23Na Nuclear Quadrupolar Relaxation as a Probe of the

Ion Diffusion within Charged Porous Network as Probed by Nuclear Quadrupolar Relaxation. Alfred Delville, Patrice Porion, and Anne Marie Faugère...
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J. Phys. Chem. B 1998, 102, 3477-3485

3477

23Na

Nuclear Quadrupolar Relaxation as a Probe of the Microstructure and Dynamics of Aqueous Clay Dispersions: An Application to Laponite Gels P. Porion, M. P. Fauge` re, E. Le´ colier, B. Gherardi, and A. Delville* Centre de Recherche sur la Matie` re DiVise´ e, UMR6619, CNRS, 1B rue de la Fe´ rollerie, 45071 Orle´ ans Cedex 02, France ReceiVed: NoVember 11, 1997; In Final Form: February 23, 1998

23Na

relaxation studies of aqueous clay dispersions showed influence of the order/disorder transition of clay gels and the quadrupolar relaxation of neutralizing sodium counterions. Detection of the fast and slow components of the 23Na transverse magnetization gives access to the microstructural and dynamical properties of colloidal dispersions. Numerical simulations of sodium diffusion, using a simplified model of clay dispersion, explain the pathology of the 23Na relaxation rates by the high sensitivity of the quadrupolar coupling to the orientational correlations of charged interfaces. Frequency variation of the NQR relaxation rates of confined counterions therefore appears as a new tool for the investigation of the spatial correlations of charged interfaces within disordered porous media.

I. Introduction Diluted colloidal dispersions of clay particles exhibit a great variety of structural and mechanical properties (spontaneous ordering, setting, swelling, gelling, thixotropy, etc.) used in many applications (cosmetics and food industry, drilling, radioactive waste management, etc.). We performed 23Na relaxation studies to investigate the local structure and dynamics of clay particles and their neutralizing sodium counterions. Phenomena such as swelling, order/disorder, and sol/gel transitions of clay dispersions occur in diluted regimes in which direct contact between the particles are rare events. As a consequence, quantitative predictions of such phenomena require a detailed description of the diffuse layers of counterions condensed in the vicinity of the clay particles. These ionic clouds propagate the interparticle interactions through the suspension up to distances related to the ionic strength of the suspension (the so-called Debye screening length1-3). It is thus important to quantify the local organization and microdynamics of condensed ions, since they are the key of the mechanical properties of diluted clay dispersions. We performed 23Na relaxation studies for that purpose, since 23Na is a sensitive dynamical probe and sodium is an abundant neutralizing counterion (nearly 1000 counterions/laponite particle). Furthermore, 23Na is a quadrupolar nucleus with 3/2 spin; under slow modulation of the quadrupolar coupling,4-8 the one quantum transitions between the 1/2 T -1/2 and (3/2 T (1/2 energy levels of the Zeeman Hamiltonian generate two signals even for a single spin population with no static quadrupolar Hamiltonian. The only source of relative shift of these two resonace lines is dynamical8 and smaller than the line width of both resonance lines.36,37 It is then possible to extract from relaxation measurements structural and dynamical information on the sodium counterions in the vicinity of macromolecules.9-13 In the presence of a residual static Quadrupolar Hamiltonian, the multiplet structure of the 23Na resonance signal is composed of a central line resulting from the -1/2 T 1/2 transition and two satellites due respectively to the (3/2 T (1/2 transitions. * To whom correspondence should be addressed.

We selected laponite clay for this preliminary study because of the good reproducibility and purity of this synthetic clay. Although aqueous suspensions of laponite have been the subject of numerous studies (using rheology;14-16 light,16,17 small-angle X-ray,15,18 and neutron14 scattering; birefringence;19 cryofracture,15 etc.), the origin of gelling and particle organization within locally ordered laponite gels is still controversial.15-17,20 Furthermore, previous 23Na relaxation studies performed with dilute laponite suspensions suggested the existence of slow modulation of the quadrupolar coupling.21 However, because of the slow recording procedure (using a time step at least equal to 10 µs), only the signal from the 1/2 T -1/2 transition was detected.21 This incomplete detection of the 23Na relaxation pathways precluded separation of the dynamical and structural parameters from 23Na relaxation measurements.9-11,22 We then performed 23Na relaxation measurements of laponite suspensions using a fast recording mode (with a time step of 1 µs) in order to simultaneously detect signals from the 1/2 T -1/2 and the (3/2 T (1/2 transitions. Our results confirm the existence of slow modulation of the quadrupolar coupling. The analysis of the different 23Na relaxation rates allows quantification of the time scale characterizing the microdynamics of the condensed sodium counterions (∼0.1 µs). Temperature and concentration studies are also performed in order to identify the mechanism of modulation of 23Na quadrupolar coupling, viz., sodium adsorption and diffusion within the network of clay particles. Simulations of molecular dynamics, using a simple model of laponite dispersions, reproduce the order of magnitude of the relaxation rates of the broad component (called fast relaxation rate [R2f]) corresponding to the (3/2 T (1/2 transition of the 23Na nuclei. Our MD simulations reproduce the long time scale characterizing the sodium microdynamics within diluted clay dispersions but neglect fluctuations of the quadrupolar coupling occurring at a short time scale. As a consequence, our approach is unable to reproduce the relaxation rate of the thin resonance line (called slow relaxation rate [R2s]) corresponding to the -1/2 T 1/2 transition. However, one should not confuse the time scale characterizing the fluctuations of the quadrupolar coupling with

