Influence of Confinement on the Long-Range Mobility of Water

J. Phys. Chem. C 2007, 111, 13117-13128

13117

Influence of Confinement on the Long-Range Mobility of Water Molecules within Clay Aggregates: A 2H NMR Analysis Using Spin-Locking Relaxation Rates 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: May 3, 2007

2H

spin-locking relaxation measurements were used to probe the dynamical properties of water molecules confined between individual montmorillonite lamellae within dense clay sediments. Because of the significant amount of structural iron in these natural clays, the dipolar paramagnetic coupling is the major mechanism responsible for the relaxation of these confined water molecules. By calculating the time evolution of the various coherences of spin 1 nuclei under the irradiation pulse implied in these spin-locking relaxation measurements, we illustrate how the paramagnetic coupling largely extends the dynamical range usually investigated by such relaxation measurements. We exploit that property to compare the water mobility predicted, at a reduced time scale, by molecular dynamics simulations with the average residence time of the water molecules inside the clay interlayer.

I. Introduction Clays are synthetic and natural materials used in many industrial applications (drilling, cosmetic, paints and food industry, cracking, heterogeneous catalysis, and waste management) exploiting their various physicochemical properties (swelling, gelling, thixotropy, high specific surface area and adsorbing power, surface acidity, etc.). A recent application of swelling clay minerals concerns the storing of nuclear waste. In that context, an accurate evaluation of the efficiency of dense clay sediments as diffusion barriers requires a detailed analysis of the mobility of the molecular and ionic fluids confined between the individual platelets constituting the elementary grains of the clay sediment. Because of the large fraction of water molecules in the interlamellar space of dense clay sediments, their average residence time in this highly confined environment is a crucial parameter for modeling the long-time diffusion of water molecules inside the clay aggregates and thus their water retention efficiency. For that purpose, various dynamical studies have been performed to quantify water mobility by using IR spectroscopy,1-3 quasi-elastic neutron scattering,4-8 NMR relaxometry,9-12 and pulsed gradient spin-echo (PGSE) NMR spectroscopy.13-15 These different experimental methods are powerful tools to investigate water mobility on various time scales: picoseconds for IR spectroscopy, nanaseconds for neutron scattering, and milliseconds for PGSE NMR spectroscopy. By contrast with these experimental methods, NMR relaxometry appears as the only suitable method to determine the water residence time since the time scale accessible by NMR relaxometry16 (10-7 s < τ < 10-2 s) perfectly matches the expected residence time of water molecules within the interlamellar space of the clay platelets whose lateral extension is * Authors to whom correspondence should be addressed. E-mail: [email protected] (A.D.) and [email protected] (P.P.). † Centre de Recherche sur la Matie ` re Divise´e. ‡ Laboratoire “Environnement et Mine ´ ralurgie”.

smaller than 1 µm. Field cycling NMR spectrometers were thus widely used to investigate water mobility10,12 in dilute clay dispersions. However, such experiments are rather useless in the case of dense clay sediments because of the significant shortening of water relaxation time that becomes much smaller than the time required to commute the magnetic field. As a consequence, only relaxation measurements under the so-called spin-locking condition9,11 can be used to investigate long-range water mobility by performing dynamical measurements in the adequate domain of angular velocities (between 102 and 105 rad‚s-1). A swelling clay (Wyoming montmorillonite), similar to the clays used for nuclear waste storage, was selected for that study. Prior to use, that natural clay was purified, exchanged, and selected according to its size. Previous 2H NMR experiments17 have successfully identified the main mechanism responsible for the fast relaxation of the deuterium of water molecules confined between the elementary clay platelets in dense clay aggregates. Because of the significant amount of structural iron in the network of this natural clay, dipolar paramagnetic coupling was shown to monitor17 the deuterium relaxation mechanisms. The diffusion of confined water molecules was also identified as the main dynamical process17 responsible for the decorrelation of that paramagnetic coupling. 2H NMR spectroscopy and relaxation measurements were also used to characterize both the average organization17 of the water molecules confined between the clay platelets and the distribution of the individual clay directors17 within the dense sediments. In the present study, 2H NMR relaxometry is used to quantify the impact of the two-dimensional (2D) diffusion of the water molecules confined in the interlamellar space of the clay by measuring the spin-locking relaxation times over a broad frequency range. Because of the short range of the dipolar coupling,16,18-19 the long time behavior of the memory function of the dipolar coupling, and thus the low-frequency variation of the deuterium relaxation rates, are both driven by the

10.1021/jp073393y CCC: $37.00 © 2007 American Chemical Society Published on Web 08/16/2007

13118 J. Phys. Chem. C, Vol. 111, No. 35, 2007

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conditional probability that the diffusing probe comes back to its initial position. As shown previously,20 that conditional probability, quantified by the initial value of the self-diffusion propagator, is highly sensitive to the effective dimension of the diffusion space. That property is exploited here to determine the average residence time of water molecules in the interlamellar space of a natural montmorillonite clay. II. Materials and Methods 1. Sample Preparation. Wyoming montmorillonite (SWy2) 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 procedures,21 and the cations were exchanged leading to mono-ionic clay samples neutralized by sodium counterions with a general formula that can be written as (Si7.76, Al0.24)(Al3.06, Mg0.48, Fe0.46)O20(OH)4Na0.77. Centrifugation was used to select the clay particles according to their size. 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 1 week. About 20 lamellae (1 cm × 0.3 cm) were cut into the film and stacked inside a cylindrical Teflon holder which fits the gap inside the solenoidal coil used for NMR measurements. The water partial pressure (p/p° ) 0.92) was selected because it corresponds to a single interlayer species22 with a period of 15.6 Å. The interlayer space is then large enough to accommodate two layers of confined water molecules.22 Furthermore, at such high water partial pressure, the interlayer space is fully saturated by water molecules.5,22 2. 2H NMR Measurements. 2H NMR spectra of heavy water were recorded on a DSX360 Bruker spectrometer operating at a field of 8.465 T. On this spectrometer, the pulse duration for the total inversion of the longitudinal magnetization is equal to 20 µs. Spectra were recorded using a fast acquisition mode with a time step of 1 µs, corresponding to a spectral width of 1 MHz. In order to reduce heterogeneities, the spectra where recorded by using a Hahn echo pulse23 sequence. The spinlocking relaxation rates24 were measured at six angular velocities: 1.57 × 105, 7.85 × 104, 3.93 × 104, 1.96 × 104, 9.81 × 103, and 4.91 × 103 rad‚s-1. Spin-locking relaxation measurements were performed by applying a high power 90° pulse (corresponding to ω1 ) 1.57 × 105 rad‚s-1) directly followed by an irradiation at reduced power (cf. above) with a phase shifted by 90° in order to lock the magnetization24 in a direction perpendicular to the static magnetic field. The maximum duration of the second irradiation pulse was 0.5 ms. The spectra and spin-locking relaxation rates were recorded for two different orientations of the film director with reference to the static magnetic field (βLF ) 0° and 90°) by using a home made sample holder and detection coil. Figure 1 exhibits the 2H NMR spectra recorded at these two orientations: in addition to the weak signal due to free water, a strong doublet results from the preferential orientation of the water molecules

