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Structural and Dynamical Properties of Water Molecules Confined within Clay Sediments Probed by Deuterium NMR Spectroscopy, Multiquanta Relaxometry, and Two-Time Stimulated Echo Attenuation 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 S Supporting Information *

ABSTRACT: The structure and multiscale dynamics of water molecules confined within dense clay sediments are investigated by deuterium (2H) NMR spectroscopy, relaxometry, and two-time correlation measurements. The splitting of the 2 H NMR resonance line quantifies the specific ordering of water molecules confined within the clay interlamellar space. The angular distribution of clay aggregates within the sediment is evaluated from variation of the transverse 2H NMR relaxation rates as a function of orientation of the clay film within the static magnetic field. The average residence time of the water molecules within clay aggregates is determined by multiquanta spin-locking relaxometry. Finally, water exchange between different aggregates is extracted from the attenuation of 2H two-time stimulated echo. These simultaneous NMR dynamical investigations cover a broad range of characteristic times (between 10 μs and 100 ms) appropriate to investigate the multiscale dynamical behavior of water molecules confined within heterogeneous porous networks.

1. INTRODUCTION Solid/liquid interfaces were recently the subject of numerous theoretical1 and experimental2 studies in order to determine how confinement modifies the structural and dynamical properties of fluids. In that context, various experimental methods calorimetry,3 atomic force microscopy,2 X-ray and neutron scattering,4−8 infrared spectroscopy,9,10 dielectric relaxation,11,12 quasi-elastic neutron scattering4,13−19 (QENS), nuclear magnetic resonance spectroscopy,3,20−26 relaxometry,27−38 and pulsed-gradient spin−echo39−43 (PGSE) NMR spectroscopy have been used to quantify the structural and dynamical properties of confined fluids in relation to their physicochemical interactions with the solid matrix. Among other interfacial systems, polyions and charged surfaces were frequently investigated since the thermodynamical properties44−51 of polar and ionized fluids are then deeply modified by the local electric field generated by the solid surface. Furthermore, that field of research is very attractive since numerous industrial and biological processes imply such charged interfacial systems. In that framework, natural and synthetic clays are ideal systems, since they are flat and atomically smooth with wellcharacterized structure and chemical composition. Furthermore, clays are ubiquitous lamellar materials involved in numerous industrial applications (drilling, civil engineering, food and cosmetic industry, heterogeneous catalysis,52 waste storing53) © 2014 American Chemical Society

that exploit their various physicochemical properties (gelling, swelling, thixotropy, high specific surface area and adsorption power, surface acidity, and ionic exchange capacity). For various applications, such as heterogeneous catalysis and waste storing, the retention capacity of clay lamellae must be monitored as a function of its composition, surface electric charge, and neutralizing counterions. For that purpose, the mobility of the confined fluids must be quantified over a broad range of diffusing time. The short-time mobility may be quantified by various methods including IR spectroscopy9 and QENS.4 By contrast, classical field-cycling NMR relaxometry28 and PGSE NMR spectroscopy39 should be useful to determine the long-time mobility of the confined ions and liquids. Unfortunately, these two last procedures are frequently useless to determine the mobility of fluids confined within dense clay sediments because their NMR relaxation rates are drastically enhanced by paramagnetic impurities54 embedded within the atomic clay network. In that context, multiquanta spin-locking relaxation55−57 measurements and two-time stimulated echo NMR spectroscopy50,57−61 appear as powerful alternative procedures since, by contrast with the previous more classical procedures, they do not Received: June 25, 2014 Revised: August 4, 2014 Published: August 6, 2014 20429

dx.doi.org/10.1021/jp506312q | J. Phys. Chem. C 2014, 118, 20429−20444

The Journal of Physical Chemistry C

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water molecules confined between two clay platelets within the same aggregate. Finally, two-time stimulated echo attenuation50 is measured to quantify the critical time characterizing water exchange between clay aggregates with different orientation.

require long delay between the preparation pulses and acquisition of the spin magnetization. Finally, dense clay sediments exhibit multiscale organization characterized by the juxtaposition of aggregates resulting from the stack of numerous (10−100) individual clay platelets3,62 (see Figure 1). As a consequence, it is

2. MATERIALS AND METHODS 2.1. Sample Preparation. Beidellite is a natural clay purchased from The Source Clay Mineral Repository of the Clay Mineral Society at Purdue University. Beidellite is a dioctahedral clay resulting from the sandwiching of one layer of octahedral aluminum oxides between two layers of tetrahedral silica. Because of the substitution of some tetrahedral silica by alumina, the clay network bears excess negative charges neutralized by interlamellar counterions, which are responsible for the hydration of dry clay sediments7 and for the swelling/ setting behavior68 of aqueous clay dispersions. Prior to use, the clay sample was purified according to classical procedures64 and the counterions were exchanged,64,69,70 leading predominantly to monoionic clay with the general formula64,71 (Si7.27Al0.73)(Al3.77Fe3+0.11Mg0.21)O20(OH)4Na0.67. Centrifugation was further used64 to select the clay particles according to their size, with average size 330 ± 50 nm. A self-supporting film (0.5 mm thick) was obtained from dilute aqueous clay dispersion (10 g/L) by ultrafiltration under nitrogen (3−5 atm) by use of a microporous membrane (Osmonics, Inc.). The clay film was finally dried under nitrogen flux before equilibration with a reservoir of heavy water at fixed water chemical potential (p/p0 = 0.92) by use of saturated salt solution (saturated anhydrous KNO3 in pure D2O). That partial pressure was selected because it corresponds mainly to a homogeneous hydration state with a period of 15.6 Å corresponding to two hydration layers of confined water molecules.6−8 A macroscopic lamella (30 × 6 mm2) was cut into the clay film and inserted into a glass cylinder that fits the gap inside a homemade solenoid coil used for the NMR measurements. 2.2. 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 required for a total inversion of the longitudinal magnetization is equal to 28 μs. NMR spectra were recorded in fast acquisition mode with a sampling time of 0.25 μs corresponding to a 4 MHz spectral window. The spectra (see Figure 2a) and relaxation rates (see Figures 7 and 10) were recorded for different orientations βLF of the film director n⃗F,L,

