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Nanostructure and Mechanical Properties of Polybutylacrylate Filled with Grafted Silica Particles Rabi Inoubli,† Sylvie Dagre´ou,† Alain Lapp,‡ Laurent Billon,† and Jean Peyrelasse*,† Laboratoire de Physico-Chimie des Polyme` res UMR 5067, UniVersite´ de Pau et des Pays de l’Adour, AVenue de l’UniVersite´ , 64000 Pau, France, and Laboratoire Le´ on Brillouin, CEA Saclay, 91191 GIF-sur-YVette Cedex, France ReceiVed March 15, 2006. In Final Form: May 16, 2006 We investigate the nanostructure and the linear rheological properties of polybutylacrylate (PBA) filled with Sto¨ber silica particles grafted with PBA chains. The silica volume fractions range from 1.8 to 4.7%. The nanostructure of these suspensions is investigated by small-angle neutron scattering (SANS), and we determine their spectromechanical behavior in the linear region. SANS measurements performed on low volume fraction composites show that the grafted silica particles are spherical, slightly polydisperse, and do not form aggregates during the synthesis process. These composites thus constitute model filled polymers. The rheological results show that introducing grafted silica particles in a polymer matrix results in the appearance of a secondary process at low frequency: for the lowest volume fractions, we observe a secondary relaxation that we attribute to the diffusion of the particles in the polymeric matrix. By increasing the silica volume fraction up to a critical value, we obtain gellike behavior at low frequency as well as the appearance of a structure factor on the scattering intensity curves obtained by SANS. Further increasing the silica particle concentration leads to composites exhibiting solidlike low-frequency behavior and to an enhanced structure peak on the SANS diagrams. This quantitative correlation between the progressive appearance of a solidlike rheological behavior, on one hand, and a structure factor, on the other hand, supports the idea that the viscoelastic behavior of filled polymers is governed by the spatial organization of the fillers in the matrix.
1. Introduction Over the last 10 years, increasing interest has been shown in the elaboration of inorganic/organic composites. These composites keep the polymer properties (lightness, ease of transformation, low cost, etc.) but exhibit enhanced properties such as better heat resistance, elastic modulus, impact resistance, and conductivity.1-6 There are several applications for them in different fields such as aeronautics and the automobile industry. Usually, the filler consists of glass, carbon,7,8 or Kevlar fibers. The increase in the composite properties is directly correlated to the surface of contact between the matrix and the filler and to the quality of the interface. More recently, interest has focused on the development of nanocomposites. In this case, the fillers can be organic or inorganic nanosized particles: silica particles,9-20 mica,21 and more recently carbon nanotubes.22-27 * Corresponding author. E-mail:
[email protected]. † Universite ´ de Pau et des Pays de l’Adour. ‡ CEA Saclay. (1) Maheri, M. R.; Adams, R. D.; Gaitonde, J. M. Compos. Sci. Technol. 1996, 56, 1425. (2) Tjong, S. C.; Meng, Y. Z. Polymer 1999, 40, 1109. (3) Ma, J.; Feng, Y. X.; Xu, J.; Xiong, M. L.; Zhu, Y. J.; Zhang, L. Q. Polymer 2002, 43, 937. (4) Anunziata, O. A.; Gomez Costa, M. B.; Sanchez, R. D. J. Colloid Interface Sci. 2005, 292, 509. (5) Hogg, P. J. Mater. Sci. Eng., A 2005, 412, 97. (6) Uotila, R.; Hippi, U.; Paavola, S.; Seppala, J. Polymer 2005, 46, 7923. (7) Ozkoc, G.; Bayram, G.; Bayramli, E. Polymer 2004, 45, 8957. (8) Wei, G.; Saitoh, S.; Saitoh, H.; Fujiki, K.; Yamauchi, T.; Tsubokawa, N. Polymer 2004, 45, 8723. (9) Castaing, G.; Allain, C.; Auroy, P.; Auvray, L. Eur. Phys. J. B 1999, 10, 61. (10) Dietrich, A.; Neubrand, A. J. Am. Ceram. Soc. 2001, 84, 806. (11) von Werne, T.; Patten, T. E. J. Am. Chem. Soc. 2001, 123, 7497. (12) Dagre´ou, S.; Kasseh, A.; Allal, A.; Marin, G.; Ait-Kadi, A. Can. J. Chem. Eng. 2002, 80, 1126. (13) Oberdisse, J. Macromolecules 2002, 35, 9441. (14) Zhang, Q.; Archer, L. Langmuir 2002, 18, 10435. (15) Bartholome, C.; Beyou, E.; Bourgeat-Lami, E.; Chaumont, P.; Zydowicz, N. Macromolecules 2003, 36, 7946. (16) Chabert, E.; Bornert, M.; Bourgeat-Lami, E.; Cavaille, J.-Y.; Dendievel, R.; Gauthier, C.; Putaux, J. L.; Zaoui, A. Mater. Sci. Eng., A 2004, 381, 320.
