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Fractal behavior and scaling law of hydrophobic silica in polyol
Journal of Colloid and Interface Science 267 (2003) 314–319 www.elsevier.com/locate/jcis
Fractal behavior and scaling law of hydrophobic silica in polyol Fabrice Saint-Michel, Frédéric Pignon, and Albert Magnin ∗ Laboratoire de Rhéologie, Université Joseph Fourier–Grenoble I, Institut National Polytechnique de Grenoble, CNRS UMR 5520, 38041 Grenoble Cedex 9, France Received 17 October 2002; accepted 24 July 2003
Abstract This article examines the rheological properties of a system composed of polyol and colloidal silica. Three types of nanosized silicas with hydrophilic and hydrophobic surfaces were studied: A200 with OH surface groups, R974 with CH3 surface groups, and R805, which is grafted with a C8 H17 alkyl chain. Rheometric measurements showed that the dispersions of R805 silicas have a yield stress at low volume fraction, unlike the R974 and A200 silicas. The plastic behavior of the hydrophobic silicas was quantified by a yield stress σ0 and an elastic modulus G . It is observed that these parameters follow scaling laws as a function of the volume fraction of silica introduced, in the form σ0 ∼ φv2.9±0.2 , G ∼ φv4.1±0.3 . Static light scattering (SLS) and small angle neutron scattering (SANS) measurements show a fractal arrangement with a fractal dimension D = 1.8 ranging from elementary particles of about 32 nm to aggregates measuring about 6 µm. Correlations were established between the theoretical scaling laws and the experimental scaling laws determined by rheometric measurements. The fractal structure observed in this system is explained by the attractive physical interaction of the octyl chains between the silica particles. Contrary to what has been observed in the past by Khan and Zoeller (J. Rheol. 37 (1993) 1225), the lower molecular weight of the polyol studied here, which has a shorter chain length, allows direct bridging of two separate silicates though alkyl chains, giving rise to the formation of a 3D gel network at volume fractions as low as φv = 2.2%. 2003 Elsevier Inc. All rights reserved. Keywords: Rheology; Polyol; Silica; Colloid; Interaction; Fractal; Yield stress; SLS; SANS
1. Introduction Colloidal silicas are used as fillers in various industrial applications to control rheological properties. They come in a wide range of sizes and with a variety of surface treatments. The considerable specific surface area of these particles (50–400 m2 /g) and the presence of functional clusters (silanol, siloxane) play a major role in their rheological behavior. In particular, they are used to give polymers viscoplastic behavior characterized by a yield stress. Their capacity to form a three-dimensional network responsible for the yield stress and the formation of a gel phase depends on the volume fraction, on the type of colloidal silica (type of surface groups), and on the polarity of the suspending phase. Indeed, interactions between the particles play a decisive role in the stability and rheological properties of the * Corresponding author.
E-mail address: [email protected] (A. Magnin). 0021-9797/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2003.07.018
dispersions. For example, in the case of aqueous dispersions of nanometric silica particles in polyethylene oxide (PEO), shear can induce gelation [1]. The concepts developed by De Gennes [2] for the formation of polymer gels can also be applied to the case of colloidal gels. Various studies have applied these concepts in order to examine the effects of interparticle forces on the elasticity of dispersions. By characterizing viscoelastic properties, it is also possible to examine particle–particle and particle–solvent interactions [3]. Many studies have looked at colloidal silica dispersions in a silicone [4,5]. These authors have shown that the viscoelastic behavior of colloidal silica suspensions in a polydimethylsiloxane (PDMS) is nonlinear above a critical strain; i.e., there is a very great drop in the elastic modulus above a certain critical strain. Aranguren et al. [4] speak of the role of bridging polymers in the behavior of these materials. They also showed that the elastic modulus increases with the volume fraction, following a power law of the type (G ∼ Kφvm ).
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Other types of polymer dispersions have similar rheological behavior. This is the case, for example, with nonpolar polymers, such as a mineral oil, or polar polymers, such as a polypropylene glycol (PPG). Khan and Zoeller [6] and Yziquel et al. [7] identified the various possible types of interaction between the solvent and particles, depending on the type of silica (hydrophilic and hydrophobic) and type of solvent (polar and nonpolar). For a dispersion of silica which is grafted with an alkyl chain, Khan and Zoeller [6] did not observe a fractal arrangement of the particles within PPG. Moreover, their work showed that when the particles interact with one another, the scaling concept may be applied to obtain the power law for the dependence on the elastic modulus and volume fraction. The power law G ∼ φvm can be linked to the fractal dimension [8,9]. Dorget [5] and Piau et al. [9] proposed a model based on the notion of semidilute fractal objects, which can be used to interpret the rheology of these compounds. They thus linked the yield stress and elastic modulus to the fractal dimension via a scaling law. Little research work deals with colloidal dispersions in polyol. Nevertheless Gulley and Martin [10] have shown a fractal type of aggregation in the case of aqueous colloidal silica dispersions with different kinds of polyols. In this article, the viscoplastic and elastic properties of silicas-polyol dispersions were studied in relation to the nature of the interactions and structures created by the silica. First of all, the thickening effects of three silicas were studied. They differ in terms of the types of their surface groups and their hydrophilic and hydrophobic nature. The hydrophobic silica with the greatest thickening power will be studied in greater detail. Second, this work involved characterizing the structure of this hydrophobic silica dispersion at rest and linking it with its mechanical behavior by a scaling law. Contrary to the results obtained by Khan and Zoeller [6], we demonstrated the formation of a fractal arrangement for the R805 silica particles within the polyol. The structural differences between the system studied in this work and that studied by Khan and Zoeller [6] reside in the polymer chain length. The work presented here clearly shows the importance of this parameter for both the silica organization and the mechanical behavior of the system. The methods used to characterize the dispersions at rest combine rheometric measurements under simple and harmonic shearing conditions and static light scattering (SLS) and small angle neutron scattering (SANS) techniques. Measurements of radiation scattering enabled the structure at rest to be characterized over a wide range of length scales. Scaling laws were established linking the yield stress and elastic modulus to the fractal dimension of the structure created by the silica network. These methods of characterization of the structure at rest were compared with the various models proposed in the literature, in order to examine the interactions between particles and polyol.
