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Article

Long-Distance Water Exchange within Dense Clay Sediments Probed by Two-Time H Stimulated Echo NMR Spectroscopy 2

Patrice Porion, Anne Marie Faugère, and Alfred Delville J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp401698b • Publication Date (Web): 10 Apr 2013 Downloaded from http://pubs.acs.org on April 16, 2013

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Long-distance water exchange within dense clay sediments probed by two-time 2H stimulated echo NMR spectroscopy

Patrice Porion*, Anne Marie Faugère, and Alfred Delville*

Centre de Recherche sur la Matière Divisée, CNRS - Université d’Orléans, FRE3520, 1b rue de la Férollerie, 45071 Orléans Cedex 02, France

*Corresponding authors: E-mail: [email protected] (A.D.) and [email protected] (P.P.).

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Abstract Two-time correlation functions are measured by deuterium (2H) NMR spectroscopy to probe the long-distance mobility of water molecules strongly confined within dense clay sediments. As checked by 2H NMR multi-quanta relaxation measurements, these strongly confined water molecules satisfy the two main requirements for a successful use of that experimental procedure, i.e. slow exchange of the water molecule between its various confining environments and slow modulation of the nuclear couplings responsible for its NMR relaxation. The exchange time-scale detected by the attenuation of the two-time 2H NMR stimulated echo is three orders of magnitude larger than the average residence time of the water molecules within the interlamellar space of individual clay platelets. Finally, numerical simulations are used to illustrate the impact of local heterogeneities of the clay orientation on the attenuation of the two-time correlation function.


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I.

Introduction Numerous theoretical1-3 and experimental4-13 studies have been recently devoted to

solid/liquids interfaces in order to predict and quantify the influence of confinement on the structural14,15, thermodynamical15-19 and dynamical20-23 properties of fluids. In that context, various

experimental

studies

(X-ray

and

neutron

diffraction14,15,

infrared

(IR)

spectroscopy14,24-26, quasi-elastic neutron spectroscopy21,23,27-32 (QENS), neutron spin-echo spectroscopy33-36 (NSE), inelastic neutron spectroscopy37 (INES), nuclear magnetic resonance (NMR) relaxometry38-46 and pulsed-gradient-echo47-52 (PGSE) NMR spectroscopy) have been used to correlate the structural and dynamical properties of the solid/liquid interfacial systems in relation with results obtained by numerical simulations.

In that framework, charged interfacial systems were frequently investigated since the properties of confined polar fluids are strongly modified by the long-range electrostatic coupling23,53 with the limiting solid surfaces and their neutralizing counterions. Among others, clay lamellae are ideal systems for two main reasons. First, from a theoretical point of view, clay platelets are flat and atomically smooth surfaces with a well characterized structure, composition and electric charge, allowing realistic modeling of charged solid/liquid interfaces54-58. Secondly, natural and synthetic clays are metallic oxides with versatile physico-chemical properties (high specific surface and ionic exchange capacity, water adsorption, swelling, gelling, thixotropy, surface acidity) used in numerous applications (food and cosmetic industry, drilling, heterogeneous catalysis, water treatment, and waste storage). For various applications, including heterogeneous catalysis and waste management, it appears crucial to carefully monitor the retention capacity of the clay lamellae by quantifying the mobility of confined probes over a broad range of diffusing time. The short-time mobility of confined liquids can be investigated by various methods including IR spectroscopy14,24-26



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(time ≤ ps) and neutron scattering experiments21,23,27-37 (ps ≤ time ≤ 100 ns), whereas the long-time

mobility

of

fluids

is

generally

investigated

by

field-cycling

NMR

relaxometry38,39,41,42,44,45 (10 ns ≤ time ≤ 0.1 ms) or PGSE NMR spectroscopy47-52 (time > ms). Unfortunately, these two latter experimental procedures are generally useless for liquids confined within the interlamellar space of clay platelets because paramagnetic impurities drastically enhance their NMR relaxation rates59,60.

Figure 1

As displayed in Figure 1, dense clay sediments exhibit complex and multi-scale structures: i) at short distances, the sediment is composed from hydrophilic and highly anisotropic (thickness ≈ 7 Å, diameter ≈ 700 Å) charged platelets neutralized by exchangeable counterions; ii) at intermediate distances, microscopic domains are formed by the stacking of numerous (10-100) parallel clay platelets60,61 (see Figure 1); iii) at largest distances, clay sediment results from the juxtaposition of micro-domains characterized by different orientations (see Figure 1). Due to this multi-scale organization, the mobility of the water molecules confined inside the sediment has been investigated by various complementary experiments21,23,27-32,40,43,49. Thanks to their short investigated time-scale (ps ≤ time ≤ 100 ns), neutron scattering experiments21,23,27-32 are perfectly appropriate to quantify the water mobility inside the interlamellar space between two parallel clay platelets. By contrast, the time-scale investigated by the frequency variation of the NMR relaxation rates24,59,60,62-66 is intrinsically limited by the transverse relaxation rate of the confined water molecules, i.e. typically 104 s-1. As a consequence, spin-locking relaxation measurements are suitable to determine the average residence time of the water molecules pertaining to the same stack of parallel platelets66. The purpose of that study is to investigate the long-distance mobility of the


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water molecule diffusing through the various micro-domains with different orientations by using two-point stimulated echo NMR spectroscopy67-72.

Due to the long time-scale (≈10-4 s) probed by NMR relaxation spectroscopy, the water molecules initially confined between clay platelets pertaining to the same oriented micro-domain explore various local environments, including bulk-like water molecules in outer- and interlayers, surface water in contact with silanol groups, water molecules physisorbed on platelet edges. As a consequence, NMR relaxometry is not able to identify the different mechanisms responsible for the exchanges between these various local environments of the confined water molecules. Furthermore, if the rate constant quantifying the exchange of the water molecules between two different micro-domains is larger than the transverse relaxation rate (104 s-1), a single spin population is detected and no additional dynamical information may be extracted. In our case, this exchange rate constant is slower than the NMR relaxation rate and various spin populations coexist, leading to a superposition of individual spectra64. We are then able to exploit the fluctuations of the orientations of the micro-domains to probe the water mobility on a large scale by using two-point stimulated echo NMR spectroscopy67-72. This method appears as a promising procedure, allowing a detailed investigation of the dynamical properties of diffusing probes over a broad time-scale (10 µs ≤ time ≤ 10 ms). Such an experimental procedure was already used67-72 to investigate the mobility of ions and molecules within solid or vitreous matrices. Its application to confined liquids is however restricted by some intrinsic requirements regarding the existence of slow exchange73,74, at the NMR time-scale, of the quadrupolar NMR probes between various environments characterized by different residual quadrupolar couplings. Furthermore, the NMR relaxation mechanisms must satisfy the so-called slow modulation condition75 for



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which the transverse NMR relaxation rate of the diffusing probe is larger than its longitudinal relaxation rate.

