Long-Time Scale Ionic Dynamics in Dense Clay Sediments Measured

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Long-Time Scale Ionic Dynamics in Dense Clay Sediments Measured by the Frequency Variation of the 7Li Multiple-Quantum NMR Relaxation Rates in Relation with a Multiscale Modeling 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: January 26, 2009; ReVised Manuscript ReceiVed: March 23, 2009

7

Li NMR relaxation measurements under spin-locking conditions are used to probe the dynamical properties of the lithium counterions within dense dispersions of charged anisotropic nanoplatelets. By simultaneously measuring the T1F and T2F relaxation times in addition to triple-quantum filtered relaxation times under the same spin-locking conditions, it is possible to separately quantify the contributions from the quadrupolar and the heterogeneous dipolar relaxation mechanisms. Thanks to the contribution from the residual quadrupolar coupling felt by the condensed lithium counterions, that procedure allows a broad dynamical range to be probed by performing spin-locking relaxation measurements using a limited number of irradiation powers. As illustrated by a multiscale modeling of the lithium diffusion and relaxation within such heterogeneous system, the frequency variation of the spectral densities characterizing the decorrelation of the quadrupolar coupling is a sensitive probe of the ionic mobility and the structure of the colloidal dispersion. I. Introduction Charged interfaces include a large class of heterogeneous materials (clay minerals,1-7 synthetic metallic oxides,8,9 cement,10 fuel cell membranes,11,12 liquid crystals,13,14 biological polyions,15-20 and membranes21) implicated in numerous industrial and biological processes. In that framework, swelling clays are natural or synthetic ionized materials that exhibit a high specific surface. In addition to their numerous industrial applications (ionic exchange, water treatment, waste storing) the clay/water interface may be considered as an ideal model to investigate the behavior of the charged interfacial systems. Because of the central role played by the counterions neutralizing these charged surfaces, many experimental studies have been devoted to the characterization of their structural and dynamical properties. In order to determine the correlation between the structure of the clay/water interfaces and the ionic mobility, it is necessary to measure the ion diffusion on a broad time-scale. Quasi-elastic neutron scattering (QENS)22-26 and dielectric relaxation measurements27 are generally used to probe the molecular motions on complementary time scales, i.e., between ps and ns for QENS and between µs and s for dielectric relaxation. However these experimental procedures are not specifically sensitive to the ionic motion. By contrast, NMR studies by using either field cycling relaxation measurements14,20,28-34 or NMR pulse gradient spin echo measurements (PGSE)35-38 can selectively extract the mobility of the selected ion on complementary times scales, i.e., varying between ps and µs for field cycling and between ms and s for NMR PGSE experiments. Unfortunately, the fast NMR relaxation rate of the confined neutralizing counterions generally restricts the applicability of these NMR dynamical studies. In order to bypass that intrinsic limitation, we selected to perform NMR relaxation measurements under spin-locking5,39-42 conditions in order to investigate ion dynamics on a time-scale varying between 10-2 * Authors to whom correspondence should be addressed. E-mail: [email protected] (A.D.); [email protected] (P.P.).

and 10-5 s. Furthermore, lithium was used as the neutralizing counterions since 7Li is a spin I ) 3/2 nucleus with a good NMR sensitivity43 and a reduced quadrupolar moment44 and thus relaxation rate. In the present study, we used a synthetic clay (Laponite) because of its high chemical purity and its reduced dispersion in size.45 After validation, it will be possible to apply the same experimental procedure to investigate the ionic mobility in the presence of other charged interfaces, including natural clay used for industrial applications or biological macromolecules (like DNA15-20). Preliminary studies of the 7Li relaxation within dense sediments of Laponite have clearly demonstrated46 the large influence of the dipolar coupling in addition to the quadrupolar coupling. The relative weight of the quadrupolar and dipolar relaxation mechanisms was then determined by performing three sets of independent NMR relaxation experiments,46 recording first 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 order to extract the frequency variation of the quadrupolar relaxation rate from spin-locking NMR relaxation measurements we perform now four sets of independent relaxation measurements under irradiation, implying the one quantum coherence (T11) during the standard T1F and T2F relaxation measurements39-42 and the triplequantum coherence (T33) during triple-quantum filtered relaxation measurements47,48 with two perpendicular directions of the irradiating magnetic field. The analysis of these experimental data is validated by the conclusions from a multiscale modeling of the structural and dynamical properties of the neutralizing lithium cations confined between Laponite clays, implying grand canonical Monte Carlo (GCMC)49 simulations and Brownian dynamics (BD).50 This multiscale modeling allows us to investigate a broad range of time and gives a better understanding of the interdependence

10.1021/jp9007625 CCC: $40.75  2009 American Chemical Society Published on Web 05/19/2009

Long-Time Scale Ionic Dynamics

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between the mobility of the neutralizing counterions, the frequency variation of their quadrupolar NMR relaxation rates and the structure of the clay dispersion. II. Material 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 104g). The excess LiCl salt was removed by another sequence of five dissolutions in aqueous solutions (LiOH 10-4 M) with separation performed by ultrafiltration under nitrogen pressure (∼3 atm) using microporous membranes (pore size 0.1 µm from Osmonics Inc.). More details may be found in the literature.46 Dense clay sediments were prepared by uniaxial compression, using the same microporous membranes. Their concentration is evaluated by water loss under vacuum. A denser Laponite sample (noted “air-dried”) is prepared by water evaporation under controlled conditions (T ) 20 °C, p/p0 ) 0.23). A small clay sample (length: 12 mm, width: 5 mm, thickness: 2-4 mm) is cut into the dense clay sediment and introduced in a cylindrical sample holder, with the compression axis oriented perpendicular to the cylinder axis. The sample holder fits the central hole of the solenoid used as an irradiation and detection coil. As shown previously,36,51,52 the clay particles within such dense sediments are partially oriented within a single nematic phase. Furthermore, the direction of their compression axis was shown to coincide with the nematic director.36,51,52 Since the sample holder rotates freely into the coil, the NMR spectra and relaxation rates can be measured for any orientation of the nematic director n of the clay sediment by reference with the static magnetic field B0. 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) to detect fast relaxation (R2 e 104 s-1) and large quadrupolar splitting (ωQ e 5 × 104 rad s-1). Pulse duration for the magnetization inversion varied between 10 and 15 µs. Preliminary NMR relaxation measurements46 of the 7Li cations within such clay sediments have already demonstrated the slow modulation of the lithium quadrupolar coupling (JQ0 (0) . JQ1 (ω0) ∼ J2Q(2 ω0)). This behavior was shown to result from the long time-scale decorrelation of the quadrupolar coupling monitored by the lithium diffusion within the clay sample. In addition to their quadrupolar coupling, heterogeneous dipolar coupling was also responsible46 for the relaxation of the lithium cations condensed near the clay particles. The paramagnetic impurities of the clay sample were clearly identified46 as the source of that heterogeneous dipolar coupling. In addition to the usual measurement of the longitudinal and transverse relaxation rates we also measured the triple-quantum filtered relaxation of the T33 coherence, in order to separately quantify the relative weight of the quadrupolar and heterogeneous dipolar relaxation mechanisms. Under the slow modulation of both mechanisms,46 the relaxation of the T33 coherence is indeed driven by the dipolar coupling (R33 ∼ 2 J0D(0)) while the fast component of the transverse relaxation is driven by the two relaxation mechanisms (R2f ∼ J0Q(0) + 2/9J0D(0)). In order to further extract dynamical information on the lithium mobility within such heterogeneous material, we selected to extend that study by performing the same relaxation measurements under spin-locking conditions in order to probe the low-frequency variations of these spectral densities (JQ0 (λ) and JD0 (λ)) which must be sensitive to the nature of the clay-ion interactions since these spectral densities are

Figure 1. Schematic view of the pulse sequences used for: (a) the T1F and T2F relaxation measurements; (b) the T33F90 and T33F0 relaxation measurements.

monitored by the long time-scale diffusion of the lithium counterions within the clay dispersion. Figure 1 illustrates the pulse sequences used to measure the four independent relaxation rates under spin locking conditions: i.e. the standard T1F and the T2F relaxation rates39-42 (here noted respectively T11F90 and T11F0) with the phase of the weak irradiation field B1 respectively perpendicular or parallel to the strong excitation pulse (Figure 1a) in addition to the triple-quantum filtered relaxation rates47,48 (noted respectively T33F90 and T33F0) with the phase of the weak irradiation field B1 respectively perpendicular or parallel to the last strong excitation pulse (Figure 1b). The angular velocity ω1 is determined by the irradiation power of the field B1. Within the standard T1F experiment, the irradiation pulse is parallel to the transverse magnetization generated by the first π/2 pulse (cf. Figure 1a). One speaks generally of spin-locking conditions. Such a sequence is used to probe the low-frequency variation of the spectral densities39-41 resulting from the long-time scale motions of the NMR probe. However, the phase of the irradiation field (φ2 in Figure 1a) may be selected along any direction by reference with the phase of the first pulse (φ1 in Figure 1a). By using φ1 ) φ2, we performed the so-called T2 F experiment.39,42 As detailed in the Appendix, the coherences describing the time evolution of spin I ) 3/2 nuclei split in two independent subsets, as described by the two matrices (eqs A13a and A13b) with different characteristic frequencies (eqs A17a-A17f). Equation A13a describes the time evolution probed by the T1F experiments (also noted T11F90 in Figures 1a and 2a) and eq A13b corresponds to the T2F experiments (noted T11F0 in Figures 1a and 2b). As a consequence, we performed two independent sets of spin-locking experiments in order to sample the whole set of characteristic frequencies and extract more dynamical information. As shown in Figure 2a,b, the standard T1F experiment in mainly sensitive of the initial characteristic frequency (λ0) while the T2F experiment mainly probes the triplet (λ4, λ5, λ6). By further performing spin-locking