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2. Materials and experimental techniques 2.1. Materials The colloidal silica consists of spherical particles with a mean diameter of 32 nm. This size is obtained from the small angle neutron scattering measurements. Three different silicas made by Degussa [11] were used: Aerosil A200, R974, and R805. Aerosil A200 is a hydrophilic silica. The surface area of silicas R974 and R805 was modified to make the particles hydrophobic. The functional clusters and specific surface areas of these three fillers are given in Table 1. The dispersions are formulated from a mixture of polyols. The polyols used are triol polypropylene. Physicochemical data are given in Table 2. A volume fraction of colloidal silica (φv ) is then incorporated. This is defined by φv =
Vsilica . Vsilica + Vmatrix
Vsilica represents the volume of colloidal silica incorporated in the matrix; Vmatrix represents the total volume of liquid formed by the polymer matrix. At a temperature of about 20 ◦ C, the dispersion obtained is then mixed using a vane stirrer revolving at 1500 rpm for 10 min. The samples are degassed in a vacuum for 24 h to eliminate all air bubbles introduced into the dispersion during mixing. This operation Table 1 Physicochemical properties of colloidal silicas used Type of colloidal Schematic representation of Density Specific area (m2 /g) silica surface clusters (g/cm−3 ) Aerosil A200
2.2
200 ± 25
Aerosil R974
2
170 ± 20
Aerosil R805
2
150 ± 25
Table 2 Physicochemical properties of chemical compounds used in the polymer matrix Chemical compound
Structure
Molar Hydroxyl Viscosity mass number at 25 ◦ C (g mol−1 ) (mg KOH/g) (Pa s)
Polyol A
450
380
0.380
Polyol B
300
570
0.500
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is intended to avoid any degassing of the sample during measurements. The time elapsing between the preparation of the dispersions and the measurements is 7 ± 3 days. 2.2. Experimental techniques 2.2.1. Rheometry Measurements in simple shear conditions were carried out at a temperature of 23 ± 1 ◦ C. The measurements are performed in a Weissenberg–Carrimed controlled strain rotating rheometer. Torsion bar torque sensors are used. The measurements are carried out with two different geometries: a coaxial cylinder configuration (external radius 10.75 mm, internal radius 8.75 mm, height 40 mm) and a cone– plate configuration with a hollowed center (external radius 25 mm, internal radius 12.5 mm, angle 3◦ ). Steady flow was obtained by setting a shear rate until stress balance was reached. By using these two configurations, it was possible to check that the observed phenomena are rheological effects due to the bulk properties of these materials and not perturbing phenomena such as flow instabilities of the slip or fracture type [12]. The instruments are covered with rough glass paper to prevent slip at the interface between the instrument wall and the sample. The measurements in dynamic conditions were carried out using a Carrimed CSL 100 controlled stress rheometer at a temperature of 23 ± 1 ◦ C. A plane–plane configuration with a diameter of 6 cm and a gap of 700 µm was chosen. The instruments were covered with glass paper. In order to confirm the validity of the measurements obtained with controlled stress, tests were performed on an ARES controlled strain rheometer. The controlled stress and controlled strain measurements provided similar results, with differences of less than 10%. 2.2.2. Small-angle neutron scattering (SANS) The measurements were performed at the Laue–Langevin Institute in Grenoble, using the D11 multidetector, at a wavelength of 6 Å. The distances between the detector and sample were fixed at 2.5, 10, and 35.7 m, with beam collimations of respectively 2.5, 20.5, and 40.5 m. The wave vector domain explored ranged from 6 × 10−4 to 10−1 Å−1 . Static condition measurements were obtained by using samples placed in a quartz cell. The sample thickness was fixed at 1 mm. The data were analyzed by using standard programs in order to remove any fundamental inconsistencies. The radially averaged total scattering intensity was then determined using classical integration software. 2.2.3. Static light scattering (SLS) The laser bench used for this study was developed and built at the “Laboratoire de Rhéologie” in Grenoble [5,9]. It consists of a 2-mW He–Ne laser beam with a wavelength of 6328 Å and a Fresnel lens acting as a scattering screen. The detector is a video camera with a CCD 752 × 582-pixel sensor. A shutter enables the acquisition time to be varied from
1/50 to 1/10,000 s. Analysis involves image processing using software that performs classical integration operations. The sample is placed between two microscope glass slides. The thickness of the sample is fixed at 0.3 mm. With this small value, the transmission measurements give satisfactory results, irrespective of the volume fraction.
3. Results and discussion 3.1. Influence of silica treatment on flow properties The steady flow curves for dispersions containing a volume fraction of 2.2% of the three different silicas are represented in Fig. 1. Adding colloidal silica results in an increase in the consistency of the dispersions in comparison with the polyol alone, irrespective of the type of silica used. The dispersions of silica A200 and R974 have Newtonian-type behavior with stress levels of the same order of magnitude. The level of viscosity increases by a factor of about 2.2 in relation to that of the polyol, which has Newtonian behavior. In the case of the suspension containing silica R805, the low-shearrate flow curve tends toward a plateau, which demonstrates the occurrence of a yield stress. A gel state is reached. The stress levels in the flow beyond 1 s−1 are multiplied by a factor of 10 in comparison with those for the polyol only and by a factor of 4 in comparison with those for the polyol containing silicas A200 and R974, at low concentration levels. Hydrophobic silica R805 is the most efficient in terms of increasing consistency. The following results concern the structural and mechanical behavior of this R805 silica dispersion at rest. 3.2. Structure of a hydrophobic silica dispersion at rest A general view of the structure is obtained from the scattering curve I (Q) of a dispersion of colloidal silica R805 at two volume fractions of 2.2% and 3% (Fig. 2). These scattering curves combine the data obtained with SANS and SALS so that the amplitudes of the slopes of the two groups of
Fig. 1. Effect of type of colloidal silica. Steady flow curves. φv = 2.2%, T = 23 ± 1 ◦ C.
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Fig. 3. Steady flow curves for suspensions of silica R805. Effect of silica concentration. T = 23 ± 1 ◦ C.