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the time evolution of the transverse magnetization, even if the occurrence of two resonance lines for a single spin population results from the slow modulation of its quadrupolar Hamiltonian. The time autocorrelation function of the quadrupolar Hamiltonian clearly illustrates how partial alignment of the laponite particles creates ordered areas in which the diffusing sodium probe retains for a long period of time (∼1 µs) the memory of its initial quadrupolar Hamiltonian. As a consequence, because of the strong electrostatic coupling responsible for ionic condensation, nuclear quadrupolar relaxation of labile counterions (with spin > 1) is a sensitive probe of the orientational correlation of charged interfaces. II. Materials and Methods (A) Clay Dispersions. We use laponite RD (from Laporte) without further treatment. Laponite clays are lamellae composed of one layer of magnesium and lithium oxides in octahedral geometry, sandwiched between two layers of silicium oxides in tetrahedral geometry. The negative charges of the clay network are neutralized by exchangeable interlamellar sodium counterions. The general formula of laponite RD is Si8Mg5.45Li0.4H4O24Na0.7.14,23 To prevent dissolution of the laponite particle, we add sodium hydroxide to set the pH of the suspension equal to 10.23 The resulting ionic strength of the suspension is nearly 10-4 M, corresponding to a Debye screening length of 300 Å. Cryofracture15 and scattering experiments14,15 were used to determine the average size of the laponite particle (diameter 300-400 Å, thickness 10 Å). Tetradecylbenzyldimethylammonium (TBDMA) chloride (from Sigma) is used without purification. The isotherm characterizing the competitive adsorption of TBDMA and sodium cations on laponite was previously determined by UV absorption at 263 nm, exploiting the high extinction coefficient of the phenyl group. (B) NMR Measurements. 23Na spectra were recorded on a DSX360 Bruker spectrometer operating at 95.28 MHz, using a 8.465 T superconducting magnet. The pulse duration for total inversion of the longitudinal magnetization is 7 µs. The dead time of the spectrometer plus the time delay before recording of the transverse magnetization is 5.5 µs. We used fast acquisition with a quadrature detection mode and a time step of 1 µs. The corresponding spectral width is 1 MHz. To avoid alteration of the relative abundance of each component of the signal, Fourier transforms of the free induction decay (FID) are performed without any preliminary treatment of the signal. A typical FID signal and a Fourier spectrum of 23Na within laponite suspension are shown in Figure 1. In addition to the slowly modulated signal from 23Na, we detect on the FID (Figure 1a) the presence of a quickly modulated signal. In addition to the 23Na spectrum centered at the origin, we observe a broad signal (line width 6.5 kHz) at 420 kHz (Figure 1b), attributed to the 63Cu nuclei of the rf coil. 63Cu resonance frequency (95.45 MHz) is indeed close to the 23Na resonance frequency (95.25 MHz). Thanks to the large separation between both resonances, the signal from copper does not interfere with the 23Na line-shape analysis. The relative fraction and relaxation rates of each component of the 23Na signal are determined from line-shape analysis. 32 768 points are used for Fourier transformation, leading to a digital resolution of 30 Hz. The chemical shift and the broadening of the sharp component of the 23Na signal are determined by direct analysis of the FID signal, improving the resolution by a factor of 10. The chemical shifts are measured with reference to a molar NaCl aqueous solution. The transverse relaxation rate deduced from the line width of

Figure 1. Typical FID signal (a) and its frequency spectrum (b) detected by 23Na NMR in a laponite suspension (4% w/w).

this reference sample (20-30 s-1) is somewhat larger than the value measured by spin-echo (18-19 s-1).24 This small difference gives an order of magnitude of the field heterogeneities. (C) Numerical Simulations. (i) Molecular Dynamics. We use Brownian dynamics to simulate sodium diffusion within suspensions of laponite particles. Langevin equations of motion25 are integrated with a time step (∆t) of 0.1 ns. Since the relaxation of hydrated sodium counterion occurs on a time scale of a few ps,24 we may assume that the time step ∆t is larger than the velocity relaxation time of bulk sodium (equal to the inverse of the friction coefficient γ). The Langevin equations of motions are then written25

xn+1 ) xn + Fn(mγ)-1 + Xn(∆t)

(1)

where Fn is the force field acting on the sodium counterion and Xn is a random force satisfying Gaussian distribution law with zero mean and fixed width

〈Xn2〉 ) 2kT(mγ)-1∆t ) 2D0∆t

(2)

where D0 is set equal to the diffusion coefficient of hydrated sodium cation (100 Å2/ns at 300 K).26 As a consequence, the average displacement of sodium cation during one elementary

