Structural and Dynamical Properties of the Water Molecules Confined

J. Phys. Chem. C 2007, 111, 5441-5453

5441

Structural and Dynamical Properties of the Water Molecules Confined in Dense Clay Sediments: a Study Combining 2H NMR Spectroscopy and Multiscale Numerical Modeling Patrice Porion,*,† Laurent J. Michot,‡ Anne Marie Fauge` re,† and Alfred Delville*,† Centre de Recherche sur la Matie` re DiVise´ e, CNRS-UniVersite´ d’Orle´ ans, 1b Rue de la Fe´ rollerie, 45071 Orle´ ans Cedex 02, France and Laboratoire EnVironnement et Mine´ ralurgie, INPL-ENSG-CNRS, AVenue du Charmois, BP40, 54501 VandoeuVre Cedex, France ReceiVed: NoVember 29, 2006; In Final Form: February 7, 2007 2

H NMR spectroscopy was used to detect the influence of confinement on the structural and dynamical properties of water molecules adsorbed in the interlamellar space of a natural clay (Montmorillonite) within partially hydrated self-supporting films. Multiscale numerical modeling (Monte Carlo simulations, molecular dynamics, and Brownian dynamics) was used to quantify the importance of the various relaxation mechanisms likely to be responsible for the NMR relaxation of the water molecules within such complex environment. Because of the significant fraction of iron present in these natural clays, the large value of the transverse relaxation rate measured for the confined water molecules is compatible with a dominant paramagnetic coupling modulated by the long-range diffusion of water molecules. Finally, the angular variation of the apparent relaxation rate can be used to extract the distribution of the directors of the clay lamellae within the selfsupporting film.

I. Introduction In the last decades, solid/liquid interfaces have been the subject of multiple theoretical1 and experimental2 studies to elucidate how confinement modifies the structural and dynamical properties of interfacial liquids. Such information is crucial to optimize any industrial application using a solid/liquid interfacial system. In that framework, clays are ubiquitous lamellar materials exploited in many industrial applications (drilling, civil engineering, food and cosmetic industry, heterogeneous catalysis, waste storage, etc.) because of their various interesting physicochemical properties (gelling, swelling, cationic exchange capacity, high specific surface area and adsorbing power, surface acidity, etc.). As an example, the optimization of dense clay sediments as diffusion barriers used for the storage of nuclear wastes requires a detailed understanding of the structural and dynamical properties of the molecular and ionic fluids confined within the elementary clay particles. In that context, a large set of experimental work has already been performed, using X-ray3-5 and neutron3 scattering, NMR spectroscopy,6-11 quasielastic neutron scattering12-16 (QENS), NMR relaxometry,17-18 and pulsed gradient spin-echo (PGSE) NMR spectroscopy,19-21 to gain molecular information on the structural and dynamical properties of the interfacial water molecules and neutralizing counterions. For confined heavy water, 2H NMR spectroscopy8,22-26 and relaxometry8-11 provide both structural and dynamical information. Despite the reduction of the sensitivity by a factor 6.5, we selected to use 2H NMR instead of 1H NMR spectroscopy because the preferential orientation8 of the water molecules, quenched at the solid/liquid interface, is directly probed by the residual quadrupolar splitting8 of the 2H NMR resonance line. In addition, the correlation time characterizing the water reorientation8-11 can be simply deduced from the analysis of * Authors to whom correspondence should be addressed. E-mail: (P.P.) [email protected]; (A.D.) [email protected]. † CNRS-Universite ´ d’Orleáns. ‡ INPL-ENSG-CNRS.

the 2H NMR relaxation time,8-11 while 1H NMR relaxation implies simultaneous contributions from the intramolecular and intermolecular dipolar couplings. However, previous 2H NMR studies, performed with water molecules in the presence of clay, yielded rather inconsistent results; indeed, the correlation time characterizing the reorientation of water molecules in contact with clay particles was found to be 3 orders of magnitude larger8 than the mobility deduced from QENS experiments12,16 or molecular dynamics (MD) simulations.12,16 To elucidate that discrepancy, we have performed a multiscale modeling of water reorientation and diffusion in the interlayer between clay particles, by using Grand Canonical Monte Carlo (GCMC), MD, and Brownian dynamics (BD) numerical simulations in relation with 2H NMR measurements of confined water molecules under well-characterized conditions. Using such an analysis, we will precisely quantify the influence of magnetic centers (natural clay minerals contain a few percent of structural FeIII) on the 2H NMR relaxation of interfacial water molecules. We will show that the presence of these centers controls to a large extent the relaxation of diffusing water molecules near clay interfaces and that, taking into account such a contribution, 2H NMR relaxometry can be used as a sensitive probe to study water mobility close to solid/liquid interfaces containing a large fraction of paramagnetic centers and/or impurities.27 II. Materials and Methods 1. Sample Preparation. Wyoming montmorillonite (SWy-2) was purchased from the Source Clay repository of the Clay Minerals Society at Purdue University. Montmorillonite is a lamellar alumino-silicate formed with two silica layers sandwiching an aluminum (dioctahedral) layer. Because of the substitution of some tetrahedral SiIV by AlIII and some octahedral AlIII by MgII, the clay particle bears an excess negative electric charge neutralized by solvated interlamellar counterions. Prior to use, the natural clay sample was purified according to classical procedures28 and the cations were exchanged leading to monoionic clay samples neutralized by sodium counterions with a general formula that can be written as: (Si7.76, Al0.24)(Al3.06,

10.1021/jp067907p CCC: $37.00 © 2007 American Chemical Society Published on Web 03/21/2007

5442 J. Phys. Chem. C, Vol. 111, No. 14, 2007 Mg0.48, Fe0.46)O20(OH)4Na0.77. Centrifugation was used to select the clay particles according to their size.28 The average diameter of the particles used in the present study was (420 ( 170) nm. A dilute aqueous clay dispersion (1.5% w/w) was dried under nitrogen flux, and self-supporting films 0.5 mm thick were thus obtained. Dried films were then equilibrated with a reservoir of heavy water at a fixed water chemical potential (p/p° ) 0.92) by using saturated salt solution (KNO3). The water uptake of the clay film was monitored by weighting and equilibrium was reached in about one week. About twenty lamellae (1 cm × 0.3 cm) were cut into the film and were superposed inside a cylindrical Teflon holder that fits the gap inside the solenoidal coil used for NMR measurements. Because the axis of the cylindrical holder coincides with the axis of the coil, which is perpendicular to the static magnetic field, the orientation of the film director by reference with the magnetic field can be controlled with an accuracy better than 1°. During NMR measurements, the clay sample was maintained in equilibrium with an external water reservoir. The water partial pressure (p/p° ) 0.92) was selected because it corresponds to a single interlayer species with a period4 of 15.6 Å. The interlayer space is then large enough to accommodate two layers of confined water molecules. Furthermore, at such high-water partial pressure, the interlayer space is fully saturated by water molecules.4,16 2. 2H NMR Measurements. 2H NMR spectra of heavy water were recorded on DSX360 Bruker spectrometer operating at a field of 8.465 T. On this spectrometer, the pulse duration for the total inversion of the longitudinal magnetization is equal to 16 µs. Spectra were recorded using a fast acquisition mode with a time step of 1 µs, corresponding to a spectral width of 1 MHz. To reduce heterogeneities, the spectra were recorded by using a Hahn echo pulse sequence.29 The spectra and transverse relaxation rates were recorded for a complete set of orientations of the film director with reference to the static magnetic field βLF varying between 0° and 90° by using a homemade sample holder and detection coil. The longitudinal relaxation rates were measured by using the classical inversion-recovery pulse sequence. 3. Multiscale Modeling. The clay/water interaction is first described at an atomic scale by the classical clay force field,30 exploiting atomic charges evaluated from ab initio quantum calculations. The water/water and ion/water interactions are described in the framework of the simple SPC model31 of rigid water molecules. The clay/water dispersion forces are restricted to the oxygen atoms of the clay network and are identified with the oxygen/oxygen dispersion force of the water molecules derived from the SPC model.31 More information is available in the literature.30 Ewald summation32 is used in addition to the three-dimensional (3D) minimum image convention to evaluate the electrostatic energy of the clay/water interface. To ensure the convergence of the electrostatic energy, 2196 replicas of the simulation cell are used to evaluate the summation in the reciprocal space and the damping parameter in the direct space is set to 0.19 Å-1, leading to an accuracy better than 0.002.33 The clay/water interface results from the stacking of three clay layers, each composed from 24 unit cells. The octahedral substitution sites were generated randomly in each clay layer, and 18 neutralizing sodium cations are distributed within each clay interlayer. The period of the clay interlayer (15.6 Å) was deduced from the analysis of high-resolution X-ray diffraction measurements4 performed with an equivalent montmorillonite clay. The initial equilibrium state of the confined water molecules is determined by GCMC simulations34 (see Figure 1a). This procedure was already successfully3,20,35-37 used to analyze the thermodynamical properties of water molecules confined be-

