Rheological Study of Two-Dimensional Very Anisometric Colloidal

Article pubs.acs.org/Langmuir

Rheological Study of Two-Dimensional Very Anisometric Colloidal Particle Suspensions: From Shear-Induced Orientation to Viscous Dissipation A. M. Philippe,*,† C. Baravian,†,⊥ V. Bezuglyy,† J. R. Angilella,‡,# F. Meneau,§ I. Bihannic,∥ and L. J. Michot∥ Laboratoire d’Énergétique et de Mécanique Théorique et Appliquée, Université de Lorraine - CNRS, UMR 7563, 2 Avenue de la Forêt de Haye, TSA 60604 54518 Vandoeuvre Cedex, France ‡ Laboratoire Environnement, Géomécanique et Ouvrages, Rue du Doyen Marcel Roubault, BP40 54501 Vandoeuvre-lès-Nancy Cedex, France § Synchrotron SOLEIL, L’Orme des Merisiers Saint-Aubin, BP48 91192 Gif-sur-Yvette Cedex, France ∥ Laboratoire Interdisciplinaire des Environnements Continentaux, Université de Lorraine - CNRS, UMR 7360, 15 Avenue du Charmois 54500 Vandoeuvre-lès-Nancy, France

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ABSTRACT: In the present study, we investigate the evolution with shear of the viscosity of aqueous suspensions of size-selected natural swelling clay minerals for volume fractions extending from isotropic liquids to weak nematic gels. Such suspensions are strongly shear-thinning, a feature that is systematically observed for suspensions of nonspherical particles and that is linked to their orientational properties. We then combined our rheological measurements with small-angle Xray scattering experiments that, after appropriate treatment, provide the orientational field of the particles. Whatever the clay nature, particle size, and volume fraction, this orientational field was shown to depend only on a nondimensional Péclet number (Pe) defined for one isolated particle as the ratio between hydrodynamic energy and Brownian thermal energy. The measured orientational fields were then directly compared to those obtained for infinitely thin disks through a numerical computation of the Fokker−Plank equation. Even in cases where multiple hydrodynamic interactions dominate, qualitative agreement between both orientational fields is observed, especially at high Péclet number. We have then used an effective approach to assess the viscosity of these suspensions through the definition of an effective volume fraction. Using such an approach, we have been able to transform the relationship between viscosity and volume fraction (ηr = f(ϕ)) into a relationship that links viscosity with both flow and volume fraction (ηr = f(ϕ, Pe)).

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suspensions of fibers with varying aspect ratios.11−13 The case of Brownian dilute suspensions of rigid nonspherical particles has also been investigated theoretically.14−17 Such work was extensively reviewed by Brenner,18 who then proposed implicit relationships between the orientational field and viscosity for various aspect ratios. When the volume fraction of anisotropic particles increases, the situation becomes complex because of the interplay among hydrodynamic interactions, translational and rotational Brownian motion, and orientation phenomena. Theoretically speaking, most studies deal with suspensions of non-Brownian fibers. In the semidilute regime, various models taking into account hydrodynamic interactions have been proposed,19−22 mostly by developing along the lines first proposed by Batchelor.23 Comparison with experiments carried out on fibers with different aspect ratios in various suspending

INTRODUCTION The suspension viscosity is mainly governed by the volume fraction of particles. In the case of monodisperse hard spheres, for low particle concentration, the relative viscosity evolves with volume fraction according to Einstein’s relation.1 Such a relation was later extended by Batchelor to take into account pair interactions in both non-Brownian2 and Brownian3 suspensions. Such analytical expressions remain valid as long as multiple hydrodynamic interactions can be neglected (i.e., for dilute and semidilute cases). This is obviously not the case for volume fractions higher than a few percent. In the latter case, analytical solutions cannot be found and various so-called effective approaches have been developed.4−7 Very early on,8 it was recognized that the presence of nonBrownian spheroidal particles strongly modifies the suspension rheology as a result of the preferred orientation of nonspherical particles under flow, which results in shear-thinning behavior. Jeffery’s work was extended to the case of any axisymetric particles by Bretherton9 and Cox.10 Experimental confirmations of such predictions were provided by experiments on dilute © 2013 American Chemical Society