Aqueous Clay Dispersions

J. Phys. Chem. B, Vol. 102, No. 18, 1998 3479 of neighboring particles (Figure 2b).15,27-28 (N,V,T) Monte Carlo simulations were also used15 to analyze the distribution of ions around two laponite particles bearing a net electric charge of 1000 electrons neutralized by sodium counterions and immersed in a 10-4 M solution of electrolyte. Ionic condensation occurs mainly on the basal faces of the laponite particle (Figure 2c), and nearly 50% of the neutralizing sodium counterions was located within the small disk (300 Å diameter and 30 Å thickness) surrounding each laponite particle (Figure 2c). Our MD simulations neglect the force field acting on the sodium cations (eq 1). Solvated sodium cations are treated as free diffusing probes, but they remain physisorbed when they hit a disk surrounding a laponite particle. When their average residence time (τB) is elapsed, immobilized cations diffuse back from the adsorbing disk. To decide when a condensed cation must desorb, we select a random number uniformly distributed within [0,1]. Desorption occurs if the random number is larger than exp(-(τ - τini)/τB), where τini is the time at which adsorption of the cation has occurred. As a consequence, the average residence time of condensed sodium (τBeff) may differ somewhat from the initial parameter τB. At equilibrium, the ratio between the average residence (τBeff) and diffusing (τF) times of the sodium cations is given by the fraction of condensed sodium (pB):

τBeff/τF ) pB/(1 - pB)

(3)

We study the simultaneous motions of 1000 noninteracting sodium probes diffusing through a disordered porous network defined by 1000 immobilized clay particles. (iii) Relaxation of Quadrupolar Nuclei. Although textbooks treating NMR relaxation are available,29-32 a simplified presentation sketching the relaxation of quadrupolar nuclei under a slow modulation regime is here appropriate. The quadrupolar Hamiltonian may be written: 2

HQ(τ) ) CQ

(-1)m F-m(t) Am(I) ∑ m)-2

(4)

where CQ ) eQ/2I(2I -1)hh, -e is the electron charge and Q the quadrupolar moment of the nucleus (0.11 × 10-28 m2 for 23Na).24 The A (I) factors in eq 4 are spin operators: m

Figure 2. Structure of the clay suspensions described at different scales by (a) a view of a fraction (1/8) of the simulation cell used for MD simulations of sodium diffusion; (b) the order parameter analyzing the spatial extension of the particle ordering; (c) a snapshot of an equilibrium configuration of sodium counterions in the vicinity of two parallel clay particles.

time step ∆t of 0.1 ns (∼8 Å) has the same order of magnitude as the thickness of the laponite particles. (ii) Model of Clay Dispersion. The laponite particles and their cloud of neighboring sodium counterions are modeled by disks with a diameter of 300 Å and a thickness of 30 Å (Figure 2a). The disks are randomly distributed and interact only through hard-core repulsion.15 Monte Carlo simulations of such organization of hard disks have already shown a local alignment

A0 ) 1/x6(3Iz2 - I(I + 1))

(5a)

A(1 ) (0.5(IzI( + I(Iz)

(5b)

A(2 ) 0.5I(2

(5c)

The Fm(τ) factors are derived from instantaneous components of the electric field gradient (efg) tensor expressed in a laboratory frame (with the z axis parallel to the magnetic field):

F0 ) (1.5)1/2VzzD(1 + γ∞)Y0DL

(6a)

F(1 ) ((VxzD ( iVyzD)(1 + γ∞)Y(1DL

(6b)

F(2 ) 0.5(VxxD - VyyD ( 2iVxyD)(1 + γ∞)Y(2DL (6c) where (1 + γ∞) is the Sternheimer antishielding factor resulting from the polarization of the innermost electronic orbitals of the nucleus by the external source of efg.33

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1/τC ) kT/h exp(-∆Gq/kT)

(10)

with ∆Hq ) 30 kJ/mol and ∆Sq ) 5 J/(mol K). Under a fast modulation regime, J0(0) ∼ J1(ω0) ∼ J2(2ω0) ∼ 2τC, and the relaxation rates coincide; they are both decreasing functions of the temperature. Under a slow modulation regime, J0(0) > J1(ω0) ∼ J2(2ω0), leading to large differences in relaxation rates: R2f > R2s ∼ R1s ∼ R1f. Note also the opposite temperature variations of the relaxation rates: R2f is still a decreasing function of the temperature, while R2s or R1R (R ) s, f) increases with the temperature. For isotropic reorientation and fast modulation of the quadrupolar coupling, eqs 9 reduce to

R2R ) 1/10(χ)2(1 + η2/3)τC Figure 3. Influence of the temperature on the fast (s) and slow (-‚-) relaxation rates of 23Na cations characterized by orientational decorrelation of their quadrupolar coupling.

