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J. Phys. Chem. B 1999, 103, 5730-5735
Microscale and Macroscale Diffusion of Water in Colloidal Gels. A Pulsed Field Gradient and NMR Imaging Investigation Franck P. Duval, Patrice Porion,* and Henri Van Damme Centre de recherche sur la Matie` re DiVise´ e, CNRS, and UniVersite´ d’Orle´ ans, 45071 Orle´ ans Cedex 02, France ReceiVed: March 17, 1999; In Final Form: May 3, 1999
Magnetic resonance imaging (MRI) has been used to determine quantitatively one-dimensional proton concentration profiles in clay gels prepared from water/heavy water mixtures, taking into account the variations of spin-spin relaxation rate and other effects on the signal intensity. The method has been used to determine the diffusion coefficient of water in Laponite gels at macroscale, by following the diffusion of water from a volume of gel prepared with water into a volume of gel prepared with heavy water. The results have been compared with the self-diffusion coefficients at microscale obtained by pulsed gradient spin-echo (PGSE) NMR. Both results are in agreement in spite of the complex structure of the gel, but the reduction in diffusion coefficient as compared to bulk water is larger than predicted from effective medium theories.
Introduction Dispersion of colloidal particles in a solvent may induce drastic rheological modifications. One of them is the sol-gel transition.1 The medium behaves as an elastic solid when a stress smaller than a critical yield stress is applied and as a viscoelastic fluid when the stress exceeds the yield value. Remarkably, this transition may be observed in both attractive and repulsive regimes. When the energetic barrier for particle aggregation is small, van der Waals attraction may lead to a highly porous but connected percolating network of particles which confers the medium its elastic properties. On the other hand, when the barrier is high due to long-range repulsive interactions, a contactless but nevertheless rigid network of particles may also be obtained at high enough concentration, each particle being trapped in the potential well generated by the repulsive interactions with its neighbors. A simple example is the case of charged spherical latex or silica particles in water which form a space-filling disordered (gel) packing of Debye spheres.1-3 Since the Debye length is a function of ionic strength, the solgel transition may be induced either by increasing the concentration of particles or by swelling the Debye spheres (i.e., lowering the ionic strength). The structure of the medium is less simple with anisotropic particles.4,5 In recent studies with rods4 or rigid platelets,5-9 it was shown that several length scales have to be considered. In particular, with charged platelets of Laponite clay, nematically ordered contact-free microdomains (“tactoids”) of isolated platelets seem to form at mesoscales (a few tens of nanometers), surrounding larger (∼micrometers) regions of lower particle density.5-10 Finally, large oriented birefringent domains may form at high concentration over macroscopic distances (∼centimeters).11 How the solvent diffuses in such complex structures is the question underlying the present study. Solvent self-diffusion coefficients of liquids in heterogeneous media (porous solids, gels, microemulsions, ...) may be measured by a number of methods: tracer methods (including isotopic labeling), quasielastic neutron scattering (QENS), and NMR. Each of these * Corresponding author. E-mail:
[email protected].
methods probes diffusion in a given range of length and time scales. Tracer methods are direct methods which are usually applied on a macroscopic slab (millimeter or centimeter scale) and which give information on diffusion at the same length scale. The associated time scales are usually at least of the order of a few hours and often much longer. On the contrary, QENS12,13 and NMR relaxation14,15 are indirect methods in which the self-diffusion coefficient, Dself, is extracted from either rotational or translational correlation times or from interjump residence times, via a microscopic model of molecular motion and requires at least as input the length of a molecular “jump”. The characteristic length and time scales of the method are those of the model, which are usually at molecular scale (nanometer or picoseconds, or less). Pulsed-gradient spin-echo NMR (PGSE-NMR)16-18 is a particularly interesting method because it is model-independent and because it probes diffusion in an intermediate range related to the spin memory loss within the imposed inhomogeneous magnetic field. For common values of the gradients in low viscosity solvents, this happens over a few micrometers in a few tens of