7Li NMR Spectroscopy and Multiquantum Relaxation as a Probe of

9808

J. Phys. Chem. C 2008, 112, 9808–9821

7Li

NMR Spectroscopy and Multiquantum Relaxation as a Probe of the Microstructure and Dynamics of Confined Li+ Cations: An Application to Dense Clay Sediments Patrice Porion,* Anne Marie Fauge`re, and Alfred Delville* Centre de Recherche sur la Matie`re DiViseé, CNRS - UniVersite´ d’Orleáns, 1b rue de la Fe´rollerie, 45071 Orleáns Cedex 02, France ReceiVed: February 4, 2008; ReVised Manuscript ReceiVed: March 20, 2008

7Li

NMR spectroscopy and relaxometry are exploited to obtain structural and dynamical information within heterogeneous systems resulting from aqueous dispersion of clay platelets. In addition to the specific orientation of the clay platelets detected by the splitting of the 7Li resonance line, multiquantum relaxation measurements are used to determine the relative contributions from the quadrupolar and dipolar relaxation mechanisms responsible for the detected broadening of the 7Li resonance line. Intrinsic spectral densities are further extracted from the variation of these apparent relaxation rates as a function of the orientation of the clay sediment within the static magnetic field. A semiquantitative interpretation of the intrinsic quadrupolar and dipolar relaxation behavior is performed by a multiscale modeling of the Li+ ions’ diffusion in relation to the structure of the clay dispersions. Equivalent structural and dynamical investigations can be performed within a broad class of material containing electrically charged solid/liquid interfaces by exploiting the sensitivity of 7Li NMR spectroscopy. I. Introduction Because of its high gyromagnetic ratio (γ7Li/ γ1H ) and natural abundance (92.58%),1 7Li is as powerful probe for experimental studies exploiting nuclear magnetic resonance. Furthermore, 7Li is also very sensitive to the asymmetry of its electrostatic environment because it is a quadrupolar nucleus with 3/2 spin.2–6 Any departure of an isotropic distribution of its quadrupolar director leads to a splitting of the 7Li NMR resonance lines with a characteristic triplet pattern. Finally, the Li+ counterions are condensed in the electrostatic well located in the vicinity of negatively charged surfaces.7,8 7Li NMR is thus a sensitive probe of the structural and dynamical properties within dispersions of charged anisotropic nanoparticles. These materials cover a large class of colloids, implying clay minerals,9,10 synthetic metallic oxides,11,12 cement13, fuel cell membranes,14 liquid crystals,15,16 interfacial systems17–19 and biological polyions (like DNA20–25 or membranes26) and biological tissues.27,28 Various numerical19,21,23,24,29–33 and experimental9–18,20,22,26,34–39 studies were devoted to these systems because the competition between the antagonistic long-range electrostatic coupling32,33 and the short-range excluded volume interaction29,30 monitors the organization of these colloidal dispersions. In that framework, we illustrate the potentiality of 7Li NMR spectroscopy and relaxometry (i.e., the frequency variation of the NMR relaxation rates) by analyzing the self-organization occurring within aqueous dispersions of a synthetic clay (Laponite).9,40,41 Laponite clay was selected because of its high chemical purity and reduced polydispersity.42 Such clay minerals are implied in various industrial applications (drilling, waste storing, heterogeneous catalysis, and cosmetic, paint, and food industries) exploiting their physicochemical properties (high specific surface and water retention capacity, ionic exchange capacity, and surface acidity). The local organization and 0.389)1

* Authors to whom correspondence should be addressed. E-mail: [email protected] (A.D.) and [email protected] (P.P.).

mobility of the confined ions are crucial parameters for optimizing applications such as waste storing or heterogeneous catalysis. The purpose of our 7Li NMR study is to selectively extract this information. Such a detailed investigation of the structural and dynamical properties of the neutralizing counterions is necessary because confinement largely modifies the ion behavior near solid/liquid interfaces. Another interesting property of the 7Li quadrupolar nucleus is its small quadrupolar moment (Q ) -0.042 × 10-28 m2),43 which is one of the smallest for alkali nuclei. As a consequence, even under a highly asymmetric environment, the quadrupolar relaxation rate of the 7Li nucleus remains tractable (R2< 104 s-1), leading to thin and thus easily detected resonance lines. In contrast, because of its reduced quadrupolar coupling, the relaxation of the 7Li nucleus generally implies more than one relaxation mechanism. In order to determine the relative weight of the quadrupolar and dipolar relaxation mechanisms, we performed three sets of NMR relaxation experiments implying separately the time evolution of the zero quantum coherence (T10) by measuring the longitudinal relaxation rate, second, the one quantum coherence (T11) by measuring the transverse relaxation rate, and finally the triple-quantum coherence (T33) by measuring the triple-quantum filtered relaxation rate. In addition to these experimental studies, we also performed multiscale numerical modeling, including grand canonical Monte Carlo44 simulations (GCMC), molecular dynamics44,45 (MD), and Brownian dynamics46 (BD) in order to better interpret the NMR measurements and investigate the interdependence between the ionic diffusion, the ionic relaxation behavior, and the organization of the colloidal dispersions. II. Materials and Methods 1. Sample Preparation. Laponite RD was purchased from Laporte. The Li+ exchanged Laponite sample was prepared by a sequence of four dissolutions in aqueous solutions (LiCl 10-1 M, LiOH 10-4 M) followed by separations performed by centrifugation (10 min at 104 g). The excess LiCl salt was

10.1021/jp8010348 CCC: $40.75  2008 American Chemical Society Published on Web 06/11/2008

7Li

NMR Spectroscopy and Multiquantum Relaxation

removed by another sequence of five dissolutions in aqueous solutions (LiOH 10-4 M) with separation performed by ultrafiltration under nitrogen pressure (3-5 atm) using microporous membranes (pore size of 0.1 µ from Osmonics Inc.). The concentration of the final Laponite dispersion (3.3% w/w) was measured by water loss under vacuum. Its total lithium concentration ((2.9 ( 0.1) × 10-2 M) was evaluated by integrating the 7Li NMR signal using added LiCl aliquots. The same procedure was used to determine the residual sodium concentration ((3.9 ( 0.3) × 10-4 M). The total lithium and sodium concentrations indicate a reduced excess of salt because the charge exchange capacity of Laponite clay corresponds, at that density, to a nearly equivalent concentration of neutralizing counterions (2.84 × 10-2 M). The iron content of the 3.3% w/w Laponite dispersion evaluated by chemical analysis is less than 100 ppm, corresponding to 0.045 atom per unit cell. Dense clay sediments were prepared by uniaxial compression, using the same microporous membranes as for ultrafiltration. A denser sediment (noted “air-dried”) was prepared by water evaporation under controlled conditions (T ) 20 °C, p/p° ) 0.23). 2. NMR Measurements. The 7Li NMR measurements were performed on a DSX360 Bruker spectrometer, with a static magnetic field Β0 of 8.465 T. A broad spectral width (100 kHz) was used, corresponding to a fast acquisition procedure (dwell time 10 µs), allowing us to detect fast relaxation (R2 e 104 s-1) and large quadrupolar splitting (ωQ e 5 × 104 rad · s-1). The pulse duration for the magnetization inversion varied between 10 and 15 µs. The longitudinal relaxation time (T1) was measured by using the classical inversion-recovery pulses sequence; the transverse relaxation time (T2) was measured by using the Hahn spin-echo procedure.47 The relaxation rates of the T31 and T33 coherences were obtained by using a triplequantum filtered pulses sequence modified48,49 in order to eliminate the effects of Β0 inhomogeneity. For all of the sequences, an adequate phase cycling50 is used to select the coherence transfer pathway as illustrated in Figure 1. The 7Li spectra were recorded by using homemade receiver. A solenoid coil was prepared by wrapping a copper wire onto the grooves cut on a Teflon tube. The clay sample is cut into the dense clay sediment and inserted in a sealed glass tube, freely rotating inside the Teflon holder. The rotation of the NMR tube within the Teflon holder is monitored with a accuracy better than 5°. The compression axis of the clay sample is selected to be perfectly perpendicular to the axis of the NMR tube, which coincides with its rotation axis into the Teflon holder. The spectra and the different relaxation-rate measurements were recorded for a complete set of orientations θLS of the compression axis (noted n) with reference to the static magnetic field Β0 varying between 0° and 90°. The initial angle (θLS ) 0°) corresponds to an orientation of the compression axis of the dense clay sample parallel to the static magnetic field. The temperature variation51 of the 7Li longitudinal relaxation rate measured within these clay dispersions (cf. Figure 2) is fully compatible with a fast exchange between the free lithium counterions and those condensed in the vicinity of the clay platelets. To quantify the relative contributions from the quadrupolar and dipolar relaxation mechanisms (see Appendix), we measure simultaneously the time evolution of the longitudinal magnetization52 (T10 coherence) (Figure 3a), the transverse magnetization47 (T11 coherence) (Figure 3b) and the triple-quantum T33 coherence48 (Figure 3c). The intensity of the T33 coherence is proportional to the T31 coherence48 initially generated by the pulse sequence (cf. Figure 1c). The maximum signal/noise ratio is

