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Rheological Determination of Interaction Potential Energy for Aqueous Clay Suspensions Christophe Baravian,*,† Delphine Vantelon,‡ and Fabien Thomas‡ Laboratoire d’Energe´ tique et de Me´ canique The´ orique et Applique´ e, UMR 7563 CNRS, BP 160, F-54504 Vandœuvre-les-Nancy Cedex, France, and Laboratoire Environnement et Mine´ ralurgie, UMR 7569 CNRS-INPL, BP 40, F-54501 Vandœuvre-les-Nancy Cedex, France Received January 31, 2003. In Final Form: June 4, 2003 Swelling clay suspensions, considered as strongly anisotropic disks, are very unstable in high shear flows. In this study of sodium montmorillonites from Wyoming and China, a hard sphere model is shown to be relevant for the rheological modeling of dilute solutions (2-20 g/L). For more concentrated systems (20-80 g/L), orientation and alignment effects seem to be preponderant. High shear measurements give complete information on the average clay size and critical percolation concentration necessary for the yield stress appearance. At low shear, when interaction effects are dominant, an electrostatic double-layer approach gives directly the average interaction pair potential, allowing good predictions of the elasticity and yield stress of clay dispersions.
Introduction Aqueous dispersions of swelling clays exhibit unique properties which confer to this type of clay an outstanding importance in natural media and industry. Even at low contents, they govern the mechanical stability and the plant nutrition in soils. Swelling clays are of great use in a number of practical applications. In the gel state, they are thickening and thixotropic agents for drilling fluids, paints, or cosmetics. In the compacted state, they are important components in underground sealing or in geotechnical constructions. Bentonite is mostly used for these purposes. The major component of bentonite is montmorillonite, the most abundant among the swelling clays, also called smectites. The unit particles of smectites are made of three atomic sheets: two silica sheets sandwiching an aluminum or magnesium hydroxide sheet. Isomorphous substitutions by lower charge metal cations in the crystal lattice result in a permanent negative charge. The charge is compensated by electrostatically attracted cations. The nature and valency of those cations govern the hydration and colloidal behavior of smectites. Numerous experimental studies have been devoted to the flow behavior of smectite dispersions during the past decades (see e.g. refs 1-4). These studies generally concern the flow parameters of the gel state, and the role of salts on these parameters. Two parameters affect the flow behavior of smectite dispersions:2 particle characteristics and interparticular forces. Smectite particles are characterized by a platy shape width of 1 nm in thickness and roughly 0.1-1 µm in diameter. Although accurate mea* Corresponding author. E-mail:
[email protected]. Telephone: 00 (33) 3 83 59 57 27. Fax: 00 (33) 3 83 59 55 51. † Laboratoire d’Energe ´ tique et de Me´canique The´orique et Applique´e, UMR 7563 CNRS. ‡ Laboratoire Environnement et Mine ´ ralurgie, UMR 7569 CNRSINPL. (1) Van Olphen, H. An Introduction to clay colloid chemistry; Interscience: London, 1977. (2) Gu¨ven, N. In Clay-Water Interface and its rheological implications; Gu¨ven, N., Pollastro, R. M., Eds.; The Clay Minerals Society: Boulder, CO, 1992. (3) Luckham, P. F.; Rossi, S. Adv. Colloid Interface Sci. 1999, 82, 43. (4) Abend, S.; Lagaly, G. Appl. Clay Sci. 2000, 16, 201.