Porion et al.

Figure 1. (a) Snapshot illustrating an equilibrium molecular configuration of water molecules and neutralizing sodium counterions confined between three layers of montmorillonite. (b) Local concentration profiles of the atoms confined in the interlamellar space of montmorillonite.

tween clay particles. As seen in Figure 1a,b, the water molecules quenched between two clay layers are not randomly distributed but exhibit positional and orientational ordering. Such layering was frequently detected by using the surface force apparatus2,38-39 for various confined liquids while their specific orientation was already evidenced by 2H NMR8,11 thanks to the residual splitting of their quadrupolar coupling. While water layering is simply induced by excluded volume interactions, the preferential orientation of the water molecules is the fingerprint of the specific interactions between the confined liquid and the clay surface. After thermalization of the GCMC simulations, one final configuration was selected and a Verlet algorithm40 was then used to determine the trajectories of water molecules and sodium counterions confined in the three clay layers. During these MD simulations, the vibrations of the clay network and water molecules were neglected. The quaternion procedure41 was used to describe water rotation in the framework of a generalized Verlet algorithm. A Berendsen thermostat42 was applied to each component of the translational and rotational kinetic energy of the water molecules. An elementary time step of 1 fs was used to integrate the trajectories of the water molecules and 0.1 fs for that of the sodium counterions. The average temperature was stabilized at 298 ( 5 K during 0.5 ns. Because the time scale investigated by these MD simulations does not exceed 0.5 ns, BD43 were further performed to determine on a larger time scale (∼0.1 µs), the influence of water confinement on its dynamical properties. In that framework, a large number (80 000) of individual-diffusing probes were randomly generated in the available interlamellar space and their trajectories were evaluated by using a generalized Langevin equation43 with a time step (∆t) of 1 ps

2H

NMR Spectroscopy and Multiscale Numerical Modeling

B Fn ∆t + B Xn(∆t) mγ

b x n+1 ) b xn +

(1a)

2kT∆t ) 2D∆t mγ

HD(t) ) CD

dW(z) d ln(c(z)/c°) *e bz ) kT *e bz B Fn ) dz dz

(1c)

where c° is the average hydrogen concentration in bulk water (i.e., 110 M). 4. Analysis of the 2H NMR Spectra and Relaxation Rates. We used the irreducible tensor operators44 as a complete basis set to simulate the time evolution of the spin density operator (noted σ) as well as the various Hamiltonians acting on the spin system. For spin 1 nuclei (like 2H), this basis set is composed IR IR IR IR of eight such operators44 (T IR 10, T 1(1, T 20, T 2(1, T 2(2). Detailed information on this orthonormal basis set may be found elsewhere.44-45 In that context, the quadrupolar Hamiltonian becomes45 2

Q Q (-1)m T 2,m F 2,-m (t) ∑ m)-2

(2a)

CQ )

x

CD ) -

x38 sin (2θ (t))e (t) ) x38 sin θ (t)e

Q and F 2,(2

LW

2

LW

(3b)

T D20 )

1 1 2IzSz - (I+ S- + I- S+) ) 2 x6 1 IR IR (2 x2 T IR 10Sz + (T 1+1S- - T 1-1S+)) (3c) x6

(

)

1 1 D IR T 2(1 ) - (IzS( + I(Sz) ) - (x2 T IR 10S( - 2T 1(1Sz) (3d) 2 2

1 D IR T 2(2 ) I(S- ) - T 1(1 S2

(3e)

D The functions F 2,m (t) in eq 3a are the same as in eq 2c-e but they describe now the reorientation of the vector joining the two coupled spin (noted brIS(t)) by reference with the static magnetic field. In addition to that angular dependency, the dipolar Hamiltonian is also very sensitive to the diffusion of the probe through the variation of the separation between the coupled spins (cf. the term r-3 IS (t) in eq 3a). In the framework of the Redfield theory,47-49 the time evolution of the various coherences satisfies the master equation

(4a)

where the coherences (σ*) are evaluated in the so-called rotating frame (as noted by the asterisk) and H /S is the static Hamiltonian including the radio frequency pulse and the various residual quadrupolar and dipolar couplings. The second term in eq 4a describes the contributions from the fluctuating part of the various couplings

f(σ*) ) -

iH τ , σ*(t)]]〉dτ ∫0t 〈[H /F(t), [e-iH τ H /+ F (t - τ)e / S

/ S

(4b)

(2c) -iφLW(t)

-2iφLW(t)

(2d) (2e)

Finally the spin operators describing the quadrupolar coupling are given by:

1 (3I 2z - I(I + 1)) ) T IR 20 x6

(2f)

1 Q IR ) - (IZI( + I(IZ) ) T 2(1 T 2,(1 2

(2g)

1 Q IR ) I 2( ) T 2(2 T 2,(2 2

(2h)

T Q2,0 )

µ0 x6 γI γS p 4π

dσ* ) -i[H /S,σ*] + f(σ*) dt (2b)

3 cos2 θLW(t) - 1 2

Q (t) ) ((1) F 2,(1

(3a)

r3IS(t)

and the spin operators become

2

is equal to x3/2 × π × 210 kHz for deuterium in bulk heavy water.46 Because the electrostatic field gradient felt by the deuterium nucleus in heavy water is directed along the OD bond, the time variation of the quadrupolar coupling is driven by the reorientation of this OD director in the static magnetic field, as described by the two Euler angles47-49 (θLW,φLW)

F Q2,0(t) )

(-1)

D D T 2,m F 2,-m (t)

where the dipolar coupling constant is given by

where the quadrupolar coupling constant, defined by

3 e qQ 8 pI(2I - 1)

∑

m

m)-2

(1b)

and the external confining force (F Bn) is determined from the mean force potential evaluated on the basis of the hydrogen concentration profile c(z), see Figure 1b