milliseconds. PGSE-NMR has been extensively applied to a variety of dispersed systems including polymeric gels,19-21 microemulsions,22 colloidal suspensions of clays,23 silica,22 or latex.22 A point of interest in several PGSE-NMR investigations is the colloid concentration dependence of Dself. The decrease of Dself is usually analyzed in terms of two effects.20,24 A first effect is the “obstruction” effect. The polymer macromolecules, the micellar assemblies, or the solid colloidal particles act as obstacles to diffusion of the solvent, leading to an increase of the effective diffusion length and to a reduction in the effective diffusion coefficient as compared to the bulk solvent. This effect, which is particle shape dependent, is usually analyzed in terms of effective medium theories. A second possible effect is the “hydration effect” which is a slowing down of the rotational and translational motions of the first layers of solvent interacting with the surface of the colloidal objects. The first effect is universal whereas the second is specific of the solvent-colloid pair. So far, macroscopic tracer techniques have only been scarcely applied to colloidal gels. In principle, magnetic resonance
10.1021/jp9909210 CCC: $18.00 © 1999 American Chemical Society Published on Web 06/17/1999
Diffusion of Water in Colloidal Gels
J. Phys. Chem. B, Vol. 103, No. 27, 1999 5731
imaging (MRI) may be a suitable technique for that purpose. It has already been applied to diffusion of paramagnetic solutes, based on the relationship between spin-lattice relaxation rates and paramagnetic ion concentration.25 In this paper, we report about the use of MRI for tracing solvent (water) diffusion. The method, which involves interdiffusion of a piece of H2O-based gel and a piece of D2O-based gel has been applied to Laponite gels and the result compared with that of the PGSE-NMR method. Considering the very heterogeneous structure of these gels over many length scales, as briefly outlined above, it was not obvious that both methods would lead to the same D values. Nevertheless, as will be shown below, both methods gave comparable results, which demonstrates that the complex structural features of the gel have little influence on solvent diffusion, at least as long as the solid volume fraction remains small. Materials and Methods (A) Laponite Gels. The gel samples were prepared by dispersing Laponite RD, a synthetic smectite clay of the hectorite type from Laporte, in H2O, D2O, or a mixture thereof. The pH was adjusted at 10 with NaOH, in order to avoid dissolution of the clay,26,27 leading to an ionic strength of the order of 10-4. The mass fraction of clay in the gel (mass of clay/total mass) was set at 5%. In these conditions, the system is in a gel state with a storage modulus of the order of 5 × 103 Pa, which makes it relatively easy to handle. The equilibrium isotopic composition of the liquid phase in the gels prepared from H2O/D2O mixtures was calculated from the initial composition and from the equilibrium constant K for the isotopic exchange reaction
H2O + D2O T 2HDO
(1)
Literature values for K ) [HDO]2/[H2O] [D2O] range from 3.26 to 4 at 298 °K.28-33 We choose arbitrarily K ) 3.5. Knowing the initial molar fractions in the liquid phase, n0(H2O) and n0(D2O) ) 1 - n0(H2O), the equilibrium molar fractions, neq(H2O), neq(D2O), and neq(HDO) are then easily calculated from eq 1 and from the molar balance equations
n0(W) ) neq(W) + (1/2)neq(HDO)
(2)
where W stands for H2O or D2O. In the following we will write n0 in short for n0(H2O). (B) NMR Equipment. Proton NMR measurements were performed on a Bruker DSX100 spectrometer operating at 100 MHz, using a 2.35 T superconducting magnet. All experiments were done at room temperature (296 K). A Bruker imaging probehead Micro-05 was used for the relaxation time measurements and for the MRI experiments. A Bruker diffusion probehead Diff-25 was used for the PGSE measurements. This probehead allows to apply pulsed gradients up to 10 T/m (1000 G/cm) by pushing the supply intensity up to 40 A. For the two probeheads, the gradients were calibrated using distilled water. (C) Relaxation Experiments. Knowing the T2 spin-spin relaxation time is important to perform a quantitative analysis of the MRI experiments. The T2 measurements were performed on all samples using the single spin-echo sequence (Hahn echo sequence: 90°-τ-180°-τ-FID) with a width for the 90° pulse equal to 2.5 µs.34 The delay 2τ was varied in the range [2 ms e 2τ e 10 s]. We preferred to use this sequence instead of the well-known CPMG sequence,35,36 even if the latter allows to eliminate effects due to magnetic field inhomogeneities, because
Figure 1. Sketch of the inter-diffusion MRI experiment.
