the orientation of the clay film within the static magnetic field B0. This difference between the intrinsic spectral densities results from the anisotropy of the confined water molecules. For isotropic liquids, the three spectral densities (see eq A8) coincide because the corresponding autocorrelation functions33,78 GQm (τ) are identical. As an example, their initial values (GQm (0)) become (see eqs A6f−A6h) GmQ (0) =

1 4π

∫0

2π

dϕLW

∫0

π

F2,Qm(0)F2,Q−m(0) sin θ LW dθ LW =

1 5 (8)

Figure 10. (a) Snapshot illustrating one GCMC equilibrium configuration of the confined water molecules and neutralizing sodium counterions; (b) concentration profiles of the sodium counterions and oxygen and hydrogen atoms pertaining to the water molecules confined between two hectorite clay lamellae.

By contrast, for anisotropic liquids, the initial values of the autocorrelation functions GQm (0) are related to the distribution of the OD directors by reference to the magnetic field B0: +1 1 (9 cos 4 θ LW − 6 cos2 θ LW + 1)f (cos θ LW ) dcos θ LW 8 −1 9f4 − 6f2 + f0

∫

G0Q (0) = =

3 +1 (cos2 θ LW − cos 4 θ LW )f (cos θ LW ) dcos θ LW 4 −1 3(f2 − f4 )

∫

G1Q (0) = =

(9b)

2

+1 3 (cos 4 θ LW − 2 cos2 θ LW + 1)f (cos θ LW ) dcos θ LW 16 −1 3(f4 − 2f2 + f0 )

G2Q (0) = =

(9a)

4

∫

8

(9c)

where f(cos θ ) is the distribution law of the OD directors and f 0, f 2, and f4 are its zero-order, second-order, and fourth-order moments, respectively. By further assuming that the three reduced autocorrelation functions gQm(τ) = GQm (τ)/GQm(0) evolve according to the same master curve, it becomes possible to extract the value of the second-order and fourth-order moments from the intrinsic spectral densities JQ,intrinsic (0) because they m become directly proportional to the corresponding initial value of the autocorrelation functions (i.e., JQ,intrinsic (0) = Cte GQm (0) m with m ∈ {0,1,2}). For the set of eqs 9a−9c, we obtain f 2/f 0 = 0.7 ± 0.1 and f4/f 0 = 0.4 ± 0.05. This result suggests a distribution law f(cos θLW) centered around |cos θLW| ≈ 1, corresponding to OD directors mainly pointing along the normal to the clay surface. GCMC50 or MD79 numerical simulations may be used to check this conclusion without any hypothesis concerning water dynamics. As clearly shown in Figure 10a,b, the oxygen atoms are strongly confined within the equatorial plane of the clay interlamellar space. By contrast, the OD directors are distributed LW

Figure 11. Histogram of the distribution laws extracted from the GCMC equilibrium configuration and used to analyze the distribution laws of |cos θLW| and its second-order f 2 and fourth-order f4 moments (see text).

along two different configurations characterized by Euler angle θLW centered either along 0 or π/2. This point is better illustrated by analyzing the distribution laws of the various moments of f(cos θLW) (Figure 11), leading to the average values f 0 = 1, f 2 = 0.40 ± 0.02, and f4 = 0.30 ± 0.02. While the numerical results are significantly smaller than the above-mentioned experimental results ( f 2/f 0 = 0.7 and f4/f 0 = 0.4), they largely exceed the values of the moments expected for isotropic liquids ( f iso 2 = 1/3 and fiso 4 = 1/5). As a consequence, GCMC numerical simulations 26125



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IR IR IR Figure 12. Time evolution of the (a) TIR 11(s), (b) T21(a), (c) T21(s), and (d) T22(a) coherences measured under spin-lock conditions, noted T11ρ(s), IR IR IR T21ρ(a), T21ρ(s), and T22ρ(a), respectively, and their Fourier transforms for (e) TIR 11(s), (f) T21(a), (g) T21(s), and (h) T22(a) coherences. For these LF 5 −1 examples, β = 20° and ω1 = 1.36 × 10 rad s .

order to extract the frequency variation of the m = 0 and m = 1 components of the quadrupolar and heteronuclear dipolar couplings. Because of reduced values of the m = 2 spectral densities (see Section III.ii.), the spin-locking relaxation measurements do not allow extraction of their frequency variations. IR Figure 12a−d displays the time-evolution of the TIR 11(s), T21(a,s), IR and T22(a) coherences under irradiation, while their Fourier transforms (Figure 12e−h) clearly illustrate the characteristic angular velocities71 (k1, k2, and k3). These angular velocities are listed in Table 1 as a function of the irradiation power ω1 and the film orientation βLF. This set of four independent spin-locking measurements is sufficient to extract the desired spectral densities, i.e., JQm(k1) for the quadrupolar coupling (eq A13) in

qualitatively validate the anisotropy of the water distribution that (0). may be extracted from the intrinsic spectral densities JQ,intrinsic m III.iii. 2H NMR Relaxation Measurements under SpinLocking. The long-distance mobility of the confined water molecules is investigated by extracting the dispersion curves of the various spectral densities characterizing the quadrupolar and heteronuclear dipolar couplings from the analysis of the frequency variations of the multiquantum relaxation rates under spin-locking conditions. As illustrated in the Appendix, spin-locking relaxation measurements allow probing of a broad domain of angular velocities by exploiting the residual quadrupolar coupling. For that purpose, we selected three different orientations of the clay film (βLF =0°, 20°, and 90°) in 26126