Figure 1. 2H NMR spectra recorded for heavy water confined between montmorillonite clay platelets within self-supporting films. The spectra were recorded at two orientations of the film directors (a, parallel; b, perpendicular) by reference with the static magnetic field B0.

confined between the individual clay platelets. Dipolar paramagnetic coupling was identified17 as the leading mechanism responsible for the relaxation of these confined water molecules. In addition to that leading relaxation mechanism, a secondary quadripolar-paramagnetic cross relaxation mechanism was identified17 as the origin of the differential broadening25-26 of the 2H doublet. As shown in Figure 1, the center of the 2H doublet does not coincide with the NMR signal of free water. Furthermore, because of some residual value of the paramagnetic coupling, the location of the 2H doublet varies as a function of film orientation, according to the classical relationship

H SP(βLF) ∝

(

)

3 cos2 βLF - 1 2

(1)

The distribution of the clay directors within the film was extracted by analyzing17 the variation of the 2H transverse relaxation rates of confined water molecules as a function of film orientation within the static magnetic field. The angle (RFC) between the film and the clay directors was shown to satisfy a normal distribution law17 FC e R FC N(R FC min e R max) )

∫RR

FC max FC min

f(RFC) sin RFC dRFC

(2a)

where

f(RFC) )

1 σ

x

(

)

(RFC)2 2 exp π 2σ2

(2b)

with a standard deviation of 25 ( 3°. Figure 2 further illustrates the distribution of the apparent orientations of the clay directors

Long-Range Mobility of Water Molecules

J. Phys. Chem. C, Vol. 111, No. 35, 2007 13119

F P2,0(t) ) P (t) ) ((1) F2,(1

3 cos2 θLW(t) - 1 2

x38 sin(2θ

LW

(3f)

(t)) e-iφ

LW(t)

(3g)

and P (t) ) F2,(2

Figure 2. Density of probability describing the distribution of the clay directors in the magnetic field for two orientations (parallel and perpendicular) of the clay film by reference with the static magnetic field B0.

by reference with the static magnetic field, resulting from both the orientation of the film director into the static field (βLF) and the distribution of clay directors inside the film (RFC in eq 2). As a consequence, the exact values of the static quadupolar and paramagnetic couplings result from averages over a broad distribution of clay orientations, determining the apparent splitting and the location of the resonance line of the confined water molecules. III. Results and Discussion 1. Water Diffusion and Paramagnetic Relaxation. Since deuterium is a spin 1 nucleus, a complete basis set of irreducible IR IR IR IR tensors27 (T IR 10, T1(1, T 20, T2(1, and T2(2) is used to describe the time evolution of all of the coherences implied in the various 2H NMR experiments. Detailed information on this orthonormal basis set may be found elsewhere.27,28 The heteronuclear dipolar Hamiltonian becomes27 2

HP(t) ) CP

∑

m

(-1)

m)-2

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

rIS3(t)

(3a)

where the dipolar coupling constant is given by

CP ) -

µ0 x6γIγSp 4π

(3b)

and the spin operators:

T P20 )

1 1 2IzSz - (I+S- + I-S+) ) 2 x6 1 IR IR (2x2T IR 10Sz + (T 1+1S- - T 1-1S+)) (3c) x6

(

)

1 P T2(1 ) - (IZS( + I(SZ) ) 2 1 IR - (x2T IR 10S( - 2T 1(1SZ) (3d) 2 1 P IR T2(2 ) I(S- ) - T1(1 S2

(3e)

P The functions F 2,m (t) in eq 3a are the spherical harmonics describing the reorientation of the vector joining the two coupled spins (noted b rIS(t)) by reference with the static magnetic field B0 by using the two Euler angles (θLW, φLW):

x38 sin θ 2

LW

(t) e-2iφ

LW(t)

(3h)

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 rIS-3(t) in eq 3a). In the framework of the Redfield theory,16,18,19 the time evolution of the various coherences satisfies the master equation:

dσ* ) -i[H /S,σ*] + f(σ*) dt

(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 paramagnetic couplings. The second term in eq 4a describes the contributions from the fluctuating part of the various couplings: iH τ , σ*(t)]]〉 dτ ∫0t〈[H /F(t), [e-iH τH /+ F (t - τ) e

f(σ*) ) -

/ s

/ s

(4b)

As shown previously, water diffusion totally drives17 the modulation of the paramagnetic coupling. Since this study focuses on the influence of confinement on the long time diffusion of water molecules through the analysis of their NMR relaxation at low frequencies, we will perform a detailed analysis of the m ) 0 component of the dipolar coupling only (cf. eq 3a) because it is not modulated by the fast rotation of the water molecules.16,18,19 In that framework, the self-diffusion propagator is very useful to quantify the influence of diffusion on the long-time behavior of the memory function describing the decorrelation of the paramagnetic coupling. For that purpose, we divide the interlamellar space of the clay, with a separation of 15.45 Å, by a set of 40 successive layers. The deuterium population within each layer (noted pi) is determined by preliminary GCMC simulations29,30 of the distribution of confined water molecules (cf. Figure 3a). The deuterium layers are sandwiched between two lamellae located in the octahedral layer of the two confining clay platelets (cf. Figure 3b). To simplify the derivation of the paramagnetic relaxation, the paramagnetic centers are uniformly distributed on these two limiting layers, reproducing the amount of iron substitution in Wyoming montmorillonite (σP ) 9.2 × 10-3 Å-2). The radial motion of the confined water molecules is described by the 2D selfdiffusion propagator31

