H NMR Multiquanta Relaxometry - American Chemical Society

Article pubs.acs.org/JPCC

Long-Time Dynamics of Confined Water Molecules Probed by 2H NMR Multiquanta Relaxometry: An Application to Dense Clay Sediments Patrice Porion,*,† Laurent J. Michot,‡ Fabienne Warmont,† 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 ‡ Laboratoire Environnement et Minéralurgie, Nancy - Université CNRS, UMR7569, 15 avenue du Charmois, BP40, 54501 Vandoeuvre Cedex, France S Supporting Information *

ABSTRACT: The structural and dynamical properties of water molecules confined within dense clay sediment are investigated by 2 H NMR spectroscopy and multiquanta relaxometry. The relative contribution of both quadrupolar and paramagnetic NMR relaxation mechanisms is evaluated by carefully analyzing the variation of 2H multiquanta NMR relaxation rates as a function of the orientation of the clay sediment within the static magnetic field. The same analysis is successfully applied to 2H multiquanta NMR spin-locking relaxation measurements, significantly increasing the probed dynamical range. That procedure leads to an accurate determination of the average residence time of the water molecule confined within the interlamellar space of the clay lamellae. implying infrared (IR) spectroscopy,23−26 quasi-elastic neutron spectroscopy (QENS),20,22,27−32 neutron spin−echo spectroscopy (NSE),33−36 inelastic neutron spectroscopy (INS),37 nuclear magnetic resonance (NMR) relaxometry,38−46 and pulsed-gradient spin−echo (PGSE)47−52 NMR spectroscopy. While the time scales investigated by IR spectroscopy ( JQm(ω), m ∈ {1, 2} and JD0 (0) > JDm(ω), m ∈ {1, 2}). Unfortunately, the experimental data (see Figure 10a) do not allow extracting precise values for all spectral densities mentioned in eqs A7 and A10. However, focusing on the dominant m = 0 components, the set of eqs A7 and A10 may be simplified:65 U R10 = 5UQ + D 3

J0X (β LF , ω) =

1 (1 − 3cos2 β LF)2 J0X,intrinsic (ω) 4

+ 3cos 2 β LF sin 2 β LFJ1X,intrinsic (ω) 3 + (1 − cos2 β LF)2 J2X,intrinsic (ω) 4 J1X (β LF , ω) =

(5a)

3 2 LF 2 LF X,intrinsic cos β sin β J0 (ω) 2

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

J2X (β LF , ω) =

(5b)

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

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

(4a)

(5c)

The fits displayed in Figure 11a,b provide the intrinsic spectral densities quantifying the order of magnitude of the

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 J (0) + UD + J1D(ωS) 9 0 3 3

(4c)

(4d)

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

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

(4e)

where ω0 and ωS are the resonance angular velocities of deuterium and iron, respectively. If one further neglects the high frequency contribution from JD1 (ω S), it is possible to plot the selected spectral densities containing all of the dynamical features corresponding to these NMR relaxation measurements (Figure 10b). Because of its low iron content (see section II.1), the NMR relaxation of the water molecules confined in hectorite is mainly driven by its quadrupolar Hamiltonian in contrast with iron-rich montmorillonite, for which heterogeneous dipolar (paramagnetic) coupling was evidenced as the dominant relaxation mechanism.55 As displayed in Figure 10b, JD0 (0) is nearly constant in the whole angular domain, while JQ0 (0) increases by 1 order of magnitude at β LF ≈ 50°. Using the Wigner rotation matrices,69 the angular variations of the apparent spectral densities JXm(ω) (with X ∈ {Q, D} and m ∈ {0,1,2}) measured in the laboratory frame, can be transformed into intrinsic the intrinsic spectral density JX, (ω) evaluated in the m frame attached to the film for each relaxation mechanism (i.e., X ∈ {Q, D}):

Figure 11. Extraction of the intrinsic spectral densities JX,intrinsic (0), 0 (0), and JX,intrinsic (0) with X ∈ {Q, D}, from the angular JX,intrinsic 1 2 variation of the apparent spectral densities JX0 (0) quantifying (a) the quadrupolar (X = Q) and (b) the heterogeneous dipolar (X = D) relaxation mechanisms. intrinsic intrinsic (0) = (2600 ± 300) s−1, JQ, (0) = quadrupolar (JQ, 0 1 −1 Q, intrinsic (20 000 ± 3000) s , and J2 (0) = (450 ± 50) s−1) and