HQ(t) ) CQ

dipolar Hamiltonian47-49 2

where the friction coefficient (γ) monitoring the random displacement of the probe (X Bn) is evaluated from the selfdiffusion coefficient (D ) 0.5 × 10-9 m2/s) of water molecules confined in a similar condition12,16

〈X2n〉 )

J. Phys. Chem. C, Vol. 111, No. 14, 2007 5443

The same basis set may be used to describe the heteronuclear

The matrices displayed in the appendix describe the time evolution of the coherences determined in a straightforward manner from the derivation of eq 4 for the classical quadrupolar and dipolar couplings. The resulting linear differential equations are solved numerically by calculating the eigenvalues and eigenvectors of these matrices. It becomes thus possible to simulate the response of the spin system to the pulse sequences used to record the 2H NMR spectra or measure the longitudinal and transverse relaxation rates (see Section III.2). More information may be found in the literature.7,45,50 In addition to the classical quadrupolar and dipolar relaxation mechanisms, one can also consider a coupled mechanism resulting from the crossed decorrelation51 of both couplings. For that purpose eq 4b must be modified to include two different fluctuating Hamiltonians. As already shown, such crossed relaxation mechanism may occur if and only if a same molecular motion (diffusion or rotation) is responsible for the decorrelation51 of both couplings. By contrast with the classical dipolar coupling, which is very local because of its sharp spatial decrease

5444 J. Phys. Chem. C, Vol. 111, No. 14, 2007

Porion et al. In addition to the 2H doublet corresponding to the highly confined water molecules, Figure 2a,b also displays a central thin line, corresponding to the chemical shift of bulk water. This result is not surprising because the analysis of the water adsorption isotherm suggests that at a partial pressure p/p° ) 0.92, a significant fraction of adsorbed water is present as free water4 in the microscopic cavities between the solid grains resulting from the stacking of individual clay lamellae. The detection of the NMR signal of this free water, separated from that of the confined water molecules, results from a slow exchange at the NMR time scale between these two spin populations. That property may be exploited to quantify the amount of free water and to remove its contribution from the water adsorption isotherm to determine the amount of water molecules really confined inside the interlamellar space of the clay. In the system under study, the fraction of free water molecules is evaluated at (10 ( 2)% of the total amount of water. By taking into account the self-diffusion coefficient of free water, it is possible to estimate the minimum size of these cavities, because the slow exchange condition requires that the diffusion time (τdiff) of free water molecules within the cavities between clay aggregates must be larger than the transverse relaxation time of these free water molecules

L2 ) τdiff > T F2 ≈ 10-3s 2D 2

Figure 2. H NMR spectra for water molecules confined within montmorillonite with an orientation of the film director n (a) parallel and (b) perpendicular to the static magnetic field B0.

according to a r-6 IS power law, the crossed quadrupolar/dipolar relaxation mechanism can propagate to large distances because 51 it decreases only according to a r-3 IS power law. III. Results and Discussion 1. 2H NMR Spectra. Figure 2a,b exhibits the 2H NMR spectra recorded for the water molecules confined within clay montmorillonite with an orientation of the film director βLF parallel and perpendicular to the static magnetic field, respectively. The maximum splitting (νQ ) 16 kHz) is detected for a parallel orientation of the film director, as expected from the cylindrical symmetry of the clay/water interface. The splitting of the 2H resonance line results from the preferential orientation of water molecules confined8 within the clay lamellae (see Figure 1a), which is a consequence of the specific clay/water interactions. The detected splitting is compatible with the residual ordering determined by our GCMC simulations (〈(3 cos2 θCW - 1)/2〉 ) 0.089) where the Euler angle θCW defines the orientation of the OD director by reference with the normal to the clay lamella. Because the static Hamiltonian satisfies (cf. Equations A3a, 2a-c, and 2f) HQS )

x23 2πν

Q

T IR 20 )

x

〈

〉

3 3 cos2 θCW - 1 IR π(210 kHz) T 20 (5) 2 2

our GCMC simulations predict a residual splitting of 14 kHz, in good quantitative agreement with the experimental data.

(6)

leading to a minimum size of 2 µm (i.e., at least 4 times the size of the individual clay particles). The same relationship may be used to evaluate the spatial extend of the domain explored by the confined water molecules under the condition of fast exchange (i.e., leading to the detection of a single spin population). By taking into account the self-diffusion coefficient of water molecules confined in the so-called two-layers regime, Deff ) 0.5 × 10-9 m2/s12,16 (see Figure 1b), and by fixing the upper limit of the diffusion time to one tenth of the transverse relaxation rate (T2 ) 1/R2 ∼ 10-4 s, cf. Table 1) we obtain a diffusion length of 100 nm. Because that distance is large compared to the average separation between the various atomic sites of the clay/water interface, the maximum period of diffusion (10 µs) is large enough to allow for a fast exchange between these different environments, leading to a single population of the water molecules confined inside the same interlamellar space. By contrast, because the exchange between water molecules confined within the interlamellar spaces pertaining to different clay aggregates necessarily proceeds through the cavities occupied by free water molecules, that exchange must be slow at the NMR time scale. Figure 2a,b also exhibits a differential broadening of the two components of the 2H doublet. Such a behavior previously detected51 for dilute dispersions of montmorillonite clays was convincingly interpreted as resulting from the cross-correlation51 between the quadrupolar and paramagnetic couplings that contribute to deuterium relaxation. This is the consequence of the significant amount of iron present in that natural clay (see Section II.1). This cross-correlation relaxation mechanism also requires that the same dynamical process (rotation or translation) induces the decorrelation of both couplings. Finally, Figures 3 displays the variation of the 2H NMR spectra as a function of the orientation of the film director with respect to the magnetic field described by the Euler angle βLF.

TABLE 1: 2H Apparent Relaxation Rates of Water Molecules R1 bulk (s-1)

R1 confined at βLF ) 0° (s-1)


R2 free (s-1)



2.5

110 ( 10

85 ( 9

1300 ( 200

13000 ( 1000

4500 ( 400

2H



Figure 3. Angular variations of the line shape detected by 2H NMR spectroscopy for water molecules confined within montmorillonite in which βLF is the angle between the static magnetic field B0 and the director n of the clay film.