Fig. 2. Small-angle neutron scattering and static light scattering of a dispersion of colloidal silica at rest, at various volume fractions. T = 23 ± 1 ◦ C.
data match each other in a log–log graph. The scattering intensity is thus expressed in arbitrary units. This scattering curve was analyzed starting with the largest wave vectors (7 × 10−2 Å−1 ), corresponding to the intraparticle dimensions (a few nanometers), and working toward the smallest (2 × 10−5 Å−1 ), corresponding to organization at the largest length scale (a few micrometers). Three regions can be identified. Between 7 × 10−2 and 2 × 10−2 Å−1 , the scattering intensity follows a Q−3.6 power law. This exponent of 3.6 is characteristic of an interface, in this case the surface of the silica. This is the Porod zone. It can be seen that there is a slight deviation in the exponent in comparison with a Porod law predicting a Q−4 power law. Between 2 × 10−2 and 1 × 10−4 Å−1 , the scattering intensity follows a Q−1.8 power law. This increase in scattering intensity is due to the variation in particle concentration within the bulk of the dispersion. It may be interpreted as a juxtaposition of dense regions (aggregates) and less dense regions. The value of the slope (−1.8) had already been observed by Dorget [5] and Piau et al. [9] on dispersions of colloidal silicas in silicone and by Schaefer et al. [13], Chen and Russel [14], and Jullien et al. [15]. There is thus a relation I (Q) ∼ Q−D in which D is the fractal dimension of the aggregates [13]. Such behavior can be observed when there are enough particles in the aggregate for it to display the same repetitive internal structure over a much greater distance than the size of the objects forming the aggregate. There are two aggregation models which depend on the sticking probability. In diffusion-limited cluster–cluster aggregation (DLCA) [16,17], the sticking probability is 1 and the fractal dimension is D = 1.75. In reaction-limited cluster–cluster aggregation (RLCA) [18], the sticking probability approaches zero and the fractal dimension is D = 2.1. The exponent (1.8) determined by our measurements re-
vealed a fractal aggregation as recommended by the DLCA model. For the dispersion at φv = 2.2% and 3.3%, this fractal organization extends over a domain ranging from a = 2π/(2 × 10−2 Å−1 ) ≈ 32 nm to about R = 2π/(1 × 10−4 Å−1 ) ≈ 6 µm. Below a scattering vector of 10−4 Å−1 , the scattering intensity tends toward a plateau, which reveals a large-scale continuous structure. Beyond a length scale of 6 µm, the gel is homogeneous. By definition, the fractal dimension D is linked to the gyration radius R of objects forming this fractal structure by D R 1/D or N ≈ . R ≈ aN a N is the number of elementary particles in the fractal. The experimental values give R ≈ 6 µm, a ≈ 32 nm, and D = 1.8, leading to N ≈ 13,000. The critical volume fraction φv∗ which represents the limit between the dilute and the semidilute is given by the following expression: 3 D−3 R a N ≈ φv∗ ≈ . 3 a R From our experimental values, we find by calculation that φv∗ ≈ 0.18%. As the volume fractions φv = 2.2% and 3% of the dispersions studied by scattering are larger than φv∗ , the structure is composed of fractal semidilute objects with a size of R. Scattering measurements revealed a fractal type of aggregation (DLCA). The overall structure may be described as consisting of elementary particles of the order of 32 nm that combine into fractal aggregates. A three-dimensional network is then formed by these aggregates of the order of 6 µm connecting. This three-dimensional network is responsible for the polyol gelling. 3.3. Rheometrical properties of silica R805 suspensions The flow curves for silica R805 suspensions at various volume fractions are represented in Fig. 3. The behavior of the resins diverges considerably from the Newtonian behavior of the polyol when the silica concentration increases.
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Fig. 4. Variation in elastic modulus G as a function of shear strain for various silica R805 concentrations. f = 1 Hz, T = 23 ± 1 ◦ C.
Above a certain concentration level, the curves tend toward a low-shear-rate stress plateau, characterized by the appearance of a yield stress and viscoplastic behavior. The yield stress level increases with the increase in the silica R805 volume fraction. The greater the space occupied by the particles in the polyol, the greater the number of particle–particle connections, which causes a rise in the yield stress. Figure 4 shows the change in elastic modulus G as a function of the strain rate at a frequency of 1 Hz for the concentration range with a yield stress. The dispersions have an elastic modulus that varies from 75 to 16,000 Pa at low strains with increasing volume fractions. The elastic modulus is constant across the entire linear domain range, which stretches from about 0.1% to 1%. When strain occurs above this range, the curve begins to fall rapidly. Figure 5 shows the effect of the density of colloidal silica on the yield stress and elastic modulus. The various yield stress values are obtained at a shear rate of 10−4 s−1 . The various variations in elastic modulus G are obtained with a strain of 0.3% and pulse frequency of 1 Hz. These parameters follow a power law. In the case of the yield stress, the curve follows a scaling law of the form σ0 ∼ φv2.9±0.2 . The elastic modulus follows a scaling law of the form G ∼ φv4.1±0.3 . These scaling laws, σ0 ∼ φvn and G ∼ φvm , have already been observed in other polymer systems. The results obtained with silica/polyol dispersions produce exponents n and m that are in good agreement with those obtained in previous studies. With regard to the change in yield stress, our experimental results (Fig. 5) give n = 2.9 ± 0.2 whereas Dorget [5] found n = 3.3 ± 0.4 for dispersions of colloidal silicas in PDMS. With regard to the change in elastic modulus, our experimental results (Fig. 5) give m = 4.1 ± 0.3 whereas Dorget [5] found 4.2 ± 0.5, and other studies found m = 4 [7,13–15]. Nevertheless our results contradict results obtained by Khan and Zoeller [6], who found m = 6 on R805 in PPG. But their system has different polymer length chains. It can be seen that the scaling laws in the different systems are similar, thus suggesting that similar structures are created.
Fig. 5. Change in rheometric parameters with volume fraction of colloidal silica. The various yield stress values are obtained at a shear rate of 10−4 s−1 . The various variations in elastic modulus G are obtained with a strain of 0.3% and pulse frequency of 1 Hz. T = 23 ± 1 ◦ C.