surements are scarce in the literature, it is commonly admitted that the shape and size of smectite particles are strongly distributed.5 Due to the strong anisotropy of the clay platelets, gelation is observed at low volume concentrations.4 Vali and Bachmann5 evidenced a relationship between ultrastructure and flow behavior of smectite dispersions. Interparticular forces between smectite particles are dominated by electrostatic interactions. The particles are negatively charged on the basal faces due to crystalline substitutions. The edges are covered with hydroxo groups coordinated to underlying silica, aluminum, and magnesium atoms, which exhibit a point of zero charge (PZC) between pH 7 (ref 1) and pH 3 (ref 6). Electrostatic repulsion and electroviscous effects exert an increase in the apparent viscosity of aqueous dispersions. The present work aims at improving the understanding of the rheological properties of aqueous clay suspensions. From this perspective, the orientation and microstructure effects induced by shear and particle concentration will be taken into account. To describe the particle’s alignment in the flow, we use an effective volume fraction approach, completed by an average interaction potential in the fully structured state. We first investigate the average disk diameter using rheology measurements at high shear for dilute systems. The model is then extended to include the average potential energy between platelets for more concentrated systems. We limit the present study to a minimum average distance between the platelets of around 50 µm, that is, far from the case for swelling systems. The clay suspension is considered as a suspension of identical rigid disks. Averaging hydrodynamic effects over the whole space leads to an average size. We then consider, as a first approximation, that all the disks have the same size. At infinite shear, all the platelets are assumed to be completely dissociated in the flow. For very dilute systems, measurable hydrodynamic effects are essentially due to the disks “flipping” motion. Indeed, because of the strong anisotropy of the disks (a . e), no disk orientation is stable (5) Vali, H.; Bachmann, L. J. Colloid Interface Sci. 1988, 126, 278291.
10.1021/la034169c CCC: $25.00 © 2003 American Chemical Society Published on Web 08/19/2003
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in a flow. Rotational Brownian motion provides the necessary perturbations inducing the flipping motion. When all disks are completely free to rotate, the suspending fluid around the disk rotates with it. For increasing clay concentrations, not all the disks can rotate freely, leading to a partial orientation of the disks in flow. For a critical concentration, hydrodynamic forces are expected to fully align the disks. At low shear, electrostatic interaction energy dominates hydrodynamic interactions. At concentrations sufficiently high, disks will therefore be in physical interaction and locally partially oriented. Assuming that they are free to move in the space limited by their neighbors, the disks can then be considered as effective truncated spheres. The Buscall interaction energy deduced from this approximation allows prediction of the high-frequency elastic modulus, which is also measured independently. As shown by recent rheological models,24 the physical approach consists of using the controlled mechanical energy to estimate the internal potential energy of the system through typical organization modifications (like shear thinning). Concentration is then used to modify the average distance between objects. Dimensional Considerations Due to their smallness (e.g. for montmorillonite: a ∼ 0.1 µm and e ∼ 1 nm), disks are strongly submitted to Brownian motion. Constructing the largest Brownian diffusion time tB, we get
tB ≈
ηf a3 kBT
(1)