Received: January 11, 2013 Revised: March 13, 2013 Published: April 1, 2013 5315

dx.doi.org/10.1021/la400111w | Langmuir 2013, 29, 5315−5324

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fluids12,24−28 reveals good agreement and confirms the crucial role played by particle orientation in defining the mechanical properties of suspensions. However, the extension of these models to concentrated suspensions underestimates both viscosity and shear-thinning when compared to experimental investigations.27−29 Other models neglecting hydrodynamic interactions and focusing on contact interactions were then developed to reproduce results in the concentrated region.29,30 A few theoretical studies focus on the behavior of Brownian fibers in the semidilute regime31,32 and predict that interactions increase the alignment of rods with the flow, which consequently enhances shear thinning. Despite extensive theoretical and experimental effort, it appears that at the present time no theory is fully able to predict the evolution of viscosity with volume fraction (from a Newtonian liquid to a gel state) in the case of anisotropic particles. In addition, whereas numerous experimental studies have been carried out with non-Brownian suspensions, wellcontrolled systems of anisotropic Brownian particles are rather scarce. The low shear viscosity of suspensions of charged boehmite rods with an aspect ratio of 22.5 was analyzed for increasing ionic strength and rationalized on the basis of effective approaches by defining effective volume fractions while taking into account the Debye length.33 Suspensions of colloidal hematite rods with two different aspect ratios were also investigated at three different ionic strengths, and their viscosity was modeled using similar effective approaches.34 Although relevant for Brownian suspensions, these two studies do not provide any detailed analysis of the influence of parameters such as the size and aspect ratio on the mechanical properties of the suspensions. In that context, we have concentrated our efforts over the past few years on aqueous suspensions of natural swelling clay minerals used as models of Brownian systems of anisotropic platelike colloids. Through careful purification and size-selection procedures, we have been able to obtain suspensions of charged platelets with aspect ratios ranging from 100 to 350. By combining small-angle X-ray scattering (SAXS), osmotic pressure, and rheological measurements, we have shown that the system was purely repulsive for ionic strength lower than 5 × 10−3 M/L.35−38 Furthermore, by adapting the effective approaches developed for spheres4−7 to the case of disklike particles, we were able to rationalize the evolution with size and volume fraction of zero shear and infinite shear viscosities.39−41 To better understand the strong shear-thinning behavior of these systems, we used a rheo-SAXS setup built around a Couette cell to analyze in detail the evolution of the orientational field with increasing shear.42−44 The present study extends these latter works. Indeed, we will analyze the shear evolution of the orientational field of four different samples with various aspect ratios at a fixed ionic strength (10−4 M/L). We will apply such an experimental strategy to suspensions evolving from quasi-Newtonian isotropic fluids to weakly gelled samples with the aim of predicting shear-thinning from a knowledge of both particle morphology and orientational field.

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tronites.36 All structural formulas can be found in ref 38. On the basis of unit-cell parameters, the density can be estimated to be around 2.9 g/cm3 for nontronites samples and 2.6 g/cm3 for montmorillonites and beidellite. To reduce the size polydispersity, successive centrifugations were performed, yielding four size fractions for each sample. The characteristic length and polydispersity of these particles were determined following the procedure described in previous articles,38,45 and all sizes are summarized in Tables 1 and 2. To scan the whole

Table 1. Characteristic Lengths of Nontronite Particles Used in This Study NAu-1 size 1 average length (nm) polydispersity (%) average width (nm) polydispersity (%) average size 2a (nm)a average thickness 2e (nm) aspect ratio (1/rp)b excluded volume fraction (range)c

580 52 61 50 320 1 320 1.18− 4.03

size 2 460 53 60 55 260 0.76 342 1.93− 2.70

size 3 350 47 54 35 202 0.76 266 0.81− 1.55

size 4 196 44 138 38 117 0.76 154 0.52− 0.96

a Calculated as the arithmetic mean of the length and width. bIn Brenner’s paper, the aspect ratio rp is defined as the polar radius divided by the equatorial one so that rp < 1 for an oblate ellipsoid. c Calculated according to eq 10.

Table 2. Characteristic Lengths of Montmorillonites and Beidellite Particles Used in This Study SAz size 2 average diameter 2a (nm) polydispersity (%) average thickness 2e (nm) aspect ratio (1/rp) excluded volume fraction (range)