it measures the real signal observed during the imaging sequence after the echo time 2τ. (D) Self-Diffusion PGSE Measurements. The self-diffusion measurements were performed using the PGSE technique originally described by Stejskal and Tanner.16 We used the Hahn echo sequence with pulsed gradients. The self-diffusion coefficient, D, is extracted by fitting the measured peak intensity integral S(g) which can be written as a function of field gradient duration δ, intensity g, separation ∆ and rf pulse interval τ
S(g) ) S0 exp(-2τ/T2) exp[-γ2δ2g2 (∆ - δ/3)D] (3) where γ is the proton gyromagnetic ratio (2.675 × 108 rads-1 T-1). The integral at zero gradient, S0 ≡ S(g)0), is proportional to the number of protons in the sample. The first exponential term is the attenuation from the spin-spin relaxation during the duration 2τ of the experiment. Since we always used the same echo time 2τ, eq 3 may be rewritten as
S(g) ) S0 exp[-kd(g)D]
(4)
with kd(g) ) γ2δ2g2(∆ - δ/3). The pulsed gradient parameters were fixed at 20 ms for ∆ and 1 ms for δ whereas g was varied from 0 to 1.2 T/m. (E) Interdiffusion Imaging Experiments. This type of experiment was performed as follows. Two cylindrical pieces of gel, in glass tubes, were put together, as sketched in Figure 1. One was prepared with H20, the other with D2O. The onedimensional proton concentration profile along the axis of the cylinders was then followed as a function of time. In MRI,17-18 spatially resolved magnetic resonance is made possible by superposing a time dependent linear field gradient G(r,t) onto the static magnetic field B0, allowing to define a local Larmor resonance frequency ω (r,t)
ω(r,t) ) γB(r,t) ) γ[B0 + G(r,t)‚r]
(5)
In these conditions, if one considers a spin density F(r), the NMR signal dS(G,t) from a dV element in a field gradient G(r,t) may be written as
dS(G,t) ) F(r) dV exp[i(γB0 + γG‚r)t]
(6)
and the integrated signal amplitude becomes
S(G,t) )
∫∫∫F(r) exp[iγ(G‚r)t] dr
(7)
Using the concept of reciprocal space vector, k ) γG‚t/2π introduced by Mansfield,37 this integrated signal amplitude may be written as
S(k) )
∫∫∫F(r) exp[i2π(k‚r)] dr
(8)
In this k-space formalism, the signal, S(k), and the spin density,
5732 J. Phys. Chem. B, Vol. 103, No. 27, 1999
Duval et al.
Figure 2. Reciprocal spin-spin relaxation time in Laponite gels (solid fraction: 5%, w/w) prepared at variable water molar fraction n0 in mixtures of H2O and D2O. The straight line is the fit using fast exchange theory.
Figure 3. Proton self-diffusion coefficient in Laponite gels (solid fraction: 5%, w/w) prepared at variable water molar fraction n0 in mixtures of H20 and D20. The straight line is the fit using fast exchange theory.
F(r), are mutually conjugate. Then, the spin density is the Fourier transform of S(k):
A similar relationship holds for the self-diffusion coefficient, Dself:
F(r) ) FT[S(k)] )
∫∫∫S(k) exp[-i2π(k‚r)] dk
(9)
This is the fundamental relationship of MRI. In practice, the experimental signal S(k) is not a perfect representation of the Fourier transform of F(r). Indeed, in eq 7, it is assumed that the NMR signal is simply proportional to the spin density F(r), whereas in reality, the signal is also affected by several parameters like the spin-spin relaxation time, dipolar and scalar coupling interactions, or the translation of the spin in the presence of magnetic field gradients. In our analysis, we took into account only the T2 effect, as will be further explained in the next section. Results (A) Relaxation and Self-Diffusion. Both the T2 relaxation time and the proton self-diffusion coefficient Dself were measured in the Laponite gel as a function of n0, the initial molar fraction of H2O in the liquid phase, which is also the atomic fraction of protons/(protons + deuterons). As shown in Figures 2 and 3, there is a continuous and linear evolution of the relaxation rate, 1/T2, and Dself with n0. Since one single relaxation time is measured for each composition, the measured 1/T2 may be interpreted as the weighted average of the relaxation rate of the protons in the H2O molecules of the gel, 1/T2(H2O), and the relaxation rate in HDO molecules, 1/T2(HDO), averaged over the two populations:
1/T2 ) 2neq(H2O)[1/T2(H2O)] + neq(HDO) [1/T2(HDO)] (10) with T2(H2O) ) 89.6 ms and T2(HDO) ) 300 ms. These are the so-called fast exchange conditions already validated in clay suspensions,38 which imply that the spin exchange rate between the two populations is faster than the reciprocal time of the sequence used to probe them.