J. Phys. Chem. C, Vol. 112, No. 26, 2008 9809

Figure 1. Schematic view of the pulse sequences used for the 7Li NMR measurements: (a) transverse relaxation T2 (Hahn echo), (b) triplequantum filtering of the evolution of the T31 coherence, and (c) triplequantum filtering of the evolution of the T33 coherence.

Figure 2. Temperature variation of the 7Li longitudinal relaxation rate R1 within Laponite dispersion (128% w/w).

obtained by optimizing the free delay δopt of that pulse sequence. In the absence of a static quadrupolar coupling (ωQ ) 0), the delay δopt ) δ0 is optimized by using the classical formula48:

δ0 )

ln (Rf2 ⁄ Rs2) Rf2 - Rs2

(1)

As illustrated in Figure 3d, that relationship largely reduces the intensity of the T33 coherence detectable in the presence of a residual quadrupolar coupling ωQ. Although the time evolution of the longitudinal magnetization (cf. Figure 3a) is described perfectly by a single-exponential law, the transverse magnetization evolves according to a biexponential law2,3

Mx(t) ) T11(t) ) 0.6 cos(ωQ t) exp(-Rf2 t) + 0.4 exp(-Rs2 t) (2) with the 60:40 ratio between the fast and slow components. This behavior is a direct consequence of a slow modulation53

9810 J. Phys. Chem. C, Vol. 112, No. 26, 2008

Porion et al.

Figure 5. 7Li NMR spectra measured within Laponite dispersions as a function of the clay concentration at the orientation θLS ) 0° by reference with the static magnetic field B0.

Figure 3. 7Li relaxation rates measured within dense clay sediment (c ) 128% w/w) as a function of the orientation θLS of the clay sample within the static magnetic field B0. The continuous lines are obtained by a simultaneous fit of the different relaxation measurements (see text): (a) for the T1 longitudinal relaxation, (b) for the transverse T2 relaxation (Hahn echo), (c) for the relaxation of the T33 coherence, and (d) for the relaxation of the T31 coherence, the delays δ0 and δopt are defined in the text.

Figure 4. Snapshot illustrating an equilibrium configuration of the water molecules and lithium cations confined between two Laponite clay surfaces.

of the quadrupolar coupling. Note that the same fast and slow components monitor the time evolution of the T31 coherence (cf. Figure 3d), the only difference being their relative weight48:

T31(t) ) 0.5 exp(-Rs2 t) - 0.5 exp(-Rf2 t) cos(ωQ t) (3) 3. Multiscale Numerical Modeling. We first performed a detailed molecular analysis of the structure and dynamics of the water molecules and neutralizing Li+ cations highly confined between two Laponite particles with a period of 30 Å, corresponding to a clay concentration of 126% w/w. Grand canonical Monte Carlo44 (GCMC) simulations are first used to

Figure 6. 7Li NMR spectra measured within Laponite dispersions (c ) 128% w/w) as a function of the orientation θLS of the clay sediment by reference with the static magnetic field B0.

determine the density of these confined water molecules in equilibrium with a reservoir of bulk water. Figure 4 illustrates a snapshot of the confined water molecules and Li+ counterions. Because of the asymmetry of the solvation sphere of the condensed lithium counterions, we detect a residual value of their electrostatic field gradient with the principal axis oriented B ) (6 ( 2) × 1018 V/m2). perpendicular to the clay surface (Vzz It corresponds to a maximum quadrupolar splitting (eq A3) of 2 kHz when all of the Li+ counterions are adsorbed on the basal surface of the Laponite clay particles, which are fully oriented in the static magnetic field (eq A6). The final equilibrium configurations of the water molecules and lithium counterions obtained by GCMC simulations are used to initialize simulations

7Li



of molecular dynamics44,45 (MD) also performed at a molecular level. The trajectories were integrated by using the Verlet algorithm44 modified to describe water rotation by using quaternions.45,54 The time step of the MD simulations was 1 fs, and a Berendsen thermostat55 was used to stabilize the temperature at 298 ( 5 K. The MD trajectories were exploited to determine the memory functions describing the decorrelation of the fluctuating part of the quadrupolar and dipolar Hamiltonian felt by the confined lithium counterions. These GCMC and MD molecular simulations of the clay/water interface were performed by using the classical clay force field.56 MD simulations are used to investigate the trajectories of the water molecules and lithium cations under highly confined conditions because the size of the spatial domain spatial domain does not exceed 30 Å, that is, 10 times smaller than the diameter of an individual Laponite platelet.42 Numerical simulations on a larger scale are required to investigate the mobility of the lithium cations around many clay particles in order to probe the influence of their relative organization on the lithium mobility and relaxation behavior. Numerical simulations of Brownian dynamics are thus performed in order to bridge the gap between the time scale (and size) accessible by MD simulations and that probed by NMR relaxation measurements. In a first approach, we remove the water molecules and consider dispersions containing only 10 platelets (diameter 300 Å), each neutralized by 1000 lithium counterions in order to mimic Laponite clay in the framework of the Primitive model.57We integrate the trajectories of these 10000 counterions (plus eventual added salt) by using a generalized Langevin algorithm:46

b xn+1 ) b xn +

b b Vn Fn 1 - e-γ ∆t (1 - e-γ ∆t) + (∆t )+b Xn γ mγ γ (4a)

xn) Fn γ(x bn+1 - b A B b f b Vn+1 ) b Vn + + (1 - e-γ ∆t)2 + V n C Cm γ C (4b)

performed with a time step of 0.1 ps, corresponding to a product γ ∆t ) 39.3, significantly larger than unity. The deterministic force b Fnin eq 4a and b includes the longrange electrostatic interaction acting on the lithium counterions plus a short-range repulsion described by a r-12 power law fitted to reproduce57the finite size of the solvated lithium cation with a radius59 of 2.5 Å. The clay/ion electrostatic and short-range repulsion is described by uniformly distributing 2828 interaction sites on a squared network located in the equatorial plane of the clay particles. Ewald summation60 is used to reproduce the long range of the Coulomb potential cut by the minimum image procedure. These BD simulations were performed to analyze the lithium trajectories in the vicinity of the adsorbing clay platelet but not to determine the memory of their quadrupolar coupling. The simulation cell is large enough (a few hundred angstroms) to contain more than one clay particle, but the diffusion time is too short (less than 10 ns) to really describe the exchange of the lithium counterions condensed on clay platelets with different orientations. Furthermore, the number of clay platelets (10) is not large enough to sample satisfactorily all the clay orientations compatible with a given order parameter of the clay dispersion. However, these BD simulations are useful to determine the average residence time of the lithium counterions in the vicinity of the basal surfaces of the clay. More qualitative information is obtained by simulating the trajectories of freely diffusing probes with elastic collisions at contact with rigid platelets mimicking the clay particles. The ion/ion interactions are ignored, and the complex clay/ion interactions are simply reproduced by an average residence time of the diffusing probe at the surface of the platelets. Thanks to these crude approximations, we can consider a large number (270) of solid platelets (diameter 300 Å, thickness 10 Å) randomly distributed in a large simulation cell (size ∼ 0.1 µm). The Langevin equation46 is then used to simulate the trajectories of 20000 probes

f b xn+1 ) b xn + X n

where

A ) 2 γ ∆t e-γ ∆t - 1 + e-2γ ∆t -2 γ ∆t

B ) γ ∆t (1 - e

(4c)