P(F,t|0,0) )

( )

1 F2 exp 4πDt 4Dt

(5a)

defining the density of probability that the diffusing probe, initially located at the origin, reaches the radial distance F at time t. In eq 5a, D is the 2D diffusion coefficient describing the transverse motion of the confined water molecules (i.e., parallel to the clay surface). The decorrelation of the m ) 0 component of the paramagnetic coupling then becomes


Porion et al.

Figure 3. (a) Average concentration profiles of the various atoms (oxygen, proton, and sodium) confined between montmorillonite clay platelets determined by GCMC simulations. (b) Schematic view illustrating the integration procedure of eq 5c. (c) Illustration of the integrand I(F) evaluated in eq 5c and used in eq 5b (see text). (d) Memory function describing the decorrelation of the m ) 0 component of the paramagnetic coupling induced by 2D diffusion of the water molecules confined between the clay lamellae. (e) Variation of the initial value of the m ) 0 component of the spectral density JP0 (0) as a function of the residence time of water molecules inside the interlamellar space of the clay (see text).

G P0 (t)

2πσPS(S + 1)CP2 ) 9

∫0∞ FI(F) P(F,t|0,0) dF

(5b)

where zij is the longitudinal separation between each of the N layers (labeled i) containing the deuterium atoms and the two layers (labeled j) containing the iron atoms (cf. Figure 3b). Equations 5b and 5c are derived in a straightforward manner from eqs 3c and 3f, respectively, since:

with 2

N

∑ ∑ pi ∫0 j)1 i)1

I(F) )

∫0

∞

2π

r2 - 2zij2 r

1 〈Sz2〉 ) NSS(S + 1) 3

dR ×

and

r2 + F2 - 2rF cos R - 2zij2

(r2 + zij2)5/2 (r2 + F2 - 2rF cos R + zij2)5/2

(6a)

dr (5c)

3 cos θij - 1 ) 2

2zij2 - r2 zij2 + r2

(6b)



(cf. Figure 3b). Previous MD simulations17 were used to determine the 2D diffusion coefficient of the confined water molecules (D ) 0.5 × 10-9 m2/s), in agreement with various experimental studies.5,6 Figure 3c displays the variation of I(F) as a function of the radial distance F, illustrating the short range of the paramagnetic coupling. After integration of eq 5b, we obtain the m ) 0 component of the memory function describing the long-time decorrelation of the paramagnetic coupling induced by water diffusion (cf. Figure 3d). Integration of that memory function finally yields the low-frequency limiting value of the spectral density corresponding to the m ) 0 component of the paramagnetic coupling (cf. eq 4b). As shown in Figure 3e, the corresponding spectral density evolves according to a logarithmic law. The maximum value of the spectral density is thus limited by the average residence time (noted τB) of the water molecules inside the interlayer space between two clay platelets. That average residence time (τB) determines the effective upper bound (tsup) for the integration of the memory function of the dipolar coupling. After desorption, the water molecule reaches environments corresponding to free water, with a three-dimensional (3D) diffusion space characterized by a much lower spectral density. As a consequence, one may represent the m ) 0 component of the spectral density describing the influence of the paramagnetic coupling on the relaxation of the confined water molecules by three different regimes (cf. Figure 4): a plateau at very low angular velocities, a logarithmic decrease in the intermediate regime, and a second plateau at high angular velocities. The time scale evaluated from the inverse of the angular velocity corresponding to the crossover (noted ωC in Figure 4) between the intermediate and the low-frequency regimes is a rough estimate of the average residence time of the water molecules in the clay interlayer. Such a logarithmic variation of the spectral density was already reported for relaxation measurements performed in confined media.32,33 It was also measured in dilute clay dispersions,9 where the low-frequency cutoff of the spectral density was then the fingerprint of the average residence time of the water molecules diffusing on the clay surface. 2. Relaxation Under Spin-Locking Conditions. In order to calculate the spin-locking relaxation rates induced by the paramagnetic coupling and the quadrupolar-paramagnetic crossed relaxation mechanism described in the second term of eq 4a, we must first evaluate the time evolution of the coherences under the influence of the various static Hamiltonians described in the first term of eq 4a by including the residual quadrupolar coupling:

HQS )

x23 ω T

IR Q 20

(7a)

HRF ) x2ω1T IR 11(a)

(7b)

the second irradiation pulse:

the Zeeman-like Hamiltonian resulting from the frequency offset:

HZeeman ) x2ωresT IR 10

(7c)

where ωQ is the residual quadrupolar splitting, ω1 is the angular velocity of the irradiating field, and ωres is the frequency offset resulting from the angular distribution (cf. eqs 1 and 2) of the static paramagnetic coupling. In eq 7b, we have used the linear combination:27

T IR lp (s) )

( )(

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

Evaluation of the first term of eq 4a leads to:

T IR 20 0 T IR 11(a) 0 T IR 21(s) -x3ω1 IR d T 22(s) )i 0 dt T IR 0 10 0 T IR (s) 11 IR 0 T 21(a) 0 IR T 22(a)

0 -x3ω1 ωQ 0 ωQ 0 -ω1 0 0 0 ωres 0 ωres 0 0 0

0 0 0 0 -ω1 0 0 0 0 0 -ω1 0 0 0 2ωres 0

)( )

T IR 20 0 0 0 IR T 11(a) ωres 0 0 T IR 21(s) ωres 0 0 IR 2ωres ‚ T 22(s) 0 0 T IR -ω1 0 0 10 ωQ 0 0 T IR 11(s) ωQ 0 -ω1 T IR(a) 21 -ω1 0 0 T IR 22(a)