In the present study, we focus on a natural clay sample (hectorite) with low water content, i.e. in conditions similar to those used for waste storage. A partially oriented clay film is hydrated by heavy water under controlled partial pressure, leading to well defined hydrated structure15. Multi-quanta 2H NMR relaxation measurements66 are first performed in order to check the above mentioned requirements, validating the use of two-point stimulated echo NMR spectroscopy to extract dynamical information on the long-distance mobility of heavy water confined within clay lamellae. The two-time correlation function extracted from the measurements of the 2H NMR stimulated echo attenuation indicates a water mobility occurring at very long time ( τ exch

45 ms), much larger than the average residence time23,66

of the confined water molecules in the clay interlamellar space ( τ B

60 µs). Numerical

modeling is finally applied to qualitatively illustrate the impact of the fluctuations of platelets orientation within the macroscopic film on the attenuation of the 2H stimulated echo.

II. Materials and Methods 1. Sample Preparation Hectorite (from Hector, CA) purchased from Ward’s Natural Science is a natural clay with the general formula: Si8 Al0.22 Fe0.05 Mg4.93 Li0.8 (OH)3.6 F0.4 O20 Na+0.6. That swelling clay60 results from the sandwiching of one layer of octahedral magnesium oxide between two layers of tetrahedral silica. Solvated interlamellar sodium cations neutralize the negative charge of the clay network due to the substitution of some octahedral Mg(II) by Li(I). Prior to use, the natural clay sample was purified according to classical procedures76, and the cations were exchanged leading to monoionic clay samples. The clay particles are further selected


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according to their size by centrifugation76. As shown previously by TEM66, individual hectorite clay particles appear as laths, with average size (700 ± 300) nm and width (70 ± 20) nm. As detected by TEM66, 7 ± 2 individual hectorite laths juxtapose themselves leading to larger platelets (500×800) nm2 in which water confinement occurs. A selfsupporting film (0.5 mm thick) is obtained from dilute aqueous clay dispersion (12 g/L) by ultrafiltration under nitrogen (3-5 atm) using a membrane with an average pore size 0.1 µm (Osmonics, Inc.). The clay film was dried under nitrogen flux before being equilibrated with a reservoir of heavy water at a fixed water chemical potential ( p p 0 = 0.33) by using saturated salt solution (MgCl2). The water partial pressure ( p p 0 = 0.33) was selected because it corresponds mainly to an interlayer space14,15,23 with a period of roughly 12 Å. This interlayer space is large enough to accommodate one layer of confined water molecules14,15,23. A macroscopic lamella (30×5.5 mm2) is cut into the self-supporting clay film and inserted into a glass cylinder which fits the gap inside the solenoidal coil used for the NMR measurements.

2.

2

H NMR Measurements Figure 2

2

H NMR spectra of heavy water were recorded on a DSX360 Bruker spectrometer

operating at a field of 8.465 T. On this spectrometer, the pulse duration for the total inversion of the longitudinal magnetization is equal to 24 µs. Spectra were recorded using a fast acquisition mode with a time step of 0.25 µs, corresponding to a spectral width of 4 MHz. The spectra and relaxation rates were recorded for different orientations β LF of the film director n F , L with reference to the static magnetic field B 0 (see Figure 2), by using a home made

sample holder and detection coil66. Figure 2 displays some 2H NMR spectra as a function of



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the orientation β LF of the film director within the static magnetic field B 0 . As shown by Monte Carlo simulations, the specific orientation of confined water molecules14,15,66 is responsible for the splitting of the deuterium resonance lines displayed in Figure 2. Furthermore, the reduced asymmetry of the water resonance line is induced by the second order relaxation mechanism resulting from the intercorrelation between the quadrupolar and heterogeneous dipolar couplings felt by the confined water molecules77,78.

Figures 3-4

A complete basis set, with eight independent operators, is required to describe the time evolution of spin I = 1 nuclei during any pulse sequence. A possible basis set is defined by the irreducible tensor operator79,80: T10IR , T11IR (a, s ) , T20IR , T21IR (a, s ) , T22IR (a, s ) . The first three operators correspond to the longitudinal ( T10IR ) and transverse ( T11IR (a, s ) ) components of the spin magnetization. The five residual operators describe the five components of the quadrupolar Hamiltonian (see Eq. A6). The relaxation rate constant of the T10IR coherence, also called longitudinal relaxation rate ( R1 ), is noted here R10 . It is measured by the classical inversion–recovery pulse sequence81 (see Figure 3a). The relaxation rate constant of the

T11IR (a, s ) coherences, also called transverse relaxation rate ( R2 ), is noted here R11 . It is measured by the Hahn echo pulse sequence82 (see Figure 3b) as illustrated in Figure 4. The relaxation rates of the T20IR and T22IR (a, s ) coherences are measured by adequate pulse sequences (see Figures 3c-d) and noted respectively R20 and R22 . This set of measurements of different 2H NMR relaxation rates is required to separately quantify66 the contributions from the quadrupolar (Eq. A7) and heterogeneous dipolar (Eq. A10) relaxation mechanisms, respectively. The time delay ( δ opt ) displayed in Figure 3c-d is selected in order to optimize


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the transfer of magnetization between the first order coherence T11IR (s ) , initially generated by the first pulse, and the second order coherence T21IR (a ) . The time delay ( ε ) displayed in Figures 3c-d and 4 is set equal to 10 µs.

Figures 5-6

The measurement of two-time correlation function is performed by using the pulse sequence illustrated in Figure 5. That pulse sequence exploits the heterogeneities of the residual quadrupolar coupling69 felt by the water molecules confined within clay lamellae with different orientations into the static magnetic field B 0 . During the first evolution period ( te ), the first order coherence T11IR oscillates (Eq. A3) according to the initial value of the residual quadrupolar coupling ( ωQ (0 ) ). During an appropriate mixing period ( τ M ), the confined water molecules diffuse within the clay sediment and exchange between clay aggregates with different orientations into the macroscopic clay film. After that mixing period, these labile water molecules sample another residual coupling ( ωQ (τ M ) ) during the second evolution period ( te ). As a consequence, the magnetization I (t e , τ M ) detected by the pulses sequence (see Figure 5) varies according to69,83:

(

I (t e , τ M ) ∝ cos (ωQ (0 ) te )× cos (ωQ (τ M ) te ) exp − R20 τ M − 2 R11 te

)

(Eq. 1)

The interferences between these various local environments lead to a net reduction of the measured echo intensity69,83 if the time-scale quantifying their exchange is longer than the transverse relaxation time, corresponding to the so-called slow exchange regime. In order to avoid artifacts resulting from the recovery of the T10IR coherence during the mixing period ( τ M ), a double-quantum filtering is applied in order to select only the contribution from the



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T20IR coherence. For that purpose, the duration of the fourth pulse of the sequence, noted ψ

(see Figure 5), is carefully set to 38 µs (see Figure 6 and Appendix) to simultaneously optimize the transfer from the T20IR coherence to the T22IR coherence (Eq. A4a) and minimize the concomitant transfer from the T10IR coherence (Eq. A4e).