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

crucial to investigate the dynamical properties of confined water molecules over a broad range of diffusing time by combining information obtained by complementary experiments such as QENS, for the local mobility of water molecules within the interlamellar space between two clay platelets; multiquanta spin-locking NMR relaxometry,63 for their average residence time within such interlamellar space; and two-time stimulated echo NMR spectroscopy,50 for their exchange between various clay aggregates. This work focuses on a natural clay64 (beidellite) under controlled hydrating conditions by using a reservoir of vapor of heavy water. 2H NMR spectroscopy is then performed to determine the specific orientation of the confined water molecules.20 These experiments are also useful to obtain information on the heterogeneity of the clay sediment54 and the dynamical regime characterizing water exchange between these various local environments.65,66 Multiquanta NMR relaxation measurements are then performed to separately quantify the contributions from quadrupolar and heterogeneous dipolar relaxation mechanisms67 monitoring the NMR relaxation behavior of the confined heavy water molecules. These results are further exploited to extract, from multiquanta spin-locking relaxation measurements,56 the average residence time of the

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. 20430



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with reference to the static magnetic field B0, by use of a homemade sample holder and detection coil. A complete basis set,27,72,73 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 given by the IR IR IR IR irreducible tensor operators 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 11(a,s)] components of the spin magnetization. The five residual operators are required to describe the quadrupolar Hamiltonian (see Appendix). The relaxation rate constant of TIR 10 coherence, also called longitudinal relaxation rate (R1), is noted here R10. It is measured by the classical inversion recovery pulse sequence.74 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 spin−echo pulse sequence39,72,75 (see Figure 3a). The relaxation rate constants of

Figure 4. Pulse sequences and coherences transfer pathways used to measure the multiquanta spin-locking relaxation rates of the (a) TIR 11ρ(s), IR IR (b) TIR 21ρ(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.

pulse56,57 whose duration was noted ψ10→21 and ψ10→22, respectively. That procedure exploits the residual static quadrupolar coupling ωQ felt by the confined water molecules. As explained previously,56,57 these pulse durations were determined by using the simplified master equation describing the time evolution of coherences under irradiation but neglecting their intrinsic relaxation (see AppendixA.1, eqs A4a−A4h). Five different irradiation powers are used, corresponding to angular velocities ω1 equal to 1.12 × 105, 5.6 × 104, 2.8 × 104, 1.4 × 104, and 7 × 103 rad·s−1. The pulse sequence illustrated in Figure 5 was used to extract two-time correlation function by measuring the attenuation of

Figure 3. Pulse sequences and coherence transfer pathways used to IR measure the NMR relaxation rate of (a) TIR 11(a,s), (b) T20, and (c) 56 (a,s) coherences. The delay δ is selected to optimize the TIR 22 opt coherence transfer, and the delay ε is set equal to 10 μs. IR IR and T22 (a,s) coherences are measured by the pulse T20 56 sequences described in Figure 3b,c and noted R20 and R22, respectively. Details on the time evolution of these coherences are given in the Appendix. As shown previously,67 this set of four independent relaxation rates (R10, R11, R20, and R22) is required to separately quantify the dominant contributions from quadrupolar and heterogeneous dipolar relaxation mechanisms. The mobility of the water molecules confined within the interlamellar space between two elementary clay platelets (see Figure 1) was quantified by use of NMR relaxometry, that is, by measuring the frequency variation of the NMR relaxation rates.55−57,63,76,77 For that purpose, we performed multiquanta spin-locking relaxation measurements.55−57,63,76,77 Three different pulse sequences (see Figure 4) are used to investigate the IR IR time evolution of TIR 11(s), T21(a,s), and T22(a) coherences under irradiation. They are noted T11ρ(s), T21ρ(a,s), and T22ρ(s), respectively. As displayed in Figure 4b,c, the transitions between the zero-order initial coherence (TIR 10) and the required secondIR order coherences [TIR 21(a,s) and T22(a)] are performed by a single

Figure 5. Pulse sequence and coherence pathway used to measure attenuation of the two-time 2H NMR echo I(te, τM) as a function of evolution period te and mixing time τM. The fourth pulse duration, noted IR ψ, is optimized in order to maximize the transfer from TIR 20 to T22(s) IR coherence and simultaneously minimize the transfer from TIR 10 to T22(a) coherence. In this study, for βLF = 90° and ω1 = 1.01 × 105 rad·s−1, the duration ψ is set equal to 44.1 μs.

two-time stimulated echo.50,57,58,61 That experiment exploits the heterogeneities of the residual quadrupolar coupling felt by water molecules confined within clay lamellae pertaining to aggregates with various orientations in the static magnetic field B0 (see Figure 1). During the first evolution period (te), the first-order coherence TIR 11 oscillates according to the initial local value of the residual quadrupolar coupling [noted ωQ(0)]. After an appropriate mixing period (τM), these confined water molecules 20431



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diffuse through the clay sediment and reach clay aggregates with a different orientation corresponding to another residual quadrupolar coupling [noted ωQ(τM)]. As a consequence, the detected magnetization varies according to50,57,58,61 I(te , τM) ∝ ⟨cos[ωQ (0)te]cos[ωQ (τM)te]⟩ × exp( −R 20τM − 2R11te)

(1)

If the time scale quantifying the water exchange between these various environments is longer than their transverse relaxation times, that is, under the so-called slow exchange regime,65 the average described in eq 1 leads to a net attenuation of the magnetization because of the interferences between magnetizations of water molecules probing various residual couplings. By contrast, if the same time scale quantifying the water exchange between its various environments is longer than the evolution period τM, no attenuation of the signal may be detected. As shown previously,50,57 the pulse sequence was optimized by carefully tuning the pulse duration ψ in order to optimize IR the transfer from TIR 20 to T22(s) coherence and simultaneously IR minimize the transfer from TIR 10 to T22(a) coherence. For that purpose, we used the same set of master equations (see eqs A4a−A4h) as for optimizing the spin-locking relaxation measurements.

3. RESULTS AND DISCUSSION 3.1. 2H NMR Spectra. As previously reported,20,22 the residual quadrupolar splitting displayed in Figure 2a is the fingerprint of the specific orientation of water molecules confined between the clay platelets, in complete agreement with predictions54,67 obtained by molecular modeling of the clay/water interfaces (see Figure 6). That phenomenon was already used to detect the isotropic/nematic transition induced by the static magnetic field within diluted clay dispersions.22,67,78 Such specific orientation of water molecules confined between clay platelets was also detected by neutron scattering.7,8 In addition to the residual splitting, Figure 2a also exhibits some asymmetry of the water NMR resonance line induced by the second-order relaxation mechanism resulting from the cross-correlation between quadrupolar and heterogeneous dipolar couplings felt by the confined water molecules.79 In addition to direct analysis of the 2H NMR resonance line, the residual quadrupolar coupling may also be extracted from 67 analysis of the time evolution of TIR via the 11(a,s) coherences simple relationship: IR IR T11 (τ ) = T11 (0) cos(ωQobsτ ) exp(−τR11)

