Langmuir 1995,11, 1361-1367
1361
Structure and Dynamics of Simple Liquids in Heterogeneous Condition: An NMR Study of the Clay- Water Interface A. Delville" and M. Letellier Centre de Recherche sur la Matiare Diviste, CNRS, 1B rue de la Ftrollerie, 45071 Orltans Cedex 02,France Received October 27, 1994. In Final Form: January 18, 1995@ We use lH NMR relaxation measurements in a broad range of frequencies to extract information on the dynamical properties of water molecules in dilute suspensions of swelling clay particles. The frequency variation of the longitudinal relaxation rate suggests the existence of a 2D diffision mechanism for water s. This observation explains molecules adsorbed on the clay surface, with a residence time longer than the paradoxical results frequently reported from NMR relaxation studies on such heterogeneous systems.
I. Introduction Because of the surface-ligand interactions, the structure of a liquid near a solifliquid interface is disturbed within the range of a few (1-3) diameters of the liquid phase m o l e ~ u l e s . l - ~These interactions modify many physical properties of the adsorbed phase (molecular ~ r i e n t a t i o nlayering: ,~ force oscillations,2z6d i f f i ~ i o netc. , ~ 1. Much e ~ p e r i m e n t a l ' - ~ , ~ and - l ' theoretica15J2-16 work has been devoted to this problem. The present study is restricted to a claylwater interface because clays are well-characterized materials involved in m a n y industrial applications. Whereas various thermodynamical measurements (calorimetry,l ~ w e l l i n g , l ~ - ' ~ force2,6 measurements, etc.) have been performed to quantify t h e claylwater interaction, little information is available on t h e dynamicsg-'l of the adsorbed water molecules. In general, NMR results can provide this information through relaxation measurements when complementary are used. @
Abstract published inAdvanceACSAbstracts,March 15,1995.
(1)Fripiat,3.;Cases, J.;Francois, M.; Letellier, M. J.Coll. Interface Sci. 1982,89,378-400. (2)Israelachvili,J. N. Intermolecularand SurfacesForces;Academic Press: London, 1985. (3)Granick, S. Science 1991,253,1374-1379. (4)Woessner, D. E.;Snowden,B. S. Ann. N.Y. Acad. Sci. 1973,204, 113-124. (5)Magda, J. J.; Tirrel, M.; Davis, H. T. J . Chem. Phys. 1985,83, 1888-1901. (6)Israelachvili, J. N.;Pashley, R. M. Nature 1983,306,249-250. (7)Liu, G.; Li, Y.; Jonas, J. J . Chem. Phys. 1991,95,6892-6901. (8)Hougardy, J.;Stone, W. E. E.; Fripiat, J. J. J . Chem. Phys. 1976, 64, 3840-3851. (9)Fripiat, J.; Letellier, M. J.Magn. Res. 1984,57,279-286. (10)Delville, A.; Grandjean,J.; Laszlo, P. J.Phys. Chem. 1991,95, 1383-1392. (11)Cebula,D. J.;Thomas,R.K.;White,J. W.J. Chem. Soc.Faraday I1980,76,314-321. (12)Schoen,M.;Diestler, D. J.;Cushman, J. H. J.Chem.Phys. 1994, 100,7707-7717. (13)Sullivan,D.E.;Levesque, D.; Weis, J. J. J. Chem. Phys. 1980, 72,1170-1174. (14)Plischke, M.;Henderson, D. J . Chem. Phys. 1986,84,28462852. (15)Snook, I. K.;van Megen, W. J . Chem. Phys. 1980,71,29072913. (16)Delville, A. J . Phys. Chem. 1993,97,9703-9712. (17)Edwards, D. G.; Quirk,J. P. J . Colloid Sci. 1962,17,872-882. (18)Callaghan, I. C.; Ottewill, R. H. Faraday Disc.Chem. SOC. 1974, 57,110-118. (19)Viani, B.E.;Roth, C. B.; Low, P. F. Clays Clay Miner. 1985,3, 244-250. (20)Petit, D.; Korb, J. P.; Delville, A.; Grandjean, J.; Laszlo, P. J. Magn. Res. 1992,96,252-279.