The instantaneous components of the efg tensor (VRβD(τ)) are generally evaluated in a frame fixed to the probing nucleus. Spherical harmonics of rank 2 use Euler angles to describe the orientation of this instantaneous local frame within the laboratory frame:29,30

Y0DL ) Y(1DL )

(8π15)

(16π5 )

1/2

Y(2DL )

1/2

(3 cos2 θ(τ) - 1)

(7a)

sin θ(τ) cos θ(τ) exp((iφ(τ))

(7b)

15 (32π )

1/2

sin2 θ(τ) exp((2iφ(τ))

(7c)

The transverse magnetization of 3/2 spin nucleus is the sum of two components, with a priori known relative abundance:4-8

R2f ) 3CQ2[J0(0) + J1(ω0)]

(9a)

R2s ) 3CQ2[J1(ω0) + J2(2ω0)]

(9b)

Jm(ω) ) FT[〈Fm(O) F-m(τ)〉]

(9c)

where ω0 is 2π times the resonance frequency of the nucleus in the static field. If the autocorrelation functions of the different components of the quadrupolar Hamiltonian are described by the same exponential law (with correlation time τC), the spectral densities are Lorentzian. The fast (ω0τC < 1) and slow (ω0τC > 1) modulation regimes are identified by the temperature variation of the relaxation rates (Figure 3). Figure 3 is drawn assuming activated decorrelation of the quadrupolar Hamiltonian following

(11)

where η (η ) (|Vxx| - |Vyy|)/|Vzz|) is the asymmetry parameter of the efg tensor formulated in the local frame defined by its principal axis (with the convention |Vzz| g |Vxx| g |Vyy|). The intensity of the quadrupolar coupling is defined by the quadrupolar coupling constant {χ ) 2πeVzz(1 + γ∞)Q/h}, where eQ is the quadrupolar moment of the nucleus. For sodium in bulk water,24 R2 ∼ 20 s-1 and τC ∼ 3.8 ps,24 leading to a quadrupolar coupling constant of 7.3 MHz. By using known values of the Sternheimer antishielding factor (1 + γ∞ ∼ 5)24 and the quadrupole moment of 23Na (Q ) 0.11 × 10-28 m2),24 one deduces the largest eigenvalue of the instantaneous efg tensor (cf. eqs 6) for 23Na in bulk water:

Vzz )

χh ) 8.7 × 1019 (SI units) 2πeQ(1 + γ∞)

(12)

(iV) Chemical Exchange. When different environments coexist (as described in eq 3), the relaxation pathways of the transverse magnetization are described by a complex asymmetrical relaxation matrix.34-35 For two-site exchange (eq 3), we obtain:

() ()

M(Ff s d M(F ) -R dt M(Bf M(Bs

Mx(t) ) Mx(0)[0.4 exp(-t/T2s) + 0.6 exp(-t/T2f)] (8) The relaxation rates (R2R ) 1/T2R, R ) f, s) are linear combinations of the spectral densities [Jm(ω)], which are Fourier transforms of the autocorrelation functions of the time fluctuating components Fm(τ) of the quadrupolar Hamiltonian (eqs 6):

R ) f, s

(

M(Ff M(Fs M(Bf M(Bs

(13)

where R) R2Ff + kF +i(ω0 - ωFs)

-kB R2Fs + kF +i(ω0 - ωFs)

-kB R2Bf + kB +i (ω0 - ωBf)

-kF -kF

R2Bs + kB +i (ω0 - ωBs)

)

where M( ) Mx ( iMy. Note that because of dynamical effects,8,36-37 the fast and slow components due to bound sodium are shifted, leading to ωBf g ωBs.

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Figure 4. Schematic view of the structural and rheogical properties of laponite suspensions at ionic strength 10-4 M.

Finally, under fast exchange conditions (R2FτF < 1, R2BτB < 1, and |ωF - ωBR| < (1/τF + 1/τB)),34 and a simple additive relationship emerges from eq 13:

R2R

)

PFR2FR

+

PBR2BR

(R ) s, f)

(14)