150 42 0.7 214 0.94− 1.45

SWy-2 size 3

95 19 0.7 136 0.49− 0.88

size 2 240 46 0.7 342 0.88− 1.68

SBId-1 size 1 320 93 1 320 1.37− 3.69

(volume fraction/ionic strength) phase diagram, homogeneous suspensions were prepared by osmotic stress46 using dialysis tubes (Visking) with a molecular weight cutoff of 14 000 Da. Solutions with varying osmotic pressures were prepared by the dilution of PEG 20 000 (Roth) in sodium chloride solutions. PEG solutions were renewed after 2 weeks, and the experiment was stopped after 1 month as it was shown previously for latexes,47 laponite,48 and nontronite,36 in which such a duration ensures osmotic equilibrium. At the end of the experiment, the suspensions were recovered and their mass concentrations were determined by weight loss upon drying, taking into account the relative humidity according to the water adsorption isotherm of Na-saponite.49 For samples studied in this Article, the ionic strength was fixed at 10−4 M/L NaCl corresponding to a Debye length of 30 nm. Rheo-SAXS measurements were performed at the SWING beamline of the SOLEIL synchrotron (Orsay, France). An MCR-501 rheometer (Anton Paar) equipped with a Couette cell was used on the SAXS beamline. This Couette cell is formed with a static outer cylinder (stator) and a rotating inner cylinder (rotor) both made of polycarbonate. Radii of the stator and the rotor are 10.25 and 9.75 mm, respectively, providing a gap of 500 μm. The total immersed height was 18 mm. A wide range of shear rates from 5000 s−1 to the static state was applied to investigate the whole shear-thinning behavior of suspensions. A full ramp of decreasing shear rates was performed in 15 min, including a 2 min preshear step (at the highest

MATERIALS AND METHODS

Four reference natural swelling clay minerals (two montmorillonites, one from Arizona (SAz-1) and one from Wyoming (SWy-2), one beidellite from Idaho (SBId-1), and one Australian nontronite (NAu1)) were purchased from the Source Clays Minerals Repository of the Clay Mineral Society at Purdue University. Clay suspensions were purified following the procedure established previously for non5316


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of q equal to that of the first peak of the structure factor, if present, or at 0.1 nm−1 in other cases. Such a choice was made after carefully checking that in the intermediate regime, where the curves scale as q−2, the q value did not influence the analysis of angular variations. As a consequence, choosing the first peak of the structure factor optimizes the signal-to-noise ratio. Recent descriptions40,42,43 have shown that in the case of platelets under shear flow it was relevant to assimilate the volume freely available to particles in suspensions as that of an oblate ellipsoid. Consequently, it is judicious, using a reference frame of (O, x, y, z) such that O is the particle center, x is the flow direction, and z is the shear direction, to express the ODF as

applied shear rate). It must be pointed out that up and down ramp experiments do no reveal any time dependence on such a time scale. Small-angle X-ray scattering (SAXS) experiments were carried out using a fixed energy of 11 keV and a sample-to-detector distance of 5 m. Two-dimensional scattering patterns were recorded on an AVIEX CCD camera formed by four detectors. The contribution from the Couette cell filled with water was subtracted after correction for transmission. For each volume fraction and applied shear rate, 2D SAXS patterns were recorded both in the radial and tangential positions (Figure 1). The curves of scattering intensity versus

ODF(θ , φ) =

axayaz 4π

1 (sin 2 θ(ax 2 cos2 φ + ay 2 sin 2 φ) + az 2 cos2 θ)3/2

(1)

where ax, ay, and az are the three axes of the ellipsoid that describes the statistical confinement of the vector normal to the particle and θ and φ are the polar and azimuthal angles of the vector normal to the particle, respectively (Figure 1). When such a formalism is used, the analysis of the angular variations of SAXS patterns obtained in the radial direction yields a first anisotropy parameter of ay/ax that describes the particle rotation along the streamline direction. An analysis of the tangential patterns yields az/ay, which allows for the calculation of az/ax that characterizes the particle deviation from the velocity streamline.

Figure 1. Schematic representation of the Couette cell and incident Xray beam orientation in tangential and radial measurements. Examples of SAXS patterns obtained in both measurement configurations are also displayed.

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RESULTS Orientation of Clay Particles under Flow. In this section, we will analyze the orientational properties of an extended data set comprising 27 clay suspensions with various particle shapes, sizes, and volume fractions. For readability reasons, we first illustrate the approach for two clay samples only (i.e., the largest beidellite and nontronite particles). Figure 2 displays the evolution of viscosity versus shear stress for nontronite (Figure 2A) and beidellite (Figure 2B) suspensions at various volume fractions. All suspensions show strong shear-thinning behavior and display a yield stress for very low volume fractions (around 1.3%). Suspensions with the lowest volume fractions exhibited an increase in viscosity at high shear stress that can clearly be assigned to the occurrence