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Table 1. Set of Angular Velocities k1, k2, and k3 (eq A5a−A5c) Detected from the Fourier Transform of the Time Evolution of the Multiquantum Coherences As a Function of the Irradiation Power ω1 and the Euler Angle βLF k1 (105 rad s−1) ω1 (105 rad s−1)

βLF = 0°

βLF = 20°

βLF = 90°

1.36 0.77 0.43 0.22 0.10 0.05 0.03

2.88 1.73 1.18 0.92 0.83 0.81 0.81

2.81 1.72 1.12 0.87 0.79 0.77 0.76 k2 (105 rad s−1)

2.74 1.63 0.97 0.60 0.46 0.42 0.41

ω1 (105 rad s−1)

βLF = 0°

βLF = 20°

βLF = 90°

1.36 0.77 0.43 0.22 0.10 0.05 0.03

1.84 1.27 0.99 0.86 0.82 0.81 0.81

1.79 1.24 0.94 0.82 0.78 0.77 0.76 k3 (105 rad s−1)

1.57 1.02 0.69 0.51 0.43 0.41 0.41

ω1 (105 rad s−1)

βLF = 0°

βLF = 20°

βLF = 90°

1.36 0.77 0.43 0.22 0.10 0.05 0.03

1.04 0.46 0.19 0.056 0.012 0.003 0.0005

1.02 0.48 0.18 0.054 0.013 0.005 0.001

1.16 0.61 0.28 0.097 0.024 0.004 0.002

Figure 13. Variation of the intrinsic spectral densities JQ,intrinsic (⟨k1⟩) m (⟨ki⟩) with m ∈ {0, 1} describing the quadrupolar coupling and JD,intrinsic m with m ∈ {0, 1} and i ∈ {2, 3} describing the heteronuclear dipolar coupling as a function of the corresponding averaged angular velocities ⟨k1⟩ (see text).

are displayed in Figure 12. The fitted parameters are collected in Figure 13. The simultaneous analysis of the dispersion curves corresponding to the quadrupolar and the heteronuclear dipolar couplings allow probing a large dynamical range by sampling the angular velocities over three decades. Because of the large difference between the orders of magnitude of the various intrinsic spectral densities, we selected to plot the reduced values of the intrinsic spectral densities (jX,intrinsic (⟨ki⟩) = jX,intrinsic (⟨ki⟩)/ m m X,intrinsic jm (0), with X ∈ {Q, D} and m ∈ {0, 1}), leading to a single master curve (Figure 13). Two different dynamical regimes are clearly identified: a low-frequency plateau and a high-frequency decrease, with a sharp transition at the critical angular velocity (ωC = (2 ± 1) × 104 rad s−1). As previously shown by numerical simulations,28 that critical angular velocity corresponds to the average residence time (τC = (50 ± 20) μs) of the water molecules confined within the interlamellar space between two clay platelets. That residence time of the water molecules strongly confined in the so-called “one-layer hydration state” is twice the residence time previously measured46 for water molecules confined within the same clay platelets in the “two-layer hydration state”. This behavior is fully compatible with dynamical results obtained at a much smaller time-scale by QENS experiments28 and MD simulations.28 Note finally that the large uncertainties of the characteristic angular velocities occurring at the weakest irradiation powers (Table 2) have limited impact on the analysis of the data because these largest uncertainties mainly occur in the plateau of the dispersion curve (Figure 13). III.iv. Two-Time Stimulated Echo Spectroscopy. In addition to the local exchange of the water molecules between the interlamellar space of the clay platelets and its free surrounding (Figure 1) probed by NMR spin-locking relaxation measurements, two-time stimulated echo NMR spectroscopy may be used to extract long-range dynamical information67 related to the exchange of the water molecules between clay aggregates characterized by different orientations of their director by reference with the static magnetic field. Such a procedure requires two complementary conditions: (i) a slow exchange, at the NMR time-scale, of the water molecules pertaining to these various aggregates and (ii) a slow modulation of the quadrupolar and dipolar coupling responsible for the relaxation of these confined water molecules.

Table 2. Set of Single Mean Angular Velocity ⟨ki⟩ (i ∈ {1, 2, 3}) as a Function of the Irradiation Power ω1 and Averaged over the Three Sets of Measurements at Various Euler Angles βLF (Table 1) ω1 (105 rad s−1)

⟨k1⟩ (105 rad s−1)

⟨k2⟩ (105 rad s−1)

⟨k3⟩ (105 rad s−1)

1.36 0.77 0.43 0.22 0.10 0.05 0.03

2.80 ± 0.06 1.70 ± 0.04 1.10 ± 0.05 0.77 ± 0.13 0.69 ± 0.13 0.67 ± 0.19 0.67 ± 0.19

1.73 ± 0.12 1.17 ± 0.11 0.87 ± 0.13 0.73 ± 0.16 0.67 ± 0.17 0.66 ± 0.18 0.66 ± 0.18