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intrinsic the heterogeneous dipolar (JD, (0) = (1100 ± 100) s−1, 0 Q, intrinsic −1 intrinsic J1 (0) = (1100 ± 100) s , and JQ, (0) = (600 ± 2 −1 60) s ) relaxation mechanisms. In the case of the quadrupolar relaxation mechanism, the intrinsic spectral intrinsic (0)) is much higher than the two other density (JQ, 1 intrinsic intrinsic components (JQ, (0) and JQ, (0)). Such a feature 0 2 was previously evidenced by measurements of 7Li and 23Na NMR relaxation71,72 in dense clay sediments and predicted by multiscale numerical modeling.72 This appears logical as, despite their a priori differences, both types of experiments display fundamental similarities. Indeed, both nuclei (2H versus 7Li and 23Na) exhibit a residual quadrupolar coupling when the corresponding probes (heavy water versus lithium or sodium counterions) are confined in the interlamellar space between two clay platelets. As shown previously,43,55−57,73,74 the only mechanism able to modulate this residual coupling is the exchange of the diffusing probes between two interlamellar domains characterized by a slight difference of their directors. This leads to long-time decorrelation of the quadrupolar Hamiltonian that significantly contributes to the various intrinsic components of the spectral densities evaluated at zero frequency.55,57 As shown by numerical simulations,72 the three intrinsic spectral densities (JQ, (0) with m ∈ {0,1,2}) are equivalent m in homogeneous clay dispersions. By contrast, when clay ordering increases, a strong enhancement72 of the differences between the m = 1 and the two other spectral densities occurs, in full agreement with the data displayed above. The above-discussed intrinsic spectral densities are obtained from an empirical analysis of the transverse relaxation R11 (see eq 1) performed independently for each angle βLF between the normal nF,L ⃗ to the self-supported film and the static magnetic field B0. However, because of the large difference between the intrinsic spectral densities intrinsic characterizing the quadrupolar coupling (JQ, (0) ≫ 1 Q, intrinsic Q, intrinsic J0 (0) ≈ J2 (0)) and the above-discussed heterogeneities of the clay directors within the film, it appears necessary to fit simultaneously the transverse relaxation rates (R11) measured for the complete set of angles (βLF) by using a modified version of eqs 5a−5c. For that purpose, let us introduce a new set of Euler angles (see Figure 12) defining the orientation of one clay platelet

XFC resulting from the intercept between the plane of the film and that of the clay platelet (see Figure 12). In the frame of the film, the clay director, noted nC,F ⃗ , is defined by its coordinates:

n ⃗C,F

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

(6a)

In the frame attached to the tube containing the clay sample, we obtain:

n ⃗C,L

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

(6b)

after the above-mentioned rotation δ of the film along the axis YL. Finally, after the rotation βLF of the NMR tube within the detection coil, we obtain the angle between the static magnetic field and the clay director, noted θLC, defined by: LF

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)

The simulation of the time evolution of the coherences is performed by sampling the angle αFC to satisfy a Gaussian distribution law, while the angle γFC is generated randomly in the interval [0,2π]. As a consequence, the set of eqs 5a−5c is still valid by simply replacing cos2 βLF and cos4 βLF formulas by their average values evaluated with the new Euler angle θLC. ⟨cos2 θ LC⟩ =

⟨cos 4 θ LC⟩ =

2 2 LF 2 FC sin β sin α + cos2 β LF cos2 δ LF cos2 α FC 3 1 + cos2 β LF sin 2 δ LF sin 2 α FC (7a) 3

8 sin 4 β LF sin 4 α FC 15 + cos 4 β LF cos 4 δ LF cos 4 α FC 1 + cos 4 β LF sin 4 δ LF sin 4 α FC 5 4 2 LF 2 LF 2 LF 4 FC + sin β cos β sin δ sin α 5 + 4sin 2 β LF cos2 β LF cos2 δ LF sin 2 α FC cos2 α FC + 2cos 4 β LF sin 2 δ LF cos 2 δ LF sin 2 α FC cos2 α FC (7b)

The new modeling of the transverse relaxation measurements R11 exploits a numerical solution of the set of eqs A3, A7, and A10 to simulate the time evolution of the various coherences during the experimental pulse sequence (see Figure 4b). Typical results are displayed in Figure 13. The set of intrinsic spectral densities is fully compatible with the previous data (see above). In addition to these results, such a simultaneous analysis of relaxation rates allows one to determine the standard deviation of the Gaussian distribution of clay directors αFC within the film (σ = (2.2 ± 0.5)°) and the angle quantifying the misalignment of the film by reference with the axis of the coil (see Figure 9: δLF = (6 ± 1)°). As was previously deduced from