3.4. Experimental and theoretical scaling laws Correlations have been established between fractal dimensions and rheological properties for various types of colloidal dispersion. Mention may be made of Shi et al. [19], who studied alumina gels, and Pignon et al. [20] in the case of clay dispersions. They showed that the mechanical properties (plasticity and elastic modulus) were governed principally by the fractal dimension of the aggregates forming the structure responsible for the rheological behavior of the dispersion. Theoretical studies based on the diffusion limited cluster–cluster aggregation (DLCA) model [8,13] predict G ∼ φv4 and I ∼ Q−1.78±0.05 . The model based on the notion of semidilute fractal objects [5,9], can be used to interpret the rheology of a colloidal silica dispersion in a sil4/(3−D) icone. It gives the following results: σ0 ∼ φv and G ∼ 5/(3−D) . With a fractal dimension D of 1.8 ± 0.1 given φv by Fig. 2, the preceding equations lead to the scaling laws σ0 ∼ φv3.3±0.3 and G ∼ φv4.1±0.4 , while our rheometry results give σ0 ∼ φv2.9±0.2 and G ∼ φv4.1±0.3 . The good agreement between the experimental scaling laws determined by rheometric measurements and the theoretical scaling laws deduced from the model [5,9] clearly demonstrates that the macroscopic mechanical properties are governed mainly by the fractal nature of the structural arrangement of the silica particles within the polyol. 3.5. Nature of interactions between particles and polymer Khan and Zoeller [6] studied the effect of silica surface treatment and polymer polarity on the rheological properties of the dispersions. Their rheological studies demonstrated three possible types of interaction between particles: primary bridging, secondary bridging, and no bridging. Primary bridging allows the formation of a structure where the interactions between the silica particles occur by hydrogen
F. Saint-Michel et al. / Journal of Colloid and Interface Science 267 (2003) 314–319
linking. These interactions are observed in all mineral oils. Secondary bridging occurs though the intermediary of attractive interactions between the hydrophobic chains which are grafted at the silica surface. This kind of interaction is observed in mineral oils and in PPG for the hydrophobic R805 silica dispersions. No bridging occurs when there is also interaction between a polar polymer and the hydrophilic silica A200. Hydrophilic silica A200 has silanol and siloxane clusters at its surface. The polyol is a polar fluid that has numerous hydroxyl functions. Yziquel et al. [7] report that particle–polymer interactions predominate over particle–particle interactions. The particles of silica A200 are therefore coated by the polyol, which hampers interactions between particles. These three kinds of interactions can explain the results obtained in Fig. 1. The surfaces of silicas R805 and R974 were modified by grafting octyl and methyl clusters onto them (Table 1). These hydrophobic silicas are distinguished from the other silicas by a lower silanol cluster density. Because of this, polyol–hydrophobic silica interactions are less significant than polyol–hydrophilic silica ones. In the case of the hydrophobic silicas (R805 and R974), the hydrophobic clusters are anchored at the surface of the particles. These create a steric screen in front of the silanol clusters that still remain and prevent the undesirable process of solvatation. The longer the hydrophobic chain, the more effective this steric screen is (Fig. 1). In the case of silica R974, the hydrophobic chain is not long enough to prevent solvatation from occurring. According to the results obtained by Khan and Zoeller [6], this steric screen hinders interactions between the particles of hydrophobic silica by the silanol clusters. Bridging between particles of silica then occurs via the hydrophobic chains grafted on the surface. Contrary to our results, Khan and Zoeller [6] do not observe a fractal arrangement of R805 silica particles in PPG. The polyol used by Khan and Zoeller [6] has a linear chain and a molecular weight of 725 g mol−1 . The polyols used during our study have shorter molecular chains because of their lower molecular weight and their ramified structure. The most important result in this article is that the length of polyol chains is a limiting factor on the particle–particle interactions through the octyl chain grafted onto the R805 silica. Different situations can arise depending on the length of polyol chains. For long chains, particle–particle interactions can be limited. If these chains are short, particle–particle interactions can take place, giving birth to a three-dimensional structural network. 4. Conclusions Studying the effect of surface treatment on mechanical behavior showed that only hydrophobic silica in the polyol resulted in thickening. A plastic type of behavior is reached at low silica concentrations. Radiation scattering (SLS, SANS) measurements showed that the structure of the dispersions of R805 hydrophobic
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silicas in the polyol corresponds to a fractal arrangement of the particles with a fractal dimension D = 1.8 ± 0.1 up to a scale of a few micrometers. The yield stress and elastic modulus of the R805 hydrophobic silica dispersions follow volume fraction scaling laws of the type σ0 ∼ φv2.9±0.3 and G ∼ φv4.1±0.4 . These experimental laws corroborate the theoretical scaling laws. The results show that the mechanical properties of a system composed of polyol and R805 hydrophobic colloidal silica are governed mainly by a fractal arrangement of the aggregates at the micrometer length scale. The interactions between the silica particles occur via the hydrophobic groups which are grafted on the R805 silica surface. Acknowledgments This work was carried out in the framework of the Rhône-Alpes Region’s 2000/2003 Priority Themes research program “Materials Engineering.” The small-angle neutron scattering measurements were carried out in association with Peter Lindner, responsible for beam line D11 at the Institut Laue–Langevin, Grenoble, and Bernard Cabane (Laboratoire de Physique et Mécanique des Milieux Hétérogènes, ESPCI). The authors express their deepest thanks for this help. References [1] B. Cabane, K. Wong, P. Lindner, F. Lafuma, J. Rheol. 41 (1997) 531. [2] P.G. De Gennes, Scaling Concepts of Polymer Physics, Cornell Univ. Press, Ithaca, NY, 1980. [3] S. Steinmann, M. Fahrländer, Nanosized Particles in Polymer Melts: Influence of Particle–Matrix Interactions and Interfacial Properties on Dynamic Rheology, presented at: XIIIth International Congress on Rheology, Cambridge, UK. [4] M.I. Aranguren, E. Mora, J.V. Degroot, C.W. Macosko, J. Rheol. 36 (1992) 1165. [5] M. Dorget, Propriétés rhéologiques des composés silice/silicone, Thèse présentée à l’Université Joseph-Fourier de Grenoble I, 1995. [6] S.A. Khan, N.J. Zoeller, J. Rheol. 37 (1993) 1225. [7] F. Yziquel, P.J. Carreau, P.A. Tanguy, Rheol. Acta 38 (1999) 14. [8] R. Buscall, P.D.A. Mills, J.W. Goodwin, D.W. Lawson, J. Chem. Soc. Faraday Trans. 1 84 (1988) 4249. [9] J.M. Piau, M. Dorget, J.F. Palierne, A. Pouchelon, J. Rheol. 43 (1999) 305. [10] G.L. Gulley, J.E. Martin, J. Colloid Interface Sci. 241 (2001) 340–345. [11] Degussa Corporation, Technical Bulletin Pigments No. 23, Allendale, NJ, 1989. [12] A. Magnin, J.M. Piau, J. Non-Newtonian Fluid Mech. 36 (1990) 85. [13] D.W. Shaefer, J.E. Martin, P. Wilzius, D.S. Cannell, Phys. Rev. A 52 (1984) 2371. [14] M. Chen, W.B. Russel, J. Colloid Interface Sci. 141 (1991) 546–577. [15] R. Jullien, R. Botet, P.M. Mors, Faraday Discuss. Chem. Soc. 83 (1987) 125–137. [16] M. Kolb, R. Botet, R. Jullien, Phys. Rev. Lett. 51 (13) (1983) 1123. [17] P. Meakin, Phys. Rev. Lett. 51 (13) (1983) 1119. [18] M. Kolb, R. Jullien, J. Phys. Lett. 45 (20) (1984) 977. [19] W.H. Shi, W.Y. Shi, S.I. Kim, J. Liu, I.A. Aksay, Phys. Rev. A 42 (1990) 4772. [20] F. Pignon, A. Magnin, J.M. Piau, B. Cabane, P. Lindner, O. Diat, Phys. Rev. E 56 (1997) 3281.
