pubs.acs.org/Langmuir © 2009 American Chemical Society
Viscoelasticity and Structure of Polystyrene/Fumed Silica Nanocomposites: Filler Network and Hydrodynamic Contributions Giovanni Filippone,† Giovanni Romeo,†,‡ and Domenico Acierno*,† †
Department of Material Engineering and Production, University of Naples Federico II, P.le Tecchio 80, 80125 Naples, Italy (INSTM Consortium - UdR Naples) and ‡Department of Pharmaceutical Sciences, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano (Sa), Italy Received July 27, 2009. Revised Manuscript Received November 18, 2009
We investigate the relationships between structure and linear viscoelasticity of a model polymer nanocomposite system based on a mixture of fumed silica nanoparticles and polystyrene. Alterations in the viscoelastic behavior are attributed to the structuring of primary silica aggregates. Above a critical filler volume fraction, a space-filling network builds up as the result of cluster aggregation, and the complex frequency-dependence of the moduli is simplified by splitting the viscoelasticity of the composites into the independent responses of the suspending polymer melt and the filler network. Specifically, we present a refinement of a two-component model recently proposed for attractive colloidal suspensions, in which hydrodynamic effects related to the presence of the filler are properly taken into account using the concept of shear stress equivalent deformation. Our approach, validated through the building of a master curve of the elastic modulus for samples of different composition, allows the estimation of the elasticity of samples in which the filler network is too tenuous to be appreciated through a simple frequency scan. In addition, the structure of the filler network is studied using both the percolation and fractal approaches, and the reliability of the critical parameters is discussed. We expect that our analysis may be useful for understanding the behavior of a wide variety of complex fluids where the elasticity of the components may be superimposed.
Introduction Adding solid particles to polymers represents a popular way to produce tailored materials with new, enhanced properties with respect to the unfilled matrix. The resulting particulate composite can be depicted as a suspension of filler particles and/or agglomerates interspersed within the polymer medium. In a filled polymer containing micrometer-sized particles, high filler volume fractions are generally required to get significant changes of the macroscopic behavior. On the other hand, when the filler size is of the order of a few nanometers, considerable enhancements of mechanical, thermal, and transport properties can be potentially obtained even for very low filler contents.1 Such improvements are the result of the high interface between the phases, resulting in micro- and mesostructures due to chain absorption to the particle surface and particle-particle interactions, which impede the modeling of the systems through continuum mechanics approaches. Since the viscoelastic response of multiphase systems is extremely sensitive to structure, particle size, shape, and surface characteristics, rheology represents a powerful investigation tool to inquire about the state of dispersion of the filler and its interactions with the suspending medium. For low amounts of noninteracting particles, an increase of the linear viscoelastic moduli is generally observed over the whole range of frequencies.2 This is due to the enhanced local deformation caused by the presence of the solid phase. An additional increase of the low frequency moduli occurs when the particles show pronounced colloidal interactions. For filler loadings high enough, attractive colloidal particles form a space-filling network or gel that builds *To whom correspondence should be addressed. E-mail:
[email protected]. Telephone: þ39 081 7682268. Fax: þ39 081 7682404. (1) Hussain, F.; Hojjati, M.; Okamoto, M.; Gorga, R. E. J. Compos. Mater. 2006, 40, 1511. (2) Barnes, H. A. Rheology Reviews 2003, 1, 1–36.