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Figure 2. Fourier transform of the time evolution of the T11 and T33 coherences under spin- locking condition illustrating the full set of eigenvalues of the static Hamiltonian (see text) as probed by: (a) a single T1F relaxation experiment, (b) a single T2F relaxation experiment, (c) a single T33F0 relaxation experiment and (d) a single T33F90 relaxation experiment.

experiments during triple-quantum filtered relaxation measurements (Figure 1b), we improve the sampling of the set of characteristic frequencies (see Figure 2c,d) and become able to separately quantify the frequency variation of the m ) 0 components of the quadrupolar and heterogeneous dipolar couplings (J0Q(λ) and J0D(λ)). As before, the delay t ) δopt used to generate the T31 coherence (Figure 1b) was optimized46 from a preliminary analysis of the triple-quantum filtered time evolution of the T31 coherence. The spin-locking relaxation measurements were performed at four different strengths of the weak irradiation pulse (noted ω1), corresponding respectively to 4.5, 9, 18, and 36 × 103 rad s-1. By comparison, the angular velocity ω1 of the strong excitation pulse corresponds roughly to 3 × 105 rad s-1. The four set of spin-locking relaxation measurements allow us to distinguish between the contributions from the dipolar and quadrupolar relaxation mechanisms and carefully extract the low-frequency variation of the spectral density describing the decorrelation of the quadrupolar coupling induced by the ionic diffusion53,54 within the dense sediment (see the Appendix). The time evolution of the various coherences induced by these two relaxation mechanisms during the various spin-locking experiments is described in the Appendix. A Fourier transform procedure (Figure 2) is used to identify the characteristic resonance angular velocities of these spin I ) 3/2 nuclei systems under irradiation (cf. eqs A17a-A17f). Depending on the relative phase shift between the last strong excitation pulse and the weak spin-locking irradiation, two subsets of characteristic angular velocities are detected corresponding either to a doublet55-58 augmented by a noticeable component at zero angular velocity (noted λ0) (see Figure 2a) or a triplet55-58

(Figure 2b). The angular velocities of both components of the doublet (with angular velocities noted λ1 and λ2) are detailed in eqs A17a and A17b. In addition to the triplet (noted λ4, λ5, and λ6 in eqs A17d-A17f), the high-frequency component55-58 (with angular velocity noted λ3 in eq A17c) is more difficult to detect experimentally. As detailed in the eqs A17e and A17f, the difference between the two external components of the triplet is used to determine the order of magnitude of the angular velocity of the weak irradiation pulse ω1. As sketched in Figure 3, spin I ) 3/2 nuclei with a residual quadrupolar coupling cover a broad range of angular velocities under spin-locking conditions thanks to the multiplicity55-58 of the characteristic angular velocities (cf. A17a-A17f). 3. Multiscale Modeling. We performed a multiscale modeling of the aqueous clay dispersions in order to understand the structural and dynamical information given by 7Li NMR spectroscopy and relaxation measurements in such heterogeneous systems. The starting point of this numerical study is a molecular modeling of the clay/water interface38,46 based on the classical clay force field.59 In that framework, we used grand canonical Monte Carlo simulations46 to determine the number of water molecules confined between two Laponite clay particles with a period of 30 Å, corresponding to a clay concentration of 126% w/w. These Monte Carlo simulations also describe any equilibrium properties of the clay/water interface like the distribution and average orientation of the water molecules located within the clay interlayer and the average hydration of the confined lithium counterions. Because of the size of the simulations cell (lateral extent ∼30 Å, period 30 Å) and the use of the periodic boundary conditions,49 these ions must be considered as condensed in the vicinity of the basal surface of



Figure 3. Distribution of the complete sets of resonance angular velocities probed by the triple-quantum filtered relaxation measurements under spin-locking condition performed at four irradiating fields. These measurements are performed with the most dense clay sample (called “air-dried” with the nematic director oriented parallel to the static magnetic field). The corresponding low-frequency dispersion of the m ) 0 component of the intrinsic spectral density characterizes the decorrelation of the quadrupolar coupling induced by the long-time ionic diffusion.

charge 1000 e). The mean-force potential (Figure 4) is evaluated by a preliminary Monte Carlo study38 of the average distribution (eq 1) of 1000 neutralizing lithium cations and 100 dissociated salt around a single clay platelet in the framework of the Primitive model:60

( )

W(F, z) ) -kT ln

Figure 4. Ionic concentration profiles of the lithium counterions condensed a single Laponite platelet and used to determine the mean force potential driving the trajectories of the cations (see text).

the clay. More details may be found in previous publications.3,23,38,46 From these equilibrium configurations of the confined water molecules and Li+ cations, we directly evaluate the various components of the tensor describing the average electrostatic field gradient (efg) felt by these condensed lithium cations, determining its magnitude (VzBz ) (6 ( 2) × 1018 V/m2) and the direction of its principal axis. Because of the cylindrical symmetry of the clay/water interface, the principal axis is oriented perpendicular to the clay surface. Since this NMR study focuses on the low-frequency variation of the quadrupolar contribution to the 7Li relaxation rate, we restrict our analysis to the long-time decorrelation of the quadruplar coupling. In that framework, we neglect the fast modulation of the quadrupolar coupling induced by the local motion of the confined cations as probed by molecular dynamics (MD) on a short time-scale (∼ns). In order to fill the gap between the time-scale accessible by MD46 and that probed by NMR relaxometry, we use Brownian dynamics50 to describe the long-time behavior of the decorrelation of the quadrupolar coupling within dense clay sediments by using a coarse-grained description of the aqueous clay dispersion. For that purpose, we remove the water molecules and replace the ion-ion and clay-ion potentials by an effective potential. In that framework, it becomes possible to simulate the trajectories of 20 000 independent counterions diffusing during on long period (∼µs) within the mean force field generated by 90 disks mimicking the Laponite clay (diameter 300 Å, thickness 10 Å, total electric

c(F, z) c0

(1)

where F and z are respectively the radial and longitudinal separation between the Li+ cation and the clay platelet and c0 is the average ionic density in the simulation cell. As shown previously,38 the weak overlap approximation (WOA)61 may then be used to approximate the effective ion/clay potential monitoring the ionic motion within a collection of charged platelets modeling a dense clay sediment. The ionic trajectories are evaluated by using the generalized Langevin algorithm:50 b x n+1 ) b xn +

b V n+1 )

(

)

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

(2a)

x n) Fn γ(x bn+1 - b Bb A Vn b Vn + + (1 - e-γ∆t)2 + b C C mγ C

(2b) where

A ) 2γ∆te-γ∆t - 1 + e-2γ∆t -2γ∆t

B ) γ∆t(1 - e

(2c)

-γ∆t 2

) - 2(1 - e -γ∆t

C ) 2γ∆t - 3 + 4e

)

-2γ∆t

-e

(2d) (2e)

The two independent random functions satisfy

bn)R〉 ) 〈(V bn)β〉 ) 〈(X bn)R(V bn)β〉 ) 0 〈(X bn)2R〉 ) kTC 〈(X mγ2 bn)2R〉 ) 2kTB 〈(V mC

(2f) (2g) (2h)

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

γ)

kT mD

(2i)

We used an intrinsic self-diffusion coefficient of the lithium cations38 (D ) 0.9 × 10-9 m2/s), corresponding to the mobility

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measured in diluted LiCl solutions by 7Li NMR exploiting echo attenuation with pulsed field gradient. The integration of eqs 2a-2i is performed with a time step of 0.1 ps, corresponding to a product γ ∆t ) 39.3, significantly larger than unity. As described above, the deterministic force b Fn in eqs 2a and 2b is evaluated from the Weak Overlap Approximation. By using such coarse grained description, it becomes possible to simulate realistic ionic trajectories within dense clay sediments during a diffusion time of 1 µs. 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 B ) modulated by the clay orientation53,54 (eq A6) simulations (Vzz when the lithium probe is physisorbed. The memory function (eq 3) describing the decorrelation of the quadrupolar coupling is not directly sensitive to these instantaneous values of the residual quadrupolar coupling felt by the diffusing lithium cations but to the product of these instantaneous quadrupolar couplings and their initial values. As a consequence, when an initially condensed lithium cation is desorbed, its instantaneous residual coupling becomes equal to zero. However, the same cation will recover 100% of that initial residual coupling53 when it hits any clay platelet with the same orientation than its initially adsorbing clay particle. Because of the fluctuation of the orientations of the clay platelets into the dispersion, the decorrelation of the quadrupolar coupling (cf. Appendix) sampled by the diffusing ions is a fingerprint of the fluctuation53,54 of the clay directors around their average orientation into a nematic dispersion: B 2 GmQ,L(τ) ) (Vzz )

[

Ni

Np

Np

2

LP ∑ ∑ ∑ ∑ P(i ∈ j|0)P(i ∈ s|τ)DlmLP D-l-m

1 Ni i)1

j

(

s

j)1 s)1 l)-2

Ni

Np

j)1 l)-2

)]

