133Cs Nuclear Magnetic Resonance Relaxometry as a Probe of the

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Cs Nuclear Magnetic Resonance Relaxometry as a Probe of the Mobility of Cesium Cations Confined within Dense Clay Sediments Patrice Porion,*,† Fabienne Warmont,† Anne Marie Faugère,† Anne-Laure Rollet,‡ Emmanuelle Dubois,‡ Virginie Marry,‡ Laurent J. Michot,‡ and Alfred Delville*,† †

ICMN, UMR 7374, CNRS-Université d’Orléans, 45071 Orléans Cedex 02, France PHENIX, UMR 8324, CNRS-Université UPMC Paris 06, 75005 Paris, France

‡

ABSTRACT: 133Cs nuclear magnetic resonance is used to quantify the mobility of cesium counterions confined within the interlamellar space of synthetic Hectorite clay under controlled hydration conditions. The degree of lamellae ordering within the macroscopic clay sediment is determined by analyzing the variation of the 133Cs NMR spectra as a function of the sample orientation in the static magnetic field. Furthermore, a detailed analysis of the 133Cs NMR spectra allows one to determine the order of magnitude of the quadrupolar and heteronuclear dipolar couplings monitoring the NMR relaxation of the cesium cations confined within clay sediments. Finally, a lower limit to the average residence time of the cesium cations within the clay interlamellar space is determined by using locking conditions.

I. INTRODUCTION Cesium radioisotopes are hazardous nuclear wastes because of their long disintegration periods and contamination capacity of soils,1−4 plants,5,6 and water resources.7 Among others (like cationic exchangers,8 electrodes,9 or cementeous material10), the clay interfacial system was suggested to be used as an adsorbent11,12 to store and immobilize such dangerous radioisotopes by exploiting its various physicochemical properties (large anisotropy, ionic exchange capacity, high specific surface area, and adsorbent power). In such context, it appears crucial to quantify the retention capacity of clay interfacial systems and determine the intrinsic mobility of cesium cations confined in the interlamellar space between individual clay platelets. An important feature of these characterizations is to determine the influence of the nature of the clay samples (chemical composition of the clay lamellae, surface charge density, and localization of the charged sites within the clay matrix) on their retention capacity in order to optimize their use as storing material. As a consequence, numerous numerical modelings13 and simulations14−16 were performed to quantify the nature of the interactions of cesium cations with clay materials15 and natural organic matter.14 In that context, various dynamical studies were performed (quasi elastic neutron scattering,17−20 dielectric relaxation,21,22 and conductivity23 measurements, nuclear magnetic resonance pulsed gradient echo attenuation,24−26 nuclear magnetic resonance relaxometry27−31) to determine the mobility of confined probes diffusing within porous media. Our study is a first step pertaining to such multiexperimental investigation of the dynamical behavior of cesium cations confined within clay © XXXX American Chemical Society

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Cs NMR relaxation measurements under spin-

sediments. For that purpose, we selected to use synthetic clay with a well-characterized structure and chemical composition, simplifying the analysis of the various experimental data. In that framework, NMR dynamical measurements are ideal investigation tools thanks to their complete selectivity. As an example, a large variety of 133Cs NMR investigations were already performed to characterize the structural and dynamical properties of cesium cation in the presence of adsorbing agents like crown-ethers32−34 or various interfacial material, including polymers,35 biological macromolecules,36,37 micelles,38,39 zeolites,40−42 and clays.43−47 Unfortunately, in the case of dense clay sediments like those used for waste storing, it becomes impossible to extract dynamical information on the cesium mobility by using NMR PGSE because of the fast relaxation of the transverse magnetization precluding the use of pulsed field gradient. Furthermore, 133Cs a priori appears as an inappropriate probe for using the field-cycling NMR relaxometer48−51 because of the small longitudinal relaxation rate measured in dilute aqueous solution52 (R1 = 7.5 × 10−2 s−1) (see the Appendix). This question is not totally settled for 133Cs relaxation measurements under confinement thanks to the expected enhancement of the NMR relaxation rates.48 In any case, spin− echo NMR relaxation measurements53−55 is a powerful alternative since it avoids the above-mentioned limitations and was successfully used previously to investigate the mobility of probes (water,27,29−31 lithium28) diffusing within dense clay Received: April 23, 2015 Revised: June 8, 2015

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DOI: 10.1021/acs.jpcc.5b03880 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C sediments. However, such investigations require a detailed modeling of the time evolution of the various quantum states56−58 accessible to the nuclei under the various steps of the pulse sequence used to perform these spin-locking relaxation measurements. The first section of this article describes briefly the preparation of the clay sample and its characterization in addition to the NMR experimental procedures. In the second section, we analyze the experimental data to extract structural and dynamical information on the cesium counterion confined within the interlamellar space of the clay sediment. By simultaneously measuring the longitudinal relaxation rate and analyzing the NMR line-shape as a function of the orientation of the clay sediment into the static magnetic field, it is possible to separately quantify the order of magnitude of the quadrupolar and heteronuclear dipolar couplings responsible for the relaxation of the confined 133Cs. In addition to the intrinsic relaxation behavior of the cesium cation under such heteronuclear conditions, we also determine the degree of alignment of the individual clay platelets within the macroscopic sediment. This information is further exploited to analyze the spin-locking relaxation measurements, leading to a lower limit of the residence time of the cesium cation confined within the interlamellar space between individual clay platelets. Finally, all the theoretical considerations at the basis of our numerical modeling of the NMR experimental data are summarized in the Appendix.