Fractal behavior and scaling law of hydrophobic silica in polyol Fabrice Saint-Michel, Frédéric Pignon, and Albert Magnin ∗ Laboratoire de Rhéologie, Université Joseph Fourier–Grenoble I, Institut National Polytechnique de Grenoble, CNRS UMR 5520, 38041 Grenoble Cedex 9, France Received 17 October 2002; accepted 24 July 2003
Abstract This article examines the rheological properties of a system composed of polyol and colloidal silica. Three types of nanosized silicas with hydrophilic and hydrophobic surfaces were studied: A200 with OH surface groups, R974 with CH3 surface groups, and R805, which is grafted with a C8 H17 alkyl chain. Rheometric measurements showed that the dispersions of R805 silicas have a yield stress at low volume fraction, unlike the R974 and A200 silicas. The plastic behavior of the hydrophobic silicas was quantified by a yield stress σ0 and an elastic modulus G . It is observed that these parameters follow scaling laws as a function of the volume fraction of silica introduced, in the form σ0 ∼ φv2.9±0.2 , G ∼ φv4.1±0.3 . Static light scattering (SLS) and small angle neutron scattering (SANS) measurements show a fractal arrangement with a fractal dimension D = 1.8 ranging from elementary particles of about 32 nm to aggregates measuring about 6 µm. Correlations were established between the theoretical scaling laws and the experimental scaling laws determined by rheometric measurements. The fractal structure observed in this system is explained by the attractive physical interaction of the octyl chains between the silica particles. Contrary to what has been observed in the past by Khan and Zoeller (J. Rheol. 37 (1993) 1225), the lower molecular weight of the polyol studied here, which has a shorter chain length, allows direct bridging of two separate silicates though alkyl chains, giving rise to the formation of a 3D gel network at volume fractions as low as φv = 2.2%. 2003 Elsevier Inc. All rights reserved. Keywords: Rheology; Polyol; Silica; Colloid; Interaction; Fractal; Yield stress; SLS; SANS
1. Introduction Colloidal silicas are used as fillers in various industrial applications to control rheological properties. They come in a wide range of sizes and with a variety of surface treatments. The considerable specific surface area of these particles (50–400 m2 /g) and the presence of functional clusters (silanol, siloxane) play a major role in their rheological behavior. In particular, they are used to give polymers viscoplastic behavior characterized by a yield stress. Their capacity to form a three-dimensional network responsible for the yield stress and the formation of a gel phase depends on the volume fraction, on the type of colloidal silica (type of surface groups), and on the polarity of the suspending phase. Indeed, interactions between the particles play a decisive role in the stability and rheological properties of the * Corresponding author.
E-mail address: [email protected] (A. Magnin). 0021-9797/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2003.07.018
dispersions. For example, in the case of aqueous dispersions of nanometric silica particles in polyethylene oxide (PEO), shear can induce gelation [1]. The concepts developed by De Gennes [2] for the formation of polymer gels can also be applied to the case of colloidal gels. Various studies have applied these concepts in order to examine the effects of interparticle forces on the elasticity of dispersions. By characterizing viscoelastic properties, it is also possible to examine particle–particle and particle–solvent interactions [3]. Many studies have looked at colloidal silica dispersions in a silicone [4,5]. These authors have shown that the viscoelastic behavior of colloidal silica suspensions in a polydimethylsiloxane (PDMS) is nonlinear above a critical strain; i.e., there is a very great drop in the elastic modulus above a certain critical strain. Aranguren et al. [4] speak of the role of bridging polymers in the behavior of these materials. They also showed that the elastic modulus increases with the volume fraction, following a power law of the type (G ∼ Kφvm ).
F. Saint-Michel et al. / Journal of Colloid and Interface Science 267 (2003) 314–319
Other types of polymer dispersions have similar rheological behavior. This is the case, for example, with nonpolar polymers, such as a mineral oil, or polar polymers, such as a polypropylene glycol (PPG). Khan and Zoeller [6] and Yziquel et al. [7] identified the various possible types of interaction between the solvent and particles, depending on the type of silica (hydrophilic and hydrophobic) and type of solvent (polar and nonpolar). For a dispersion of silica which is grafted with an alkyl chain, Khan and Zoeller [6] did not observe a fractal arrangement of the particles within PPG. Moreover, their work showed that when the particles interact with one another, the scaling concept may be applied to obtain the power law for the dependence on the elastic modulus and volume fraction. The power law G ∼ φvm can be linked to the fractal dimension [8,9]. Dorget [5] and Piau et al. [9] proposed a model based on the notion of semidilute fractal objects, which can be used to interpret the rheology of these compounds. They thus linked the yield stress and elastic modulus to the fractal dimension via a scaling law. Little research work deals with colloidal dispersions in polyol. Nevertheless Gulley and Martin [10] have shown a fractal type of aggregation in the case of aqueous colloidal silica dispersions with different kinds of polyols. In this article, the viscoplastic and elastic properties of silicas-polyol dispersions were studied in relation to the nature of the interactions and structures created by the silica. First of all, the thickening effects of three silicas were studied. They differ in terms of the types of their surface groups and their hydrophilic and hydrophobic nature. The hydrophobic silica with the greatest thickening power will be studied in greater detail. Second, this work involved characterizing the structure of this hydrophobic silica dispersion at rest and linking it with its mechanical behavior by a scaling law. Contrary to the results obtained by Khan and Zoeller [6], we demonstrated the formation of a fractal arrangement for the R805 silica particles within the polyol. The structural differences between the system studied in this work and that studied by Khan and Zoeller [6] reside in the polymer chain length. The work presented here clearly shows the importance of this parameter for both the silica organization and the mechanical behavior of the system. The methods used to characterize the dispersions at rest combine rheometric measurements under simple and harmonic shearing conditions and static light scattering (SLS) and small angle neutron scattering (SANS) techniques. Measurements of radiation scattering enabled the structure at rest to be characterized over a wide range of length scales. Scaling laws were established linking the yield stress and elastic modulus to the fractal dimension of the structure created by the silica network. These methods of characterization of the structure at rest were compared with the various models proposed in the literature, in order to examine the interactions between particles and polyol.