Dself ) 2neq(H2O)[Dself(H2O)] + neq(HDO)[Dself(HDO)] (11) where Dself(H2O) refers to self-diffusion of water molecules in a gel prepared from pure (H2O) and Dself (HDO) to self-diffusion of HDO molecules in a gel prepared from almost pure D2O. As expected, Dself (HDO) (= 1.44 × 10-9 m2/s) is smaller than Dself(H2O) (= 1.75 × 10-9 m2/s). The ratio is 0.82, which is somewhat smaller than expected from Stokes-Einstein relationship. On the other hand, Dself(H2O) is significantly smaller than the pure water value at the same temperature (=2.1 × 10-9 m2/s). The ratio is 0.83. This may be interpreted in terms of the obstruction and hydration effect mentioned in the Introduction. The simplest effective medium theory for the generalized conductivity of a mixture is that of Maxwell Garnett.39 The generalized conductivity may be the electrical conductivity, the self-diffusion coefficient, the complex dielectric permittivity, the thermal conductivity,... When applied to self-diffusion in a suspension of compact spherical particles in a suspending medium,40 this model predicts a decrease of the measured selfdiffusion coefficient, Dself, with respect to self-diffusion in the bulk suspending medium, D0, according to
Dself /D0 ) 1/(1 + p/2)
(12)
where p is the compositional phase volume fraction of particles. In the case of the Laponite gel of this study (5% w/w, p ) 0.02), eq 12 predicts a Dself/D0 ratio of about 0.97, which represents a much smaller reduction of D than what has been measured. This is a frequent situation in colloidal suspensions, which is usually interpreted by doing the assumption that the effective excluded volume fraction for diffusion is larger than p, due to the presence of a tightly bound layer of solvent molecules.22 In that case, eq 12 takes the form
Dself/D0 ) 1/(1 + p′/2)
(13)
Diffusion of Water in Colloidal Gels
J. Phys. Chem. B, Vol. 103, No. 27, 1999 5733 A series of profiles were recorded on homogeneous gels prepared with H2O/D2O mixtures characterized by their light water molar fraction n0 to which the proton density is directly proportional. This is the first factor entering the signal intensity. On the other hand, the NMR signal decreases during the 2τ echo time introduced by the imaging sequence, according to a decreasing exponential function with time constant T2. This is a second factor controlling signal intensity. The local signal intensity may thus be written as
I(z) ) An0 exp[-2τ/T2(n0)]
(14)
where A is a constant depending on the experimental conditions such as the temperature. In these calibration experiments, I(z) is in fact a constant since the gels have a homogeneous isotopic composition. When compared to the experimental results, eq 14 was found to not correctly describe the data. In order to take this empirically into account, we considered that A is not a constant anymore and depends on n0. Its value was determined for a series of compositions and fitted to the following analytical expression: Figure 4. One-dimensional MRI intensity profiles for interdiffusion of protons from a Laponite-H2O gel (left) into a Laponite-D2O gel (right). The profiles have been recorded at intervals of 600 s. The signal perturbations are due to the presence of a small air bubble at the interface (middle) and at the bottom of the left container.
in which p′ is the actual excluded volume for diffusion taking into account the tightly bound layer. The left hand side represents the reduction of the actual volume for diffusion. Introducing the actual measured value of Dself/D0 in eq 13 leads to a value of p′ of the order of 0.12, which seems to be unrealistically high compared to the actual solid volume fraction (0.02) since it would correspond to a temporarily immobilized water film 3 nm thick, on each side of the clay platelets. (B) Imaging Profiles and Macroscopic Inter-Diffusion. When the two cylindrical tubes containing the gel (one prepared with H2O and the other with D2O) are put in contact at time t ) 0, an interface is generated, as evidenced by some light scattering. Small air bubbles may remain trapped at the junction. Despite these imperfections, diffusion of water molecules starts immediately. Figure 4 displays 10 signal intensity profiles along the z-axis in a typical experiment, recorded at 600 s intervals. Due to the time delay between t ) 0 and the end of the first MRI recording, the first profile does not exactly look like a step function, which also means that the interface is not a barrier for diffusion. As time goes on, diffusion of protons from one compartment to the other is quite evident. The influence of two small air bubbles at the interface and on the bottom of the left tube may be clearly detected on the figure as they deeply perturb the profiles. This effect, due to local magnetic inhomogeneity, is surprisingly large compared to the small size of the bubble (less than 1 mm, i.e., one-tenth of the tube diameter). Looking at the profiles, it may be noticed that the area between the last and the first profile on the right side of the interface is smaller than the area between the first and last profile on the left. If the signal intensity profiles were the direct reflect of the proton density profiles, this would violate the proton conservation balance in the system. In fact, the signal intensity is related to the proton density weighted by the spin-spin relaxation time. Since T2, as was established above (eq 10 and Figure 2), is a function of the isotopic composition of the liquid phase, a z-dependent correction factor has to be applied to the signal intensity profiles in order to recover the proton concentration profiles. This correction factor was taken into account by combining an analytical and a numerical approach.