Figure 12. Schematic view of the Euler angles characterizing the orientation of an individual clay platelet within the macroscopic film (see text).

within the film. The angle between the clay director (ZC) and the film director (ZF) is noted αFC, while γFC is the angle between the axis XF attached to the film and the axis 17688



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the perfect canceling of the residual quadrupolar coupling at the magic angle, the amplitude of both defects remains limited, thus validating the previous analysis of the relaxation rates. 3. 2H NMR Multiquanta Spin-Locking Measurements. Spin-locking relaxation measurements can be performed to extract the long-time dynamical behavior of the confined water molecules. As shown by eq A4c, the time variation of the locked TIR 21(s) coherence, that is, T21ρ(s), can be easily measured because it contains a single oscillating component (see Figures 14a and 15a) corresponding to the k1 angular velocity, by contrast

Figure 13. Fits of the Hahn spin echo attenuation obtained from a complete simulation of the time evolution of the TIR 11 coherences during the NMR pulse sequence for few orientation βLF of the clay film into the static magnetic field B0.

IR Figure 15. Time evolution of the (a) TIR 21(a), (b) T21(s), and (a) coherences measured under spin-lock conditions, noted (c) TIR 22 T21ρ(a), T21ρ(s), and T22ρ(a), respectively. For these examples, βLF = 0° and ω1 = 1.38 × 105 rad·s−1.

with the more classical TIR 11(a) coherence (see eq A4b), which also contains a constant component. The range of probed angular velocities is further extended by additionally measuring the time evolution of the locked TIR 21(a) (see eq A4g) and TIR 22(a) (see eq A4h) coherences, that is, T21ρ(a) and T22ρ(a), that oscillate according to the k2 and k3 angular velocities (see Figures 14b,c and 15b,c). This set of three independent spinlocking measurements is then sufficient for extracting the desired spectral densities, that is, JQ0 (k1) for the quadrupolar coupling (see eq A13) in addition to JD0 (k2) and JD0 (k3) for the heterogeneous dipolar coupling (eq A15). As displayed in Figures 14b,c and 15b,c, the time evolution of the TIR 21(a) and TIR 22(a) coherences probes the same angular velocities (k2 and k3) but with different initial condition a priori allowing one to

Figure 14. Fourier transforms of the time evolution of the (a) TIR 21(a), IR (b) TIR 21(s), and (c) T22(a) coherences measured under spin-lock conditions, noted T21ρ(a), T21ρ(s), and T21ρ(a), respectively. For these examples, βLF = 0° and ω1 = 1.38 × 105 rad·s−1. 17689



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distinguish both values of the spectral densities JD0 (k2) and JD0 (k3). As above, the set of eqs A3, A7, A10, A13, and A15 can be used to simulate the time evolution of the various coherences during the pulse sequences used to perform these multiquanta spin-locking relaxation measurements. Here also, because of the intrinsic (ω) coherence, we simultaneously magnitude of the JQ, 1 analyze the three spin-locking measurements, each performed at three different angles βLF equal to 0°, 30°, and 90°, respectively. We avoid working at values close to the magic angle, becauase in that region, the pulse sequence becomes inefficient. Table 1 exhibits the set of angular velocities (k1, k2, Table 1. Set of Angular Velocities k1, k2, and k3 (See Eqs A5a−A5c) Detected from the Fourier Transform of the Time Evolution of the Coherences as a Function of the Irradiation Power ω1 and the Euler Angle βLF 5

Figure 16. Distribution law quantifying the heterogeneity (ω1 − ⟨ω1⟩)/⟨ω1⟩ within the clay sample of the irradiation field B1 generated by the solenoid used for these spin-locking relaxation measurements at the angular velocity ω1.