1.08 ± 0.06 0.52 ± 0.07 0.22 ± 0.05 0.07 ± 0.02 0.016 ± 0.005 0.004 ± 0.001 0.001 ± 0.0005

addition to JDm(k2) and JDm(k3) for the heteronuclear dipolar coupling (eq A15). To reduce the number of fitted parameters, we selected to average,46 at each irradiation power ω1, the characteristic angular velocities ⟨ki⟩ over the three film orientations βLF. As displayed in Table 2, this approximation is valid at high-irradiation power but leads to large deviations of the characteristic angular velocities ⟨ki⟩ at the weakest irradiation power. Using the set of eqs A3, A7, A10, A13, and A15, it is possible to simulate the time evolution of the coherences during each step of the pulse sequences (Figure 4) by taking into account the abovementioned distribution of the clay directors (see Section III.ii.). This analysis allows extracting the frequency variation of the intrinsic spectral densities of both the quadrupolar and heteronuclear dipolar relaxation mechanisms. Typical results 26127



Article

dated by the analysis of the relaxation measurements of the transverse magnetization (see Section III.ii.). The measurements of the two-time echo attenuation have been performed at two different orientations of the clay film (βLF = 0° and 90°). The enhancement of the transverse relaxation rate (R11) at intermediate orientations (Figure 8a) restricts the range of applicability of that experimental procedure. Figure 14 exhibits the echo attenuation as a function of the mixing time τM of the pulse sequence (see Figure 6a and eq 1). Before analysis, the raw data are renormalized67 (Figure 14b,d) to remove the attenuation of the magnetization induced by the intrinsic relaxation of the TIR 20 coherence during the same mixing time (i.e., the factor exp(−R20 τM) in eq 1). To improve the signal/ noise ratio, the echo attenuation displayed in Figure 15 is

Figure 15. Two-time correlation function extracted from the normalized stimulated echo attenuation as a function of the mixing time τM and the clay film orientations βLF = 0° and βLF = 90° (see text).

evaluated by recording the intensity of the second maximum67 in Figure 14b,d. As displayed in Figure 15, a single correlation function characterizes the water exchange between different clay aggregates independent of the orientation of the self-supporting clay film in the static magnetic field. Here also two different dynamical regimes are clearly identified (a plateau at short mixing times and a sharp decrease at long mixing times) leading to an exchange time of τexch ≈ (50 ± 15) ms, i.e., 3 orders of magnitude larger than the characteristic time detected by spin-locking relaxation measurements (τC ≈ (50 ± 20) μs). This coexistence of two separated exchange times is the fingerprint of the multiscale organization of the clay sediment. While spin-locking relaxation measurements are sensitive to the local exchange between confined and free water molecules, the two-time stimulated echo attenuation quantifies the long-time exchange between clay aggregates characterized by different orientations.67,80 Note that such a coexistence of various environments also contributes to the relaxation of the coherences (see Section III.ii.) but without any possibility of quantification of the corresponding exchange time. Finally, an upper limit of the water displacement may be evaluated using the self-diffusion coefficient of bulk water

Figure 14. Variation of the two-time stimulated echo attenuation I(te,τM) as a function of the mixing time τM for the film orientation βLF = 0°: (a) raw data and (b) normalized data to take into account the relaxation of the TIR 20 coherence during the mixing time τM (i.e., the factor exp(−R20τM), see eq 1). For βLF = 90°: (c) raw data and (d) normalized data.

As illustrated in Figure 8a,b, this last requirement is fully satisfied, while coexistence of clay platelets with different orientation is suggested by the angular variation of the residual quadrupolar couplings displayed in Figure 2b and further vali-

Lmax =

2D bulk τexch ≈ (13 ± 1) μm

(10)

i.e., 1 order of magnitude larger than the size of the individual confining platelets. 26128



■

IV. CONCLUSIONS We exploited the various possibilities of 2H NMR spectroscopy to determine the structural and dynamical properties of water molecules confined within the multiscale structure of dense clay sediment. By combining 2H NMR spectroscopy, multiquantum relaxation measurements, multiquantum spin-locking relaxometry, and two-time stimulated echo attenuation, we totally characterize the NMR relaxation mechanisms of these confined water molecules, allowing for the extraction of dynamical information on the long-distance water mobility within the multiscale porous network of the clay sediment. The residual quadrupolar coupling of the water molecules detected by 2H NMR spectroscopy is the fingerprint of the specific orientation induced by water confinement within the clay interlamellar space. Multiquantum relaxation measurements are used to quantify the degree of ordering of the individual clay platelets within the sediment. Spin-locking multiquantum relaxation measurements are sensitive to the average residence time of the confined water molecules pertaining to the first hydration layer of the swelling clay. Finally, two-time stimulated echo attenuation probes the long-distance water mobility within the porous network between the various clay microdomains. Because these experimental procedures are appropriate to quantify the mobility of a large class of NMR quadrupolar probes, they can be successfully applied to study the mobility of confined fluids within different porous networks, including natural or synthetic membranes or zeolytic networks. ⎛ T IR ⎞ ⎛ ⎜ 20 ⎟ ⎜ ⎜T IR (a)⎟ ⎜ ⎜ 11 ⎟ ⎜ ⎜ T IR (s) ⎟ ⎜− ⎜ 21 ⎟ ⎜ ⎜ T IR (s) ⎟ ⎜ d ⎜ 22 ⎟ = i⎜ dt ⎜ T IR ⎟ ⎜ ⎜ 10 ⎟ ⎜ ⎜ IR ⎟ ⎜ ⎜ T11 (s) ⎟ ⎜ ⎜ IR ⎟ ( ) T a ⎜ ⎜ 21 ⎟ ⎜ ⎜ IR ⎟ ⎝ ⎝T22 (a)⎠