II. MATERIALS AND METHODS II.1. Sample Preparation. The clay used in the present study is a synthetic fluoro-hectorite with a unit cell formula that can be written as Si8(Mg5.2Li0.8)O20F4Cs0.8. It was prepared from a melt at high temperature, which yields a homogeneous material in terms of composition and charge density.59 The structural60 and dynamical18,61 features of its sodium-exchanged form were previously analyzed in details, confirming this high homogeneity. The size of the Hectorite clay particles was evaluated by using transmission electronic microscope (TEM) CM20 Philips operating at 200 kV. Typical TEM micrographs are displayed in Figure 1 (panels a and b). Histogram of the size distribution (Figure 1c) was obtained by visual analysis of 24 micrographs, leading to an apparent average size of 0.4 μm. Oriented films of Cs-fluoro-hectorite were prepared starting from a dilute dispersion (≈ 20g/L) that was air-dried on a flat support. Each preparation yielded a few tens of microns thick films, and successive deposits were then accumulated until a total thickness of around 3 mm was reached. In view of the low hydration energy of Cs cations, the dry films were then wetted with liquid water and equilibrated in a desiccator in the presence of saturated potassium sulfate to maintain a high relative humidity close to 97%. At such high relative humidity, the water content of the clay sample is 3.9 mmol per gram of dry clay, corresponding to 4.2 water molecules per cesium cation. As displayed in Figure 2, the clay film is made of partially oriented aggregates, each resulting from the stacking of individual clay platelets. As determined by high-resolution X-ray diffraction,60 the average size of coherent scattering domains corresponds to roughly 20 platelets per clay aggregate. As a consequence, 133Cs NMR spectroscopy is sensitive only to cesium cations pertaining to the interlamellar space inside the clay aggregate. The clay films were cut to the desired geometry (20 mm long, 3 mm wide, see Figure 3) and placed in a homemade glass

Figure 1. (a and b) TEM micrographs. (c) Histogram of the size distribution of the synthetic Hectorite clay dispersion.

Figure 2. Schematic view of the distribution of clay aggregates within the self-supporting film.

cell that could directly be inserted into a glass tube adapted to the NMR probe. To avoid film damage and ensure fixed film position upon sample rotation, the glass cell was covered with a thin Teflon film. To avoid possible clay film dehydration, a reservoir of heavy water was placed in the NMR tube at a position out of the radiofrequency irradiation zone. II.2. NMR Measurements. The 133Cs NMR measurements were performed on a DSX360 Bruker spectrometer, with a static magnetic field B0 of 8.465 T. Typical pulse duration for the total inversion of the longitudinal magnetization is roughly equal to 16 μs. The NMR measurements are performed by using a homemade sample holder and detection coil. The clay film is immobilized in a tube containing a reservoir of bulk B


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significantly as a function of the clay sample orientation in the static magnetic field. Because of the long relaxation time (T1 ∼ 1 s), a rather long relaxation delay (typically 5 s) is introduced within each pulse sequence. The NMR spectra were recorded with a broad spectral width (1 MHz), corresponding to a fast acquisition procedure (dwell time = 1 μs). A line broadening of 100 Hz is used to improve the signal/noise ratio. In order to reduce artifacts resulting from acoustic ringing, a reduced delay (typically 20 μs) is introduced between the detection pulse and the free induction decay (FID) recording. By contrast with the longitudinal relaxation rates, the 133Cs NMR spectra vary strongly as a function of the film orientation βFL in the static magnetic field B0 (see Figure 4). A detailed analysis of these spectra requires simulating the evolution of the various quantum states available to such I = 7/2 spin system under irradiation and relaxation. For that purpose, a complete basis set is used (see the Appendix), allowing to quantify the relative contributions of the quadrupolar and heteronuclear dipolar coupling responsible for the relaxation of the 133Cs confined within clay sediments. Finally, relaxation measurements under the double tilted rotating frame, also called spin−locking relaxation measurements,53,54 were performed to determine the frequency variation of the 133Cs NMR relaxation rates. The pulse sequence used to perform these spin-locking relaxation measurements is detailed in Figure 5. To improve the signal-to-noise ratio, the free induction decay is recorded after irradiation by

Figure 3. (a) Schematic view of the film orientation within the NMR tube used to insert the clay sample into the detection coil. (b) Schematic view of the Euler angles characterizing the orientation of an individual clay platelet within the macroscopic film (see text).

water in its sealing cap (see Figure 3). The tube diameter fits the cavity of the detection coil, allowing its free rotation along the coil axis (see Figure 3). The orientation of the film director nF,L ⃗ in the static magnetic field B0 is quantified by the βFL Euler angle. The longitudinal relaxation rate R1 is recorded by the classical inversion−recovery pulse sequence.62 It does not vary

Figure 4.

Figure 5. Schematic view of the pulse sequence used for the spinlocking relaxation measurements. The angular velocity ω1 is applied during the delay τ with various irradiation powers (see text). The delay δ is set equal to 40 μs to remove artifacts induced by acoustic ringing.

Cs NMR spectra recorded as a function of the film orientation βFL into the static magnetic field B0.

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The Journal of Physical Chemistry C using a spectral width of 140 kHz, corresponding to a dwell time of 7.14 μs. A delay δ of 40 μs between the irradiation pulse and FID recording is now required to remove artifacts induced by acoustic ringing. As illustrated in Figure 5, a single pulse is required to perform these nutation experiments.54 The angular velocities ω1 corresponding to the various irradiation powers (2.1 × 105, 1.1 × 105, 6.1 × 104, 3.2 × 104, 1.6 × 104, 7.0 × 103, and 2.7 × 103 rad s−1) were selected to roughly cover two decades. Thanks to the large residual quadrupolar coupling, the angular velocities potentially sampled by the quadrupolar and heteronuclear dipolar relaxation mechanisms cover respectively more than two and four decades (see Figure 12, panels a and b in the Appendix). II.3. Numerical Simulations. Molecular dynamics simulations were performed with the LAMMPS package63 (http:// lammps.sandia.gov). Interactions between atoms were calculated using CLAYFF force field,64 adapted to fluoro-hectorite.18 The TIP4P/2005 model of H2O molecules was chosen because it is one of the best models for water, especially concerning dynamical properties.65 The simulation box contained two clay layers of lateral dimensions 41.92 × 45.47 Å2. The number of water molecules was fixed to 4 H2O per cation, what is close to the experimental water uptake (see Section II.1). After a short phase of equilibration in the NVT ensemble with a time step of 1 fs, a NPT simulation of 200 ps was achieved in order to obtain the interlayer spacing at equilibrium. A value of 12.2 Å was found. A 15 ns trajectories was performed in the NVT ensemble, with a relaxation time of 1 ps.

