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How Do Colloidal Aggregates Yield to Compressive Stress? Caroline Parneix,*,†,‡ Jacques Persello,†,§ Ralf Schweins,^ and Bernard Cabane*,‡ LCMI, Universit e de Franche-Comt e, 16 route de Gray, 25030 Besanc- on Cedex, France, ‡ PMMH, CNRS UMR 7636, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France, § LPMC, CNRS UMR 6622, Universit e de Nice, place Valrose, 06108 Nice, France and ^ DS/LSS Group, Institut Laue-Langevin, BP156, 38042 Grenoble Cedex 9, France †
Received October 31, 2008. Revised Manuscript Received December 22, 2008 Aqueous dispersions of silica nanoparticles have been aggregated through the addition of Al13 polycations and then submitted to osmotic compression. The structures of these dispersions have been determined through smallangle neutron scattering, before and after compression. Some dispersions consisted of mixtures of aggregated and nonaggregated particles;actually a few aggregates dispersed in a “sea” of nonaggregated particles. In these dispersions, it was found that the resistance to osmotic compression originated from the ionic repulsions of the nonaggregated particles; the compression law that related the applied osmotic pressure Π to the silica volume fraction Φ was Π ∼ [Φ/(1 - Φ)]2. Other dispersions were fully aggregated, with all particles forming a fractal network that extended throughout the available volume. In these dispersions, it was found that the resistance to compression originated from surface-surface interparticle bonds. The application of low osmotic pressures (300 nm), while the structure of the network at local and mesoscopic scales was unchanged. Accordingly, few interparticle bonds were broken, and the deformation was primarily elastic. The compression law for this elastic deformation was in agreement with the predicted scaling law Π ∼ Φ4. The application of higher osmotic pressures (>50 kPa) resulted in compression at macroscopic and mesoscopic scales (30-300 nm), while the local structure was still retained. Accordingly, many more interparticle bonds were broken. The compression law for this plastic deformation was in agreement with a scaling prediction of Π ∼ Φ1.7. The location of the elastic-plastic transition indicated that the strength of the interparticle bonds was on the order of 5 times the thermal energies at ambient temperature.
Introduction Colloidal aggregates are assemblies of very small particles that are held together by surface-surface forces. Because these forces are noncentral, the particles may be held in configurations such that the aggregates are not dense. Indeed, many colloidal aggregates have structures that are bushy, dendritic, or fractal. When they are dispersed in a volume of liquid, such aggregates can span across the volume and create mechanical connections between remote points of the liquid, thereby controlling the mechanical or flow properties. In this paper, we examine the extent to which such structures are maintained (or yield) when the aggregates are submitted to external forces. This is a problem of fundamental interest because the effects of noncentral forces on colloidal particles have rarely been studied quantitatively. It is also of practical relevance because colloidal aggregates are used as gelling agents, retention aids, and coating agents. In these applications, the aggregates may be submitted to large external forces such as mechanical compression or shear, hydrodynamic drag forces, or else capillary forces. These forces may cause the aggregates to break up, collapse, or deform in other ways. Consequently, the response of the aggregates to applied forces may determine how they perform in applications. In this work we have chosen to study colloidal aggregates made of silica nanoparticles. Aqueous silica dispersions can be synthesized in a very reproducible way, with particles that * Corresponding authors. E-mail:
[email protected] (C.P.);
[email protected] (B.C.).
(1) Iler, R. K. The Chemistry of Silica; John Wiley: New York, 1979. (2) Foissy, A.; Persello, J. In The Surface Properties of Silicas; Legrand, A. P., Ed.; Wiley: New York, 1998; Chapter 4B, p 365.
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have narrow size distributions in the 10-20 nm range and well-defined surface chemistry.1,2 These particles were aggregated by the addition of Al13 polycations (see Flocculating Agent section) which bind to the silica surfaces and compensate or reverse their surface charges.3 It is already well-known that such aggregates have fractal structures, with sizes that exceed 1000 nm and self-similarity exponents close to df = 2.4 There are many types of external forces that can be applied to such aggregates: hydrodynamic forces such as the drag forces exerted by shear flows,5-7 mechanical forces that can be applied to macroscopic networks,8-10 and capillary forces that are generated by surface tension when such aggregates are located at a water/air interface.12 However, the simplest force that can be applied is the compressive stress that results from extraction of the solvent from the volume that contains the aggregates. In this work, we have chosen to apply this force through osmotic stress.9,11 For this purpose, the dispersion (3) Lartiges, B. S.; Bottero, J. Y.; Derrendinger, L. S.; Humbert, B.; Tekely, P.; Suty, H. Langmuir 1997, 13, 147. (4) Madeline, J. B.; Meireles, M.; Bourgerette, C.; Botet, R.; Schweins, R.; Cabane, B. Langmuir 2007, 23, 1645. (5) Buscall, R.; McGowan, I.; Mills, P.; Stewart, R.; Sutton, D.; White, L.; Yates, G. J. Non-Newtonian Fluid Mech. 1987, 24, 183. (6) Buscall, R.; Mills, P.; Goodwin, J.; Lawson, D. J. Chem. Soc., Faraday Trans. 1 1988, 84, 4249. (7) Tang, S.; Preece, J.; McFarlane, C.; Zhang, Z. J. Colloid Interface Sci. 2000, 221, 114. (8) Buscall, R. Colloids Surf. 1982, 5, 269. (9) Miller, K. T.; Melant, R.; Zukoski, C. F. J. Am. Ceram. Soc. 1996, 79, 2545. (10) Antelmi, D.; Cabane, B.; Meireles, M.; Aimar, P. Langmuir 2001, 17, 7137. (11) Bonnet-Gonnet, C.; Belloni, L.; Cabane, B. Langmuir 1994, 10, 4012. (12) Brinker, C. J.; Scherer, G. W. Sol-Gel Science. The Physics and Chemistry of Sol-Gel Processing; Academic Press: San Diego, 1990.
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containing colloidal aggregates is placed in a dialysis bag, which is immersed in a polymer solution of known osmotic pressure. Water is then extracted from the dispersion, and the aggregates are compressed until their compression resistance matches the applied osmotic pressure. In the present work, we report the results of experiments that were designed to address the following questions: What was the initial state of the colloidal aggregates dispersed into the aqueous phase, i.e., what were their structures and how were they organized? What is the macroscopic compression law of the dispersion, i.e., what pressure is required to extract water and compress the aggregates in a smaller volume? What are the new structures of the aggregates in the compressed state? Can this resistance to compression be predicted from the strength of surface to surface forces and from the structures of the aggregates?
Materials Aqueous Silica Dispersions. Colloidal silica particles were synthesized by neutralization of aqueous sodium silicate solutions with sulfuric acid, according to the method described by Iler.1 The overall reaction scheme is 3:4SiO2 =Na2 O þ H2 SO4 3:4SiO2 þ Na2 SO4 þ H2 O
ð1Þ
During the first stage of this nucleation-and-growth process, nucleation of silica particles was triggered by adding sulfuric acid ([H2SO4] = 17 g kg-1) to a dilute sodium silicate solution (Silmaco 35-37, diluted to [SiO2] = 2.5 g kg-1) until the pH was lowered to 9;13 the mixture was constantly stirred at 250 rpm, and the nucleation rate was controlled by temperature in the range 60-90 C. Then, particles were grown from these nuclei by the simultaneous addition of a sodium silicate solution ([SiO2] = 39 g kg-1) and of a sulfuric acid solution ([H2SO4] = 17 g kg-1), with addition rates controlled so that the pH was maintained at 9 while the temperature was kept at 90 C. Finally, the dispersion was cooled to room temperature, and it was washed in a tangential ultrafiltration setup where the permeate (sodium sulfate solution) was constantly replaced by deionized water. This washing procedure removed sodium sulfate and other ions that would otherwise screen the surface charges of the particles and cause them to aggregate prematurely. Finally, the dispersion was concentrated through tangential ultrafiltration without permeate replacement, up to a weight fraction equal to 0.05. The final pH was in the range 9-9.5, and the final ionic strength was about 5 10-3 M. Two silica dispersions, S1 and S2, were prepared following this procedure. They were fully characterized, regarding particle size distributions, surface chemistry, and ionic composition of the solvent. The average particle diameters were determined through small-angle neutron scattering (SANS); they were 2a1 = 13.0 nm for S1 and 2a2 = 16.6 nm for S2. These particle diameters were in good agreement with the hydrodynamic diameters (15.0 and 18.1 nm, respectively) measured through dynamic light scattering and with the specific surface areas measured through cetyltrimethylammonium bromide adsorption (202 and 158 m2 g-1, respectively). The original dispersions were prepared at pH = 9. Parts of them were brought to pH = 5 through exchange with an Amberlite IRN77 (Rohm and Haas) sulfonic ion-exchange resin. Another dispersion, S3, was prepared at a higher concentration ([H2SO4] = 20 g kg-1 and [SiO2] = 80 g kg-1 at pH = 9 and (13) Persello, J. Silica sol, composition containing the same, method for treating said silica sol and uses thereof. Patent WO0069976, 2000.