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up as the result of particle aggregation. Since such a gel confers to the system new scattering, conductive, and elastic properties, a considerable interest exists in understanding the network structure in order to control it for engineering applications. Rheological data can be used to study the elasticity of aggregate networks essentially in two ways. By using a percolation model, the network elasticity is expected to exhibit critical behavior around the percolation threshold volume fraction.3 On the other hand, modeling the aggregates as fractal objects leads to scaling relations predicting power-law dependencies of the elastic properties on particle volume fraction, and the powerlaw exponents are related to the fractal dimensions of the filler network.4 Among different additives, fumed silica is a widely used filler made of primary particles of few nanometers fused together in aggregates, whose dimensions span from tens to hundreds of nanometers. If the surface of the particles is untreated, attractive interparticle forces are important when the particles are dispersed in a polymer melt. If the viscosity of the suspending medium is low enough, Brownian motion of the clusters becomes relevant and leads to formation of agglomerates, which eventually assemble to form a space-spanning network of clusters. As a consequence, the appearance of a low frequency plateau of the storage modulus, G0 , is generally observed. Although a wide range of literature exists dealing with the linear viscoelastic moduli of polymer-based nanocomposites, general physical models able to describe their frequency response as function of filler volume fraction are still scarce. This is not surprising, since continuum rheological models, successfully capturing the main features of polymers filled with micrometer-sized particles, generally fail when applied to (3) Stauffer, D.; Aharony, A. Introduction to Percolation Theory; Taylor & Francis: London, 1992. (4) Shih, W. H.; Shih, W. Y.; Kim, S. I.; Liu, J.; Aksay, I. A. Phys. Rev. A 1990, 42, 4772.
Published on Web 12/11/2009
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nanostructured systems.5 On the other hand, rheological models represent the key relating the structure to the final macroscopic properties of the material. A two-component model, first proposed to describe the linear viscoelasticity of aggregating Newtonian suspensions,6,7 has been recently extended to a polymerbased nanocomposite with negligible polymer-particle interactions.8 The underlying physics of this model lies on the independent rheological responses of the polymer and the particle network.9 For times longer than the polymer relaxation time, the only contribution to G0 comes from the filler network. This elastic contribution increases with particle volume fraction Φ, and its crossing with the matrix loss modulus allows for the scaling of the frequency-dependent moduli of samples at different filler contents on a single pair of master curves. Although the scaling works well, there are still unresolved issues regarding the interpretation and correctness of the values used to scale the curves. In the present Article, we extend such an approach to describe the frequency and particle volume fraction dependences of the viscoelastic moduli of a model nanocomposite system obtained by adding small amounts (Φ e 4.1%) of fumed silica (SiO2) nanoparticles into a polystyrene (PS) matrix. As negligible polymer-particle interactions characterize this system, alterations in its linear viscoelastic behavior are attributed to the structuring of the pristine SiO2 aggregates. Once hydrodynamic contributions related to the presence of the filler are removed, the physical meaning of the two-component model becomes evident. This is reflected in the excellent scaling of the G0 curves of samples at different Φ. In addition, the scaling allows estimating the elasticity of networks too tenuous to be appreciated through a frequency scan. Finally, the structure and elasticity of the particle network are described using both percolation and fractal approaches. We discuss the reliability of the parameters of the models, emphasizing their remarkable sensitivity to the accuracy of the experimental data.
were dried under vacuum for 16 h at T = 70 °C. The pure PS used as reference material for rheological investigations was extruded under the same conditions. The internal morphology of the filled systems was inspected using a transmission electron microscope (TEM, model EM 208, Philips). The thickness of the samples, randomly cut from the extruded pellets using a diamond knife at room temperature, was ∼150 nm. Rheological tests were carried out by means of a stresscontrolled rotational rheometer (ARG2, TA Instruments) using parallel plate geometry (plate diameter = 25 mm). All the measurements were performed in dry nitrogen at T = 200 °C. Preliminary oscillatory strain scans were performed on each sample at a fixed frequency ω = 0.063 rad/s in order to determine the maximum strain amplitude for linearity, γcr. Low-frequency (ω = 0.063 rad/s) time-sweep experiments were performed to investigate the temporal evolution of the linear viscoelastic properties. The frequency-dependent viscoelastic moduli, G0 and G00 , were measured through oscillatory shear frequency scans in the linear regime (γ < γcr). Due to the marked sensitivity of the sample rheology on filler content, thermogravimetrical analyses (TGA) were performed on each sample at the end of rheological tests to evaluate the actual amount of the inorganic phase. The samples were heated in a dry nitrogen atmosphere from room temperature up to 600 at 10 °C/min, and the residuals at 600 °C were recorded. Fourier transform infrared (FT-IR) spectra were collected in the MIR wavenumber range (4000-400 cm-1) at a resolution of 4 cm-1 by using a Perkin-Elmer System 2000 spectrometer. The measurements on the filler were carried out on disks obtained by compressing the pristine SiO2 (5 wt %) with potassium bromide (KBr). The spectra of the neat PS and the composite at 5 wt % SiO2 were performed on 25 μm thick disks prepared through compression molding at 200 °C. The polymer-based samples were annealed for 3 h at 200 °C under nitrogen to allow meaningful comparison with the samples used for rheological analyses.