Figure 6. (a) Snapshot illustrating one grand canonical Monte Carlo (GCMC) equilibrium configuration of confined water molecules and neutralizing sodium counterions. (b) Concentration profiles of sodium counterions and oxygen and hydrogen atoms pertaining to water molecules confined between two beidellite clay lamellae. (c) Histogram of distribution law of |cos θLW| extracted from GCMC equilibrium configurations and illustrating the specific orientation of confined water molecules by reference with a fully random orientation (see text).

not reproduce perfectly the data displayed in Figure 2b. Such discrepancy results from the simultaneous occurrence of two phenomena: first, the coexistence of various water environments with slightly different residual splittings, and second, a slow exchange, at the NMR time scale, between these various environments. As a consequence, 2H NMR spectra of the confined water molecules (Figure 2a) are characteristic of powder spectra of a partially oriented sample34,56,57 under the so-called slow exchange regime.65 These two conditions are required to successfully probe long-distance water dynamics by measuring two-time stimulated echo attenuation. Surprisingly enough, the maximum apparent splitting [ωQmax = (82 ± 6) × 103 rad·s−1] detected here for heavy water pertaining to the two hydration layers of beidellite clay differs significantly from the value reported previously56 [ωQmax = (55 ± 3) × 103 rad·s−1] for heavy water confined within hectorite clay under the same hydration conditions. Such a difference largely exceeds the experimental error while both clays bear nearly the same

(2)

Figure 2b displays the variation of residual splitting as a function of the orientation of film βLF into static magnetic field B0. If a single spin population exists, the maximum value of the detected residual splitting results from the specific orientation of the confined water molecules (see Figure 6). Under such conditions, Figure 2b perfectly matches the theoretical relationship: ωQobs(β LF) = ωQmax P2(cos(β LF)) = ωQmax

3 cos2(β LF) − 1 2 (3)

where P2 is the second-order Legendre polynomial and ωQmax is the maximum value of the measured residual coupling detected for a perfect alignment of the film director nF,L ⃗ into the static magnetic field B0. However, the predicted behavior (eq 3) does 20432



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electric charge density and are both neutralized by the same sodium counterions. Such discrepancy originates from the structural difference between these two clays: hectorite is a trioctahedral clay with negative charges induced by atomic substitutions occurring mainly within its central octahedral layer.70 By contrast, beidellite is a dioctahedral clay with negative charges resulting from atomic substitutions occurring mainly in its sandwiching tetrahedral layers (see section 2.1). Thanks to Gauss’s law, the average electric field is the same within both clay interfaces; by contrast, the corresponding local electric fields differ significantly.80,81 As a consequence, the same confined molecules are more oriented by the local electrostatic field generated by the atomic network81 of beidellite clay than that generated by hectorite clay. This result clearly illustrates the great sensitivity of 2H NMR spectroscopy to the influence of the exact composition of the clay network on average organization of confined water molecules. Further experimental studies are necessary to investigate the correlations between the detected 2 H residual quadrupolar coupling of confined water molecules and the structural properties of the clay network under wellcontrolled hydration conditions. 3.2. 2H NMR Multiquanta Relaxation Measurements. NMR relaxometry is a powerful tool for extracting dynamical information on the long-time mobility of confined fluids once their relaxation mechanisms are clearly identified. For that purpose, we perform a set of preliminary multiquanta relaxation measurements67 in order to quantify the contribution from both intrinsic quadrupolar coupling and heterogeneous dipolar coupling induced by paramagnetic impurities embedded within the clay network. As shown in the Appendix, such a separation is easily performed under the so-called slow modulation condition,67 that is, when the various spectral densities (see sections A.2 and A.3, eqs A7 and A10) satisfy the relationships J0X(0) > JmX(ω) for m ∈ {1, 2} and X ∈ {Q, D}. As displayed in Figure 7a, such a condition is fully satisfied since the transverse relaxation rate (R11) is at least 2 orders of magnitude larger than the longitudinal relaxation rate (R10). As a consequence, the detailed equations describing the various apparent relaxation rates (see eqs A7 and A10 in the Appendix) may be simplified, leading to U R10 = 5UQ + D 3

Figure 7. Influence of film orientation βLF into the static magnetic field IR IR B0: (a) on the apparent multiquanta relaxation rates of TIR 10, T11(a,s), T20, (a,s) coherences, noted R , R , R , and R , respectively; and and TIR 22 10 11 20 22 (b) on the apparent spectral densities J0Q(0), J0D(0), UQ, and UD extracted from these Rij values (see eqs 4a−4f).

by simple matrix inversion, quantifying the contributions of both quadrupolar and heterogeneous dipolar relaxation mechanisms.67 The data displayed in Figure 7b fully validate the abovementioned approximation concerning slow modulation of quadrupolar and heterogeneous dipolar couplings. Furthermore, the m = 0 component of the quadrupolar coupling totally dominates the relaxation behavior of the confined water molecules and appears very sensitive to the orientation of the clay film within the static magnetic field. By contrast, the m = 0 component of the heterogeneous dipolar coupling is nearly independent of film orientation and is reduced by 1 order of magnitude. Such behavior was already reported for water molecules confined within other clay sediments.56,57 By using the Wigner rotation matrices,82−84 it should be possible to extract the intrinsic spectral densities from the angular variation of the apparent spectral densities of the various relaxation mechanisms (X ∈ {Q, D}) as a function of the film orientation (βLF) into the static magnetic field B0: 1 J0X (β LF , ω) = (1 − 3 cos2 β LF)2 J0X,intr (ω) 4

(4a)

R11 = R 21 3 5 2 1 1 = J0Q (0) + UQ + J0D(0) + UD + J1D(ωS) 2 2 9 2 3 (4b)

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

8 D 1 4 J0 (0) + UD + J1D(ωS) 9 3 3

(4c) (4d)

with UQ = J1Q (ω0) ≈ J2Q (2ω0)

UD =

1 D J (ωS − ω0) + J1D(ω0) + 2J2D(ωS + ω0) 3 0

(4e) (4f)

where ω0 and ωS are respectively the angular velocities (rad·s−1) of deuterium and unpaired electron of iron. When the highfrequency contribution from J1D(ωs) is neglected, the set of four independent relaxation measurements (R10, R11, R20, and R22) may be used to extract the four parameters [UQ, UD, J0Q(0), and J0D(0)]

+ 3 cos2 β LF sin 2 β LFJ1X,intr (ω) 3 + (1 − cos2 β LF)2 J2X,intr (ω) 4 20433

(5a)



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J1X (β LF , ω) =

3 cos 2 β LF sin 2 β LFJ0X,intr (ω) 2 1 + (1 − 3 cos2 β LF + 4 cos 4 β LF)J1X,intr (ω) 2 1 + (1 − cos 4 β LF)J2X,intr (ω) (5b) 2