where individual relaxation rates are given by eqs 9. The variation of the relaxation rates as a function of temperature and concentration of adsorbate help to determine the nature and the time scale of the exchange mechanisms of the sodium cations.38,39 III. Results and Discussion (A) NMR Experiments. 23Na NMR spectra are recorded for aqueous dispersions of laponite particles (pH ) 10 and ionic strength ∼ 10-4 M) and for clay weight fractions varying between 0.4 and 4%. Under these conditions, laponite RD clay froms repulsive aqueous dispersions,15 with a rheological sol/ gel transition occurring at a clay fraction 1.85%.15 Below this concentration, laponite RD suspensions are isotropic sols. Between 1.85 and 3%, an isotropic gel is observed,15 and above 3% the formation of a nematic gel is detected.15 Similar critical concentrations were recently reported for aqueous dispersions of laponite B.19 However, despite the diversity of structural information, no clear explanation of these phenomena have been published up to now. What is the reason for the long-distance propagation of nematic order within dilute clay dispersions of repulsiVe clay particles? At the sol/gel transition, the average interparticle separation is of the order of 450 Å, i.e., larger than the size of the laponite particle (300 Å diameter and 10 Å thickness). This illustrates the central role played by the diffuse layer of condensed counterions (Figure 2) on the local organization of the clay particle. The general purpose of this NQR relaxation study is to detect the fingerprints of the sol/gel and order/disorder transitions by exploiting the dynamical and structural information offered by the 3/2 spin nucleus under slow modulation condition (cf. section IICiii). Figure 1b shows a typical 23Na NMR spectrum of a laponite gel. Two components of the transverse magnetization are detected with relative abundance always agreeing with the theoretical predictions (eq 8). This result proves the existence of slow modulation of the quadrupolar coupling, at least for a fraction of the spin population in fast exchange with bulk sodium (eqs 13 and 14). The fast and slow components of the transverse magnetization are shown in Figure 5 as a function of the clay concentration. Two concentration ranges are identified from the fast component: (i) below 2.5%, we observe a continuous increase of R2f, from 9000 ( 500 s-1 (at 0.4%) to 18000 ( 500 s-1; (ii) above 2.5%, no pronounced variation of R2f is detected.

Figure 5. Variations of the fast (9) and slow (b) transverse relaxation rates of 23Na as a function of laponite concentration. The corresponding empty marks (0 and O) are obtained by MD simulations (see text).

The increase of the slow component is nearly parallel, but its maximum value (240 ( 10 s-1) is smaller by 2 orders of magnitude. The large value of the fast component of the transverse magnetization (R2f ∼ 104 s-1) explains the previously reported lack of visibility of 60% of the spin population21 since the FID signal was detected under a slow acquisition regime. The continuous increase of the relaxation rates as a function of clay concentration detected below 2.5% suggests the existence of fast exchange between two sodium populations: free sodium ions in the suspension, and condensed sodium ions in the vicinity of the clay particle. The critical concentration for the detection of the plateau (Figure 5) corresponds to the lower limit of the propagation of the nematic order within the clay gel (Figure 4). This observation suggests that NQR relaxation of the counterions is sensitive to the local ordering of the clay particles, well before the long-range propagation of the nematic order. This conclusion is somewhat surprising, since quadrupolar relaxation is generally considered to imply only very local mechanisms such as ion reorientation.29,30 However, the decorrelation of the quadrupolar Hamiltonian of the condensed counterions is monitored by the reorientation of their proximate clay particle, which is the source of the electric field gradient (efg). The influence of the clay concentration on the fast and slow components of the 23Na transverse magnetization is expected to result from an increase of the correlation time, because of the loss of orientational mobility of the clay particle during gel formation. By assuming (i) that two different environments coexist in fast exchange regime, with condensed (noted B) and free (noted F) sodium, (ii) that the spectral densities corresponding to the same chemical environment of the cation are described by a single Lorentzian, and (iii) that free/condensed sodiums are respectively in fast/slow modulation regime, it is then possible, by using eqs 8, 9, 11, and 14 to extract the reorientational correlation time of condensed sodium counterions (τCB) from the ratio

R2f - R2F R2s

- R2F

)

J0(0) + J1(ωB) J1(ωB) + J2(2ωB)

1 1 + (ωBτCB)2 ) 1 1 + 2 1 + (ωBτCB) 1 + (2ωBτCB)2

(15a)

1+

(15b)

The relative fraction of condensed counterions (pB) is then

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Figure 6. Analysis of the data shown on Figure 5, assuming orientational decorrelation of the quadrupolar coupling (eqs 15 and 9a), leading to the quadrupolar coupling constant (a) and the correlation time (b) of condensed sodium counterions.

derived from the value of the fast component, assuming R2F ) 50 s-1 (twice the relaxation rate of sodium in bulk water).9-11,22 As shown in Figures 6, the structural (pBχB2) and dynamical (τCB) parameters deduced from the relaxation of condensed sodium counterions vary differently as a function of the clay concentration. The increase of the fraction of condensed sodium (Figure 6a) is in agreement with the fast exchange hypothesis, but the decrease of the correlation time (Figure 6b) is in complete disagreement with our expectation. We performed two complementary experiments to test the validity of the hypothesis used to analyze the experimental results. We first added aliquots of TBDMA, to englobe the laponite particles within uniform neutralizing hydrophobic layers and remove condensed sodium counterions from the polyion surface.11,40,41 Our analysis of 23Na NQR relaxation rates (Figure 7a) shows a continuous decrease of the relative fraction of condensed sodium cation (Figure 7b) as a function of the fraction of added quaternary ammonium cations necessary to form a monolayer at the clay surface. The cooperativity11,40 of the adsorption mechanism of TBDMA on laponite particles is responsible for the sigmoidal shape in Figure 7b. The correlation between the fraction of condensed sodium counterions and the residual charge of the laponite particles (not fully neutralized by the aggregated quaternary amonium) illustrates the central role played by ionic condensation on the NQR relaxation of sodium cations within aqueous clay dispersions. Second, we studied the temperature variation of the relaxation rates in order to test the regime of modulation of the quadrupolar Hamiltonian (Figure 3). The results shown in Figure 8 totally disagree with the temperature variations expected for both R2f and R2s under slow modulation regime (cf. Figure 3). It follows that another modulation mechanism of the quadrupolar Hamil-

Figure 7. Influence of the addition of highly hydrophobic quaternary amonium (TBDMA) on the fast (9) and slow (b) transverse relaxation rates of 23Na (a) and the residual relative fraction of condensed sodium counterions (b).