2 2

2

∑ ∑ ∑ P(i ∈

1 Ni i)1

LPj j|0)Dlm

(3)

where m ){ 0, 1, 2 }, Ni is the number of diffusing ions, Np the number of platelets, and P(i ∈ j | τ) the probability that the ion labeled i is adsorbed on platelet labeled j at time τ. The modulation of the quadrupolar coupling induced by the orientation of the platelet labeled j into the static magnetic field B0 is described by the Wigner rotation matrices30 Dl,LP(j m(θ,φ,ψ) (see eq A4 in the Appendix). III. Results and Discussion 1. Numerical Modeling. In dilute regime (less than 8.7% w/w) the Laponite dispersions were shown to be isotropic.36,51,52 Numerical simulations of BD (eq 2a) are first used to simulate the ionic trajectories within such dilute (5.4% w/w) and isotropic dispersion. Equation 3 is used to evaluate the corresponding memory function of the quadrupolar coupling felt by the condensed lithium counterions. The three components describing the decorrelation of the quadrupolar coupling are equal and decrease slowly: after 1 µs, the initial quadrupolar coupling is just reduced by a factor 5 (see Figure 5). The simulation time is too short to detect the long-time asymptotic decrease of the quadrupolar coupling. By contrast, the memory function describing the desorption of the initially condensed lithium counterions exhibits a fast decrease with a characteristic timescale of 100 ps. The discrepancy between these two behaviors was previously explained by the later adsorption of the desorbed53,54 and diffusing cations on a clay particle nearly parallel to the initial one. However, that interpretation was

Figure 5. Average value of the normalized memory function gm(τ) defined as the ratio Gm(τ)/Gm(0) describing the decorrelation of the quadrupolar coupling within isotropic dispersion of Laponite clays in relation with the memory function describing the residual number of initially condensed counterions with and without intermediate desorption and diffusion into the dispersion.

subject to caution because of the large separation between the clay particles in such dilute regime and the relatively short diffusion time: after 100 ns, the mean squared displacement of the lithium counterions does not exceed 100 Å,38 while the average separation between the clay platelets reaches 330 Å in the same dilute dispersion (5.4% w/w). However, by including into the memory function of the fraction of condensed cations those adsorbed back on their initially adsorbing particle after some intermediate diffusion period,62 we obtain a memory function fully compatible with that of the quadrupolar coupling. These multiple loops within the trajectories of the desorbed cations result from the strong attraction exerted by the clay platelet as quantified by the mean force potential (see Figure 4). That strong attraction also efficiently reduces the apparent ionic mobility38 even in dilute clay dispersions. In order to probe the influence of the size of the clay platelets and reach the long-time asymptotic regime within the same simulation period (1 µs), we have performed simulations of BD by reducing the size of the clay platelets by a factor 2. In addition to the previous isotropic dispersion, we also generated dense dispersions (18% w/w) with two different degrees of nematic ordering characterized by their residual order parameter (P2(cos θ) ) 0.40 and 0.71). As displayed in Figure 6a, size reduction roughly accelerates by a factor 10 the decrease of the memory functions. Within nematic dispersions (Figure 6, panels b and c), we detect a different behavior of the memory functions describing the various components of the quadrupolar relaxation mechanism. That difference was shown to result from the partial alignment of the clay platelets in the nematic phase.52 In the intermediate time regime (100 ps < t < 100 ns), the memory functions decrease according to a logarithmic law (see Figure 6a). As a consequence, the corresponding spectral densities decrease according to a ω-1 power law for angular velocities varying between 107 and 1010 rad s-1. Because of the size reduction, the same power law variation of the spectral densities is expected to occur in the range 106 and 109 rad s-1 within Laponite dispersions. That behavior was previously54 detected in the high frequency variation of the 23Na longitudinal relaxation rate (ω ∼ 108 rad s-1) measured within dilute dispersions of Laponite clay. The effective potential describing the clay/ion interactions induces the logarithmic decrease of the memory function characterizing the decorrelation of the qua-


Figure 6. Same results as in Figure 5 characterizing the dynamical behavior of the lithium counterions diffusing within dispersions of charged nanoplatelets with a reduced size (see text): (a) in isotropic dspersion; (b) in weakly ordered nematic dispersion; (c) in strongly ordered nematic dispersion.

drupolar coupling in that intermediate time regime. Previous numerical simulations of BD of the ionic diffusion within clay dispersions were indeed performed54 without using such a longrange potential (eq 1) but simply by including an average residence time54 of the condensed cations at the surface of the clay particles. These simplified simulations predicted a powerlaw decrease54 of the memory functions describing the decorrelation of the quadrupolar coupling instead of the logarithmic behavior displayed in Figure 6. The long-time asymptotic behavior of the memory functions is better displayed by using a logarithmic scale (Figure 7a-c). Despite of the numerical uncertainties, the memory functions describing the decrease of the three components of the quadrupolar coupling within dilute dispersions (Figure 7a) are roughly equivalent and evolve according to a t-1 power law for diffusion time longer than 102 ns. Within dense nematic dispersions


Figure 7. Same results as in Figure 6, focused on the long time decorrelation of the same memory functions by using a logarithmic scale.

(Figure 7b,c), the asymptotic behavior of the various memory functions is the same even if their attenuation in the intermediate time regime differs significantly (see Figure 7c). As a consequence, the spectral densities describing the quadrupolar relaxation mechanism within Laponite dispersions are expected to exhibit a logarithmic decrease in the same low-frequency regime (ω < 106 rad s-1) as that probed by our spin-locking relaxation measurements (cf. Figure 3). 2. Multiquantum Relaxation Measurements under SpinLocking. In the presence of quadrupolar nuclei with a high quadrupolar moment like 23Na, dynamical information on the ion dynamics within clay dispersions can be extracted from the frequency variation of T1F alone53,54 since the quadrupolar coupling is then strong enough to totally drive the relaxation of the condensed counterions. However, the strength of this

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quadrupolar coupling may become an intrinsic limitation to these relaxation measurements when the quadrupolar relaxation rate exceeds the experimentally accessible dynamical range (R2 > 105 s-1). Exchanging the initial counterion by another quadrupolar cation of the same valence but with a lower quadrupolar moment, like 7Li, greatly facilitates the relaxation measurements in such heterogeneous systems because of the corresponding decrease of the quadrupolar relaxation rate of the condensed counterions.46 Unfortunately, under the same conditions the heterogeneous dipolar coupling becomes an efficient relaxation mechanism partially masking46 the initially investigated quadrupolar relaxation mechanism. By simultaneously measuring the time evolution of the T10, T11, and T33 coherences we were able to quantify the relative weight46 of both relaxation mechanisms. As a consequence, one may reasonable expect to be able to extract now the low-frequency variation of the quadrupolar coupling and thus the long-time ion dynamics from simultaneous relaxation measurements of the T11 and T33 coherences under spin-locking conditions with two different phases (i.e., perpendicular or parallel) of the weak irradiation field ω1 by reference with the last strong excitation pulse (see Figure 1, panels a and b). While the first set of relaxation measurements are well-known (T1F and T2F),39-42 this is, to our knowledge, the first use of triple-quantum filtered relaxation measurements under spin-locking conditions. By using the whole set of eqs A13, A14, A23, A25, and A28, it is possible to simulate the time evolution of any coherence during each step of the pulse sequence used for our spin-locking relaxation measurements (cf. Figure 1). A Simplex procedure63 is used to simultaneously fit the time evolution of the magnetization measured during these 16 independent spin-locking experiments (i.e., four pulse sequences each performed at four different irradiation powers). In order to reduce the number of fitted parameters, we first perform multiquantum relaxation measurements by recording the time evolution of the longitudinal magnetization (T10 coherence), the transverse magnetization (T11 coherence) and the triple-quantum filtered T33 coherence. As previously,46 we extract from these preliminary relaxation measurements a set of parameters including the residual quadrupolar coupling and the spectral densities characterizing the quadrupolar relaxation mechanism (J0Q,L(0) and J1Q,L(ωI)) and the dipolar relaxation mechanism (J0D,L(0),J0D,L(ωI - ωS) and JD,L 1 (ωS)). As before, the number of these preliminary fitted parameters is reduced by assuming J2Q,L(2ωI) ) J1Q,L(ωI) and J2D,L(ωI + ωS) ) J1D,L(ωI) ) J1D,L(ωS). All these parameters remain constant during the simultaneous fitting of the spinlocking relaxation measurements. The Simplex procedure is used then to fit 12 parameters corresponding to the exact values of the power of the weak irradiation fields (4 parameters), the frequency dispersion of the spectral density (JD,L 0 (λ)) describing the dipolar coupling (4 parameters), the low frequency variation of the spectral density (J0Q,L(λ)) describing the quadrupolar coupling (2 parameters) and the heterogeneity of the weak irradiation field (2 parameters). In order to reduce the number of fitted parameters, the spectral density describing the dipolar coupling (J0D,L(λ)) is fitted only at four angular velocities uniformly spread between the minimum and the maximum values of the probed resonance angular velocities (see Figure 3). The spectral density describing the quadrupolar coupling (JQ,L 0 (λ)) is fitted on a logarithmic law with only two parameters: the slope of the logarithmic decrease (noted η) and the angular velocity describing the cutoff (noted ωC) between a low frequency plateau (with the spectral density equal to J0Q,L(0)) and the beginning of the logarithmic variation:

Porion et al. Q,L JQ,L 0 (ω) ) J0 (0)

J0Q,L(ω) ) J0Q,L(0) - η ln

JQ,L 0 (ω) ) 0

( ) ω ωC

if ω e ωC

( )

if ωC < ω < ωC exp

( )

if ω > ωC exp

JQ,L 0 (0) η

(4a)

J0Q,L(0)

η (4b)

(4c)

That logarithmic decrease of the spectral density is fully justified by the numerical simulations of Brownian dynamics (see above). Finally, because of the size of the clay sample and its location within the coil, we must take into account the variation ∆ω1 of the irradiation power ω1 within the sample. For that purpose, we have numerically evaluated the local magnetic field generated by a solenoid similar to that used in our experimental setup (length: 35 mm; diameter: 12 mm; number of turns of the wire: 6). As illustrated in Figure 8, the field heterogeneity depends on the location of the sample by reference with the center of the coil and is well described by an asymmetric Lorentzian distribution. The two last parameters are used to reproduce the width of the two Lorientzians describing the heterogeneity of the weak irradiation field ω1. As displayed in Figure 9a-d, we obtain a good agreement between the spin-locking relaxation measurements and the fitted data despite the numerous approximations implied in our analysis. While the experimental data are fitted as a function of the time, their Fourier transform better exhibits the contribution from the various characteristic angular velocities (cf. eqs A17). As shown in Figure 10, panels b and c, even the weak contribution from the high-frequency component is well described. Such an agreement results from a complete treatment of the time evolution of the coherences under the influence of the quadrupolar (eqs A14 and A23) and dipolar (eqs A25 and A28) relaxation mechanisms in addition to the static Hamiltonian including the spin-locking irradiation and the residual quadrupolar Hamiltonian (cf. eq A13). Thanks to the broad range of characteristic angular velocities (cf. eq A17) probed by the spinlocking relaxation measurements performed only at four irradiation fields, we obtain a good sensitivity on the dispersion of the spectral density describing the m ) 0 component of the quadrupolar coupling (see Figure 3). One may thus be confident in the position of the crossover between the low-frequency plateau and the high-frequency logarithmic decrease of that spectral density.

Figure 8. Distribution law quantifying the heterogeneity ∆ω1/〈ω1〉 within the clay sample of the irradiation field B1 generated by the solenoid used for these spin-locking relaxation measurements. In the experimental setup (see text), the center of the clay sample is located along the axis of the coil and shifted by 2 mm from its center.



Figure 9. Comparison between the experimental and fitted time evolution of the coherences during to the spin-locked relaxation measurements performed for the most concentrated clay sample (called “air-dried”) with an irradiation field corresponding, for the 7Li nucleus, to a resonance angular velocity of 18 × 103 rad s-1. Four spin-locking relaxation measurements are performed leading to: (a) the standard T1F relaxation time; (b) the T2F relaxation time; (c) the spin-locked relaxation of the triple-quantum filtered T33F90 coherence; (d) the spin-locked relaxation of the triplequantum filtered T33F0 coherence.

Figure 10. Fourier transforms of the panels in Figure 9a-d.

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TABLE 1: Intrinsic Parameters WQ, J0Q,S (0), J1Q,S (0), J2Q,S (0), η, and ωC (See Text) Extracted from the Analysis of the Set of T1G, T2G, T33G0, and T33G90 Experiments for the Different Samples clay sample

VQ (Hz)

J0Q,S (0) (s-1)

η (s-1)

56% w/w 112% w/w “air-dried”

680 1140 5000

160 210 400

11 18 80

ωC (rad s-1)

J1Q,S (0) (s-1)

η (s-1)

360 340 1300

38 37 260

300 310 1.9 × 104

In nematic dispersions, the relaxation measurements may be performed for any orientation of the nematic director n by reference with the static magnetic field B0. We simply avoid to orient the nematic director along the so-called magic angle since, because of the corresponding cancelation of the residual quadrupolar coupling (cf. eq A6), the five characteristic angular velocities (cf. eqs A17-A17f) reduce then to zero and 2 ω1, strongly limiting the dynamical range accessible by the spinlocking relaxation measurements. By using the Wigner rotation matrices30 (see the Appendix), it is possible to relate64 the apparent correlation functions evaluated in the laboratory frame (noted L) to their intrinsic 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 cos2 θLS sin2 θLSGQ,S 1 (τ) + (1 - cos θ ) G2 (τ) 4

(5)

In the same manner, the frequency variation of the apparent spectral density (J0Q,L(ω)) describing the quadrupolar coupling results form an average of three intrinsic spectral densities characterizing the ion dynamics in the frame of the dispersion: Q,S Q,S JQ,S 0 (ω), J1 (ω) and J 2 (ω). By selecting three different orientations of the clay sample by reference with the static magnetic field B0, it is possible to extract these three intrinsic spectral densities. First, by aligning the nematic director n parallel to the static magnetic field B0 (i.e., θLS ) 0°), we directly measure the m ) 0 component (cf. eq 5). The m ) 1 component is determined from the relaxation measurements performed at the angle θLS ) 30° after removing the contribution from the m ) 0 component while the contribution from the m ) 2 component is neglected because of its reduced weight (less than 5%, as displayed in eq 5). Finally, measurements are performed with a perpendicular orientation of the nematic director n (i.e., θLS ) 90°) to give access to m ) 2 component after subtraction of the contribution from the m ) 0 component (cf. eq 5). The data are summarized in Table 1. As shown previously, the difference between the initial values of the intrinsic spectral densities (J 1Q,S(0) > J 0Q,S(0) > J 2Q,S(0)) is the fingerprint of the nematic ordering52 of the clay dispersions. The angular velocity (ωC) describing the crossover between the two dynamical regimes (i.e., the low-frequency plateau and the logarithmic decrease of the spectral density) can be used to estimate the size of the domain necessary for the decorrelation of the quadrupolar coupling. While the spectral densities corresponding to the m ) 0 and m ) 1 components of the quadrupolar relaxation mechanism have roughly the same cutoff frequency, the value reported for the m ) 2 component is larger by 1 order of magnitude. This result is somewhat surprising since we expect some similitude between the contributions from the m ) 1 and 2 components of the quadrupolar coupling. These two components include a contribution from the azimuthal Euler angle30 (i.e., φ in eq A4) describing the orientation of the clay platelet in the static magnetic field B0 in addition to the colatitude (i.e., θ in eq A4) which contributes alone to the m ) 0 component of the quadrupolar coupling (cf. eq A6). The very small value of the slope describing the logarithmic decrease of the m ) 2

ωC (rad s-1) 440 510 1.1 × 104

J2Q,S (0) (s-1)

η (s-1)

80 97 280

3 × 10-4 9 × 10-4 100

ωC (rad s-1) 5100 5500 1.5 × 104

component further sketches the limitation of our analysis. Because of the weakness of the initial value of the m ) 2 spectral density, our analysis fails to detect a noticeable frequency variation of that spectral density. By focusing on the m ) 0 and 1 components of the quadrupolar relaxation mechanism, we obtain an average diffusion time of about 2.6 × 10-3 s. That diffusion time is used to estimate the length required for the decorrelation of the quadrupolar coupling for the Li+ cations in the two less concentrated clay samples. From the order of magnitude of the lithium self-diffusion tensor previously measured38 within dense Laponite dispersions (i.e., DF ∼ 3 × 10-10 m2/s in the direction perpendicular to the nematic director and Dz ∼ 5 × 10-11 m2/s in the direction parallel to the nematic director), we obtain respectively lF ) 1.2 × 104 Å and lz ) 5 × 103 Å assuming a Gaussian diffusion regime35 (lR ) (2DRt)1/2). The size of such elementary volume largely exceeds the diameter of the Laponite particle (∼300 Å).45 As a consequence, the diffusing lithium cation must probe various clay platelets with different relative orientations in order to totally decorrelate its initial quadrupolar coupling, i.e. lose the memory associated with the orientation of its initially adsorbing clay particle. By extrapolating the same ionic mobility to the so-called “air-dried” clay sample, we detect an increase of the cutoff frequency by a factor 50 and thus a decrease of the size of this elementary volume by a factor 7. IV. Conclusions We have studied the mobility of the Li+ counterions within sediments of synthetic clay by analyzing the frequency variation of the 7Li NMR relaxation rate. By focusing on the low-frequency regime (ω < 105 s-1) it was possible to investigate the long-time ionic mobility within such aqueous dispersions of charged nanoplatelets. The analysis of the NMR relaxation measurements was validated by a multiscale numerical modeling of the ion diffusion and relaxation within aqueous dispersions of charged anisotropic particles. We succeed to separately quantify the contributions from the two dominant NMR relaxation mechanisms of the 7Li nuclei in such heterogeneous media, i.e. the quadrupolar and the heterogeneous dipolar couplings, by simultaneously measuring the standard T1 F and T2 F relaxation times in addition to the triplequantum filtered relaxation times measured under spin-locking condition. This is, to our knowledge, the first use of multiquantum filtered relaxation measurements under spin-locking conditions. Furthermore, these complementary spin-locking relaxation measurements allow investigating a broad range of frequencies by using a limited number of irradiation powers. The same experimental procedure, validated within aqueous dispersions of synthetic and well-characterized clay, should be useful to determine the ionic mobility within a large class of heterogeneous material resulting from the dispersion of charged solid/liquid interfaces. Acknowledgment. The DSX360 Bruker spectrometer used for that 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).