−1

k1 (10 rad·s ) ω1 (105 rad·s−1)

βLF = 0°

βLF = 30°

βLF = 90°

1.44 0.82 0.41 0.23 0.089

2.93 1.75 1.01 0.748 0.616

2.65 1.55 0.898 0.515 0.383 k2 (105 rad·s−1)

2.72 1.57 0.905 0.528 0.358

ω1 (105 rad·s−1)

βLF = 0°

βLF = 30°

βLF = 90°

1.44 0.82 0.41 0.23 0.089

1.75 1.15 0.823 0.647 0.591

1.50 0.936 0.635 0.478 0.408 k3 (105 rad·s−1)

1.48 0.924 0.591 0.408 0.327

ω1 (105 rad·s−1)

βLF = 0°

βLF = 30°

βLF = 90°

1.44 0.82 0.41 0.23 0.089

1.17 0.584 0.270 0.101 0.0308

1.12 0.584 0.289 0.119 0.0402

1.26 0.641 0.333 0.138 0.0390

and k3, see eqs A5a−A5c) detected from the Fourier transform of the time evolution of the coherences as a function of the irradiation power (ω1) and the Euler angle βLF. By summing their specific dynamical ranges, the whole set of spin-locking relaxation measurements covers nearly two decades, while the irradiation power varies only within a single decade (see Table 1). To simplify the numerical analysis of the experimental data, we take into account a single angular velocity ki (i ∈ {1,2,3 }) for each irradiation power ω1, independent of the Euler angle βLF. As displayed in Table 2, this approximation is valid at high irradiation power, but becomes less valid at weak irradiation Table 2. Set of Single Mean Angular Velocity ⟨ki⟩ (i ∈ {1,2,3}) for Each Irradiation Power ω1, Independent of the Euler Angle βLF ω1 (105 rad·s−1) 1.44 0.82 0.41 0.23 0.089

⟨k1⟩ (105 rad·s−1) 2.76 1.62 0.96 0.60 0.45

± ± ± ± ±

0.12 0.09 0.08 0.11 0.11

⟨k2⟩ (105 rad·s−1) 1.58 1.01 0.68 0.51 0.44

± ± ± ± ±

0.12 0.10 0.10 0.10 0.11

Figure 17. Variation of the intrinsic (a) J 0Q,intrinsic (k 1 ) and (k2,k3) spectral densities as a function of the corresponding (b) JD,intrinsic 0 angular velocities ki.

⟨k3⟩ (105 rad·s−1) 1.18 0.60 0.30 0.12 0.036

± ± ± ± ±

0.06 0.03 0.03 0.02 0.004

power because it leads to relative errors of the order of 25%. Note however that the relative error induced by such approximation does not exceed 10% of the value of k3 at the weakest irradiation power. Finally, we take into account the 17690



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heterogeneity of the irradiation field probed by the water molecules located in the film.58 For that purpose, we numerically evaluate the local magnetic field inside the whole volume of the film generated by a solenoid similar to that used in our experimental setup (length 35 mm, diameter 11 mm, number of turns of the wire 6). As displayed in Figure 16, the broad range of the ω1 values felt by the water molecules within the film is roughly reproduced by a Lorentzian distribution law. As displayed in Figures 14 and 15, this numerical analysis IR reproduces fairly well the time evolution of the TIR 21(s), T21(a), IR and T22(a) coherences during these spin-locking relaxation measurements (i.e., T21ρ(s), T21ρ(a), and T22ρ(a)). Unfortunately, because of the above-mentioned approximation concerning the angular variation of the characteristic angular velocities ki, it is not possible to extract analytical laws describing the low frequency variation of the intrinsic spectral intrinsic (ω), m ∈ densities quantifying the quadrupolar (i.e., JQ, m intrinsic (ω), m ∈ {0,1,2}) and heterogeneous dipolar (i.e,. JD, m {0,1,2}) relaxation mechanisms. Nevertheless, the intrinsic spectral densities displayed in Figure 17a,b clearly exhibit a transition between a plateau, at low angular velocities, and a marked decrease at high angular velocities. The complementarities between the intrinsic quadrupolar and dipolar spectral densities allow one to define the critical angular velocity (ωC = (4 ± 1) × 104 rad·s−1) corresponding to the transition between these two dynamical regimes. Let us recall that the long-time memory of both the quadrupolar and the heteregeneous dipolar couplings is modulated by the residence55,57,74 of the diffusing water molecules inside the interlamellar space limited by the clay platelets. As a consequence, the critical angular velocity extracted from Figure 17a,b may be used to determine this residence time τC = (25 ± 6) μs. Numerical simulations of Brownian dynamics75 were then performed to compare such a time scale with the local mobility measured by QENS.22,28−31,34 Taking into account the water self-diffusion coefficient measured by QENS22 on equivalent clay samples (D = 7 × 10−10 m2·s−1), such a residence time leads to an apparent platelets size of (500 × 800 nm2) (see Figure 18) that would correspond on average to the juxtaposition of 7 ± 2 individual

detected by 2H NMR multiquanta spin-locking relaxation measurements is fully compatible with the local self-diffusion coefficient measured by QENS.