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T = 90 C). In these conditions the silica particles aggregate during their growth. This aggregated dispersion was then washed and filtered, producing a wet silica cake that was redispersed in water. This dispersion had the same surface chemistry as S1 and S2 and the same surface area (170 m2 g-1). It was used exclusively for the determination of the adsorption isotherms, since it was easier to separate the large aggregates from the supernatant through centrifugation. The surfaces of precipitated silica particles are made of silanol groups, with a surface density of 5 SiOH/nm2.14 These sites :: behave as Bronsted acids according to the following reactions: Si -O - þ H þ h Si -OH Si -OH þ H þ h Si -OH2þ
ðK1 Þ ðK2 Þ
ð2Þ
where K1 and K2 are the reaction constants for the first and second protonation of the silanol groups. The theoretical values for these constants are log K1 = 9.70 and log K2 = -4.61.2 Accordingly, the Si-OH+ sites are present only 2 at pH < 1, whereas the Si-O- sites become numerous at pH > 6. The number of Si-O- sites is then called the structural surface charge number Zs; this charge is compensated by Na+ counterions, which may be either condensed on the surface or distributed in a diffuse layer. Each particle with its condensed counterions can be identified to an effective particle of radius aeff ≈ a and effective charge number Zeff, which corresponds to the number of counterions in the diffuse layer. This number is related to the particle radius a and to the Bjerrum length LB by the condensation criterion:15 4a ð3Þ Zeff ¼ LB For silica dispersion S1 (a = 6.5 nm), this yields Zeff = 36, and a charge number density Zeff/4πa2 = 0.07 nm-2. The surface charge number Zs of the silica particles was measured through potentiometric titrations2 at pH varying between 3 and 10, whereas their effective charge number Zeff was calculated from electrophoretic mobility measurements.16 At low pH, we find Zeff = Zs, as expected since the surface charge potential is too low to cause counterion condensation; at basic pH, Zs increases rapidly whereas Zeff remains locked by counterion condensation at the value predicted by eq 3 (Figure 1). Flocculating Agent. Aggregation of the colloidal dispersions was triggered by addition of an aqueous “flocculant” solution. This solution was prepared by partial neutralization of an aluminum chloride solution ([AlCl3] = 0.25 M) with sodium hydroxide ([NaOH] = 0.25 M) at 70 C.17 The final ratio rOH/Al = [OH]/[Al] was 2.4, and the final pH was 4.5. Under these conditions, it is known that the hydrolysis of the aluminum salt leads to the formation of soluble polynuclear species, among which the poly7+ cation [AlIVO4AlVI (hereafter abbreviated Al13) 12 (OH)24(H2O)12] is the predominant one.18 The structure of this polycation consists of 12 six-coordinated aluminum ions, surrounding a central aluminum ion with tetrahedral coordination. At the hydrolysis ratio rOH/Al = 2.4, the Al13 units can be either isolated or slightly aggregated, yielding linear-shaped clusters.19,20 Figure 2 shows the 27Al NMR spectrum of the flocculant solution, the chemical shifts being measured with respect to that (14) Zhuravlev, L. T. Langmuir 1987, 3, 316. (15) Belloni, L. Colloids Surf., A 1998, 140, 227. (16) Persello, J. In Adsorption on Silica Surfaces; Papirer, E., Ed.; Marcel Dekker: New York, 2000; Chapter 10, p 297. (17) Wang, M.; Muhammed, M. Nanostruct. Mater. 1999, 11, 1219. (18) Bottero, J. Y.; Cases, J. M.; Fiessinger, F.; Poirier, J. E. J. Phys. Chem. 1980, 84, 2933. (19) Axelos, M.; Tchoubar, D.; Bottero, J. Y.; Fiessinger, F. J. Phys. (Paris) 1985, 46, 1587. (20) Bottero, J. Y.; Axelos, M.; Tchoubar, D.; Cases, J. M.; Fripiat, J. J.; Fiessinger, F. J. Colloid Interface Sci. 1987, 117, 47.
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Figure 1. Comparison of the effective charge number density (number of counterions in the diffuse layer, per nm-2 of silica surface) calculated from electrophoretic mobility measurements at ionic strengths I = 10-2 M (O) and 10-1 M (b) with the structural charge number density determined from potentiometric titrations performed at the same ionic strengths (4 and 2).
Figure 2. 27Al liquid-state NMR spectrum of the flocculant solution ([Al] = 7.2 10-2 M as measured by inductively coupled plasma spectroscopy). Al(H2O)3+ 6
of ion (δ = 0 ppm), given by an aluminum nitrate solution with concentration 0.1 M. The spectrum exhibits three resonances: (i) a sharp line at 63 ppm, corresponding to the central tetrahedral aluminum of the Al13 structure,18 located in a high-symmetry environment; (ii) a broad line at 8 ppm, originating from the 12 outer octahedral aluminums of Al1321 (this last resonance is quite broad because of the asymmetric environment of these atoms);22 (iii) a very weak signal at 0 ppm, originating from a few octahedral Al atoms that are in a symmetric environment (unreacted Al(H2O)3+ ions or octahedral Al13 6 atoms in aggregated polycations).
Methods 27
NMR Spectroscopy. Al NMR spectra were acquired on a Bruker ASX 500 spectrometer operating at 130 MHz. Both liquid (Al13 original solution) and powder (silica flocculated (21) Bradley, S. M.; Kydd, R. A.; Howe, R. F. J. Colloid Interface Sci. 1993, 159, 405. (22) Fu, G.; Nazar, L. F.; Bain, A. D. Chem. Mater. 1991, 3, 602.
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with Al13) samples were analyzed. The powders were prepared from silica suspensions aggregated with Al13, by centrifugation, removal of the supernatant in order to eliminate excess aluminum ions that were not adsorbed on the silica surfaces, and finally freeze-drying. In this last case, magic angle spinning (MAS-27Al NMR) of the rotors was used, at a rotation speed of 14.5 kHz. The chemical shifts are reported with respect to the resonance of the Al(H2O)3+ ion (δ = 0 ppm) in a 0.1 M 6 aluminum nitrate solution. The values of chemical shifts and integrated intensities associated with each resonance were determined from simulation and decomposition of the spectra into Gaussian peaks using Dmfit2005 software developed by Massiot et al.23 Adsorption Isotherms. Various amounts of Al13 solution were added to silica dispersion S3, and the part of Al that had reacted with silica in each sample was determined through the depletion method: after 48 h of equilibration, the silica particles were separated through centrifugation, and the concentration of Al in the supernatants was measured through inductively coupled plasma (ICP) spectroscopy after filtration of the samples through 0.22 μm filters. Osmotic Stress. The osmotic stress technique is based on water exchange between the sample (i.e., a colloidal dispersion) and a reservoir of known osmotic pressure. A complete description of this method can be found in the publications of Parsegian et al. and Bonnet-Gonnet et al.11,24 Briefly, the sample is placed in a dialysis bag which, in turn, is immersed in a reservoir that contains a solute (generally a polymer) for which the relation between osmotic pressure and concentration is known. The cutoff of the dialysis bag is chosen so that it only retains the polymer and the colloidal matter of the sample. Conversely the solvent, i.e., water, ions, and small organic molecules, can exchange between both compartments. At equilibrium, the chemical potentials of water on either side of the membrane are equal, and therefore the osmotic pressure of the sample equals that of the polymer in the reservoir. This technique makes it possible to play with interactions in a colloidal system over a very wide range of pressure. A poly(ethylene glycol) with a molar mass of 35 000 Da (Fluka, Switzerland) was used as the “stressing” polymer. We have determined the osmotic pressures of this polymer in water using a membrane osmometer (Knauer, Germany) for concentrations up to 20% (w/w) at 20 C. The same pressures were found at pH ranging from 2 and 11 and at ionic strengths up to 0.3 M. They were fitted to the following expression for the osmotic pressure Π (Pa) as a function of PEG concentration [PEG] (%, w/w): log Π ¼ a þ b½PEGc
ð4Þ
with a = 0.49, b = 2.50, and c = 0.24. Solutions of PEG at osmotic pressures from 0.3 kPa to 0.2 MPa were prepared by dissolving the polymer in aqueous solutions at concentrations ranging from 0.4 to 15% (w/w). The pH was adjusted to match that of the aggregated dispersions introduced in the dialysis bags (i.e., the final pH of aggregated silica dispersions after reaction with Al13 polycations). Standard regenerated cellulose Visking 8/32 dialysis bags with a molecular weight cutoff of 12 000-14 000 Da were used (Medicell International Ltd., UK). These bags were chosen so as to allow exchange of ions but not silica particles or PEG. Prior to the experiments, the bags were washed in deionized water. Then, the silica dispersions were placed in the bags and immersed in the polymer solutions kept at 20 C. The content of the bags was (23) Massiot, D.; Fayon, F.; Capron, M.; King, I.; Le Calve, S.; Alonso, B.; Durand, J.-O.; Bujoli, B.; Gan, Z.; Hoatson, G. Magn. Reson. Chem. 2002, 40, 70. (24) Parsegian, V. A.; Rand, R. P.; Fuller, N. L.; Rau, D. C. Methods Enzymol. 1986, 127, 400.