Figure 8. Influence of the temperature on the fast (9) and slow (b) transverse relaxation rates of 23Na within laponite gel.

tonian must be more efficient than particle reorientation and is the main cause of the large difference between R2f and R2s. (B) Numerical Simulations. Sodium diffusion through a network of immobilized clay particles is another source of modulation of the quadrupolar coupling probed by the condensed sodium counterions. In the vicinity of the clay particle, condensed sodium ions are embedded in a strong efg. Diffusion of condensed cations at the clay surface may be able to average only partially the different components of the efg tensor. More precisely, we expect residual static contribution of the efg tensor along the normal to the clay particle. This mechanism was already proposed42 to interpret the 23Na NQR relaxation of sodium counterions in dilute suspensions of DNA9 at low ionic

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J. Phys. Chem. B, Vol. 102, No. 18, 1998 3483 condensed and free sodium probes

GQ(τ) )

〈F0static(0)

F0static(τ)〉

)

15(Vzzstatic (1 + γ∞))2 32πNat

×

Nat Npt Npt

p(i∈j|0) p(i∈k|τ) (3 cos2 θj - 1)(3 cos2 θk - 1)} ∑ ∑ ∑ i)1 j)1 k)1

{

(16)

Figure 9. Snapshot of one equilibrium configuration of the water molecules and sodium counterions determined by GCMC simulations of laponite hydration (see text).

strength. From a formal point of view, the cylindrical symmetry of elongated polymeric chains is similar to the symmetry of laponite disks. We used grand canonical Monte Carlo (GCMC) molecular simulations of clay hydration43,44 to prove the existence of such static residual quadrupolar coupling. We use a molecular model of the clay/water interface based on quantum calculations of the clay/water and clay/ion interactions43,44 and GCMC simulations43,44 to determine the average number of water molecules hydrating a fragment of clay particle included in a periodic simulation cell (31 × 36 × 30 Å; see Figure 9). We calculated the average field gradient probed by 16 sodium cations confined within the hydration layers (Figure 9). Three sets of electric charges contribute to the average efg probed by each condensed sodium; they originate from the clay particle, the other cations, and the confined water molecules. Because of the residual structure of the diffuse layers of condensed counterions45,46 and adsorbed water molecules43-44,47,48 (Figure 9), we detected a residual static value of the efg aligned along the direction normal to the clay particle (noted ez). By averaging the results of three independent GCMC simulations, we obtain Vzzstatic ) (7 ( 3) × 1019 (SI units), i.e., of the same order of magnitude as its instantaneous value for sodium in bulk water (cf. eq 12). As described in section IIC, we performed simulations of Brownian dynamics to estimate the trajectories of 1000 solvated ions diffusing througth a network of adsorbing laponite particles. The average residence time (τB) is the only parameter of this limiting two-state model. These simulations probe the longtime behavior of the diffusion of sodium cations and neglect local structures or events occurring on short time scales. The large difference between measured R2f and R2s for sodium ions in the presence of laponite clay is indeed related to the lowfrequency variation of the spectral densities (eqs 8 and 9; ω < 109 s-1), i.e., to the long-time (τ > 10-9 s) evolution of the autocorrelation function of the quadrupolar Hamiltonian (eq 9). Because of the long-time step of these MD simulations (0.1 ns), we evaluate only the decorrelation of the residual static component of the efg tensor (Vzzstatic) felt by condensed sodium counterions and neglect both instantaneous contributions from