Although the splitting detected at 0 and 90° varies according to the expected relationship P2(cos βLF) ) (3 cos2 βLF - 1)/2, no cancellation of the apparent splitting can be observed at the magic angle (βLF M ) 54.7°). Such a deviation can be assigned to a distribution of the clay directors within the film. Because of the slow exchange of the water molecules confined in the interlamellar spaces pertaining to different clay aggregates, such a distribution induces NMR spectra characteristic of an oriented powder. 2. 2H NMR Relaxation Rates. Analyzing the NMR relaxation rates is a powerful tool to extract dynamical information on the molecular motions of the probe. As shown in eqs 4a,b, A4-A5, A7-A8, and A10-A11, the relaxation of the longitudinal and transverse magnetizations (i.e., respectively the IR T IR 10 and T 11(a/s) coherences) is monitored by the decorrelation of the various Hamiltonians. Molecular motions (diffusion or rotation) are themselves responsible for these decorrelations. As a consequence, two different time scales must be clearly identified47-49 corresponding to the relaxation time and correlation time, respectively. The first one quantifies the relaxation of the magnetizations with a time scale varying typically between 80 µs (for confined water), 0.8 ms (for free water) and 0.4 s (for bulk water). By contrast, the molecular motions are faster with orientation correlation times varying between 1 ps (for bulk water) and 50 ps (for confined water).16 The use of the Redfield theory47-49 to analyze the relaxation mechanisms requires such a gap between these two dynamical regimes to extend to infinity the upper bound of the integral49 in eq 4b. Table 1 displays the relaxation rates measured for parallel and perpendicular orientations of the film directors. The apparent transverse relaxation rates are determined by fitting the Hahn echo attenuation by a simple exponential law (see Figure 4a,b), describing the attenuation of the signal corresponding to free water molecules, and an exponentially damped oscillation, corresponding to the contribution of confined water molecules. The residual quadrupolar coupling felt by these confined water molecules is responsible for the detected oscillation. The 2 orders of magnitude difference between the longitudinal (110 s-1) and transverse (13 000 s-1) relaxation rates suggest a relaxation mechanism under the so-called slow modulation regime: the time scale characterizing the decorrelation of the Hamiltonian

Figure 4. Attenuation of the Hahn echo measured for an orientation of the director n of the clay film (a) parallel and (b) perpendicular to the static magnetic field B0.

Figure 5. Autocorrelation functions of the quadrupolar coupling G Qm(τ) predicted by MD simulations.

must be larger than the inverse of the resonance angular velocity (τC > 1/ω0 ) 2.9 ns). If the NMR relaxation of the deuterium nuclei in heavy water is induced by their quadrupolar coupling, which is only monitored by the reorientation of the OD director, the slow modulation condition requires a reorientation time of the confined water molecules much larger8 than the value measured by QENS12,16 on equivalent systems (50 ps). To clarify that point, we have performed MD simulations to simulate the trajectories of the confined water molecules and quantify the relative importance of the various relaxation mechanisms. As shown in Figure 5, the correlation times of the various components of the quadrupolar coupling do not exceed 5 ps. Furthermore, the integration of the three correlation functions leads to spectral densities of the same order of magnitude (J Q0 (0) ) 0.14 s-1, J Q1 (0) ) 0.10 s-1, and J Q2 (0) ) 0.09 s-1) and 5 orders of magnitude smaller than the measured transverse relaxation rate. In the same manner, MD simulations predict a fast decorrelation of the paramagnetic coupling between the deuterium of confined water molecules and the iron atoms of the octahedral network of the clay. That time scale (see Figure 6) varies between 5 and 50 ps, leading to smaller spectral densities (J D0 (0) ) 0.04 s-1, J D1 (0) ) 0.19 s-1, and J D2 (0) )


Figure 6. Autocorrelation functions of the dipolar paramagnetic coupling G Pm(τ) predicted by MD simulations.

Figure 7. Autocorrelation function of the m ) 0 component of the dipolar paramagnetic coupling G P0 (τ) predicted by BD simulations.

0.05 s-1). As a consequence, the reorientation of the confined water molecule is too fast to significantly contribute to its relaxation. What is now the origin of such a fast transverse relaxation mechanism? To determine the order of magnitude of the contribution resulting from the long-time water diffusion on the paramagnetic relaxation of the confined water molecules, we have performed simulations of 3D BD by using an external potential to confine the water trajectories between the two clay layers (see Section III.3). The width of the simulation cell is equal to 200 Å, and 800 paramagnetic centers (FeIII) are spread over two-squared networks located at the center of each octahedral layer of the confining clay lamellae. The surface density of these paramagnetic entities corresponds to the average concentration of FeIII detected in the SWy-2. It must be pointed out that in the case of SWy-2, infrared and extended X-ray absorption fine structure spectroscopy have revealed an ordered distribution of octahedral iron atoms,52,53 which fully justifies the model used in the present study for the distribution of iron. We focus that analysis on the m ) 0 component of the paramagnetic coupling because, as shown by the MD simulations (cf. Figure 6), the two other components are averaged to zero by the local motions of confined water molecules after a reduced diffusion time (τ < 100 ps). By contrast, long-time diffusion is required to achieve the average of the residual value of the m ) 0 component, partially averaged during these MD simulations. Figure 7 displays the long-time decrease of the autocorrelation function of the m ) 0 component of the paramagnetic coupling (cf. eq 3a-e) felt by the deuterium nuclei. The integration of this autocorrelation function (Figure 8) helps in quantifying the contribution of the paramagnetic relaxation mechanism to the spectral density (J D0 (0) ∼ 2700 s-1) felt by the deuterium nuclei of confined water molecules. By contrast to the paramagnetic contribution induced by the fast water reorientation occurring at a time scale of a few ps, the decorrelation induced

Porion et al.

Figure 8. Evaluation of the m ) 0 spectral density describing the order of magnitude of the paramagnetic relaxation mechanism J P0 (0) deduced from the integration of the memory function displayed in Figure 7.

Figure 9. Schematic view of the diffusing surface used to simulate the trajectories of the water molecules in a 2D space along a curved surface (see text).

by the long-time water diffusion better matches the order of magnitude of the 2H NMR relaxation rates (cf. Table 1). This result solves the previously mentioned controversy; indeed, by interpreting the large value of the transverse relaxation rate detected by 2H NMR for the water molecules confined within clay lamellae by the modulation of its paramagnetic coupling induced by the water diffusion, our analysis remains compatible with the fast reorientation of these water molecules detected by QENS12,16 or predicted by MD simulations12,16 (τR < 50 ps). However, that interpretation raises another question. In that context, how is it possible to obtain a crossed relaxation mechanism between the quadrupolar and paramagnetic couplings because these Hamiltonians are modulated by different molecular motions (i.e., reorientation and diffusion) with completely different time scales? The existence of a common mechanism, responsible for the modulation of both couplings, is indeed a prerequisite for the existence of such a crossed relaxation mechanism. An explanation may come from the exact shape of the clay particles inside the aggregates. As observed by transmission electron microscopy,54-55 natural clay particles are not perfectly flat but exhibit a curvature with radii ranging around a few hundred of Å.54-55 As a consequence, water diffusion along such curved surfaces induces small fluctuations of the residual quadrupolar coupling, fast-averaged by water orientation down to a local static value. Water diffusion is thus a molecular mechanism responsible for a partial modulation of the quadrupolar coupling and thus is able to generate an additional quadrupolar relaxation mechanism, which can lead to intercorrelation with the paramagnetic coupling. To validate such an interpretation, we have simulated the trajectories of water molecules in a two-dimensional (2D) space along a curved surface. The diffusing surface is located at half separation between two parallel and curved surfaces (see Figure 9). The separation between surfaces is fixed at 15.6 Å and the molecular probes always remain at a separation of 7.8 Å from each surface. Both limiting surfaces are flat in the X-direction with a length

2H


Figure 10. Evaluation of the long-time evolution of the memory functions of the quadrupolar, paramagnetic, and crossed quadrupolar/ paramagnetic couplings evaluated by BD simulations of water diffusion within a curved interlamellar space (see text).