Article

APPENDIX

A.i. Time Evolution of the Coherences

In the framework of the Redfield theory,81 the time evolution of the coherence is described by the complete master equation33,82,83 dσ * = −i[HS*, σ *] + f (σ *) dt

where all calculations are performed in the Larmor frequency rotating frame, as indicated by the asterisk. The first contribution to the master equation describes the influence of the static Hamiltonian HS* that includes the residual static quadrupolar Hamiltonian (H*QS = (2/3)1/2 ωQ TIR 20) and the Hamiltonian corresponding to the radio frequency pulse (H*1S = Ix ω1 = √2 ω1TIR * = √2 ωres TIR 11(a)). The Zeeman-like Hamiltonian (HZS 10) results from the frequency offset (ωres). The second contribution to the master equation describes the contribution from the fluctuating components of the Hamiltonians (H*F (t)) monitoring the relaxation of the coherences:33,82,83 f (σ *) = −

− 3 ω1

0

0

0

0

0

0

ωQ

0

0

ωres

0

0

− ω1

0

0

ωres

0

− ω1

0

0

0

0

0

0

0

0

0

− ω1

0

0

ωres

0

0

− ω1

0

ωQ

0

0

ωres

0

0

ωQ

0

0

0

0

2ωres

0

0

− ω1

= iωQ

sub-sets:71

+ +

k12 3 ω12

3 ω1ωQ

+ ω1ωQ

+ (A4a)

ωQ2 cos(k1τ ) + 4ω12 k12

+

IR T11 (a)

3 ω12

(A4c)

k12 − ω12(1 − cos(k1τ )) IR T22 (s) k12

− iω1 −i

(A4b) 26129

1 − cos(k1τ ) IR T11 (a) k12

cos(k1τ ) − 1 IR sin(k1τ ) IR T20 − iω1 T21 (s) k1 k12

IR exp(iHS*τ ) T10 exp(− iHS*τ ) =

1 − cos(k1τ ) IR sin(k1τ ) IR T20 + iωQ T21 (s) k1 k12

1 − cos(k1τ ) IR T22 (s) k12

(A3)

sin(k1τ ) IR T22 (s) k1

IR exp(iHS*τ ) T22 (s) exp(− iHS*τ ) = ω1ωQ

sin(k1τ ) IR T21 (s) k1

cos(k1τ ) − 1 IR T22 (s) k12

IR exp(iHS*τ ) T11 (a) exp(− iHS*τ ) =

+

1 − cos(k1τ ) IR T11 (a) k12

IR − i 3 ω1 T20

⎛ T IR ⎞ 0 ⎞ ⎜ 20 ⎟ ⎟ ⎜ IR ⎟ 0 ⎟ ⎜T11 (a)⎟ ⎟ ⎜ IR ⎟ 0 ⎟ ⎜ T21 (s) ⎟ ⎟ ⎜ IR ⎟ 2ωres ⎟ ⎜ T22 (s) ⎟ ⎟· IR ⎟ 0 ⎟ ⎜ T10 ⎟ ⎜ ⎟ ⎜ IR ⎟ 0 ⎟ T (s) ⎜ 11 ⎟ − ω1 ⎟⎟ ⎜T IR (a)⎟ 21 ⎟ ⎟⎜ 0 ⎠ ⎜ IR ⎟ ( ) T a ⎝ 22 ⎠

sin(k1τ ) IR sin(k1τ ) IR IR T11 (a) − i 3 ω1 T20 + cos(k1τ ) T21 (s) k1 k1

− iω1

ωQ2 + ω12(1 + 3 cos(k1τ ))

(A2)

IR exp(iHS*τ ) T21 (s) exp(− iHS*τ )

the time evolution of the coherences splits into two independent

3 ω1ωQ

⟨[HF*(t ), [exp( −iHS*τ ) HF* +(t − τ )

Let us, in a first approximation, neglect the contribution from the relaxation mechanisms. The resulting simplified master equation may be written in a matrix form:71

0

3 ω1 ωQ

∞

× exp(iHS*τ ), σ *(t )]]⟩ dτ

By carefully operating at the resonance frequency (ωres= 0),

IR exp(iHS*τ ) T20 exp(− iHS*τ ) =

∫0

0

0

(A1)

(A4d)

k 2 cos(k 3τ ) + k 3 cos(k 2τ ) IR T10 k1

sin(k 3τ ) + sin(k 2τ ) IR cos(k 3τ ) − cos(k 2τ ) IR T11 (s) + ω1 T21 (a) k1 k1

k 2 sin(k 3τ ) − k 3sin(k 2τ ) IR T22 (a) k1

(A4e)


The Journal of Physical Chemistry C IR exp(iHS*τ ) T11 (s) exp(− iHS*τ ) = − iω1

+

Article

along the OD bond, the functions FQ2,m(t) are related to the two Euler angles (θLW, ϕLW):

sin(k 3τ ) + sin(k 2τ ) IR T10 k1

k 2 cos(k 2τ ) + k 3 cos(k 3τ ) IR T11 (s) k1

+i

Q F2,0 (t ) =

k 2 sin(k 2τ ) − k 3 sin(k 3τ ) IR cos(k 3τ ) − cos(k 2τ ) IR T21 (a) + ω1 T22 (a) k1 k1

F2,Q±1(t ) = ( ±1)

(A4f) exp(iHS*τ ) +i +

IR T21 (a)

cos(k 3τ ) − cos(k 2τ ) IR exp(− iHS*τ ) = ω1 T10 k1

(A4g)

where the characteristic angular velocities defined by k1 = k2 =

(A4h) 71

k1, k2, and k3 are

ωQ2 + 4ω12 ωQ +

ωQ2

(A5a)