scattering vector modulus q (q = 4π sin θ/λ, where 2θ is the scattering angle and λ is the wavelength) were obtained by the angular integration of the data with a fixed opening angle of 5°. For all investigated samples, the q−2 dependence of the scattering intensity in the so-called intermediate regime (where 2π/2a < q < 2π/2e, with 2a and 2e being the characteristic lengths of clay particles reported in Tables 1 and 2) has been checked to confirm the bidimensional nature of individual scattering objects.50 In this domain, the scattering of a single platelet is strongly directed along its normal. As shown by Van der Beek et al.,51 this implies that the orientation distribution function (ODF) of the vector normal to the particle can be obtained directly from the scattering data at constant q in the intermediate regime. In the present case, we chose to plot the angular variations at a fixed value

Figure 2. Evolution with shear stress of the viscosity of (A) size 1 NAu-1 and (B) size 1 SBId-1 suspensions at various volume fractions. 5317


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Figure 3. Evolution with shear rates of (A) ay/ax and (B) az/ax (details in the text) for size 1 NAu-1 and size 1 SBId-1 suspensions at various volume fractions. The symbols and color codes are those of Figure 2.

Figure 4. Evolution with shear stress of (A) ay/ax and (B) az/ax (details in the text) of size 1 NAu-1 and size 1 SBId-1 suspensions at various volume fractions. The symbols and color codes are those of Figure 2.

large particles it is rather difficult to reach an isotropic state once the system has been oriented. Plotting the same parameters as a function of shear stress (Figure 4A,B) does not significantly modify the curves corresponding to ay/ax. In contrast, using such a treatment, the evolution of az/ax for both beidellite and nontronite falls on a master curve. This shows that, for samples with similar phase behavior,38 the exact particle shape (i.e., disk vs lath) does not significantly influence the orientational field. Furthermore, comparing the evolution of az/ax with that of viscosity (Figure 2) strongly suggests that shear-thinning results from increasing particle orientation under flow and that the rotation of the particle along the vertical direction (i.e., deviation from the streamline) is strongly involved in viscous dissipation. Figure 4 also reveals that the orientational field of bidimensional particles is mainly governed by the local shear stress that takes into account not only the shear rate applied to the medium but also the suspension’s viscosity. Therefore, the orientational field is almost independent of volume fraction for

of the Taylor−Couette instability, the onset of which varies with particle anisotropy.44 Figure 3 presents the evolution with shear rate of anisotropy parameters ay/ax and az/ax for different suspensions of nontronite and beidellite. Whatever the clay and volume fraction, ay/ax values (Figure 3A) are always higher than 0.75 and almost independent of shear rate. Such high values indicate that the particles spend most of their time with their normal in the velocity gradient direction (z). Still, because ay/ax is always lower than 1, a non-negligible population of particles stands with their normal slightly out of the shear plane. Such a feature is more pronounced for nontronite particles. As shown in Figure 3B, az/ax is much more sensitive to the hydrodynamic field as its value decreases with increasing shear rate, thus revealing the increasing confinement of the particles in the flow direction (x). For all shear rates, more concentrated suspensions lead to lower values of az/ax. Even for the leastconcentrated suspensions, az/ax does not reach a value of 1 for the lowest shear rate investigated, which shows that for such 5318


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Figure 5. Evolution with shear stress of the viscosity of (A) size 2 NAu-1, (B) size 3 NAu-1, (C) size 4 NAu-1, (D) size 2 SAz montmorillonite, (E) size 3 SAz montmorillonite, and (F) size 2 SWy-2 montmorillonite suspensions at various volume fractions.

Figure 6. Evolution with shear stress of (A) ay/ax and (B) az/ax for all of the swelling clay suspensions used in the present study. The symbols and color codes are those of Figures 2 and 5.

a wide range of solid contents that includes suspensions behaving either as Newtonian fluids or as yield stress materials. It is then worthwhile to determine if such a conclusion can be extended to a larger data set. The rheological curves corresponding to the six additional clay samples (Tables 1 and 2) are presented in Figure 5. Figure 6A,B displays the evolution with shear stress of anisotropy parameters ay/ax and az/ax for various particle sizes

of four different clay minerals. To enhance the readability, each symbol shape corresponds to a clay nature whereas its color is associated with the particle size. Qualitatively, all samples exhibit a significant decrease in both parameters with increasing shear stress, with such a tendency being more pronounced for parameter az/ax. For each clay sample, plotting az/ax and ay/ax as a function of shear stress yields a single curve as observed previously. However, the curves corresponding to the different 5319