It is now well established that particle-filled polymers in concentrated solutions and melts present a liquid-solid transition due to the formation of a gellike structure (percolating network) even at low solid fraction. Similar behavior is observed for styreneisoprene block copolymers.16,28 The solid behavior is characterized by the appearance of a constant elastic modulus at low frequencies and is attributed to the bridging of particles. For bare silica particles dispersed in transdecalin29 or in molten EVA,30 the suspension exhibits an elastic response due to the formation of a gellike structure through hydrogen bonding of the silanol groups. In the case of silica particles dispersed in a polymer, the experiments support the hypothesis that the bonds between the particles are often due to the adsorption of polymer on the silica surface.14,31,32 Unfortunately, in this case, it is not possible to control the thickness of the surface polymer layer and to correlate the experimental (17) Hu, Y.-H.; Chen, C.-Y.; Wang, C.-C. Polym. Degrad. Stab. 2004, 84, 545. (18) Huang, S.-L.; Chin, W.-K.; Yang, W. P. Polymer 2005, 46, 1865. (19) Inoubli, R.; Dagreou, S.; Khoukh, A.; Roby, F.; Peyrelasse, J.; Billon, L. Polymer 2005, 46, 2486. (20) Oberdisse, J.; El Harrak, A.; Carrot, G.; Jestin, J.; Boue, F. Polymer 2005, 46, 6695. (21) Kuelpmann, A.; Osman, M. A.; Kocher, L.; Suter, U. W. Polymer 2005, 46, 523. (22) Thostensen, E. T.; Ren, Z.; Chou, T.-W. Compos. Sci. Technol. 2001, 61, 1899. (23) Lau, K.-T.; Hui, D. Composites: Part B 2002, 33, 263. (24) Andrews, R.; Weisenberger, M. C. Curr. Opin. Solid State Mater. Sci. 2004, 8, 31. (25) Mamalis, A. G.; Vogtlander, L. O. G.; Markopoulos, A. Precis. Eng. 2004, 28, 16. (26) Datsyuk, V.; Guerret-Pie´court, C.; Dagre´ou, S.; Billon, L.; Dupin, J.-C.; Flahaut, E.; Peigney, A.; Laurent, C. Carbon 2005, 43, 873. (27) Ramirez, A. P. AT&T Bell Lab. Techn. J. 2005, 10, 171. (28) Cazenave, M. N.; Derail, C.; Leonardi, F.; Marin, G.; Kappes, N. J. Adhes. 2005, 81, 623. (29) Kawaguchi, M.; Mizutani, A.; Matsushita, Y.; Kato, T. Langmuir 1996, 12, 6179. (30) Cassagnau, P. Polymer 2003, 44, 2455-2462. (31) Vignaux-Nassiet, V.; Allal, A.; Montfort, J.-P. Eur. Polym. J. 1998, 34, 309. (32) Zaman, A. Colloid Polym. Sci. 2000, 278, 1187.