In this context, we studied the lH longitudinal relaxation r a t e in a broad frequency domain and proved the existence of a 2D diffision of adsorbed water molecules which remain for at least s on t h e surface of the clay particles. This observation provides a new interpretation of t h e paradoxica121results reported from N M R relaxation measurements under heterogeneous conditions. 11. Materials and Methods 1. Clay Suspensions. Clay particles result from the sandwiching of one layer of octahedral metallic oxides between two layers of tetrahedral metallic oxides.22~23In this study we used hectorite (from Hector, CA), a trioctehadra122magnesiosilicates clay [theoretical formula, S4011Mg3(OH)2]. However, this perfect clay does not exist: some MgI1atoms in the octahedral network are replaced by, for example, Lil and FeIII.22 The iron content of hectorite (250 ppm) is small compared to that in other natural clays (for montmorillonite it may be as high as 4 wt %).22 This parameter is crucial for NMR relaxation measurements since paramagnetic impurities may contribute significantly to the relaxation of adsorbed water molecules, leading to a complex relaxation b e h a v i ~and r ~ shortening ~ ~ ~ ~ of the apparent relaxation times of the absorbed molecules. Because of the chemical substitutions, the clay network bears a negative charge (-7 x electrodA2)22neutralized by interlamellar solvatable counterions. Na homoionic clay is prepared by dialysis. 2. NMR Measurements. Details on the theory of NMR relaxation can be found in the Appendix. Three relaxation times can be measured by NMR: (1)the longitudinal relaxation time (TI),measured by an inversion recovery pulse sequence; (2) the transverse relaxation time (Tz),measured by the spin-echo sequence; and (3) the longitudinal relaxation time under spinlocking conditions (TI&giving acess to the spectral densities at low angular frequency ( w 1 ) (see Appendix I). The field inhomogeneity determines the lower limit of angular frequencies w1 compatible with spin-locking conditions: the intensity of the locking field must be 1 order of magnitude larger than the apparent broadening resulting from the field inhomogeneity. The N M R measurements were performed on a Bruker MSL360 spectrometer using an 8 T superconducting magnet and a nonsuperconducting magnet with an induction varying continuously measurements were performed between 0.2and 2.1 T. The TI@ with the resistive magnet at 2.1 T. (21)Letellier, M.; Tinet,D.; Maggion,R.; Fripiat,J.Magn. Res. Imag. 1991,9,709-716. (22)Grim, R. E.Clay Mineralogy; McGraw-Hill: New York, 1953. (23)van Olphen, H. An Introduction to Clay Colloid Chemistry; Wiley: New York, 1977. (24)Sur, S. IC;Heinsbergen, J. F.; Bryant, R. G. J . Magn. Res. A 1993,103,8-12.
0743-7463/95/2411-1361$09.00/00 1995 American Chemical Society
Delville and Letellier
1362 Langmuir, Vol. 11, No. 4, 1995 3. Numerical Simulations. We used Monte Carlo simulat i o n in ~ the ~ ~grand ensemblez6(GCMC simulations) to determine the number and the organization ofthe water molecules confined between two hectorite clay particles (periodicity, 25.4A). The atomic model used for these simulations is based on quantum calculations of the clay-water and clay-counterion interacto an empirical d e s c r i p t i ~ nof~the ~ . ~water~ t i o n ~in, ~ addition ~ water and counterion-water interactions. A detailed presentation of this atomic model may be found e l s e ~ h e r e . ~ ~ , ~ O ~ ~ ~ To illustrate the relaxation behavior of a confined liquid, we performed molecular dynamics (MD) simulation^^^ of LennardJones (W) fluids confined in a slit-shaped pore modeled by two LJ smooth walls. The fluid-surface potential is
V ( Z )= 2~~12[2/5(dz)’O - (o/z)~]
20
1
i5t .