Results and Discussion Materials and Methods The polymer matrix of our systems is an atactic polystyrene (PS, kindly supplied by Polimeri Europa) in the entangled regime, having average molecular weight Mw = 125 KDa, polydispersity index Mw/Mn = 2, zero-shear rate viscosity η0 = 1.7 103 Pa 3 s at temperature T = 200 °C, and glass transition temperature Tg = 100 °C. The filler is fumed silica nanoparticles (SiO2 Aerosil 150 by Degussa) with density F = 2.2 g/cm3, specific surface area 150 ( 15 m2/g, and average diameter of the primary particle d = 14 nm. The particles are prepared through a flame hydrolysis process, giving rise to spherical particles of a few nanometers in diameter which collide while hot and irreversibly fuse into aggregates with sizes of the order of ∼100 nm. Samples at different composition (filler volume fractions up to Φ = 0.041) were prepared by melt compounding the constituents using a counter-rotating intermeshing twin-screw extruder (Minilab Microcompounder by Thermohaake) equipped with a capillary die (diameter = 2 mm). A feedback chamber allowed accurate control of the residence time, which was set to 4 min for all the samples. The extrusions were all performed at T = 200 °C in nitrogen atmosphere to prevent thermo-oxidative degradation phenomena, and the screw speed was 100 rpm, corresponding to shear rates of the order of ∼50 s-1. Prior to mixing, the materials (5) Zhang, Q.; Archer, L. A. Langmuir 2002, 18, 10435. (6) Trappe, V.; Weitz, D. A. Phys. Rev. Lett. 2000, 85, 449. (7) Prasad, V.; Trappe, V.; Dinsmore, A. D.; Segre, P. N.; Cipelletti, L.; Weitz, D. A. Faraday Discuss. 2003, 123, 1. (8) Romeo, G.; Filippone, G.; Fernandez-Nieves, A.; Russo, P.; Acierno, D. Rheol. Acta 2008, 47, 989. (9) Trappe, V.; Prasad, V.; Cipelletti, L.; Segre, P. N.; Weitz, D. A. Nature 2001, 411, 772.
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The FT-IR spectra of the neat PS, the sample PS/SiO2 at Φ = 2.2%, and the pristine SiO2 are shown in Figure 1. In order to inquire about polymer-filler interactions, we employed difference spectroscopy subtracting out the spectrum of the neat PS from that of the composite. The overall shape of the difference spectrum PS/SiO2-PS is remarkably similar to that of the pristine SiO2, and only the peaks related to the vibrational modes of the oxygen atoms with respect to the silicon atom pairs which they bridge can be noticed (rocking, 475 cm-1; symmetrical stretching, 812 cm-1; antisymmetrical stretching, 1110-1200 cm-1).10 As noticeable distortions are expected in the difference spectrum when relevant interactions establish between the polymer and the filler surface,11 we can assert that the interactions between the residual silanol groups at the surface of SiO2 nanoparticles and the benzene ring of the PS chains are negligible with respect to those setting up among the particles. The quenched internal morphology of the composite at Φ = 3.6% soon after the extrusion process is shown in the TEM micrographs of Figure 2a. Branched flocs from tens to hundreds of nanometers appear well distributed on the microscale. The magnification of one of these clusters shown in Figure 2b reveals an open arrangement of the particles, which is typical of fractal structures. In such structures the mass M scales with floc size L as M ∼ Ldf, where df is the fractal dimension.12 The value of df (10) Kirk, C. T. Phys. Rev. B 1988, 38, 1255. (11) Musto, P.; Abbate, M.; Lavorgna, M.; Ragosta, G.; Scarinzi, G. Polymer 2006, 47, 6172. (12) Weitz, D. A.; Oliveira, M. Phys. Rev. Lett. 1984, 52, 1433.