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adjusted (and readjusted) so that the bags were neither too full nor too flat, because in either case there would be a contribution to the actual pressure from the elasticity of the bag. After equilibrium was reached (30 days), a sample of the concentrated silica dispersion was taken from each bag and dried at 120 C in order to determine the silica concentration. The dispersion left in each bag was immersed again in the PEG solutions for structural analysis by small-angle neutron scattering. Small-Angle Neutron Scattering (SANS). Scattering methods measure the distributions of distances within the sample. These experiments may be described on the basis of interactions of incident radiation with individual scatterers and interferences between scattered rays.25 In neutron scattering, the scattering pattern results from interferences between rays scattered by nuclei located at different positions.26 The phase differences that control these interferences are determined by the scalar product Q 3 r, where r is the vector joining two nuclei and Q is the scattering vector. If the sample is isotropic as a whole, then the scattered intensity only depends on the magnitude of the scattering vector, which varies with the neutron wavelength and scattering angle according to Q ¼
4π θ sin λ 2
ð5Þ
If the sample is made of identical particles dispersed in a homogeneous solvent, then the intensity can be decomposed as a product of the intensity scattered by a single particle and a structure factor that describes the interferences between rays scattered by different particles:26-28 IðQÞ ¼ ðFp -Fs Þ2 Vp 2 Np PðQÞSðQÞ
ð6Þ
where (Fp - Fs) is the difference in scattering density between the particles and the solvent, Vp the volume of a particle, Np the number of particles per unit volume, P(Q) the form factor of particle, and S(Q) the structure factor that describes the pair correlations between the positions of all particles. For spherical particles of radius a, the form factor is simply determined by the particle radius a: PðQÞ ¼ ½3ðsin Qa -Qa cos QaÞ=ðQaÞ3 2
ð7Þ
Since P(Q) is known and I(Q) is measured, eq 6 makes it possible calculate S(Q). This structure factor is related to the pair correlation function of the particles, g(r), through
Results
Z
¥½gðrÞ -1r2 sinQrQr dr
SðQÞ ¼ 1 þ 4πNp
ð8Þ
0
In real particle dispersions the particles are polydisperse in sizes, and therefore eqs 6-8 are not strictly valid. Nevertheless, the study of colloidal dispersions in which the distribution of particle sizes is not too broad shows that a very good approximation consists in taking into account the polydispersity of particle sizes through the form factor P(Q) only.29 In practice, P (Q) is measured on dilute dispersions, and it is fitted by inserting a suitable distribution of particle diameters into eq 7. This effective form factor is then used within eq 6 to extract an (25) Champeney, D. C. Fourier Transforms and Their Physical Applications; Academic Press: New York, 1973. (26) Lindner, P. In Neutrons, X-rays, and Light: Scattering Methods Applied to Soft Condensed Matter; Lindner, P., Zemb, T., Eds.; Elsevier: Amsterdam, 2002; Chapter 2, p 23. (27) Hayter, J. B. Faraday Discuss. Chem. Soc. 1983, 76, 7. (28) Cebula, D. J.; Goodwin, J. W.; Jeffrey, G. C.; Ottewill, R. H.; Parentich, A.; Richardson, R. A. Faraday Discuss. Chem. Soc. 1983, 76, 37. (29) Chang, J.; Lesieur, P.; Delsanti, M.; Belloni, L.; Bonnet-Gonnet, C.; Cabane, B. J. Phys. Chem. 1995, 99, 15993.
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effective structure factor S(Q), from which the pair correlations can be determined. The structure factor has a simple behavior in a few limiting cases, depending on the types of interparticle correlations: No correlations. For a perfect gas of noninteracting “phantom” particles, g(r) = 1 and S(Q) = 1 (no correlation) at all Q values. Associative correlations. If the particles are associated, forming large aggregates, then g(r) has a long positive tail at large r, which yields a large contribution to the integral of eq 8 for low Q values, where the contributions from all particles in the aggregate are in phase. At higher Q values, interferences between particles located at different positions within the aggregate cause oscillations in sin(Qr) and a decay in S(Q). If the aggregate is self-similar with a fractal dimension df, then the number of particles n(r) located at a distance r from a reference particle decays as n(r) rdf, and the pair correlation function g(r) must decay as g(r) rdf-3; therefore, the structure factor S(Q), which is a Fourier transform of g(r), must decay as S(Q) Q-df.30 Repulsive correlations. If the particles repel each other as in an equilibrium hard sphere liquid, then g(r) can be related to the pair potential through the Percus-Yevick approximation31,32 and calculated through the analytical expression given by Wertheim.32,33 This pair correlation function has an oscillation at large r due to the pseudoperiodic correlations caused by repulsions of the particles. At low Q values, this oscillation of g (r) causes a compensation in the integral of eq 8, resulting in a depression of S(Q). The depth of this depression is a measure of the strength of correlations between the positions of particles. Conversely, at Q = 2π/d, where d is the particle diameter, the oscillations of g(r) are in phase with those of sin(Qr), which gives a peak in S(Q). The magnitude of this peak is a measure of the organization of the first coordination shell of a particle: a sizable peak is obtained if the coordination shell contains 8-10 neighbors, as in equilibrium liquids, whereas this peak vanishes if each particle has only 3-4 neighbors. For experiments on compressed dispersions, wet dialysis bags were placed on the neutron beam of the instrument D11 at ILL. We checked that the water loss due to evaporation was negligible during the acquisition of the spectra. Intensities were collected at each detector cell, radially averaged, and arranged as a function of the scattering vector Q. Spectra were obtained using different detector positions and wavelengths and then combined to yield scattering curves that extended from Q = 2 10-3 A˚-1 (real space distances on the order of 300 nm) to Q = 8 10-2 A˚-1 (distances on the order of 10 nm).
This section presents the results of experiments that determine (a) the reaction of Al13 polycations with silica surfaces, (b) the extent of silica aggregation caused by this reaction, and (c) the resistance of the aggregates to osmotic compression. Reaction of Al13 Polycations with Colloidal Silica. In the first experiment, aqueous solutions of Al13 polycations were added to silica dispersions, and the amount of Al that reacted with silica was determined through the depletion method (see Methods). The results are presented in Figure 3 as sorption isotherms, where the amount of reacted Al is plotted vs the equilibrium concentration of free Al in the dispersion. These isotherms have the shape expected for the case of high affinity, with a steep rise followed by saturation at a surface (30) Jullien, R.; Botet, R. Aggregates and Fractal Aggregates; World Scientific: Singapore, 1987. (31) Hansen, J. P.; MacDonald, I. R. Theory of Simple Liquids; Academic Press: New York, 1986. (32) Egelstaff, P. A. An Introduction to the Liquid State; Clarendon Press: Oxford, 1994. (33) Ashcroft, N. W.; Leckner, J. Phys. Rev. 1966, 145, 83.