(8)

(9)

which simplifies in two sets of independent coherences when the resonance condition is satisfied (ωres ) 0). Under such a condition, the eigenvalues and eigenvectors of the matrix displayed in eq 9 are evaluated analytically.27 Three different eigenvalues {0, 0, and

IR IR IR ( ik1 ) (ixωQ2 + 4ω21}27 drive the time evolution of the first set of coherences [T IR 20, T 11(a), T 21(s), and T 22(s)]. By contrast, the IR IR IR IR time evolution of the second set of coherences [T 10, T 11(s), T 21(a), and T 22(a)] is driven by four eigenvalues {(ik2 ) (i(ωQ +

xωQ2 + 4ω21)/2 and (ik3 ) (i(ωQ - xωQ2 + 4ω21)/2}.27 By treating the off-resonance Zeeman-like Hamiltonian as a perturbation

of the evolution matrix displayed in eq 9, we still obtain the same eigenvalues by using a first-order approximation. Numerical calculations of the eigenvalues and corresponding eigenvectors of the matrix displayed in eq 9 under the so-called off-resonance


Porion et al.

condition

Figure 4. Schematic view of the spectral densities used to describe the frequency variation of the paramagnetic and crossed quadrupolar/ paramagnetic contributions in the analysis of the spin-locking relaxation measurements.

condition totally validate that approximation for evaluating the order of magnitude of the various angular velocities explored by our spin-locking relaxation measurements. By using the apparent distribution of the clay directors displayed in Figure 2, Figures 5 exhibits the distribution of the eigenvalues as a function of film orientations (βLF) at the six irradiation powers used in that study (cf. section II.2). Since the m ) 0 component of the quadrupolar coupling (see eq A1C in Appendix) implies the TIR 20 coherence, its time evolution described in eq 4b will be evaluated on the basis of the three first eigenvalues (0 and (ik1). More details may be found in the literature.27 The spin-locking relaxation measurements are generally used in addition to the transverse relaxation measurements to obtain additional information on the lowfrequency variation of the spectral density.32,33 In that context, spin-locking relaxation measurements driven by the quadrupolar coupling alone, probe the spectral density at a single additional angular velocity (|k1|) since the zero angular velocity is already probed by the transverse relaxation rate. By contrast, the m ) 0 component of the paramagnetic coupling, which in our case is the main relaxation mechanism, implies the T IR 10 coherence (cf. eq 3c), whose time evolution requires the four eigenvalues ((ik2 and (ik3).27 As a consequence, two additional angular velocities are probed by the spin-locking relaxation induced by the paramagnetic coupling, thus extending by 2 orders of magnitude the dynamical range investigated (see Figure 5c,e) by reference to the frequency range probed by the quadrupolar coupling alone which is restricted to the first angular velocity (|k1|) (cf. Figure 5a,d). More details on the derivation of the spin-locking relaxation rate induced by the paramagnetic and the crossed quadrupolar-paramagnetic couplings are given in Appendix. The highest irradiation power (ω1 ) 1.57 × 105 rad‚s-1) used in these spin-locking experiments is high enough to generate angular velocities (|k1|, |k2|, |k3|) nearly independent of the clay orientation into the magnetic field (cf. Figure 5). By contrast, at weaker irradiation power, the angular velocities |k2| and |k3| exhibit a strong angular variation (i.e., covering 3 orders of magnitude). As a consequence, analyzing the spin-locking relaxation measurements requires the a priori knowledge of the analytical form of the m ) 0 component of the spectral density (noted J P0 (ω)), since, even at a single irradiation power and

film orientation, a large set of effective angular velocities are probed because of the broad angular distribution of the clay platelets in the film. 3. Comparison with Experimental Data. After a 90° pulse along IY, the equilibrium magnetization is transferred along the T IR 11(a) or IX/x2 coherence. Figures 6 and 7 display the time evolution of the intensity of the TIR 11(a) coherence as a function of the duration of the second irradiation pulse. According to the first order approximation considered above, since the TIR 11(a) coherence pertains to the first set of coherIR IR IR , ences [TIR 20 T11(a), T21(s), and T22(s)], in the absence of a relaxation mechanism, it oscillates at the angular velocity |k1| which is easily identified in Figures 6 and 7. By contrast, the calculation of the contribution from the paramagnetic relaxation mechanism requires considering the time evolution of the TIR 10 coherence (see Appendix) which pertains to the second set of IR IR IR coherences [TIR 10, T11(s), T21(a), and T22(a)] characterized by angular velocities (|k2| and |k3|). As a consequence, the effective angular velocities explored by the spin-locking relaxation measurements vary with the residual value of the quadrupolar coupling and thus with the clay orientation in the magnetic field (cf. Figure 2). Furthermore, since the clay directors are not aligned within the film along a single direction,17 the resonance frequencies of the confined water molecules are also distributed according to a distribution law that varies as a function of film orientation into the magnetic field (cf. Figure 2). In that context, even if the resonance frequency is fixed at the center of the apparent doublet corresponding to the confined water molecules, the resonance condition is not satisfied by the water molecules located within each interlayer because of the slow exchange17 between these different environments. The resonance condition cannot be applied to calculate the eigenvalues of the static Hamiltonian (cf. eq 9), and a numerical treatment is required to determine the exact contribution of the different angular velocities to the relaxation of the TIR 11(a) coherence. For that purpose, 400 angular segments are selected in the distribution law, and their time responses are superimposed to simulate the apparent variation of the magnetization displayed in Figures 6 and 7. A generalized least squared fitting procedure is used to simultaneously fit the data displayed in Figures 6 and 7 on the basis of a simulation of the time evolution of the various coherences during the pulses and evolution period of the spinlocking relaxation measurement. In order to reduce the total number of fitted parameters, we assume that both the m ) 0 component describing the paramagnetic and the crossed quadrupolar-paramagnetic relaxation mechanisms evolve according to the same analytical function(cf. Figure 4). We have also fixed the maximum quadrupolar coupling (νQ ) 17 kHz),17 the zero limiting value of the m ) 0 component of the spectral density (0) ) 104 s-1)17 and the describing the paramagnetic (JP,intrinsic 0 (0) ) 2 × 103 crossed quadrupolar-paramagnetic (JQP,intrinsic 0 s-1)17 relaxation mechanisms determining the value of the plateau at low frequencies (see Figure 4). In the same manner, (ω0), the high-frequency value of the spectral densities (JP,intrinsic 0 P,intrinsic P,intrinsic P,intrinsic (2ω JP,intrinsic ), J (0), J (ω ), J (2ω 0 0 0), 0 1 1 1 P,intrinsic P,intrinsic QP,intrinsic (0), J (ω JP,intrinsic ), J (2ω ), J (ω ), and 0 0 0 2 2 2 0 JQP,intrinsic (2ω0))17 are set equal to 500 s-1, fixing the value 0 of the plateau at high frequencies (see Figure 4). These parameters were previously determined from the variation of the transverse magnetization17 as a function of the orientation