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2. Materials and experimental techniques 2.1. Materials The colloidal silica consists of spherical particles with a mean diameter of 32 nm. This size is obtained from the small angle neutron scattering measurements. Three different silicas made by Degussa [11] were used: Aerosil A200, R974, and R805. Aerosil A200 is a hydrophilic silica. The surface area of silicas R974 and R805 was modified to make the particles hydrophobic. The functional clusters and specific surface areas of these three fillers are given in Table 1. The dispersions are formulated from a mixture of polyols. The polyols used are triol polypropylene. Physicochemical data are given in Table 2. A volume fraction of colloidal silica (φv ) is then incorporated. This is defined by φv =
Vsilica . Vsilica + Vmatrix
Vsilica represents the volume of colloidal silica incorporated in the matrix; Vmatrix represents the total volume of liquid formed by the polymer matrix. At a temperature of about 20 ◦ C, the dispersion obtained is then mixed using a vane stirrer revolving at 1500 rpm for 10 min. The samples are degassed in a vacuum for 24 h to eliminate all air bubbles introduced into the dispersion during mixing. This operation Table 1 Physicochemical properties of colloidal silicas used Type of colloidal Schematic representation of Density Specific area (m2 /g) silica surface clusters (g/cm−3 ) Aerosil A200
2.2
200 ± 25
Aerosil R974
2
170 ± 20
Aerosil R805
2
150 ± 25
Table 2 Physicochemical properties of chemical compounds used in the polymer matrix Chemical compound
Structure
Molar Hydroxyl Viscosity mass number at 25 ◦ C (g mol−1 ) (mg KOH/g) (Pa s)
Polyol A
450
380
0.380
Polyol B
300
570
0.500
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is intended to avoid any degassing of the sample during measurements. The time elapsing between the preparation of the dispersions and the measurements is 7 ± 3 days. 2.2. Experimental techniques 2.2.1. Rheometry Measurements in simple shear conditions were carried out at a temperature of 23 ± 1 ◦ C. The measurements are performed in a Weissenberg–Carrimed controlled strain rotating rheometer. Torsion bar torque sensors are used. The measurements are carried out with two different geometries: a coaxial cylinder configuration (external radius 10.75 mm, internal radius 8.75 mm, height 40 mm) and a cone– plate configuration with a hollowed center (external radius 25 mm, internal radius 12.5 mm, angle 3◦ ). Steady flow was obtained by setting a shear rate until stress balance was reached. By using these two configurations, it was possible to check that the observed phenomena are rheological effects due to the bulk properties of these materials and not perturbing phenomena such as flow instabilities of the slip or fracture type [12]. The instruments are covered with rough glass paper to prevent slip at the interface between the instrument wall and the sample. The measurements in dynamic conditions were carried out using a Carrimed CSL 100 controlled stress rheometer at a temperature of 23 ± 1 ◦ C. A plane–plane configuration with a diameter of 6 cm and a gap of 700 µm was chosen. The instruments were covered with glass paper. In order to confirm the validity of the measurements obtained with controlled stress, tests were performed on an ARES controlled strain rheometer. The controlled stress and controlled strain measurements provided similar results, with differences of less than 10%. 2.2.2. Small-angle neutron scattering (SANS) The measurements were performed at the Laue–Langevin Institute in Grenoble, using the D11 multidetector, at a wavelength of 6 Å. The distances between the detector and sample were fixed at 2.5, 10, and 35.7 m, with beam collimations of respectively 2.5, 20.5, and 40.5 m. The wave vector domain explored ranged from 6 × 10−4 to 10−1 Å−1 . Static condition measurements were obtained by using samples placed in a quartz cell. The sample thickness was fixed at 1 mm. The data were analyzed by using standard programs in order to remove any fundamental inconsistencies. The radially averaged total scattering intensity was then determined using classical integration software. 2.2.3. Static light scattering (SLS) The laser bench used for this study was developed and built at the “Laboratoire de Rhéologie” in Grenoble [5,9]. It consists of a 2-mW He–Ne laser beam with a wavelength of 6328 Å and a Fresnel lens acting as a scattering screen. The detector is a video camera with a CCD 752 × 582-pixel sensor. A shutter enables the acquisition time to be varied from
1/50 to 1/10,000 s. Analysis involves image processing using software that performs classical integration operations. The sample is placed between two microscope glass slides. The thickness of the sample is fixed at 0.3 mm. With this small value, the transmission measurements give satisfactory results, irrespective of the volume fraction.
3. Results and discussion 3.1. Influence of silica treatment on flow properties The steady flow curves for dispersions containing a volume fraction of 2.2% of the three different silicas are represented in Fig. 1. Adding colloidal silica results in an increase in the consistency of the dispersions in comparison with the polyol alone, irrespective of the type of silica used. The dispersions of silica A200 and R974 have Newtonian-type behavior with stress levels of the same order of magnitude. The level of viscosity increases by a factor of about 2.2 in relation to that of the polyol, which has Newtonian behavior. In the case of the suspension containing silica R805, the low-shearrate flow curve tends toward a plateau, which demonstrates the occurrence of a yield stress. A gel state is reached. The stress levels in the flow beyond 1 s−1 are multiplied by a factor of 10 in comparison with those for the polyol only and by a factor of 4 in comparison with those for the polyol containing silicas A200 and R974, at low concentration levels. Hydrophobic silica R805 is the most efficient in terms of increasing consistency. The following results concern the structural and mechanical behavior of this R805 silica dispersion at rest. 3.2. Structure of a hydrophobic silica dispersion at rest A general view of the structure is obtained from the scattering curve I (Q) of a dispersion of colloidal silica R805 at two volume fractions of 2.2% and 3% (Fig. 2). These scattering curves combine the data obtained with SANS and SALS so that the amplitudes of the slopes of the two groups of
Fig. 1. Effect of type of colloidal silica. Steady flow curves. φv = 2.2%, T = 23 ± 1 ◦ C.
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Fig. 3. Steady flow curves for suspensions of silica R805. Effect of silica concentration. T = 23 ± 1 ◦ C.
Fig. 2. Small-angle neutron scattering and static light scattering of a dispersion of colloidal silica at rest, at various volume fractions. T = 23 ± 1 ◦ C.