where ηf is the viscosity of the suspending medium, kB is the Boltzmann constant, and T is the absolute temperature. The hydrodynamic force FH exerted on the free rotating disks is given by FH ≈ ηf γ˘ a2 for dilute solutions, where γ˘ is the shear rate. For concentrated systems, the surrounding medium has an effective viscosity that must take into account the presence of other disks. The hydrodynamic force should then be expressed as (6) Thomas, F.; Michot, L. J.; Vantelon, D.; Montarge`s, E.; Pre´lot, B.; Cruchaudet, M.; Delon, J. F. Colloids Surf. 1999, 159, 351. (7) Krieger, I. Adv. Colloid Interface Sci. 1972, 3, 111. (8) Buscall, R. J. Chem. Soc., Faraday Trans. 1991, 87, 1365. (9) Vantelon, D. Re´partition des cations dans le couche octae´drique des montmorillonites: re´percussions sur les proprie´te´s colloı¨dales. Ph.D. Dissertation INPL, Nancy, France, 2001; p 253. (10) Baravian, C.; Quemada, D. Rheol. Acta 1998, 37, 223. (11) Bihannic, I.; Michot, L. J.; Lartiges, B. S.; Vantelon, D.; Labille, J.; Thomas, F.; Susini, J.; Salome´, M.; Fayard, B. Langmuir 2001, 17, 4144. (12) Zwanzig, R.; Mountain, R. D. J. Chem. Phys. 1965, 43, 4464. (13) Berli, C.; Quemada, D. Langmuir 2000, 16, 10509. (14) Berli, C.; Quemada, D. Langmuir 2000, 16, 7968. (15) Mewis, J.; D’Haene, P. Makromol. Chem. Macromol. Symp. 1993, 68, 213. (16) De Genne, P. G. Adv. Colloid Interface Sci. 1987, 27, 189. (17) Israelachvili, J. Intermolecular and Surfaces Forces, 3rd ed.; Academic Press: London, 1997. (18) Russel, W. B.; Saville, D. A.; Schowamlter, W. R. Colloidal dispersions, 2nd ed.; Cambridge University Press: Cambridge, U.K., 1991. (19) Ben Ohoud, M.; Van Damme, H. C. R. Acad. Sci. Paris 1990, 311, 665. (20) Perkins, R. S. J. Phys. I 1994, 4, 357. (21) Williams, D. E. G. Phys. Rev. E 1998, 57, 7344. (22) Brandenburg, U.; Lagaly, G. Appl. Clay Sci. 1988, 3, 263. (23) Lott, M. P.; Williams, D. J. A.; Williams, P. R. Colloid Polym. Sci. 1996, 274, 43. (24) Quemada, D.; Berli, C. Adv. Colloid Interface Sci. 2002, 98, 51.
FH ≈ σa2
(2)
where σ is the shear stress. The shear rate will be the pertinent independent variable for hydrodynamic effects in dilute systems, but this variable changes to shear stress in concentrated ones.7 Comparison between hydrodynamic effects and Brownian motion leads to a hydrodynamic Peclet number that can be evaluated as
Pe ≈
σa3 kBT
(3)
Apart from hydrodynamic and Brownian energies, the interaction potential between platelets will occur for sufficiently concentrated systems. Taking the interaction energy into account, a new Peclet number can be defined, comparing the hydrodynamic energy to Brownian and interaction energies E:
Pe ≈
σa3 σ ≡ E σc
(4)
E represents the pair interaction potential energy U for concentrated systems.8 σc then represents a critical shear stress: when σ = σc, hydrodynamics effects are comparable to interaction energy and/or thermal Brownian effects. The Peclet number is therefore comparing the mechanical energy (σa3) with the average interaction pair potential energy U. Knowing the average particle dimension (section below), a typical shear stress value (σc) for the shear thinning behavior is assumed to be directly related to the average potential energy of the system. Extended Hard Sphere Model The viscosities of hard sphere suspensions follow master curves, well represented by Quemada model:
ηr )
φ η ) 1ηf φ*
(
)
-2
(5)
where ηr is the relative viscosity of the suspension, η is its viscosity, and ηf is the viscosity of the suspending fluid; φ represents the particle volume fraction, and φ* represents the packing fraction. If the particle volume fraction is high enough, a Peclet dependency appears, and the suspension shows a non-Newtonian behavior. This shear dependency is induced by a variation of the packing fraction, from random packing fraction φ/0 = 0.64 for the system at rest (Pe ) 0) to a more compacted one φ/∞ = 0.72 at infinite shear (Pe f ∞). For anisotropic systems, we define the effective hard sphere volume fraction φHS as the whole volume trapped by the system elements. For disks, we will therefore have
4 a φHS ≈ c 3 e
(6)
where c is the volume concentration of disks. For a suspension of strongly anisotropic disks, a concentration dependency should appear at a critical effective hard sphere volume fraction, expected before the cubic conformation (φcubic = 0.52) where a physical contact between all effective spheres constructed on the disk’s diameter appears. The effective volume fraction occupied by partially oriented disks can then be expressed as RφHS with R ) 1 for free rotating disks at low concentration and R ) 3/4e/a for fully oriented disks at high concentration.
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Figure 1. Viscosity-shear stress variation for montmorillonite Na+ suspension at various concentrations: solid curve, adjustment of the viscosity model (eqs 7 and 4); black symbols, cone and plate experiments; grey symbols, parallel plate experiments.