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results to the volume fraction of the dispersed matter. These problems can in principle be avoided by grafting a polymer layer onto the surface of the silica particles, but it is necessary to be able to control the grafting density as well as the length of the grafted chains. It should be noticed that polymer adsorption is still possible for grafted silica balls if the grafting density is too low.9 In addition, the grafting of a polymer layer does not always prevent particles clustering. Berriot et al.33 obtain aggregation numbers of grafted silica particles ranging from 12 to 18 depending on the volume fraction. In addition, it should be noted that by controlling the aggregation of particles the size of the hard region can be modified continuously from a few nanometers to a few micrometers.13,19-21 The volume fraction at which the liquid-solid transition is observed in filled polymers is also a controversial subject. For hard sphere dispersions, the liquid-solid transition is obtained at the close packing volume fraction of φp) 0.64,34 but the increase in the effective particle volume fraction due to the swelling of the corona or clustering due to interparticle interactions 9,12,35-41 leads to a decrease in φp. φp can be very low when the dispersed particles have a high length/diameter ratio.16 The aim of this work is to investigate the structure/properties relationship of polybutylacrylate filled with grafted silica particles. The grafted polymer is the same as the polymer constituting the matrix, and the method used to synthesize the particles can be used to vary the length, density, and molecular topology of the polybutylacrylate layer.19,42 First we carry out a small-angle neutron scattering (SANS) study of our composites as a function of silica volume fraction. Measurements at low and moderate volume fractions provide information about the particles’ shapes and sizes. They can show whether the particles are well dispersed and ensure that there are no aggregates. Our composite can thus be regarded as a model system. We then subject the same nanocomposites to spectromechanical analysis to study the evolution of linear viscoelastic behavior with silica concentration. The different relaxation processes are identified. The analysis of the results shows a direct correlation between the structure, as determined by SANS, and the mechanical properties. 2. Experimental Section 2.1. Samples. We initially prepared, by the Sto¨ber method,43 a colloidal suspension of silica nanoparticles in a mixture of absolute ethanol, ammonium hydroxide, and water using the so-called solgel process. The final particle size depends on the initial concentrations of the reactants, and we expect a narrow size distribution. The second step is the grafting of polybutylacrylate (PBA) onto the surface of the silica particles. An initiator of radical-chain-controlled polymerization is first grafted onto the silica surface as explained in previous studies.19,42 The PBA is then grafted by living free radical polymerization, a method that makes it possible to control the molecular weight and the polydispersity. The Ip index measured by SEC is 1.5 for bulk polymer and 1.35 for grafted polymer. (33) Berriot, J.; Montes, H.; Martin, F.; Mauger, M.; Pyckhout-Hintzen, W.; Meier, G.; Frielinghaus, H. Polymer 2003, 44, 4909. (34) Duits, M.; Nommensen, P.; Van Den Ende, D.; Mellema, J. Colloids Surf., A 2001, 183, 335. (35) Boonstra, B. B. Polymer 1979, 20, 691. (36) Russel, W.; Gast, A. J. Chem. Phys. 1986, 84, 1815. (37) Elliot, S.; Russel, W. J. Rheol. 1997, 42, 361. (38) Russel, W. B.; Grant, M. C. Colloids Surf., A 2000, 161, 271. (39) Chen, Y.; Gohr, K.; Schaertl, W.; Schmidt, M.; Yezek, L.; Lagaly, G. Prog. Colloid Polym. Sci. 2002, 121, 28. (40) Yezek, L.; Schaertl, W.; Yongming, C.; Gohr, K.; Schmidt, M. Macromolecules 2003, 36, 4226. (41) Carriere, P.; Feller, J.-F.; Dupuis, D.; Grohens, Y. J. Colloid Interface Sci. 2004, 272, 218. (42) Inoubli, R. Ph.D. Thesis. Universite´ de Pau et des Pays de l’Adour, Pau, France, 2005. (43) Sto¨ber, W.; Fink, A. J. Colloid Interface Sci. 1968, 26, 62.
Inoubli et al. These grafted silica particles are then dispersed in a polybutylacrylate matrix of the same molecular weight as that of the grafted polymer. The dispersions of grafted silica particles into the polybutylacrylate matrix are prepared as follows: at the end of polymerization, the grafted silica particles are suspended in acetone; the amount of polybutylacrylate necessary to obtain the desired volume fraction is then added. The dispersion is kept under magnetic stirring for 24 h and then poured into a 25-mm-diameter mould. The residual solvent is removed in two steps: first at room temperature and ordinary pressure for 6 h to evaporate the majority of the solvent and then under vacuum at room temperature for 24 h to remove all of the solvent. It is important to notice that it is not possible to disperse bare silica particles in the PBA matrix by this method, apart from low silica volume fractions. For this study, we prepared two types of samples with very different silica particle sizes: about 50 and 12 nm. The grafting density and the molecular weight Mn of polybutylacrylate are 0.25 molecules/nm2 and 33.8 kg/mole for 12 nm silica particles and 0.5 molecules/nm2 and 61.2 kg/mole for 50 nm silica particles. The radius of gyration of a PBA molecule with Mn ) 33 800 g mol-1 as used is this work is about 8.5 nm, and the corresponding cross section is 230 nm2. With a grafting density of 0.25 molecules/ nm2, the available surface for each molecule on the silica surface is only 4 nm2. This surface is much smaller than the coil cross section, so the grafted polymer stretches to form polymer brushes, as schematized on Figure 10. In the following text, we will call Si1 and Si2 the dispersions in PBA of bare silica particles with radii of 50 and 12 nm, respectively, and Si1g and Si2g the dispersions of the grafted particles with radii of 50 and 12 nm, respectively. 