III. Results and Discussion 1.

2

H NMR Spectra Figure 7

Figure 2 displays some 2H NMR spectra of heavy water molecules strongly confined within the dense clay sediment. Because of the broadening of the 2H resonance line in the presence of clay, the residual quadrupolar coupling is difficult to extract from the lineshape analysis for the whole set of orientations of the clay film into the static magnetic field. An alternate approach is given by the time evolution of the T11IR coherence (see Figure 4) by using the simple relationship:

(

)

T11IR (τ ) = T11IR (0) cos ωQobs τ exp(− τ R11 )

(Eq. 2)

The resulting apparent splitting of the 2H resonance lines are displayed in Figure 7 as a function of the film orientation β LF . If a single spin population is detected, Figure 7 should exhibit a perfect agreement between the observed residual splitting and the theoretical relationship84:

( )

( ( ))

ωQobs β LF = ωQMax P2 cos β LF

= ωQMax

( )

3 cos 2 β LF − 1 2


(Eq. 3)

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where P2 is the second order Legendre polynomial and ωQMax is the maximum value of the residual quadrupolar coupling detected for a perfect alignment between the static magnetic field B 0 and the film director n F , L . As already discussed in the Introduction, the detected residual quadrupolar coupling results from an average between various configurations accessible to the confined water molecules as quantified by numerical simulations exploiting a molecular model of the clay-water interfaces2,85. As shown in Figure 7, experimental data do not follow perfectly the theoretical expression. Such a discrepancy indicates that two conditions are fulfilled in this experiment:

-

firstly, various water environments, with different intrinsic residual quadrupolar coupling coexist;

-

secondly, the time-scale ( τ exch ) characterizing the exchange of water molecules between these various spin-environments is longer than the transverse relaxation time R11−1 .

As a consequence, the 2H NMR spectra of confined water molecules (see Figure 2) correspond to powder spectra of partially oriented sample under the so-called slow exchange regime73. Such conditions are ideal for performing two-time correlation measurements by exploiting 2H stimulated echo NMR spectroscopy67-72.

2.

2

H NMR Multi-Quanta Relaxation Measurements

Figure 8

Further evidence for the fulfillment of conditions required for performing two-time 2H stimulated echo NMR experiments are provided by measuring multi-quanta relaxation rates



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R10 , R11 , R20 and R22 . Indeed, as detailed in the Appendix, in the presence of quadrupolar (Eq. A7) and heterogeneous dipolar (Eq. A10) couplings, numerous spectral densities contribute to the 2H NMR relaxation rates. Fortunately, under the slow modulation regime of both relaxation mechanisms ( J 0Q (0 ) > J mQ (ω) , m ∈ {1 , 2 } and J 0D (0 ) > J mD (ω), m ∈ {1, 2 }), the set of equations A7 and A10 may be simplified66, leading to:

R10 = 5U Q +

R11 = R21 =

UD 3

(Eq. 4a)

3 Q 5 2 1 1 J 0 (0 ) + U Q + J 0D (0 ) + U D + J1D (ωS ) 2 2 9 2 3

(Eq. 4b)

R20 = 3U Q + U D

(Eq. 4c)

8 1 4 R22 = 3U Q + J 0D (0 ) + U D + J1D (ωS ) 9 3 3

(Eq. 4d)

with

U Q = J1Q (ω0 ) ≈ J 2Q (2 ω0 ) and U D =

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

(Eq. 4e)

where ω0 and ωS are the angular velocities of deuterium and paramagnetic centers (iron), respectively. As shown in Figure 8, both the R10 and R20 relaxation rates are two or three orders of magnitude smaller than the R22 and R11 relaxation rates. Consequently, the condition of slow modulation of the quadrupolar and dipolar relaxation mechanisms is fully satisfied, which is ideal to perform two-time 2H stimulated echo NMR measurements because of the −1 ) while the R20 relaxation rate is the limiting factor of slow exchange requirement ( R11 > τ exch

these two-time correlation measurements (see Eq. 1) leading to the sensitivity requirement:


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−1 τ exch > R20 , and thus R11 > R20 . Since the smallest transverse relaxation rate ( R11 ) is detected

for a perpendicular orientation of the film director n F , L in the static magnetic field B 0 (i.e. β LF =90°), we selected that specific orientation to perform two-time 2H stimulated echo NMR measurements because of the corresponding increase of the accessible evolution period ( te ) (see Eq. 1).

3. Two-Time Correlation Functions

Figures 9-11

The measured intensity of the two-time 2H stimulated echoes are reported in Figure 9 as a function of the mixing time τ M . As displayed in Eq. 1, these data must be renormalized to take into account the intensity attenuation resulting from the R20 relaxation during the various mixing times τ M . The normalized intensities are reported in Figure 10. No noticeable echo attenuation is observed for mixing time shorter than 10 ms. By contrast, drastic attenuation is detected for mixing times longer than 30 ms. A quantitative evaluation of the influence of mixing time τ M on the echo attenuation is given in Figure 11 that displays the evolution with τ M of the height of the first maximum displayed in Figure 10. Data displayed in Figure 11 easily cover four decades and clearly identify texch

(45 ± 5) ms as the time-

scale characterizing the water exchange between clay aggregates corresponding to different orientations within the static magnetic field B 0 . That time period is three orders of magnitude larger than the average residence time of water molecules confined within the hectorite lamellae ( τ B

60 ± 15 µs) as evaluated from the water self-diffusion coefficients obtained

either by Molecular Dynamics simulations23 or from QENS measurements23,31 performed on equivalent systems. The exchange time-scale extracted from these two-time 2H stimulated



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echo measurements indicates a long-distance mobility of the water molecules strongly confined within dense clay sediments. An upper-limit of that water displacement is given by the self-diffusion coefficient of bulk water:

Lmax =

2 Dbulk τ exch ≈ (13 ± 1) m

(Eq. 5)

i.e. one order of magnitude larger than the average size of the confining platelets66.

Such two-time stimulated echo measurements were already performed to quantify the ultra-slow mobility of ionic and molecular probes, such as Li+, Ag+, water or polyethylene in complex systems including glassy solids or ionic conductors67-72. To our knowledge, the present study is the first that used this method to identify water long-distance exchange within dense clay sediments similar to those used as diffusion barrier in waste disposal facilities.

4. Numerical Modeling

Numerical simulations were used to access the influence of the heterogeneities of the residual quadrupolar coupling felt by confined water molecules on the attenuation of the twotime 2H stimulated echo. For that purpose, a cubic simulation cell was divided in 153 individual sub-cells, each characterized by its residual quadrupolar coupling. A Gaussian random field86,87, with a priori field-field correlation, was used to generate the distribution of the 3375 local values of the residual quadrupolar coupling ωQ . We selected to use a Gaussian field-field correlation function, with a standard deviation corresponding to the length of two elementary sub-cells. The initial Gaussian random field has zero mean and standard deviation equal to one86,87. Before use, its mean is shifted to unity and its standard deviation is multiplied by a factor 0.15. The resulting random field is used to generate a distribution of cos(β LF ) thus monitoring the local values of the residual quadrupolar coupling (see Eq. 3).