lo
. . . ,
0
(1)
0 where E and u are the classical W parameters, taken to be identical for the fluid-fluid and fluid-surface interactions. The multiplicative parameter (A) takes into account the density of the U sites at the surface of the solid; its increase permits a countinuous variation of the fluidlsurface interaction. A nonwetting surface corresponds to a low value of the coupling parameter (1 RZ 01, while perfect wetting occurs when A % 1. The U fluids are composedof 400 particles. Their trajectories are calculated by a fourth-order Gear algorithm3z with a time step of 0.01 ps. The fourth-order Gear algorithm ensures the stability of the temperature of the system during at least 300 ps without any regulation of the v e l o c i t i e ~if, ~the ~ initial configuration is well thermalized. The configuration of the particles used by the fourth-order Gear algorithm includes the knowledge of the positions and velocities of the particles, plus the forces acting on each particle at the present and previous time step. The initial positions are thermalized by preliminary Monte Carlo simulations in the canonical ensemble.z5 The initial velocities are thermalized by some iterations of Brownian dynamics.34The forces at a given time step are directly calculated, while the initial value of the forces at the previous time step are determined by one iteration using a classical fourth-order Runge-Kutta procedure. The temperature is then checked and stabilized during ,~~ a few MD iterations, by multiplication of the v e l o c i t i e ~and allowed to fluctuate during the rest of the MD simulations. The coefficients of diffusion are determined by integration of the velocity autocorrelation function^.^^^^^ The NMR relaxation rates are calculated from the Fourier transform of the autocor~~ the dipolar relation functions of the H a m i l t ~ n i a ndescribing coupling between the molecules (see Appendix I).
111. Results and Discussion Figures 1and 2 show the variation of the longitudinal ( R J and transverse (Rz) lH relaxation rates of water molecules as a function of clay concentration in hectorite suspensions. Rz is always larger than R1, and both relaxation rates are linear functions of clay concentration. This behavior12,21results from the coexistence of two environments available for the protons, pertaining either to free bulk water molecules (noted F) or to bound (25) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J . Chem. Phys. 1953,21,1087-1092. (26) Adams, D. J. Mol. Phys. 1974,28,1241-1252. (27) Delville, A. Langmuir 1991,7,547-555. (28) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. J . Chem. Phys. 1983,79, 926-935. (29) Bounds, D. G. Mol. Phys. 1984,54,1335-1355. (30) Delville, A. Langmuir 1992,8, 1796-1805. (31) Delville, A.; Sokolowski, S. J . Phys. Chem. 1993,97,62616271. (32) Berendsen, H. J. C.; van Gunsteren, W. F. Proc. Inter. School Phys. “Enrico Fermi”, course 97; Cicotti, C., Hoover, W. G., Eds.; pp 43-65, 1986. (33)Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; DiNola, A.; Haak, J. R. J . Chem. Phys. 1984,81,3684-3690. (34) van Gunsteren, W. F.; Berendsen, H. J. C.; Rullman, J. A. C. Mol. P h p . 1981,44,69-95. (35) Diestler, D. J.;Schoen, M.; Hertzner, A. W.; Cushman, J. H. J . Chem. Phys. 1991,95,5432-5436. (36) Berne, B. J. Physical Chemistry-An Advanced Treatise: Liquid State; Eyring, H., Henderson, D., Jost, W., Eds.; 1971;Vol. 8B, Chapter 9.
?
.
.
.
I
5
.
.
0
.
.
10
15
Clay concentration (%) Figure 1. Variation ofthe longitudinal (0)and transverse (D) relaxation rates at 300 K as a function of the hectorite/water weightlweight ratio.