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Figure 1. FT-IR spectra of PS, PS/SiO2 at Φ = 2.2%, neat SiO2, and difference spectrum obtained by subtracting the spectrum of the neat PS from that of the filled sample.
Figure 2. (a) TEM micrograph of the sample at Φ = 3.6% soon after the extrusion. (b) Magnification of an aggregate.
depends on the mechanism of particle aggregation.13 The fractal dimension of flame-made silica aggregates is rather invariant on process parameters, and recent ultrasmall angle X-ray scattering analyses indicate df ≈ 2.2 for flame-made silica nanoparticles such those used in this work.14 The mesostructure shown in Figure 2a, resulting from the shear and elongational stresses experienced by the materials during extrusion, is far from equilibrium, and the clusters tend to flocculate in the melt. In Newtonian dispersions of aggregating colloids, the linear viscoelastic moduli increase during time, reflecting the rearrangements of the particles, eventually resulting (13) Weitz, D. A.; Huang, J. S.; Lin, M. Y.; Sung, J. Phys. Rev. Lett. 1985, 54, 1416. (14) Kammler, H. K.; Beucage, G.; Mueller, R.; Pratsinis, S. E. Langmuir 2004, 20, 1915.
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in the formation of a whole space-spanning filler network above a critical volume fraction.15,16 A similar network is well-known to build up in filled elastomers17 as well as in polymer-based nanocomposites above the matrix melting or glass transition temperature.18-20 We monitor the clustering in our systems by measuring the temporal evolutions of G0 and G00 during an oscillatory shear in the linear regime at ω = 0.063 rad/s. At this frequency, the polymer can be considered fully relaxed, so that changes in the moduli during time are related to structural rearrangements of the solid phase. The results at 200 °C are plotted in Figure 3a for samples at different compositions. The elastic modulus increases with time at all volume fractions, and the growth rate progressively decreases until a time-independent value is reached. The aging reflects the flocculation of the primary SiO2 aggregates resulting from the extrusion process. This is shown in the TEM micrograph of Figure 3b, which was taken on a sample at the same composition as in Figure 2a after 4 h of aging at 200 °C. The pristine aggregates of Figure 2a assemble into bigger structures. Flocculation causes the enhancement of the elasticity due to two distinct mechanisms: first, the ramified character of flocs increases the effective filler volume fraction; in addition, an extra effect due to the interactions between contiguous flocs occurs at Φ above a critical threshold for the structuring of a space-spanning, stress-bearing filler network.2,21 To address the effect of nanoparticle concentration over the viscoelasticity of the samples, we measured G0 and G00 as a function of ω for 0 e Φ e 4.1% and show the results in Figure 4. The pure PS is predominantly viscous, with the loss modulus being higher than the elastic one throughout the frequency range investigated. The neat polymer exhibits Maxwellian behavior (G0 PS ∼ ω2, G00 PS ∼ ω1) at low ω, with a terminal relaxation time τ = limωf0(G0 /ωG00 ) = 10-1 s. Both moduli of the filled samples monotonically increase with Φ in the whole range of frequencies. (15) Russel, W. B.; Saville, D. A.; Schowalter W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (16) Cipelletti, L.; Manley, S.; Ball, R. C.; Weitz, D. A. Phys. Rev. Lett. 2000, 84, 2275. (17) Heinrich, G.; Kl€uppel, M. Adv. Polym. Sci. 2002, 160, 1. (18) Galgali, G.; Ramesh, C.; Lele, A. Macromolecules 2001, 34, 852. (19) Reichert, P.; Hoffman, B.; Bock, T.; Thomann, R.; M€ullhaupt, R.; Friedrich, C. Macromol. Rapid Commun. 2001, 22, 519. (20) Huang, Y. Y.; Ahir, S. V.; Terentjev, E. M. Phys. Rev. B 2006, 73, 125422. (21) Acierno, D.; Filippone, G.; Romeo, G.; Russo, P. Macromol. Mater. Eng. 2007, 292, 347.