A(n0) ) A0[1 + 7.7 × 10-2 ln(n0)]
(15)
where A0 is A(n0)1). Equations 14 and 15 together yield the signal intensity of a homogeneous 5% w/w Laponite gel prepared from a light/heavy water mixture at n0, with a T2 value given by the fast exchange model (eq 10). Returning to a situation where the isotopic composition is not homogeneous, as in the mutual diffusion experiments, the above equations allow for the computation of the signal intensity profile from the local isotopic compositions. In order to determine the interdiffusion coefficient using these equations, we assumed that, as a first approximation, H20 and D20 molecules diffuse with the same constant coefficient, D. The theoretical evolution of the concentration profile may then be obtained by solving the one-dimensional diffusion equation in a finite volume of width l, with an initial steplike concentration profile of width h. The solution is41
{ (
1 n(z,t) ) n(0) 2
i)+∞
∑ erf i)-∞
)
h + 2il - z 2xDt
+
i)+∞
(
∑ erf i)-∞
)}
h - 2il - z 2xDt
(16)
where n(z,t) is the concentration of the diffusing species. This equation is used to calculate the local n0(H2O) and n0(D2O) values function of time and space. The actual local concentrations neq(H2O), neq(D2O), and neq(HDO) after local isotopic equilibration may then be obtained by solving eq 2 with the appropriate equilibrium constant K. Finally, the predicted MRI signal intensity profile at time t may then be calculated from eqs 14 and 15. For each experiment, the recorded intensity profiles were fitted using two adjustable parameters:xDtexp/l and a constant prefactor, which is needed because the MRI intensity is given in absolute value (Figure 5). However, for the best fits, we noticed a small decrease of this prefactor as time elapses, which points to the limits of our approach. Knowing/xDtexp/l for different profiles (i.e., different times), D may be extracted from the/xDtexp/l vs t plot (Figure 6). We obtained D ) (1.50 ( 0.02) × 10-9 m2/s for the macroscopic diffusion coefficient of the proton from water molecules in the presence of heavy water,
5734 J. Phys. Chem. B, Vol. 103, No. 27, 1999
Figure 5. An example intensity profile (dotted line) and its fit (continuous line).
Duval et al. 15%) of D in Laponite gels as compared to bulk water is much larger than what is predicted (∼1%) by a simple effective medium theory, taking into account the geometrical obstruction effect of the clay particles. At the time scale of PGSE this might eventually be interpreted by admitting the temporary binding of a fraction of the solvent molecules population to the colloidal particles (the so-called “hydration” effect). However, in order to account for the observed effect, this fraction of molecules in a more bound environment would have to be much larger than the solid volume fraction itself, corresponding to a layer thickness of several nanometers, which is incompatible with previous NMR and neutron scattering data.6,38 In addition, this explanation does not hold anymore at the time scale of the MRI, which is much too long to admit that a molecule close to a particle surface would not be able to diffuse a few nanometers away and enter the bulk solvent phase. A quantitative analysis of the multiscale tortuosity for random walks in the actual gel structure is necessary. Finally, our result demonstrates the feasibility of measuring a macroscopic interdiffusion coefficient using a MRI-based technique. However, in order to be quantitative, this requires a careful analysis of all the parameters influencing the relationship between the MRI signal and the concentration of diffusing species. Among those parameters, the local spin-spin relaxation rate is the most important but magnetic susceptibility inhomogeneities may also deeply perturb the observed signal intensity profiles. Acknowledgment. We gratefully thank Pierre Levitz for his help in the analysis of the concentration profiles. References and Notes
Figure 6. xDtexp/l versus time plot and its fit for a MRI interdiffusion experiment.
in reasonably good agreement with the value obtained by PGSE at n0 f 0 (Figure 3). The uncertainty was been calculated over four experiments. Conclusion Using two NMR techniques, PGSE and MRI, we have shown that water diffusion in Laponite gels occurs with basically the same rate at short and long time and length scales (micrometers and milliseconds on one hand, centimeters and hours on the other hand), in spite of the complex hierarchical structure of the gel. Furthermore, the water diffusion coefficient is only marginally smaller than that of bulk water. A possible reason for that could be the very open structure at all length scales of this gel made from small platelets. This may not be a general feature of other types of smectite gels formed from much larger and anisotropic particles (montmorillonite gels for instance) in which, due to the larger deformability, a cellular structure develops.42 Nevertheless, the small decrease (of the order of
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