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Figure 7. Evolution with Péclet number of (A) ay/ax and (B) az/ax for all of the swelling clay suspensions used in the present study. The symbols and color codes are those of Figures 2 and 5. The solid black lines correspond to the theoretical calculations (details in the text).

should be possible to predict a particle orientation field for any shear flow. Still, such a conclusion appears to be fully valid only in the case of strongly repulsive particles. Indeed, a close examination of the curve displayed in Figure 7B shows that Wyoming montmorillonite, the less repulsive of the swelling clays that we investigated,38,41 appears to diverge from the general trend and to present a weaker confinement along the velocity streamlines, a conclusion that was already reached in a previous study comparing the behavior of beidellite and Wyoming montmorillonite.43 The curves at high Péclet number are truly independent of the clay nature, clay size, and volume fraction. It must be pointed out that in the treatment we implemented the particle size considered is that of the bare particle without taking into account the Debye length. Increasing the particle size by adding a Debye length strongly degrades the master curve displayed in Figure 7B. The independence with respect to volume fraction revealed by Figure 7B does not appear to be predicted in the theoretical studies dealing with the mechanical properties of semidilute suspensions of anisotropic particles. Indeed, such works31,32 based on solving pair interactions show that the presence of neighboring particles enhances the orientation, which should lead to a dependence of the orientational field on the interparticle distance (i.e., on volume fraction). It must be pointed out that, considering the aspect ratio of the particles, all of the suspensions investigated in the present study are above the semidilute regime (Tables 1 and 2), which prevents a direct comparison with most currently available theories. As a consequence and on the basis of the independence of the orientational field with respect to the volume fraction, it may be relevant to compare our experimental result with theoretical hydrodynamic calculations carried out for isolated anisometric particles to try to assess the influence of multiple hydrodynamic interactions. Orientation under Flow of an Isolated Infinitely Thin Disk. We start with the expression describing the time evolution of the statistical distribution of the orientation of the vector normal to the particle, P,

samples are scattered along the y axis. This is not surprising considering the size differences between the various samples investigated (Tables 1 and 2). To take into account size differences, it is relevant to define a dimensionless Péclet number, Pe. This parameter should compare hydrodynamic energy with Brownian rotational diffusion because these two energies are involved in particles orientation. As shown previously,18 the rotary diffusion coefficient of an infinitely thin disk in a suspending fluid of viscosity μ0 is given by Dr =

3 kBT 32 μ0 a3

(2)

where a is the particle characteristic length, kB is the Boltzmann constant, and T is the medium temperature. Therefore, Pe is given by Pe ≡

γ̇ Dr

(3)

with γ̇ being the shear rate. As already pointed out, in the suspensions we investigate, disks can never be considered to be isolated and, as proposed by Quemada52 to semi-quantitatively account for interactions of a disk with his neighbors, it is appropriate to consider that Brownian diffusion occurs in an effective medium whose viscosity can be assimilated to that of the suspension (ηs). Consequently, the expression of the Péclet number can be written as Pe ≈

3 32 ηsγȧ 32 σa3 = 3 kBT 3 kBT

(4)

with σ being the local shear stress (viscosity of the suspension ηs multiplied by the applied shear rate γ̇). Figure 7A,B presents the evolution of ay/ax and az/ax as a function of the above-defined Péclet number. When such a representation is used, most samples exhibit very similar trends, and as far as az/ax is concerned, almost all samples gather on a single curve for Pe > 1. In view of the variability in size, clay nature, and volume fraction, such a result is remarkable. It validates the definition of the Péclet number (eq 4) and strongly suggests that on the sole basis of particle dimensions it 5320


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individual particles is rather difficult. Such an interpretation is further confirmed by a close examination of the data in Figure 7A, where a clear size effect is observed for all of the disklike particles, with smaller particles leading to lower ay/ax values. Furthermore, a shape effect can be observed, with nontronites displaying low ay/ax values considering their large size. In view of their lath shape, the rotation of one particle can be less hampered by the presence of a parallel neighboring one. Despite the fact that the suspensions investigated are very distant from an assembly of individual bidimensional particles, it appears that the comparison with a computation carried out on isolated disks is not irrelevant for understanding the evolution of orientation under increasing shear. It is then natural to extend such a comparison to the macroscopic level (i.e., to derive links between the existence of such oriented microstructures and the viscosity of the suspensions). From Microscopic Organization to Rheological Properties. Theoretical approaches to dilute suspension rheology are based on the calculation of energy dissipation resulting from both rotational and translational motions of isolated particles. Whatever the particle shape and size,1,8,16,53,55,56 such methods yield relationships between the viscosity and volume fraction that can be written as