Figure 5. Density of probability describing the distribution of the apparent angular velocities (|k1|, |k2|, |k3|) monitoring the time evolution of the various coherences under the influence of the irradiation field used for the spin-locking relaxation experiments. The distributions are evaluated at two orientations of the film directors (a,b,c; parallel, and d,e,f; perpendicular) by reference with the static magnetic field B0.

of the film director in the static magnetic field. As for the analysis of the transverse relaxation rates, the Wigner rotation matrices16,18 were used to determine the apparent value of the spectral densities from the orientation of each clay particle into the static magnetic field (described by Euler angle θLC) and the intrinsic values of the spectral densities34 listed above

1 J apparent (θLC,ω) ) (1 - 3 cos2 θLC)2 J intrinsic (ω) + 0 0 4 2 LC 2 LC intrinsic (ω) + 3 cos θ sin θ J 1 3 2 LC 2 intrinsic (ω) (10a) (1 - cos θ ) J 2 4 3 (θLC,ω) ) cos2 θLC sin2 θLC J intrinsic (ω) + J apparent 1 0 2 1 (1 - 3 cos2 θLC + 4 cos4 θLC) J intrinsic (ω) + 1 2 1 (1 - cos4 θLC) J intrinsic (ω) (10b) 2 2 3 (θLC,ω) ) (1 - cos2 θLC)2 J intrinsic (ω) + J apparent 2 0 8 1 (1 - cos4 θLC) J intrinsic (ω) + 1 2 1 (1 + 6 cos2 θLC + cos4 θLC) J intrinsic (ω) (10c) 2 8 Although such an analysis is rather simplified, we obtain a fair agreement between experimental and fitted data. By

simultaneously analyzing the time evolution of the TIR 11(a) coherence at various irradiation powers, we investigate a broad range of angular velocities by performing spin-locking relaxation measurements mainly driven by the paramagnetic coupling (cf. Figure 5). This then significantly improves the accuracy of the location of the crossover between the various dynamical regimes. From that analysis, the transition between the low-frequency plateau and the logarithmic regime of the m ) 0 component of the spectral density is observed at ωcrit ) (2 × 104) ( (1 × 104) rad‚s-1. Such a value corresponds to an average residence time of 5 × 10-5 s for confined water molecules inside the interlayer between two clay platelets, in good agreement with the average radius ((2.1 × 10-7) ( (0.8 × 10-7) m) of the montmorillonite clay used in that study and the transverse water mobility (D ) 0.5 × 10-9 m2/s) determined by MD simulations17

τB )

radius2 4.4 × 10-14 m2 1 ) ) ) 4.4 × 10-5 s -9 2 -1 ωC 2D 10 m s

(11)

In view of the good quantitative agreement between the average residence time of the confined water molecules estimated from the low-frequency cutoff of the spin-locking relaxation rates and the size of the clay platelets, one may be tempted to claim that the water self-diffusion coefficient evaluated by MD simulations during a short time scale (less than 1 ns)17 perfectly describes the water mobility on a much larger time scale (∼50 µs). To validate that conclusion, we performed BD35 simulations of the mobility of water


Porion et al.

Figure 6. Comparison between the time evolution of the TIR 11(a) coherence measured during the spin-locking experiments and derived from our numerical analysis (see text) for a parallel alignment of the film director with reference to the static magnetic field B0.

molecules diffusing at the surface of disks with a 2D diffusion coefficient of 0.5 × 10-9 m2/s. The diffusing molecular probes are randomly generated on the disks surface with radii distributed along a normal distribution law (average, 210 nm, and standard deviation, 85 nm) mimicking the montmorillonite clays. The average residence time of these water molecules is estimated from the diffusion time required to reach the border of the disks. As shown in Figure 8, this average residence time is not unique but covers one decade (between 5 × 10-6 s and 5 × 10-5 s). As a consequence, because of the crude approximations implied in the evaluation of the spectral density J0(ω) with clear-cut transitions between the different dynamical regimes (cf. Figure 4), the above-mentioned quantitative agreement must be considered as only qualitative. From this analysis, we expect nevertheless to observe a strong influence of clay size and water mobility on the location of the crossover between the logarithmic decrease and the lowfrequency plateau displayed by the spin-locking relaxation measurements. We are currently working along that guideline by studying the mobility of water molecules under stronger confinement in the so-called one layer regime, corresponding to interlamellar separations of 12.5 Å. In a more general sense, the experimental procedure described in the present paper should be useful to determine average residence times of confined liquids diffusing within porous media in equilibrium with a reservoir.