Of course, the average disk diameter should be interpreted for clays as a typical persistent length more than a real diameter, since it is determined hydrodynamically. Interacting DiskssViscosity Model When disks are interacting, the system is modified to create structures that should depend on interaction, concentration, and shear (Peclet number). Building a viscosity-shear relationship inspired from eq 5, the volume fraction can then be replaced by the effective volume fraction of the flowing structures. Using Quemada’s approach,14 shape, compactness, and behavior in shear of these objects can be normalized using 0-shear and ∞-shear viscosities (η0 and η∞, respectively). Equation 5 can be rewritten as
ηr )
η∞ 1 + Pe ηf χ + Pe
(
12
)
with χ ) 1-
RφHS φ/0 RφHS
(7)
φ/∞
When a yield stress appears at rest, the χ parameter becomes negative (RφHS > φ/0). The yield stress is then given by σy ) -χσc. If only shear thinning behavior occurs, then χ ) (η∞/η0)1/2. The viscosity model (eqs 5 and 7) then represents shear thinning and yield stress behavior by adjustment of a set of three parameters, that is, η∞, χ, and σc, whether a yield stress exists or not. Preparation and Measurements Two montmorillonites were selected for this study due to their strongly differing rheological behavior:9 montmorillonite from Wyoming (Swy2) and a China deposit which exhibits an almost 1 order of magnitude higher yield value and viscosity than those of the Swy2 at an equal weight fraction. The montmorillonite phase is purified from the raw clay samples. Wyoming montmorillonite was supplied by the Clay Minerals Society (SWy2, Crook County deposit, WY). China montmorillonite was supplied by IKO Erbslo¨h (Germany). The clays were first purified by dispersion in deionized water and sedimentation to remove impurities such as quartz and feldspar.
Homoionization to Na+ includes two exchanges with 1 mol L-1 NaCl, washing by centrifugation (30 000 g, 15 min) with Milli-Q water until free of salt. The clays are then dried at room temperature. Montmorillonite suspensions are prepared by dispersing various amounts of dry powder in MilliQ water. The dispersions are then allowed to equilibrate under magnetic stirring during 24 h. Due to a low amount of salt residue from the purification procedure, the aqueous medium contains about 0.001 mol/L NaCl, which generates a Debye length close to 10 nm. Rheological measurements are performed on an AR1000 TA Instruments apparatus. To measure very low viscosities, the air bearing resistance has been fully modeled and its contribution removed from viscosity measures. Two geometries are used: cone and plate (6 cm, 1°) and parallel plates (6 cm, 100 µm gap) to reach higher shear rates for flow experiments. Parallel plate measurements are corrected for nonNewtonian behavior. Creep measurements, performed in parallel plate geometry (6 cm, 200 µm gap), are used for determination of the highfrequency modulus of montmorillonite Na+ Wyoming suspensions in the gel state. Using the coupling between apparatus inertia and viscoelastic properties of the studied system, a technique described elsewhere10 allows higher frequencies than that available in classical oscillating control stress rheometry (Figure 4). The studied systems have been shown to be very stable in time for at least 2 weeks. Two completely independent preparations have been performed, showing an identical rheological behavior.
Results and Discussion Figure 1 shows adjustment of the viscosity model for the Wyoming suspension at various concentrations (for clarity reasons, not all experimental curves are represented). As expected, a measurable yield stress appears for φHS = 0.52, corresponding to a mass concentration close to 20 g/L. The infinite shear viscosity limit dependence versus concentration (Figure 2a) follows a general hard sphere increase at low concentrations (up to 15 g/L). By adjustment of eq 5, the only open parameter is the average hydrodynamic disk radius a. We find a radius of 46 nm for the Wyoming clay and 33 nm for the China clay (using the Krieger-Dougherty equation leads to similar values: 45 nm for Wyoming clay and 35 nm for China clay). This
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Figure 2. Infinite shear viscosity limit versus (a) mass concentration and (b) hydrodynamic volume fraction (equivalent sphere): closed symbols: Wyoming clay; open symbols: China clay. Lines represent adjustment of the hard sphere model (eq 5) with φ* ) 0.72: solid line, a ) 0.046 µm (Wyoming); dashed line, a ) 0.033 µm (China).