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The time evolution of the various coherences is then evaluated during each step of the pulse sequence used to perform these two-time correlation measurements (see Figure 5). The exchange between every neighboring sub-cell is described, for each local sub-set of coherences, using the generalized Bloch equations73,74

d σi , j ,k = ( R i , j , k − 6 kexch I ) σi , j ,k + kexch I ( σi +1, j ,k + σi −1, j ,k + σi , j +1,k + σi , j −1,k + σi , j ,k −1 + σi , j ,k +1 ) dt (Eq.6)

where indexes ( i, j , k ) describe the location of the sub-cell, R i , j , k contains the contribution from the relaxation mechanisms, pulses and residual quadrupolar coupling (see Appendix, Eqs A3, A7 and A11) and I is the identity matrix. Three different approaches can be used to determine the time evolution of the coherences:

-

an analytical approximation (see Appendix, Eqs A4 a-h) is applied during each pulse, by neglecting the relaxation and exchange phenomena;

-

an analytical treatment is applied during the two evolution periods ( te ) by neglecting only the exchange phenomenon;

-

an iterative numerical approach is applied during the mixing period ( τ M ) in order to reproduce the contribution from the relaxation, exchange and residual quadrupolar coupling.

These approximations are justified by the short duration of the applied pulses (a few µs) and the long time required for detecting any influence of the water exchange (i.e. 10 ms as displayed in Figure 10) by comparison with the evolution period ( te ≤ 0.3 ms). The time step



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used to numerically solve the complete differential equation (third procedure above) is set equal to 0.1 µs.

Figure 12-13

The results from these numerical simulations are displayed in Figure 12, after compensation of the attenuation of the two-time echoes induced by the R20 relaxation phenomenon occurring during the mixing period τ M (see Eq. 1). Figure 12 clearly illustrates the influence of the low-rate water exchange on the additional attenuation of the intensity of these simulated two-time stimulated echoes. The major discrepancy between numerical (Figure 12) and experimental data (Figure 10) arises from a phase difference in the oscillatory behavior of the echo amplitude during the evolution period te . Such a phase is related (see Appendix, Eq. A3) to the initial populations of the T11IR (0) and T21IR (0) coherences induced by the first and fifth pulses of the sequence displayed in Figure 5. Since our modeling neglects the contributions of relaxation during the pulses (see above), the balance of these two initial populations is not carefully reproduced. As a consequence, our simulations will provide only a qualitative illustration of the impact of water exchange on the attenuation of the twotime stimulated echo. As previously (see Figure 10), the contribution of water exchange is further analyzed by reporting in Figure 13 the normalized height of the secondary maximum occurring in Figure 12 for an evolution time te around 100 µs. By selecting an exchange rate kexch = 10 s-1 (see Eq. 6), we reproduce the order of magnitude of the time-scale (45 ms) quantifying the echo attenuation. By contrast, the variations of the echo attenuation as a function of the mixing time τ M differ significantly. For the experimental data (see Figure 11), the attenuation is very sharp and the transition between 0.9 and 0.1 occurs in less than one


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decade of the mixing time τ M . By contrast, numerical data yield a much smoother transition that extends over two decades of mixing time τ M (see Figure 13). This shows that a simple Gaussian distribution law cannot quantitatively describes the spatial decorrelation of the local values of the residual quadrupolar coupling felt by the water molecules confined within clay platelets. Nevertheless, such numerical modeling qualitatively illustrates the influence of lowrate water exchange on the attenuation of the two-time stimulated echo detected by 2H NMR spectroscopy.

IV. Conclusions Two-time 2H stimulated echo NMR measurements were performed to detect the slowrate exchange of the water molecules confined within dense clay sediment. Preliminary 2H multi-quanta NMR relaxation measurements were used to quantify the relative contribution from the quadrupolar and heterogenous dipolar couplings monitoring the relaxation of the strongly confined water molecules. That preliminary study also illustrates the slow modulation of both quadrupolar and dipolar NMR relaxation mechanisms. Furthermore, the angular variation of the residual quadrupolar coupling felt by the confined water molecules is compatible with a slow exchange, at the NMR time-scale, between water molecules pertaining to clay aggregates with a different orientation within the self-supporting film. These two conditions (slow modulation and slow exchange) are required for a successful use of two-time stimulated echo NMR measurement of the diffusing probe. The analysis of the experimental two-time correlations exhibits an exchange time-scale texch

45 ms, much

longer than the average residence time of the water molecules confined within individual clay platelets τ B

60 µs. The qualitative agreement between experimental and numerical data

validates our interpretation based on the spatial correlations between local values of the



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residual quadrupolar coupling felt by the confined water molecules. That study helps apprehending the impact of long-distance mobility of the water molecules confined within the interlamellar space of dense clay sediments similar to those used as diffusion barriers. Such investigations can also easily be extended to quantify the long-range mobility of fluids diffusing within other porous networks or membranes.

Acknowledgements The DSX360 Bruker spectrometers used for that NMR study were purchased thanks to grants from Région Centre (France). We thank Dr Laurent J. Michot (LEM, Nancy) for providing us with the hectorite sample used in the present study. We cordially thank Dr. Joël Puibasset (CRMD, Orléans) for helpful discussions.


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Appendix 1. Time Evolution of the Coherences

In the framework of the Redfield theory88, the time evolution of the coherences is described by the complete master equation41,75,89:

[

]

dσ* = −i H S* , σ* + f (σ* ) dt

(Eq. 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 Q* S =

H S*

that

includes

the

residual

static

quadrupolar

Hamiltonian

2 ωQ T20IR ) and the Hamiltonian corresponding to the radio frequency pulse 3

( H1*S = I x ω 1 = 2 ω1 T11IR (a ) ). The second contribution to the master equation describes the contribution from the fluctuating components of the Hamiltonians41,75 ( H F* (t ) ) monitoring the relaxation of the coherences:

f ( σ* ) = −

∞ 0

[H

* F

[

*

*

(t ) , e −iH S τ H F*+ (t − τ) eiH S τ , σ* (t )

]]

dτ

(Eq. A2)

Let us, in a first approximation, neglect the contribution from the relaxation mechanisms. The resulting simplified master equation may be written in a matrix form80:



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T20IR T11IR (a ) T21IR ( s ) d T (s) dt T10IR T11IR ( s ) T21IR (a ) T22IR (a ) IR 22

0 0

0 0

− 3 ω1 ωQ 0 0 =i 0 0 0 0 0 0 0 0

− 3 ω1

ωQ 0 − ω1 0 0 0 0

0 0

0 0

0 0

0 0

0 0

− ω1 0 0 0 0 0

0 0 0

0 0

0 0 0

0 0 0 0

− ω1 0 0

− ω1 0

ωQ

ωQ

0

0

− ω1

T20IR T11IR (a ) T21IR ( s )

T22IR ( s ) ⋅ T10IR T11IR ( s ) − ω1 T21IR (a ) T22IR (a ) 0

(Eq. A3)

As a consequence, the time evolution of the coherences splits in two independent sub-sets80:

*

ω + ω1 (1 + 3 cos(k1τ) ) IR 1 − cos(k1τ) IR T20 T11 (a) + Q 2 k12 k1 2

*

eiH S τ T20IR e −iH S τ = 3 ω1 ωQ

2

sin(k1τ) IR cos(k1τ) − 1 IR T22 ( s ) T21 ( s ) + 3 ω12 − i 3 ω1 k12 k1

e

iH S* τ

IR 11

T

(a ) e

−iH S* τ

=

ωQ2 cos(k1τ) + 4 ω12 2 1

k

T11IR (a) + 3 ω1 ωQ

1 − cos(k1τ) IR T20 k12

sin(k1τ) IR 1 − cos(k1τ) IR T22 ( s ) T21 ( s ) + ω1 ωQ + i ωQ k12 k1 eiH S τ T21IR (s ) e −iH S τ = i ωQ *

*

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

sin(k1τ) IR T22 ( s ) + cos(k1τ) T ( s ) − i ω1 k1

(Eq. A4a)

(Eq. A4b)

(Eq. A4c)

IR 21

eiH S τ T22IR (s ) e −iH S τ = ω1 ωQ *

*

1 − cos(k1τ) IR cos(k1τ) − 1 IR T11 (a ) + 3 ω12 T20 2 k1 k12

sin( k1τ) IR k 2 − ω12 (1 − cos(k1τ) ) IR T22 ( s ) − i ω1 T21 ( s ) + 1 k12 k1 *

*

eiH S τ T10IR e −iH S τ =

sin( k3τ) + sin(k 2 τ) IR k 2 cos(k3τ) + k3 cos( k 2 τ) IR T11 ( s) T10 − i ω1 k1 k1

cos( k3 τ) − cos( k 2 τ) IR k sin(k3τ) − k3 sin( k 2 τ) IR T22 (a) T21 (a ) − i 2 + ω1 k1 k1


(Eq. A4d)

(Eq. A4e)

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eiH S τ T11IR (s ) e −iH S τ = − i ω1 *

*

sin( k3τ) + sin(k 2 τ) IR k 2 cos(k 2 τ) + k3 cos(k3τ) IR T11 ( s ) T10 + k1 k1

cos(k3τ) − cos(k 2 τ) IR k sin(k 2 τ) − k3 sin(k3 τ) IR T22 (a ) T21 (a ) + ω1 +i 2 k1 k1 eiH S τ T21IR (a )e −iH S τ = ω1 *

*

cos(k3τ) − cos(k 2 τ) IR k sin(k 2 τ) − k3 sin( k3 τ) IR T10 + i 2 T11 ( s ) k1 k1

sin(k3τ) + sin( k 2 τ) IR k cos(k 2 τ) + k3 cos(k3τ) IR T22 (a ) T21 (a ) − i ω1 + 2 k1 k1

eiH S τ T22IR (a ) e −iH S τ = − i *

*

k 2 sin(k3τ) − k3 sin( k 2 τ) IR cos(k3τ) − cos(k 2 τ) IR T10 + ω1 T11 ( s) k1 k1

sin( k3 τ) + sin( k 2 τ) IR k cos( k3 τ) + k3 cos(k 2 τ) IR − i ω1 T21 (a ) + 2 T22 (a ) k1 k1

(Eq. A4f)

(Eq. A4g)

(Eq. A4h)

where the characteristic angular velocities80 k1 , k 2 and k3 are defined by:

k1 = ωQ2 + 4 ω12

k2 =

k3 =

(Eq. A5a)

ωQ + ωQ2 + 4 ω12

(Eq. A5b)

2 ωQ2 + 4 ω12 − ωQ

(Eq. A5c)

2

2. Quadrupolar Relaxation

In bulk water, quadrupolar coupling is the main mechanism responsible for relaxation of heavy water41,75,89:

H Q (t ) = CQ

2

(− 1) m T2Q, m F2Q, − m (t )

m = −2

where the quadrupolar coupling constant, defined by:


(Eq. A6a)


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p. 22 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

3 CQ = 8

e2q Q

(Eq. A6b)

I (2 I − 1)

3 × π × 210 kHz for deuterium in bulk heavy water90. The spin operators 2

is equal to

describing quadrupolar coupling are given by:

T2Q, 0 =

1 ( 3 I z2 − I (I + 1) ) = T20IR 6 1 1 ( IZ I± + I± I Z ) = ( T21IR (s ) T21IR (a ) ) 2 2

T2Q, ±1 =

and

T2Q, ± 2 =

(Eq. A6c)

(Eq. A6d)

1 2 1 I± = ( T22IR (s ) T22IR (a ) ) 2 2

(Eq. A6e)

Functions F2Q, m (t ) in Equation A6a are the second order spherical harmonics describing the reorientation of the OD director in the static magnetic field B 0 by using the two Euler angles ( θ LW , φ LW ) since 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 (Eq. A1-2) may also be written in a matrix form80:

T20IR T11IR (a) T21IR ( s ) d T22IR ( s ) = − diag ( AQ , B Q , C Q , D Q , E Q , B Q , C Q , D Q ) ⋅ IR dt T10 T11IR ( s ) T21IR (a) T22IR (a)

T20IR T11IR (a ) T21IR ( s ) T22IR ( s ) T10IR T11IR ( s ) T21IR (a ) T22IR (a )


(Eq. A7)

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p. 23 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

with

AQ = 3 J1Q (ω0 ) ,

B Q = 3 2 J 0Q (0) + 5 2 J1Q (ω0 ) + J 2Q (2ω0 ) ,

C Q = 3 2 J 0Q (0) + 1 2 J1Q (ω0 ) + J 2Q (2ω0 ) ,

D Q = J1Q (ω0 ) + 2 J 2Q (2ω0 ) ,

E Q = J1Q (ω0 ) + 4 J 2Q (2ω0 ) .

The spectral densities ( J mQ (m ω0 ) ) used in Equation A7 are the Fourier transform of the autocorrelation functions GmQ (τ) describing the loss of memory of the fluctuating part of the quadrupolar coupling:

J mQ (mω0 ) = (−1) m CQ2

∞

0

(F

Q 2, − m

(0) −

F2Q, − m

)( F

Q 2, m

(τ) −

F2Q,m

)e

i m ω0 τ

dτ

(Eq. A8)

3. Paramagnetic Relaxation

Because of the presence of iron within the natural hectorite clay particle, the heterogeneous dipolar coupling is also responsible64 for the NMR relaxation of the confined heavy water, in addition to the intrinsic quadrupolar relaxation mechanism. The corresponding Hamiltonian41,75,80,89 is defined by:

H D (t ) = C D

2

(− 1) m

m = −2

T2D, m F2D, −m (t ) rI3S (t )

where the dipolar coupling constant is given by :


(Eq. A9a)


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p. 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

CD = −

µ0 6 γI γS 4π

(Eq. A9b)

and the spin operators become64:

T20D =

T2D±1 =

T2D± 2 =

1 6

2I z S z −

(

1 I + S− + I − S+ 2

(

)

1 I Z S± + I ± SZ = 2 1 I± S = 2

(

)

=

1 6

(

1 T10IR S ± 2

2 2 T10IR S z +

[T

IR 11

(s )

[

(

)

(

)]

1 IR T11 (s ) S − − S + − T11IR (a ) S − + S + 2 (Eq. A9c)

] )

T11IR (a ) S Z

)

1 T11IR (s ) T11IR (a ) S 2

(Eq. A9d)