Figure 6. Evaluation of the longitudinal relaxation time T1 from the inversion−recovery pulses experiment (see text).

performed to determine the order of magnitude of the high frequency spectral densities66 monitoring the quadrupolar [JQ1 (ω0) and JQ2 (2ω0)] and heteronuclear dipolar [JD0 (ω0 − ωS), JD1 (ω0), JD1 (ωS), and JD2 (ω0 − ωS)] couplings responsible for the longitudinal relaxation of the 133Cs magnetization. As explained in the Appendix (see Section A2), the heteronuclear dipolar relaxation mechanism is expected to contribute significantly to the NMR relaxation of 133Cs magnetization because of the weakness of its quadrupolar mechanism. Quantitative agreement with experimental data (see Figure 6) is obtained by setting these high frequency values to 0.7 and 0.07 s−1 for the quadrupolar and dipolar spectral densities, respectively. The measured longitudinal relaxation rate (R1 ∼ 1 s−1) remains too small to efficiently exploit field-cycling NMR spectrometers for extracting dispersion curves from the frequency variation of the NMR longitudinal relaxation rate. As a consequence, spin-locking relaxation measurements appears as the unique alternative to obtain such dispersion curves, leading to dynamical information on the mobility of the cesium counterion confined within clay platelets. III.2. Angular Variation of the 133Cs NMR Spectra. Figure 4 displays 133Cs NMR spectra recorded for various orientations βFL of the clay director into the static magnetic field B0 (see Figure 3). While the half-width broadening of the central resonance line reaches 600 Hz, the first set of satellites appears strongly broadened, and the two second sets of satellites are practically undetectable. By taking into account the 100 Hz broadening used to enhance the signal/noise ratio (see Section II.2), the width of the thin central resonance line is compatible with a transverse relaxation rate of the order of magnitude of 3000 s−1, much larger than the longitudinal relaxation rate (R1 ∼ 1 s−1, see Section III.1). Furthermore, the shape of the spectrum displayed in Figure 4 assuredly indicated that the broadening of the satellites largely exceeds the broadening of the central resonance line. This result requires a slow modulation of the quadrupolar coupling [i.e. JQ0 (0) ≫ JQ1 (ω0) ∼ JQ2 (2ω0)], since the broadening of the central line is driven only by the high-frequency spectral densities of the quadrupolar relaxation mechanism [JQ1 (ω0) and JQ2 (2ω0)], while the low frequency spectral density [JQ0 (0)] also contributes to the broadening of the satellites.68 This result is encouraging since the above-mentioned inequality is required to detect a frequency variation of the spectral densities. However, if the quadrupolar coupling alone monitors the relaxation of the confined cesium cations, the transverse relaxation rate evaluated

III. RESULTS AND DISCUSSION III.1. NMR Relaxation Rate. As explained in the Appendix, two mechanisms (i.e., quadrupolar and heterogeneous dipolar) are expected to significantly contribute to the NMR relaxation of 133Cs under heterogeneous conditions. In the framework of the NMR relaxation theory,48,66,67 the measured NMR relaxation rates are linear combinations of spectral densities that are Fourier transformed of the autocorrelation functions of the magnetic couplings (see the Appendix). Under fast modulation conditions, all spectral densities corresponding to the same relaxation mechanism are identical, thus precluding the extraction of dynamical information from the analysis of NMR relaxation measurements. Such conditions usually occur for various NMR probes, implying solvent molecules or ions within isotropic solutions. By contrast, the same probes within heterogeneous dispersions or confined environments frequently exhibit strong differences of their spectral densities, leading to useful and selective dynamical probes.48,49 Under such conditions, the longitudinal relaxation rate gives information on the high-frequency values of the various spectral densities (see the Appendix). Spectral densities at zero angular velocity are further extracted from the NMR line shape analysis or the measurement of the transverse relaxation rate48,67 (see the Appendix). Finally measurements of the longitudinal relaxation rate using field cycling spectrometer or spin-locking relaxation measurements (see the Appendix) may be used to extract the dispersion curve of the spectral densities, leading to dynamical information on the mobility of the confined NMR probes. The recovery of the longitudinal magnetization is displayed in Figure 6, corresponding to a single relaxation time of roughly 1 s. It does not vary significantly as a function of the film orientation in the static magnetic field. A complete modeling of the time evolution of the various coherences (see the Appendix) during the inversion−recovery pulse sequence is D


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The Journal of Physical Chemistry C from the width of the central resonance line (3000 s−1) should have the same order of magnitude than the longitudinal relaxation rate69−72 (1 s−1). That discrepancy originates from the heteronuclear dipolar coupling (see the Appendix) containing the same dominant contribution [JD0 (0)] for the central line and the various satellites. This result will be used to separately quantify the dominant contribution from both quadrupolar [JQ0 (0)] and heteronuclear dipolar [JD0 (0)] relaxation mechanisms from the line-shape analysis of the 133Cs NMR spectra. Figure 7 exhibits the variation of the apparent residual quadrupolar coupling directly measured on the experimental