IV. CONCLUSIONS 2 H NMR multiquanta spin-locking relaxation measurement is a powerful tool for extracting the long-time dynamical behavior of the water molecules confined within the interlamellar space of dense clay sediments. Because of the low iron content of the natural hectorite clay used in the present study, both the quadrupolar and the heterogeneous dipolar (paramagnetic) couplings significantly contribute to the relaxation of confined water molecules. The relative contribution of both relaxation mechanisms can then be carefully evaluated by a simultaneous analysis of the variation of the 2H multiquanta relaxation rates as a function of the orientation of the clay film into the static magnetic field. The same separation of both quadrupolar and paramagnetic contributions to the 2H multiquanta spin-locking relaxation measurements doubles the range of frequencies probed by these NMR measurements. The complementarities between both set of intrinsic spectral densities characterizing the quadrupolar and paramagnetic relaxation mechanisms allow an accurate determination of the average residence time of the water molecules confined within the interlamellar space of the hectorite dense sediment. In contrast with the local water mobility determined by QENS experiments, such measurements provide information about water mobility during its complete traveling through the interlamellar space between clay platelets. The experimental procedure presented in our study could certainly be extended to other solid/liquid interfacial systems, including a large class of metallic oxides (like cement particles), liquid crystals, or biological membranes, where information about dynamical properties of confined water is of prime importance.

■

APPENDIX

1. Time Evolution of the Coherences

In the framework of the Redfield theory,76 the time evolution of the coherence is described by the complete master equation:41,70,77 dσ* = −i[H *S , σ *] + f (σ *) dt

(A1)

where all calculations are performed in the Larmor frequency rotating frame, as indicated by the asterisk. The first contribution to the master equation describes the influence of the static Hamiltonian H*S that includes the residual static quadrupolar Hamiltonian (H*QS = (2/3)1/2ωQTIR 20) and the Hamiltonian corresponding to the radio frequency pulse (H*1S = Ixω1 = 2 ω1TIR 11(a)). The Zeeman-like Hamiltonian IR (H*ZS = 2 ωresT10) results from the frequency offset (ωres). The second contribution to the master equation describes the contribution from the fluctuating components of the Hamiltonians41,70 (H*F(t)) monitoring the relaxation of the coherences:

Figure 18. Average residence time of the confined water molecules obtained from Brownian dynamics numerical simulations.

f (σ *) = −

∫0

∞

⟨[H *F (t ), [e−iH *S τH *+F (t − τ )eiH *S τ , σ *(t )]]⟩dτ

(A2)

hectorite laths within the clay sediment. Such a value concurs with the pattern observed by TEM (see Figure 1), where parallel alignments of a few individual laths are mainly identified. As a consequence, the long-time water mobility

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:63 17691



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⎛ T IR ⎞ ⎛ ⎜ 20 ⎟ ⎜ ⎜T IR (a)⎟ ⎜ 11 ⎜ ⎟ ⎜ ⎜ T IR (s) ⎟ ⎜− 21 ⎜ ⎟ ⎜ ⎜ T IR (s) ⎟ ⎜ d ⎜ 22 ⎟ = i⎜ dt ⎜ T IR ⎟ ⎜ ⎜ 10 ⎟ ⎜ ⎜ T IR (s) ⎟ ⎜ ⎜ 11 ⎟ ⎜ ⎜T IR (a)⎟ ⎜ 21 ⎜ ⎟ ⎜ ⎜ IR ⎟ ⎝ ( ) T a ⎝ 22 ⎠

0

0

− 3 ω1

0

0

0

ωQ

0

3 ω1 ωQ

0

−ω1

0

0

−ω1

0

0

0

0

0

0

ωres

0

0

0

0

ωres

0

0

0

0

2ωres

By carefully operating at the resonance frequency (ωres = 0), the time evolution of the coherences splits in two independent subsets:63 e

iH *S τ

IR −iH *S τ T20 e

=

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

0

= −iω1

ωQ2 + ω12(1 + 3cos(k1τ ))