2.2. Experimental Methods. 2.2.1. Small-Angle Neutron Scattering. SANS experiments were performed at the Laboratoire Le´on Brillouin, CEA of Saclay (France) on the PAXY spectrometer. In a neutron-scattering experiment, the scattering intensity I is measured as a function of the scattering angle θ, which is related to the scattering wave vector q ) 4π/λ sin θ/2. Here λ is the wavelength of the neutron beam. Data are collected on a 2D detector that counts the number of neutrons scattered during a given period of time. A 2D scattering pattern is obtained, and the 1D I(q) data set is generated by circular integration. In this study, we work with two geometries: a sampledetector distance of 6.7 m with a wavelength of 15 Å and a sampledetector distance of 3.2 m with a wavelength of 6 Å. These two geometries yield a scattering wave vector ranging from 3 × 10-3 to 0.1 Å-1. Data correction allowed for sample transmission and detector efficiency. The efficiency of the detector was taken into account with the scattering of H2O. Absolute intensities were obtained by reference to the attenuated direct beam, and the scattering of the pure polymer was subtracted. Finally, the intensities were corrected for a small incoherent contribution. Quartz cells with a 2 or 1 mm path length were used. 2.2.2. Rheology. Rheological measurements were carried out to determine the complex shear modulus G* ) G′ + jG′′ as a function of angular frequency ω. The complex viscosity η* ) η′ - jη′′, where η′ ) G′′/ω and η′′ ) G′/ω, can also be determined. The measurements were performed with a TA Instruments AR 2000 imposed stress rheometer equipped with a cone-and-plate cell (25 mm in diameter, with a 4° angle truncated at 104 µm). The airoven-imposed temperatures ranged from 15 to 55 °C, and the timetemperature superposition principle was applied to construct master curves at 25 °C. All experiments were performed within the angular frequency range from 10-2 to 102 rad‚s-1, and the stress was set to work in the linear region (i.e., the region where the moduli are independent of the imposed stress).
3. SANS Studies 3.1. Results. We used small-angle neutron scattering to study samples Si1 and Si1g of various volume fractions. The I(q) curves are similar, and by plotting I(q)/φ versus q, a master curve is obtained as shown in Figure 1. The scattering intensity must tend toward a limit at low q, but this limit could not be reached with
Polybutylacrylate Filled with Silica Particles
Figure 1. Scattering intensities of Si1 and Si1g samples.
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Figure 4. Scattering intensities of Si2g samples. The full lines are the theoretical curves.
It should be noted that this peak is not highlighted for similar silica volume fractions with the Si1g samples in the available q range. 3.2. Discussion. A great deal of work has been devoted to the interpretation of results from neutron-scattering experiments with interacting colloidal systems.44-48 When the particle volume fraction is low, the scattering intensity is mainly governed by the form factor P(q). When the concentration increases, interparticle correlations become significant, and for a monodisperse system, the scattering intensity I(q) can be written as the product of the particle form factor and the structure factor S(q)
I(q) ) NP(q) S(q) ) N∆F2F(q)2 S(q) Figure 2. Scattering intensity of the Si2g sample; φ ) 0.018.
where N is the number density of scatterers, ∆F2 is the contrast factor, and F(q) is the form factor. Note that S(q) ) 1 if the particle volume fraction is small and that S(q) tends toward 1 when q is large. All of the SANS diagrams show that, at large q, the scattering intensity I(q) scales as q-4, which is characteristic of spherical particles. This confirms that the silica balls that we synthesized by the Sto¨ber method are spherical. We will thus restrict the following discussion to the special case of spherical particles. Our particles consist of a silica core surrounded by a grafted polymer layer. For a core-and-shell dispersion, ∆FF(q) is given by the following equation
∆FF(q) ) Figure 3. TEM image of the Si2g sample; φ ) 0.047.
our experimental setup. This is one of the reasons that we prepared samples with smaller silica particles. The results obtained with samples Si2 and Si2g are similar to those obtained with Si1 and Si1g, but in this case, the limit of I at small q is reached. Figure 2 shows, for example, the result obtained with a Si2g sample of small volume fraction. One can note that there is no significant increase in the intensity in the low-q range. Thus, we can reasonably assume that silica particles do not form aggregates during sample preparation. This was checked by TEM, and Figure 3 shows that the grafted silica particles are well dispersed in the PBA matrix. With the Si2g samples, a peak appears on the I(q) curves as soon as the silica volume fraction reaches 0.025. The intensity of this peak increases with the volume fraction of silica, and its position moves slightly toward greater values of q. (Figure 4).