-γ ∆t 2

) - 2 (1 - e -γ ∆t

C ) 2 γ ∆t - 3 + 4 e

)

-2 γ ∆t

-e

(4d) (4e)

The two independent random functions satisfy

bn) R 〉 ) 〈 (V bn) β 〉 ) 〈 (X bn) R (V bn) β 〉 ) 0 〈 (X bn) 2R 〉 ) k T C 〈 (X m γ2 bn) 2R 〉 ) 2 k T B 〈 (V mC

(4f) (4g) (4h)

with R,β ∈ {x,y,z}, where the friction coefficient is defined by

γ)

kT mD

(4i)

We used two different values of the self-diffusion coefficient of the lithium cations (D ) 0.7 × 10-9 and 0.9 × 10-9 m2/s), corresponding to the range of the lithium mobility measured in diluted LiCl solutions by 7Li NMR exploiting echo attenuation with pulsed field gradient.58 The integration of eq 4a-i is

(5a)

where the random force satisfies

bn) 2R 〉 ) 2 k T ∆t ) 2 D ∆t 〈 (X mγ

(5b)

Integration of eq 5a and b is performed with a diffusion coefficient D ) 9 × 10-10 m2/s and a time step of 1 ps during a time period (τ) larger than 1 µs. The corresponding mean free path (L ) 2 D τ ) 0.04 µm) is now large enough to really probe the organization of the clay particles within a partially ordered nematic dispersion and its influence on the mobility and relaxation behavior of the ionic diffusing probes. The quadrupolar Hamiltonian felt by these diffusing probes is also very simplified and assumed to be either zero in the bulk or the maximum splitting previously detected by GCMC simulations (VBzz) modulated by the clay orientation (eq A4) when the lithium probe is physisorbed.5,41 Because of the fluctuation of the orientations of the clay platelets into the dispersion, the decorrelation of the quadrupolar coupling of the diffusing probes is a fingerprint of the fluctuation of the clay directors around their average orientation into a nematic dispersion:41


Porion et al.

B 2 GmQ,L(τ) ) (Vzz ) ×

[

Ni

Np Np

2

∑∑∑ ∑

1 LPj LPs P(i ∈ j|0) P(i ∈ s|τ) Dlm D-l-m Ni i)1 j)1 s)1 l)-2

( ∑∑ ∑ Ni

Np

2

1 LPj P(i ∈ j|0) Dlm Ni i)1 j)1 l)-2

)]

2 2

(6)

where Ni is the number of diffusing probes, Np is the number of platelets, and P(i ∈ j|τ) is the probability that the probe labeled i is adsorbed on the platelet labeled j at time τ. The orientation of the platelet labeled j in the static magnetic field is described by the Wigner rotation matrices.61 This treatment is fully justified by the fast exchange (cf. Figure 2) between the two spin populations corresponding, respectively, to the free and condensed lithium counterions. III. Results and Discussion 1. 7Li NMR Spectra and Quadrupolar Splitting. Figure 5 displays the 7Li NMR spectra recorded within sediments at various clay concentrations (55.8, 93.5, 112, and 128% w/w) prepared by oedometric compression with an external pressure varying between 0.56 and 2 MPa. As shown by the molecular modeling of the clay/water interface (cf. Figure 4), the splitting of the 7Li NMR resonance line is the fingerprint of the asymmetry of the electric environment of the condensed lithium counterions modulated by two parameters (cf. eq A6): the fraction of lithium cations at contact with the clay surface and the ordering of the individual clay particles within the dense sediments. The residual splitting varies according to the (3 cos2 θLS-1)/2 relationship (cf. Figure 6) as expected for macroscopic nematic dispersions. The perfect cancelation of the 7Li splitting detected at the so-called magic angle results from the existence of a single nematic phase within the whole clay sample. The maximum measured splitting (1.3 kHz in Figures 5 and 6) is significantly smaller than the value predicted by our molecular modeling (2 kHz) when all Li+ cations are condensed on the clay surface and all of the clay directors are aligned along the same direction. In order to further check the order of magnitude agreement between the predicted and measured splitting, we have prepared a more concentrated sample by water evaporation from a previously compressed sediment (noted “air-dried” sample in the figures). As expected, the measured splitting (4 kHz in Figure 5) is twice the predicted value because of the

() (

reduction of the interlamellar separation. However, in order to remove the possible contribution from the heterogeneities of the static magnetic field, we do not exploit these NMR spectra and rather perform relaxation measurements by using Hahn Echo attenuation.47 2. 7Li NMR Relaxation. Table 1 exhibits the apparent relaxation rates describing the time evolution of the T10, T11, and T33 coherences as a function of the clay concentration. Because two components of the transverse magnetization are detected with the appropriate 60:40 ratio (see eq 2), the time evolution of the transverse magnetization is compatible with a slow modulation of the quadrupolar coupling (cf. Appendix). The same behavior was reported by 23Na NMR relaxation studies5,6 within the aqueous dispersions of Laponite clay neutralized by sodium counterions. As confirmed by the frequency variation of the 23Na relaxation rates,6,40 the selfdiffusion of the neutralizing counterions (cf. eq 6) is responsible for the slow modulation of their quadupolar coupling. By contrast with the predictions from the theory of NMR relaxation of these 3/2 spin nuclei (cf. Appendix), a single longitudinal relaxation time is detected because of the slow modulation of Q,S Q,S the quadrupolar coupling (JQ,S 0 (0) > J1 (ω0) > J2 (2ω0)). Finally, the time evolution of the T33 coherence is monoexponential48 (cf. Figure 3c), with a relaxation rate much larger than the slow component of the transverse magnetization (see Table 1). As illustrated in the Appendix, this is the fingerprint of a relaxation mechanism acting in competition with the quadrupolar coupling (see eq A14a and b and eq A18a and b). To further illustrate the efficiency of the heterogeneous dipolar coupling and explain that behavior, let us simplify the set of eq A21a-c by assuming D,L a slow modulation of the dipolar coupling (JD,L 0 (0) . J0 (ωI D,L D,L - ωS) ≈ J1 ≈ J2 ), leading to the master equation describing the time evolution of the set of pertinent coherences under the combined influence of the quadrupolar and dipolar couplings as shown in eq 7 with JmQ ≡ JmQ,L for m ) 0-2, and JD0 (0) ≡ JD,L 0 (0). The apparent relaxation rate of the T33 coherence (RL33 ) JQ,L 1 + JQ,L + 2 JD,L 2 0 ) may become larger than the slow component Q,L D,L 2 of the transverse magnetization (Rs2 ) JQ,L 1 + J2 + /9 J0 ) or f Q,L Q,L Q,L 2 even the fast component (R2 ) J0 + J1 + J2 + /9 JD,L 0 ) if Q,L JD,L 0 (0) is large enough compared to J0 (0). By numerically evaluating the eigenvalues and corresponding eigenvectors of the matrices displayed in eqs A13a and b, A14a and b, and A21a and b, it is possible to simulate the time