+

(A7)

AQ = 3 J1Q (ω0)

(A5b)

BQ =

3 Q 5 J (0) + J1Q (ω0) + J2Q (2ω0) 2 0 2

A.ii. Quadrupolar Relaxation

CQ =

In bulk water, the quadrupolar coupling is the main mechanism responsible for relaxation of heavy water:33,82,83

3 Q 1 J0 (0) + J1Q (ω0) + J2Q (2ω0) 2 2

DQ = J1Q (ω0) + 2 J2Q (2ω0)

k3 =

ωQ2 + 4ω12 − ωQ (A5c)

2

2

HQ (t ) = CQ

∑

( −1)m T2,QmF2,Q−m(t )

m =−2

E Q = J1Q (ω0) + 4 J2Q (2ω0)

(A6a)

The spectral densities (JQm (mω0) used in eq A7 are the Fourier transform of the autocorrelation functions GQm(τ) describing the loss of memory of the fluctuating part of the quadrupolar coupling:

where the quadrupolar coupling constant, defined by CQ =

e 2qQ 3 8 ℏI(2I − 1)

(A6b)

JmQ (mω0) = (−1)m CQ2

is equal to (3/2) π·210 kHz for deuterium in bulk heavy water.84 The spin operators describing the quadrupolar coupling are given by 1 Q IR T2,0 = (3Iz2 − I(I + 1)) = T20 (A6c) 6 1/2

∫0

∞

(F2,Q−m(0) − ⟨F2,Q−m⟩)(F2,Qm(τ )

− ⟨F2,Qm⟩) exp(imω0τ ) dτ

Because of the presence of iron within the natural hectorite clay particle, the hetero-nuclear dipolar coupling may be responsible for the NMR relaxation of the confined heavy water, in addition to the intrinsic quadrupolar relaxation mechanism. The corresponding Hamiltonian33,82,83 is defined by

(A6d)

2

and 1 2 1 IR IR I± = (T22 (s) ∓ T22 (a)) 2 2

(A8)

A.iii. Hetero-Nuclear (Paramagnetic) Dipolar Relaxation

1 1 IR IR T2,Q±1 = ∓ (IZI± + I±IZ) = (T21 (s) ∓ T21 (a)) 2 2

T2,Q±2 =

(A6h)

with

4ω12

2

(A6g)

⎛ T IR ⎞ ⎛ T IR ⎞ ⎜ 20 ⎟ ⎜ 20 ⎟ ⎜T IR (a)⎟ ⎜T IR (a)⎟ ⎜ 11 ⎟ ⎜ 11 ⎟ ⎜ T IR (s) ⎟ ⎜ T IR (s) ⎟ ⎜ 21 ⎟ ⎜ 21 ⎟ ⎜ IR ⎟ ⎜ T IR (s) ⎟ 22 d ⎜ T22 (s) ⎟ ⎟ = − diag(AQ , BQ , C Q , DQ , EQ , BQ , C Q , DQ )·⎜ ⎜ T IR ⎟ dt ⎜ T IR ⎟ 10 10 ⎜ ⎟ ⎟ ⎜ ⎜ IR ⎟ ⎜ IR ⎟ ( ) T ( s ) T s 11 11 ⎜ ⎟ ⎟ ⎜ ⎜ IR ⎟ ⎜ IR ⎟ ⎜T21 (a)⎟ ⎜T21 (a)⎟ ⎜ IR ⎟ ⎜ IR ⎟ ⎝T22 (a)⎠ ⎝T22 (a)⎠

cos(k 3τ ) − cos(k 2τ ) IR sin(k 3τ ) + sin(k 2τ ) IR T11 (s) − iω1 T21 (a) k1 k1

k 2 cos(k 3τ ) + k 3 cos(k 2τ ) IR T22 (a) k1

3 sin 2θ LW (t ) exp(∓ iφLW (t )) 8

The contribution from the quadrupolar relaxation mechanism to the complete master equation (eq A1) may also be written in a matrix form:

k 2 cos(k 2τ ) + k 3 cos(k 3τ ) IR sin(k 3τ ) + sin(k 2τ ) IR T21 (a) − iω1 T22 (a) k1 k1

k sin(k 3τ ) − k 3 sin(k 2τ ) IR IR T10 exp(iHS*τ ) T22 (a) exp(− iHS*τ ) = − i 2 k1

+

(A6f)

3 sin 2 θ LW (t ) exp(∓ 2iφLW (t )) 8

F2,Q±2(t ) =

k 2 sin(k 2τ ) − k 3 sin(k 3τ ) IR T11 (s) k1

+ ω1

3 cos2 θ LW(t ) − 1 2

HD(t ) = C D

∑

( −1)m

T2,DmF2,D−m(t )

m =−2

(A6e)

rIS3(t )

where the dipolar coupling constant is given by μ C D = − 0 6 γIγSℏ 4π

The functions FQ2,m(t) in eq A6a are the second-order spherical harmonics33 describing the reorientation of the OD director in the static magnetic field B0. Because the electrostatic field gradient felt by the deuterium nucleus in heavy water is directed

(A9a)

(A9b)

and the spin operators become 26130



Article

⎞ 1 ⎛⎜ 1 2IzSz − (I+S− + I −S+)⎟ ⎝ ⎠ 2 6 1 1 IR IR = Sz + [T11 2 2 T10 (s)(S− − S+) 2 6

D T20 =

JmD(ω) = ( −1)m NSS(S + 1)C D2

{

−

}

IR − T11 (a)(S− + S+)]

T2D± 2 =

rIS3

(A9c)

∫

⎞ ⎛ F D (τ ) ⎟⎜ 2, m ⎟ ⎜ r 3 (τ ) − ⎠⎝ IS

F2,Dm rIS3

⎞ ⎟ exp(imω0τ ) dτ ⎟ ⎠ (A11)

where NS is the total number of paramagnetic spin coupled to the deuterium nucleus.