+

k12

sin(k1τ ) IR T21 (s) + k1

3 ω12

ωQ2 cos(k1τ ) + 4ω12

IR (a)e−iH *S τ = e iH *S τT11

k12

IR T20 − i 3 ω1

cos(k1τ ) − 1 IR T22 (s) k12 (A4a)

+ iωQ

+ ω1

= ω1

sin(k1τ ) IR T21 (s) k1

(A4b)

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

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

+

IR −iH *S τ = eiH *S τT10 e

= −i

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

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

(A4d)

cos(k 3τ ) − cos(k 2τ ) IR T11 (s) k1

− iω1 +

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

−i

(A4g)

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

+ ω1

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

+ ω1

sin(k 3τ ) + sin(k 2τ ) IR T22 (a) k1

IR eiH *S τT22 (a)e−iH *S τ

1 − cos(k1τ ) IR T11 (a) k12

3 ω12

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

− iω1

(A4c)

+

(A4f)

cos(k 3τ ) − cos(k 2τ ) IR T10 k1

+i

sin(k1τ ) IR IR + cos(k1τ )T21 (s) − iω1 T22 (s) k1

IR eiH *S τT22 (s)e−iH *S τ = ω1ωQ

cos(k 3τ ) − cos(k 2τ ) IR T22 (a) k1

IR eiH *S τT21 (a)e−iH *S τ

+ IR e iH *S τT21 (s)e−iH *S τ = iωQ

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

IR T11 (a)

1 − cos(k1τ ) IR + ω1ωQ T22 (s) k12

sin(k 3τ ) + sin(k 2τ ) IR T10 k1

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

+i

1 − cos(k1τ ) IR 3 ω1ωQ T20 k12

+

(A3)

IR eiH *S τT11 (s)e−iH *S τ

1 − cos(k1τ ) IR 3 ω1ωQ T11 (a) k12 +

0

sin(k 3τ ) + sin(k 2τ ) IR T21 (a) k1

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

(A4h)

where the characteristic angular velocities63 k1, k2, and k3 are defined by:

cos(k 3τ ) − cos(k 2τ ) IR T21 (a) k1

k1 =

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

k2 =

(A4e) 17692

ωQ2 + 4ω12 ωQ +

(A5a)

ωQ2 + 4ω12 2

(A5b)


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ωQ2 + 4ω12 − ωQ

Q T2,0 =

(A5c)

2

1 IR (3Iz2 − I(I + 1)) = T20 6

(A6c)

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

2. Quadrupolar Relaxation

In bulk, the quadrupolar coupling is the main mechanism responsible for relaxation of heavy water:41,63,70

(A6d)

and 2

HQ (t ) = CQ

∑

( −1)m T2,QmF2,Q−m(t )

m =−2

T2,Q±2 = (A6a)

e 2qQ 3 8 ℏI(2I − 1)

(A6e)

FQ2,m(t) 41,77

The functions in eq A6a are the second-order spherical harmonics describing the reorientation of the OD director in the static magnetic field B0 by using the two Euler angles (θLW, ϕLW) because the electrostatic field gradient felt by the deuterium nucleus in heavy water is directed along the OD bond. The contribution from the quadrupolar relaxation mechanism to the complete master equation (eqs A1−A2) may also be written in a matrix form:63

where the quadrupolar coupling constant, defined by: CQ =

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

(A6b)

is equal to (3/2) × π × 210 kHz for deuterium in bulk heavy water.78 The spin operators describing the quadrupolar coupling are given by: 1/2

⎛ 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(AQ , BQ , C Q , DQ , E Q , BQ , C Q , DQ )·⎜ ⎜ T IR ⎟ dt ⎜ T IR ⎟ ⎜ 10 ⎟ ⎜ 10 ⎟ ⎜ T IR (s) ⎟ ⎜ T IR (s) ⎟ ⎜ 11 ⎟ ⎜ 11 ⎟ ⎜T IR (a)⎟ ⎜T IR (a)⎟ ⎜ 21 ⎟ ⎜ 21 ⎟ ⎜ IR ⎟ ⎜ IR ⎟ ⎝T22 (a)⎠ ⎝T22 (a)⎠

with Q

A =

addition to the intrinsic quadrupolar relaxation mechanism. The corresponding Hamiltonian is defined by:

3J1Q (ω0)

BQ = 3/2J0Q (0) + 5/2J1Q (ω0) + J2Q (2ω0)