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Figure 3. Sorption isotherms for the reaction of Al13 polycations with colloidal silica. The initial pH of the silica dispersions was set to 5 (a) or 9 (b) before addition of the Al13 solution, and was not readjusted during or after the reaction (see Figures 5 and 7 for pH changes). The weight fraction Φw of the silica dispersions was 0.01 or 0.04. Vertical scale: number of adsorbed Al atoms per nm2 of surface area. Horizontal scale: equilibrium concentration of Al in the supernatant of the silica dispersion.
coverage corresponding to 3 Al atoms per nm2 of surface area. Surprisingly, the sorbed amounts are nearly the same for silica dispersions prepared either at pH = 5 or at pH = 9, even though the silica surface is weakly ionized at pH = 5 and strongly ionized at pH = 9 (Figure 3). There is, however, a slight difference that shows up in the effect of different silica concentrations. Indeed at pH = 5, the density of Al adsorbed on the silica surface depends only on the equilibrium concentration of free Al, as it should for an adsorption isotherm. However, at pH = 9, the measured amount of bound Al is higher at the lower silica concentration. This concentration dependence of the isotherm indicates that not only surface but also bulk solution chemistry is at work here. Indeed, the bound amount actually measures not only Al that reacted with silica surfaces but also Al that reacted with dissolved silicates, precipitated as an aluminosilicate, and was centrifuged along with the silica particles. The concentration of soluble silicates in equilibrium with silica is about 2 10-3 mol L-1 in the pH range 2-9; however, their reactivity and the dissolution rate of silica both increase with pH.1 Consequently, the contribution of these soluble silicates to the consumption of Al13 polycations is larger at pH = 9. Since this contribution is independent of the concentration of silica particles, it becomes relatively larger at the lower silica concentration, giving a higher total of “bound” Al per unit surface area. The comparison of the two isotherms in Figure 3b shows that the apparent increase in “bound” Al is about 0.5 Al/nm2 when the weight fraction of silica particles decreases from 4 to 1%. Since the silica specific surface area is 170 m2 g-1, this indicates that the concentration of aluminum atoms that reacted with dissolved silicates is about 2 10-3 mol L-1, which is comparable to the concentration of dissolved silicates. NMR experiments were performed in order to determine the chemical environment of the adsorbed Al atoms. For this purpose, silica dispersions were equilibrated with different concentrations of the aqueous solution containing Al13 as for the determination of adsorption isotherms. They were then centrifuged, separated from the supernatant, freeze-dried, and examined through 27Al MAS NMR. Figure 4 shows the spectra of the powders obtained from silica dispersions at 4696
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pH = 5 and at pH = 9 to which various amounts of Al13 have been added. In both experiments (at pH 5 and 9) the lowest spectrum corresponds to an adsorbed amount of 0.3 Al/nm2 according to the adsorption isotherm (Figure 3). It shows a main peak at δ = 53-55 ppm, corresponding to the resonance of tetrahedral Al atoms in Al-(OSi)4 units such as those found in aluminosilicates,34,35 and a weaker peak at δ = 2-4 ppm, which is associated with octahedral Al sites similar to those of Al13 polycations. With further addition of Al13 (0.8 Al/nm2), the relative intensity I3/I54 (ratio of the integrated intensities of peaks at 2-4 and 53-55 ppm) increases, and a third peak becomes visible at δ = 32 ppm, which is characteristic of five-coordinated Al atoms in aluminosilicates.34 Finally, at the highest Al content (2 Al/nm2), the intensity ratio I3/I54 still increases, and an additional narrow peak comes up at δ = 63 ppm, which is the location expected for the central (tetrahedral) Al atom in Al13 polycations. These observations are consistent with those of Lartiges et al.3 These authors propose that the interaction of Al13 with the surface of colloidal silica takes place in two stages. In a first stage, the Al13 polycations react with the surface silanols, and the Al atoms change their coordination to form a layer of negatively charged36 tetrahedral aluminosilicate sites on the silica surface (broad peak at 53 ppm). Then, additional Al13 polycations adsorb on this negative aluminosilicate surface layer and compensate its charge (increase of the 2 ppm resonance and appearance of the 63 ppm peak that is characteristic of the central tetrahedral Al atom of Al13 polycation). The 32 ppm resonance observed on the upper spectra probably reveals the presence in the Al13 polycations of Al atoms with a tetrahedral configuration that was distorted due to the removal of water by freezedrying. Surprisingly, and as noticed by Lartiges et al.,3 the interaction sequence of Al13 with silica is practically the same :: (34) Lippmaa, E.; Samoson, A.; Magi, M. J. Am. Chem. Soc. 1986, 108, 1730. (35) Engelhardt, G.; Michel, D. High-Resolution Solid-State NMR of Silicates and Zeolites; Wiley: New York, 1987. (36) Stone, W. E. E.; El Shafei, G. M. S.; Sanz, J.; Selim, S. A. J. Phys. Chem. 1993, 97, 10127.
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Figure 4.
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Al MAS NMR spectra of powders obtained from the reaction of silica particles with Al13 polycations. The silica dispersion was S2, at a weight fraction of 0.01, and initially at pH = 5 (a) or pH = 9 (b). The amounts of reacted aluminum atoms were calculated from the adsorption isotherms. The spectra have been scaled to a constant height for the peak at 53-55 ppm, corresponding to Al atoms in a tetrahedral coordination (aluminosilicates).
at pH = 5 and 9, even though the silica surface is weakly ionized at pH = 5 and strongly ionized at pH = 9. The effect of Al13 addition on the surface charge of the silica particles was evaluated through measurements of their electrophoretic mobility, which are presented in Figure 5. At low amounts of Al13, the mobility is negative, as expected from the sign of the surface charge of the silica particles (Figure 1). At higher amounts of Al13, the mobility reverses its sign and becomes positive. For the dispersion made at pH = 9, the amount that causes the reversal is 0.9 Al/nm2. This corresponds to 0.07 Al13 polycations/nm2, or a deposited positive charge of 0.5 elementary charge/nm2, which is of the same magnitude as the structural surface charge of silica at the initial pH. Surprisingly, at pH = 5, the charge reversal takes place at 0.6 Al/nm2, which correspond to 0.3 elementary charge/nm2, much higher than the actual surface charge density of silica at this pH, which is only 0.05 charge/nm2. Thus, when silica particles are equilibrated with solutions of Al13 polycations, their surface properties (amount of adsorbed Al and net surface charge) are nearly the same at pH 5 and 9, even though the bare particles have very different surface charge densities. Moreover, the amount of Al that is needed to reverse the surface charge density of silica has no relation to the original surface charge. These results are consistent with those of the NMR experiments: indeed, the first adsorbed Al13 polycations (up to 0.3 Al/nm2) react with the surface silanols to form an aluminosilicate layer, which is still negatively charged; then the subsequent Al13 polycations bind to this layer and reverse its electrical charge. Aggregation of Colloidal Silica in the Presence of Al13 Polycations. The kinetics of silica aggregation, caused by the addition of Al13 polycations, was estimated through turbidity measurements (Figure 6). There was a major difference in the behaviors of dispersions prepared at either pH = 5 or pH = 9. With the silica dispersion prepared at pH = 5, the addition of Al13 caused an immediate drop in pH (Figure 7a) and a very slow rise in turbidity, at a rate that was approximately proportional to the total concentration of Al13. Typical aggregation times were 1-6 h, depending on the availability of Al13 polycations (Figure 6a). On the other hand, with an aqueous silica dispersion prepared at pH = 9, the drop in pH was limited Langmuir 2009, 25(8), 4692–4707
(Figure 7b), but the turbidity appeared immediately upon mixing, regardless of the amount of Al13 (Figure 6b). It is interesting to note that this aggregation was not caused by the increase in the concentration of monovalent ions. Indeed, this concentration was 0.01 M in the present experiments, much below the salt concentration (0.1 M) that caused aggregation of this silica dispersion. The structures of the silica aggregates have been determined through SANS. Figure 8 shows the structure factors of silica dispersions that have been aggregated by addition of various amounts of Al13 polycations. The results show different aggregation behaviors at both pH values. At pH = 5, the dispersion with the lowest amount of added Al gave a structure factor that was nearly flat, indicating that most particles remained independent, with a small amount of finite aggregates. All the dispersions with at least 0.3 adsorbed Al/nm2 gave structure factors that were typical of fractal aggregates, with a self-similarity exponent equal to 2.2; the magnitude of S(Q) was nearly the same regardless of the amount of added Al13. Thus, at pH = 5, there was a clear transition between additions of Al13 that did not cause any aggregation and additions that did. At pH = 9, the aggregates had the same fractal structure as those made at pH = 5. However, the magnitude of the structure factor grew progressively with the amount of added Al13. It was remarkable to find that the different aggregation kinetics at pH = 5 and 9 (Figure 7) yielded different populations of aggregated and nonaggregated particles, but with the same aggregate structures at both pH values (Figure 8). Compression Resistance of the Silica Aggregates. After aggregation was completed, the aggregated dispersions were submitted to osmotic stress. When osmotic equilibrium was reached, a fraction of the compressed dispersion was analyzed through gravimetric methods in order to determine the volume fraction of silica in the compressed dispersion. At each osmotic pressure Π there is one volume fraction Φ that corresponds to the osmotic equilibrium that was reached in these conditions, and the complete set of pressures and volume fractions is the “compression curve” Π = f(Φ) of the dispersion. Figure 9 presents the compression curves obtained with dispersions prepared either at pH = 5 or at pH = 9 and with various amounts of added Al13. DOI: 10.1021/la803627z
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Figure 5. Electrophoretic mobility of colloidal silica particles (weight fraction 0.01) with adsorbed Al13 polycations. Dispersions (a) initially at pH = 5 and (b) initially at pH = 9. Vertical scales: electrophoretic mobility (left-hand scales) and solution pH (right-hand scale). Horizontal scales: surface density of adsorbed Al atoms, according to the adsorption isotherms presented in Figure 3.