Figure 11. Evaluation of the spectral densities describing the order of magnitude of the quadrupolar, paramagnetic, and crossed relaxation mechanisms deduced from the integration of the memory functions displayed in Figure 10.

of 400 Å whereas in the Y-direction, they appear as a succession of linear segments (length 100 Å) with an angular deviation of 20° (see Figure 9). Such a simple model can be used to estimate the influence of the curvature of clay particles on the relaxation behavior of confined water molecules. As shown in Figure 9, our simple model is consistent with a curvature radius of 280 Å. The two limiting surfaces are covered by paramagnetic centers with the same density as before (surface density ) 10-2 Å-2). To take into account the long range of the quadrupolar/ dipolar crossed relaxation mechanism (cf. eq A10), we also considered 40 supplementary layers4 on both sides of the central interface with the same period and the same coverage by paramagnetic centers. Figure 10 displays the autocorrelation function of the m ) 0 components of the quadrupolar, dipolar, and crossed relaxation mechanisms evaluated by simple BD diffusion of the probes along the surface illustrated in Figure 9. As before, the corresponding spectral densities J Q0 (0) are simply evaluated in Figure 11 by integrating the memory functions G Q0 (τ), leading again to a negligible contribution of quadrupolar coupling. One may wonder about that result because the memory of the quadrupolar coupling propagates now up to nanoseconds (cf. Figure 10) instead of the previous picoseconds (cf. Figure 5). While this enhancement of the time scale results from the change of the molecular mechanism responsible for the decorrelation of the quadrupolar coupling (i.e., diffusion versus rotation), the initial value of the quadrupolar coupling modulated by the water diffusion is not the instantaneous quadrupolar coupling felt by the deuterium within heavy water (i.e., 210 kHz) but the residual value of that coupling locally averaged by the fast reorientation of confined water molecules (i.e., roughly 9% of the initial

J. Phys. Chem. C, Vol. 111, No. 14, 2007 5447 value), leading to a simple enhancement by a factor of 10 of the efficiency of the quadrupolar relaxation mechanism (from 0.14 to 3 s-1). Despite the supplementary 80 layers4 of paramagnetic centers, the properties of the paramagnetic relaxation mechanism displayed in Figures 10 and 11 do not deviate significantly when compared to the previous predictions performed with two lamellae only (see Figures 7 and 8). This results from the short range of the autocorrelation function of the paramagnetic coupling, which decreases according to a r-6 IS power law (cf. eq A7). Finally, our model predicts a noticeable contribution of the crossed relaxation mechanism resulting from the intercorrelation between the quadrupolar and paramagnetic couplings (cf. Figure 11). Because of the numerous oversimplifications, these BD simulations of water diffusion confined along a simple model of curved clay surfaces cannot reproduce the exact value of the measured relaxation rates but can provide a sound basis for a qualitative interpretation of this complex dynamical phenomenon. 3. Modeling of the Angular Variation of the Transverse Rates. As shown in Figures 2-4 and in Table 1, the orientation of the clay film by reference with the static magnetic field largely modifies the NMR spectra and transverse relaxation rates of confined water molecules. A systematic study of that variation should allow the extraction of information on the orientation of the clay directors within the film. To avoid artifacts resulting from the heterogeneities of the static magnetic field, we focus that study on the variation of the transverse relaxation as measured by the Hahn echo attenuation procedure (see Section III.2). Because the F2,m(t) functions describing the angular variation of the quadrupolar and dipolar couplings (cf. eq 2c-e) are proportional to the second-order spherical harmonics, the influence of the clay orientation in the static field on the variation of these Hamiltonians may be described by using a (2) Wigner rotation matrix,47-49 noted R n,m (θLC, φLC, ψLC) New Old -1 LC F 2,m ) R(θLC, φLC, ψLC)F 2,m R (θ , φLC, ψLC) ) 2

Old (2) F 2,n R n,m (θLC, φLC, ψLC) ∑ n)-2

(7)

where (θLC, φLC, ψLC) are the three Euler angles describing the rotation transforming the old frame (with the clay director parallel to the static magnetic field) into the new one. A straightforward use of eq 7 leads to a simple description of the influence of clay orientation on the various spectral densities56 (cf. eqs A4, A7, and A10) 1 LC 2 LC 2 Old J New 0 (θ ,ω) ) (1 - 3 cos θ ) J 0 (ω) + 4 3 2 LC 2 Old 3 cos2 θLC sin2 θLC J Old 1 (ω) + (1 - cos θ ) J 2 (ω) (8a) 4 3 LC 2 LC J New sin2 θLC J Old 1 (θ ,ω) ) cos θ 0 (ω) + 2 1 (1 - 3 cos2 θLC + 4 cos4 θLC) J Old 1 (ω) + 2 1 (1 - cos4 θLC) J Old 2 (ω) (8b) 2 3 LC 2 LC 2 Old J New 2 (θ ,ω) ) (1 - cos θ ) J 0 (ω) + 8 1 (1 - cos4 θLC) J Old 1 (ω) + 2 1 (1 + 6 cos2 θLC + cos4 θLC) J Old 2 (ω) (8c) 8 for systems with cylindrical symmetry. In eqs 8a-c, the terms


Porion et al.

Figure 12. Schematic view of the different directors of the clay particles generated randomly within an angular segment in the film and used to simulate the time evolution of the 2H coherences during the Hahn echo attenuation experiment as a function of film orientation by reference with the static magnetic field (see text).

J Old m (ω) describe the intrinsic spectral densities corresponding to a parallel orientation of the clay director with respect to the static magnetic field. In a similar way, the apparent residual quadrupolar splitting varies according to

νNew ) Q

3 cos2 θLC - 1 Old νQ 2

(9)

where νOld Q is the intrinsic residual quadrupolar splitting detected initially when the clay director is parallel to the static magnetic field. We use the relaxation (cf. eqs A5, A8, and A11) and evolution (cf. eq A3c) matrixes to simulate the time evolution of the coherences during the classical pulse sequence of the Hahn echo attenuation experiment. To analyze the influence of the orientation of the clay directors within the film on the time evolution of the 2H coherences, we consider the clay directors to be distributed inside a cone (attached to the film) with a total angular aperture of 70°. This whole cone is divided into 35 elementary angular segments, each with a reduced 2° aperture (cf. Figure 12a). Within each angular segment, 10 000 clay directors are randomly generated. The initial orientation of the film is parallel to the static magnetic field (βLF ) 0°). Nine supplementary orientations are generated by a gradual rotation of the film director up to a perpendicular orientation of the film by reference with the static magnetic field (βLF ) 90°) (cf. Figure 12b). The corresponding new 10 000 clay directors are simply deduced from the initial ones by applying the adequate rotation matrix

b n

New

(

)(

)

1 0 0 sin θLCcos ψLC LF LC LC 0 ) sin θ sin ψ ) -sin βLF ‚b n Old cos β 0 sin βLF cos βLF cos θLC (10a)

where the initial clay director is defined by its two Euler angles (RFC,γFC)

(

sin R cos γ b n Old ) sin RFCsin γFC cos RFC FC

FC

)