1 ∂ 1 ∂ (J sin θP) − (J P ) sin θ ∂θ θ sin θ ∂φ φ

⎡ 1 ∂ ⎛ 1 ∂ 2P ⎤ ∂P ⎞⎟ ⎜sin θ + D0⎢ + ⎥ ∂θ ⎠ sin 2 θ ∂φ 2 ⎦ ⎣ sin θ ∂θ ⎝

(5)

in which θ and φ are the polar and azimuthal angles of the vector n⃗ normal to the particle, which satisfies dn/⃗ dt = J(⃗ n)⃗ in the absence of noise, and of noninertial particles at zero Reynolds number. Vector J ⃗ depends only on the aspect ratio of the inclusion and on the velocity gradient of the flow.8 Equation 5 is the corresponding Fokker−Planck equation obtained when rotational noise with diffusion coefficient D0 is added to the dynamics of n⃗.17,18 In the present study, because we are interested in a direct comparison with experimental SAXS data, we focus only on stationary situations53 (i.e., when ∂P/∂t = 0). In the case of shear flow, even in the stationary case, this equation cannot be solved analytically. It was then computed numerically with a central finite difference method for the space derivatives. Once P is known, ay/ax and az/ax can be retrieved numerically for any aspect ratio. Figure 7 presents the evolution with Pe of both ay/ax and az/ ax together with that calculated following eq 5. As far as az/ax is concerned (Figure 7B), qualitative agreement between measurements and calculations is obtained as az/ax significantly decreases with increasing Pe, showing an increasing confinement of the particles in the flow direction (x). Still, the theoretical evolution is sharper than the experimental evolution. At high Péclet number, the theoretical anisotropy parameter is close to zero, indicating a nearly perfect alignment of the particles that is never reached in the experiments as the lowest az/ax values obtained are around 0.35. It must be pointed out that such imperfect alignment has been repeatedly observed in experiments dealing with anisometric particles.27,29,54 Furthermore, in our case three additional effects may increase the discrepancy: (i) because of the curvature of the Couette cell used in rheo-SAXS measurements, along its path through the gap the X-ray beam probes particles with slightly different orientations (Figure 1); (ii) the clay platelets are not perfect rigid objects and any flexibility would degrade the orientation; (iii) as shown by Van der Beek et al.,51 the approximation used for SAXS pattern angular variation analysis could underestimate the particle orientation. For low Péclet numbers, experimental az/ax values reach 1 (isotropic state) for Pe < 1 whereas theoretical calculations predict random orientation at Pe = 1. This suggests that in the concentration regime investigated the presence of neighboring particles hampers the disorientation of particles to an isotropic state through rotation. As far as parameter ay/ax is concerned (Figure 7A), the low Péclet value region reveals trends similar to those described for az/ax (i.e., the isotropic state is experimentally reached for Pe values lower than those predicted theoretically). For high Péclet values, calculations predict a strong decrease in ay/ax, whereas experimental results display only a moderate dependence. According to theory, at high Péclet numbers an infinitely thin disk will experience so-called tumbling motions and consequently will rotate significantly along the velocity streamlines. The reason that this is not observed experimentally is likely due to the presence of surrounding platelets that impair such a rotation. Indeed, in the case of swelling clay minerals, the distances between platelets that can be deduced from the analysis of SAXS patterns are of the same order of magnitude as the particle characteristic lengths;36,38 therefore, the rotation of

lim

ϕp → 0

ηs ηf

= 1 + [η]ϕp

(6)

where ηs is the suspension viscosity, ηf is the suspending fluid viscosity, ϕp is the volume fraction of particles, and [η] is the intrinsic viscosity. The last parameter depends on the particle shape and flow characteristics. For a given particle shape (or aspect ratio), one can calculate the evolution of particle orientation with Péclet number. The link to viscosity is then obtained by calculating some moments of the particle orientation distribution that are related to viscous dissipation. In the case of rigid, infinitely thin disks, Brenner showed that such an approach requires knowing five material constants that depend only on the aspect ratio rp of the particle and three goniometric factors that are ⟨sin2 θ⟩, ⟨sin2 θ cos 2φ⟩, and ⟨sin2 θ sin 2φ⟩. Following Brenner’s description (eq [8.5] on p 241 in ref 18), the intrinsic viscosity [η] can be calculated from the expression 15 5 Q 2⟨sin 2 θ ⟩ − B−1(3Q 2 + 4Q 30) 4 4 15 −1 2 ⟨sin θ cos 2φ⟩ + λ (3Q 2 + 4Q 3)⟨sin 2 θ sin 2φ⟩ 2