hydrodynamic size determination is taken as the physical clay diameter, although (i) the hydrodynamic size can overestimate the physical size, (ii) the clay may not be perfectly rigid, and (iii) the electrical cloud around the platelet is moving with the particle. This size difference between the two systems is nevertheless significant and important for the rheological behavior, since it is the effective volume that matters. When the concentration increases, disks are progressively oriented. The orientation is essentially induced by spatial limitation, as shown by superposition of high shear viscosity for China and Wyoming clay suspensions using the hydrodynamic volume fraction (Figure 2b). This master curve should therefore be relevant for other anisotropic disklike suspensions, until complete alignment is reached, since particle thickness will then have to be taken into account. The plateau region observed after φHS ≈ 0.5 can be explained by two effects. On one hand, when all disks are in contact, an alignment effect is produced, which corresponds to a measurable yield stress. Increasing concentration then corresponds to adding oriented platelets in the system, which is of little influence for the viscosity of the system. On the other hand, Quemada also suggests that little groups of parallel platelets may rotate together to minimize the global hydrodynamic dissipation. Adding new platelets then corresponds to larger groups of parallel platelets, which will also be of very little influence for the system viscosity. This phenomenon stops when the “sandwich” like groups of platelets reach the size of the platelet diameter. Optical measurements may give information about the relative importance of these two phenomena. For the highest studied concentrations, the pair potential energy must dominate the Brownian thermal energy due to (i) space limitations to free disk movement, (ii) entrapment of most of the disks in mesoscopic structures,11 and (iii) negligible Brownian motion of these mesoscopic structures due to their large size. The pair potential energy is then directly related to the critical shear stress σc through eq 4 by U = a3σc. Since the average distance between disks D is related to the volume fraction via D ) 2a(0.64/φHS)1/3 - 2e, the σc(φ) data can directly be converted to the explicit dependence of the pair potential energy versus mean disk distance (Figure 3). The de-
Figure 3. Experimental data for the average pair potential energy obtained from U = a3σc: solid line, eq 8; dashed line, exponential Ae-D/L with L ) 7 nm and A ) 224 764.
creasing exponential variation confirms that sodium montmorillonite suspensions are stabilized by electrostatic repulsion. The relevant interaction potential for sufficiently concentrated systems is therefore the electrostatic repulsion between two parallel charged plates, that can be approximated in the weak overlap approximation (eq 12.47 of ref 17) and for high surface charge density by
U S -D/L = 0.482[NaCl]1/2 e kBT kBT
(8)
where U represents the pair potential energy, D the average distance between two platelets, L the characteristic Debye length, [NaCl] the salt concentration (estimated as 10-3 M, leading to L = 0.304/([NaCl])1/2 = 9.6 nm), and S a characteristic exchange surface, taken as πa2. We can see in Figure 3 that, for long distances, experimental results are in good agreement with this simple double-layer approach. Since, for relatively short distances, other platelets and multiple, more complicated conformations have an influence, we adjust a simple exponential to the experimental data points of Figure 3 given by U/kBT ) Ae-D/L with A ) 224 764 and L ) 7 nm. Although face-face interaction dominates,3 it is important
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Figure 4. Creep experiment for Wyoming clay: concentration, 30 g/L; closed symbols, 5 Pa solicitation; open symbols, 10 Pa solicitation; solid curve, adjustment of the Kelvin-Voight model taking apparatus inertia into account with η ) 0.047 Pa‚s and G ) 117.48 Pa; apparatus inertia, 41.87 µN‚m‚s-1.