data match each other in a log–log graph. The scattering intensity is thus expressed in arbitrary units. This scattering curve was analyzed starting with the largest wave vectors (7 × 10−2 Å−1 ), corresponding to the intraparticle dimensions (a few nanometers), and working toward the smallest (2 × 10−5 Å−1 ), corresponding to organization at the largest length scale (a few micrometers). Three regions can be identified. Between 7 × 10−2 and 2 × 10−2 Å−1 , the scattering intensity follows a Q−3.6 power law. This exponent of 3.6 is characteristic of an interface, in this case the surface of the silica. This is the Porod zone. It can be seen that there is a slight deviation in the exponent in comparison with a Porod law predicting a Q−4 power law. Between 2 × 10−2 and 1 × 10−4 Å−1 , the scattering intensity follows a Q−1.8 power law. This increase in scattering intensity is due to the variation in particle concentration within the bulk of the dispersion. It may be interpreted as a juxtaposition of dense regions (aggregates) and less dense regions. The value of the slope (−1.8) had already been observed by Dorget [5] and Piau et al. [9] on dispersions of colloidal silicas in silicone and by Schaefer et al. [13], Chen and Russel [14], and Jullien et al. [15]. There is thus a relation I (Q) ∼ Q−D in which D is the fractal dimension of the aggregates [13]. Such behavior can be observed when there are enough particles in the aggregate for it to display the same repetitive internal structure over a much greater distance than the size of the objects forming the aggregate. There are two aggregation models which depend on the sticking probability. In diffusion-limited cluster–cluster aggregation (DLCA) [16,17], the sticking probability is 1 and the fractal dimension is D = 1.75. In reaction-limited cluster–cluster aggregation (RLCA) [18], the sticking probability approaches zero and the fractal dimension is D = 2.1. The exponent (1.8) determined by our measurements re-
vealed a fractal aggregation as recommended by the DLCA model. For the dispersion at φv = 2.2% and 3.3%, this fractal organization extends over a domain ranging from a = 2π/(2 × 10−2 Å−1 ) ≈ 32 nm to about R = 2π/(1 × 10−4 Å−1 ) ≈ 6 µm. Below a scattering vector of 10−4 Å−1 , the scattering intensity tends toward a plateau, which reveals a large-scale continuous structure. Beyond a length scale of 6 µm, the gel is homogeneous. By definition, the fractal dimension D is linked to the gyration radius R of objects forming this fractal structure by D R 1/D or N ≈ . R ≈ aN a N is the number of elementary particles in the fractal. The experimental values give R ≈ 6 µm, a ≈ 32 nm, and D = 1.8, leading to N ≈ 13,000. The critical volume fraction φv∗ which represents the limit between the dilute and the semidilute is given by the following expression: 3 D−3 R a N ≈ φv∗ ≈ . 3 a R From our experimental values, we find by calculation that φv∗ ≈ 0.18%. As the volume fractions φv = 2.2% and 3% of the dispersions studied by scattering are larger than φv∗ , the structure is composed of fractal semidilute objects with a size of R. Scattering measurements revealed a fractal type of aggregation (DLCA). The overall structure may be described as consisting of elementary particles of the order of 32 nm that combine into fractal aggregates. A three-dimensional network is then formed by these aggregates of the order of 6 µm connecting. This three-dimensional network is responsible for the polyol gelling. 3.3. Rheometrical properties of silica R805 suspensions The flow curves for silica R805 suspensions at various volume fractions are represented in Fig. 3. The behavior of the resins diverges considerably from the Newtonian behavior of the polyol when the silica concentration increases.
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Fig. 4. Variation in elastic modulus G as a function of shear strain for various silica R805 concentrations. f = 1 Hz, T = 23 ± 1 ◦ C.
Above a certain concentration level, the curves tend toward a low-shear-rate stress plateau, characterized by the appearance of a yield stress and viscoplastic behavior. The yield stress level increases with the increase in the silica R805 volume fraction. The greater the space occupied by the particles in the polyol, the greater the number of particle–particle connections, which causes a rise in the yield stress. Figure 4 shows the change in elastic modulus G as a function of the strain rate at a frequency of 1 Hz for the concentration range with a yield stress. The dispersions have an elastic modulus that varies from 75 to 16,000 Pa at low strains with increasing volume fractions. The elastic modulus is constant across the entire linear domain range, which stretches from about 0.1% to 1%. When strain occurs above this range, the curve begins to fall rapidly. Figure 5 shows the effect of the density of colloidal silica on the yield stress and elastic modulus. The various yield stress values are obtained at a shear rate of 10−4 s−1 . The various variations in elastic modulus G are obtained with a strain of 0.3% and pulse frequency of 1 Hz. These parameters follow a power law. In the case of the yield stress, the curve follows a scaling law of the form σ0 ∼ φv2.9±0.2 . The elastic modulus follows a scaling law of the form G ∼ φv4.1±0.3 . These scaling laws, σ0 ∼ φvn and G ∼ φvm , have already been observed in other polymer systems. The results obtained with silica/polyol dispersions produce exponents n and m that are in good agreement with those obtained in previous studies. With regard to the change in yield stress, our experimental results (Fig. 5) give n = 2.9 ± 0.2 whereas Dorget [5] found n = 3.3 ± 0.4 for dispersions of colloidal silicas in PDMS. With regard to the change in elastic modulus, our experimental results (Fig. 5) give m = 4.1 ± 0.3 whereas Dorget [5] found 4.2 ± 0.5, and other studies found m = 4 [7,13–15]. Nevertheless our results contradict results obtained by Khan and Zoeller [6], who found m = 6 on R805 in PPG. But their system has different polymer length chains. It can be seen that the scaling laws in the different systems are similar, thus suggesting that similar structures are created.
Fig. 5. Change in rheometric parameters with volume fraction of colloidal silica. The various yield stress values are obtained at a shear rate of 10−4 s−1 . The various variations in elastic modulus G are obtained with a strain of 0.3% and pulse frequency of 1 Hz. T = 23 ± 1 ◦ C.