IV. Conclusions In the framework of the use of dense clay sediments for the safe storage of organic or nuclear wastes, the average residence time of confined fluids is a crucial parameter for developing large scale dynamical simulations to predict the time scale corresponding to the dispersion of the contaminants within the storing sites and thus optimize their retention efficiency. In that context, we have used 2H nuclear magnetic resonance relaxation measurements under the so-called spinlocking conditions to probe the low-frequency variation of the relaxation rate of the water molecules confined between the individual platelets of a montmorillonite clay, similar to the natural clays used for the storage of nuclear wastes. The lowfrequency variation of the NMR relaxation rates of confined fluids is indeed the fingerprint of the reduced dimension of their diffusion space. Confinement induces large differences between the spectral densities characterizing the NMR relaxation of bulk and confined liquids. The average residence time of the confined water molecules is determined by the analysis of the low-frequency dispersion curve of the spectral density monitoring the paramagnetic coupling. Because of the significant amount of structural iron in these natural clays, paramagnetic coupling is the main relaxation mechanism of the water molecules in contact with the clay surface. As demonstrated by a careful analysis of the time evolution of the spin 1 coherences under spin-locking conditions, we illustrate how the



Figure 7. Comparison between the time evolution of the TIR 11(a) coherence measured during the spin-locking experiments and derived from our numerical analysis (see text) for a perpendicular alignment of the film director with reference to the static magnetic field B0.

solids and could be considered in the preparation of synthetic porous solids. Acknowledgment. We cordially thank Ms. S. Maddi for the purification of the clay. The Bruker DSX360 spectrometer used in that study was purchased thanks to a grant from Re´gion Centre (France). The BD simulations were performed locally at CRMD on workstations purchased thanks to grants from Re´gion Centre. Appendix

Figure 8. Evaluation of the average residence time of the confined water molecules deduced from the Brownian dynamics simulations of 2D water diffusion within the interlamellar space between montmorillonite platelets.

1. Quadrupolar Hamiltonian. Since the relaxation of the water molecules confined between clay platelets is driven by two mechanisms17 (i.e., paramagnetic and crossed quadrupolarparamagnetic couplings), in addition to the paramagnetic Hamiltonian described in eq 3a-h, we must also consider the quadrupolar Hamiltonian27 2

paramagnetic coupling significantly extends the dynamical range usually investigated by the spin-locking relaxation measurements of such quadrupolar nuclei. This experimental procedure can certainly be used to investigate the dynamical behavior of a large class of fluids confined within porous media as long as paramagnetic coupling is the dominant relaxation mechanism. Such a condition is often satisfied for natural porous

HQ(t) ) CQ

Q Q (-1)mT 2,m F 2,-m (t) ∑ m)-2

(A1a)

where the quadrupolar coupling constant, defined by

CQ )

x

3 e2qQ 8 pI(2I - 1)

(A1b)


Porion et al.

is equal to x3/2 × π × 210 kHz for deuterium in bulk heavy water.36 The spin operators describing the quadrupolar coupling are given by

1 ) (3Iz2 - I(I + 1)) ) T IR 20 x6

(A1c)

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

(A1d)

T Q2,0

/

/

Q T 2,(2

1 IR ) I(2 ) T 2(2 2

(A1e)

Q As in eq 3f-h, the functions F 2,m (t) are the second-order spherical harmonics describing now the reorientation of this OD director in the static magnetic field B0 by using the two Euler angles (θLW, φLW) since the electrostatic field gradient felt by the deuterium nucleus in heavy water is directed along the OD bond. Details on the calculation of the quadrupolar, paramagnetic, and crossed quadrupolar-paramagnetic longitudinal and transverse relaxation rates may be found in the literature. 2. Paramagnetic Contribution to the Spin-Locking Relaxation Rate. Let us treat in more details the derivation of the spin-locking relaxation rate under the general condition corresponding to off-resonance detection (ωres * 0) since, to our knowledge, that problem is not discussed in the literature. As a consequence, one must derive the time evolution of the coherences (cf. eq 4a) under the influence of the static Hamiltonian including the residual quadrupolar coupling, the irradiation pulse, and the Zeeman-like Hamiltonian (cf. eq 9) in addition to the contribution from the fluctuating Hamiltonian (cf. eq 4b). Since the contribution from the irradiation pulse (with angular velocity ω1 < 106 rad‚s-1) introduces small perturbations of the nominal angular velocity (ω0 ∼ 108 rad‚s-1), we focus that analysis on the m ) 0 components of the paramagnetic and crossed quadrupolar-paramagnetic relaxation mechanisms.27 The contributions from the m ) (1 and m ) (2 components in eq 4b are evaluated by neglecting27 the static Hamiltonian since their angular velocities are (ω0 and (2ω0 respectively and thus much higher than ω1. For the paramagnetic relaxation, the m ) 0 component of eq 4b becomes

4 P (σ*) ) - C2PS(S + 1) f m)0 9

(

F P20(t - τ) r3iS(t - τ)

-

〈 〉) F P20 r3iS /

(

P ∞ F 20(t) 0 r3iS(t)

∫

-

-iH /s τ IR+ [T IR T 10 10,[e

/

〈 〉) F P20 r3iS

×

iH /s τ

e

, σ*]] dτ (A2)

+ iH s τ inside the double commutator where the term e- iH s τ T IR 10 e describes the time evolution of the coherence T IR 10 under the static Hamiltonian HS. That evolution is evaluated beforehand to construct the relaxation matrix describing the time evolution of all the coherences in the complete master equation now including the contribution from the paramagnetic relaxation (cf. eq 4a). IR Thanks to the completeness of the basis set [T IR 20, T 11(a), IR IR IR IR IR IR T 21(s), T 22(s), T 10, T 11(s), T 21(a), T 22(a)], we can express the / IR iH /s τ e as a linear combination term e-iH s τT 10

8

∑ cp(τ)T IRp p)1

(A3)

Note that, under the resonance condition (ωres ) 0),27 only the four last coherences contribute to the summation in eq A3 because of the separation of the complete basis in two independent subsets (cf. eq 9). In that case, an analytical solution of the problem was obtained by using /

iH s τ ) e-iH s τ T IR 10 e

and

/

iH s τ e-iH s τ T IR ) 10 e

8

∑ p)5

8

d0p b V p exp(iλpτ) )

cq(τ) T IR ∑ q q)5

(A4)

where iλp are the eigenvalues of the matrix describing the static Hamiltonian and b νp the corresponding eigenvectors. The coefficients d 0p are evaluated by numerically solving the equation destcribing the initial condition 8