[η] = 5Q 1 −

(7)

Q1, Q2, and Q3 are three of the five material constants whose definitions can be found in Brenner’s original paper. Q03 is a constant that can be expressed as a function of Q3 and N, another material constant. Finally, λ is a Péclet-dependent parameter such that λ = BPe with B =

rp2 + 1 rp2 − 1

(8)

Starting from a completely different perspective, various authors6,57−59 have advocated the use of effective approaches to link suspension viscosity and volume fraction. In that context, on the basis of the minimization of the energy dissipated in the 5321


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flow by viscous effects, Quemada5,52 provided an expression for hard spheres that can be written as −2 ⎛ ϕ ⎞ ηs = ηf ⎜1 − ⎟ ϕ* ⎠ ⎝

(9)

with ϕ* being the packing volume fraction that evolves between 0.58 and 0.72 depending on the investigated system. We recently showed38−41,43 that such an expression could be adapted to the case of very anisometric particles by using a value of ϕ* = 1 and replacing ϕ with ϕeff. ϕeff represents the effective volume fraction accounting for the fluid volume trapped by the particles through their average motion and can be defined as αϕexcl.vol with α being a Péclet-dependent function and ϕexcl.vol being the volume fraction of spheres with excluded volume encompassing the particle. For very anisometric particles such as those considered in this study, ϕexcl . vol ≈

21 ϕ 3 rp p

Figure 8. Evolution with Péclet number of parameter α (details in the text) for all of the swelling clay suspensions used in this study. The symbols and color codes are those of Figures 2 and 5. The solid line correspond to the theoretical calculations.

(10)

α can be easily determined experimentally from viscosity measurements if the particle dimensions and volume fractions are known. Indeed, when eq 9 is used, it becomes α=

⎛ ⎜1 − ϕexcl . vol ⎜⎝ 1

ηf ⎞ ⎟ ηs ⎟⎠

Péclet numbers. For high Péclet numbers, experimental α values are slightly higher than theoretical values, which can be linked to the imperfect alignment of particles along the velocity streamlines that could be tentatively assigned to irregular edges of real particles or to their slightly flexible nature. At low Péclet numbers (i.e., when Brownian rotational motion dominates shear), experimental α values reach the theoretical one (α = 0.5) for values significantly lower than 1. Such a small discrepancy could be assigned either to the presence of collective effects that hamper the disorientation of particles toward an isotropic state or to the fact that experiments were carried out for time periods that may be too short compared to the Brownian rotational characteristic time, which can be calculated to be around 30 s for the large particles considered here. A clear size-dependent stratification appears on that graph, with the smallest particles (green symbols) exhibiting much higher α values. Such a trend is consistent with what we previously observed for the size dependence of the sol−gel transition for various clay particles,35,37,41 with suspensions of smaller particles gelling at lower volume fractions. The influence of size on α is particularly marked at low Péclet numbers. Indeed, a visual extrapolation of all data suggests that such a dependence would vanish toward “infinite” Péclet number. It must be pointed out that it could even yield a reverse trend as shown by the modeling of infinite shear viscosities that revealed that the smaller the particles, the lower the infinite shear viscosity.39,41 For low Péclet numbers, the smallest particles exhibit α values that reach 1.40, a value that appears to be anomalously high. Indeed, in eq 6, [η] takes a value of 2.5 for hard spheres. Accordingly, the maximum value of α should be 1.25 (eq 12). However, with the maximum value of α being linked to the choice of ϕ*, the packing volume fraction that was taken as equal to 1 for anisotropic particles, calculated α values are likely slightly overestimated. Consequently, for the smallest particles, the low Péclet value limit can be considered to correspond to the hard sphere case. It then appears that, for swelling clay minerals, moderate variations of particle anisotropy (Tables 1 and 2) lead at very

(11)

Equation 9 applies to the whole range of volume fraction (i.e., to a wide range of rheological behaviors ranging from a Newtonian fluid to a yield stress gel). As a consequence, its first-order development to the limit of zero volume fraction yields a volume fraction dependence that is equivalent to that of eq 6. In the present case, when the definition of the effective volume fraction is used, it becomes η lim s = 1 + 2αϕexcl . vol ϕp → 0 η (12) f Comparing eq 6 with eq 12 yields