2

HD(t ) = C D

C Q = 3/2J0Q (0) + 1/2J1Q (ω0) + J2Q (2ω0)

and the spin operators become:

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

T2,DmF2,D−m(t ) rIS3(t )

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

E Q = J1Q (ω0) + 4J2Q (2ω0)

∫0

∑

( −1)m

m =−2

DQ = J1Q (ω0) + 2J2Q (2ω0)

JmQ (mω0) = ( −1)m CQ2

(A7)

(A9b)

55

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

D T20 =

(A9c)

(F2,Q−m(0) − ⟨F2,Q−m⟩)

·(F2,Qm(τ ) − ⟨F2,Qm⟩)eimω0τ dτ

(A9a)

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

(A8)

3. Paramagnetic Relaxation

Because of the presence of iron within the natural hectorite clay particle, the heterogeneous dipolar coupling may be responsible55 for the NMR relaxation of the confined heavy water, in

T2D± 2 = 17693

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

(A9d)

(A9e)



Article

The functions FD2,m(t) in eq A9a are the same as in eq A6a, but they describe now the reorientation of the vector joining the two coupled 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 the diffusion of

the probe through the variation of the separation between the coupled spins (cf., the term r−3 IS (t) in eq A9a). The contribution from the heterogeneous dipolar coupling to the complete master equation (eqs A1−A2) may also be written in a matrix form:55

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

(A10)

The corresponding spectral densities are:

with 1 A = J0D(ω0 − ωS) + J1D(ω0) + 2J2D(ω0 + ωS) 3 D

BD =

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

CD =

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 3 3 9 0 2 D + J2 (ω0 + ωS) 3

∞

JmD(ω) = (− 1)m NSS(S + 1)C D2 ×

∫0

⎛ F D (τ ) 2, m × ⎜⎜ 3 − ⎝ rIS(τ )

⎞ ⎟dτ ⎟ ⎠

F2,Dm rIS3

⎛ F D (0) 2, −m − eimω0τ ⎜⎜ 3 ⎝ rIS(0)

⎞ ⎟ ⎟ ⎠

F2,D−m rIS3

(A11)

where NS is the total number of paramagnetic spins coupled to the deuterium nucleus. 4. Contributions to the Spin-Locking Relaxation Rate

The relaxation of the coherences during the spin-locking experiments is described by the complete master eq A1. We focus only on the m = 0 contribution of eq A2, because the m = ±1 and m = ±2 components are only slightly modified63 by the weak irradiation field. For the quadrupolar coupling, the corresponding m = 0 component is given by: f mQ= 0 (σ *) = − CQ2

∫0

∞

Q Q Q Q (F20 (t ) − ⟨F20 (t )⟩) × (F20 (t − τ ) − ⟨F20 (t )⟩)

IR IR + iH *S τ × [T20 , [e−iH *S τT20 e , σ *]]dτ

1 1 2 E = J0D(ω0 − ωS) + J1D(ω0) + J2D(ω0 + ωS) 9 3 3 D

(A12)

leading to the contribution from the quadrupolar relaxation mechanism to the master equation:63

⎛ T IR ⎞ ⎛ T IR ⎞ 20 ⎟ ⎛ AQ − 3 K Q 0 ⎞ ⎜ 20 ⎟ 0 0 0 0 0 ⎜ ⎜ ⎟ ⎜T IR (a)⎟ ⎜T IR (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 0 DQ 0 0 0 0 ⎟ ⎜ T22 (s) ⎟ −K Q = −⎜⎜ ⎟ ·⎜ IR ⎟ Q Q dt ⎜ T IR ⎟ 0 0 0 0 0 0 E K − 10 ⎜ ⎟ ⎜ T10 ⎟ ⎜ ⎟ Q ⎜ 0 ⎜ T IR (s) ⎟ ⎟ IR 0 0 0 0 0 0 ⎟ ⎜ T11 L ⎜ ⎟ ⎜ (s ) ⎟ ⎜ 11 ⎟ ⎜ 0 ⎜T IR (a)⎟ 0 0 0 0 0 0 ⎟ ⎜T IR (a)⎟ MQ 21 ⎜ ⎟ ⎜ 21 ⎟ ⎜ ⎟ ⎝ 0 0 0 0 0 −K Q 0 DQ ⎠ ⎜ IR ⎟ ⎜ IR ⎟ ⎝T22 (a)⎠ ⎝T22 (a)⎠

17694

(A13)