(10b)

where the Euler angle RFC is randomly generated within the

FC FC varies uniinterval [RFC min,Rmax] and the azimuthal angle γ formly between 0 and 2π (cf. Figure 12a,b). Of course, if βLF is equal to zero, due to the small aperture of each angular segment, the clay lamellae pertaining to the same angular segment have nearly the same Euler angle θLC defining their orientation with respect to the static magnetic field, leading to the same apparent spectral densities (cf. eqs 8a-c) and apparent residual quadrupolar splitting (cf. eq 9). By contrast, after some rotation of the film (cf. eqs 10a,b), the resulting Euler angle θLC varies markedly as a function of the random azimuthal angle γFC

cos θLC ) sin βLF sin RFC sin γFC + cos βLF cos RFC (11) leading to a large distribution of the apparent spectral densities and apparent residual quadrupolar splitting, even for the clay lamellae pertaining to the same angular segment. For each orientation of the film and within each angular segment, the time response of the 2H coherences is evaluated for each clay director by calculating the eigenvalues and eigenvectors of the evolution and relaxation matrices reproducing the Hahn pulse sequence. For that purpose, eqs 8a-c are used to determine the apparent spectral densities from their initial intrinsic value when the clay director is parallel to the static magnetic field. The results from these simulations are averaged to determine the characteristic time response of the angular segments for a given orientation of the film. A leastsquare fitting procedure is used to determine the distribution of the angular segments by modeling the variation of the Hahn echo attenuation as a function of film orientation (cf. Figure 13). As shown in Figure 14, the density of distribution of the clay directors satisfies a normal distribution law FC e RFC N(RFC min e R max) )

where f(RFC) )

∫RR 1 σ

FC max FC min

f(RFC) sin RFC dRFC (12a)

x

(12b)

(

)

(RFC)2 2 exp π 2σ2

with a standard deviation of (25 ( 3)°. Such a large dispersion of the clay directors was already detected for oriented clay powders by analyzing high-resolution X-ray diffraction patterns4,57 and 2H NMR spectroscopy.58 The data displayed in Figure 13 are obtained by using the P,Old (0) ) (10 ( 3) set of parameters: νOld Q ) (17 ( 1) kHz, J 0 PQ,Old × 103 s-1 and J 0 (0) ) (2 ( 1) × 103 s-1. These

2H



Figure 13. Results of the modeling of the Hahn echo attenuation as a function of the orientation of the clay film βLF.

Figure 14. Populations of the different angular segments describing the orientation of the clay directors resulting from the fit displayed in Figure 13.

parameters are in qualitative agreement with the predictions of our GCMC and BD numerical simulations. To simplify the (ω0), fitting procedure, all other spectral densities (J P,Old 0 P,Old P,Old P,Old P,Old P,Old J P,Old (2ω ), J (0), J (ω ), J (2ω ), J (0), J (ω0) 0 0 0 0 1 1 1 2 2 -1. As shown by eq A8, and J P,Old (2ω )) are set equal to 500 s 0 2 the major contribution to the transverse relaxation rate is then (0) (2220 s-1) (i.e., 5 times smaller than the equal to 2/9 J P,Old 0 apparent transverse relaxation rate recorded for confined water molecules when the film director is parallel to the static magnetic field (cf. Figure 2a and Table 1). This enhancement of the apparent relaxation rate results from interferences between the contributions to the transverse magnetization generated by the water molecules confined between clay lamellae with various orientations of their director. Furthermore, the distribution of clay directors within the film (cf. Figure 15a,b) provides a basis for the interpretation of the evolution of the 2H NMR spectra as a function of the orientation of the film by reference with the static magnetic film (noted βLF) (cf. Figure 3). For that purpose, Figure 15 displays the density of probability of the apparent order parameter resulting from the lamellae distribution within the film, P2(cos θLC) ) (3 cos2 θLC - 1)/2. Equation 11 describes the influence on the apparent residual quadrupolar coupling (cf. eq 9) of the film orientation in the static magnetic field. These distribution functions are rather smooth and cover a broad range of accessible order parameters despite the

Figure 15. Density of probability of the order parameter P2(cos θLC) ) (3 cos2 θLC - 1)/2 defining the residual quadrupolar coupling as a function of the film orientation βLF.

detection of a single apparent quadrupolar splitting in the analysis of the Hahn echo attenuation (cf. Figure 4) recorded at βLF ) 0 and 90°, respectively. Figure 15 exhibits a continuous transition between these two limiting cases: (i) for βLF in the range 0-20°, the maximum of the distribution occurs for the upper values of the order parameter; (ii) for βLF in the range 60-90°, the maximum of the distribution occurs for the lower values of the order parameter with a more pronounced peak; (iii) for βLF in the range 30-50°, the distribution is very broad with nearly equiprobable order parameters. The same qualitative trends are detected in the analysis of the 2H NMR spectra (cf. Figure 3) and Hahn echo attenuations (cf. Figure 13).


Porion et al.

T IR lp (s) )

IV. Conclusions 2H

We have used NMR spectroscopy and relaxometry to determine the structural and dynamical properties of water molecules confined within a purified natural clay (sodium montmorillonite) under unsaturated conditions (p/p° ) 0.92). Two resonance lines are clearly identified, corresponding respectively to the water molecules adsorbed in the interlamellar space between two clay lamellae and the free water molecules located in the microcavities between the aggregates resulting from the stacking of clay lamellae. Such a separation may be exploited to distinguish between these two populations of water molecules in water adsorption isotherms. The detection of a residual quadrupolar splitting in the 2H resonance line of confined water molecules is due to specific interactions between these water molecules and the clay surfaces neutralized by solvated sodium counterions. This residual splitting is compatible with the GCMC predictions of the organization of confined water molecules in the interlamellar space of sodium montmorillonite. Furthermore, the relaxation measurements of the same confined water molecules exhibit a large difference between the longitudinal and transverse relaxation rates. Such a difference generally results from the slow modulation of the coupling mechanism responsible for the relaxation of the 2H nuclei. Multiscale dynamical simulations (MD and BD) performed to identify the 2H relaxation mechanism reveal that because of the large fraction of FeIII located within the octahedral network of montmorillonite the paramagnetic dipolar coupling, modulated by the long-range diffusion of water molecules within the interlamellar space of the clay, monitors the relaxation of these confined water molecules. This dynamical information is fully compatible with recent dynamical data obtained by QENS: despite their strong confinement, water molecules adsorbed in the interlamellar space of the clay in the so-called two layers regime. are highly mobile with a diffusion coefficient of the same order of magnitude than in bulk water and an reorientation time simply larger by 1 order of magnitude or even less. As a consequence, the confinement of water molecules in the interlamellar space of the clay, under the condition of the so-called two water layers regime (d001 ∼ 15.6 Å), does not drastically reduce the mobility of adsorbed water molecules but mainly induces some anisotropy of the water structural and dynamical properties. A larger influence of confinement on the water mobility should be detected in the so-called one water layer regime (d001 ∼ 12.6 Å). This will be the subject of further investigations. Finally, by analyzing the variation of the 2H transverse relaxation rate as a function of the orientation of the clay film within the static magnetic field, we are able to extract the distribution of the clay directors within the self-supporting film. Acknowledgment. We cordially thank Ms. S. Maddi for the purification of the clay sample and Drs. J. Fripiat, J. W. Stucki, and J. Puibasset for helpful discussions. The Bruker DSX360 spectrometer used in that study was purchased thanks to a grant from Re´gion Centre (France). The GCMC simulations were performed on a Nec supercomputer (Idris, Orsay, France). The MD and BD simulations were performed locally at CRMD on workstations purchased thanks to grants from Re´gion Centre. Appendix 1. Time Evolution of the Coherences. The various operators of the basis set are combined to yield symmetrical and unsymmetrical operators

1 IR (T l-p + T IR lp ) x2

and T IR lp (a) )

1 IR - T IR (T l-p lp ) (A1) x2

which are easily related to the spin operators

IX ) x2 T IR 11(a)