Tˆ11(s) Tˆ (a)

d 21 )dt Tˆ31(s) Tˆ (s) 33

3 / 5JQ0 + JQ1 + 2 / 5JQ2 + 2 / 9JD0 (0) i √3 / 5 ωQ √6 / 5 (JQ0 - JQ2 ) 0

√6 / 5 (JQ0 - JQ2 ) i √2 / 5 ωQ

i √3 / 5 ωQ JQ0 + JQ1 + 2

JQ2 + 2 /

i √2 / 5 ωQ 0

9JD0 (0) 2

/

5JQ0 + JQ1 + 3 / 0

5JQ2 + 2 /

0 0 9JD0 (0)

0

)

·

()

JQ1 + JQ2 + 2 JD0 (0)

Tˆ11(s) Tˆ (a) 21

Tˆ31(s) Tˆ (s) 33

(7)

7Li



TABLE 1: Apparent 7Li NMR Relaxation Rates R1, Rf2, Rs2, and R33 and Static Quadrupolar Splitting νQ as a Function of the Laponite Concentration, at the Orientations θLS ) 0° and θLS ) 90° by Reference with the Static Magnetic Field B0 sample 3.3% w/w isotropic 55.8% w/w 93.5% w/w 112% w/w 128% w/w air-dried

θLS angle (degree)

R1(s-1)

Rf2 (s-1)

Rs2 (s-1)

R33 (s-1)

νQ (Hz)

0 90 0 90 0 90 0 90 0 90

5.0 6.6 5.3 7.5 5.6 8.1 6.2 8.5 6.5 21.4 15.5

104 222 156 213 149 286 192 278 179 1111 435

26 64 55 76 64 90 74 94 72 313 159

208 139 270 164 313 182 357 196 1667 500

0 673 333 936 480 1137 572 1314 641 4079 2221

Q,S D,S TABLE 2: Intrinsic Parameters (Jm (0), Jm (0), and νQ) Extracted from the Angular Variations of the T10, T11, and T33 Relaxation Rates

sample

JQ,S (0) (s-1) 0

JQ,S (0) (s-1) 1

JQ,S (0) (s-1) 2

JD,S (0) (s-1) 0

JD,S (0) (s-1) 1

JD,S (0) 2

νQ (Hz)

55.8% w/w 93.5% w/w 112% w/w 128% w/w Air dried

160 ( 20 150 ( 20 210 ( 20 180 ( 20 750 ( 100

330 ( 40 220 ( 40 350 ( 40 260 ( 40 3000 ( 300

80 ( 20 90 ( 20 120 ( 20 90 ( 20 250 ( 40

100 ( 20 115 ( 20 170 ( 25 170 ( 25 860 ( 100

70 ( 20 90 ( 20 70 ( 20 85 ( 20 500 ( 100

50 ( 10 50 ( 10 65 ( 10 60 ( 10 100 ( 20

680 ( 10 940 ( 15 1140 ( 15 1320 ( 15 4200 ( 80

evolution of the T10, T11, and T33 coherences during the elementary steps of the pulse sequence and the phase cycling (cf. Figure 1) used to determine the longitudinal relaxation rate (T10), the Hahn echo attenuation (T11) and the triple-quantum relaxation (T33), respectively. A Simplex procedure62 is used to simultaneously fit the experimental data by exploiting a generalized mean-squared minimization. For each clay sample and orientation, the same minimum set of parameters is fit by Q,L including JQ,L 0 (0), J1 (ωI) for the quadrupolar mechanism and D,L D,L J0 (0), J0 (ωI - ωS), JD,L 1 (ωS) for the dipolar mechanism. In order to reduce the total number of fitted parameters, we assume Q,L D,L D,L D,L JQ,L 2 (2ωI) ) J1 (ωI) and J2 (ωI + ωS) ) J1 (ωI) ) J1 (ωS). That approximation is validated by the slow modulation of the quadrupolar and dipolar couplings, which leads to spectral densities at zero frequency much larger than the same spectral densities at higher frequencies. Typical results of this fit are displayed in Figure 3a-d. No additional parameter is required to also reproduce the time evolution of the T31 coherence obtained by triple-quantum filtering. Because the analysis of the time evolution of the T31 coherence does not add new dynamical information to the set of data extracted from the time evolution of the T10, T11, and T33 coherences, this last measurement is not treated in the fitting procedure. This measurement (T31) is simply exploited to optimize the delay δopt used for the detection of the T33 coherence (cf. Figure 1c). Generally, the optimum delay δopt is much smaller than the delay δ0 evaluated by eq 1, which assumes the cancelation of the static quadrupolar coupling, largely enhancing the intensity of the detected T33 coherence (cf. Figure 3d). Figure 7 displays the variation of the fitted 7Li splitting as a function of the clay concentration and sample orientation, in perfect agreement with the expected relationship (cf. eq A6). The maximum 7Li splitting increases as a function of the clay concentration because of two simultaneous phenomena: the increase of the fraction of condensed lithium counterions and the improvement of the alignment of the clay platelets along their nematic director. In addition, Table 2 illustrates the angular variation of the two dominant contributions to the 7Li NMR D,L relaxation, that is, JQ,L 0 (0) and J0 (0). As detailed in eq A17a, the angular variation of these apparent spectral densities may be used to extract the set of corresponding intrinsic16,63 spectral X,S X,S densities (JX,S 0 (0), J1 (0), J2 (0), with X ∈ {Q, D}) evaluated

in the frame of the sediment (noted S). In the framework of the two-state model, both these intrinsic spectral densities and the detected splitting are proportional to the fraction of condensed lithium counterions. Although the three intrinsic spectral densities must coincide in homogeneous samples, we detect a large difference between these intrinsic spectral densities in the dense Q,S Q,S D,S D,S clay sediments: JQ,S 1 (0) > J0 (0) > J2 (0) and J0 (0) > J1 (0) D,S 41 > J2 (0). As reported previously, the increase of the difference Q,S (JQ,S 1 (0) - J2 (0)) as a function of the detected splitting is the fingerprint of the increase of the ordering within the nematic dispersion as a function of the clay concentration. Unfortunately, because of its reduced order of magnitude -1 (40 s-1 < JD,L 0 (ωI - ωS) < 100 s ), the m ) 0 component of the dipolar heterogeneous coupling detected at high frequency appears to be independent of the clay orientation. Finally, the other components of the quadrupolar (JQ,L 1 (ωI)) and dipolar -1 (JD,L 1 (ωS)) couplings are strongly reduced ( G0 (τ) > G2 (τ)), in agreement with the intrinsic spectral densities extracted from data analysis (cf. Table 2). That difference between the intrinsic memory functions of the quadrupolar coupling was already reported in the case of the quadrupolar relaxation of 23Na counterions within nematic dispersions41and was shown to be strongly dependent on the degree of orientational ordering of the platelets. Because the m ) 1 and m ) 2 components decrease according to a τ-1.5 power law (cf. Figure 12b), their contribution to the intrinsic spectral densities evaluated at zero frequency are evaluated easily, -1 and JQ,S(0) ≈ 52 s-1. The leading to JQ,S 1 (0) ≈ 400 s 2 determination of the m ) 0 component of the spectral density evaluated at zero frequency is more delicate to perform because

Rf2 ≈ JQ,S 0 (0) ≈

1 tsup

)