. 1
0.5
I.,:, ,
,
e 0 ,
,
,
,
,
,
,
,
,
e
0
0
5
10
15
Clay concentration (“YO) Figure 2. Influence of temperature on the longitudinal relaxation rates of hectorite suspensions: (0)300 K and (MI 280 K.
physisorbed water (noted B). The protons of the two environments exchange rapidly on the NMR time scale (ms), and a single population of spins is detected. What is the origin of the large difference between Rz and R1? Water and many simple liquids, under standard conditions, are characterized by equal longitudinal and transverse relaxation rates. Large differences were however frequently reported for the same liquids in the presence of solid interface^.^^*^^ A possible explanation may be a slow modulation of the dipolar relaxation of the protons of the adsorbed water molecules. Using an autocorrelation function described by a single exponential function (see Appendix I),one generally extracts from these data a correlation time varying between and s,20,37 corresponding to the reorientation of the adsorbed water molecules. The slowing down of the motion of water (sc s for bulk free water) is attributed to watersurface interaction^.^^,^^ A proof of the slow modulation of the dipolar coupling is generally obtained by the temperature variation of the longitudinal relaxation rates (see Appendix I). The problem here arises from the fact that R1 is a decreasing function of temperature (cf. Figure 2) whereas the longitudinal relaxation rate of nuclei under slow modulation conditions should be an increasing function of the tem-
-
(37) Piculell, L. J . Chem. Soc. Faraday Trans. I1986,82,387-399.
An NMR Study of the Clay-Water Interface
Langmuir, Vol. 1 1 , No. 4,1995 1363
0 102
IO‘
106
108
10‘0
Angular frequency ( C ’ ) Figure 3. Variation of the longitudinal (full symbol)and spinlocking longitudinal (open symbol)relaxation rates at 300 K as a function of angular frequencies w and 01, respectively. The samples studied are pure water (A)and hectorite suspensions at concentrations of 7% (u)and 12% (01,respectively. Table 1. Influence of the Angular Frequency on the Longitudinal and Transverse Relaxation Rates of a Dilute Hectorite Suspension (12%w/w) 6.1 x lo8 5.9 107
2.04 3.25
17.9 19.8
perature (see Figure 8a) if the correlation function is described by an exponential. The classical analysis ofthe NMR results thus leads to a paradox.21 Nevertheless, the expected influence oftemperature on the longitudinal relaxation rate only concerns bound protons (RIB). A direct detection of the signal of the adsorbed water molecules is not possible and the measured relaxation rate is the average of the relaxation rates of free and bound water molecules (eq A8). In eq A8, the relaxation rate of bound water (RIB)and the fraction of free water ( p ~both ) increase with temperature, while R ~ F andpB decrease. An a priori prediction of the temperature variation of R1 thus becomes impossible. The regime of slow modulation may be evidenced by the angular frequency variation of the relaxation rates (cf. Figure 8b), since the protons of free water are in fast modulation conditions, with relaxation rates insensitive to angular frequency. The results reported in Table 1 show a parallel decrease of R1 (1.2 s-l) and RZ(0.9 s-l) when the angular frequency increases. This observation confirms the slow modulation regime (see Figure 8b), characterized by large differences between Rz and R1. We performed R1 measurements in a broad range of angular frequencies (6 x lo7 s-l w 2.3 x lo9 s-l) to determine the frequency variation of the spectral density for two dilute hectorite suspensions (7 and 12 wt %). In contrast with the expected Lorentzian shape, Figure 3 shows a logarithmic decrease of the longitudinal relaxation rates in the whole range of angular frequencies. Some measurements of Rl,, performed at angular frequencies 4 x I O 3 s-l w1 < 3 x lo5 s-l, are compatible with an extension of the logarithmic decrease of the spectral densities for angular frequencies as low as 107-106 s-l. Below that limit, the relaxation rate becomes independent of the angular frequency. The spectral density of bulk water was also determined to check the experimental procedure: in agreement with the fast modulation condis), no variation of the spectral density tions ( t ~ was detected in the whole range of angular frequencies. Clearly, the spectral densities of clay suspensions cannot be described by a single Lorentzian function with an intermediate regime extending over 4 orders of magnitude
-
I
1 o7
I
I
i0’
10‘
1 013
0 Figure 4. Spectral densities JO(-) and Jz (- - -) calculated from MD simulations of the trajectories of a U fluid confined into a nonwetting narrow pore (3 molecular diameter). The insert shows the local concentration profile of the U fluid.