IY ) i x2 T IR 11(s)

IZ ) x2 T IR 10 (A2)

The master equation (eq 4a) may be used to describe the time evolution of the coherences under the influence of the static Hamiltonian resulting from a residual quadrupolar coupling

HQS )

x23 ω T Q

IR 20

(A3a)

in addition to a radio frequency (RF) pulse along the x-axis IX (where ω1 is related to the power of the RF pulse)

HRF ) x2 ω1T IR 11(a)

(A3b)

and a Zeeman shift of the resonance frequency

HZeeman ) x2 ωres T IR 10

()

(A3c)

Thanks to the completeness of the basis set composed from the irreducible tensor operators, the first term of eq 4a may be translated into a linear equation: T IR 10

T IR 11(s)

T IR 11(a)

IR d T 20 dt T IR 21(s)

(

)

T IR 21(a) T IR 22(s)

T IR 22(a)

-iω1 0 0 0 -iω1 0 iωres 0 iωres 0 0 0

0 0 iωQ

0

0

- x3 iω1 0

0

iωQ 0 iωQ 0 0 0 0 0

0 0 0

0

0

- x3 iω1 0 iωres 0 -iω1 0 0

0

0 0 iωQ 0 0 0 0

0 0 0 0

iωres -iω1 0 -iω1 0 0 2iωres 0 0 -iω1 2iωres 0

)( ) T IR 10

T IR 11(s)

T IR 11(a) T IR 20

T IR 21(s)

T IR 21(a) T IR 22(s)

T IR 22(a)

(A3d)

By defining the spectral densities as the Fourier transform of the autocorrelation functions G Qm(τ) describing the loss of memory of the fluctuating part of the coupling, one obtains

J Qm(mω0) ) (-1)m C2Q

Q Q (0) - 〈F 2,-m 〉) × ∫0∞ (F 2,-m Q Q (F 2,m (τ) - 〈F 2,m 〉)eimω0τ dτ (A4)

which can be used to describe the time evolution of the coherences induced by the quadrupolar relaxation mechanism

2H

()

()



T IR 10

T IR 10

T IR 11(s)

T IR 11(s)

T IR 11(a)

T IR 11(a)

IR T IR 20 d T 20 Q Q Q Q Q Q Q Q ) diag(A , B , B , C , D , D , E , E )‚ dt T IR(s) T IR 21 21(s)

T IR 21(a)

(A5)

T IR 21(a)

T IR 22(s)

T IR 22(s)

T IR 22(a)

T IR 22(a)

where

AQ ) J Q1 (ω0) + 4J Q2 (2ω0) BQ ) 3/2 J Q0 (0) + 5/2 J Q1 (ω0) + J Q2 (2ω0) CQ ) 3J Q1 (ω0) DQ ) 3/2 J Q0 (0) + 1/2 J Q1 (ω0) + J Q2 (2ω0) EQ ) J Q1 (ω0) + 2J Q2 (2ω0)

In the case of the NMR relaxation induced by dipolar paramagnetic coupling, we proceed to a first-order approximation to evaluate the average values of the paramagnetic magnetization

〈SZ〉 ) 〈S+ 〉 ) 〈S- 〉 ) 0

(A6a)

2 〈S+ S- 〉 ) 〈S- S+ 〉 ) NS S(S + 1) 3

(A6b)

1 〈S2Z〉 ) NS S(S + 1) 3

(A6c)

where NS is the total number of paramagnetic spin coupled with the deuterium nucleus. These approximations are useful to derive the spectral densities describing the paramagnetic relaxation

J Dm(ω)

) (-1) NS S(S + m

1)C2D

∫0

∞

e

imω0τ

(

D F 2,-m (0)

-

〈 〉)( D F 2,-m

D F 2,m (τ)

-

() () r3IS(0)

r3IS

r3IS(τ)

〈 〉) D F 2,m

r3IS

dτ

(A7)

which are used to describe the time evolution of the coherences induced by the dipolar paramagnetic relaxation mechanism

T IR 10

T IR 10

T IR 11(s)

T IR 11(s)

T IR 11(a)

T IR 11(a)

IR T IR 20 d T 20 P P P P P P P P ) - diag(A , B , B , C , D , D , E , E )‚ IR dt T IR(s) T 21(s) 21

T IR 21(a)

T IR 21(a)

T IR 22(s)

T IR 22(s)

T IR 22(a)

T IR 22(a)

(A8)


Porion et al.

where

AP ) 1/9 J P0 (ω0 - ωS) + 1/3 J P1 (ω0) + 2/3 J P2 (ω0 + ωS) BP ) 2/9 J P0 (0) + 1/18 J P0 (ω0 - ωS) + 1/6 J P1 (ω0) + 1/3 J P1 (ωS) + 1/3 J P2 (ω0 + ωS) CP ) 1/3 J P0 (ω0 - ωS) + J P1 (ω0) + 2 J P2 (ω0 + ωS) DP ) 2/9 J P0 (0) + 5/18 J P0 (ω0 - ωS) + 5/6 J P1 (ω0) + 1/3 J P1 (ωS) + 5/3 J P2 (ω0 + ωS) and EP ) 8/9 J P0 (0) + 1/9 J P0 (ω0 - ωS) + 1/3 J P1 (ω0) + 4/3 J P1 (ωS) + 2/3 J P2 (ω0 + ωS) To derive the influence of crossed quadrupolar/dipolar relaxation mechanism on the time evolution of the coherence, one must proceed to a second-order approximation of the longitudinal magnetization of the paramagnetic spins

pωSS(S + 1)NS 3kT

〈Sz〉 ) 〈Seq z 〉)-

(A9a)

which results from a linearization of the Boltzmann distribution law [exp(-HS/kT)] with the Zeeman Hamiltonian HS ) pωS SZ. The resulting coupling constant is thus defined by

CQD ) CQCD〈Seq z 〉

(A9b)

where CQ and CD are defined in eqs 2b and 3b, respectively. The corresponding spectral densities become m J QD m (ω) ) (-1) CQD

()(

∫0∞ eimωt

((

D F 2,-m (0)

r3IS(0)

-

〈 〉) D F 2,-m

r3IS

Q Q (F 2,m (τ) - 〈F 2,m 〉) +

(

D F 2,m (τ)

r3IS(τ)

-

〈 〉) D F 2,m

r3IS

)

Q Q (F 2,-m (0) - 〈F 2,-m 〉) dτ

which are used to determine the time evolution of the coherences

d dt

T IR 10

0

0

0

x12 J QP 1 (ω0) 0

T IR 11(s)

0

0

0

0

T IR 11(a) T IR 20 T IR 21(s) T IR 21(a) T IR 22(s) T IR 22(a)

0 0

QP 2 J QP 0 (0) + J 1 (ω0) 0

0 0 J QP 0 (0)

+

J QP 1 (ω0)

0

0

0

0

2

x12 J QP 1 (ω0) 0

0

0

0

0

0 0

0

QP 2 J QP 0 (0) + J 1 (ω0) 0

0

0

0

0 0

0 0 0

0 0 0

QP 2 J QP 0 (0) + J 1 (ω0) 0 0 0 0 0

0 0 0

0 0 0

0 0 0 0 0 0

0

)-

0

0 0

(A10)

)( ) T IR 10

T IR 11(s)

T IR 11(a)