Figure 6. Aggregation kinetics of silica suspension S2 at pH = 5 (a) and pH = 9 (b), according to turbidity measurements. The weight fraction of the suspension was 0.04. The densities of adsorbed Al atoms were determined from the adsorption isotherms. Vertical scale: optical transmittance of the silica dispersion. Horizontal scale: time elapsed since the addition of the Al13 solution. The experimental compression curves of dispersions made with low or intermediate amounts of Al follow a power law Π ∼ [Φ/(1-Φ)]2 with an exponent b ≈ 2, over the pressure range 0.1-10 kPa. Deviations observed at high pressure (above 10 kPa) reflect a change to an incompressible state at high volume fractions. On the other hand, a much steeper compression law is observed with the dispersions prepared with a large amount of added Al13 (exponent ≈ 4). Throughout these different stages of compression, the structures of the dispersions change considerably. A quantitative view of these changes is provided by the SANS spectra of the compressed dispersions. Figure 10 presents the sequence of spectra for the compression of dispersions prepared either at pH = 5 or at pH = 9 and modified through addition of small amounts of Al13 (0.1 Al/nm2). At pH = 5, the structure factors have a depression followed by a weak peak. Upon compression, the depression gets deeper, and the peak position shifts to higher Q values. These features reflect the correlations of nonaggregated, repelling 4698
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particles, with an average center-center distance that shifts from 50 to 25 nm.28,29 At the lowest Q values, there is a slight upturn of S(Q), which must originate from a small number of large objects, e.g., silica aggregates that were already present in the noncompressed dispersion (Figure 8). At pH = 9 the structure factors are higher at low Q (S(Q) = 2-5) and then decay to a similar depression that is again followed by a weak peak. These features indicate that there is a larger fraction of silica in large aggregates, coexisting with nonaggregated particles. Again, this is consistent with the results obtained with the noncompressed dispersion at pH = 9, which show the same high values of S(Q) at low Q, indicating that large aggregates coexist with nonaggregated particles. Figure 11 presents the sequence of structure factors for the compression of dispersions that were modified through addition of larger amounts of Al13 (0.6 Al/nm2). According to the results obtained at Π = 0 (Figure 8), the dispersions made at pH = 5 were fully aggregated, whereas those made at pH = 9 contained mixtures of aggregated and nonaggregated particles. The compression behaviors of both disperLangmuir 2009, 25(8), 4692–4707
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Figure 7. Effect of Al13 addition on the pH of silica suspension S2 initially at pH = 5 (a) and pH = 9 (b). The weight fraction of the suspension was 0.04. The densities of adsorbed Al atoms were determined from the adsorption isotherms. Vertical scale: pH. Horizontal scale: time elapsed since the addition of the Al13 solution.
Figure 8. SANS spectra of silica dispersions S2 after addition of various amounts of Al13 polycations. (a) Dispersion at pH = 5, with amounts of adsorbed aluminum as indicated in the graph. (b) Dispersion at pH = 9, with amounts of adsorbed aluminum as indicated in the graph.
sions are not at all the same. In dispersions made at pH 5, the structures remained unchanged up to 10 kPa, indicating that the aggregates were able to withstand the applied pressure; at higher pressures, the intensity was depressed mostly at the lowest Q values, indicating compression of the largest voids only (diameters above 300 nm). In dispersions made at pH 9, even small pressures (0.7 kPa) produced a depression and a small peak at the average interparticle distance. At higher pressures the depression became so deep and so broad that S (Q) was below 0.1 at all Q values that correspond to interparticle distances. This depression indicated that small pressures were enough to obtain a uniform distribution of silica particles throughout the sample volume. Figure 12 presents the sequence of spectra for the compression of dispersions that were modified through addition of very large amounts of Al13 (2 adsorbed Al atoms per nm2 of silica surface area). In both sequences (pH 5 or pH 9) (37) Bocquet, L.; Trizac, E.; Aubouy, M. J. Chem. Phys. 2002, 117, 8138. (38) Trizac, E.; Bocquet, L.; Aubouy, M.; von Grunberg, H. H. Langmuir 2003, 19, 4027.
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the aggregates resist deformation at pressures up to about 10 kPa. At higher pressures, the same type of deformation was observed in both sets of dispersions. Indeed, in both cases the intensity was depressed mostly at the lowest Q values, indicating compression of the largest voids in the range experienced by SANS (200-300 nm).
Discussion The aim of this discussion is to examine the questions raised in the introduction: (i) What is the initial state of the aggregates (bonds, coordination, structure)? (ii) What is their resistance to compression (which compression law)? (iii) What is their mode of deformation (relation between the new structure and the old structure)? Initial State of the Colloidal Aggregates. All the aggregates made by addition of Al13 to silica dispersions had branched self-similar structures. This is demonstrated by the structure factors presented in Figure 8, which follow the power law S (Q) Q-df with df = 2.2. This behavior has been observed quite generally for Brownian aggregation of clusters in the DOI: 10.1021/la803627z
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Figure 9. Compression curves Π = f(Φ) of S1 dispersions prepared with various amounts of added Al13 (amounts of adsorbed aluminum atoms
per unit of silica surface area: 0.1 Al/nm2 (a), 0.6 Al/nm2 (b), 2.0 Al/nm2 (c)) and at an initial pH value 5 or 9. Each data point corresponds to one sample equilibrated at one osmotic pressure. The two lines in (a) and the full line in (b) are the pressures calculated through the Poisson-Boltzmann cell model37,38 for independent, nonaggregated particles (see Appendix). In (b) the dashed line is a power law with exponent 3.0. In (c) the two lines are power laws with exponents 3.5 and 4.4. Note that the data span 1 decade in volume fractions and 3 decades in pressures.
case where the success rate of collisions is low (reaction limited cluster aggregation39,40). Nevertheless, Figure 8 shows that the structure factors depend on the pH and on the amount of added Al13. It is easy to see that these changes reflect the different fractions of silica particles that are either aggregated or nonaggregated. At pH = 5 and very low amounts of Al13 (0.1 Al/nm2), S(Q) is close to unity over the whole range of Q values, indicating that most particles remain independent (nonaggregated). At all higher amounts of Al13 (g0.3 Al/nm2), S(Q) follows the power law over all Q values that correspond to interparticle distances, indicating that all particles are aggregated (Figure 8a). At pH = 9 and lower Al13 contents, the structure factors follow the power law only at the lowest Q values and then go to unity at Q values that are still in the interparticle range; it is only for amounts of adsorbed aluminum higher than 0.9 Al/nm2 that the dispersions are fully aggregated (39) Jullien, R.; Kolb, M. J. Phys. A 1984, 17, L639. (40) Meakin, P. Annu. Rev. Phys. Chem. 1988, 39, 237.
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(Figure 8b). This behavior can be reproduced by linear combinations of the structure factors for repulsive particles and for fractal aggregates, as shown in Figure 13a. It indicates that at pH = 9 the dispersions contain mixtures of fractal aggregates and independent particles; the proportion of particles belonging to aggregates increases with increasing aluminum contents (Figure 13b), until a density of adsorbed aluminum 0.9 Al/nm2 at which no independent particles remain. These observations suggest that the aggregation process must be homogeneous at pH = 5 and heterogeneous at pH = 9. Aggregation Mechanisms at Acidic and Basic pH. In order to explain these differences in the aggregation processes at pH 5 and 9, one may look into the surface chemistry of either the Al13 polycations or the silica particles. The Al13 polycations bear a structural charge +7 at pH = 5 and become almost (41) Teixeira, J. In On Growth and Form: Fractal and Non-fractal Patterns in Physics; Stanley, H. E., Ostrowski, N., Eds.; Martinus Nijhoff Publishers: Dordrecht, 1986; p 145.
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Figure 10. SANS spectra of silica dispersions S1 after addition of small amounts of Al13 (0.1 Al/nm2) and osmotic compression. Each spectrum corresponds to a dispersion that was equilibrated for 30 days at the osmotic pressure indicated in the graph. Dispersions (a) prepared at pH = 5 and (b) prepared at pH = 9. In both cases the depression at Q < 10-2 A˚-1 and the peak at Q = 2 10-2 A˚-1 reflect the spatial correlations of repelling particles. The excess intensity at very low Q originates from a few aggregates that coexist with the repelling particles.