of angular frequency. As a consequence, the dominant mechanism for NMR relaxation of the hydrogen atoms of water molecules in the presence of clay is not the intramolecular proton-proton dipolar coupling modulated by molecular rotation, since the autocorrelation function of the dipolar Hamiltonian is not exponential (see Appendix I). Diffusion of adsorbed water at the clay surface is another relaxation mechanism, able to modulate the interparticular dipolar Hamiltonian and to induce relaxation of the macroscopic transverse and longitudinal magnetizations. We use MD simulations of a simple Lennard- Jones fluid confined between smooth LennardJones planes as a guide to understand the relative role of “geometrical and energetical confinement” on the relaxation behavior of the claylwater interface. As previously reported, liquids confined in small pores, with a diameter comparable to the molecular size (geometrical confinement), behave in a way intermediate between the 2D and 3D bulk l i q ~ i d(Figure ~ a ~ ~ 4)whatever their interaction with the solid. The reason for this behavior is the anisotropy of the diffusion tensor.35 A direct consequence of this partial 2D diffusion is a logarithmic decrease of R1 and R1, as a function of the angular frequencies wo and 01 (Figure 4), and a large difference between R1 and Rz.7 Another consequence of this “geometricalc ~ n f i n e m e n tis” an ~ ~important ~~~ layering of the (see insert of Figure 4). In addition to this short-range “geometrical confinement”,surface-fluid interactions may induce an “energetic c ~ n f i n e m e n t ”even ~ ~ , ~in~ large pores (more than a few molecular diameter), with adsorbed molecules also characterized by 2D diffusion during their residence on the surface. MD simulations of confined U fluids with increasing surface-liquid interaction clearly show the occurrence of this energetic confinement (Figure 5): layering of the fluid and partial 2D diffusion (logarithmic decreases of the spectral densities) appear when the coupling parameter (A)(eq 1)increases. As a consequence, a strong fluid-solid interaction induces a partial 2D diffusion of the adsorbed liquid, even a t large separation of pore walls. At times larger than the mean residence time, adsorbed molecules start t o desorb and diffuse in a 3D space. The corresponding spectral densities show a break of their logarithmic variation a t an angular frequency equal to the inverse of the residence time. At (38)Korb, J. P.;Delville, A.; Xu,S.; Jonas, J. Mat. Res. SOC.Symp. Proc. 1993,290,107-112. (39)Korb, J.P.;Delville, A.;Xu,S.;Jonas, J. Mugn. Res. Imug. 1994, 12,179-181. (40)Korb, J. P.; Delville, A.; Demeulenaere. G.: Costa. P.: Jonas. J. J. Chem. Phys. 1994,101,7074-7081.
1364 Langmuir, Vol. 11, No. 4,1995
Delville and Letellier c(z*)/cO 3
0.002
h = 0.01
h = 0.01 2
0,001
1
0 1.
10
h = 0.05 0.002
h = 0.05
8
6
0,001 4
0 2
15
0.02
0.01 10
0 108
100
loio
loii
loi2
ioi3
5
Angular frequency (C') 0 0
2
6
4
a
10
Z*
Figure 6. Influence of the fluid/solid attraction (see text) on the spectral densities [Jo(-), Jl(.-*), and Jz (- - -)] (Figure 5a) and local concentration profiles [c(z*)/col(Figure 5b) of a U fluid confined within a large pore (10 molecular diameter).