J. Phys. Chem. C, Vol. 113, No. 24, 2009 10589 In the framework of a two state model with a fast exchange,46 at the NMR time-scale, between the two spin populations, one deduces the measured quadrupolar splitting by using eqs A3 and A4

Appendix 1. NMR Relaxation Theory of Spin I ) 3/2 Nuclei a. Static Quadrupolar Coupling

ωQapp ) pFωQF + pBωQB )

The quadrupolar Hamiltonian65 is defined by

2

HQ )

2 3 eQ(1 - γ∞) Q,L IR (-1)pF2,-p T2,p 2 I(2I - 1)p p)-2

∑

(A1a)

where Q is the quadrupolar moment43 of the nuclei (-0.042 × 10-28 m2 for 7Li), γ∞ is the Steinheimer antishielding factor44 (0.26 for lithium),

1 L Q,L F2,0 ) Vzz , 2

1 L (Vxz ( iVLyz), √6 1 ) (VLxx - VLyy ( 2iVLxy) 2√6

Q,L F2,(1 )Q,L F2,(2

(A1b)

and IR T2,0 )

1 2 (3Iz - I(I + 1)), √6 1 IR ) - (IzI( + I(Iz), T2,(1 2

IR T2,(2

1 2 ) I( 2

3eQ(1 - γ∞) L 〈V 〉(1 - 2m), 4I(2I - 1)p zz for m varying between I and - I + 1

(A2)

(A3)

2

)

∑

Q,P LP F2,p Dp,q(θ, φ, ψ)

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

〈

3 cos2 θLS - 1 3 cos2 θSP - 1 2 2

〉

(A6)

b. Quadrupolar Relaxation

During a change of frame, the components of the field gradients transform like the second order spherical harmonics30 Q,L F2,q

(A5)

p)-2

(A1c)

leading to a quadrupolar splitting of the resonance lines according to

ωm-1,m )

LS SP SP (θLS, φLS, ψLS)〈D0,p (θ , φSP, ψSP)〉 ∑ Dp,0

B ωQapp ) AQpBVzz

where VLR β are the components of the electrostatic field gradient IR (for p ) -2 evaluated in the laboratory frame (noted L), T2,(p to +2) are the second order irreducible tensors operators, Ix, Iy, and Iz are the spin operators and I( ) Ix ( iIy. 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

eQ(1 - γ∞) L j Q1 ) H 〈V 〉(3m2 - I(I + 1)) 4I(2I - 1)p zz

B AQpBVzz

(A4)

In the framework of the Redfield theory, the relaxation of the quadrupolar nuclei is related to the fluctuating part of the quadrupolar coupling, by using the master equation65,66

dσ* ) -i[H*S , σ*] + f(σ*) dt

where all calculations are performed in the Larmor frequency rotating frame, as indicated by the asterisk. The first contribution to the master equation describes the influence of the static Hamiltonian H*S that includes the residual static quadrupolar Hamiltonian [H*QS ) (ωQ/6)(3I2z - I(I + 1) )] and the Hamiltonian corresponding to the radio frequency pulse (H*1 S ) Ixω1). The second contribution to the master equation results from the fluctuating components64 of the quadrupolar Hamiltonian H*QF(t)

f(σ*) ) -

∫0∞ 〈[H*QF(t), [e-iH τH*QF+(t - τ)eiH τ, σ*(t)]]〉 dτ * S

* S

(A8) For spin I ) 3/2 nuclei, this fluctuating component of the quadrupolar Hamiltonian becomes

p)-2 LP where Dp,q (θ,φ,ψ) are the components of the Wigner rotation 30 matrix defined by the set of (θ,φ,ψ) Euler angles describing the orientation, into the static magnetic field B0, of the principal axis of the tensor describing the electrostatic field gradient felt by the quadrupolar nucleus. Since the clay director coincides with the direction of the principal axis of the residual field gradient, three 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 director (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).

(A7)

2

H*QF(t) ) CQ

∑

IR imω0t Q,L Q,L (-1)mT2,m e (F2,-m(t) - 〈F2,-m (t)〉)

m)-2

(A9) with CQ ) e Q (1 - γ∞)/(6p). IR , with a0 ) The orthonormal tensor operators (Tˆl,m ) al Tl,m 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 introduced55-58 1 1 Tˆl,m(s) ) (Tˆl,-m + Tˆl,m) and Tˆl,m(a) ) (Tˆl,-m - Tˆl,m) √2 √2

(A10) simplifying the notations of the magnetization

10590


Porion et al.

Ix ) √5Tˆ1,1(a);

Iz ) √5Tˆ1,0

(A11)

while the residual quadrupolar Hamiltonian reduces to

HQS )

ωQ 2 (3Iz - I(I + 1)) ) ωQTˆ2,0 6

(A12)

For spin I ) 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,55-58 leading to

)( )

( )(

√3/5ωQ 0 0 0 0 0 0 Tˆ11(a) Tˆ11(a) ˆT20 0 0 0 0 0 0 Tˆ20 - √3ω1 -ω1 √2/5ωQ Tˆ21(s) Tˆ21(s) √3/5ωQ - √3ω1 0 0 0 d ˆ -ω1 ωQ T (s) ) i 0 0 0 0 0 · Tˆ22(s) dt 22 ˆT31(a) √2/5ωQ 0 0 0 0 0 - √5/2ω1 Tˆ31(a) ωQ - √5/2ω1 Tˆ32(a) 0 0 0 0 - √3/2ω1 Tˆ32(a) Tˆ33(a) Tˆ33(a) 0 0 0 0 0 0 - √3/2ω1

(A13a)

-ω1 0 0 0 0 0 0 0 Tˆ10 Tˆ10 -ω √ 0 0 0 0 0 3/5ωQ 0 1 Tˆ11(s) Tˆ11(s) -ω1 ˆT21(a) √2/5ωQ 0 0 0 0 0 √3/5ωQ Tˆ21(a) ωQ -ω1 0 0 0 0 0 0 Tˆ22(a) d Tˆ22(a) )i 0 · √ 0 0 0 0 0 0 6ω1 dt Tˆ30 Tˆ30 ˆT31(s) ˆT31(s) √2/5ωQ 0 √6ω1 0 0 0 0 - √5/2ω1 ωQ 0 0 0 0 0 - √5/2ω1 - √3/2ω1 Tˆ32(s) Tˆ32(s) Tˆ33(s) Tˆ33(s) 0 0 0 0 0 0 0 - √3/2ω1

(A13b)

( )(

)( )

during an irradiation along Ix. The second contribution to the master equation (eq A7) results from the fluctuating part of the quadrupolar Hamiltonian. In the lack of irradiation (ω1 ) 0), it leads to the matrices55-58

() (

Tˆ11(a) Tˆ20 ˆT21(s) d ˆ T (s) ) dt 22 Tˆ31(a) Tˆ32(a) Tˆ33(a) (3/5)J0Q + J1Q + (2/5)J2Q

-

and

0

0 2J1Q

+

2J2Q

0

0

(√6/5)(J0Q - J2Q)

0

0

0

0

0

0

0 0

0

0

J0Q + J1Q + 2J2Q

0

0

0

0

0

0

J0Q + 2J1Q + J2Q

0

0

0

(√6/5)(J0Q - J2Q)

0

0

0

(2/5)J0Q + J1Q + (3/5)J2Q

0

0

0

0

0

0

0

J0Q + J2Q

0

0

0

0

0

0

0 J1Q

+

J2Q

)( ) Tˆ11(a) Tˆ20 ˆT21(s) · Tˆ22(s) Tˆ31(a) Tˆ32(a) Tˆ33(a)

(A14a)

() (

Long-Time Scale Ionic Dynamics Tˆ10 Tˆ11(s) Tˆ21(a) d Tˆ22(a) dt Tˆ30 Tˆ31(s) Tˆ32(s) Tˆ33(s)

)

(2/5)J1Q + (8/5)J2Q

-


0 (3/5)J0Q

0 0

+

J1Q

+

(2/5)J2Q

0

0

0

(4/5)(J1Q - J2Q)

0

0

0

0

0

0

(√6/5)(J0Q - J2Q)

0

0

0

0

0

0

J0Q + J1Q + 2J2Q

0 J0Q

+

2J1Q

+

J2Q

0

0

0

0

0

0

0

(4/5)(J1Q - J2Q)

0

0

0

(8/5)J1Q + (2/5)J2Q

0

0

0

0

(√6/5)(J0Q - J2Q)

0

0

0

(2/5)J0Q + J1Q + (3/5)J2Q

0

0

0

0

0

0

0

0

0

0

0

0

0

0

()

J0Q + J2Q 0

Tˆ10 Tˆ11(s) Tˆ21(a) Tˆ22(a) Tˆ30 Tˆ31(s) Tˆ32(s) Tˆ33(s)

with JmQ ≡ JmQ,L for m ){ 0, 1, 2 }. The spectral densities displayed in eqs A14a and A14b are evaluated in the laboratory frame

JmQ,L(mω0) ) 6CQ2

∫0∞ GmQ,L(τ)eimω τ dτ 0

0 J1Q

+

J2Q

)

·

(A14b)