J2X (β LF , ω) =

3 (1 − cos2 β LF)2 J0X,intr (ω) 8 1 + (1 − cos 4 β LF)J1X,intr (ω) 2 1 + (1 + 6 cos2 β LF + cos 4 β LF)J2X,intr (ω) 8

One may be tempted to apply the same approach to analyze the angular variation of quadrupolar coupling felt by the confined water molecule (see Figure 8b). Unfortunately, this treatment is useless because of the strong angular variation of the m = 0 component of apparent spectral density J0Q(0). In order to obtain realistic intrinsic components, such analysis of the quadrupolar coupling must take into account the abovementioned distribution of clay directors within the clay film (see section 3.1). Moreover, we exploit the high angular sensitivity of the apparent quadrupolar relaxation mechanism to quantify the angular distribution of clay directors within the self-supporting film. For that purpose, the orientations of the various clay platelets within the film are characterized by two Euler angles, αFC (colatitude) and γFC (azimuth) (see Figure 9a,b).

(5c)

Figure 8a displays the intrinsic spectral densities extracted from the raw experimental data [J0D(0) in Figure 7b] by using the

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

In the frame attached to the clay film, the normal to the clay platelet (n⃗C,F) becomes Figure 8. Extraction of intrinsic spectral densities J0X,intr(0), J1X,intr(0), and J2X,intr(0) with X ∈ {D, Q}, from angular variation of the apparent spectral densities J0X(0) quantifying (a) heterogeneous dipolar (X = D) and (b) quadrupolar (X = Q) relaxation mechanisms.

n ⃗C,F

set of eqs 5a−5c. These intrinsic spectral densities describe the heterogeneous dipolar coupling evaluated at zero angular velocity in the frame of the clay lamella [J0D,intr(0) = 1200 ± 100 s−1, J1D,intr(0) = 1700 ± 200 s−1, and J2D,intr(0) = 550 ± 60 s−1]. These intrinsic spectral densities describing the heterogeneous dipolar relaxation mechanism of heavy water confined within beidellite clay are slightly larger than those previously detected for hectorite clay under the same hydration conditions.56

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

(6a)

In the frame attached to the NMR tube containing the clay film (see Figure 9b), we obtain

n ⃗C,L

20434

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

(6b)



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by including a rotation δLF of the film along the YL axis (see Figure 9b). Finally, after the βLF rotation of the NMR tube along the XL axis of the coil, we obtain the net angle θLC between the clay director and the static magnetic field B0 by the relationship: 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) (6c)

By using the set of equations A3, A7, and A10 in the Appendix, we simulate the time evolution of magnetization detected during the spin−echo pulse sequence (see Figure 3a) used to measure the transverse relaxation rate R11. In these simulations, we assume a Gaussian distribution law with zero average of the first αFC Euler angle, while the second Euler angle γFC is distributed uniformly in the [0, 2π] interval. As a consequence, the set of equations 5a−5c must be modified by taking into account the distribution of the clay directors into the film (see eq 6c). For that purpose the cos2 βLF and cos4 βLF formulas, based on the orientation of the clay film into the static magnetic field (βLF), are respectively replaced by the average values of ⟨cos2 θLC⟩ and ⟨cos4 θLC⟩, based now on the orientation θLC of each platelet evaluated into the static magnetic field B0: ⟨cos2 θ LC⟩ =

2 2 LF 2 FC sin β sin α 3 + cos2 β LF cos2 δ LF cos2 α FC 1 + cos 2 β LF sin 2 δ LF sin 2 α FC 3

Figure 10. Raw data and fits of Hahn spin−echo attenuation obtained from complete simulation of time evolution of TIR 11 coherences during the NMR pulse sequence for several clay film orientations βLF into the static magnetic field B0: (a) βLF = 0°, (b) βLF = 30°, and (c) βLF = 90°.

(7a)

and

on the mobility of the confined water molecules. As detailed in section 2.2, spin-locking relaxation measurements are performed IR IR with four independent coherences [TIR 11(s), T21(a,s), and T22(a)] by use of five irradiation powers. Each experiment is performed at three different orientations βLF of the clay film within the static magnetic field in order to extract the frequency variation of the six intrinsic spectral densities [i.e. JmX,intr(ω) with X ∈ {Q, D} and m ∈ {0, 1, 2}]. However, because of the reduced values of m = 2 components of intrinsic quadrupolar [J2Q,intr(0) = 200 ± 100 s−1] and heterogeneous dipolar [J2D,intr(0) = 550 ± 60 s−1] relaxation mechanisms, we refrain from extracting their frequency variations. Typical experimental results are displayed in Figure 11 in conjunction with their Fourier transforms. These Fourier transforms are very useful to identify the characteristic angular velocity (k1, k2, and k3) monitoring the time evolution of these coherences under irradiation (see section A.1, eqs A4a−A4h and A5a−A5c). Table 1 displays the characteristic angular velocities as a function of irradiation power and orientation (βLF) of the clay film. As noted previously,57 the angular velocities are nearly independent of film orientation at high irradiation power (see Tables 1 and 2). By contrast, their angular variation becomes more pronounced at weak irradiation power. Nevertheless, our numerical analysis of the coherences evolution is performed by using the set of equations A3, A7, A10, A13 and A15 of the Appendix, by neglecting the angular variation of the characteristic angular velocities (k1, k2, and k3). Typical results are displayed in Figure 11. The fitted parameters are collected in Figure 12, leading to the dispersion curves of m = 0 and m = 1 components of quadrupolar and heterogeneous dipolar relaxation mechanisms. By use of the reduced spectral densities noted:

8 ⟨cos 4 θ LC⟩ = sin 4 β LF sin 4 α FC + cos 4 β LF cos 4 δ LF cos 4 α FC 15 1 + cos 4 β LF sin 4 δ LF sin 4 α FC 5 4 2 LF + sin β cos2 β LF sin 2 δ LF sin 4 α FC 5 + 4 sin 2 β LF cos2 β LF cos2 δ LF sin 2 α FC cos2 α FC + 2 cos 4 β LF sin 2 δ LF cos2 δ LF sin 2 α FC cos2 α FC (7b)