Figure 11. SANS spectra of silica dispersions S1 after addition of larger amounts of Al13 (0.6 Al/nm2) and osmotic compression. (a) Dispersions prepared at pH = 5. The high values of S(Q) at low Q and the power-law decay are characteristic of fractal aggregates that withstand the applied pressure. (b) Dispersions prepared at pH = 9. The low values of S(Q) show that the spatial distribution of silica particles becomes homogeneous and therefore that the aggregates yield to the applied pressures.
uncharged at pH values above 6.5.42 Further transformations may occur at these high hydrolysis ratios, through aggregation of the Al13 units, but these reactions take a few days,20 much beyond the aggregation times of our experiments. The silica particles carry a negative charge, with a density that increases from 0.05 e-/nm2 at pH = 5 to 1 e-/nm2 at pH = 9 (Figure 1). However, in spite of the variations of the surface properties of both species with pH, no substantial differences appeared either in the amount of aluminum that reacted with the silica particles (Figure 3) or in the structural environment of these Al atoms (Figure 4). Indeed, at both pH values the Al13 polycations first reacted with the silica surfaces to form a negatively charged aluminosilicate layer and then further adsorption of Al13 polycations induced a charge reversal of (42) Furrer, G.; Ludwig, C.; Schindler, P. W. J. Colloid Interface Sci. 1992, 149, 56.
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the particles. Consequently, the characterization of the surface species fails to explain the differences found in the extent of aggregation at pH = 5 and pH = 9. On the other hand, there are enormous differences in the aggregation kinetics of the suspensions. Indeed, at pH = 9, addition of Al13 polycations triggers instant aggregation of some or all of the dispersed silica particles (Figure 6b), depending on the Al13 concentration: thus, when the silica particles are highly charged, the reaction of Al13 with silica surfaces is extremely fast. Consequently, all the added Al13 reacts locally, at the point of addition, with a fraction of the silica particles. The surface charge of those particles is reversed, and they aggregate with a fraction of the remaining negatively charged particles, until all aggregates are terminated with particles that have no accessible polycation on their surfaces (Figure 14). This aggregation mechanism is consistent with the findings that, at pH = 9, first the number DOI: 10.1021/la803627z
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Figure 12. SANS spectra of silica dispersions S1 after addition of very large amounts of Al13 (2.0 adsorbed Al/nm2) and osmotic compression. (a) Dispersions prepared at pH = 5. (b) Dispersions prepared at pH = 9. The aggregates resist deformation at all pressures up to 10 kPa at pH = 5 and 40 kPa at pH = 9. At higher pressures compression takes place at the largest scales only (lowest Q values).
Figure 13. (a) Calculated structure factors for dispersions containing both repulsive nonaggregated particles (PY) and large aggregates (F) with a fractal dimension df = 2.2; comparison with the experimental structure factors from Figure 8b. The structure factor of the repulsive particles has been calculated through the Percus-Yevick approximation for monodisperse spheres with average diameter 16.6 nm. The structure factor of the aggregates has been calculated through the Teixeira formula41 with an effective diameter adjusted to take into account the short-range order of the particles. (b) Fractions of aggregated particles, according to the fit of the experimental structure factors by the calculated ones.
of aggregated particles increases with the amount of added Al13 (Figure 13) and then total aggregation occurs at a density of adsorbed aluminum (0.9 Al/nm2; see Figure 13b) for which the charge of the colloidal particles is globally compensated (Figure 5b). To the contrary, at pH = 5, turbidimetric measurements (Figure 6a) indicate that the aggregation of silica particles induced by addition of Al13 is quite slow (times on the order of hours). The adsorption of polycations on the silica surfaces and their reaction with the surface silanols to form the aluminosilicate layer are presumably the limiting steps of the destabilization process at this pH because of the low number of ionized silanols on the silica surface. Consequently, the suspension is homogenized before initiation of the reaction between the two species, and the added Al13 is equally distributed among all the silica particles. If there is too little Al13 (for example, 0.1 Al/nm2), no free polycations are left in the suspension after the formation of the first 4702
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aluminosilicate layer: consequently, all particles remain negatively charged and repel each other during collisions (Figure 15a). If there is enough Al13 ([Al]ads g 0.3 Al/nm2), some polycations are still available to adsorb on the particles after formation of the negatively charged aluminosilicate layer: consequently, all particles become reactive and take part in the aggregation process (Figure 15b), as observed. The fact that, at pH = 5, total aggregation of the dispersions takes place at Al contents lower than that required for complete charge neutralization (see Figure 5a) indicates that the aggregation mechanism involves bridging of particles by Al13 polycations. Interparticle Bonds. Another important characteristic of the silica aggregates is the number of bound neighbors of a silica particle and the strength of these interparticle bonds. The average coordination of a silica particle can be evaluated from the shape of S(Q) in the range of Q values that correspond to nearest-neighbor distances. Indeed, if most Langmuir 2009, 25(8), 4692–4707
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Figure 14. Schematic representation of the destabilization mechanism accounting for the aggregation of colloidal silica suspensions at pH = 9 after addition of Al13 polycations. (1) Reaction of all the added Al13 with a fraction of the silica particles. (2) Aggregation of the particles until no accessible Al13 is left.
Figure 15. Schematic representation of the destabilization mechanism accounting for the aggregation of colloidal silica suspensions at pH = 5 after addition of Al13 polycations. (a) Low Al13 concentrations: (step 1) homogeneous adsorption of the polycations on the silica particles; (step 2) formation of the negatively charged the aluminosilicate layer. (b) High Al13 concentrations: (1) and (2) are unchanged; (step 3) adsorption of remaining free polycations on the aluminosilicate layer; (step 4) aggregation.
silica particles had a dense coordination shell with 8-10 neighbors, as in dense colloidal liquids, then S(Q) would have a strong peak at those Q values. In fact, the structure factors of aggregates only show a faint hump at those Q values (Figure 8). As in most colloidal aggregates, this absence of a nearest-neighbor peak indicates that the coordination number is on the order of 3-4 only.30 Regarding the nature of interparticle bonds, some information is provided by the 27Al MAS NMR analyses. These results suggest that the silica particles can be linked in two ways: (i) chemically via the aluminosilicate phase formed from the reaction of Al13 with two facing silica surfaces or (ii) physically via bridging polycations that are adsorbed on the surface aluminosilicate layer and have retained the Al13 structure. When comparing the spectra obtained at pH = 5 and pH = 9 (Figure 4a,b), it appears that, for small Al13 amounts, the AlIV/AlVI ratio is higher at pH = 9 than at pH = 5: indeed, the ratio of integrated intensities I54/I3 decreases between pH = 9 and pH = 5 (from 2.9 to 1.0 when [Al]ads = 0.3 Al/nm2 and from 2.0 to 0.7 when [Al]ads = 0.9 Al/nm2). This indicates, as previously noticed by Lartiges et al.,3 that the fraction of aluminum incorporated in the aluminosilicate layer is higher at pH = 9, which is consistent with the Langmuir 2009, 25(8), 4692–4707
increased reactivity of the silica surface at this pH. As a consequence, we can expect the aggregates prepared at pH = 9 to contain a larger fraction of chemical bonds than those prepared at pH = 5. Resistance to Compression. Now we proceed to the next question, which deals with the resistance of these aggregates to osmotic compression. According to the previous discussion, we have applied osmotic pressure to dispersions that may contain both aggregated and nonaggregated particles, depending on the amount of Al13 that was added to the silica dispersions. Looking at the results of compression experiments (Figure 9), we can see that two types of compression curves Π = f(Φ) were obtained. At lower Al13 concentrations (0.1 and 0.6 Al/nm2) the pressure required to compress the dispersions varied as Φb with b between 2 and 3, whereas this exponent increased to b ≈ 4 for higher Al13 concentrations (2.0 Al/nm2). This indicates that the dispersions containing lower flocculant concentrations are more easily compressible. The relation to the fractions of aggregated vs nonaggregated particles is as follows. (i) When the density of adsorbed aluminum on the silica surfaces is very low (0.1 Al/nm2), at both pH values, the dispersions consist of nonaggregated repelling particles; DOI: 10.1021/la803627z
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therefore, the resistance of the dispersions to compressive stress originates from the ionic repulsions between particles. These repulsions can be calculated through the PoissonBoltzmann cell (PBC) model (see Appendix).37,38 The compression curve Π = f(Φ) of the dispersion at pH = 9 (Figure 9a) was fitted through the PBC model with an effective charge Z = 36 e- per particle (which corresponds to the effective charge calculated through eq 3) and an ionic strength I = 3 10-3 M, consistent with the composition of the dispersions in the dialysis bags. The compression curve of the dispersion at pH = 5 was fitted with the same ionic strength and a lower effective charge, Z = 20 charges per particle, which corresponds to a surface charge number density of 0.04 nm-2, in good agreement with the results of electrophoretic mobility measurements (Figures 1 and 5). (ii) At intermediate amounts of added aluminum (corresponding to 0.6 Al/nm2), there is an inversion of the relative magnitudes of the resistances to compression according to the pH value. Indeed, at pH = 5 the pressures are higher than they were at the lower coverage, and they rise more steeply (Figure 9b). SANS experiments indicate that in these conditions the dispersions were fully aggregated (Figure 8). If the aggregates are connected in such a way that they form a continuous fractal network extending throughout the dialysis bag, then the resistance to compression depends on the mechanical strength of this network. The structure factors presented in Figure 11a show that this silica network was not deformed (at least over distances up to 300 nm) until the applied pressure reached 10 kPa; at higher pressures the decrease in the structure factor at low Q values indicates that large scale heterogeneities (voids) were compressed.4,10 On the other hand, at pH = 9 the pressures were lower than at the lower coverage (Figure 9b). In this case SANS experiments indicate that only 35% of the particles were aggregated (Figure 13). Moreover, the structure factors of the compressed dispersions could be fitted as linear combinations of the structure factor of fractal aggregates and that of a compressed dispersion of repelling particles (Figure 16a). As the dispersion was compressed, the average density of the colloidal liquid of repelling particles rose and matched that of the aggregates, which cause the aggregates to become “invisible” in the total scattering (Figure 16b). Accordingly, the fractal aggregates did not form a continuous network inside the dialysis bag, and the resistance to compression was that of the nonaggregated repelling particles. Therefore, the compression curve was fitted using the PBC model, with an effective charge of the particles Z = 20, consistent with the decreased electrophoretic mobility of the silica particles compared to lower aluminum contents (Figure 5b), and the same ionic strength as previously (3 10-3 M). (iii) At high amounts of added aluminum (corresponding to 2.0 Al/nm2) and at both pH values, the dispersions were fully aggregated. Consequently, the resistance that opposed this compression originated from the stiffness of the columns of bound particles within the fractal networks. We found that this resistance was a steeply increasing function of silica volume fraction (Figure 9c). This law (Π ≈ Φb with b ≈ 4) is characteristic of a fractal network that has an elastic response to applied forces, i.e., with interparticle bonds that can stretch, bend, or twist but do not break under the effect of these forces.43 This resistance of the particle network is confirmed by the structure factors of the compressed dispersions: indeed, (43) Botet, R.; Cabane, B. Phys. Rev. E 2004, 70, 031403.