(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 〉)〉

(A16)

c. Quadrupolar Relaxation under Spin Locking Condition The derivation of the time evolution of the various coherences during spin-locking conditions was already developed55-58 for the quadrupolar relaxation of the spin I ) 3/2 nuclei when the relaxation measurements are performed at the resonance (ω0 ) 0). Under such conditions, the whole basis set splits in two independent subsets (cf. eqs A13-A14), leading to an analytical treatment of the problem. The starting point is the calculation55-58 of the eigenvalues and eigenvectors of the two matrices implied in eqs A13a and A13b. The eigenvalues are λ0 ) 0, ( i λ1 and ( i λ2 for the matrix displayed in eq A13a and ( i λ3, ( i λ4, ( i λ5 and ( i λ6 for eq A13b where55-58

λ1 ) √ωQ2 + 2ω1ωQ + 4ω21

(A17a)

λ2 ) √ωQ2 - 2ω1ωQ + 4ω21

(A17b)

λ3 ) ω1 + λ5 ) ω1 λ4 ) ω1 + λ6 ) ω1 -

ωQ2 + 4ω21 + λ1λ2 2

(A17c)

ωQ2 + 4ω21 + λ1λ2 2

(A17d)

ωQ2 + 4ω21 - λ1λ2 2

(A17e)

ωQ2 + 4ω21 - λ1λ2 2

(A17f)

Because of the separation of the two subsets of coherences, the time evolution of the T20 coherence under the influence of the static Hamiltonian is written formally:

10592


Porion et al.

eiHs tTˆ20e-iHs t ) *

*

7

∑

7

cp(t)Tˆp )

p)1

∑ b0pbV p exp(iλpt)

(A18)

p)1

where the summation is performed for the first subset of coherences to which Tˆ20 pertains (cf. eq A13a), i λp are the corresponding eigenvalues (see above) and b Vp the corresponding eigenvectors. The coefficients b0p are evaluated by applying the initial condition55-58 7

Tˆ20 )

∑ b0pbV p

(A19)

p)1

The matrices displaying the time evolution of the coherences under spin-locking condition are derived by analytically evaluating the double commutator for the m ) 0 contribution to the quadrupolar coupling55-58 (cf. eqs A8-A9)

[Tˆ20, [eiH tTˆ20e-iH t, σ*]] * S

* S

(A20)

where σ* stems for any coherence from the two subsets. Equation A20 is used to describe the time evolution of various coherences under the action of the m ) 0 component of the quadrupolar coupling during the application of the static Hamiltonian (eq A13a and A13b) which implies spin-locking in addition to the static quadrupolar coupling. The two other components (m ) 1 and 2) of the relaxation matrices (eq A14a and A14b) remain unchanged.55-58 But in the present study, the 16 spin-locking relaxation measurements are time-consuming (∼150 h). Because of the drift (∼0.13 Hz/h) of the unlocked static magnetic field B0, the offset of the resonance reaches significant values that cannot be neglected in the derivation of the time evolution of the coherences (especially for the triple-quantum coherences). As the consequence, eqs A13 becomes

A C dσ* σ* )i D B dt

( )

(A21a)

σ˜ * ) (σ˜ *1 , σ˜ *2 )

(A21b)

σ˜ *1 ) (Tˆ11(a), Tˆ20, Tˆ21(s), Tˆ22(s), Tˆ31(a), Tˆ32(a), Tˆ33(a))

(A21c)

σ˜ *2 ) (Tˆ10, Tˆ11(s), Tˆ21(a), Tˆ22(a), Tˆ30, Tˆ31(s), Tˆ32(s), Tˆ33(s))

(A21d)

where

with

and

A and B are subdiagonal matrices already displayed in eqs A13a and A13b, respectively, whereas C and D are the matrices introducing the off-resonance coupling between the two subsets of coherences

(

0 ω0 0 0 0 0 0 0 0 0 ω0 0 C ) 0 0 0 2ω0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω0 0 0 0 2ω0 0 0 0 0 3ω0

)(

0 ω0 0 0 D) 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ω0 0 0 0 0 0 2ω0 0 0 0 0 0 0 0 ω 0 0 0 0 0 0 0 0 0 0 2ω0 0 0 0 0 0 0 3ω0

)

(A21e)

As a consequence of that coupling, the eigenvalues and eigenvectors of eq A19 are now evaluated numerically, and the summation in eqs A18 and A19 is performed for the whole set of eigenvectors


*

15

∑

p)1

15

cp(t)Tˆp )

∑ b0pbV p exp(iλpt)

(A22)

p)1

After straightforward calculations, eq A20 gives

( )

* * I K σ* [Tˆ20, [eiHs tTˆ20e-iHs t, σ*]] ) L J

where

(A23a)

(


I)

( (

3 c + √3 c 52 5 4 -3 c - √6 c 5 1 5 5 0

-√3 c + 3 c + 3c 5 1 √50 5 10 7 √6 c - 3 c 5 2 √50 4

0

0

0

0

0

c2

0

0

0

0

c2

-√2 c + √3 c + 1 c 5 1 5 5 √5 7

103 103

-

-1 c + 1 √5

-

0

c4

-1 c 1 √2

c6

c5 +

1c 6 √2

1c 7 √2

0

(

2

2 c - √3 c 52 5 4

-

103

c1 -

1c 5 √5

-2 c 6 √5

0

-√2 c + √3 c - 1 c 5 1 5 5 √5 7

-√12 c - √8 c 5 1 5 5 3c 10 6 -1 c + 3c - 1c 1 10 5 √2 7 √5 -1 c 6 √2

1c 6 √5

0

-√12 c 5 3 √6 c + 3 c 5 2 √50 4

0

-

103

c4

√3 c

5

11

8

+

√12 c

5

12

+

-

-1 c - 1 c 1 7 √2 √5

0

0

0

0

103

1c + 1c 13 14 √2 √5

√6 c 3 -c10 5 5 13 √3 c + √12 c 0 5 8 5 12 √3 c - 3 c + 3c 0 5 9 √50 13 10 15 -3 c √2 c + √8 c 3c 11 √50 5 8 5 12 10 14 1c 3c + 1c 0 9 10 13 √2 15 √5 1c 3c 14 √2 10 11

-√3 c 5 10 -√3 c 5 11 √3 c + √12 c 5 8 5 12 √3 c - 3 c 3c 5 9 √50 13 10 15 √12 c 5 10 3 c 11 √50

0

103

0

0

0

0

0

0

0

-1 c + 1 c 9 15 √2 √5 2c - 1c 8 12 √5 √5 3c + 1c 10 9 √5 13

0

0

0

-√12 c - √8 c 5 9 5 13 √2 c + √8 c + 3c 5 8 5 12 10 14 1c 3c - 1c 9 10 13 √2 15 √5 1c 14 √2

-

0

103

c3

-2 c 4 √5 1c 3 √5 3c - 1c 10 5 √2 7

-1 c + 1 √5

) ) )

0

0

0

0

1c 4 √5 3c 10 3

0

c2

0

-1 c 4 √5

-1 c 3 √2

0

5

11

√6 c

1c 10 √5

2 -c14 - c 5 9 5 13 -1 c + 3c + 1c √2 c + √8 c 9 10 13 √2 15 5 8 5 12 √5 2c - 1c √2 c - √3 c + 1 c 8 12 5 9 5 13 √5 15 √5 √5 3c -√3 c 10 10 5 11

0

0

0

0

0

0

0

0

1c 11 √5

1c 10 √2

0

-2 c -√2 c 11 √5 5 10 1c -√2 c 10 √5 5 11 -1 c + 3c - 1c √2 c + √8 c 9 10 13 √2 15 5 8 5 12 √5 2c - 1c √2 c - √3 c - 1 c 8 12 5 9 5 13 √5 15 √5 √5 1c -√8 c 11 √5 5 10 √3 c 3c 10 10 5 11

(A23c)

0

0

-

(A23b)

0

√2 c -

0

0

1c 14 √5 -1 c - 1 c 9 15 √2 √5 2c - 1c 8 12 √5 √5 3c + 1c 10 9 √5 13

0

0

0

0

0

0

-2 c -√3 c - √2 c 14 √5 5 9 5 13 √3 c + √12 c - 1 c 1c 3c 10 15 5 8 5 12 √5 14 √2 13 0

-√8 c 5 3 2 c + √3 c 52 5 4

c15 0

-c11

0

0

1c 5 √5

103

0

0

c11

c7 -

1c 7 √2

0

c2

c7

0

c5 +

-1 c 3 √2

0

103

103

-1 c 4 √5

0

-√3 c + 3 c 5 1 √50 5

0

0

0

c7

-c6

-√2 c 5 3 √6 c - √2 c 5 2 5 4

0

103

0

c2

0

1c 5 √2

-1 c + 1 √5

1c 3 √5

0

0

c2

0

L)

0

- c9 -

0

0

-c4

√2 c

+

5 4 √6 c - 2 c 5 1 55 5

0

5

0

0

-c3

√6 c

c7

-√3 c - √2 c 5 1 5 5 -1 c 6 √5

√3 c

0

0

1c + 5 √2

-√3 c 5 3 3 c - √3 c 52 5 4

0

0

1c 6 √5

0

103

103

0

0

0

0

J)

K)

-

)


0

0

0

0

0

0

0

0

0

1c 11 √5

1c 10 √2

0

(A23d)

(A23e)

10594


Porion et al.