As detailed in the Appendix, a detailed modeling of the coherence evolution is performed during the detection pulse and the FID acquisition, by including contributions from the quadrupolar and heteronuclear dipolar relaxation mechanisms. Since the broadening of the central resonance line appears unchanged by the film orientation, we use a single spectral density [JD0 (0)] to quantify the dominant contribution to the heteronuclear dipolar relaxation mechanism. By contrast, the broadening of the satellites is largely modified by the orientation of the clay film, thus requiring a careful analysis of the apparent spectral density quantifying the quadrupolar coupling as a function of the film orientation. For that purpose, we take into account the distribution of the directors of the individual clay lamellae by reference with the film director (n⃗C,F). Thanks to the clay cylindrical symmetry, the orientation of the clay lamellae within the film is characterized by two Euler angles: αCF (the colatitude) and γCF (the azimuth) (see Figure 3b). In the frame attached to the sample holder containing the clay film, we obtain n ⃗C , F

⎛ sin αCF cos γ CF ⎞ ⎜ ⎟ = ⎜ sin αCF sin γ CF ⎟ ⎜⎜ ⎟⎟ ⎝ cos αCF ⎠

(1)

After the rotation βFL of the NMR tube within the detection coil, we obtain the angle between the clay director and the static magnetic field, noted θCL and defined by cos θ CL = sin β FL sin αCF sin γ CF + cos β FL cos αCF

Figure 7. Variation of the residual quadrupolar coupling νQ extracted from the 133Cs NMR spectra as a function of the clay film orientation βFL into the static magnetic field B0. The red line corresponds to the best fit using the P2(cos βFL) function (see text).

(2)

Finally, the numerical simulations are performed by sampling the angle αCF in order to satisfy a Gaussian distribution law with a standard deviation noted σ while the angle γCF is generated randomly in the interval [0,2π]. By using the Wigner rotation matrices,77 it is then possible to relate the apparent spectral density detected in the laboratory frame and characterizing the quadrupolar relaxation mechanism [JQ,app (0)] to its intrinsic 0 components (i.e., JQ,intrinsic (0) with m ∈ {0,1,2}) evaluated in m the frame of the individual clay lamellae: 1 J0Q ,app (0) = ⟨(1 − 3 cos2 θ CL)2 ⟩J0Q ,int rinsic (0) 4 3 + ⟨(sin 2θ CL)2 ⟩J1Q ,intrinsic (0) 4 3 + ⟨(sin θ CL)4 ⟩J2Q ,aintrinsic (0) (3) 4

NMR spectra as a function of the orientation βFL of the clay film. A large gap is reported for clay orientations βFL varying between 30° and 60° [i.e., at clay orientations around the magic angle (βFL ≈ 54.7°)] because of the important broadening of the satellites at such orientations. A detailed numerical simulation of the coherence evolution during the detection pulse, time delay, and FID recording is required to extract structural and dynamical information from these 133Cs NMR spectra. Before proceeding to a systematic analysis of the line-shape as a function of the orientation of the clay film βFL into the static magnetic field B0, we simplify the 133Cs NMR spectra. First, the spectrum displayed in Figure 4 exhibits some differential broadening of its first satellites. Such behavior was already reported for quadrupolar27,29−31,73,74 nuclei (2H, 7Li) confined within clay lamellae and was shown to result from a crosscorrelation between the quadrupolar and heteronuclear dipolar coupling.75,76 This second-order relaxation mechanism has minor contribution to the detected NMR spectra. In order to neglect its contribution, we remove it by symmetrizing the measured NMR spectra (see Figure 8). Finally, the 133Cs NMR spectra exhibit a secondary thin resonance line induced by anomalous orientation of a reduced fraction of the clay film. This artifact results from a break of the self-supported clay film occurring during its insertion in the sealing tube. It is also suppressed by smoothing the baseline in the vicinity of this supplementary thin resonance line. The comparison between simulated and experimental spectra is performed simultaneously for a set of clay orientations βFL into the static magnetic field B0, varying between 0° and 90°.

In the same manner, the apparent quadrupolar splitting ωobs Q is related to its intrinsic value evaluated in the frame attached to the individual clay lamellae: ωQobs = ωQmax |P2[cos(θ CL)]| = ωQmax

3 cos2(θ CL) − 1 2 (4)

As displayed in Figure 8, we obtain a perfect agreement between experimental and numerical spectra by optimizing the 3 −1 residual quadrupolar coupling [ωmax Q = (97 ± 5) × 10 rad s ], the standard deviation of the Gaussian distribution law of the Euler angle (see above) [σ = (19 ± 3)°], the unique spectral density characterizing the heteronuclear dipolar relaxation [JD,intrinsic (0) ≈ JD,intrinsic (0) ≈ JD,intrinsic (0) = [4.0 ± 0.5) × 103 s−1], 0 1 2 and the three intrinsic spectral densities characterizing the quadrupolar relaxation: [JQ,intrinsic (0) = (1.2 ± 0.2) × 103 s−1, 0 E


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along a single director in the self-supporting clay film. This equality between the orders of magnitude of both relaxation mechanisms is important for spin-locking relaxation measurements since it allows investigating a broad dynamical range (see Figure 12, panels a and b, in the Appendix). Because of the large standard deviation quantifying the distribution of the lamellae directors within the clay film and the drastic enhancement of the m = 1 spectral density of the quadrupolar relaxation mechanisms [JQ,intrinsic (0) = (30 ± 5) × 104 s−1], the 1 apparent spectral density quantifying the quadrupolar relaxation (0)] becomes generally 1 order of magnitude mechanism [JQ,app 0 larger than the corresponding spectral density of the heteronuclear dipolar relaxation mechanism [JD,app (0)]. Under 0 such conditions, the contribution of the heteronuclear dipolar coupling [JP0 (λPp )] becomes impossible to extract from the spinlocking measurements, thus reducing the investigated dynamical range (see Figure 12, panels a and b, in the Appendix). Figures 9 and 10 display the experimental data obtained by the spin-locking relaxation measurements for seven irradiation

Figure 8. Direct comparison between the 133Cs NMR spectra and the results from the numerical modeling for several clay film orientations βFL into the static magnetic field B0 (see text).