angular frequencies smaller than this cutoff, the spectral densities and the relaxation rate become characteristic of a 3D fluid, i.e. are independent of the angular frequency. Obviously, the simple model of LD fluid confined between two smooth planes cannot quantitatively predict the dynamics of water molecules in the presence of clay particles. Nevertheless, the large differences between the transverse and longitudinal relaxation rates are due to the low-frequency variation of the spectral densities (cf. eq A5), corresponding to the long-time evolution of the autocorrelation functions of the dipolar Hamiltonian (see Appendix I). While the molecular structure ofthe solvent and the solid surface greatly influence the short-time evolution of the dipolar autocorrelation functions, these details are averaged a t a larger time scale. From a qualitative point of view, this long time evolution depends only on the dimension of the diffusion space and the intensity of the surface-solvent interactions. A detailed description of the structure of the water molecule aggregates induced by the clay particles must take into account the directional character of the solvent-
solvent and solvent-solid interactions. As an example, the surface potential of the clay particle is highly h e t e r o g e n e o ~ and s ~ ~water molecules near the clay surface adopt preferential o r i e n t a t i o n ~ ~not , ~ considered J~ by the simple LJ model. A consequence of this local ordering of adsorbed water is the inefficiency of water reorientation for the relaxation of magnetization through the fluctuations of intramolecular dipolar coupling. In this case, diffusion becomes a possible relaxation mechanism inducing fluctuations of intennolecualr dipolar coupling. In that context, dynamical simulations of a simple U fluid illustrate how clay-water interactions are responsible for the partial 2D diffusion of water molecules in dilute suspensions, even with an interparticle separation as large as 103 A. A molecular model based on an atomic description of the montmorillonite-water interface16z30 also shows a strong layering of liquid water in contact with separate hectorite particles (Figure 6). Because of the clay-water interactions, the structure of bulk water is modified in the vicinity of the clay particle. Water molecules in the
An NMR Study of the Clay-Water Interface
.
1
I
I
T
Langmuir, Vol. 11, No. 4, 1995 1365
-
40 c)
0
c
1
e
b
I
8-.
0 1 u)
0
5
10
15
20
(A) Figure 6. Local concentration profiles (M) of water molecules (I) and sodium counterions (0)confined between two hectorite particles.
first hydration layer are strongly connected to the clay particles and their condensed sodium counterions, neutralizing the negative charges of the clay particles. Their average energy, compared to highly diluted vapor (perfect gas), is (-95 f 3) kJ/mol, i.e. 55 kJ/mol lower than the cohesion energy ofbulk water. Hectorite possesses a high surface energy, which explains its high wettabi1ity.l Previous simulations of the montmorillonite/water interface showed a uniform orientation of the water molecules close to the surface of the particle.16 The water molecules point at least one proton in the direction of the clay surface, because dioctahedral clays appear as a regular network of nucleophilic oxygen atoms. These oxygen atoms from the surface tetrahedral layer form a hexagonal cavity with a single hydrogen atom at its center. This hydrogen originates from a hydroxyl group of the inner octahedral layer. For dioctahedral clays (like montmorillonite), the OH director is parallel to the clay surface, leading to a fully nucleophilic surface. For trioctahedral clays (like hectorite), the OH director is perpendicular to the surface of the particle, and the preferential orientation of the adsorbed water molecule depends on its location relative to the clay surface (Figure 7). Near a hexagonal cavity, some water molecules direct a lone electron pair toward one of the clay protons, leading to a less ordered configuration of the first hydration layer. Our GCMC simulations of the clay-water interface are able to describe this hydrogen bond, since the clay-water potential is derived from,MO calculation^.^^ In this context, the observationof a logarithmicvariation of the spectral densities for angular frequencies as low as lo6 s-l requires a residence time as long as s for water molecules in the direct vicinity of the clay particles. This longtime appears excessivefor water molecules under standard conditions. Using Eyring formalism, it corresponds to an activation free energy of 40 kJ/mol (equal to the cohesion energy of bulk water). Whatever the nature of the transition state, the activation enthalpy for water desorptionis greater or equal to the enthalpy of desorption (55 kJ/mol), as calculated by GCMC simulations of hectorite hydration (see above). Since desorption does not require the breaking of strong chemical bonds, but simply of relatively weak hydrogen bonds, the activation enthalpy is probably of the same order of magnitude as the desorption enthalpy. However, the entropic contributions are difficult to determine: in addition to the gain of one translation degree of freedom, desorption implicates a complex balance between the contributions resulting from the structure of bulk and adsorbed water molecules.