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from Π = 0.3 to 10 kPa at pH = 5 and to 40 kPa at pH = 9, they remain identical to the structure factor of the uncompressed dispersion (Figure 12). Accordingly, the network is compressed (the volume fraction rises by a factor of 3; therefore, the overall dimensions shrink by a factor of 1.7), but this compression takes place at scales that are not accessible in SANS (i.e., beyond 300 nm), and smaller voids are not compressed. However, at the highest pressures, the structure factors start to be depressed at low Q values, indicating the beginning of a compression of voids in the 30-300 nm range. This takes place sooner for the dispersions aggregated at pH = 5 compared to those aggregated at pH = 9 (Figures 9c and 12), which could be a consequence of a larger fraction of chemical interparticle bonds formed at pH = 9. We have not examined whether the osmotic compression of the aggregates was reversible. However, there are two effects that may prevent reversibility. First, a few bonds may break even in the elastic regime. Second, new bonds are created as silica particles are pushed closer together. The Transition from Elastic to Plastic Deformation. Compression experiments performed by Madeline et al.4 on the same system but at still higher pressures (100-400 kPa) showed that, at those high pressures, the structure factors of the aggregated dispersions were strongly depressed, indicating a collapse of all voids in the 30-300 nm range and complete loss of the fractal structure (see Figure 9 of Madeline et al.4). This behavior is characteristic of a plastic deformation of the silica network, in which interparticle bonds are broken, rearranged or shifted. In this regime the compression law has a lower exponent (Π ≈ Φb with b ≈ 1.7).43 Hence, the combination of our results with those of Madeline et al. allows us to determine the pressure range for the transition from elastic to plastic behavior and therefore to estimate the strength of the interparticle bonds in the silica network. The experimental results indicate that the silica aggregates have a transition from elastic response to plastic response when the applied pressure reaches about 100 kPa. The energy required to break the interparticle bonds can be estimated from this value of the osmotic pressure at the transition between the two regimes. The bond-breaking energy per particle, Ud, is given by4 Ud ¼
Πð2aÞ2 ðld -l0 Þ Φ
ð9Þ
where a is the radius of the particles, Φ is their volume fraction, and l0 and ld are respectively the length at rest and the breaking length of the spring to which each interparticle bond is identified.43 If we assume (ld - l0) = 0.1 nm, we find that the bond-breaking energy is in the order of Ud = 20 10-21 J or Ud = 5kT. However, we have to note that this value of Ud is underestimated, since we assumed here that the applied pressure was distributed between all the colloidal particles, whereas the resistance to compression in fact only originates from the stiffness of the continuous columns of bound particles extending throughout the dispersion.
Conclusions In this work we have produced a large number of colloidal dispersions that differed by the extent of particle aggregation. Some dispersions contained only repulsive nonaggregated particles, some contained particles that were fully aggregated, and some contained small aggregates immersed in a “sea” of nonaggregated particles. Yet, when we applied osmotic stress Langmuir 2009, 25(8), 4692–4707
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Figure 16. (a) Calculated structure factors (solid lines) for compressed dispersions containing fractal aggregates and repulsive nonaggregated particles; comparison with experimental structure factors from Figure 11b. The structure factor of repulsive particles was calculated through the Percus-Yevick approximation for monodisperse spheres of diameter 13.0 nm. The structure factor of aggregates was calculated through the Teixeira formula41 with an effective diameter adjusted to take into account the short-range order of the particles. The volume fraction of repulsive particles and the relative contrast of the aggregates were taken as fitting parameters. (b) The intensity from the aggregates varies as the square of the contrast between the aggregates and the colloidal liquid of repulsive particles.
in order to extract water and pack the particles into a smaller volume, we found only two behaviors. The different initial states (before compression) were obtained by adding different amounts of Al13 polycations to silica dispersions, prepared either at pH = 5 or at pH = 9. Surprisingly, we found that the extent of aggregation is determined mostly by the kinetics of this reaction. In acidic conditions the reaction was very slow; consequently, the silica particles reached a homogeneous surface state, and they ended up either completely nonaggregated or else fully aggregated. In basic conditions, the reaction was much faster than the homogenization process, and it took place entirely at the point of addition. This yielded dispersions that contained mixtures of aggregated and nonaggregated particles, depending on the Al13 content. In all cases, however, the aggregates had self-similar structures with voids at all scales from the particle scale (each particle has 3-4 neighbors) to the largest scales probed by SANS (300 nm). When we applied osmotic stress up to 50 kPa to dispersions that were fully aggregated, we found that the particle network retained its structure at local scales (the average coordination of a particle was unchanged) and at mesoscopic scales (the fractal exponent remained the same). At macroscopic scales, the network was compressed, but the compression law, Π ≈ Φ4, was quite stiff. This power law is identical with that found by Zukoski9 for dispersions of zirconia particles that have been aggregated at the iep. It is the predicted behavior for the elastic response of a fractal network, according to the theory of Ball et al.44 At higher pressures (g50 kPa), the network was compressed at macroscopic scales and also at mesoscopic scales. This deformation indicates that a significant number of interparticle bonds were broken or rearranged, as in a plastic deformation.43 According to this threshold, the strength of an interparticle bond must be on the order of 5kT in the case of silica particles that are bridged by Al13 polycations. Finally, the dispersions that were not fully aggregated had a very different behavior in a compression experiment: the (44) Brown, W. D.; Ball, R. C. J. Phys. A 1985, 18, L517.
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spatial distribution of silica particles became homogeneous, and the resistance to compression was produced by the ionic repulsions of neighboring silica particles. The compression law was Π ≈ [Φ/(1 - Φ)]2. These different compression behaviors may have important consequences for the practical applications of aggregated colloidal dispersions. In some applications, the aggregates are subjected to weak shear forces only. This is the case for colloidal aggregates that are added to aqueous or oily liquids in order to turn them into gels. In such cases, aggregates such as the silica-Al13 aggregates studied here may withstand the applied forces and provide the desired resistance to flow. On the other hand, in applications that involve grinding of drying, the applied pressures are on the order of megapascals, and such forces will destroy the fractal structures of the colloidal aggregates. In such cases, it may be necessary to consolidate the aggregate structures through the formation of stronger (chemical) interparticle bonds or through the condensation of dissolved minerals at the junctions of colloidal particles. Acknowledgment. It is a pleasure for us to thank Bruno Lartiges for his insightful comments on the aggregation of silica by Al13, Jean-Baptiste d’Espinose de Lacaillerie for help and guidance with the NMR experiments, and Sylvie Henon for the Poisson-Boltzmann cell model and Christian Buron for ICP titrations.