∫0t

sup

GQ,S 0 (τ) dτ

(8)

-1 leading to a rough estimate (JQ,S 0 (0) ≈ 300 s ). Despite of the simplicity of our numerical model, the predicted data reach the orders of magnitude of the intrinsic quadrupolar spectral densities extracted from the experimental data and also their Q,S Q,S sequence (JQ,S 1 (0) > J0 (0) > J2 (0)) (cf. Table 2). This agreement is only semiquantitative because our BD simulations are somewhat simplified. Furthermore, the evaluation used in eq 8 introduces an overlap between the time scales characterizing the decorrelation of the quadrupolar coupling on one hand, and the time evolution of the coherences on the other hand. We reach the limit of the validity of the Redfield theory52 for treating NMR relaxation (cf. the Appendix). Heterogeneous Dipolar Relaxation. In addition to the quadrupolar coupling (cf. Table 2), the analysis of our 7Li NMR relaxation measurements within dense clay sediments clearly exhibits a significant contribution from heterogeneous dipolar coupling. By comparison with the quadrupolar relaxation detected previously by 23Na NMR spectroscopy37,38 under similar conditions, this result is not surprising because of two complementary properties: (1) the relative weakness of the 7Li quadrupolar relaxation mechanism ((Q7Li(1 + γ∞)7Li/Q23Na(1 + γ∞)23Na)2 ≈ 3.3 × 10-3);1,43 (2) the relative strength of the 7Li heterogeneous dipolar relaxation mechanism ((γ7Li/γ23Na))2 ) 2.15).1 From the previous MD trajectories, one can determine the contribution to the NMR relaxation of the neutralizing lithium counterions resulting from the 7Li-1H dipolar coupling between the condensed lithium cations and the confined water molecules. The three components describing that heterogeneous dipolar coupling are equal and decrease at long time according to a t-1.5 power law (cf. Figure 13), as expected for short-time diffusion within a 3D space.64–66 The resulting intrinsic spectral densities are negligible (JmD,S(0) ≈ 0.015 s-1, m ) 0-2) compared to the intrinsic dipolar spectral densities extracted


Figure 14. Average residence time of the water molecules pertaining to the first hydration shell of the neutralizing lithium counterions molecules as predicted by the MD simulations.

Porion et al.

Figure 16. Memory functions GmD,S(τ) describing the heterogeneous dipolar coupling of the 7Li counterions induced the iron atoms of the clay particle as predicted by the MD trajectories.

densities describing the lithium heterogeneous dipolar coupling induced by the iron atoms of the Laponite clays (∼0.045 atom per unit cell). Surprisingly enough, the order of magnitude of -1 D,S the predicted values (cf. Figure 16) (JD,S 0 (0) ≈ 290 s , J1 (0) D,S -1 -1 ≈ 180 s , J2 (0) ≈ 30 s ) reaches the order of magnitude of the intrinsic dipolar coupling extracted from the analysis of the experimental data (cf. Table 2) and also reproduces the sequence D,S D,S of the spectral densities (JD,S 0 (0) > J1 (0) > J2 (0)). IV. Conclusions

Figure 15. Memory functions GmD,S(τ) describing the heterogeneous dipolar coupling of the 7Li counterions induced the protons of the clay particle as predicted by the MD trajectories.

from the experimental data (cf. Table 2). One may be tempted to apply the same multiscale modeling as in the case of the quadrupolar coupling. Such an interpretation of the gap between the measured and predicted dipolar relaxation rates is less obvious because the dipolar coupling is modulated not only by the local orientation of the clay platelets within the dispersion but also by the instantaneous separation between the condensed lithium cation and the protons of its initially solvating water molecule. If the residence time of these water molecules within the first hydration shell of the confined lithium cations is much larger than the average residence time of the lithium cations on the clay basal surface (i.e., 3 ns), then we can describe the longtime modulation of the lithium-proton heterogeneous dipolar coupling in the same manner as the 7Li quadrupolar coupling. Unfortunately, as displayed in Figure 14, this approach is hopeless because the average residence time of the water molecules in the first hydration layer of the condensed lithium counterions does not exceed 20 ps! The protons located within the hexagonal cavities at the surface of the Laponite clay are another source of heterogeneous dipolar relaxation of the confined lithium counterions. As displayed in Figure 15, the three components are now different but still evolve at long time according to a t-1.5 power law. Here also, the corresponding intrinsic spectral densities are -3 s-1,JD,S(0) ≈ 1.4 × 10-3 s-1 negligible (JD,S 0 (0) ≈ 2.2 × 10 1 D,S -4 -1 and J2 (0) ≈ 2.5 × 10 s ). As a consequence, one may neglect the contribution from all of the protons of the clay/ water interface as a possible source of efficient dipolar relaxation of the condensed lithium counterions. We finally exploit the same molecular trajectories to evaluate the intrinsic spectral

We have exploited the great sensitivity of the 7Li NMR spectroscopy and the multiquantum relaxation measurements to investigate the organization of individual clay platelets within dense aqueous dispersions and determine the mobility of the Li+ counterions diffusing through the porous network limited by the solid/liquid clay/water interfaces. Such structural and dynamical studies may be performed within a large class of heterogeneous material involving electrically charged solid/ liquid interfaces (like porous glasses, colloidal dispersions, solutions of DNA or synthetic polyions, and fuel cell membranes). Because of the weakness of the 7Li quadrupolar coupling, heterogeneous dipolar coupling was shown to significantly contribute to the lithium NMR relaxation in such heterogeneous media. Multiquantum relaxation measurements were performed to separately quantify the contributions from the quadrupolar and dipolar relaxation mechanisms by implying the zero-quantum T10 coherence, the single-quantum T11 coherence, and the triple-quantum T33 coherence. The intrinsic components of these apparent quadrupolar and dipolar relaxation rates were further extracted by carefully analyzing the variations of these apparent relaxation rates as a function of the orientation of the nematic dispersion within the static magnetic field. Finally, multiscale modeling was performed to semiquantitatively interpret these relaxation measurements by focusing on the intercorrelations between the microstructure of the clay dispersion, the dynamics of the Li+ cations, and their NMR relaxation behavior. Appendix NMR Relaxation Theory of 3/2 Spin Nuclei. Static Quadrupolar Coupling. The quadrupolar Hamiltonian is defined by52,61,67–69

HQ )

3 e Q (1 + γ∞) 2 I (2I - 1) p

2

Q,L TIR ∑ (-1)p F2,-p 2,p

(A1a)

p)-2

where Q is the quadrupolar moment of the nuclei (-0.042 × 10-28 m2 for 7Li)43, (1 + γ∞) is the Steinhermer antishielding factor (0.74 for lithium43),

7Li



1 L 1 V , FQ,L ) (VLx z ( i VLy z) 2 z z 2,(1 √6 1 FQ,L (VLx x - VLy y ( 2 i VLx y) (A1b) 2,(2 ) 2√6 1 2 and TIR (3Iz - I(I + 1)), 2,0 ) √6 1 1 2 IR TIR 2,(1 ) - (IzI( + I(Iz), T 2,(2 ) I( (A1c) 2 2

dispersion. The environment of the free lithium cations is isotropic (ωFQ ) 0). The first Wigner rotation matrix describes the orientation of the macroscopic clay sample with respect to the magnetic field, and the last factor characterizes the average orientation of the clay particles within the dispersion. For clay dispersions with cylindrical symmetry, only the component p ) 0 contributes to eq A5, which reduces to

where VLR β are the components of the electrostatic field gradient evaluated in the laboratory frame (noted L), TIR 2,(p (for p ) -2 to 2) are the second order irreducible tensors operators,61,67Ix, Iy, and Iz are the spin operators and I( ) Ix ( i Iy. In the presence of a static quadrupolar coupling, the equidistant Zeeman energy levels are modified by the residual quadrupolar coupling whose first-order approximation gives

leading to a quadrupolar splitting of the resonance lines according to

As a consequence, the intrinsic splitting characterizing the clay dispersion is the maximum splitting measured when the nematic director of the dispersion coincides with the direction of the static magnetic field. However, it is not an absolute measurement of the degree of ordering of particles into the clay dispersion (〈(3 cos2 θSP - 1)/2〉) because of the possible variation of the fraction of condensed lithium counterions ( pB) as a function of the clay concentration. b. Quadrupolar Relaxation. In the framework of the Redfield theory,52,61,67–69the relaxation of the quadrupolar nuclei is related to the fluctuating part of the quadrupolar coupling, by using the master equation