(1)
3j1(qR) 4π 2 3j1(qRc) 4π 2 R (FSi - Fc) + Rc (Fc - Fm) 3 qR 3 qRc (2)
where R and RC are the silica and silica + shell radii, respectively, FSi and Fc are the scattering-length densities (SLDs) of the core and corona, respectively, and Fm is the SLD of the continuous medium. j1 is the first-order spherical Bessel function defined by
j1(x) )
sin x - x cos x x2
(3)
Because the SLD of the polymer corona is the same as the SLD of the surrounding medium (Fc ) Fm), ∆FF(q) reduces to the first term of eq 2. It is obvious that we will not obtain any information on the layer of grafted polymer with our samples, (44) Griffith, W. L.; Triolo, R.; Compere, A. L. Phys. ReV. A 1986, 33, 2197. (45) Griffith, W. L.; Triolo, R.; Compere, A. L. Phys. ReV. A 1987, 35, 2200. (46) Kotlarchyk, M.; Chen, S. H. J. Chem. Phys. 1983, 79, 2461. (47) Pedersen, J. S. J. Appl. Crystallogr. 1994, 27, 595-608. (48) Sun, Z.; Tomlin, C. D.; Sevick-Muraca, E. M. Langmuir 2001, 17, 6142.
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Table 1. Fitting Parameters of the Low Volume Fraction Nanocomposites Si1 Si1g Si1g Si2g
R h (Å)
z
σ (Å)
φ
502.9 505.8 507.8 109.3
53.4 35.8 29.6 17.5
68 83 91 25
0.0053 0.0458 0.144 0.0178
this synthesis, the polydispersity of silica particles increases with decreasing particle size.43 When increasing the volume fraction, we can no longer consider that S(q) ) 1. Using the Ornstein-Zernike and Perkus-Yevick approximations, the structure factor S(q) may be calculated analytically in some cases by assuming an interaction potential. The result is well known for the hard sphere model
but it will be possible to analyze the results without making assumptions about the structure of this layer. In the future, we plan to graft a layer of deuterated PBA to investigate its structure. We will first discuss the case where S(q) ) 1. If the particles are homogeneous but polydisperse, then the scattering intensity is
∫0∞ F2(q, R) N(R) dR
I(q) ) ∆F2
(4)
N(R) dR is the number density of silica particles with a radius between R and R + dR. To calculate I(q), the analytical form of the distribution function must be postulated. Here, we assume that the size distribution is given by the Schulz equation, which expresses the probability density function as
f(R) )
Rc - 1e-R/b bcΓ(c)
(5)
R is the particle radius. Parameters b and c are given by b ) R h /(z + 1) and c ) z + 1, where R h is the mean radius value and z is the Schulz width factor. Γ(c) is the gamma function. The standard deviation is σ ) R h /(xz + 1). The number of particles per unit volume can be calculated from the volume fraction
Np )
φ
∫0∞ 34πR3f(R) dR
S(q) )
1 1 + 24ΦhsG(x, Φhs)/x
(8)
where
x ) 2qRhs in which Rhs is the hard sphere interaction radius and Φhs is the hard sphere volume fraction
G(x, Φhs) ) A(x) + B(x) + C(x) A(x) ) B(x) )
R [sin x - x cos x] x2
β [2x sin x + (2 - x2)cos x - 2] x3
C(x) ) γ [-x4 cos x + 4(3x2 - 6)cos x + 4(x3 - 6x)sin x + 24] 5 x R)
β)
(1 + 2 Φhs)2 (1 - Φhs)4
-6Φhs(1 + Φhs/2)2
(6) γ)
(1 - Φhs)4 Φhs (1 + 2Φhs)2 2 (1 - Φ )4 hs
The number density of silica particles having a radius between R and R + dR is thus given by
N(R) dR ) Np f(R) dR
(7)
In this model, there are only two adjustable parameters: the mean radius of the silica particles R h and the Schulz width factor z. It should be noted that, for constant mean radius and distribution width, the scattering intensity is proportional to the volume fraction (eq 6). This is what we observed for the Si1 and Si1g samples since the I/φ curves are superimposed. The preparation method of our samples is such that it is not possible to determine the volume fraction φ of the silica particles with a high degree of accuracy. We thus consider φ to be a third adjustable parameter. We report in Table 1 the values obtained for the different parameters. One can note that the values of the volume fractions are very close to the approximate experimental values. Figure 1 shows that there is excellent agreement between the model and the results for Si1 and Si1g samples. The theoretical curve is calculated with the average values obtained from the fitting with the model (i.e., R h ) 505 Å, z ) 40). For a similar system, Berriot et al.33 also obtain good fittings of their neutron-scattering data with a Schultz distribution function. For the Si2g sample, the agreement with the model is also very good, as shown in Figure 2. It seems, however, that the polydispersity is larger than for the Si1 samples. That can be explained by the specific properties of the Sto¨ber synthesis: with