∑ d 0p bν p p)5

T IR 10 )

(A5)

leading to the analytical solution27 /

k2 cos(k3τ) - k3 cos(k2τ) IR T 10 + k1 sin(k3τ) - sin(k2τ) IR iω1 T 11 (s) + k1 cos(k3τ) - cos(k2τ) IR ω1 T 21 (a) + k1 k2 sin(k3τ) - k3 sin(k2τ) IR T 22 (a) (A6) i k1 /

IR iH s τ e ) e-iH s τ T 10

illustrating the contributions from the |k2| and |k3| angular velocities (cf. section III.2) which will introduces spectral densities J P0 (k2) and J P0 (k3) in the master equation after integration of the autocorrelation function of the m ) 0 component of the paramagnetic coupling

4 J P0 (ki) ) C 2P S(S + 1) 9

∫0∞

(

F P20(0) r3iS(0)

〈 〉)( 〈 〉)

-

F P20

F P20(τ)

r3iS

r3iS(τ)

F P20 r3iS

-

cos(kiτ) dτ (A7)

Note that our derivation of the master equation (eq 4a) neglects the contributions from the dynamical shifts, that is, the imaginary part of eq A6. As discussed above, the contribution of the irradiation pulse in the master equation splits the J P0 (0) contribution of the paramagnetic relaxation mechanism into J P0 (k2) and J P0 (k3). However, in our case, the resonance condition (ωres ) 0) is never fulfilled. We must determine numerically the eigenvectors and eigenvalues of the matrix generated by the full static Hamiltonian (cf. eq 9). The numerical solution is totally equivalent to the analytical one, except for the extension of the summation in eqs A4 and A5 on the eight coherences. After straightforward calculations, the double commutator of eq A2 is described in a matrix form by using the full basis set IR IR IR IR IR IR IR [T IR 20, T 11(a), T 21(s), T 22(s), T 10, T 11(s), T 21(a), T 22(a)]:

(

Long-Range Mobility of Water Molecules /

/

-iH Sτ IR+ iHSτ [TIR T 10 e ,σ*]] ) 10, [e

0 -x3c7(τ) 0 c5(τ)

0 1 0 2 0 0

c8(τ) c7(τ) -c2(τ) 0

)


-x3c6(τ) c8(τ)

0 2c7(τ)

0 -x3c3(τ)

0 -x3c2(τ) x 0 3c1(τ) + c4(τ) 2c3(τ) -2c2(τ) x3c1(τ) + c4(τ) 0 -c3(τ) -c2(τ) 0 σ* (A8) -c6(τ) -c7(τ) -4c8(τ) c5(τ) -c8(τ) 2c7(τ)

c5(τ) c6(τ) -c3(τ) x3c1(τ) - c4(τ)

-2c6(τ) 4c3(τ) -4c4(τ) 2c3(τ)

0 0 0 0

0

-2c2(τ) 0 -c8(τ) 0 0 c7(τ)

0 x3c1(τ) - c4(τ) 0 -c2(τ) 0 -c3(τ)

-2c6(τ) 4c5(τ)

c5(τ) c6(τ)

In order to derive the m ) 0 component of the relaxation contribution to the master equation (cf. eq A2) as for eq A6, we finally neglect the dynamical shifts by only taking into account the real part of the exponential laws included in the cq(τ) functions, leading to the spectral densities J P0 (λp). During our fitting procedure (see section III.2), these spectral densities are evaluated at each angular velocity from an a priori analytical law (cf. Figure 4). 3. Contribution from the Crossed Quadrupolar-Paramagnetic Mechanism. The m ) 0 component of the influence of the fluctuations of the crossed quadupolar-paramagnetic relaxation mechanism on the evolution of the coherences is given by QP (σ*) f m)0

2 )CPCQ 〈Szeq〉 x3

{∫ ( ∞

0

F P20(t) riS3(t)

-

〈 〉) F P20

×

riS3

/

/

-iH s τ IR+ iH s τ (F Q20(t - τ) - 〈F Q20〉) [T IR T 20 e , σ*]] dτ + 10,[e

∫0

∞

(

F P20(t - τ)

(F Q20(t) - 〈F Q20〉)

riS (t - τ) 3

-

〈 〉) F P20 riS

3

/

/

-iH s τ IR+ iH s τ [T IR T 10 e , σ*]] dτ 20, [e

}

(A9)

where the equilibrium paramagnetic magnetization is evaluated by a second order approximation (〈S eq z 〉 ≈ -(pωSS(S + 1) NS)/ / iH /s τ e term, previously implied in the paramgnetic relaxation mechanism, we must consider now 3kT).17 In addition to the e- iH s τ T IR+ 10 IR -iH /s τ IR+ iH /s τ the e T 20 e term, describing the time evolution of the T 20 coherence under the influence of the static Hamiltonian alone. As above, the complete basis set is used to formulate the time evolution of the T IR 20 coherence /

/

iH s τ ) e-iH s τ T IR 20 e

8

∑ gp(τ) T IRp p)1

(A10)

Here also, an analytical solution is available under the resonance condition27

e

-iH /s τ

IR iH /s τ T 20 e )

ωQ2 + ω12(1 + 3 cos(k1τ)) IR 1 - cos(k1τ) IR sin(k1τ) IR x3 ω1ωQ T 11(a) + T 20 - ix3ω1 T 21(s) + 2 2 k1 k1 k1

x3ω12

cos(k1τ) - 1 k12

T IR 22(s) (A11)

IR IR IR IR Note that, again, T IR 20 evolves inside the first subset of coherences [T 20, T 11(a), T 21(s), T 22(s)] and oscillates at a single angular velocity (|k1|). Because of the off-resonance condition, we must again exploit the eigenvalues and eigenvectors of the static Hamiltonian to numerically formulate the time evolution of the T IR 20 coherence

e

-iH /s τ

T IR 20

iH /s τ

e

)

8

∑ p)1

8

k 0p b νp

exp(iλpτ) )

gq(τ) T IR ∑ q q)1

(A12)

where the coefficients k 0p are evaluated by numerically solving the equation describing the initial condition 8

T IR 20 ) /

k 0p b νp ∑ p)1

(A13)

-iH s τ IR+ iH Sτ Finally, the first double commutator of eq A9 ([T IR T 20 e , σ*]]) is evaluated by using the same matrix as in eq A8, 10, [e simply replacing all coefficients cq(τ) by gq(τ). Straightforward calculations lead to the matrix formulation of the second double commutator: /

(


0 -x3c2(τ) -x3c3(τ) 0 x3c1(τ) + c4(τ) 0

[T IR 20,

[e

-iH /Sτ

iH /Sτ T IR+ , 10 e

0 0 x3 0 -c2(τ) σ*]] ) 2 0 -c7(τ) 0 -c8(τ) 0 c5(τ) 0 c6(τ)

)

Porion et al.