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Figure 5. Amplification factor B as a function of Φ (b) or Φ h (O). The amplification factors for silicon-oil-based suspensions filled with micrometer-sized glass beads ( ), quartz (þ), and limestone (Δ) particles are reported as a function of Φ for comparison (data taken from Gleissle and Hochstein in ref 22). The line represents the increase of the relative zero-shear rate viscosity of a Newtonian suspension of hard spheres as a function of Φ.24 The complex modulus G* of the samples is shown in the inset as a function of frequency; the symbols are the same as those in Figure 4.
Figure 3. (a) Evolution during time of G0 of the neat PS (;) and
filled samples at different compositions; from bottom to top: Φ = 1%, 1.7%, 2.2%, 2.9%, and 3.6%. (b) TEM micrographs of the hybrids at Φ = 3.6% after 4 h of aging at 200°C.
Figure 4. G0 (a) and G00 (b) of the neat PS (;) and filled samples at different compositions; from bottom to top: Φ = 1%, 1.3%, 1.7%, 2.2%, 2.3%, 2.4%, 2.9%, 3%, 3.6%, and 4.1%.
The faster relaxation modes, however, are essentially those of the unfilled polymer, the filler only causing a vertical shift of the curves for ω greater than ∼101 rad/s. Gleissle and Hochstein Langmuir 2010, 26(4), 2714–2720
accounted for a similar behavior in non-Newtonian liquids filled with micrometer-sized particles by introducing the concept of shear stress equivalent deformation: the rigid particles reduce the effective gap distance available for the suspending medium by an amount proportional to the filler content.22 As a consequence, the inner shear amplitude is higher than that externally imposed, and the measured complex modulus, G* = (G0 2 þ G00 2)1/2, increases for all frequencies by an amplification factor B = B(Φ). We argue that a similar hydrodynamic approach holds true for our systems for frequencies greater than ∼20 rad/s, that is, the range of ω in which the polymer governs the rheological response. Hence, we evaluate B in this range as B(Φ) = G*(Φ)/G*PS. In order to test the consistency of the adopted procedure, in Figure 5, we compare our B(Φ) values with those reported by Gleissle and Hochestein for silicon oils filled with micrometersized particles.22 The latter data result in much lower values than those we estimated for our nanocomposites. To account for such discrepancy, we observe that the fractal feature of the SiO2 aggregates implies an effective filler volume fraction Φ h higher than Φ. Therefore, quantitative agreement with the amplification factor by Gleissle and Hochstein can be attained assuming the aggregates as nondraining objects, and recasting the hydrodynamic contribution in terms of Φ h, which can be estimated as the ratio of Φ to the solid fraction of the fractal clusters, Φ h =Φ/ (d/L)3-df.23 Hence, using df = 2.2 as a typical fractal dimension of our fumed silica aggregates14 and taking d/L ≈ 0.1 as deduced from the analysis of several TEM micrographs, we get amplification factors comparable with those by Gleissle and Hochstein. This supports the assumption that the polymer does not flow through the pores of the aggregates but rather moves around them. In this sense, the aggregates behave as hard spheres, causing an increase of the viscosity proportional to their volume fraction Φ h. As a matter of fact, the empirical relationship η0(Φ)/μ = (1 - Φ/0.63)-2, (22) Gleissle, W.; Hochstein, B. J. Rheol. 2003, 47, 897. (23) Wolthers, W.; van den Ende, D.; Bredveld, V.; Duits, M. H. G.; Potanin, A. A.; Wientjens, R. H. W.; Mellema, J. Phys. Rev. E 1997, 56, 5726.