Figure 7. Lateral (a)and top (b)views showing an equilibrium configuration of the water molecules and sodium counterions in direct contact with a hectorite particle.
These results are certainly not specific to the claywater interface. However, the clear appearance of partially 2D diffusionthrough NMR relaxation measurements is restricted to carefully selected samples, in which dipolar coupling modulated by spin diffusion is the dominant relaxation mechanism. As already ~ h 0 ~ nsamples , 2 ~ ~ ~ ~ with a large amount of paramagnetic impurities display complex relaxation behavior.
IV. Conclusions We used lH NMR to study the dynamical properties of water molecules in contact with hectorite clay particles. Relaxation measurements in a broad frequency domain showed a logarithmic variation of the spectral densities due to a partial 2D diffusion of the adsorbed water molecules. The correspondingresidence time is extremely long ( s)but agrees with adsorption energy determined by Monte Carlo simulations of the clay/water. interface. Appendix I. NMR Relaxation of Water I. Dipolar Coupling. This Appendix sketches some features of the theory of NMR relaxation; a more detailed presentation is not our purpose and may be found e l ~ e w h e r e .Dipolar ~~ relaxation is the main relaxation mechanism of the protons of the water molecule. The Hamiltonian describing the time evolution of the population of spins is the sum of the Zeeman interaction and the dipolar coupling between the nuclei:
Out-of-equilibrium macroscopic magnetizations are obtained by applying excitation pulses. Relaxation to equilibrium results from dissipation of the magnetic (41)Abragam, A. The principle of Nuclear Magnetism; Clarendo: Oxford, 1961.
Delville and Letellier
1366 Langmuir, Vol. 11, No. 4, 1995 energy (TI measurements) or order (2'2 measurements) induced by the fluctuations of the dipolar coupling. The dipolar coupling between two nuclear spins (I)is described by the dipolar Hamiltonian [&(t)], which may be decomposed into a sum of products of time-independent spin (quantum) operators [ A ~ ~ ( and l l l time-dependent (classical) functions [F~~(t)l: 2
HD(t)= -2y2
C FD-m(t)ADmQ
1 o4 u)
c1 P)
e
102
C
.-w0
-m
loo
X
(A2)
m=-2
P)
U
10'2
The time-dependent orientation functions [FDm(t)ldepend on the relative positions of two interacting nuclei 1 10''
2 10"
3 10"
s
4 10''
10''
1/T 200
The autocorrelation functions [G"Tt)l quantify the fluctuations of the dipolar Hamiltonian:
G Y z ) = (F"(O)T"(t))- I(F"(O>)12
(A4)
The spectral densities [ J D m ( o ) ] are the Fourier transforms of these autocorrelation functions; they are directly related to the relaxation rates:
1 2 R, = = -y:h2N~I(I + l > [ J ~ l ( W o )+ 4JD2(2Wo)]
Tl
3
150
100
50
0
(Ma)
1 o6
1 o7
1 OB
1 o9
1 O'O
0 (s-' 1 Figure8. Temperature (a)and angular frequency(b)variations of the longitudinal (-) and transverse (- -) relaxation rates due to an exponential decorrelation of the dipolar Hamiltonian (see the text).