α=

3 rp[η] 4

(13)

which allows a direct comparison between the effective approach and the theoretical one. Figure 8 compares the evolution of the experimental and theoretical values of parameter α with the Péclet number. The symbols represent the evolution of α obtained by applying eq 11 to experimental rheological measurements. The solid line corresponds to eq 13 where the intrinsic viscosity [η] and the goniometric factors (eq 7) were calculated using the numerically computed orientation distribution function. Whatever the clay nature and volume fraction, experimental α values (symbols in Figure 8) display a significant decrease with increasing Péclet numbers, revealing a “shrinking” of the effective volume of particles. This can be linked to the increasing confinement of the particles along the velocity streamlines and is in agreement with the strong shear-thinning behavior of the suspensions. For the largest particles (blue symbols), the decrease in α obtained experimentally (with eq 11) is rather close to that derived from calculations carried out for isolated infinitely thin particles (line in Figure 8). However, the theoretical evolution is sharper than the experimental one; therefore, differences are observed for very low and very high 5322


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dimension. Consequently, continuum fluid mechanics may not be fully applicable, and a discrete approach may be required.

low shear to a spectrum of behavior extending from the infinitely thin case to the spherical one.

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CONCLUSION AND PERSPECTIVES Studying in detail the evolution of the orientational field of nonspherical platelike particles provides a synthetic description of the phenomena that are at the origin of the strong shearthinning of the suspensions of such particles. We have first shown that for a given suspension (one clay mineral of a given size), one shear stress leads to a single orientational state, independently of the volume fraction. This relationship can be extended to most clay suspensions investigated in this study through the definition of one dimensionless Péclet number that allows us to take into account the effect of particle size. It must be pointed out that the definition that is used validates an effective approach because Brownian diffusion is supposed to occur in a medium whose viscosity is that of the suspension and not that of the suspending fluid. We have then used an effective approach to assess the viscosity of these suspensions through the definition of an effective volume fraction. Using such an approach, we have been able to transform the relationship between viscosity and volume fraction (ηr = f(ϕ)) into a relationship that links the viscosity to both the flow and volume fraction (ηr = f(ϕ, Pe)) in which the link with flow is introduced through a Péclet-dependent parameter, α(Pe). Using such a treatment strategy, the effective volume fraction appears to depend solely on the Péclet number, which reveals a link between the uniqueness of the orientational field and macroscopic rheological properties. For the clay systems investigated in this work, one given Péclet number corresponds to one microscopic state (independent of the particle concentration) that can be expressed at the macroscopic level by an effective volume fraction associated with the confinement of anisotropic particles mainly along the flow direction. Such a description provides a continuous microscale/macroscale link between zero shear and infinite shear conditions that are classically analyzed in the literature. We finally evidenced, for regimes where Brownian disorientation dominates, a strong effect of particle size on the effective volume fraction. Such a size effect does not concur with purely classical hydrodynamic considerations. It could be related to the presence of a cloud of counterions around the particle that may modify the actual size to be considered. However, adding the Debye length to particle sizes does not eliminate the size dependence. It then clearly appears that in repulsive aqueous suspensions of clay particles at low Péclet number, one or a combination of various unknown mechanisms lead to a stronger viscous dissipation than what could be calculated on the basis of particle morphology. Possible mechanisms could be linked to electroviscous phenomena that were shown to be more important for smaller particle size.39 In that regard, performing experiments in apolar liquids could provide valuable information. The occurrence in the suspensions of large-scale structures that are not probed by SAXS experiments could also play a role. Such hypothetical structures could be investigated by either USAXS or soft-X-ray microscopy experiments. It would also be extremely relevant, for low Péclet numbers, to measure rotational and translational diffusion coefficients jointly to determine if size effects also appear in such measurements. This could be achieved by carrying out advanced light-scattering experiments60 and dynamical birefringence measurements. Finally, it must be pointed out that the thickness of all of the particles investigated in the present study is of a molecular

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address #

LUSAC, University of Caen UCBN, Cherbourg, France.

Notes ⊥

Deceased. The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This Article is dedicated to the memory of Prof. Christophe Baravian, who died at the beginning of December 2012 at the age of 44. This work is mainly based on the ideas he developed all along his too-short career. We gratefully acknowledge the French ANR for financial support under the ANR-07-BLANC0194 program.

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REFERENCES

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