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describing the increase of the relative zero-shear rate viscosity of a Newtonian suspension of hard spheres due to hydrodynamic interactions,24 follows the same trend as our amplification factor B(Φ h). Such agreement is quite surprising, since our samples at the highest Φ are characterized by the presence of a space-spanning filler network, as will be discussed soon. This result suggests that, at least in the frequency region where the response is dominated by the suspending medium, the hydrodynamic disturbance depends more on the effective volume fraction Φh than on the way the clusters are arranged in the melt. Over time scales long enough, the matrix relaxes and the filler dominates the viscoelastic response (shaded zone of the inset of Figure 5). In this region, both G0 and G00 exhibit diminished frequency-dependences when compared to the PS, indicating the occurrence of non-hydrodynamic stress contributions. The filler mainly affects the storage modulus, which exhibits a clear plateau G00(Φ) at low ω for Φ g 2.2%. Due to the negligible polymerparticle interactions, such a truly solidlike behavior supports the idea of an elastic, space-spanning filler network setting up above a critical volume fraction Φc. Interspersed throughout this structure is the suspending polymer phase, whose intrinsic viscoelastic feature mixes with the network dynamics giving rise to the complex ω- and Φ-dependences shown in Figure 4. As a consequence of the different relaxation time scales, the rheological properties of the samples for Φ g Φc can be rationalized with a two-component model, first proposed for colloidal gels in Newtonian fluids, which combines the elasticity of the particle network and the viscosity of the suspending medium.6,7 For these systems, the linear viscoelastic moduli of samples at different Φ can be scaled onto a single pair of master curves, and the analysis of the shift factors reveals that the Φ-dependent elastic modulus scales along the viscosity of the neat matrix. We have recently shown that the two-component model allows describing the elasticity and dynamics of titanium dioxide nanoparticle gels in a polypropylene melt.8 Until now, however, hydrodynamic effects have not been taken into account, and the curves have been generally scaled with respect to the point at which the network elasticity, identified by the low-frequency elastic modulus G00(Φ), equals the viscous feature of the suspension, G00 (Φ) (point (a0 , b) of Figure 6).6,7 This leads to a quite good collapse of the data onto a single master curve. Despite this, such a procedure does not reflect the physical meaning of the model, as the elasticity of the filler network should be scaled along the merely ω-dependent loss modulus of the neat matrix. In addition, for an accurate scaling, we argue that the hydrodynamic effects related to the presence of the solid phase cannot be eluded. Consequently, we multiply the loss modulus of the neat polymer by the B(Φ) of Figure 5; then we identify the horizontal (a) and vertical (b) shift factors to build the master curve of G0 as the coordinates of the point in which the network elasticity G00(Φ), conventionally defined as the elastic modulus at the lowest frequency investigated, equals the viscous feature of the matrix corrected for the hydrodynamic effects, B(Φ)G00 PS. An example of how a and b are obtained for the sample at Φ = 2.9% is shown in Figure 6. We test the validity of our approach by scaling the G0 curves of the samples at Φ g 2.2%, in which the existence of the particle network can be suggested by the presence of a clear low-frequency plateau of the elastic modulus. The resulting master curve shown in Figure 7 (open symbols) supports the approach adopted, and (24) De Kruif, C. G.; van Iersel, E. M. F.; Vrij, A.; Russel, W. B. J. Chem. Phys. 1986, 83, 4717.
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Figure 6. Example of evaluation of the scaling factors to build the master curve of G0 . The triangles represent G0 (solid symbols) and G00 (open symbols) of the sample at Φ = 2.9%; the dashed line is the loss modulus of the neat PS, G00 PS; the solid line is the amplified loss modulus of the PS, B(Φ)G00 PS. The horizontal (a, a0 , and a00 ) and vertical (b) shift factors to build the master curve of G0 are pointed out.