3 e Q (1 + γ∞) L ωm-1,m ) 〈V 〉 (1 - 2 m) 4 I (2 I - 1) p z z

dσ* ) -i [HS*, σ*] + f(σ*) dt

FQ,L 2,0 )

HQ1 )

e Q (1 + γ∞) 〈VL 〉 ( 3 m2 - I (I + 1) ) 4 I (2 I - 1) p z z

(A2)

(A3)

for m varying between I and -I + 1 During a change of frame, the components of the field gradients transform like the second-order spherical harmonics61 2

FQ,L 2,q )

∑

LP FQ,P 2,p Dp,q(θ, φ, ψ)

(A4)

p)-2

where DLP p,q(θ,φ,ψ) are the components of the Wigner rotation matrix61defined by the set of (θ,φ,ψ) Euler angles describing the orientation, into the static magnetic field, of the principal axis of the tensor describing the electrostatic field gradient felt by the quadrupolar nucleus. Because the clay director coincides with the direction of the principal axis of the residual field gradient,40three sets of frames must be considered: the laboratory frame (noted L), a frame attached to the dense clay sediment (noted S) and a frame attached to the individual clay particles (noted P). The ez directors of these different frames are, respectively, the direction of the static magnetic field B0 (laboratory frame L), the direction of the compression axis n, which is parallel to the nematic director38,41(sediment frame S), and the director of the principal component of the electrostatic field gradient parallel to the director of the clay particle (particle frame P). In the framework of a two-state model with a fast exchange, at the NMR time scale, between the two spin populations, one deduces the measured quadrupolar splitting by using eqs A3 and A4

ωQapp ) pF ωQF + pB ωQB 2

) AQ pB VzBz

ωQapp ) AQ pB VzBz

∑ DLSp,0(θLS, φLS, ψLS) 〈DSP0,p(θSP, φSP, ψSP)〉

p)-2

(A5) where pF and pB are the fraction of free and condensed lithium counterions, respectively, and AQ ) (e Q (1 + γ∞))/(2 p) for spin 3/2 nuclei. The angular average is evaluated in eq A5 over all of the orientations of the clay particles within the

〈

〉

3 cos 2 θLS - 1 3 cos 2 θSP - 1 (A6) 2 2

(A7)

where all calculations are performed in the Larmor frequency rotating frame, as indicated by the asterisk. The first contribution to the master equation describes the influence of the static Hamiltonian HS* that includes the residual static * quadrupolar Hamiltonian HQS ) ωQ / 6(3Iz2 - I(I + 1)) and the Hamiltonian corresponding to the radio frequency pulse (H*1 S ) Ix ω1). The second contribution to the master equation results from the fluctuating components of the quadrupolar * Hamiltonian HQF (t):

f(σ*) ) -

∫0t

sup

* 〈 [HQF (t), *+ [ e-iHSτ HQF (t - τ) eiHSτ, σ*(t) ] ] 〉 dτ (A8) *

*

If the time scale characterizing the decorrelation of the quadrupolar Hamiltonian is much smaller than the time scale used to study the time evolution of the coherences, then the upper limit of the integral in eq A8 may be set equal to infinity.52That hypothesis restricts the validity of the Redfield theory of NMR relaxation. For spin 3/2 nuclei, this fluctuating component of the quadrupolar Hamiltonian becomes 2

* HQF (t) ) CQ

∑

m ) -2

IR Q,L (-1)m T2,m ei m ω0 t( F2,-m (t) Q,L 〈F2,-m (t)〉 ) (A9)

with CQ ) eQ(1 + γ∞) / √6p IR , with a The orthonormal tensor operators2–4 (Tˆ l,m ) al Tl,m 0 ) 1/2, a1 ) 1/5, a2 ) 1/6 and a3 ) 2/3 for I ) 3/2) are used as a complete basis set to solve eqs A7–A9 in order to calculate the time evolution of the different coherences. Symmetric and antisymmetric combinations of the operators are also introduced2,3

1 ˆ 1 ˆ Tˆl,m(s) ) (Tl,-m + Tˆl,m) and Tˆl,m(a) ) (Tl,-m - Tˆl,m) √2 √2 (A10) simplifying the notations of the magnetization


Porion et al.

Ix ) √5 Tˆ1,1(a);

Iz ) √5 Tˆ1,0

(A11)

while the residual quadrupolar Hamiltonian reduces to

HQS )

ωQ 2 ( 3Iz - I(I + 1) ) ) ωQ Tˆ2,0 6

(A12)

For spin 3/2 nuclei, 16 spin operators represented by irreducible tensors are required to fully describe the spin dynamics by including the identity (Tˆ0,0) , the longitudinal (Tˆ1,0) , and transverse (Tˆ1,1(s), Tˆ1,1(a)) components of the magnetization, the quadrupolar spin polarization (Tˆ2,0), the second-order single-quantum coherences (Tˆ2,1(s), Tˆ2,1(a)), the second-order double-quantum coherences (Tˆ2,2(s), Tˆ2,2(a)), the octopolar spin polarization (Tˆ3,0), the third-order single-quantum coherences (Tˆ3,1(s), Tˆ3,1(a)), the third-order double-quantum coherences (Tˆ3,2(s), Tˆ3,2(a)), and the third-order triple-quantum coherences (Tˆ3,3(s), Tˆ3,3(a)). By using this complete set of coherences (noted σ), the first component of the master equation (eq A7) may be translated into a matrix form, leading to2,3

( )( Tˆ11(a) Tˆ

0

0

√3 ⁄ 5 ωQ - √3 ω1

0

0

0

)( ) Tˆ11(a) Tˆ

0

0 0 0 0 0 0 20 20 ˆT (s) ˆ T21(s) -ω1 √2 ⁄ 5 ωQ √3 ⁄ 5 ωQ - √3 ω1 0 0 0 21 d ˆ ω -ω 0 0 0 · Tˆ22(s) 0 0 T (s) ) i 1 Q dt 22 √2 ⁄ 5 ωQ 0 0 0 0 0 - √5 ⁄ 2 ω1 Tˆ31(a) Tˆ31(a) ωQ - √5 ⁄ 2 ω1 0 0 0 0 - √3 ⁄ 2 ω1 Tˆ32(a) Tˆ32(a) 0 0 0 0 0 0 - √3 ⁄ 2 ω1 Tˆ33(a) Tˆ33(a)

( )( Tˆ10

Tˆ11(s) Tˆ (a) 21

ˆ d T22(a) )i dt Tˆ30 Tˆ (s) 31

() (

Tˆ32(s) Tˆ (s) 33

0 -ω1

-ω1 0

0 √3 ⁄ 5ωQ

0 0 0 0 0

√3 ⁄ 5ωQ 0

0 -ω1

0

0 0

0 0

0 0

0 0

0

0 0

√2 ⁄ 5ωQ 0

0 ωQ

0 0

0

0

0

√6ω1

0

0

0

√2 ⁄ 5ωQ

√6ω1

0

- √5 ⁄ 2ω1

0

0

0

0 ωQ

0

- √5 ⁄ 2ω1

0

- √3 ⁄ 2ω1

0

- √3 ⁄ 2ω1

0

0

0

)( ) Tˆ10

0 0

0

0 0 -ω1

(A13a)