JQ,intrinsic (0) = (30 ± 5) × 104 s−1, and JQ,intrinsic (0) = (4.0 ± 0.5) 1 2 × 103 s−1]. This set of six independent parameters [ωmax Q ,σ, JmD,intrinsic(0), J0Q,intrinsic(0), J1Q,intrinsic(0) and J2Q,intrinsic(0)] is extracted from simultaneous fitting of all the spectra displayed in Figure 4. Figure 7 further exhibits fair agreement between the apparent quadrupolar splitting resulting from visual analysis of the NMR spectra (see Figure 4) and some empirical fit using eq 4. Because of the distribution of the clay directors within the film and the large enhancement of the NMR relaxation rate near βFL = 45° (see Figure 4), Figure 7 exhibits a large gap between residual quadrupolar couplings recorded around βFL = 0° and βFL = 90°. That enhancement of NMR relaxation rate around βFL = 45° results from the order of magnitude of the m = 1 component of the quadrupolar relaxation mechanism [JQ,intrinsic (0) = (30 ± 5) × 104 s−1] whose contribution 1 becomes dominant near βFL = 45° (see eq 3), totally masking the satellites of the NMR resonance lines (see Figure 4). Such behavior was already reported for various quadrupolar nuclei confined within dense clay sediments.27,29−31,78 Under isotropic conditions (i.e., aqueous solutions), the three spectral densities quantifying the quadrupolar relaxation mechanism are identical,48,67 leading to the same NMR relaxation rate, whatever the sample orientation into the static magnetic field (see eq 3). By contrast, within heterogeneous samples, the relative enhancement of the m = 1 component of the quadrupolar relaxation mechanism may be used to probe the degree of nematic ordering even in the lack of detectable residual quadrupolar splitting.78,79 III.3. Spin-Locking Relaxation Measurements. In order to minimize the transverse relaxation rate of the confined 133Cs, the spin-locking relaxation measurements are performed with the director of the clay film nF,L ⃗ oriented perpendicular to the static magnetic field B0. As illustrated by eq 4, that orientation reduces roughly the residual quadrupolar splitting by a factor two, thus extending significantly the range of angular velocities probed by the m = 0 component [JQ0 (λQp )] of the quadrupolar relaxation mechanism (see Figure 12a in the Appendix). Furthermore, under such film orientation, the apparent spectral densities quantifying the quadrupolar [JQ,app (0)] and hetero0 nuclear dipolar [JD,app (0)] couplings have the same order of 0 magnitude (see Section III.2) if the clay lamellae are oriented

Figure 9. Time evolutions of the spin-locking relaxation measurements for seven irradiation powers corresponding to various angular velocities ω1: (a) decreasing from 2.1 × 105 to 6.1 × 104 rad s−1; (b) decreasing from 3.2 × 104 to 2.7 × 103 rad s−1. The clay film orientation is set at βFL = 90°.

powers corresponding to angular velocities ω1, decreasing gradually from 2.1 × 105 rad s−1 to 2.7 × 103 rad s−1. The simulated data are obtained by numerical solving the set of eqs A1 and A2, using the set of eqs A5−A7 to describe the influence of the irradiation on the time evolution of the coherences. In the numerical treatment, we assume that the spectral densities F


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Figure 11. Schematic view of the spectral density JQ0 (ω) for the quadrupolar relaxation mechanism used for the numerical modeling of the spin-locking relaxation measurements.

continuous decrease of the dispersion curve (see Figure 11) is certainly much lower than 1 × 103 rad s−1, leading to a lower limit to the cesium residence time within the interlamellar space of Hectorite platelets (τB > 1 ms). This residence time helps to quantify the retention power of synthetic Hectorite for the cesium counterions. By taking into account the average size of Hectorite platelets (L ≈ 0.4 μm, see Section II.1), the detected limit is compatible with a radial self-diffusion coefficient [Dρ = L2/(2τB)] of the confined cesium counterions smaller than 8 × 10−11 m2 s−1. We performed numerical simulations of molecular dynamics (see Section II.3) to quantify the cesium self-diffusion coefficient in the direction parallel to the clay surface. For that purpose, we determine the slope of the radial mean-square displacement as a function of time80 between 2 and 4 ns. At small times, the cation motion could not be described by a simple diffusive motion, since Cs+ remains a few hundreds of picoseconds above a hexagonal cavity before jumping to another one. The average diffusion coefficient parallel to the clay layers was found to be (1.4 ± 0.1) × 10−11 m2 s−1, in complete agreement with the upper limit (8 × 10−11 m2 s−1) extracted from our 133Cs NMR relaxometry measurements (see above). Furthermore, various MD simulations16,81−85 of the mobility of the cesium cation confined within Montmorillonite platelets under the same hydration condition lead to equivalent results (2 × 10−11 m2 s−1 ≤ DMD ≤ 2 × 10−10 m2 s−1). By contrast with such local dynamical properties, tracer diffusion measurements may be used to determine cesium mobility within macroscopic clay sediments, leading to lower apparent cesium mobility86−88 (Dtracer ≤ 1 × 10−11 m2 s−1). In the lack of detailed analysis of the structural and topological properties of the macroscopic porous network limited by the clay aggregates (see Figure 2), these two sets of measurements of cesium mobility performed at different length-scales cannot be directly compared. Furthermore, multiscale analysis of ionic mobility within dense clay sediments is required to take into account complementary diffusion pathways,89 such as cation diffusion inside the clay interlamellar space90 and through the macroscopic porous network surrounding clay aggregates.91 In that framework, the residence time evaluated by these spin-locking relaxation measurements give information on the ionic mobility within the interlamellar space of clay sediments27−29,90 measured over a reduced time period. Furthermore, two-time stimulated echo attenuation30,31,92