where Nat and Npt are respectively the total number of sodium ions and laponite particles and p(i∈k|τ) is the probability of finding ion i condensed on particle k at time τ. Note that, because of the partial alignment of neighboring clay particles,14,19 the correlation function is not totally described by the self-diffusion propagator of the sodium probes,49-51 since crossed terms also contribute to the correlation function (eq 16). As a consequence, diffusing sodium ions are sensitive to the spatial extension of oriented microdomains,15 and they must retain for a long period of time the orientational memory of their initial adsorbing polyion when they diffuse within a nematic gel. Results from MD simulations performed for three clay concentrations (1, 2.7, and 7%) are summarized in Table 1. As expected from our simple exchange model (eq 3), the fraction of condensed sodium increases as a function of clay concentration. The average residence time of condensed sodium cations (τBeff ) 1.05 ns) is independent of the clay concentration but significantly smaller than the parameter (τB ) 7 ns) selected for the dynamical simulations (see section IICii). As expected,42 the average diffusing time of free sodium (τF) is reduced when the interparticle separation decreases. The fraction of condensed sodium (pB) is also derived from eq 3; the perfect matching with the direct derivation of pB results from the occurrence of a stationary regime during the MD simulations (the first trajectory is used to thermalize the ionic distribution within the clay network). The ratio Deff/D0 measures the tortuosity of the porous network resulting from the dispersion of adsorbing clay particles. Finally, Table 1 shows large differences between the Fourier transforms of eq 16 (the spectral densities) at zero and sodium resonance frequencies, leading to large differences between the fast and slow components of the transverse magnetization (eqs 9). The slow and fast transverse relaxation rates are drawn on Figure 5; despite the simplicity of our two-state model, we reproduce the order of magnitude of experimental fast relaxation rates; however, the calculated slow relaxation rates are 1 order of magnitude smaller than the experimental values. This discrepancy is the consequence of the neglect of events occurring at time scales smaller than 0.1 ns, like sodium/clay and sodium/ sodium collisions, sodium diffusion at the clay surface, etc. The autocorrelation functions of the residual quadrupolar coupling and their corresponding spectral densities are shown in Figure 10. Although sodium desorption occurs at a short time scale (τBeff ) 1 ns), the residual quadrupolar Hamiltonian keeps its memory for a long period of time: 200 ns are required in order to observe a reduction of GQ(τ) by a factor of 10. The corresponding average displacement of sodium diffusors during that period of time is of the order of magnitude of the laponite diameter. Figure 10a also exhibits an asymptotic long-time decrease of GQ(τ) according to a power law (GQ(τ) ∼ τ-3/2), as expected for sodium diffusion within 3D space.52,53 This asymptotic regime appears within the same time scale (200300 ns), whatever the clay concentration. Below that time limit, GQ(τ) exhibits a pathological decrease resulting from the

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TABLE 1: Dynamical Properties of Sodium Counterions in Laponite Dispersions clay concn (w/w)

pB

τeffB (ns)

τF (ns)

pB (from eq 3)

Deff/D0

J(0) × 1030

J(ω0) × 1030

J(2ω0) × 1030

7% 2.7% 1%

0.47 ( 0.01 0.24 ( 0.01 0.11 ( 0.01

1.05(0) 1.05(0) 1.05(0)

1.19 ( 0.01 3.26 ( 0.04 8.77 ( 0.08

0.47 0.24 0.11

0.55 ( 0.01 0.77 ( 0.01 0.88 ( 0.01

1180 610 180

0.4 0.2 0.1

0.2 0.1 0.05

Figure 10. Correlation functions of the 23Na quadrupolar coupling (a) and their corresponding spectral densities (b) determined by MD simulations of sodium diffusion within homogeneous (s, 7%; - - -, 2.7%; -‚-‚:1%).

tortuosity and the orientational correlation of the dispersed laponite particles (cf. eq 12). Such pathological behavior of the autocorrelation function was already reported for dipolar relaxation of confined liquids,49-51 when the diffusing probes are coupled to highly magnetized nuclei (protons, paramagnetic impurities) embedded within the solid matrix. At a time scale smaller than the transition time, the autocorrelation function GQ(τ) is modulated by the local structure of the diffusing space.49-51 However, beyond that time limit, no structural information may be deduced from GQ(τ). The frequency variation of the corresponding spectral densities (Figure 10b) is spread over a broad domain. At angular velocity ω g 107 s-1, the spectral densities decrease according to a power law (J(ω) ∝ ω-1.5(0.2). This pathological variation of the spectral density is another fingerprint of the local organization of the clay particles.49-51 Below that critical angular velocity, the spectral densities decrease according to -ω1/2, as expected for sodium diffusion within 3D isotropic space.52,53 Obviously the results of these MD simulations depend totally on the assumptions implied in our model and the parameters characterizing the microdynamics of the system: the sodium residence time (τB), the residual field gradient (Vzzstatic) and the sodium diffusion constant (D0). The last parameter is taken