Appendix. Simplified Poisson-Boltzmann Cell Model Model. Consider an aqueous dispersion of monodisperse spherical particles. The aqueous solution that disperses the particles is called the interstitial solution. The particle diameter is 2a, the volume fraction of particles is φP, and the number of particles per unit volume is nP. Each particle has Z ionic surface groups, which release Na+ counterions, and monovalent salt (NaCl) is added to the dispersion. The dispersion is in equilibrium through a dialysis membrane with a large volume of aqueous solution, called the bulk solution, which sets the chemical potentials of all ions. In the cell model, each particle is located at the center of the DOI: 10.1021/la803627z
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spherical cell of radius R, such that the volume of the cell equals the average volume per particle in the dispersion. This makes it possible to predict the behavior of the whole dispersion from that of a single cell. The simplified model is based on the notion of effective charge, as defined by Belloni15 and Trizac.38 Among the Z counterions that have been released by the particle, a number Zeff are actually dispersed in the interstitial solution, while a number (Z - Zeff) are condensed in the Stern layer near the particle surface. The value of Zeff can be estimated through the equation proposed by Belloni, which is Zeff = 4a/LB (eq 3), or that proposed by Trizac, which is Zeff = (a/LB) (1 + κa). In the comparison of the model with experimental data, Zeff is taken as a fitting parameter (and it is the only adjustable parameter of the model). The other variables and parameters of the model are defined are as follows: • Ψ(r) is the electrical potential at a distance r from the center of the particle, and φ(r) = eΨ(r)/kBT is a reduced, dimensionless potential (e is the elementary charge = 1.6 10-19 C ; kB is the Boltzmann constant, T is the temperature, and kBT = 4.1 10-21 J at room temperature). • n+(r) (n-(r)) is the number of Na+ (Cl-) ions per unit volume in the interstitial solution at a distance r from the center of the particle (beyond the Stern layer). • nS is the number of Na+ ions per unit volume in the bulk solution. • LB = e2/(4πε0εrkBT) is the Bjerrum length (ε0 dielectric constant of vacuum, εr relative dielectric constant of water; LB = 0.7 nm at room temperature). With these definitions the relation between the geometrical parameters of the cell, and the volume fraction or number density of particles, is as follows: ða=RÞ3 ¼ φP ;
φP ¼ nP ð4π=3Þa3 ;
1=nP ¼ ð4π=3ÞR3 ðA1Þ
Moreover, the electric field E(r) is known at the particle surface and at the boundary of the cell: EðaÞ ¼ -Ze=ð4πε0 εr a2 Þ;
EðRÞ ¼ 0
ðA2Þ
Donnan Equilibrium. At equilibrium, the chemical potentials of all ions in the interstitial solution and in the bulk solution are equal: μ0þ þ NAvogadro eΨðrÞ þ RT ln n þ ðrÞ ¼ μ0þ þ RT ln nS ðA3Þ μ0-
-NAvogadro eΨðrÞ þ RT ln n - ðrÞ ¼
μ0-
þ RT ln nS ðA4Þ
n þ ðrÞ ¼ nS exp½ -jðrÞ and
n - ðrÞ ¼ nS exp½jðrÞ
situation, the cell is not much larger than the particle, and the ionic concentrations cannot vary significantly through the cell, unless the screening length is extremely short. For instance, in the present work, for nonaggregated dispersions, the ionic strength was 3 10-3 M, which corresponds to a Debye screening length of 5.6 nm. If the particle radius is a = 8 nm and the volume fraction is φP = 0.3, then the cell radius is 12 nm and the thickness of the interstitial solution is 4 nm, smaller than the Debye screening length. The volume of this region of the interstitial solution is v ¼ ð4π=3Þa3 ½ð1 -φÞ=φ
In these conditions the electrical potential and the ionic concentrations in the interstitial solution can be taken as uniform and will therefore be designated as φ, n+, and n-. Thus, the positive ions can be in two states only, i.e., the Stern layer and the interstitial solution. Both types of ions may enter or leave the cell, and therefore the effective charge Zeff is not linked to n+ only; instead, since the cell is electroneutral, the ionic concentrations are related to the effective charge Zeff through Z eff ¼ ðn þ -n - Þv
n þ ðrÞ þ n - ðrÞ ¼ 2nS ch½jðrÞ
and n þ ðrÞn - ðrÞ ¼ nS 2
ðA8Þ
Using eqs A5 and A7 yields a set of equations that describe the Donnan equilibrium of ions across the membrane: n - ðn - þ Zeff =vÞ ¼ nS 2
ðA9Þ
n - ¼ f -Z eff =v þ ½4nS 2 þ ðZeff =vÞ2 1=2 g=2
ðA10Þ
n þ ¼ fZ eff =v þ ½4nS 2 þ ðZ eff =vÞ2 1=2 g=2
ðA11Þ
These relations impose n- < nS, since 4n2S + (Zeff/v)2 < (2nS+ Zeff/v)2 (this is the Donnan effect: the concentration of co-ions inside the cell is always below that in the interstitial solution because counterions are repelled by the particle surface). Similarly, (n+ - Zeff/v) < nS because each co-ion that is transferred from the interstitial solution to the bulk solution must be accompanied by a counterion. Excess Osmotic Pressure of the Interstitial Solution Compared with the Bulk Solution. The excess osmotic pressure ΔΠ of the interstitial solution is equal to the thermal pressure that would be exerted by all ions and by the particle on a virtual membrane that would be located at the cell boundary (r = R) where the electric field vanishes (eq A2): ðA12Þ ΔΠ=kB T ¼ nP þ n þ ðRÞ þ n - ðRÞ -2nS This pressure may be calculated using the Donnan equilibrium equations (A10) and (A11): ΔΠ=kB T ¼ nP þ ½4nS 2 þ ðZ eff =vÞ2 1=2 -2nS
ðA13Þ
This can also be transformed as follows: ΔΠ=kB T ¼ nP þ 2nS f½1 þ ððn þ -n - Þ=2nS Þ2 1=2 -1g ðA14Þ
ðA5Þ Consequently, the following relations exist between the ionic concentrations in the interstitial solution:
ðA7Þ
Using (A5), this osmotic pressure can be expressed as ΔΠ=kB T ¼ nP þ 2nS f½1 þ sh2 ðjðRÞÞ1=2 -1g ¼ nP þ 2nS ½chðjðRÞÞ -1 ðA15Þ
ðA6Þ Two-State Model. In most practical situations, the most interesting case is that of high volume fractions. In this 4706
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High Salt and Low Salt Limits. In the limit where the salt concentration is high, i.e., when 2nS > (Zeff/v), the excess osmotic pressure of the cell, given in (A13), Langmuir 2009, 25(8), 4692–4707
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simplifies to ΔΠ=kB T ¼ nP þ ðZ eff =vÞ2 =4nS
ðA16Þ
In the limit of extremely high salt concentrations, the excess osmotic pressure would be that of the particles only. In the opposite limit where the salt concentration is low, i.e., 2nS < (Zeff/v), the excess osmotic pressure of the cell, given in (A13), simplifies to ΔΠ=kB T ¼ nP þ ðZ eff =vÞ -2nS þ 2nS 2 ðv=Z eff Þ
ðA17Þ
In the limit of extremely low salt concentrations, the excess osmotic pressure would be that of the particles, nP, plus that of the counterions, Zeff/v. Comparison with Experimental Data. The osmotic pressure of an aqueous dispersion of nonaggregated silica nanoparticles have been measured through osmotic stress experiments, as described in Methods. Two sets of data are presented in Figure 17; they have been measured in different laboratories with silica dispersions of similar characteristics. The good agreement between both sets of data demonstrates that they really measure the equation of state of the dispersion. The model calculations have been made using eq A16 with the following parameters: particle radius a = 10 nm; effective charge number Zeff = 29; salt concentration nS = 3.91 1023 m-3 (corresponding to an ionic strength equal to 6.5 10-4 M). The agreement with the data is surprisingly good (over 4 decades in pressure), considering that the volume fraction is not always high and that the salt concentration is not high either. This good agreement presumably originates from
Langmuir 2009, 25(8), 4692–4707
Figure 17. Compression curves of aqueous dispersions of silica nanoparticles, obtained through the osmotic stress technique: (b) data from Chang et al.;29 (O) data from Persello (private communication). Full line: calculation according to the cell model (eq A16), with parameters given in the text.
the fact that the effective charge has been taken as an adjustable parameter and that the variation with volume fraction originates from the variation of the volume of interstitial solution, v, which is determined by geometry according to eq A7. This yields Π ∼ [Φ/(1-Φ)]2, which is a consequence of the 2-state model, and fits the data presented in Figures 9 and 17.
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