‚

T IR 20

T IR 21(s)

T IR 21(a) T IR 22(s)

T IR 22(a)

(A11)

which is described by a nondiagonal relaxation matrix. References and Notes (1) Henderson, D. Fundamentals of Inhomogeneous Fluids; M. Dekker Inc.: New York, 1992. (2) Israelachvili, J. N. Intermolecular and Surface Forces: with Applications to Colloidal and Biological Systems; Academic Press: Orlando, FL, 1992. (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) Ferrage, E.; Lanson, B.; Malikova, N.; Planç on, A.; Sakharov, B.; Drits, V. A. Chem. Mater. 2005, 17, 3499. (5) Drits, V. A.; Tchoubar, C. X-Ray Diffraction by Disordered Lamellar Structures; Springer: Berlin, 1990. (6) Porion, P.; Al-Mukhtar, M.; Fauge`re, A. M.; Delville, A. J. Phys. Chem. B 2004, 108, 10825. (7) Porion, P.; Fauge`re, A. M.; Delville, A. J. Phys. Chem. B 2005, 109, 20145. (8) Delville, A.; Grandjean, J.; Laszlo, P. J. Phys. Chem. 1991, 95, 1383. (9) Delville, A.; Letellier, M. Langmuir 1995, 11, 1361. (10) Fripiat, J.; Cases, J.; Franç ois, M.; Letellier, M. J. Colloid Interface Sci. 1982, 89, 378. (11) Woessner, D. E.; Snowden, B. S., Jr. J. Chem. Phys. 1969, 50, 1516. (12) Malikova, N.; Cade`ne, A.; Marry, V.; Dubois, E.; Turq, P. J. Phys. Chem. B 2006, 110, 3206.

(13) Swenson, J.; Bergman, R.; Howells, W. S. J. Chem. Phys. 2001, 115, 11299. (14) Skipper, N. T.; Lock, P. A.; Titiloye, J. O.; Swenson, J.; Mirza, Z. A.; Howells, W. S.; Fernandez-Alonso, F. Chem. Geol. 2006, 230, 182. (15) Poisignon, C. Solid State Ionics 1997, 97, 399. (16) Michot, L. J.; Delville, A.; Humbert, B.; Plazanet, M.; Levitz, P. J. Phys. Chem. C, submitted for publication. (17) Sur, S. K.; Heinsgergen, J. F.; Bryant, R. G. J. Magn. Reson., Ser. A 1993, 103, 8. (18) Korb, J. P.; Delville, A.; Xu, Sh.; Demeulenaere, D.; Costa, P.; Jonas, J. J. Chem. Phys. 1994, 101, 7074. (19) Porion, P.; Rodts, S.; Al-Mukhtar, M.; Fauge`re, A. M.; Delville, A. Phys. ReV. Lett. 2001, 87, 208302. (20) Nakashima, Y.; Mitsumori, F. Appl. Clay Sci. 2005, 28, 209. (21) Porion, P.; Al-Mukhtar, M.; Fauge`re, A. M.; Pellenq, R. J. M.; Meyer, S.; Delville, A. J. Phys. Chem. B 2003, 107, 4012. (22) Chang, T. H.; Cheng, C. P.; Yeh, C. T. J. Phys. Chem. 1991, 95, 5239. (23) Hakala, M. R.; Wong, T. C. Langmuir 1986, 2, 83. (24) Jacobsen, J. P.; Schaumburg, K. J. Magn. Reson. 1976, 24, 173. (25) Xiong, J. C.; Maciel, G. E. J. Phys. Chem. B 1999, 103, 5543. (26) Sparrman, T.; Westlund, P.-O. J. Phys. Chem. B 2001, 105, 12524. (27) Godefroy, S.; Korb, J. P.; Fleury, M.; Bryant, R. G. Phys. ReV. E 2001, 64, 021605. (28) Michot, L. J.; Bihannic, I.; Porsch, K.; Maddi, S.; Mougel, J.; Bavarian, C.; Levitz, P. Langmuir 2004, 20, 10829. (29) Hahn, E. L. Phys. ReV. 1950, 80, 580.

2H


(30) Cygan, R. T.; Liang, J. J.; Kalinichev, A. G. J. Phys. Chem. B 2004, 108, 1255. (31) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J. Phys. Chem. 1987, 91, 6269. (32) Heyes, D. M. Phys. ReV. B 1994, 49, 755. (33) Hummer, G. Chem. Phys. Lett. 1995, 235, 297. (34) Adams, D. J. Mol. Phys. 1974, 5, 1241. (35) Delville, A. Langmuir 1991, 7, 547. (36) Delville, A. J. Phys. Chem. 1993, 97, 9703. (37) Delville, A. J. Phys. Chem. 1995, 99, 2033. (38) Christenson, H. K.; Horn, R. G. J. Colloid Interface Sci. 1985, 103, 50. (39) Christenson, H. K. J. Phys.: Condens. Matter 2001, 13, R95. (40) Allen, M. P.; Tildesley, D.J. Computer Simulation of Liquids; Clarendo Press: Oxford, 1994. (41) Evans, D. J.; Murad, S. Mol. Phys. 1977, 34, 327. (42) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. J. Chem. Phys. 1984, 81, 3684. (43) van Gunsteren, W. F.; Berendsen, H. J. C.; Rullmann, J. A. C. Mol. Phys. 1981, 44, 69. (44) Mu¨ller, N.; Bodenhausen, G.; Ernst, R. R. J. Magn. Reson. 1987, 75, 297. (45) van der Maarel, J. R. C. J. Chem. Phys. 1993, 99, 5646. (46) Edmonds, D. T.; Mackay, A. L. J. Magn. Reson. 1975, 20, 515.

J. Phys. Chem. C, Vol. 111, No. 14, 2007 5453 (47) Mehring, M. Principles of High Resolution NMR in Solids, 2nd ed.; Springer-Verlag: Berlin, 1983. (48) Kimmich, R. NMR: Tomography, Diffusometry, Relaxometry; Springer-Verlag: Berlin, 1997. (49) Abragam, A. The Principles of Nuclear Magnetism; Clarendon Press: Oxford, 1961. (50) 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. (51) Petit, D.; Korb, J. P.; Delville, A.; Grandjean, J.; Laszlo, P. J. Magn. Reson. 1992, 96, 252. (52) Vantelon, D.; Pelletier, M.; Michot, L. J.; Barre`s, O.; Thomas, F. Clay Miner. 2001, 36, 369. (53) Vantelon, D.; Montarge`s-Pelletier, E.; Michot, L. J.; Briois, V.; Pelletier, M.; Thomas, F. Phys. Chem. Miner. 2003, 30, 44. (54) Hetzel, F.; Teyssier, D.; Jaunet, A. M.; Doner, H. Clays Clay Miner. 1994, 42, 242. (55) Cebula, D. J.; Thomas, R. K.; Middleton, S.; Ottewill, R. H.; White, J. W. Clays Clay Miner. 1979, 27, 39. (56) Barbara, Th. M.; Vold, R. R.; Vold, R. L. J. Chem. Phys. 1983, 79, 6338. (57) So, H.; Jung, H.; Choy, J. H.; Belford, R. L. J. Phys. Chem. B 2005, 109, 3324. (58) Woessner, D. E. Mol. Phys. 1977, 34, 899.

Structural and Dynamical Properties of the Water Molecules Confined

Recommend Documents