Figure 10 exhibits typical fits of the time evolution of the transverse magnetization TIR 11(a,s) leading to the set of parameters J0Q,intr(0) = 3800 ± 400 s−1, J1Q,intr(0) = 32 000 ± 3000 s−1, J2Q,intr(0) = 200 ± 100 s−1, and ωQmax = 84 000 ± 4000 rad·s−1. The same analysis is also used to determine the standard deviation of the Gaussian distribution law of the αFC Euler angle [i.e., σ = (7 ± 1)°] and the angle δLF quantifying the misalignment of the film inside the NMR tube (see Figure 9b) [i.e., δLF = (9 ± 1)°]. These spectral densities describing the quadrupolar relaxation mechanism of heavy water confined within beidellite clay are slightly larger than those previously detected for hectorite clay under the same hydration conditions.56 The large enhancement of the m = 0 intrinsic spectral density of quadrupolar coupling was already reported previously.56,57 It results from the specific orientation of water molecules confined within the interlamellar space of the clay platelets (see Figure 6). 3.3. 2H NMR Relaxation Measurements under SpinLocking. Once the leading contributions of the quadrupolar and heterogeneous dipolar relaxation mechanisms are carefully quantified, it becomes possible to exploit multiquanta spinlocking relaxation measurements to extract dynamical information

jmX,intr (⟨ki⟩) = 20435

JmX,intr (⟨ki⟩) JmX,intr (0)

(8)



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IR IR IR Figure 11. Time evolution of (a) TIR 11(s), (b) T21(a), (c) T21(s), and (d) T22(a) coherences measured under spin-lock conditions, denoted 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 examples, βLF = 0° and ω1 = 1.12 × 105 rad·s−1.

with X ∈ {Q, D} and m ∈ {0, 1}, all data merge into a single master curve covering two decades. Two different dynamical regimes are clearly identified: a low-frequency plateau and a highfrequency decrease, with a sharp transition at the critical angular velocity [ωC = (6 ± 1) × 104 rad·s−1]. As previously shown by numerical simulations,56 that critical angular velocity corresponds to an average residence time (τc = 17 ± 3 μs) of the water molecules confined within the interlamellar space of beidellite clay, in agreement with data obtained by QENS and MD simulations at a much smaller time scale.18 That residence time is very similar to that previously detected for hectorite clay (τc = 25 ± 6 μs) with nearly the same size (500 ± 100 nm) and under the same hydration conditions.56 Finally, it is possible to justify a posteriori our hypothesis concerning the angular variation of the characteristic angular velocities (k1, k2, and k3) since the results extracted at the lowest irradiation power are located in the plateau of the dispersion curve (see Figure 12). 3.4. Two-Time Stimulated Echo Spectroscopy. While the time scale accessible by spin-locking relaxation measurements perfectly matches the average residence time of water molecules confined between two clay platelets, another experimental procedure is required to probe the long-time exchange

between water molecules pertaining to clay aggregates characterized by different orientations within the static magnetic field (see Figure 1). Two-time stimulated echo spectroscopy may be used to investigate such long-range dynamical processes.50,59,61,85−87 For that purpose, two conditions must be fulfilled: (i) a slow exchange of the probe, at the NMR time scale, between the various environments and (ii) a slow modulation of the relaxation mechanisms. As discussed above (see sections 3.1 and 3.2), these two conditions are satisfied in the case of water molecules confined between dense clay aggregates.50,57 In order to minimize the relaxation rates, the two-time echo attenuation is measured by orienting the clay film at 90° by reference with the static magnetic field B0 (see Figure 9b). Before analysis, the raw data are normalized in order to take into account the reduction of magnetization induced by intrinsic relaxation of TIR 20 coherence during the mixing time τM (see eq 1), leading to the data displayed in Figure 13. In order to optimize the signal/noise ratio, the attenuation is measured from the intensity of the secondary maximum displayed in Figure 13. These measurements lead to the two-time correlation function (Figure 14) quantifying the exchange time (τexch ≈ 33 ± 15 ms) between water molecules initially pertaining to clay aggregates characterized by different 20436



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Table 1. Set of Angular Velocities k1, k2, and k3a Detected from Fourier Transform of Time Evolution of Multiquanta Coherences as a Function of Irradiation Power ω1 and Euler Angle βLFb) ω1 (105 rad·s−1)

βLF = 0°

βLF = 30° 5

1.122 0.561 0.280 0.140 0.070 1.122 0.561 0.280 0.140 0.070 1.122 0.561 0.280 0.140 0.070 a

βLF = 90°

−1

k1 (10 rad·s ) 2.463 1.477 1.049 0.861 0.798 k2 (105 rad·s−1) 1.602 1.106 0.924 0.798 0.798 k3 (105 rad·s−1) 0.861 0.371 0.182 0.063 0.063

2.281 1.294 0.798 0.553 0.553

2.218 1.357 0.798 0.553 0.434

1.357 0.861 0.679 0.616 0.553

1.232 0.861 0.616 0.490 0.434

0.861 0.371 0.182 0.063 0.063

0.861 0.434 0.094 0.094 0.031

Figure 13. Variation of two-time stimulated echo attenuation I(te,τM) as a function of mixing time τM for film orientation βLF = 90°. The data are normalized to take into account relaxation of TIR 20 coherence during the mixing time τM [i.e., the factor exp(−R20τM), see eq 1].

See eqs A5a−A5c. bSee Figure 11e−h.

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

⟨k1⟩ (105 rad·s−1)

⟨k2⟩ (105 rad·s−1)

⟨k3⟩ (105 rad·s−1)

1.122 0.561 0.280 0.140 0.070

2.321 ± 0.104 1.376 ± 0.076 0.882 ± 0.118 0.656 ± 0.145 0.595 ± 0.152

1.397 ± 0.154 0.942 ± 0.116 0.739 ± 0.133 0.635 ± 0.126 0.595 ± 0.152

0.861 ± 0.000 0.392 ± 0.030 0.153 ± 0.041 0.073 ± 0.015 0.052 ± 0.015

a

Figure 14. Two-time correlation function extracted from the normalized stimulated echo attenuation as a function of mixing time τM for clay film orientation βLF = 90°. The red line corresponds to the best fit of experimental data to a stretched exponential function, f(t) = A exp(−t/τexch)λ, to determine the exchange time τexch (τexch = 33 ± 5 ms with an exponent λ set equal to 1.5), and the green dots are obtained by numerical modeling (see text).

See Table 1.

Since this exchange time is much larger than the relaxation times, it cannot be determined by NMR relaxation measurements but leads to interferences between the different magnetic responses of water molecules pertaining to various aggregates. Such interferences are responsible for anomalous variation of the residual quadrupolar splitting as a function of the film orientation (see Figure 2b) and are carefully taken into account in our analysis of relaxation measurements (see sections 3.2 and 3.3). By using the self-diffusion coefficient of bulk water, one can estimate the order of magnitude of the correlation length characterizing the angular distribution of clay aggregates: L≈ Figure 12. Variation of intrinsic spectral densities JmQ,intr(⟨k1⟩) with m ∈ {0, 1}, describing quadrupolar coupling, and JmD,intr(⟨ki⟩) with m ∈ {0, 1} and i ∈ {2, 3}, describing heteronuclear dipolar coupling, as a function of the corresponding averaged angular velocities ⟨ki⟩ (see text).