For concentrated dispersions of silica particles stabilized by electrostatic repulsions, Qiu et al.49 use the Hayter-PenfoldYukawa potential. For polydisperse particles, the calculation of the scattering intensity was solved only in some particular cases. In the case of polydisperse homogeneous hard spheres with Schulzdistributed diameters, Griffith et al.44,45 obtained an analytical scattering function. This solution is not usable in our case because our particles are not homogeneous. Because of the complexity of the fully polydisperse hard sphere model, approximations are often used to analyze the scattering data. Kotlarchyk and Chen46 proposed the decoupling approximation (DA) in which the structure factor is assumed to be independent of size and is taken as that of a monodisperse system of particles with the average size R h . A different approximation was suggested by Pedersen,47 in which the total scattering can be calculated as the sum of the scattering of monodisperse subsystems, weighted by the size distribution. This approximation is known as the local monodisperse approximation (LMA), and the expression of I for homogeneous particles is as follows
∫0∞F(q, R)2 S(q, R, Φhs)N(R) dR
I(q) ) ∆F2
(9)
where S(q, R, Φhs) is the structure factor of a monodisperse (49) Qiu, D.; Dreiss, C.; Cosgrove, T. Langmuir 2005, 21, 9964.
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Figure 5. Scattering intensity of the Si2g sample; φ ) 0.058. The full line is the theoretical curve.
Figure 6. Spectromechanical measurements of Si2g-based nanocomposites as a function of volume fraction.
Table 2. Fitting Parameters of Nanocomposites Presenting a Structure Factor Si2g 5.8% Si2g 4.7% Si2g 2.5%
R h (Å)
z
σ (Å)
φ
∆ (Å)
Φhs
114.8 110.3 125.5
19.5 15.0 26.2
25 27 24
0.057 0.046 0.025
104.9 111.1 111.7
0.366 0.332 0.157
system of particles with radius R at the same volume fraction Φhs as for the full system. Sun et al.48 compared these two approximations, and they found that the LMA model gives better results than the HSPY model at high volume fractions. Pedersen47 obtained the same results. We used the LMA approximation to analyze our experimental results. ∆FF(q, R) is the same as in the case of low volume fractions. For the structure factor, we used the hard sphere analytical solution (eq 8). Given the grafting method, it is possible to assume that the length of the grafted chains is independent of the radius of the silica particles. This is all the more probable because the distribution of sizes is narrow. We thus assume that Rhs ) R + ∆, where R is the radius of the silica particles calculated with the Schulz distribution and ∆ is a constant. ∆ is not necessarily equal to the length of the grafted chains because they can interpenetrate. 2Rhs must be regarded as the minimal distance of approach of the centers of two silica particles. The hard sphere volume fraction may be calculated from the number density of particles Np (eq 6):
Φhs ) Np
∫0∞ 34πRhs3f(R) dR
(10)
In this case, there are four parameters to determine: the mean radius of the silica particles, the Schulz width factor z, the volume fraction of the silica particles, and the thickness of the impenetrable layer ∆. Figure 4 shows that the positions of the peaks and their intensities are well described by the model. The representation of intensity on a linear scale does not display the part of the curve where I is small. On a double logarithmic scale, Figure 5 shows that the experimental results are also well described when q is large. Table 2 gives the values of the parameters obtained for three silica volume fractions. We observe that the mean radii of the silica particles, as well as the Schulz parameters, are roughly equal to those calculated in the case of samples with low volume fractions. We also observe that ∆ is quite independent of the silica volume fraction. As already mentioned, in the case of the Si1g samples, no peak is observed in the I(q) curves for comparable silica volume
Figure 7. Variations of the storage modulus as a function of the angular frequency of Si2g-based nanocomposites.
fractions. That is simply due to the larger size of the particles. This peak should exist for q values outside the accessible experimental range.