k, H$F
where wo and 0 1 are the resonance angular frequencies of the nuclei immersed in the inductions Bo and B1, respectively (oi= yB,). If the autocorrelation functions are described by a single exponential,
the spectral densities are Lorentzian
as observed for the intramolecular dipolar coupling, modulated by molecular rotation. Parts a and b of Figure 8 respectively show the influence of temperature and angular frequency on the relaxation rates. The inverse of the correlation time follows an Eyring law, with a constant activation free energy (27.53kJImol), leading to s a t 300 K. The angular a correlation time of frequency is equal to 2.26 lo9 s-l in Figure 8a, while the temperature is equal to 300 Kin Figure 8b. Three domains may be distinguished, corresponding respectively to fast ( w t c < l), slow ( w t ~> l),and intermediate ( w t ~ 1) regimes of modulation. In the fast modulation regime, R1 and R2 are equal, independent of the angular frequency
+ SF
___)
H2°~
k, Figure 9. Rate constant describing the exchange between free bulk and physisorbed water molecules.
and decreasing functions of the temperature. In the slow modulation regime, R2 is larger than R1; the difference is independent of the angular frequency but decreases with the temperature since R1 and Rz are respectively increasing and decreasing functions of the temperature. The intermediate regime covers 2 orders of magnitude of the angular frequency and presents complex behaviors ofthe relaxation rates. 11. Contribution of Chemical Exchange. A second possibility for obtaining frequency-dependent relaxation rates, with R2 larger than R1, is chemical exchange. WoessneflZmodified the Bloch equation to incorporate the contribution of the chemical exchange. To simplify, let us consider the exchange of water molecules between ~ two sites (Figure 9). In the fast exchange regime ( k and k~ >> R l ~R,B , and /OF - OBI), a single population of spins is detected and the observed relaxation rates are the average of the relaxation rates characteristic of each chemical environment
-
(42) Woessner,
D.E.J.Chem. Phys. 1€%1,35,41-48.
An NMR Study of the Clay-Water Interface However, in the intermediate exchange regime ( ( W F - W B ) ~ >> (RZF - R~B)') the longitudinal relaxation rate still follows eq A8 while the transverse relaxation rate is given by
leading to a frequency-dependent transverse relaxation rate and larger than the longitudinal relaxation rate. 111. Contribution of Diffusion Mechanisms. The exponential function correctly describes the autocorrelation of the intramolecular proton-proton dipolar coupling modulated by the rotation of the water molecule. This fast mechanism (ZC s) overcomes the relaxation of bulk water. Diffusion is another relaxation mechanism,characterized by a lower efficiency. However, in heterogeneous conditions, with a lack of isotropy of the physisorbed water molecules, the diffusion of the adsorbed molecules may become an efficient mechanism for the modulation of the intermolecular dipolar coupling. For 3D bulk liquids, the three autocorrelation functions (eqs A4) are equal and decrease asymptotically as t-312 for
Langmuir, Vol. 11, No. 4, 1995 1367 long time^.^^^^^ The resulting spectral densities are equal and vary as A - Bwln as w tends to zero,43but for 2D bulk liquids, the three autocorrelation functions differ totally. If the plane of motion is perpendicular to the magnetic field, GI is zero, while GOand Gz decrease asymptotically as t-l and r2, r e s p e c t i ~ e l y . ~As ~ ,a~ consequence, the spectral density, J d w ) ,diverges logarithmically as w tends to zero while Jz(o) remains finite.44-46
Acknowledgment. We cordially thank Mr. F. Obrecht for hectorite purification and Drs. J. P. Korb, R. Setton, and H. Van Damme for interesting discussions. The MD calculations were performed locally at the CRMD on minicomputers (HP 9000-720 and 712/80) and the GCMC simulations were performed on a Cray YMP C98 supercomputer (IDRIS, Orsay, France). LA9408477 (43)Fries, P. H. J.Phys. 1981,42, L 257-261. (44) Avogadro, A.; Villa, M. J . Chem. Phys. 1977, 66, 2359-2367. (45) Korb,J. P.; Ahadi, M.; McConnell, H. M. J . Phys. Chem. 1987, 91, 1255-1259. (46) Chachaty, C.; Korb, J. P. J . Phys. Chem. 1988,92,2834-2841.