2D bulk τexch = 11 ± 2 μm

(9)

that is, 1 order of magnitude larger than the size of the beidellite platelets. The decrease of the two-time correlation function reported here for beidellite in the so-called two-hydrate state appears more progressive than the data previously reported for hectorite in the so-called one-layer hydration state (see Figure 11 in ref 50). That behavior may be induced by different positional correlations between clay aggregates characterized by various orientations. As a consequence, we obtain a quantitative agreement (see Figure 14)

orientations in the static magnetic field. As previously reported for another clay sediment,50,57 that time scale is 3 orders of magnitude larger than the average residence time of water molecules within the interlamellar space between two clay platelets located within the same aggregate (see Figure 1). 20437



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with the predictions from numerical modeling50 used to describe, within a large cubic box, the exchange of water molecules between neighboring small cubes labeled by the three indices (i, j ,k):

by use of multiquanta spin-locking relaxometry. Finally, the exchange time of the water molecules between different microdomains of the clay sediment is determined by extracting a two-time correlation function from the attenuation of two-time stimulated echo NMR spectroscopy. This multiscale analysis is a powerful tool to characterize the mobility of heavy water molecules within dense clay sediments. It should be successfully applied to investigate the dynamical properties of a large class of quadrupolar NMR probes diffusing within various porous networks including zeolites, lyotropic liquid crystals and natural or synthetic membranes.

d σi , j , k = (R i , j , k − 6kexchI )σi , j , k + kexchI(σi + 1, j , k + σi − 1, j , k dt + σi , j + 1, k + σi , j − 1, k + σi , j , k − 1 + σi , j , k + 1)

(10)

where Ri,j,k contains contributions from pulses, local residual quadrupolar couplings, and relaxation mechanisms (see eqs A3, A7, and A10 in the Appendix).

■

4. CONCLUSIONS Structural and dynamical properties of water molecules confined within dense sediments are determined by exploiting various possibilities of 2H NMR spectroscopy. By using simultaneously 2 H NMR spectroscopy, multiquanta relaxation measurements and spin-locking relaxometry in conjunction with two-time stimulated echo attenuation mesasurements, we were able to quantify the specific orientation of confined water molecules and the structural heterogeneities of clay sediment in addition to the multiscale mobility of water molecules within such heterogeneous porous material. The residual quadrupolar splitting of the 2 H resonance line is the fingerprint of the specific orientation of water molecules confined within two clay platelets. Due to the interactions of confined water molecule with its surrounding (clay lamellae and neutralizing sodium counterions), thermal motion cannot average to zero its quadrupolar coupling. Both water rotational motions and self-diffusion processes inside the interlamellar space contribute to that thermal average process. The different intrinsic contributions to the NMR relaxation mechanism are carefully evaluated by using multiquanta relaxation measurements. Furthermore, the angular distribution of clay aggregates within the sediment is extracted from variation of the transverse relaxation rate as a function of orientation of the clay into the static magnetic field. The average residence time of the water molecules confined within clay aggregates is evaluated ⎛ 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)⎠

A.1. Time Evolution of Coherences

In the framework of the Redfield theory,88 the time evolution of coherence is described by the complete master equation:28,84,89 dσ * = −i[HS*, σ *] + f (σ ∗) dt

f (σ *) = −

*

*

3 ω1ωQ +

k12 k12

3 ω12

− 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

*

k12

IR T22 (s)

⎛ 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 ⎠

*

IR eiHS τ T11 (a)e−iHS τ =

IR T20

(A3)

ωQ 2cos(k1τ ) + 4ω12 k 12 +

3 ω1ωQ

+ iωQ

sin(k1τ ) IR T21 (s) k1 cos(k1τ ) − 1

* * ⟨[HF*(t ), [e−iHS τ HF* +(t − τ )eiHS τ , σ *(t )]]⟩ dτ

(A2)

0

3 ω1 ωQ

∞

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:27

IR T11 (a)

ωQ 2 + ω12[1 + 3cos(k1τ )]

− i 3 ω1 +

1 − cos(k1τ )

∫0

0

0

(A1)

where all calculations are performed in the Larmor frequency rotating frame, as indicated by the asterisk. The first term of the master equation describes the influence of the static Hamiltonian Hs* that includes the residual static quadrupolar Hamiltonian [HQS* = √(2/3)ωQTIR 20] and the Hamiltonian corresponding to the radio frequency pulse [H1S* = Ixω1 = √2ω1TIR 11(a)]. The Zeeman-like Hamiltonian (Hzs* = √2ωresTIR 10) results from the frequency offset (ωres). The second term of the master equation describes the contribution from fluctuating components of the Hamiltonians [HF*(t)] monitoring relaxation of coherences:28,84,89,90

By carefully operating at the resonance frequency (ωres = 0), the time evolution of coherences splits in two independent subsets:27 IR −iHS τ = eiHS τ T20 e

APPENDIX

IR T11 (a)

1 − cos(k1τ ) k 12

IR T20

sin(k1τ ) IR T21 (s) k1

+ ω1ωQ

1 − cos(k1τ ) k 12

IR T22 (s )

(A4b)

(A4a) 20438


The Journal of Physical Chemistry C *

Article

sin(k1τ ) IR sin(k1τ ) IR T11 (a) − i 3 ω1 T20 k1 k1

*

IR eiHS τ T21 (s)e−iHS τ = iωQ

IR + cos(k1τ )T21 (s) − iω1

*

*

IR eiHS τ T22 (a)e−i HS τ = −i

sin(k1τ ) IR T22 (s) k1

k 2 sin(k 3τ ) − k 3 sin(k 2τ ) IR T10 k1

+ ω1

(A4c)

cos(k 3τ ) − cos(k 2τ ) IR T11 (s) k1

− iω1

e

iHS*τ

IR T22 (s)e

−iHS*τ

= ω1ωQ

1 − cos(k1τ ) k 12

cos(k1τ ) − 1 k 12

IR T20

IR T11 (a) +

k1

3 ω12

2

(A4h)

where the characteristic angular velocities defined by

k2 =

e

IR −iHS*τ T10 e

k cos(k 3τ ) + k 3 cos(k 2τ ) IR = 2 T10 k1 − iω1 + ω1

2

Q T2,0 =

cos(k 3τ ) − cos(k 2τ ) IR T22 (a) k1

+

(A6c)

(A6d)

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

(A6e)

FQ2,m(t)