1 1 IR T2D± 1 = ∓ (IzS± + I±Sz) = ∓ [T10 S± 2 2 IR IR ∓ (T11 (s) ∓ T11 (a))Sz]

F2,D−m

∞ ⎛ F D (0) ⎜ 2, −m ⎜ 3 0 ⎝ rIS(0)

A.iv. Contributions to the Spin-Locking Relaxation Rate

(A9d)

1 1 IR IR I±S± = ∓ (T11 (s) ∓ T11 (a))S± 2 2

The relaxation of the coherences during the spin-locking experiments is described by the complete master eq A1. We focus only on the m = 0 contribution of eq A2 because the m = ± 1 and m = ± 2 components are only slightly modified71 by the weak irradiation field. For the quadrupolar coupling, the corresponding m = 0 component is given by

(A9e)

where Sα stems for the various components of the electronic spin. The functions FD2,m(t) in eq A9a are the same as in eq A6a but they describe now the reorientation of the vector joining the two coupled spin (noted rI⃗ S(t)) by reference with the static magnetic field B0. In addition to that angular dependency, the dipolar Hamiltonian is also very sensitive to the diffusion of the probe through the variation of the separation between the coupled spins (cf. the term r−3 IS (t) in eq A9a). The contribution from the hetero-nuclear dipolar coupling to the complete master equation (eq A1) may also be written in a matrix form:63

f mQ= 0 (σ *) = −CQ2

∫0

∞

Q Q (F20 (t ) − ⟨F20 (t )⟩)

Q Q IR × (F20 (t − τ ) − ⟨F20 (t )⟩)[T20 , [exp(iHS*τ ) IR+ × T20 exp( − iHS*τ ), σ *]] dτ

(A12)

leading to the contribution from the quadrupolar relaxation mechanism to the master equation:71

⎛ T IR ⎞ ⎛ T IR ⎞ ⎜ 20 ⎟ ⎜ 20 ⎟ ⎜T IR (a)⎟ ⎜T IR (a)⎟ ⎜ 11 ⎟ ⎜ 11 ⎟ ⎜ T IR (s) ⎟ ⎜ T IR (s) ⎟ 21 ⎜ ⎟ ⎜ 21 ⎟ ⎜ T IR (s) ⎟ ⎜ T IR (s) ⎟ 22 d ⎜ 22 ⎟ ⎟ = − diag(AD , BD , C D , DD , ED , BD , C D , DD)·⎜ ⎜ T IR ⎟ dt ⎜ T IR ⎟ 10 10 ⎜ ⎟ ⎟ ⎜ ⎜ IR ⎟ ⎜ IR ⎟ ⎜ T11 (s) ⎟ ⎜ T11 (s) ⎟ ⎜ IR ⎟ ⎜ IR ⎟ ⎜T21 (a)⎟ ⎜T21 (a)⎟ ⎜ IR ⎟ ⎜ IR ⎟ ⎝T22 (a)⎠ ⎝T22 (a)⎠

⎛ T IR ⎞ ⎛ T IR ⎞ 20 ⎟ ⎛ AQ − 3 K Q 0 ⎜ 20 ⎟ 0 0 0 0 0 ⎞⎜ ⎜ ⎟ ⎜ IR ⎟ ⎜T IR (a)⎟ (a) T ⎜ 0 ⎜ 11 ⎟ 0 0 0 0 0 0 ⎟ ⎜ 11 ⎟ BQ ⎜ ⎟ ⎜ IR ⎟ ⎜ T IR (s) ⎟ T (s) ⎜ 0 ⎜ 21 ⎟ 0 0 0 0 0 ⎟ ⎜ 21 ⎟ CQ 0 ⎜ ⎟ ⎜ IR ⎟ ⎜ IR ⎟ Q Q ( ) T s d ⎜ 22 ⎟ −K 0 D 0 0 0 0 ⎟ ⎜ T22 (s) ⎟ ⎜ 0 = −⎜ ⎟ ·⎜ IR ⎟ Q Q dt ⎜ T IR ⎟ −K 0 0 0 E 0 0 ⎟ ⎜ T10 ⎟ ⎜ 0 ⎜ 10 ⎟ ⎜ ⎟ ⎜ IR ⎟ ⎟ IR 0 0 0 0 0 0 ⎟ ⎜⎜ T11 LQ (s) ⎟ ⎜ 0 ⎜ T11 (s) ⎟ Q ⎜ ⎟ ⎜ IR ⎟ 0 0 0 0 0 0 0 ⎜T IR (a)⎟ M ⎜⎜ ⎟⎟ ⎜ 21 ⎟ ⎜T21 (a)⎟ ⎝ 0 ⎜ IR ⎟ 0 0 0 0 −K Q 0 DQ ⎠ ⎜ IR ⎟ ⎝T22 (a)⎠ ⎝T22 (a)⎠

(A13)