(Eq. A9e)

Functions F2D, m (t ) in Equation A9a are the same as in Equation A6a but they now describe the reorientation of the vector joining the two coupled spin (noted rI S (t ) ) by reference with the static magnetic field B 0 . 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 rI−S3 (t ) in Eq. A9a). The contribution from the heterogeneous dipolar coupling to the complete master equation (Eq. A1-2) may also be written in a matrix form64:

T20IR T11IR (a) T21IR ( s ) d T22IR ( s ) = − diag ( A D , B D , C D , D D , E D , B D , C D , D D ) ⋅ dt T10IR T11IR ( s ) T21IR (a) T22IR (a)

T20IR T11IR (a ) T21IR ( s ) T22IR ( s ) T10IR T11IR ( s ) T21IR (a ) T22IR (a )


(Eq. A10)

Page 25 of 45


p. 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

with

AD =

1

3

J 0D (ω0 − ωS ) + J1D (ω0 ) + 2 J 2D (ω0 + ωS ) ,

B D = 2 9 J 0D (0) + 118 J 0D (ω0 − ωS ) + 1 6 J1D (ω0 ) + 13 J1D (ωS )° + 1 3 J 2Q (ω0 + ωS ) , C D = 2 9 J 0D (0) + 518 J 0D (ω0 − ωS ) + 5 6 J1D (ω0 ) + 13 J1D (ωS ) + 5 3 J 2D (ω0 + ωS ) , D D = 8 9 J 0D (0) + 1 9 J 0D (ω0 − ωS ) + 13 J1D (ω0 ) + 4 3 J1D (ωs ) + 2 3 J 2D (ω0 + ωS ) , ED =

1

9

J 0D (ω0 − ωS ) + 13 J1D (ω0 ) + 2 3 J 2D (ω0 + ωS ) .

The corresponding spectral densities are:

J mD (ω) = (− 1) N S S (S + 1) C D2 m

∞ 0

e i m ω0 τ

F2D, − m (0 ) 3 IS

r

(0 )

−

F2D, − m 3 IS

r

F2D, m (τ ) 3 IS

r

(τ)

−

F2D, m rI3S

dτ (Eq. A11)

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



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(47) Dvinskikh, S. V.; Furo, I. Cross-Relaxation Effects in Stimulated-Echo-Type PGSE NMR Experiments by Bipolar and Monopolar Gradient Pulses. J. Magn. Reson. 2000, 146, 283-289. (48) Nakashima, Y. Pulsed Field Gradient Proton NMR Study of the Self-Diffusion of H2O in Montmorillonite Gel: Effects of Temperature and Water Fraction. Am. Mineral. 2001, 86, 132-138. (49) Porion, P.; Rodts, S.; Al-Mukhtar, M.; Faugère, A. M.; Delville, A. Anisotropy of the Solvent Self-Diffusion Tensor as a Probe of Nematic Ordering within Dispersion of Nano-Composite. Phys. Rev. Lett. 2001, 87, 208302. (50) Porion, P.; Al-Mukhtar, M.; Faugère, A. M.; Pellenq, R. J. M.; Meyer, S.; Delville, A. Water Self-Diffusion within Nematic Dispersion of Nanocomposites: A Multiscale Analysis of 1H Pulsed Gradient Spin-Echo NMR Measurements. J. Phys. Chem. B 2003, 107, 4012-4023. (51) Nakashima, Y.; Mitsumori, F. H2O Self-Diffusion Restricted by Clay Platelets with Immobilized Bound H2O Layers: PGSE NMR Study of Water-Rich Saponite Gels. Appl. Clay Sci. 2005, 28, 209-221. (52) de Azevedo, E. N.; Engelsberg, M.; Fossum, J. O.; de Souza, R. E. Anisotropic Water Diffusion in Nematic Self-Assemblies of Clay Nanoplatelets Suspended in Water. Langmuir 2007, 23, 5100-5105. (53) Delville, A. Toward a Detailed Molecular Analysis of the Long-Range Swelling Gap of Charged Rigid Lamellae Dispersed in Water. J. Phys. Chem. C 2012, 116, 818-825. (54) Delville, A. Structure and Properties of Confined Liquids: Molecular-Model of the Clay Water Interface. J. Phys. Chem. 1993, 97, 9703-9712. (55) Boek, E. S.; Coveney, P. V.; Skipper, N. T. Monte Carlo Molecular Modeling Studies of Hydrated Li-, Na-, and K-Smectites: Understanding the Role of Potassium as a Clay Swelling Inhibitor. J. Am. Chem. Soc. 1995, 117, 12608-12617. (56) Chang, F. R. C.; Skipper, N. T.; Sposito, G. Computer-Simulation of Interlayer Molecular Structure in Sodium Montmorillonite Hydrates. Langmuir 1995, 11, 27342741. (57) Smith, D. E. Molecular Computer Simulations of the Swelling Properties and Interlayer Structure of Cesium Montmorillonite. Langmuir 1998, 14, 5959-5967. (58) Marry, V.; Turq, P.; Cartailler, T.; Levesque, D. Microscopic Simulation of Structure and Dynamics of Water and Counterions in a Monohydrated Montmorillonite. J. Chem. Phys. 2002, 117, 3454-3463. (59) Woessner, D. E.; Snowden, B. S. NMR Doublet Splitting in Aqueous Montmorillonite Gels. J. Chem. Phys. 1969, 50, 1516-1523. (60) Fripiat, J.; Cases, J.; François, M.; Letellier, M. Thermodynamic and Microdynamic Behavior of Water in Clay Suspensions and Gels. J. Colloid Interface Sci. 1982, 89, 378400. (61) Hetzel, F.; Tessier, D.; Jaunet, A.-M., J.; Doner, H. The Microstructure of Three Na+ Smectites: The Importance of Particle Geometry on Dehydration and Rehydration. Clays Clay Miner. 1994, 42, 242-248.