The functions in eq A6a are the second-order spherical harmonics28 describing the reorientation of the OD director in the static magnetic field B0. Since the electrostatic field gradient felt by the deuterium nucleus in heavy water is directed along the OD bond, the functions FQ2,m(t) are related to the two Euler angles (θLW, ϕLW):

cos(k 3τ ) − cos(k 2τ ) IR = ω1 T10 k1 +i

1 IR [3Iz 2 − I(I + 1)] = T20 6

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

T2,Q±2 =

e

(A6b)

is equal to (3/2) × π × 210 kHz for deuterium in bulk heavy water.91 The spin operators describing the quadrupolar coupling are given by

sin(k 3τ ) + sin(k 2τ ) IR T10 k1

(A4f)

* IR T21 (a)e−iHS τ

e 2qQ 3 8 ℏI(2I − 1) 1/2

k sin(k 2τ ) − k 3 sin(k 3τ ) IR +i 2 T21 (a) k1

iHS*τ

(A6a)

where the quadrupolar coupling constant, defined by

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

+ ω1

( −1)m T2,QmF2,Q−m(t )

∑ m =−2

CQ =

+

(A5c)

2

HQ (t ) = CQ

(A4e)

*

ωQ 2 + 4ω12 − ωQ

In bulk water, quadrupolar coupling is the main mechanism responsible for relaxation of heavy water:28,84,89,90

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

*

(A5b)

2

A.2. Quadrupolar Relaxation

cos(k 3τ ) − cos(k 2τ ) IR T21 (a) k1

IR eiHS τ T11 (s)e−iHS τ = −iω1

(A5a)

ωQ 2 + 4ω12

ωQ +

k3 =

sin(k 3τ ) + sin(k 2τ ) IR T11 (s) k1

k1, k2 and k3 are

ωQ 2 + 4ω12

k1 =

IR T22 (s )

(A4d)

iHS*τ

k 2 cos(k 3τ ) + k 3 cos(k 2τ ) IR T22 (a) k1 27

sin(k1τ ) IR − iω1 T21 (s) k1

k12 − ω12[1 − cos(k1τ )]

+

+

sin(k 3τ ) + sin(k 2τ ) IR T21 (a) k1

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

k 2 cos(k 2τ ) + k 3 cos(k 3τ ) IR T21 (a) k1

Q F2,0 (t ) =

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

3 cos2 θ LW(t ) − 1 2

F2,Q±1(t ) = ( ±1)

(A4g) 20439

LW 3 sin 2θ LW(t ) e∓iϕ (t ) 8

(A6f)

(A6g)



Article

LW 3 2 LW sin θ (t ) e∓2iϕ (t ) 8

F2,Q±2(t ) =

1 T2D± 1 = ∓ (IZS± + I±SZ) 2 1 IR IR IR =∓ {T10 S± ∓ [T11 (s) ∓ T11 (a)]SZ} 2

(A6h)

The contribution from the quadrupolar relaxation mechanism to the complete master equation (eqs A1 and A2) may also be written in a matrix form:

T2D± 2 =

⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜T IR (a)⎟ ⎜T IR (a)⎟ ⎜ 11 ⎟ ⎜ 11 ⎟ ⎜ T IR (s) ⎟ ⎜ T IR (s) ⎟ ⎜ 21 ⎟ ⎜ 21 ⎟ ⎜ T IR (s) ⎟ ⎜ IR ⎟ 22 d⎜ ⎟ = − diag(AQ , BQ , C Q , DQ , EQ , BQ , C Q , DQ )⎜ T22 (s) ⎟ ⎜ T IR ⎟ dt ⎜ T IR ⎟ ⎜ 10 ⎟ ⎜ 10 ⎟ ⎜ IR ⎟ ⎜ IR ⎟ ⎜ T11 (s) ⎟ ⎜ T11 (s) ⎟ ⎜ IR ⎟ ⎜ IR ⎟ ⎜T21 (a)⎟ ⎜T21 (a)⎟ ⎜ IR ⎟ ⎜ IR ⎟ T ( a ) ⎝ 22 ⎠ ⎝T22 (a)⎠ IR T20

IR T20

(A7)

CQ =

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

(A9e)

⎛ 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 ⎟ ( ) ( ) T s T s 11 11 ⎜ ⎟ ⎟ ⎜ ⎜ IR ⎟ ⎜ IR ⎟ ( ) ( ) T a T a 21 21 ⎜ ⎟ ⎟ ⎜ ⎜ IR ⎟ ⎜ IR ⎟ ⎝T22 (a)⎠ ⎝T22 (a)⎠

AQ = 3J1Q (ω0) 3 Q 5 J0 (0) + J1Q (ω0) + J2Q (2ω0) 2 2

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

where Sα stems from various components of the electronic spin. The functions FQ2,m(t) in eq A9a are the same as in eq A6a but now they describe the reorientation of the vector joining the two coupled spins [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 diffusion of the probe through variation of the separation between the coupled spins [cf. the term rIS−3(t) in eq A9a]. The contribution from the heteronuclear dipolar coupling to the complete master equation (eqs A1 and A2) may also be written in a matrix form:54

with

BQ =

(A9d)

DQ = J1Q (ω0) + 2J2Q (2ω0)

(A10)

E Q = J1Q (ω0) + 4J2Q (2ω0)

with

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

AD =

1 D J (ω0 − ωS) + J1D(ω0) + 2J2D(ω0 + ωS) 3 0

BD =

2 D 1 D 1 J (0) + J (ω0 − ωS) + J1D(ω0) 9 0 18 0 6 1 D 1 D + J1 (ωS) + J2 (ω0 + ωS) 3 3

A.3. Heteronuclear (Paramagnetic) Dipolar Relaxation

CD =

Because of the presence of iron within the natural beidellite clay particle, the heteronuclear dipolar coupling may be responsible for NMR relaxation of the confined heavy water, in addition to the intrinsic quadrupolar relaxation mechanism. The corresponding Hamiltonian28,84,90 is defined by

2 D 5 D 5 1 J (0) + J (ω0 − ωS) + J1D(ω0) + J1D(ωS) 9 0 18 0 6 3 5 D + J2 (ω0 + ωS) 3

DD =

8 D 1 1 4 J (0) + J0D(ω0 − ωS) + J1D(ω0) + J1D(ωS) 9 0 9 3 3 2 D + J2 (ω0 + ωS) 3

ED =

1 D 1 2 J0 (ω0 − ωS) + J1D(ω0) + J2D(ω0 + ωS) 9 3 3

JmQ (mω0) = (−1)m CQ 2 ×

∫0

∞

[F2,Q−m(0) − ⟨F2,Q−m⟩][F2,Qm(τ ) − ⟨F2,Qm⟩]eimω0τ dτ

(A8)