Figure 10. Fourier transforms of the time evolutions (see Figure 9) of the spin-locking relaxation measurements for seven irradiation powers corresponding to various angular velocities ω1: (a) decreasing from 2.1 × 105 to 6.1 × 104 rad s−1; (b) decreasing from 3.2 × 104 to 2.7 × 103 rad s−1. The clay film orientation is set at βFL = 90°.

probed at all nonzero angular velocities by these spin-locking measurements have nearly the same order of magnitude as their values probed at high angular velocities [i.e., JQ0 (λQp ≠ 0) = JQ0 (ω0) ≈ 0.7 rad s−1 and JP0 (λPp ≠ 0) = JP0 (ω0) ≈ 0.07 rad s−1]. The agreement between experimental and numerical data is satisfactory, in spite of the approximations used in our numerical approach. As reported by previous experimental and numerical modeling,27,29−31 the dispersion curves detected by NMR relaxometry exhibit three dynamical regimes: a plateau at low angular velocities, a continuous decrease, and a second plateau at high angular velocities (see Figure 11). As shown by numerical simulations, the characteristic angular velocity ωc corresponding to the transition between the low-frequency plateau and the continuous decrease (ωc = 1/τB) describes the average residence time τB of the cesium cation within the individual clay lamellae.27,29−31 Since all the spectral densities, used to reproduce spin-locking relaxation measurements, are set equal to their high-frequency values [JQ0 (λQp ≠ 0) = JQ0 (ω0) and JP0 (λPp ≠ 0) = JP0 (ω0)], the transition between the high frequency plateau and the continuous decrease of the dispersion curve (see Figure 11) must occur at angular velocity smaller than 5 × 103 rad s−1 (i.e., the lowest nonzero angular velocity probed here by the quadrupolar relaxation mechanism) (see Figure 12a in the Appendix). As a consequence, the transition between the low-frequency plateau and the same G


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coupling, quadrupolar, and heteronuclear dipolar relaxation mechanisms). To our knowledge, this study is the first use of 133 C spin-locking NMR relaxation measurements to obtain dynamical information on the mobility of confined cesium cations. These experimental and numerical procedures are not limited to the study of hydrated clay particles but may be exploited to investigate the dynamical behavior of a large class of interfacial systems (including biopolymers, zeolite, and disordered porous media) able to adsorb or confine cesium cations.

■

APPENDIX

A.1. Time Evolution of the Coherences

In the framework of the Redfield theory,48,66,67,93 the time evolution of the various spin states may be described by the master equation: dσ * = −i[HS*, σ *] + f (σ *) dt

(A1)

As noted by the asterisk, all calculations are performed in the Larmor frequency rotating frame. The first contribution to eq A1 results from the static Hamiltonian HS* describing the residual quadrupolar and dipolar couplings in addition to the radio frequency pulses. The second contribution to eq A1 originates from the fluctuations of the quadrupolar and heteronuclear dipolar couplings [HF*(t)] that are responsible for the relaxation of the spin states. At a first order approximation,48,66,67 these relaxation contributions may be evaluated by f (σ *) = −

∫0

∞

*

*

⟨[HF*(t ), [e−iHS τ HF* +(t − τ )eiHS τ , σ *(t )]]⟩dτ (A2)

A complete basis set containing 64 spin operators, also called coherences, is required to describe all the quantum states available to the I = 7/2 spin system. For that purpose, we selected to use an irreducible representation evaluated from the irreducible tensor operators.94−97 As detailed in Tables 1 and 2, this complete set of coherences splits in two independent subsets from which the trivial identity (T00) has been removed. The first index of these coherences describes their rank and the second their order. The interest of this basis set originates from its direct relation to the spin states: the three first rank coherences [T10, T11(a), and T11(s)] describe respectively the three components of the spin magnetization (Iz, Ix, and Iy), while the five second-order coherences [T20, T21(a), T21(s), T22(a), and T22(s)] are required to describe the quadrupolar Hamiltonian. Thanks to the normalization and orthogonality of this basis set,57,58 the matrix formulation of eqs A1 and A2 becomes straightforward by using formal calculation (Mathematica 5.1 software98).

Figure 12. Histograms of the distributions of the angular velocities that may are probed by the spin-locking experiment of I = 7/2 spin for various irradiation powers. (a) Distribution of the λQp angular velocities corresponding to the quadrupolar relaxation mechanism; (b) Distribution of the λPp angular velocities corresponding to the dipolar relaxation mechanism.

may be used to investigate the exchange of the counterions confined within the interlamellar space of clay platelets pertaining to different clay aggregates thanks to their diffusion within the macroscopic porous network.30,31,92

IV. CONCLUSIONS 133 Cs NMR spectroscopy and spin-locking NMR relaxation measurements are used to obtain dynamical information on the mobility of cesium counterions confined within the interlamellar space of synthetic Hectorite. For that purpose, a selfsupporting clay film is conditioned under controlled hydration condition. The relative contributions of the quadrupolar and heteronuclear dipolar relaxation mechanisms, responsible for the NMR relaxation of the confined cesium cations, are quantified by carefully analyzing the 133Cs NMR longitudinal relaxation rate and the variation of the 133Cs NMR resonance line as a function of the orientation of the clay film into the static magnetic field. The same analysis is also used to determine the distribution of the clay directors with the selfsupporting film. In addition to these NMR measurements, a numerical modeling is required to reproduce the time evolution of all the quantum state available to the I = 7/2 spin system under the various experimental conditions used in this study (i.e., detection and irradiation pulses, static residual quadrupolar

A.2. Relaxation Contributions

The quadrupolar Hamiltonian may be written: 2

HQ (t ) = CQ

∑

( −1)m T2,QmF2,Q−m(t )