4. Rheology 4.1. Results. The results of spectromechanical measurements, performed on Si2g samples of various volume fractions, are presented in Figure 6. For clarity, the curves have been arbitrarily shifted along the y axis. Figure 7 represents G′ versus ω for the different samples studied. This Figure shows very clearly that independently of the volume fraction the nanocomposites show high-frequency behavior close to that of the polymer matrix. Major differences are observed at intermediate and low frequencies. For the lowest volume fraction (1.8%), a secondary relaxation process appears at low frequencies and is characterized by an inflection in the storage modulus G′. The 2.5% sample exhibits intermediate properties. However, it is important to note that the secondary relaxation seems to be strongly enhanced, with gellike behavior (G′ ) G′′ ∝ ω0.5) in the intermediate frequency region. The 4.7% suspension no longer exhibits a flow domain, but a predominantly elastic behavior appears in the low-frequency region. This solidlike behavior has already been observed for suspensions of grafted latex particles,13 for filled polymers and suspensions of grafted silica particles,9,12,30 and for diblock copolymers.16,28 It is usually attributed to the formation of a 3D network. The Cole-Cole representation (η′′ vs η′) provides a better display of the various relaxations. Figure 8 shows the ColeCole curves of the samples investigated by spectromechanics in Figure 6. It gives further indications of the appearance of the secondary relaxation processes.
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Figure 8. Cole-Cole representation of spectromechanical results of Si2g-based nanocomposites. The solid line corresponds to the double Cole-Cole fitting. Table 3. Parameters of the PBA and Secondary Relaxations PBA relaxation samples
η1 (Pa s)
PBA PBA + 1.8% of Si2g PBA + 2.5% of Si2g
2026
τ1 (s)
secondary relaxation R1
0.0386 0.715
η2 (Pa s) τ2 (s) 1998 2470
R2
87.8 0.632 0.397 0.524
The comparison between the Cole-Cole diagrams of PBA and the 1.8% Si2g suspension clearly shows that the lowfrequency relaxation is distinct from that of the matrix and has a longer characteristic time. It is related to the presence of the grafted particles in the matrix. When increasing the volume fraction to 2.5%, the relaxation of the matrix is screened by the process related to the presence of particles. This complex process can be divided into two parts: a high-frequency contribution, corresponding to the arc of the circle part of the curve, and a low-frequency contribution, corresponding to the straight line part of the diagram. We assumed that the relaxation of the matrix can be represented by the Cole-Cole equation (eq 11)
η* )
η 1 + (jωτ)R
(11)
where η is the amplitude of the process, τ is the relaxation time, and R is a parameter related to the width of the distribution of the relation times. The result of the fitting is very good, and the parameters obtained are displayed in Table 3. For the 1.8% sample, we assumed a double Cole-Cole relaxation. If, in the iteration process, the characteristic parameters of the first relaxation (matrix) are left free, then the results are those of the pure PBA, except for uncertainties. This is consistent with Figure 8, which shows that the relaxation of the matrix is only slightly modified. To decrease the number of adjustable parameters, we thus decided to impose for the first relaxation the results obtained for the pure PBA. For the 2.5% composite, a third process appears that is due to the existence of a percolative network. In addition, the characteristic time of the secondary process decreases, resulting in the superposition of the secondary relaxation with that of the matrix. The high-frequency part of the curve can be deconvoluted into two Cole-Cole relaxations, one of them being that of the PBA. In Figure 9, we plotted η′′ versus ω for the different relaxations: this representation gives the best image of the decoupling of the three processes.
Figure 9. Relaxation processes of the Si2g-based nanocomposite; φ ) 0.025%.