Article

with Q

A = Q

B

=

3ωQ2 J0Q (0) + 4ω12J0Q (k1) 2k12 +

CQ =

MQ =

3J1Q (ω0) +

KQ =

2k12

2k12 + 1/2J1Q (ω0) + J2Q (2ω0)

5/2J1Q (ω0)

J2Q (2ω0)

3ωQ2 J0Q (0) + 4ω12J0Q (k1)

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

3ω1ωQ (J0Q (0) − J0Q (k1)) 2k12

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

+ 1/2J1Q (ω0)

+ J2Q (2ω0)

4 fmD= 0 (σ *) = − C D2NSS(S + 1) 9

DQ = J1Q (ω0) + 2J2Q (2ω0)

∫0

⎛ F D (t − τ ) × ⎜⎜ 20 − 3 ⎝ rIS(t − τ )

E Q = J1Q (ω0) + 4J2Q (2ω0) 2 Q 2 Q Q 3 ωQ J0 (0) + 2ω1 (J0 (0) + J0 (k1)) L = 2 k12

∞ ⎛ F D (t ) ⎜⎜ 20 3 ⎝ rIS(t )

−

D ⎞ F20 ⎟⎟ 3 rIS ⎠

D ⎞ F20 ⎟⎟ 3 rIS ⎠

IR IR + iH *S τ × [T10 , [e−iH *S τT10 e , σ *]]dτ

Q

leading to the contribution from the heterogeneous dipolar relaxation mechanism to the master equation:57

+ 5/2J1Q (ω0) + J2Q (2ω0)

⎛ 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 ⎟ ⎟ 0 2K D 0 0 0 0 ⎟⎜ BD ⎜ ⎟ ⎜ T IR (s) ⎟ ⎜ T IR (s) ⎟ ⎜0 0 0 0 0 0 ⎟ ⎜ 21 ⎟ CD 0 ⎜ 21 ⎟ ⎜ ⎟ ⎜ IR ⎟ ⎜ IR ⎟ d ⎜ T22 (s) ⎟ 0 0 DD 0 0 0 0 ⎟ ⎜ T22 (s) ⎟ KD ⎜ =− · ⎜ ⎟ ⎜ IR ⎟ D D dt ⎜ T IR ⎟ − 0 0 0 0 0 0 E K 10 ⎜ ⎟ ⎜ T10 ⎟ ⎜ ⎟ D D ⎟ ⎜ IR ⎜0 ⎜ T IR (s) ⎟ ⎟ 0 0 0 0 L 0 2K ⎜ ⎟ ⎜ T11 (s) ⎟ ⎜ 11 ⎟ ⎜0 ⎜T IR (a)⎟ 0 0 0 0 0 MD 0 ⎟ ⎜T IR (a)⎟ ⎜ ⎟ ⎜ 21 ⎟ ⎜ 21 ⎟ D ⎝0 0 0 0 0 K 0 DD ⎠ ⎜ IR ⎟ ⎜ IR ⎟ ⎝T22 (a)⎠ ⎝T22 (a)⎠

with

LD =

1 A = J0D(ω0 − ωS) + J1D(ω0) + 2J2D(ω0 + ωS) 3 D

D

B =

CD =

DD =

ED =

2(k 2J0D(k 3) + k 3J0D(k 2))

1 D J (ω0 − ωS) 18 0 1 1 1 + J1D(ω0) + J1D(ωS) + J2Q (ω0 + ωS) 6 3 3 9k1

2(k 2J0D(k 3) + k 3J0D(k 2))

2(k 2J0D(k 3) + k 3J0D(k 2))

(A15)

1 D J (ω0 − ωS) 9k1 18 0 1 1 1 + J1D(ω0) + J1D(ωS) + J2Q (ω0 + ωS) 6 3 3 +

+

MD =

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

(A14)

+

KD =

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

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

■

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

Pulse sequences NMR and detailed phase cycling procedures. This material is available free of charge via the Internet at http://pubs.acs.org.

ASSOCIATED CONTENT

S Supporting Information *

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Article

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (A.D.); porion@cnrs-orleans. fr (P.P.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS

■

REFERENCES

The DSX360 Bruker spectrometers used for the NMR study were purchased thanks to grants from Région Centre (France). P.P. wants to thank Drs. J.-F. Fontaine, E. Denamur, I. Dufetel, and A. Villeneuve.

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H NMR Multiquanta Relaxometry - American Chemical Society

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