0 0 -x3c6(τ) 0 0 c8(τ)

x3c1(τ) + c4(τ) -c3(τ) -c6(τ) c5(τ)

0 0 0 0

-c8(τ) c7(τ)

0 0 0 0 0 -c2(τ)

References and Notes (1) Sposito, G.; Prost, R. Chem. ReV. 1982, 82, 553. (2) Pelletier, M.; Thomas, F.; de Donato, P.; Michot, L. J.; Cases, J. M. Clays for Our Future. Proceedings of the 11th International Clay Conference; Kodama, H., Mermut, A. R., Torrance, J. C., Eds.; Ottawa, Canada, 1999; p 555. (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) Malikova, N.; Cade`ne, A.; Marry, V.; Dubois, E.; Turq, P. J. Phys. Chem. B 2006, 110, 3206. (5) Michot, L. J.; Delville, A.; Humbert, B.; Plazanet, M.; Levitz, P. J. Phys. Chem. C 2007, 111, 9878. (6) Poinsignon, C. Solid State Ionics 1997, 97, 399. (7) Swenson, J.; Bergman, R.; Longeville, S. J. Chem. Phys. 2001, 115, 11299. (8) 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. (9) Delville, A.; Letellier, M. Langmuir 1995, 11, 1361. (10) Sur, S. K.; Heinsbergen, J. F.; Bryant, R. G. J. Magn. Reson. A 1993, 103, 8. (11) 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. (12) Levitz, P. J. Phys.: Condens. Matter 2005, 17, S4059. (13) Porion, P.; Rodts, S.; Al-Mukhtar, M.; Fauge`re, A.-M.; Delville, A. Phys. ReV. Lett. 2001, 87, 208302. (14) Nakashima, Y.; Mitsumori, F. Appl. Clay Sci. 2005, 28, 209. (15) Porion, P.; Al-Mukhtar, M.; Fauge`re, A. M.; Pellenq, R. J. M.; Meyer, S.; Delville, A. J. Phys. Chem. B 2003, 107, 4012. (16) Kimmich, R. NMR: Tomography, Diffusometry, Relaxometry; Springer-Verlag: Berlin, 1997. (17) Porion, P.; Michot, L. J.; Fauge`re, A.-M.; Delville, A. J. Phys. Chem. C 2007, 111, 5441.

0 0 0 0

c5(τ) c6(τ) -c3(τ) x3c1(τ) - c4(τ)

-x3c7(τ) c5(τ)

0

c8(τ) c7(τ) -c2(τ)

0 0 σ* 0 0

0

0

x3c1(τ) - c4(τ) 0 -c3(τ) 0

(A14)

(18) Mehring, M. Principles of High Resolution NMR in Solids, 2nd ed.; Springer-Verlag: Berlin, 1983. (19) Ernst, R. R.; Bodenhausen, G.; Wokaun, A. Principles of Nuclear Magnetic Resonance in One and Two Dimensions; Clarendon Press: Oxford, 1987. (20) Pasquier, V.; Levitz, P.; Delville, A. J. Phys. Chem. 1996, 100, 10249. (21) Michot, L. J.; Bihannic, I.; Porsch, K.; Maddi, S.; Baravian, C.; Mougel, J.; Levitz, P. Langmuir 2004, 20, 10829. (22) Ferrage, E.; Lanson, B.; Malikova, N.; Planc¸ on, A.; Sakharov, B. A.; Drits, V. A. Chem. Mater. 2005, 17, 3499. (23) Hahn, E. L. Phys. ReV. 1950, 80, 580. (24) Fukushima, E.; Roeder, S. B. W. Experimental Pulse NMR: A Nuts and Bolts Approach; Addison-Wesley: Reading, MA, 1981. (25) Delville, A.; Grandjean, J.; Laszlo, P. J. Phys. Chem. 1991, 95, 1383. (26) Petit, D.; Korb, J. P.; Delville, A.; Grandjean, J.; Laszlo, P. J. Magn. Reson. 1992, 96, 252. (27) van der Maarel, J. R. C. J. Chem. Phys. 1993, 99, 5646. (28) Mu¨ller, N.; Bodenhausen, G.; Ernst, R. R. J. Magn. Reson. 1987, 75, 297. (29) Delville, A. J. Phys. Chem. 1993, 97, 9703. (30) Delville, A. J. Phys. Chem. 1995, 99, 2033. (31) Callaghan, P. T. Principles of Nuclear Magnetic Resonance Microscopy; Clarendon Press: Oxford, 1991. (32) Liu, G.; Li, Y.; Jonas, J. J. Chem. Phys. 1991, 95, 6892. (33) Korb, J. P.; Delville, A.; Xu, S.; Demeulenaere, G.; Costa, P.; Jonas, J. J. Chem. Phys. 1994, 101, 7074. (34) Barbara, T. M.; Vold, R. R.; Vold, R. L. J. Chem. Phys. 1983, 79, 6338. (35) van Gunsteren, W. F.; Berendsen, H. J. C.; Rullmann, J. A. C. Mol. Phys. 1981, 44, 69. (36) Edmonds, D. T.; Mackay, A. L. J. Magn. Reson. 1975, 20, 515.

Influence of Confinement on the Long-Range Mobility of Water

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