Tˆ11(s) Tˆ (a) 21

Tˆ22(a) · Tˆ

(A13b)

30

Tˆ31(s) Tˆ (s) 32

Tˆ33(s)

0

during an irradiation along Ix. The second contribution to the master equation (eq A7) resulting from the fluctuating part of the quadrupolar Hamiltonian leads to the matrices2,3

Tˆ11(a) Tˆ 20

Tˆ21(s) d ˆ T (s) ) dt 22 Tˆ (a) 31

Tˆ32(a) Tˆ (a) 33

3 / 5JQ0 + JQ1 + 2 / 5JQ2

0

0

2JQ1 + 2JQ2

0

0

0

0

JQ0 + JQ1 + 2JQ2

0

0

0

0

JQ0 + 2JQ1 + JQ2

-

√6 /

and,

5(JQ0 - JQ2 )

0

0

0

√6 / 5(JQ0 - JQ2 ) 0 0 0

0

0

2

Q / 5JQ / 0 + J1 + 3

5JQ2

0

0

0

0

0

0

0

0

0

0 0 JQ1 + JQ2

0

0

0

0

0

JQ0 + JQ2

0

0

0

0

0

0

)( ) Tˆ11(a) Tˆ 20

Tˆ21(s) · Tˆ22(s) Tˆ (a) 31

Tˆ32(a) Tˆ (a) 33

(A14a)

() (

7Li



Tˆ10 ˆT (s) 11 Tˆ (a) 21

ˆ d T22(a) ) dt Tˆ30 Tˆ (s) 31

Tˆ32(s) Tˆ (s) 33

2 / 5J1Q + 8 / 5J2Q

-

0 3

0

/ 5J0Q + J1Q + 2 /

0

0 5J2Q

0

4 / 5(J1Q - J2Q)

0

0

√6 /

5(J0Q - J2Q)

0

0

0

0

0

0

0

J0Q + J1Q + 2J2Q

0

0

0

0

0

0

0

0

J0Q + 2J1Q + J2Q

0

0

0

0

4 / 5(J1Q - J2Q)

0

0

0

8 / 5J1Q + 2 / 5J2Q

0

0

0

0

√6 / 5(J0Q - J2Q)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

2

0

/

5J0Q + J1Q + 3 / 0

5J2Q

()

J0Q + J2Q

0

0

J1Q + J2Q

0

Tˆ10 ˆT (s) 11 Tˆ (a)

)

21

Tˆ22(a) · Tˆ

(A14b)

30

Tˆ31(s) Tˆ (s) 32

Tˆ33(s)

with JmQ ≡ JmQ,L for m ) 0-2. The spectral densities displayed in eq A14a and b are evaluated in the laboratory frame

JmQ,L(mω0) ) 6CQ2

∫0∞ GmQ,L(τ)eimω τdτ 0

(A15)

by using the memory functions describing the apparent decorrelation of the various component of the quadrupolar Hamiltonian evaluated in the laboratory frame: Q,L Q,L Q,L Q,L Q,L Q,L Q,L Q,L GmQ,L(τ) ) 〈(F2,m (0) - 〈F2,m 〉)(F2,-m (τ) - 〈F2,-m 〉)〉 + 〈(F2,-m (0) - 〈F2,-m 〉)(F2,m (τ) - 〈F2,m 〉)〉

By using the Wigner rotation matrices61(cf. eq A4), it is possible to relate the derivation of these apparent correlation functions evaluated in the laboratory frame (noted L) to their intrinsic63 value evaluated in the frame attached to the clay dispersion (noted S):

1 2 LS 2 Q,S GQ,L 0 (τ) ) (1 - 3 cos θ ) G0 (τ) + 4 3 2 LS 2 Q,S 3 cos 2 θLS sin 2 θLSGQ,S 1 (τ) + (1 - cos θ ) G2 (τ) 4 (A17a) GQ,L 1 (τ) )

3 1 2 LS cos 2 θLS sin 2 θLS GQ,S 0 (τ) + (1 - 3 cos θ + 2 2

1 Q,S 4 LS 4 cos 4 θLS) GQ,S 1 (τ) + (1 - cos θ ) G2 (τ) (A17b) 2

(A16)

3 2 LS 2 Q,S GQ,L 2 (τ) ) (1 - cos θ ) G0 (τ) 8 1 1 2 LS + (1 - cos 4 θLS) GQ,S 1 (τ) + (1 + 6 cos θ + 2 8 cos 4 θLS) GQ,S 2 (τ) (A17c) In the presence of a residual quadrupolar splitting, eqs A13 and A14 lead to a triplet in the NMR resonance line (cf. Figure 5). Under slow modulation of the quadrupolar coupling, the spectral densities satisfy the inequality JQ,L 0 (0) Q,L > JQ,L 1 (ω0) ≈ J2 (2ω0) and the time evolution of the transverse magnetization detected during a Hahn echo experiment is biexponential2,3,53 (cf. Figure 3b). The two relaxation rates Rf2 and Rs2, respectively, the fast and the slow transverse relaxation rates, are then given by the equations


Rf2 )

Porion et al.

1 Q,L Q,L ) JQ,L 0 (0) + J1 (ω0) + J2 (2ω0) f T2

(A18a)

1 Q,L ) JQ,L 1 (ω0) + J2 (2ω0) Ts2

(A18b)

Rs2 )

while the fast component oscillates at the angular velocity ωQ. That biexponential behavior results from a coupling2,3 between the T11, T21, and T31 coherences described above (eqs A13 and A14). The longitudinal magnetization also evolves according to a biexponential behavior53with two relaxation rates satisfying:

Rf1 )

1 ) 2 JQ,L 1 (ω0) f T1

(A19a)

Rs1 )

1 ) 2 JQ,L 2 (ω0) Tf1

(A19b)

HD(t) ) CD

∑

(-1) m

m)-2

D D,L T2, m F2, -m(t)

(A20a)

rI3 S(t)

where the dipolar coupling constant is given by

CD ) -

µ0 γ γ p 4π I S

and the spin operators become61

(A20b)

(

1 1 1 2IzSz - (I+S- + I-S+) ) 2 TIR 10Sz + 2 √6 √6 1 IR (A20c) ( T1+1S- - TIR 1-1S+ ) √2 1 1 D S - √2 TIR T2, ( I S + I S ) ) - ( TIR (1 ) 1(1SZ ) 2 Z ( ( Z 2 10 ( (A20d)

TD2, 0 )

(

)

D and T2, (2 )

1 1 IR I S )T1(1S2 ( √2

)

(A20e)

D,L The functions F2, m(t) in eq A20a are related to the second spherical harmonics describing the reorientation of the vector joining the two coupled spin (noted rbIS (t)) by reference with the static magnetic field61

D,L F2,-m )

24π5 Y

Tˆ11(a) Tˆ 20

2,-m(ϑ, φ)

(A20f)

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. -3 the term rIS (t) in eq A20a).