d. Heterogeneous Dipolar Relaxation In addition to the quadrupolar coupling, the dipolar coupling may also be responsible for the NMR relaxation of 7Li. The heterogeneous dipolar Hamiltonian becomes30

TD FD,L (t) m 2,m 2,-m CD (-1) r3IS(t) m)-2

(A24a)

µ0 γγ p 4π I S

(A24b)

2

HD(t) )

∑

where the dipolar coupling constant is given by

CD ) and the spin operators become D T2,0 )

(

)

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

(

)

(A24c) (A24d)

and

1 1 IR D T2,(2 ) I(S- ) - T1(1 S2 √2

(A24e)

The functions F2,D,Lm(t) in eq A24a are related to the second spherical harmonics describing the reorientation of the vector joining the two coupled spin (noted b rIS(t)) by reference with the static magnetic field30

()

D,L F2,-m )

24π5 Y

2,-m(ϑ, φ)

(A24f)

()

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-A9), leading to the matrices

Tˆ11(a) Tˆ20 ˆT21(s) d ˆ T (s) ) -diag(a1, a2, a3, a4, a5, a6, a7) · dt 22 Tˆ31(a) Tˆ32(a) Tˆ33(a)

()

Tˆ11(a) Tˆ20 ˆT21(s) Tˆ22(s) Tˆ31(a) Tˆ32(a) Tˆ33(a)

Tˆ10 Tˆ11(s) Tˆ21(a) d Tˆ22(a) ) -diag(a8, a1, a3, a4, a9, a5, a6, a7) · dt Tˆ30 Tˆ31(s) Tˆ32(s) Tˆ33(s) where

() Tˆ10 Tˆ11(s) Tˆ21(a) Tˆ22(a) Tˆ30 Tˆ31(s) Tˆ32(s) Tˆ33(s)

2 1 1 1 1 a1 ) JD0 (0) + JD0 (ωS - ωI) + JD1 (ωS) + JD1 (ωI) + JD2 (ωS + ωI) 9 18 3 6 3 1 a2 ) JD0 (ωS - ωI) + JD1 (ωI) + 2JD2 (ωS + ωI) 3 2 5 1 5 5 a3 ) JD0 (0) + JD0 (ωS - ωI) + JD1 (ωS) + JD1 (ωI) + JD2 (ωS + ωI) 9 18 3 6 3 8 1 4 1 2 a4 ) JD0 (0) + JD0 (ωS - ωI) + JD1 (ωS) + JD1 (ωI) + JD2 (ωS + ωI) 9 9 3 3 3 2 11 1 11 11 a5 ) JD0 (0) + JD0 (ωS - ωI) + JD1 (ωS) + JD1 (ωI) + JD2 (ωS + ωI) 9 18 3 6 3

(A25a)

(A25b)

(A25c) (A25d) (A25e) (A25f) (A25g)



8 4 4 4 8 a6 ) JD0 (0) + JD0 (ωS - ωI) + JD1 (ωS) + JD1 (ωI) + JD2 (ωS + ωI) 9 9 3 3 3 1 1 a7 ) 2JD0 (0) + JD0 (ωS - ωI) + 3JD1 (ωS) + JD1 (ωI) + JD2 (ωS + ωI) 6 2 a2 a8 ) , and a9 ) 2a2 3

(A25h) (A25i) (A25j)

e. Heterogeneous Dipolar Relaxation under Spin Locking Condition The same procedure is used to quantify the influence of the heterogeneous dipolar coupling on the time-evolution of the coherences under spin-locking conditions. By contrast with the quadrupolar coupling, the low frequency component of the heterogeneous dipolar relaxation mechanism is driven by the T10 coherence (cf. eq A24c) instead of the T20 coherence (cf. eq A18). The time evolution of that T10 coherence under spin-locking condition is formally described by


*

15

15

p)1

p)1

∑ dp(t)Tˆp ) ∑ h0pbV p exp(iλpt)

(A26)

The coefficients hp0 are evaluated numerically by applying the initial condition 15

Tˆ10 )

∑ h0pbV p

(A27)

p)1

The low frequency components of the heterogeneous dipolar coupling under spin-locking condition requires again the derivation of the double commutator

(

( )

* * M O [Tˆ10, [eiHs tTˆ10e-iHs t, σ*]] ) σ* P N

leading, after straightforward calculations, to the four matrices

M)

1d 5 8 -√3 d 5 10 1d 5 11 1d 5 10 √6 d - 1 d 5 12 √10 14 -1 d + 3d 13 50 15 √10 3d 50 14

N)

(

0 0

0

0

0

0

0

0

1d 5 11

0

0

0

0

0

-√3 d - √2 d 5 9 5 13 1d + 2d 5 8 5 12 1d 3d + 1 d 5 9 50 13 √10 15 3d 50 11

0

0

0

1 d 11 √10

0

d10

d15

-√6 d 5 10

52

1 d 14 √10 -√2 d 5 10 3d 50 11 3d 50 10 1d - 1d 5 8 5 12 1 d + 1d 9 5 15 √10 1d 5 14

5

52

-1 d -1 d -4 d 5 9 5 10 5 11 1d -1 d 2d 5 8 5 11 5 10 -1 d 1d + 2d -2 d + √6 d 5 11 5 8 5 12 5 9 5 13 1d 3d - 1 d 1d 4d - 2d 5 9 50 13 √10 15 5 10 5 8 5 12 -2 d 2d -√6 d 5 10 5 11 5 13 √6 d + 1 d 3d -√6 d 5 12 √10 14 50 11 5 10 -1 d 3d 0 0 13 50 15 √10 1 d 3d 2d 11 √10 50 14 5 10

√6 d

2d 5 10 -2 d 14 √5 -2 d + √6 d + 5 9 5 13 4d - 2d 5 8 5 12

0 0

2d 5 15

0

0

0

0

0

0

12

+

52

d13 +

5027

√6 d

5

15

-2 d 11 √5

-1 d 5 13 √6 d - 1 d 5 12 √10 14 3d 50 11 3d 50 10 -√6 d + 1 d 5 9 5 13 1d - 1d 5 8 5 12 1 d - 1d 9 5 15 √10 1d 5 14

)

(A28a)

0

-3 d 11 √10 3 d 10 √10 3d 5 14

0

0

-

52

d9 +

2d 5 15

4d - 2d 5 8 5 12 √6 d

9

5

+

-

-4 d 5 14 2 d - √6 d 5 13 5 15

5

9

+

2d 5 13

3d 5 13

√6 d

d9 -

-9 d 5 15 27 d 50 14 -3 d 11 √10 3 d 10 √10 -3 d 5 15 3d 5 14

0

2d 5 14 2d - 2d 5 9 5 15 4d - 2d 5 8 5 12

5027

9d + 3d 5 8 5 12

2d 5 13

0

-

d14

-

27 d - 3 d 50 9 5 13 9d + 3d 5 8 5 12

)

(A28b)

(A28c)

10596

( (


O)

P)

0

√3 d

√3 d

-

0

√3 d

5

2

5

+

3

1d 5 4

0

0

0

-

d5 +

d2 -

503

0

1 d + 5 √10

503 √110 √2 503 5

-1 d + 5 1

-1 d 5 3 -1 d 6 √10

0

3d 50 7

-1 d 5 1

0

0

0

1 - d 5 2 5 4 -1 d 5 3 √6 d 5 5 1 d 6 √10 1 d 3d 5 50 7 √10 3d 50 6

√6 d

0

-1 d 5 3 √3 d - 1 d 5 2 5 4

-4 d 5 4 2d 5 3 -2 d + √6 d 5 1 5 5

503

d5 -

-2 d 5 3

√2 d

5

0

2

+

0

0

0

-1 d 4 √10

1 d 7 √10

3d 50 4

5

2

-

-1 d + 1 d 1 5 7 √10 -1 d 5 6 -1 d 5 5 -1 d 6 √10

2d 5 7

2d + 5 2

2d 5 4 √6 d 5 3 2d 2 √5 2d 5 3

-

503

√6 d

5

1

+

7

-

0

-

√6 d

1

5

-

-

-9 d 5 7 27 d 50 6 -3 d 4 √10 3 d 3 √10 -3 d 5 7 3d 5 6

2d 2 √5 2d 5 6 2d - 2d 5 1 5 7

-

0

-

√6 d

5

1

-

0

0

1d 5 5

-3 d 4 √10 3 d 3 √10 3d 5 6 27 d - 3 d 50 1 5 5

2d 5 5

-4 d 5 6 2 d - √6 d 5 5 5 7

d3

2d 5 5

d6

0

2d 2 √5 2d + 2d 5 1 5 7

-1 d - 1 d 1 5 7 √10 -1 d 5 6

5

2d 4 √5

3d 50 4

0

5027

√6 d

0

-

0

0

d5 +

d4

d3

0

-

503 503

√2 d

0

0

-1 d + 5 1

-

d7 0

0

-1 d 4 √10

0

0

d4

0

0

0

d7

0

52

52

1 d 6 √10 √2 d 5 3

0

5 3 2d 2 √5 2d 5 3

d6

√3 d

2d 5 3 -2 d 6 √5 -2 d + √6 d + 5 1 5 5

1 + d 5 2 5 4 √3 d - √2 d 5 1 5 5

0

0

) )

Porion et al.