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(62) Woessner, D. E.; Snowden, B. S.; Meyer, G. H. A Tetrahedral Model for Pulsed Nuclear Magnetic Resonance Transverse Relaxation: Application to the Clay-Water System. J. Colloid Interface Sci. 1970, 34, 43-52. (63) Hougardy, J.; Stone, W. E. E.; Fripiat, J. J. NMR Study of Adsorbed Water. I. Molecular Orientation and Protonic Motions in the Two-Layer Hydrate of a Na Vermiculite. J. Chem. Phys. 1976, 64, 3840-3851. (64) Porion, P.; Michot, L. J.; Faugère, A. M.; Delville, A. Structural and Dynamical Properties of the Water Molecules Confined in Dense Clay Sediments: A Study Combining 2H NMR Spectroscopy and Multiscale Numerical Modeling. J. Phys. Chem. C 2007, 111, 5441-5453. (65) Porion, P.; Michot, L. J.; Faugère, A. M.; Delville, A. Influence of Confinement on the Long-Range Mobility of Water Molecules within Clay Aggregates: A 2H NMR Analysis Using Spin-Locking Relaxation Rates. J. Phys. Chem. C 2007, 111, 13117-13128. (66) Porion, P.; Michot, L. J.; Warmont, F.; Faugère, A. M.; Delville, A. Long-Time Dynamics of Confined Water Molecules Probed by 2H NMR Multiquanta Relaxometry: An Application to Dense Clay Sediments. J. Phys. Chem. C 2012, 116, 17682-17697. (67) Böhmer, R.; Jörg, T.; Qi, F.; Titze, A. Stimulated 7Li Echo NMR Spectroscopy of Slow Ionic Motions in a Solid Electrolyte. Chem. Phys. Lett. 2000, 316, 419-424. (68) Wilkening, M.; Heitjans, P. New Prospects in Studying Li Diffusion - Two-Time Stimulated Echo NMR of Spin-3/2 Nuclei. Solid State Ionics 2006, 177, 3031-3036. (69) Böhmer, R.; Jeffrey, K. R.; Vogel, M. Solid-State Li NMR with Applications to the Translational Dynamics in Ion Conductors. Prog. Nucl. Magn. Reson. Spectrosc. 2007, 50, 87-174. (70) Schildmann, S.; Nowaczyk, A.; Geil, B.; Gainaru, C.; Böhmer, R. Water Dynamics on the Hydrate Lattice of a Tetrabutyl Ammonium Bromide Semiclathrate. J. Chem. Phys. 2009, 130, 104505. (71) Faske, S.; Koch, B.; Murawski, S.; Küchler, R.; Böhmer, R.; Melchior, J.; Vogel, M. Mixed-Cation LixAg1-xPO3 Glasses Studied by 6Li, 7Li, and 109Ag Stimulated-Echo NMR Spectroscopy. Phys. Rev. B 2011, 84, 024202. (72) Storek, M.; Böhmer, R.; Martin, S. W.; Larink, D.; Eckert, H. NMR and Conductivity Studies of the Mixed Glass Former Effect in Lithium Borophosphate Glasses. J. Chem. Phys. 2012, 137, 124507. (73) Woessner, D. E. Nuclear Transfert Effects in Nuclear Magnetic Resonance Pulse Experiments. J. Chem. Phys. 1961, 35, 41-48. (74) Cuperlovic, M.; Meresi, G. H.; Palke, W. E.; Gerig, J. T. Spin Relaxation and Chemical Exchange in NMR Simulations. J. Magn. Reson. 2000, 142, 11-23. (75) Abragam, A. The Principles of Nuclear Magnetism; Clarendon Press: Oxford, 1961. (76) Michot, L. J.; Bihannic, I.; Porsch, K.; Maddi, S.; Baravian, C.; Mougel, J.; Levitz, P. Phase Diagrams of Wyoming Na-Montmorillonite Clay. Influence of Particle Anisotropy. Langmuir 2004, 20, 10829-10837. (77) Delville, A.; Grandjean, J.; Laszlo, P. Order Acquisition by Clay Platelets in a Magnetic Field. NMR Study of the Structure and Microdynamics of the Adsorbed Water Layer. J. Phys. Chem. 1991, 95, 1383-1392.


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(78) Petit, D.; Korb, J. P.; Delville, A.; Grandjean, J.; Laszlo, P. Theory of Nuclear Spin Relaxation in Heterogeneous Media and Application to the Cross Correlation between Quadrupolar and Dipolar Fluctuations of Deuterons in Clay Gels. J. Magn. Reson. 1992, 96, 252-279. (79) Müller, N.; Bodenhausen, G.; Ernst, R. R. Relaxation-Induced Violations of Coherence Transfer Selection Rules in Nuclear Magnetic Resonance. J. Magn. Reson. 1987, 75, 297-334. (80) van der Maarel, J. R. C. The Relaxation Dynamics of Spin I=1 Nuclei with a Static Quadrupolar Coupling and a Radio-Frequency Field. J. Chem. Phys. 1993, 99, 56465653. (81) Fukushima, E.; Roeder, S. B. W. Experimental Pulse NMR: A Nuts and Bolts Approach; Addison-Wesley: Reading, MA, 1981. (82) Hahn, E. L. Spin Echoes. Phys. Rev. 1950, 80, 580-594. (83) Böhmer, R.; Qi, F. Spin Relaxation and Ultra-Slow Li Motion in an Aluminosilicate Glass Ceramic. Solid State Nucl. Magn. Reson. 2007, 31, 28-34. (84) Barbara, T. M.; Vold, R. R.; Vold, R. L. A Determination of Individual Spectral Densities in a Smectic Liquid-Crystal from Angle Dependent Nuclear Spin Relaxation Measurements. J. Chem. Phys. 1983, 79, 6338-6340. (85) Porion, P.; Faugère, A. M.; Michot, L. J.; Paineau, E.; Delville, A. 2H NMR Spectroscopy and Multiquantum Relaxometry as a Probe of the Magnetic-Field-Induced Ordering of Clay Nanoplatelets within Aqueous Dispersions. J. Phys. Chem. C 2011, 115, 14253-14263. (86) Berk, N. F. Scattering Properties of the Leveled-Wave Model of Random Morphologies. Phys. Rev. A 1991, 44, 5069-5079. (87) Levitz, P. Off-Lattice Reconstruction of Porous Media: Critical Evaluation, Geometrical Confinement and Molecular Transport. Adv. Colloid Interface Sci. 1998, 77, 71-106. (88) Redfield, A. G. On the Theory of Relaxation Processes. IBM J. Res. Develop. 1957, 1, 19-31. (89) Mehring, M. Principles of High Resolution NMR in Solids, 2nd ed.; Springer-Verlag: Berlin, 1983. (90) Edmonds, D. T.; Mackay, A. L. Pure Quadrupole-Resonance of Deuteron in Ice. J. Magn. Reson. 1975, 20, 515-519.



Page 32 of 45

p. 32 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 1

Micro-domains

Clay platelets

Figure 1. Schematic view of the multi-scale structure of dense clay sediments (see text).


Page 33 of 45


p. 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 2 (on two columns)

β

LF

2H

NMR

B0

n F ,L

90° 80° 70° 60°

β LF

50°

2νQ

40° 30° 20° 10° 0°

80

60

40

20

0

-20 -40 -60 -80

Frequency (kHz)

Figure 2. Variation of the 2H NMR spectra of confined water molecules as a function of the

orientation β LF of the film director n F , L into the static magnetic field B 0 (see text).



Page 34 of 45

p. 34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 3

π

π2

(a)

τ

Acq. time

+2 +1 p=0 -1 -2

π

π2

(b)

echo

τ2

τ2

Acq. time

+2 +1 p=0 -1 -2

π2

(c)

π

δ opt 2

π

π2

π4 τ

δopt 2

ε/2

π4 ε/2

Acq. time

+2 +1 p=0 -1 -2

π2

(d)

δ opt 2

π

π

π2 δ opt 2

τ2

π2

τ2

Acq. time

+2 +1 p=0 -1 -2

Figure 3. Pulse sequences and coherence transfer pathways used to measure the 2H NMR

relaxation rate of the (a) T10IR , (b) T11IR (a, s ) , (c) T20IR , and (d) T22IR (a, s ) coherences, respectively (see text).