() Tˆ11(a) Tˆ 20

Tˆ21(s) Tˆ21(s) d ˆ T (s) ) - diag (a1, a2, a3, a4, a5, a6, a7) · Tˆ22(s) dt 22 Tˆ31(a) Tˆ31(a) Tˆ (a) Tˆ (a)

() () 32

Tˆ33(a)

In contrast, if the relaxation mechanism is purely quadrupolar, then the time evolution of the T33 coherences is totally decoupled from those of the other coherences (eqs A13 and A14) in the absence of an irradiation. The T33 coherence evolves then according to a single exponential law48,70 with a relaxation rate exactly equal to the slow component of the transverse magnetization (eq A18b). Dipolar Relaxation. In addition to the quadrupolar coupling, the dipolar coupling may also be responsible for the NMR relaxation68,69,71 of 7Li. In addition to the paramagnetic impurities, protons from the water molecules or the clay network are also possible relaxation agents because of their proximity to the condensed and solvated lithium cations (cf. Figure 4). The corresponding heterogeneous dipolar Hamiltonian becomes52,61,67 2

()

By using the same basis set, it is possible to describe the contribution to the master equation (eq A7) of the fluctuating part of the dipolar coupling in the same manner as for the quadrupolar coupling (eqs A8 and A9), leading to the matrices

32

Tˆ33(a)

(A21a)

Tˆ10

Tˆ10

Tˆ11(s) Tˆ (a)

Tˆ11(s) Tˆ (a)

21

21

ˆ Tˆ22(a) d T22(a) ) - diag (a8, a1, a3, a4, a9, a5, a6, a7) · dt Tˆ30 Tˆ30 Tˆ31(s) Tˆ31(s) Tˆ (s) Tˆ (s) 32

Tˆ33(s)

32

Tˆ33(s)

(A21b)

where

2 1 1 1 a1 ) JD0 (0) + JD0 (ωS - ωI) + JD1 (ωS) + JD1 (ωI) + 9 18 3 6 1 D J (ω + ωI) (A21c) 3 2 S 1 a2 ) JD0 (ωS - ωI) + JD1 (ωI) + 2 JD2 (ωS + ωI) (A21d) 3 2 5 1 5 a3 ) JD0 (0) + JD0 (ωS - ωI) + JD1 (ωS) + JD1 (ωI) + 9 18 3 6 5 D J (ω + ωI) (A21e) 3 2 S 8 1 4 1 a4 ) JD0 (0) + JD0 (ωS - ωI) + JD1 (ωS) + JD1 (ωI) + 9 9 3 3 2 D J (ω + ωI) (A21f) 3 2 S 2 11 1 11 a5 ) JD0 (0) + JD0 (ωS - ωI) + JD1 (ωS) + JD1 (ωI) + 9 18 3 6 11 D J (ω + ωI) (A21g) 3 2 S 8 4 4 4 a6 ) JD0 (0) + JD0 (ωS - ωI) + JD1 (ωS) + JD1 (ωI) + 9 9 3 3 8 D J (ω + ωI) (A21h) 3 2 S 1 1 a7 ) 2 JD0 (0) + JD0 (ωS - ωI) + 3 JD1 (ωS) + JD1 (ωI) + 6 2 JD2 (ωS + ωI) (A21i) a8 ) a2 / 3 , and a9 ) 2a2. The spectral densities characterizing the dipolar coupling are defined by

7Li


{(

JmD,L(ω) ) (-1)mNS S(S + 1) CD2

(

+

D,L F2,-m (0)

rI3 S(0) D,L F2,m (0)

rI3 S(0)

-

〈 〉)( 〈 〉)(

∫0∞ ei m ω τ

D,L F2,-m

D,L F2,m (τ)

rI3 S

rI3 S(τ)

D,L F2,m

D,L F2,-m (τ)

rI3 S

rI3 S(τ)

0

-

〈 〉) 〈 〉) D,L F2,m

rI3 S

D,L F2,-m

rI3 S

}

dτ (A21j)

A direct consequence of eq A21a-c is to break down the equality between the relaxation rate of the T33 coherence and the slow component of the transverse magnetization. As a consequence, a simultaneous measurement of the relaxation of the transverse magnetization and the T33 coherence is a powerful procedure to separately quantify the dipolar and quadrupolar relaxation mechanisms. Acknowledgment. The DSX360 Bruker spectrometer used for the NMR study was purchased thanks to grants from Re´gion Centre (France). Monte Carlo, MD, and BD numerical simulations were performed locally at CRMD on workstations purchased thanks to grants from Re´gion Centre (France). References and Notes (1) Harris, R. K.; Mann, B. E. NMR and the Periodic Table; Academic Press: London, 1978. (2) van der Maarel, J. R. C.; Jesse, W.; Hancu, I.; Woessner, D. E. J. Magn. Reson. 2001, 151, 298. (3) van der Maarel, J. R. C. Concepts Magn. Reson. Part A 2003, 19A, 97and 117. (4) Jaccard, G.; Wimperis, S.; Bodenhausen, G. J. Chem. Phys. 1986, 85, 6282. (5) Porion, P.; Fauge`re, A. M.; Lećolier, E.; Gherardi, B.; Delville, A. J. Phys. Chem. B 1998, 102, 3477. (6) Porion, P.; Al-Mukhtar, M.; Meyer, S.; Fauge`re, A. M.; van der Maarel, J. R. C.; Delville, A. J. Phys. Chem. B 2001, 105, 10505. (7) Delville, A. J. Phys. Chem. B 1999, 103, 8296. (8) Delville, A. Langmuir 2003, 19, 7094. (9) Levitz, P.; Lecolier, E.; Mourchid, A.; Delville, A.; Lyonnard, S. Europhys. Lett. 2000, 49, 672. (10) Michot, L. J.; Bihannic, I.; Maddi, S.; Funari, S. S.; Baravian, C.; Levitz, P.; Davidson, P. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 16101. (11) van der Beek, D.; Petukhov, A. V.; Davidson, P.; Ferre´, J.; Jamet, J. P.; Wensink, H. H.; Vroege, G. J.; Bras, W.; Lekkerkerker, H. N. W. Phys. ReV. E 2006, 73, 041402. (12) Mamontov, E.; Vlcek, L.; Wesolowski, D. J.; Cummings, P. T.; Wang, W.; Anovitz, L. M.; Rosenqvist, J.; Brown, C. M.; Sakai, V. G. J. Phys. Chem. C 2007, 111, 4328. (13) FitzGerald, S. A.; Neumann, D. A.; Rush, J. J.; Kirkpatrick, R. J.; Cong, X.; Livingston, R. A. J. Mater. Res. 1999, 14, 1160. (14) Saito, M.; Hayamizu, K.; Okada, T. J. Phys. Chem. B 2005, 109, 3112. (15) Halle, B.; Quist, P. O.; Furo, I. Phys. ReV. A 1992, 45, 3763. (16) Halle, B.; Quist, P. O.; Furo, I. Liq. Cryst. 1993, 14, 227. (17) Furo, I.; Halle, B.; Wong, T. C. J. Chem. Phys. 1988, 89, 5382. (18) Furo, I.; Halle, B. J. Chem. Phys. 1989, 91, 42. (19) Linse, P.; Halle, B. Mol. Phys. 1989, 67, 537. (20) Delville, A.; Laszlo, P.; Schyns, R. Biophys. Chem. 1986, 24, 121. (21) Reddy, M. R.; Rossky, P. J.; Murthy, C. S. J. Phys. Chem. 1987, 91, 4923. (22) Einarsson, L.; Nordenskio¨ld, L.; Rupprecht, A.; Furo, I.; Wong, T. C. J. Magn. Reson. 1991, 93, 34. (23) Chen, S. W. W.; Rossky, P. J. J. Phys. Chem. 1993, 97, 10803. (24) Mocci, F.; Laaksonen, A.; Lyubartsev, A.; Saba, G. J. Phys. Chem. B 2004, 108, 16295. (25) Victor, K. G.; Teng, C.-L.; Dinesen, T. R. D.; Korb, J.-P.; Bryant, R. G. Magn. Reson. Chem. 2004, 42, 518. (26) Quist, P. O.; Halle, B. Phys. ReV. Lett. 1997, 78, 3689.

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