5027 0

d1 -

3d 5 5

(A28d)

(A28e)

The contributions from the spectral densities are implicitly included in eqs A22 and A23 for the quadrupolar coupling and in eqs A26 and A28 for the dipolar coupling. In these equations, the expected spectral densities (J0Q(λp) and J0D(λp)) will emerge after Fourier transform of the corresponding exponential laws (i.e., exp( iλpt) in eqs A22 and A26). Because of the complexity of these sets of equations, we did not succeed to extract analytical laws and we must proceed to a numerical treatment. After numerical bp) of the matrix described in eq A21a, we determine the derivation of the eigenvalues (λp) and the corresponding eigenvectors (V coefficients bp0 (eq A22) and hp0 (eq A26) by applying the initial conditions: 15 15 bp0b Vp for the quadrupolar coupling (cf. eq A22) and Tˆ10 ) ∑p)1 hp0b Vp for the dipolar coupling (cf. eq A27). Tˆ20 ) ∑p)1 These inversions are again performed numerically. From eqs A22 and A26, we numerically evaluates the coefficients cp(t) and dp(t) as linear combination of the exponential laws (exp(iλpt)). By using the Redfield theory,65,66 these exponential laws are translated into equivalent spectral densities after neglecting the dynamical shifts, leading to a numerical evaluation of the m ) 0 contributions to f(σ*) (see eq A8). References and Notes (1) Duval, F. P.; Porion, P.; Van Damme, H. J. Phys. Chem. B 1999, 103, 5730. (2) Levitz, P.; Lećolier, E.; Mourchid, A.; Delville, A.; Lyonnard, S. Europhys. Lett. 2000, 49, 672. (3) Rinnert, E.; Carteret, C.; Humbert, B.; Fragneto-Cusani, G.; Ramsay, J. D. F.; Delville, A.; Robert, J.-L.; Bihannic, I.; Pelletier, M.; Michot, L. J. J. Phys. Chem. B 2005, 109, 23745. (4) 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. (5) Porion, P.; Michot, L. J.; Fauge`re, A. M.; Delville, A. J. Phys. Chem. C 2007, 111, 13117. (6) de Azevedo, E. N.; Engelsberg, M.; Fossum, J. O.; de Souza, R. E. Langmuir 2007, 23, 5100. (7) Bourg, I. C.; Sposito, G.; Bourg, A. C. M. Appl. Geochem. 2008, 23, 3635. (8) 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. (9) 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. (10) FitzGerald, S. A.; Neumann, D. A.; Rush, J. J.; Kirkpatrick, R. J.; Cong, X.; Livingston, R. A. J. Mater. Res. 1999, 14, 1160. (11) Saito, M.; Hayamizu, K.; Okada, T. J. Phys. Chem. B 2005, 109, 3112.

(12) Perrin, J.-C.; Lyonnard, S.; Guillermo, A.; Levitz, P. J. Phys. Chem. B 2006, 110, 5439. (13) Halle, B.; Quist, P. O.; Furo, I. Phys. ReV. A 1992, 45, 3763. (14) Vilfan, M.; Apih, T.; Sebastiao, P. J.; Lahajnar, G.; Zumer, S. Phys. ReV. E 2007, 76, 051708. (15) Delville, A.; Laszlo, P.; Schyns, R. Biophys. Chemist. 1986, 24, 121. (16) Reddy, M. R.; Rossky, P. J.; Murthy, C. S. J. Phys. Chem. 1987, 91, 4923. (17) Einarsson, L.; Nordenskio¨ld, L.; Rupprecht, A.; Furo, I.; Wong, T. C. J. Magn. Reson. 1991, 93, 34. (18) Chen, S. W. W.; Rossky, P. J. J. Phys. Chem. 1993, 97, 10803. (19) Mocci, F.; Laaksonen, A.; Lyubartsev, A.; Saba, G. J. Phys. Chem. B 2004, 108, 16295. (20) Victor, K. G.; Teng, C.-L.; Dinesen, T. R. D.; Korb, J.-P.; Bryant, R. G. Magn. Reson. Chem. 2004, 42, 518. (21) Quist, P. O.; Halle, B. Phys. ReV. Lett. 1997, 78, 3689. (22) Malikova, N.; Cade`ne, A.; Marry, V.; Dubois, E.; Turq, P. J. Phys. Chem. B 2006, 110, 3206. (23) Michot, L. J.; Delville, A.; Humbert, B.; Plazanet, M.; Levitz, P. J. Phys. Chem. C 2007, 111, 9818. (24) Poinsignon, C. Solid State Ionics 1997, 97, 399. (25) Swenson, J.; Bergman, R.; Longeville, S. J. Chem. Phys. 2001, 115, 11299. (26) Skipper, N. T.; Williams, G. D.; de Siqueira, A. V. C.; Lobban, C.; Soper, A. K. Clay Miner. 2000, 35, 283.

Long-Time Scale Ionic Dynamics (27) Rotenberg, B.; Dufreˆche, J. F.; Bagchi, B.; Giffaut, E.; Hansen, J. P.; Turq, P. J. Chem. Phys. 2006, 124. (28) Sur, S. K.; Heinsbergen, J. F.; Bryant, R. G. J. Magn. Reson. A 1993, 103, 8. (29) Stapf, S.; Kimmich, R. Macromolecules 1996, 29, 1638. (30) Kimmich, R. NMR: Tomography, Diffusometry, Relaxometry; Springer-Verlag: Berlin, 1997. (31) Zavada, T.; Kimmich, R. J. Chem. Phys. 1998, 109, 6929. (32) 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. (33) Levitz, P. J. Phys.: Condens. Matter 2005, 17, S4059. (34) Perrin, J.-C.; Lyonnard, S.; Guillermo, A.; Levitz, P. Magn. Reson. Imaging 2007, 25, 501. (35) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Clarendon Press: Oxford, 1991. (36) Porion, P.; Rodts, S.; Al-Mukhtar, M.; Fauge`re, A. M.; Delville, A. Phys. ReV. Lett. 2001, 87, 208302. (37) Nakashima, Y.; Mitsumori, F. Appl. Clay Sci. 2005, 28, 209. (38) Porion, P.; Fauge`re, A. M.; Delville, A. J. Phys. Chem. C 2008, 112, 11893. (39) Blicharski, J. S. Acta Phys. Pol., A 1972, 41, 223. (40) Korb, J. P.; Delville, A.; Xu, S.; Demeulenaere, G.; Costa, P.; Jonas, J. J. Chem. Phys. 1994, 101, 7074. (41) Delville, A.; Letellier, M. Langmuir 1995, 11, 1361. (42) Hwang, D. W.; Jhao, W.-J.; Hwang, L.-P. J. Magn. Reson. 2005, 172, 214. (43) Harris, R. K.; 2Mann, B. E. NMR and the Periodic Table; Academic Press: London, 1978. (44) Sternheimer, R. M. Phys. ReV. 1966, 146, 140. (45) Balnois, E.; Durand-Vidal, S.; Levitz, P. Langmuir 2003, 19, 6633. (46) Porion, P.; Fauge`re, A. M.; Delville, A. J. Phys. Chem. C 2008, 112, 9808. (47) Jaccard, G.; Wimperis, S.; Bodenhausen, G. J. Chem. Phys. 1986, 85, 6282. (48) Ashbrook, S. E.; Wimperis, S. J. Chem. Phys. 2004, 120, 2719.

J. Phys. Chem. C, Vol. 113, No. 24, 2009 10597 (49) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1994. (50) van Gunsteren, W. F.; Berendsen, H. J. C.; Rullmann, J. A. C. Mol. Phys. 1981, 44, 69. (51) Porion, P.; Al-Mukhtar, M.; Fauge`re, A. M.; Delville, A. J. Phys. Chem. B 2004, 108, 10825. (52) Porion, P.; Fauge`re, A. M.; Delville, A. J. Phys. Chem. B 2005, 109, 20145. (53) Porion, P.; Fauge`re, A. M.; Lećolier, E.; Gherardi, B.; Delville, A. J. Phys. Chem. B 1998, 102, 3477. (54) Delville, A.; Porion, P.; Fauge`re, A. M. J. Phys. Chem. B 2000, 104, 1546. (55) Hancu, I.; van der Maarel, J. R. C.; Boada, F. E. J. Magn. Reson. 2000, 147, 179. (56) van der Maarel, J. R. C.; Jesse, W.; Hancu, I.; Woessner, D. E. J. Magn. Reson. 2001, 151, 298. (57) van der Maarel, J. R. C. Concepts Magn. Reson. Part A 2003, 19A, 117. (58) van der Maarel, J. R. C. Concepts Magn. Reson. Part A 2003, 19A, 97. (59) Cygan, R. T.; Liang, J. J.; Kalinichev, A. G. J. Phys. Chem. B 2004, 108, 1255. (60) Carley, D. D. J. Chem. Phys. 1967, 46, 3783. (61) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1985. (62) Levitz, P. E. Magn. Reson. Imaging 2005, 23, 147. (63) Deming, S. N.; Morgan, S. L. Anal. Chem. 1973, 45, 278A. (64) Barbara, T. M.; Vold, R. R.; Vold, R. L. J. Chem. Phys. 1983, 79, 6338. (65) Abragam, A. The Principles of Nuclear Magnetism; Clarendon Press: Oxford, 1961. (66) Redfield, A. G IBM J. Res. DeVelop. 1957, 1, 19.

JP9007625

Long-Time Scale Ionic Dynamics in Dense Clay Sediments Measured

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