Page 35 of 45


p. 35

Hahn echo intensity (a. u.)

Hahn echo Intensity (a. u.)

Figure 4

Hahn echo intensity (a. u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.0

(a)

βLF = 0°

0.5 0.0

0.8

≈ 79400 rad.s

Q

R

≈ 7100 s

-4

6 10

-1

11

-1.0 0 0 10 1.0

-1

obs

ω -0.5

2 10

-4

-4

4 10

(b)

8 10

-4

1 10

-3

βLF = 55°

0.6

obs

ω

-1

≈ 150 rad.s

Q

0.4

R

0.2

11

≈ 48000 s

-1

0.0 -0.2 0 0.0 10 1.0

-4

1.0 10

2.0 10

(c)

β

0.5

LF

-4

= 90°

0.0 obs

ω

-0.5

Q

R

11

-1.0 0 0 10

2 10

-4

4 10

-4

-1

≈ 40500 rad.s

≈ 3400 s -4

6 10

Echo time τ (s)

-1

8 10

-4

1 10

-3

Figure 4. Typical evolutions of the T11IR coherence (i.e. the transverse magnetization M X )

measured during the Hahn echo attenuation NMR sequence and used to extract the residual quadrupolar splitting ωQobs and the transverse relaxation rate R11 (see Eq. 2) as a function of the clay film orientation β LF into the static magnetic field B 0 : (a) β LF =0°, (b) β LF =55°, and (c) β LF =90°.



Page 36 of 45

p. 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 5

π

π2 te 2

ψ

π4 te 2

τM

π2 ε

te 2

π te 2 Acq. time

+2 +1 p=0 -1 -2

Figure 5. Pulse sequence and coherence pathway used to measure the attenuation of the two-

time 2H NMR echo I (t e , τ M ) as a function of the evolution period te and the mixing time τ M (see text).


Page 37 of 45


p. 37

Figure 6

1.0

Transfert of coherences (T ij)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

T20 --> T22(s) T10 --> T22(a) 0.5

Optimization of the pulse duration : ψ = 38 µs 0.0

-0.5

-1.0 0 0 10

-6

10 10

-6

20 10

-6

30 10

-6

40 10

-6

50 10

Pulse duration (s)

Figure 6. Illustration of the optimization of the duration of the fourth pulse, noted ψ in the

sequence illustrated in Figure 5, and used to generate two-time 2H NMR stimulated spin echo.



Page 38 of 45

p. 38

100

obs

3

-1

(10 rad.s )

Figure 7

Quadrupolar splitting ωQ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Hahn sequence echo 80

60

40

20

0 -20

0

20

40

60

80

100

120

Angle βLF (degree)

Figure 7. Variation of the residual quadrupolar coupling felt by the confined water molecules

ωQobs as a function of the clay film orientation β LF into the static magnetic field B 0 , extracted from the Hahn echo NMR sequence.


Page 39 of 45


p. 39

Figure 8

10

6

10

5

10

4

10

3

10

2

10

1

10

0

ij

R

10

20

R (a,s)

R (a,s)

11

-1

(s )

R

Relaxation rates R

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

-20

0

20

22

40

Angle β

60 LF

80

100

120

(degrees)

Figure 8. Variation of the apparent multi-quanta relaxation rates R10 , R11 , R20 and R22 as a

function of the film orientation β LF into the static magnetic field B 0 .



Page 40 of 45

p. 40


30 25

Intensity (a. u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

βLF = 90°

τ = 10 µs

τ = 30 ms

τ = 100 µs

τ = 35 ms

τ = 1 ms

τ = 40 ms

τ = 5 ms

τ = 45 ms

τ = 10 ms

τ = 50 ms

M

M

M

M

M

20

M

M

15

M

M

M

τ = 20 ms M

10 5 0 -5 0 0 10

50 10

-6

100 10

-6

150 10

-6

200 10

-6

-6

250 10

Evolution time t (s) e

Figure 9. Variation of the measured two-point 2H NMR stimulated echo as a function of the

mixing time τ M , for the film orientation β LF =90°.


Page 41 of 45


p. 41


1.2

Nomalized intensity (a. u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.0

β

LF

= 90°

0.8 0.6

τ = 10 µs M

τ = 30 ms M

τ = 100 µs M

τ = 35 ms M

τ = 1 ms M

τ = 40 ms M

τ = 5 ms M

τ = 45 ms M

τ = 10 ms M

τ = 50 ms M

τ = 20 ms M

0.4 0.2 0.0 -0.2 0 0 10

-6

50 10

-6

100 10

150 10

-6

200 10

-6

250 10

-6

Evolution time te (s)

Figure 10. Variation as a function of the mixing time τ M of the measured two-point 2H NMR

stimulated echo normalized to take into account the relaxation of the T20IR coherence during the mixing time (i.e. the factor exp (− R20 τ M ) , see Eq. 1), for the film orientation β LF =90°.



Page 42 of 45

p. 42

Figure 11

Two-time stimulated echo attenuation (a. u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.2 1.0 0.8

τ ≈ 60 ± 15 µs B

0.6 0.4

t

exch

≈ 45 ± 10 ms

0.2 0.0 -5 10

-4

10

-3

-2

10

Mixing time τ

M

10

-1

10

(s)

Figure 11. Two-point correlation function extracted from the attenuation as a function of the

mixing time τ M of the intensity of the normalized 2H NMR stimulated echo.


Page 43 of 45


p. 43

Figure 12

1.2

Normalized echo intensity (a. u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.0 0.8

τ = 10 µs M

τ = 10 ms M

τ = 100 µs M

τ = 20 ms M

τ = 1 ms

τ = 30ms

τ = 5 ms

τ = 40ms

M M

M M

0.6 0.4 0.2 0.0 -0.2 0 0 10

100 10

-6

-6

200 10

300 10

-6


Figure 12. Variation as a function of the mixing time τ M of the simulated two-point 2H NMR

stimulated echo normalized to take into account the relaxation of the T20IR coherence during the mixing time (i.e. the factor exp (− R20 τ M ) , see Eq. 1).



Page 44 of 45

p. 44

Figure 13

Two-time stimulated echo attenuation (a. u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -5 10

-4

10

10

-3

-2

10

Mixing time τM (s)

10

-1

0

10

Figure 13. Two-point correlation function extracted from the numerical simulations (see text)

of the attenuation of the two-point 2H NMR stimulated echo as a function of the mixing time τ M .


Page 45 of 45


p. 45

Figure for Table of Contents (TOC) Graphic

1.2

τ = 30 ms

1.0

M

τ = 35 ms M

τ = 40 ms

0.8

M

τ = 45 ms M

τ = 50 ms M

0.6 0.4 0.2 0.0 -6 60 10

1.2

τ = 10 µs M

τ

M

100 10-6


140 10-6

Echo attenuation (a. u.)

Echo intensity (a. u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1.0 0.8

τ ≈ 60 ± 15 µs

0.6 0.4

B

t

exch

≈ 45 ms

0.2 0.0 10-5

10-4

10-3

10-2

Mixing time τM (s)


10-1

Long-Distance Water Exchange within Dense Clay ... - ACS Publications

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