In the case of the 4.7% sample, the Cole-Cole diagram shows the beginning of an incomplete relaxation process, represented by the arc of circle, followed by solidlike behavior, represented by the straight line. In this case, it is not possible to deconvolute the high-frequency part of the curve because the solidlike behavior is predominant. 4.2. Discussion. The existence of secondary relaxation processes, similar to those which we highlight here, has already been observed in the literature in the case of various dispersions. It appears in dispersions of bare silica particles in polymers, where bulk polymer chains are adsorbed on the silica particles,31,32,50 in dispersions of grafted silica particles in a polymer matrix9,12,30,40 but also for blends of two viscoelastic polymers,51 for micelles of diblock copolymers dispersed in an elastomer,52 and for emulsions.53 Two interpretations are usually suggested to explain them. The first supposes that it is due to the form relaxation of the dispersed matter (droplets, micelles, etc.) or to the form relaxation of the grafted (adsorbed) polymer layer.31,51 The second assumption attributes this secondary relaxation to the diffusion of the dispersed objects in the matrix.12,40,52,53 From Table 3, one can notice that the characteristic time of the secondary relaxation decreases very rapidly when the silica volume fraction increases. However, if this secondary relaxation were due to the form relaxation of the layer, then a constant characteristic time should be observed because the thickness of the grafted layer is independent of the volume fraction. Moreover, we studied a dispersion of silica particles on which we grafted a low thickness polymer layer. The grafting density is high enough to prevent the adsorption of polymer on the surface of the spheres. In this case, we also observed a low-frequency relaxation. This supports the hypothesis that in our case the low-frequency process is not related to the relaxation of the polymer layer and could thus be explained in terms of the diffusion of the particles. As already mentioned, for the 2.5% Si2g sample, G′ and G′′ vary as ω0.5 in the intermediate part of the spectrum. Even if G′ is still slightly lower than G′′ at the lowest frequencies, we can reasonably estimate that 2.5% of the silica is very close to the percolation threshold. Knowing that the geometrical percolation threshold for hard spheres is 0.297,54 it is thus possible to calculate (50) Walberer, J. A.; McHugh, A. J. J. Rheol. 2001, 45, 187. (51) Graebling, D.; Muller, R.; Palierne, J.-F. Macromolecules 1993, 26, 320. (52) Watanabe, H.; Sato, T.; Osaki, K.; Hamersky, M. W.; Chapman, B. R.; Lodge, T. P. Macromolecules 1998, 31, 3740. (53) Dagre´ou, S.; Mendiboure, B.; Allal, A.; Marin, G.; Lachaise, J.; Marchal, P.; Choplin, L. J. Colloid Interface Sci. 2005, 282, 202. (54) Consiglio, R.; Baker, D. R.; Paul, G.; Stanley, H. E. Physica A 2003, 319, 49.
Polybutylacrylate Filled with Silica Particles
Langmuir, Vol. 22, No. 15, 2006 6689
at low frequency. Our filled polymers exhibit a strong correlation between the appearance of a structure factor as revealed by SANS experiments and the transition between liquidlike and solidlike behavior at low frequency. We can thus conclude that the existence of a structure factor seems to be the signature of 3D organization of the particles into the matrix. It is also important to underline that the strong correlation between the spatial organization and the rheological properties of suspensions is here demonstrated quantitatively.
5. Conclusions
Figure 10. Structure of the grafted silica particles.
the thickness of the layer giving this effective volume fraction when the silica volume fraction is 0.025. For a silica particle with a mean radius of 11.5 nm, we obtain a layer thickness of 14.7 nm. From the SANS measurements, we calculated the hard sphere radius and a thickness of the impenetrable layer of about 11 nm. This implies that, at the percolation threshold, the polymer shell of our grafted silica particles presents a penetrable part of 3.7 nm thickness. A scheme of the structure of the grafted silica particles is given in Figure 10. The scattering curves obtained by SANS do not display any structure factor for silica volume fractions lower than 1.8%. All of these samples exhibit liquidlike behavior at low frequency. When the silica volume fraction is increased to 2.5%, we observe both the appearance of a structure factor in SANS experiments and a transition toward gellike behavior. Finally, at 4.7% grafted silica particles, we observe a strong structure peak in the SANS results, whereas the suspension behaves like a viscoelastic solid
In this work, we studied by SANS and rheological measurements the structure of dispersions of grafted silica particles in PBA. From the SANS results, we showed that the silica particles are slightly polydisperse spheres, and we determined their average radius (11.5 nm). In the low-q range, we showed that there is no significant increase in the scattering intensity: we can thus reasonably assume that there is no formation of aggregates during the sample preparation. From the results of spectromechanical experiments, presented both in direct (G′(ω), G′′(ω)) and ColeCole (η′′(η′)) representations, we can assert that the addition of grafted spherical silica particles to the PBA matrix results in the appearance of a low-frequency secondary relaxation, distinct from that of the matrix. We show that this relaxation is not due to the shape relaxation of the grafted layer but could rather be explained in terms of the diffusion of the particles. For a volume fraction corresponding to the geometrical percolation threshold of hard spheres, the composites present a transition from liquidlike to solidlike behavior. This transition is accompanied by the appearance of a structure peak in I(q) curves: this last result supports the idea of the existence of a strong correlation between the 3D organization of the dispersed particles in the matrix and the rheological properties of the suspensions. Acknowledgment. We thank Dr. M. H. Deville from the ICMCB (Universite´ de Bordeaux I, France) for the TEM image. R.I. also thanks the Conseil Re´gional d’Aquitaine for allowing him to perform this work. LA0607003