to note that the average pair potential also takes into account edge-face or edge-edge neighbor to neighbor configurations around a given platelet. So this adjusted potential will be used for further calculation. To validate the average disk size and the potential energy obtained through this rheological analysis, we try to predict the infinite frequency elastic modulus G′∞ obtained by creep measurements at small deformation for the Wyoming system. Buscall8 proposed a general relationship between U and G′∞ derived from the work of Zwanzig and Mountain:12
G′∞ =
2 Nφ* ∂ U(D) 5πD ∂D2
(9)
where N is the average neighbor number and φ* is the corresponding maximum packing fraction. This model has been successfully applied to various microgel like systems.13-15 It can be noted that when two disks are getting closer when the concentration increases, the ionic clouds around the disks will “interpenetrate”, as for deformable particles. Of course, this is realistic only if the disks are sufficiently close to each other for the decreasing exponential dependence of the potential energy to be valid.16-18 For clay suspensions, since elasticity experiments are performed at rest, platelets are assumed to be randomly oriented and we consider that the face to face configuration gives the major repulsive contribution.3 We therefore use at first approximation N ) 2 and φ* ) 1 in eq 9. The good prediction (obtained without any adjustment parameter) for high concentrations validates the whole rheological approach (Figure 5). It should be noted that a small disk diameter variation or a slightly different potential energy has a dramatic effect on eq 9. The low shear suspension viscosity limit dependence with hydrodynamic volume fraction does not follow a classical hard sphere model (Figure 6). This can be interpreted as a modification of cluster compactness with concentration, apart form the cation influence.19 Although the compactness is different for suspensions of China and Wyoming montmorillonites, the compactness dependence with concentration is the same in both cases, as shown in Figure 6b: the two viscosity curves are superimposed when a modified volume fraction variable is used for the China suspension (0.72φHS). Further analysis of this phenomenon could be performed by taking into account the attractive part of the interaction potential, the aggregation process, and the random disk organization, which is a difficult task in three dimensions.20,21
Figure 5. Solid line: Prediction of the high-frequency elastic shear modulus from eq 9 and Figure 3. Closed symbols: Highfrequency shear modulus. Each experimental point is obtained by the technique presented in Figure 4.
As a final remark, Figure 7 shows a prediction of yield stress (σy) from interaction potential, as cited by Quemada: 24
σy =
6U(D) π(2a)3
(10)
The difference observed at high concentration may be due to local orientation and easier propagation of fractures, showing a yield stress value overestimated by the homogeneous isotropic model. Summary We propose a simple dimensional approach for clay suspensions assuming that (i) platelets have a disk shape, (ii) the system is monodisperse, and (iii) the stabilized system is essentially repulsive. At rest, platelets are randomly distributed, and for sufficiently high concentrations, the dominant interaction energy is responsible for formation of meso- or macroscopic structures (Figure 6). At very high shear, mechanical energy dominates, and all particles are dissociated. The anisotropy and smallness of these disks then seems to present a general high shearhydrodynamic volume fraction dependence due to alignment essentially induced by the space filling effect. At low concentration, this curve allows determination of the average hydrodynamic disk diameter, identified as the average platelet persistent length. When the concentration is sufficiently high (i.e. when a gel state exists at rest), the critical shear stress in the
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Figure 6. 0 shear limit viscosity dependence versus (a) concentration and (b) hydrodynamic volume fraction: closed symbols, Wyoming clay; open symbols, China clay.
(low shear limit). This consideration allows direct determination of the average pair potential energy. The coherence of these general considerations is fully tested by prediction of the high-frequency elastic modulus and, to a certain extend, the yield stress determination. This dimensional analysis approach may be useful for understanding of swelling clays’ rheology (refs 22 and 23, for example). Of course, this study should be further extended considering a more rigorous local approach25 and different particle anisotropy.
Figure 7. Solid line: Prediction of yield stress value from eq 10. Symbols: Experimental data (from flow curves, Figure 1).
non-Newtonian behavior of stabilized clay suspensions is interpreted as the equilibrium point between hydrodynamic energy (high shear limit) and pair interaction forces
Acknowledgment. We would like to thank Dr. Franc¸ ois Caton, Prof. Daniel Quemada and Prof. Christian Moyne for constructive remarks and discussions. LA034169C (25) Delville, A.; Levitz, P. J. Phys. Chem. B 2001, 105, 663.