from experimental data26 and Vzzstatic is evaluated from preliminary GCMC simulations of laponite hydration.43,44 The boundaries of the last parameter (τB) is deduced a posteriori from the effective residence time of condensed sodium (τBeff; cf. eq 3), since the fraction of condensed counterions was previously calculated by Monte Carlo simulations of ionic distribution around two laponite particles (cf. Figure 2).15 We introduced in this model the minimal set of assumptions required to obtain tractable calculations, leading only to partial and qualitative results. Soon, we shall perform relaxation measurements at various field strengths to probe the shape of the spectral densities. Despite their intrinsic oversimplifications, our MD simulations illustrate how the local organization of the charged laponite particles is responsible for the large differences between the fast and slow transverse relaxation rates of diffusing sodium probes. Because of the occurrence of crossed terms in the autocorrelation function of the residual quadrupolar coupling (eq 16), its long-time evolution probes the orientational correlations of the clay particles. As a consequence, 23Na relaxation measurement performed at different frequencies50,53 is a sensitive probe of the local structure of disordered interfacial media.32 Such investigations are not limited to laponite particles neutralized by sodium counterions, but one may consider all charged interfaces (including surfaces of porous materials50-52) neutralized by quadrupolar counterions. This analysis also suggests an interpretation of the paradoxical temperature variation of the fast tranverse relaxation rate (Figure 8). Any increase of the temperature is expected to speed kinetics processes controlled by diffusional or desorption mechanisms, leading to a decrease of the fast transverse relaxation rate (Figure 3). However, we also expect a large influence of temperature on the spatial extension of nematic microdomains within the laponite suspension. Since order/disorder transition of discotic particles is entropy driven,27,28 any increase of temperature will also increase the effective correlation time of the quadrupolar coupling (cf. eq 16 and Figure 10a) because of the propagation of the local nematic order to larger distances. We finally note that, because of the fast decay of 60% of the spin population (T2f g 50 µs [Figure 1a]), the time period necessary to obtain a complete decorrelation of the quadrupolar Hamiltonian (τeff ∼ 1 µs [Figure 10a]) reaches the upper limit of validity of eqs 8 and 9, because these equations, which describe the time evolution of the transverse magnetization, stem from a solution of the differential equations relating the time evolution of the transverse magnetization to the fluctuations and the decorrelations of the quadrupolar coupling by assuming a complete separation7,29 of the time scales characterizing both processes. IV. Conclusion By using both experimental and numerical studies of 23Na NQR relaxation properties within laponite gels, we showed the influence of the local ordering of clay particles on the dynamics and relaxation of diffusing sodium cations. We used 23Na NMR with a fast acquisition mode to detect the fast and slow components of the transverse magnetization of sodium counterions within laponite gels. We performed GCMC simulations

Aqueous Clay Dispersions of clay hydration and molecular dynamics simulations of sodium diffusion to determine how the local structure of laponite gels modulate the long-time evolution of the residual quadrupolar coupling of condensed counterions. These results showed that the frequency variation of the spectral density of quadrupolar nuclei is a sensitive probe of the interfacial correlation of an ionized solid matrix. Acknowledgment. It is a pleasure to cordially acknowledge Drs. H. van Damme, P. Levitz, A. Mourchid, R. J. M. Pellenq and R. Setton (CRMD, Orleans, France), J. Grandjean (Universite´ de Lie`ge, Belgium), J. P. Korb and D. Petit (PMC, Ecole Polytechnique, Palaiseau, France), and Prs. C. Detellier (University of Ottawa, Canada) and J. C. Leyte (University of Leiden, Holland) for helpful discussions. The Bruker spectrometer and minicomputers used in this study were purchased thanks to grants from CNRS and Re´gion Centre (France). References and Notes (1) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1985. (2) Lyklema, J.Fundamentals of Interface and Colloid Science; Academic Press: London, 1981. (3) Dubois, M.; Zemb, Th.; Belloni, L.; Delville, A.; Levitz, P.; Setton, R. J. Chem. Phys. 1992, 96, 2278. (4) McLachlan, A. D. Proc. R. Soc. London 1964, A280, 271. (5) Hubbard, P. S. J. Chem. Phys. 1970, 53, 985. (6) Bull, T. E. J. Magn. Reson. 1972, 8, 344. (7) Werbelow, L. G. J. Chem. Phys. 1979, 70, 5381. (8) Petit, D.; Korb, J. P. Phys. ReV. B 1988, 37, 5761. (9) Levij, M.; De Bleijser, J.; Leyte, J. C. Chem. Phys. Lett. 1981, 83, 183. (10) Urry, D. W.; Trapane, T. L.; Venkatachalam, C. M.; Prasad, K. U. J. Am. Chem. Soc. 1986, 108, 1448. (11) Delville, A.; Laszlo, P.; Schyns, R. Biophys. Chem. 1986, 24, 121. (12) Jang, H. M.; Fuerstenau, D. W. Langmuir 1987, 3, 1114. (13) Zwetsloot, J. P. H.; Leyte, J. C. J. Colloid Interface Sci. 1996, 181, 351. (14) Ramsay, J. D. F.; Lindner, P. J. Chem. Soc., Faraday Trans. 1993, 89, 4207. (15) Mourchid, A.; Delville, A.; Lambard, J.; Le´colier, E.; Levitz, P. Langmuir 1995, 11, 1942. (16) Pignon, F.; Piau, J. M.; Magnin, A. Phys. ReV. Lett. 1996, 76, 4857. (17) Kroon, M.; Wegdam, G. H.; Sprik, R. Phys. ReV. E 1996, 54, 6541. (18) Morvan, M.; Espinat, D.; Lambard, J.; Zemb, Th. Colloids Surf. A: Physiochem. Eng. Aspects 1994, 84, 193. (19) Gabriel, JC P.; Sanchez, C.; Davidson, P. J. Phys. Chem. 1996, 100, 11139. (20) Dijkstra, M.; Hansen, J. P.; Madden, P. A. Phys. ReV. Lett. 1997, 55, 3044.

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