2

HD(t ) = C D

∑

( −1)m

T2,DmF2,D−m(t )

m =−2

rIS3(t )

(A9a)

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

The corresponding spectral densities are

(A9b)

and the spin operators become: D T20

JmD(ω) = (− 1)m NSS(S + 1)C D2

⎤ 1 ⎡ 1 1 IR = 2 2 T10 Sz ⎢2IzSz − 2 (I+S− + I −S+)⎦⎥ = ⎣ 6 6 1 IR IR + ⎡⎣T11 (s)(S− − S+) − T11 (a)(S− + S+)⎦⎤ 2

{

⎡ F D (τ ) 2, m ×⎢ 3 − ⎢⎣ rIS (τ )

}

F2,Dm rIS

3

∫0

∞ ⎡ F D (0) ⎢ 2, −m ⎢⎣ rIS3(0)

⎤ ⎥eimω0τ dτ ⎥⎦

−

F2,D−m rIS

3

⎤ ⎥ ⎥⎦

(A11)

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

(A9c) 20440



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A.4. Contributions to Spin-Locking Relaxation Rate

f mQ= 0 (σ *) = − CQ 2

The relaxation of coherences during the spin-locking experiments is described by the complete master equation A1. We focus only on the m = 0 contribution of equation A2, since the m = ± 1 and m = ± 2 components are only slightly modified27 by the weak irradiation field. For quadrupolar coupling, the corresponding m = 0 component is given by

∫0

∞

Q Q Q Q [F20 (t ) − ⟨F20 (t )⟩] × [F20 (t − τ ) − ⟨F20 (t )⟩] *

*

IR IR + iHS τ × [T20 , [e−iHS τ T20 e , σ *]] dτ

(A12)

leading to the contribution from the quadrupolar relaxation mechanism to the master equation:27

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

with KQ =

AQ = 3J1Q (ω0) 3ωQ 2J0Q (0) + 4ω12J0Q (k1)

BQ =

2k12 3ωQ 2J0Q (0) + 4ω12J0Q (k1)

CQ =

2k12

+

5 Q J (ω0) + J2Q (2ω0) 2 1

E

J1Q (ω0)

=

LQ =

+

2k12

heteronuclear dipolar coupling is given by

4 fmD= 0 (σ *) = − C D2NSS(S + 1) 9

4J2Q (2ω0)

⎡ F D (t − τ ) × ⎢ 203 − ⎢⎣ rIS (t − τ )

2 Q 2 Q Q 3 ωQ J0 (0) + 2ω1 [J0 (0) + J0 (k1)] 2 k 12 5 + J1Q (ω0) + J2Q (2ω0) 2

MQ =

3ω1ωQ [J0Q (0) − J0Q (k1)]

In the same manner, the m = 0 contribution of the

1 + J1Q (ω0) + J2Q (2ω0) 2

DQ = J1Q (ω0) + 2J2Q (2ω0) Q

(A13)

∫0

∞⎡

F D (t ) ⎢ 20 − ⎢⎣ rIS3(t )

D ⎤ F20 ⎥ 3 rIS ⎥⎦

D ⎤ F20 ⎥ 3 rIS ⎥⎦

* IR + iHS*τ IR × ⎡⎣T10 , [e−iHS τ T10 e , σ *]⎤⎦ dτ

(A14)

3ωQ 2J0Q (0) + 2ω12[J0Q (0) + J0Q (k1)] 2k12

leading to the contribution from the heteronuclear dipolar

1 + J1Q (ω0) + J2Q (2ω0) 2

relaxation mechanism to the master equation:63

⎛ 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 C 0 0 0 0 0 0 ⎟⎜ 21 ⎟ ⎜ 21 ⎟ ⎟⎜ IR ⎟ ⎜ ⎜ T IR (s) ⎟ D D d ⎜ 22 ⎟ K 0 0 D 0 0 0 0 ⎟⎜ T22 (s) ⎟ = − ⎜⎜ ⎟⎜ IR ⎟ dt ⎜ T IR ⎟ 0 0 0 ED 0 − K D 0 ⎟⎜ T10 ⎟ ⎜0 ⎜ 10 ⎟ ⎜0 ⎟ ⎜ IR ⎟ IR 0 0 0 0 LD 0 2K D ⎟⎜ T11 ⎟⎜ (s) ⎟ ⎜ ⎜ T11 (s) ⎟ ⎜0 ⎜ IR ⎟ 0 0 0 0 0 MD 0 ⎟⎜T IR (a)⎟ ⎟⎜ 21 ⎟ ⎜ ⎜T21 (a)⎟ D ⎝0 DD ⎠⎜ IR ⎟ 0 0 0 0 K 0 ⎜ IR ⎟ ⎝T22 (a)⎠ ⎝T22 (a)⎠

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with 1 D J (ω0 − ωS) + J1D(ω0) + 2J2D(ω0 + ωS) 3 0

AD =

+

2[k 2J0D(k 3) + k 3J0D(k 2)]

+

CD =

5 D J (ω0 − ωS) 9k1 18 0 5 1 5 + J1D(ω0) + J1D(ωS) + J2D(ω0 + ωS) 6 3 3

DD =

8 D 1 1 4 J0 (0) + J0D(ω0 − ωS) + J1D(ω0) + J1D(ωS) 9 9 3 3 2 D + J2 (ω0 + ωS) 3

ED =

1 D 1 2 J0 (ω0 − ωS) + J1D(ω0) + J2D(ω0 + ωS) 9 3 3

LD =

KD =

2[k 2J0D(k 3) + k 3J0D(k 2)]

1 D J (ω0 − ωS) 18 0 1 1 1 + J1D(ω0) + J1D(ωS) + J2D(ω0 + ωS) 6 3 3

MD =

■

2[k 2J0D(k 3) + k 3J0D(k 2)]

1 D J (ω0 − ωS) 9k1 18 0 1 1 1 + J1D(ω0) + J1D(ωS) + J2D(ω0 + ωS) 6 3 3

BD =

9k1

2[k 2J0D(k 3) + k 3J0D(k 2)]

+

5 D J (ω0 − ωS) 18 0 5 1 5 + J1D(ω0) + J1D(ωS) + J2D(ω0 + ωS) 6 3 3 9k1

+

2ω1[J0D(k 3) − J0D(k 2)] 9k1

ASSOCIATED CONTENT

* Supporting Information S

Complete author list for ref 64. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Authors

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

The authors declare no competing financial interest.

■

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 (PHENIX, UPMC, Paris) for providing us with the beidellite clay sample used in the present study.

■

REFERENCES

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