(A3a)

m =−2

where CQ =

2 3 e Qq(1 − γ∞) 2 2I(2I − 1)

(A3b)

is the quadrupolar coupling constant, γ∞ is the Sternheimer anti-shielding factor,52 eQ and eq are the principal components of the tensor describing the intrinsic electrostatic quadrupole of the nucleus and the electric field gradient generated by its electronic surrounding, respectively. TQ2,m are 48,66,67

H


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Table 1. First Sub-Set of 32 Independent Coherences, Including the T10 Coherence and Constituting a First Based Set Used to Describe the Time Evolution of Spin I = 7/2 T10

T11(s) T21(a) T31(s) T41(a) T51(s) T61(a) T71(s)

T30 T50 T70

T22(a) T32(s) T42(a) T52(s) T62(a) T72(s)

T33(s) T43(a) T53(s) T63(a) T73(s)

T44(a) T54(s) T64(a) T74(s)

T55(s) T65(a) T75(s)

T66(a) T76(s)

T77(s)

Table 2. Second Sub-Set of 31 Independent Coherences, Including the T20 Coherence, and Constituting a Second Based Set Used to Describe the Time Evolution of Spin I = 7/2 T11(a) T21(s) T31(a) T41(s) T51(a) T61(s) T71(a)

T20 T40 T60

T22(s) T32(a) T42(s) T52(a) T62(s) T72(a)

T33(a) T43(s) T53(a) T63(s) T73(a)

T44(s) T54(a) T64(s) T74(a)

equal to the five second-order coherences required to describe the quadrupolar couping (see above). Usual values of the quadrupolar moment and the Sternheimer anti-shielding factor are displayed in Table 3 for alkali nuclei.52,99 The second-order

7

Li Na 39 K 85 Rb 133 Cs 23

spin I

Q (barn)

γ∞

3/2 3/2 3/2 5/2 7/2

0.042 0.11 0.09 0.31 0.004

0.74 5.1 18.3 48.2 111

[Q(1 − γ∞)/ 2I(2I − 1)]2 (barn2) 2.7 8.8 7.3 5.6 1.2

× × × × ×

R1 (s−1 in water)

10−5 10−3 10−2 10−1 10−4

0.027 16.2 24 420 0.075

2

HD(t ) = C D

Q (t) are used to describe the spherical harmonics F2,−m orientation, into the static magnetic field, of the principal component of the second-order traceless tensor, quantifying the electrostatic field gradient felt by the confined cesium cations. The spectral densities are derived from the Fourier transform of the autocorrelation functions (see eq A2) of these second-order spherical harmonics

JmQ (mω0) = ( −1)m CQ2

∫0

∞

∑ m =−2

T77(a)

( −1)m

T2,DmF2,D−m(t ) rIS3 (t )

(A4a)

where CD is the dipolar coupling constant,48,66,67 and the spin operators become D T20 =

(F2,Q−m(0) − ⟨F2,Q−m⟩)

× (F2,Qm(τ ) − ⟨F2,Qm⟩)eimω0τ dτ

T66(s) T76(a)

among a set of quadrupolar nuclei. As displayed in Table 3 for some alkali nuclei, we obtain some correlation between the orders of magnitude of their relaxation rate measured in dilute aqueous solution52 and the intrinsic parameter extracted from the quadrupolar coupling constant ([Q(1 − γ∞)]2/[2I(2I − 1)]2). As illustrated in Table 3, the quadrupolar relaxation mechanism appears very efficient for monitoring the NMR relaxation of some alkali cations (85Rb, 39K, and 23Na) in dilute aqueous solutions. By contrast, in the presence of paramagnetic impurities, the heteronuclear dipolar coupling may significantly contribute to the NMR relaxation of other quadrupolar nuclei (7Li and 133Cs) because of the weakness of their quadrupolar contribution. The above mentioned heteronuclear dipolar Halmiltonian is written:

Table 3. Intrinsic Parameters Monitoring the Order of Magnitude of the Quadrupolar Relaxation Mechanism of Some Alkali Cations (See Text) isotope

T55(a) T65(s) T75(a)

⎛ 1 7 ⎜2T10IRSz + [T11IR(s)(S− − S+) ⎝ 2 2 ⎞ − T11IR(a)(S− + S+)]⎟ ⎠

(A4b)

(A3c)

These spectral densities are directly used in eqs A1−A2 to describe the contribution of the quadrupolar relaxation mechanism to the time evolution of the various coherences. Predicting the efficiency of the quadrupolar relaxation mechanism is difficult to perform a priori since the spectral densities vary according to the fluctuations of the electrostatic field gradient probed by the quadrupolar nucleus (see eq A3c). As a consequence, a detailed molecular modeling of ion hydration100,101 is required to satisfactorily treat that problem. Nevertheless, thanks to the large variation of the intrinsic parameters monitoring the quadrupolar coupling constant (i.e., eQ, γ∞, and I in eq A3b), it becomes possible to predict the relative strength of the quadrupolar relaxation mechanism

T2D± 1 = ∓

21 IR (T10 S± ∓ [T11IR(s) ∓ T11IR(a)]Sz) 2

(A4c)

T2D± 2 = ∓

21 IR [T11 (s) ∓ T11IR(a)]S∓ 2

(A4d)

The function FD2,m(t) are also evaluated by using the secondorder spherical harmonics describing now the orientation of the vector joining the two coupled spin [noted rI⃗ S(t)] by reference with the static magnetic field. Here also, spectral densities are derived from the Fourier transform of the autocorrelation functions of these second-order spherical harmonics I


Article


Thanks to the above mentioned separation of the coherences (see Tables 1 and 2), a specific subset is selected to analyze the time evolution of the T10 and T20 coherences. By noting SX, the matrices describing the eigenvectors in the corresponding independent sub-set, we can extract the set of coefficients bXp by inverting numerically the relationship:

JmD (ω) = ( −1)m NSS(S + 1)C D2

∫0

∞ ⎛ F D (0) ⎜ 2, −m ⎜ r 3 (0) ⎝ IS

−

F2,Dm rIS3

−

F2,D−m rIS3

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

⎞ ⎟eimω0τ dτ ⎟ ⎠

⎛b X ⎞ ⎜ 1⎟ T XSL0(0) = ∑ bpX νp⃗ X = S X ⎜ ⋮ ⎟ ⎜⎜ X ⎟⎟ p=1 ⎝ bn ⎠ n

(A4e)

A.3. Line Shape Analysis

After some analysis of the numerical solution of eqs A1−A4, one easily concludes that the coherences located in the different columns of Tables 1 and 2 are coupled together by the static quadrupolar Hamiltonian, responsible for the splitting of the NMR resonance line, and the contributions of the quadrupolar relaxation mechanism, while the coherences located in the different lines of Tables 1 and 2 are coupled only through irradiation pulses along T11(a). Furthermore, the heteronuclear dipolar coupling leads to the same diagonal contributions to the relaxation of the coherences located in the same columns within Tables 1 and 2. As a consequence, seven coherences of order one [T11(s), T21(a), T31(s), T41(a), T21(s), T61(a), T71(s)] are required to describe the free induction decay of I = 7/2 spin under the influence of the residual quadrupolar coupling. A matrix analysis of the quadrupolar and heteronuclear dipolar couplings leads to seven eigenstates with one purely real eigenvalue and three complex eigenvalues in addition to their complex conjugate. The real eigenvalue leads to the central line of the NMR spectrum, corresponding to the −1/2 ↔ 1/2 spin transition, and the three complex conjugate eigenvalues lead to the set of three symmetric satellites of the NMR resonance line, corresponding, respectively, to the ±7/2 ↔ ±5/2, ± 5/2 ↔ ±3/2, and ±3/2 ↔ ±1/2 spin transitions. The broadenings of these four different resonance lines are given by the real part of the corresponding eigenvalue, leading to a single relaxation mode per spin transition. By contrast, in the lack of residual quadrupolar coupling, the NMR resonance line of I = 7/2 spin may result from the superposition of four independent relaxation modes.70

By inserting eq A5 in eq A2, we can evaluate numerically the time evolution of any coherence under spin-locking condition. Due to eq A5, the spectral densities JX0 (0), previously evaluated at zero angular velocity, are now replaced by a linear combination of JX0 (λXp ). Since the m = 1 and m = 2 components of the quadrupolar and dipolar relaxation mechanisms remain unchanged, we can focus on their m = 0 components, leading to n p=1

where the vector (ap⃗ X )k

*

■

(A7a)

k=1

satisfies the relationship:

bpX SkpX

(A7b)

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 and the CM20 Philips TEM used for this study were purchased thanks to grants from Région Centre (France). We acknowledge the contribution from the NEEDS interdisciplinary project (MIPOR, MULTIDYN). We cordially thank Dr. Joël Puibasset (ICMN, Orléans) for helpful discussions and Dr. Joseph Breu (Institut für Anorganische Chemie der Universität Regensburg) for providing the Hectorite clays. The authors are grateful to José C. Gomes and Santiago Braley for their help in the design and the machining of the glass cell.

n

∑ bpX νp⃗ X exp(iλpX t ) p=1

=

aXp⃗

Figure 12 (panels a and b) display histograms of the distributions of the angular velocities that may be probed by spin-locking experiments of I = 7/2 spin under our experimental conditions (see Section III.2) [i.e., a residual quadrupolar coupling ωmax = (97 ± 5) × 103 rad s−1, an Q FL orientation of the clay film β =90° and a Gaussian distribution of the clay lamellae with a standard deviation σ = (19 ±2)°]. Furthermore, seven angular velocities ω1 are selected for the irradiation, decreasing gradually from 2.1 × 105 rad s−1 to 2.7 × 103 rad s−1. In addition to the classical zero angular velocity JX0 (0), the m = 0 component of the quadrupolar coupling [i.e., JQ0 (λQp ) in equation A7] covers two decades (see Figure 12a). The lowest non-zero angular velocity is monitored by the residual quadrupolar coupling felt by the confined cesium cations. By contrast, the angular velocities probed by the m = 0 component of the heteronuclear dipolar coupling [i.e., JP0 (λPp ) in eq A7a] are nonzero but decrease to lower values, thus covering four decades (see Figure 12b).

Spin-locking relaxation measurements are used to investigate the frequency variation of the m = 0 components of the quadrupolar and dipolar spectral densities (see Section III) by varying the irradiation power ω1. The main difficulty arises from the time evolution of the m = 0 components of the quadrupolar and dipolar couplings under the influence of the corresponding static Hamiltonian HS* (see the term e−iHS*τ HF*+(t − τ)eiHS*τ in eq A2). Thanks to the lack of relaxation mechanisms, we calculate numerically the purely imaginary eigenvalues iλp and corresponding eigenvectors ν⃗p of the static Hamiltonian H*S in order to determine the time evolution of the T20 and T10 coherences monitoring the m = 0 components of the quadrupolar (see eq A3a) and dipolar heteronuclear (see eq A4c) couplings, respectively. Since the other contributions imply high angular velocities (ω0,2ω0,ω0 + ωS or ω0 − ωS), we may reasonable neglect102 the variation of their spectral densities induced by the weak angular velocity, noted ω1, corresponding to the applied irradiation. These time evolutions are given by *

m

f (σ *) = − ∑ J0X (λpX ) ∑ (ap⃗ X )k [TX 0 , [Tk+ , σ ]]

A.4. Spin-Locking Relaxation Measurements

T XSL0(t ) = e−iHS t TX 0(0)eiHS τ =

(A6)

(A5)

with X = 1 or 2. J


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