(A10)

with

with

1 D J (ω0 − ωS) + J1D(ω0) + 2J2D(ω0 + ωS) 3 0 1 D 1 2 BD = J0D(0) + J0 (ω0 − ωS) + J1D(ω0) 9 18 6 1 1 + J1D(ωS)° + J2Q (ω0 + ωS) 3 3 2 5 D 5 C D = J0D(0) + J (ω0 − ωS) + J1D(ω0) 9 18 0 6 1 5 + J1D(ωS) + J2D(ω0 + ωS) 3 3 8 1 1 4 DD = J0D(0) + J0D(ω0 − ωS) + J1D(ω0) + J1D(ωS) 9 9 3 3 2 + J2D(ω0 + ωS) 3 1 1 2 E D = J0D(ω0 − ωS) + J1D(ω0) + J2D(ω0 + ωS) 9 3 3

AQ = 3J1Q (ω0)

AD =

BQ = CQ =

3ωQ2 J0Q (0) + 4ω12J0Q (k1) 2k12 3ωQ2 J0Q (0) + 4ω12J0Q (k1) 2k12

+

5 Q J (ω0) + J2Q (2ω0) 21

+

1 Q J (ω0) + J2Q (2ω0) 21

DQ = J1Q (ω0) + 2J2Q (2ω0) EQ = J1Q (ω0) + 4J2Q (2ω0) LQ = MQ = KQ =

2 Q 2 Q Q 3 ωQ J0 (0) + 2ω1 (J0 (0) + J0 (k1)) 5 + J1Q (ω0) + J2Q (2ω0) 2 2 k12

3ωQ2 J0Q (0) + 2ω12(J0Q (0) + J0Q (k1)) 2k12

+

1 Q J (ω0) + J2Q (2ω0) 21

3ω1ωQ (J0Q (0) − J0Q (k1)) 2k12

In the same manner, the m = 0 contribution of the heteronuclear dipolar coupling is given by

The corresponding spectral densities are 26131


The Journal of Physical Chemistry C 4 fmD= 0 (σ *) = − C D2NSS(S + 1) 9 ⎛ F D (t − τ ) × ⎜⎜ 20 − 3 ⎝ rIS(t − τ )

∫0

∞ ⎛ F D (t ) ⎜⎜ 20 3 ⎝ rIS(t )

D ⎞ F20 ⎟⎟ 3 rIS ⎠

−

(A14)

leading to the contribution from the hetero-nuclear dipolar relaxation mechanism to the master equation:44

(A15)

with

BD =

1 D J (ω0 − ωS) + J1D(ω0) + 2J2D(ω0 + ωS) 3 0 2(k 2J0D(k 3) + k 3J0D(k 2)) 9k1

+ CD =

+

1 D 1 J (ω0 − ωS) + J1D(ω0) 18 0 6

1 D 1 J (ωS) + J2Q (ω0 + ωS) 31 3

2(k 2J0D(k 3) + k 3J0D(k 2)) 9k1

+

5 D 5 J (ω0 − ωS) + J1D(ω0) 18 0 6

1 D 5 J (ωS) + J2D(ω0 + ωS) 31 3 8 1 1 DD = J0D(0) + J0D(ω0 − ωS) + J1D(ω0) + 9 9 3 2 + J2D(ω0 + ωS) 3 1 1 2 ED = J0D(ω0 − ωS) + J1D(ω0) + J2D(ω0 + 9 3 3 +

LD =

2(k 2J0D(k 3) + k 3J0D(k 2)) 9k1

+ MD =

KD =

■

4 D J (ωS) 31

ωS)