3.4. Experimental and theoretical scaling laws Correlations have been established between fractal dimensions and rheological properties for various types of colloidal dispersion. Mention may be made of Shi et al. [19], who studied alumina gels, and Pignon et al. [20] in the case of clay dispersions. They showed that the mechanical properties (plasticity and elastic modulus) were governed principally by the fractal dimension of the aggregates forming the structure responsible for the rheological behavior of the dispersion. Theoretical studies based on the diffusion limited cluster–cluster aggregation (DLCA) model [8,13] predict G ∼ φv4 and I ∼ Q−1.78±0.05 . The model based on the notion of semidilute fractal objects [5,9], can be used to interpret the rheology of a colloidal silica dispersion in a sil4/(3−D) icone. It gives the following results: σ0 ∼ φv and G ∼ 5/(3−D) . With a fractal dimension D of 1.8 ± 0.1 given φv by Fig. 2, the preceding equations lead to the scaling laws σ0 ∼ φv3.3±0.3 and G ∼ φv4.1±0.4 , while our rheometry results give σ0 ∼ φv2.9±0.2 and G ∼ φv4.1±0.3 . The good agreement between the experimental scaling laws determined by rheometric measurements and the theoretical scaling laws deduced from the model [5,9] clearly demonstrates that the macroscopic mechanical properties are governed mainly by the fractal nature of the structural arrangement of the silica particles within the polyol. 3.5. Nature of interactions between particles and polymer Khan and Zoeller [6] studied the effect of silica surface treatment and polymer polarity on the rheological properties of the dispersions. Their rheological studies demonstrated three possible types of interaction between particles: primary bridging, secondary bridging, and no bridging. Primary bridging allows the formation of a structure where the interactions between the silica particles occur by hydrogen
F. Saint-Michel et al. / Journal of Colloid and Interface Science 267 (2003) 314–319
linking. These interactions are observed in all mineral oils. Secondary bridging occurs though the intermediary of attractive interactions between the hydrophobic chains which are grafted at the silica surface. This kind of interaction is observed in mineral oils and in PPG for the hydrophobic R805 silica dispersions. No bridging occurs when there is also interaction between a polar polymer and the hydrophilic silica A200. Hydrophilic silica A200 has silanol and siloxane clusters at its surface. The polyol is a polar fluid that has numerous hydroxyl functions. Yziquel et al. [7] report that particle–polymer interactions predominate over particle–particle interactions. The particles of silica A200 are therefore coated by the polyol, which hampers interactions between particles. These three kinds of interactions can explain the results obtained in Fig. 1. The surfaces of silicas R805 and R974 were modified by grafting octyl and methyl clusters onto them (Table 1). These hydrophobic silicas are distinguished from the other silicas by a lower silanol cluster density. Because of this, polyol–hydrophobic silica interactions are less significant than polyol–hydrophilic silica ones. In the case of the hydrophobic silicas (R805 and R974), the hydrophobic clusters are anchored at the surface of the particles. These create a steric screen in front of the silanol clusters that still remain and prevent the undesirable process of solvatation. The longer the hydrophobic chain, the more effective this steric screen is (Fig. 1). In the case of silica R974, the hydrophobic chain is not long enough to prevent solvatation from occurring. According to the results obtained by Khan and Zoeller [6], this steric screen hinders interactions between the particles of hydrophobic silica by the silanol clusters. Bridging between particles of silica then occurs via the hydrophobic chains grafted on the surface. Contrary to our results, Khan and Zoeller [6] do not observe a fractal arrangement of R805 silica particles in PPG. The polyol used by Khan and Zoeller [6] has a linear chain and a molecular weight of 725 g mol−1 . The polyols used during our study have shorter molecular chains because of their lower molecular weight and their ramified structure. The most important result in this article is that the length of polyol chains is a limiting factor on the particle–particle interactions through the octyl chain grafted onto the R805 silica. Different situations can arise depending on the length of polyol chains. For long chains, particle–particle interactions can be limited. If these chains are short, particle–particle interactions can take place, giving birth to a three-dimensional structural network. 4. Conclusions Studying the effect of surface treatment on mechanical behavior showed that only hydrophobic silica in the polyol resulted in thickening. A plastic type of behavior is reached at low silica concentrations. Radiation scattering (SLS, SANS) measurements showed that the structure of the dispersions of R805 hydrophobic
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silicas in the polyol corresponds to a fractal arrangement of the particles with a fractal dimension D = 1.8 ± 0.1 up to a scale of a few micrometers. The yield stress and elastic modulus of the R805 hydrophobic silica dispersions follow volume fraction scaling laws of the type σ0 ∼ φv2.9±0.3 and G ∼ φv4.1±0.4 . These experimental laws corroborate the theoretical scaling laws. The results show that the mechanical properties of a system composed of polyol and R805 hydrophobic colloidal silica are governed mainly by a fractal arrangement of the aggregates at the micrometer length scale. The interactions between the silica particles occur via the hydrophobic groups which are grafted on the R805 silica surface. Acknowledgments This work was carried out in the framework of the Rhône-Alpes Region’s 2000/2003 Priority Themes research program “Materials Engineering.” The small-angle neutron scattering measurements were carried out in association with Peter Lindner, responsible for beam line D11 at the Institut Laue–Langevin, Grenoble, and Bernard Cabane (Laboratoire de Physique et Mécanique des Milieux Hétérogènes, ESPCI). The authors express their deepest thanks for this help. References [1] B. Cabane, K. Wong, P. Lindner, F. Lafuma, J. Rheol. 41 (1997) 531. [2] P.G. De Gennes, Scaling Concepts of Polymer Physics, Cornell Univ. Press, Ithaca, NY, 1980. [3] S. Steinmann, M. Fahrländer, Nanosized Particles in Polymer Melts: Influence of Particle–Matrix Interactions and Interfacial Properties on Dynamic Rheology, presented at: XIIIth International Congress on Rheology, Cambridge, UK. [4] M.I. Aranguren, E. Mora, J.V. Degroot, C.W. Macosko, J. Rheol. 36 (1992) 1165. [5] M. Dorget, Propriétés rhéologiques des composés silice/silicone, Thèse présentée à l’Université Joseph-Fourier de Grenoble I, 1995. [6] S.A. Khan, N.J. Zoeller, J. Rheol. 37 (1993) 1225. [7] F. Yziquel, P.J. Carreau, P.A. Tanguy, Rheol. Acta 38 (1999) 14. [8] R. Buscall, P.D.A. Mills, J.W. Goodwin, D.W. Lawson, J. Chem. Soc. Faraday Trans. 1 84 (1988) 4249. [9] J.M. Piau, M. Dorget, J.F. Palierne, A. Pouchelon, J. Rheol. 43 (1999) 305. [10] G.L. Gulley, J.E. Martin, J. Colloid Interface Sci. 241 (2001) 340–345. [11] Degussa Corporation, Technical Bulletin Pigments No. 23, Allendale, NJ, 1989. [12] A. Magnin, J.M. Piau, J. Non-Newtonian Fluid Mech. 36 (1990) 85. [13] D.W. Shaefer, J.E. Martin, P. Wilzius, D.S. Cannell, Phys. Rev. A 52 (1984) 2371. [14] M. Chen, W.B. Russel, J. Colloid Interface Sci. 141 (1991) 546–577. [15] R. Jullien, R. Botet, P.M. Mors, Faraday Discuss. Chem. Soc. 83 (1987) 125–137. [16] M. Kolb, R. Botet, R. Jullien, Phys. Rev. Lett. 51 (13) (1983) 1123. [17] P. Meakin, Phys. Rev. Lett. 51 (13) (1983) 1119. [18] M. Kolb, R. Jullien, J. Phys. Lett. 45 (20) (1984) 977. [19] W.H. Shi, W.Y. Shi, S.I. Kim, J. Liu, I.A. Aksay, Phys. Rev. A 42 (1990) 4772. [20] F. Pignon, A. Magnin, J.M. Piau, B. Cabane, P. Lindner, O. Diat, Phys. Rev. E 56 (1997) 3281.