1 D 1 J (ω0 − ωS) + J1D(ω0) 18 0 6

1 D 1 J1 (ωS) + J2Q (ω0 + ωS) 3 3

2(k 2J0D(k 3) + k 3J0D(k 2)) 9k1

+

+

+

■

REFERENCES

(1) Henderson, D. Fundamentals of Inhomogeneous Fluids; M. Dekker: New York, 1992. (2) Cygan, R. T.; Greathouse, J. A.; Heinz, H.; Kalinichev, A. G. Molecular Models and Simulations of Layered Materials. J. Mater. Chem. 2009, 19, 2470−2481. (3) Skelton, A. A.; Wesolowski, D. J.; Cummings, P. T. Investigating the Quartz (101̅0)/Water Interface Using Classical and Ab Initio Molecular Dynamics. Langmuir 2012, 27, 8700−8709. (4) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1985. (5) Terzis, A. F.; Snee, P. T.; Samulski, E. T. Orientational Order of Water Confined in Anisotropic Cavities. Chem. Phys. Lett. 1997, 264, 481−486. (6) Pelletier, O.; Sotta, P.; Davidson, P. Deuterium Nuclear Magnetic Resonance Study of the Nematic Phase of Vanadium Pentoxide Aqueous Suspensions. J. Phys. Chem. B 1999, 103, 5427−5433. (7) Bellissent-Funel, M.-C. Status of Experiments Probing the Dynamics of Water in Confinement. Eur. Phys. J. E 2003, 12, 83−92. (8) Nakashima, Y. Nuclear Magnetic Resonance Properties of WaterRich Gels of Kunigel-V1 Bentonite. J. Nucl. Sci. Technol. 2004, 41, 981− 992. (9) Bowers, G. M.; Bish, D. L.; Kirkpatrick, R. J. H2O and Cation Structure and Dynamics in Expandable Clays: 2H and 39K NMR Investigations of Hectorite. J. Phys. Chem. C 2008, 112, 6430−6438. (10) O’Hare, B.; Grutzeck, M. W.; Kim, S. H.; Asay, D. B.; Benesi, A. J. Solid State Water Motions Revealed by Deuterium Relaxation in 2H2OSynthesized Kanemite and 2H2O-Hydrated Na+-Zeolite A. J. Magn. Reson. 2008, 195, 85−102. (11) Tenorio, R. P.; Alme, L. R.; Engelsberg, M.; Fossum, J. O.; Hallwass, F. Geometry and Dynamics of Intercalated Water in NaFluorhectorite Clay Hydrates. J. Phys. Chem. C 2008, 112, 575−580. (12) Reinholdt, M. X.; Babu, P. K.; Kirkpatrick, R. J. Proton Dynamics in Layered Double Hydroxides: A 1H T1 Relaxation and Line Width Investigation. J. Phys. Chem. C 2009, 113, 10623−10631. (13) Bowers, G. M.; Singer, J. W.; Bish, D. L.; Kirkpatrick, R. J. Alkali Metal and H2O Dynamics at the Smectite/Water Interface. J. Phys. Chem. C 2011, 115, 23395−23407. (14) Maheshwari, P.; Pujari, P. K.; Sharma, S. K.; Dutta, D.; Sudarshan, K.; Mithu, V. S.; Madhu, P. K.; Deshpande, S. K.; Patil, P. N.; Raje, N. Phase Transition of Nanoconfined Water in Clay: Positron Annihilation, Nuclear Magnetic Resonance, and Dielectric Relaxation Studies. J. Phys. Chem. C 2013, 117, 14313−14324. (15) Sato, K.; Fujimoto, K.; Dai, W.; Hunger, M. Molecular Mechanism of Heavily Adhesive Cs: Why Radioactive Cs Is Not Decontaminated from Soil. J. Phys. Chem. C 2013, 117, 14075−14080. (16) Tuck, J. J.; Hall, P. L.; Hayes, M. H. B.; Ross, D. K.; Poinsignon, C. Quasi-Elastic Neutron-Scattering Studies of the Dynamics of Intercalated Molecules in Charge-Deficient Layer Silicates. 1. Temperature Dependence of the Scattering from Water in Ca2+ Exchanged Montmorillonite. J. Chem. Soc., Faraday Trans. I 1984, 80, 309−324. (17) Poinsignon, C. Protonic Conductivity and Water Dynamics in Swelling Clays. Solid State Ionics 1997, 97, 399−407. (18) Swenson, J.; Bergman, R.; Longeville, S. A Neutron Spin-Echo Study of Confined Water. J. Chem. Phys. 2001, 115, 11299−11305. (19) Malikova, N.; Cadéne, A.; Marry, V.; Dubois, E.; Turq, P.; Zanotti, J.-M.; Longeville, S. Diffusion of Water in Clays − Microscopic Simulation and Neutron Scattering. Chem. Phys. 2005, 317, 226−235.

⎛ T IR ⎞ ⎛ T IR ⎞ 20 ⎟ ⎛ AD − 3 K D 0 ⎜ 20 ⎟ 0 0 0 0 0 ⎞⎜ ⎜ ⎟ ⎜T IR (a)⎟ ⎜T IR (a)⎟ 11 ⎜0 ⎜ 11 ⎟ ⎟ BD 0 2K D 0 0 0 0 ⎟⎜ ⎜ ⎟ ⎜ IR ⎟ ⎜ T IR (s) ⎟ T ( s ) D ⎜0 ⎜ 21 ⎟ 0 C 0 0 0 0 0 ⎟ ⎜ 21 ⎟ ⎜ ⎟ ⎜ IR ⎟ ⎜ IR ⎟ d ⎜ T22 (s) ⎟ KD 0 DD 0 0 0 0 ⎟ ⎜ T22 (s) ⎟ ⎜0 = −⎜ ⎟ ·⎜ IR ⎟ ⎜ ⎟ IR dt T 0 0 0 E D 0 − K D 0 ⎟ ⎜ T10 ⎟ ⎜0 ⎜ 10 ⎟ ⎜ ⎟ ⎜ IR ⎟ ⎟ IR 0 0 0 0 LD 0 2K D ⎟ ⎜⎜ T11 (s) ⎟ ⎜0 ⎜ T11 (s) ⎟ D ⎜0 ⎟ ⎜ IR ⎟ ⎜ ⎟ IR 0 0 0 0 0 M 0 ⎜⎜ ⎟⎟ ⎜T21 (a)⎟ ⎜T21 (a)⎟ D D ⎝0 ⎜ IR ⎟ 0 0 0 0 K 0 D ⎠ ⎜ IR ⎟ ⎝T22 (a)⎠ ⎝T22 (a)⎠

AD =

ACKNOWLEDGMENTS

The DSX360 Bruker spectrometer used for the NMR study was purchased thanks to grants from Région Centre (France). We thank Dr. Laurent J. Michot (PECSA, Paris) for providing us with the hectorite clay sample used in the present study. We cordially thank Dr. Joël Puibasset (CRMD, Orléans) for helpful discussions.

D ⎞ F20 IR ⎟⎟[T10 , [exp(iHS*τ ) 3 rIS ⎠

IR+ × T10 exp(− iHS*τ ), σ *]] dτ

■

Article

5 D 5 J (ω0 − ωS) + J1D(ω0) 18 0 6

1 D 5 J (ωS) + J2D(ω0 + ωS) 31 3

2ω1(J0D(k 3) − J0D(k 2)) 9k1

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest. 26132



Article

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