Figure 7. Empty symbols: master curve of G0 of the samples at Φ g 2.2%, built using a and b as shift factors; symbols are the same as those in Figure 4. Solid tilted squares are the elastic modulus of the sample at Φ = 1.7% shifted to allow partial superposition with the master curve. In the inset, we compare the magnifications at low scaled frequencies of the curves obtained using as shift factors (a, b) (b), (a0 , b) (]), and (a00 , b) (4).
the scaled moduli lie on top of each other over more than five decades of scaled frequencies. The slight deviations at ω/a greater than ∼101 do not invalidate the consistency of the scaling, being a consequence of the intrinsic viscoelastic feature of the polymer matrix, which dominates the high-frequency behavior and whose relaxation spectrum is Φ-independent. The accuracy of our approach is supported by the analysis of the inset of Figure 7, where we compare the magnifications at low scaled frequencies of our master curve with those obtained by shifting the G0 curves with respect to the points at which the network elasticity equals the viscous modulus of the suspension (point (a0 , b) of Figure 6) and that of the neat matrix (point (a00 , b) of Figure 6). The points of both of these master curves are more scattered, confirming the importance of properly accounting for hydrodynamic contributions in order to emphasize the fundamental physics underlying the two-phase model.6 Physical insights about the scaling procedure arise once the inter-relations between the scaling factors a, b, and B(Φ) are analyzed. As shown in Figure 6, the elasticity of the network equals the viscous modulus of the suspending medium at the frequency ω = a, that is, b = G00 PS(ω)*B(Φ)|ω=a. As a consequence, the plot of b/B(Φ) versus a collapses onto the curve of G00 PS versus ω as shown in Figure 8. Langmuir 2010, 26(4), 2714–2720
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Article Table 1. Parameters K and ν Obtained for Different Values of Φc by Fitting the Relationship G00 = K(Φ - Φc)ν to the Experimental G00 Data of the Samples at Φ g 2.2%a Φc [%]
Figure 8. Comparison between the plots of G00 PS versus ω (;) and
K [Pa]
ν
R2
G00 @ Φ = 1.7% [Pa]b
0 20.2 ( 11 4.9 ( 0.5 0.9880 280 0.5 97 ( 35 4.2 ( 0.4 0.9875 215 1 413 ( 86 3.5 ( 0.3 0.9867 120 1.5 1527 ( 124 2.8 ( 0.2 0.9850 26 2 4670 ( 86 2.0 ( 0.2 0.9806 2.2 6771 ( 124 1.8 ( 0.1 0.9768 a 2 R is the regression of the fitting procedure. The estimations of G00 for the sample at Φ = 1.7%, only possible for the samples at Φc < 1.7%, are reported in the last column. b Estimated lining the relationship G00 = K(Φ - Φc)ν.
b/B(Φ) versus a (O).
Figure 10. Experimental Figure 9. Master curve of G00 of the samples at Φ g 2.2%, built using a and b as shift factors; symbols are the same as those in Figure 4.
The loss modulus of filled samples was directly involved in the scaling procedure used in our previous paper, in which the shift factors represented the coordinates of the low-frequency crossover point of the moduli of the filled samples.8 As a consequence, G0 and G00 shared the same scaling factors and collapsed quite well onto a pair of master curves. A similar approach was used in the papers by Trappe and Weitz6 and Prasad et al.,7 allowing for the building of a master curve for both moduli. Conversely, the scaling procedure used in this work does not involve the loss moduli of the filled samples, and the Φ-dependence of G00 cannot be described adequately by means of the same shift factors a and b used for G0 . This is shown in Figure 9, where, apart from the high frequency behavior dominated by the polymer, the scarce quality of the scaling of G00 emerges especially in the nonhydrodynamic regime, that is, at ω/a lower than ∼10-1. In this region, the loss modulus of the filled samples represents the dissipation of the filler network, and a targeted modeling of this phenomenon would be required in order to identify another couple of possible scaling factors for G00 . The two-component model only applies for samples with filler volume fractions Φ greater than a threshold value Φc, representing the minimum particle volume fraction necessary for the formation of a space-spanning network. Looking at the G0 curves of Figure 4a, we can assert that the samples at Φ g 2.2%, all exhibiting a clear low frequency plateau G00, are above this critical threshold, but for a thorough detection of Φc a more rigorous approach is required. The liquid-solid transition for suspensions of aggregating particles shares many peculiar features of chemical gelation, namely, the divergence of the longest relaxation time and power law relaxation spectrum.25 Using dynamic-mechanical (25) Winter, H. H.; Mours, M. Adv. Polym. Sci. 1997, 134, 165.
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low-frequency plateau modulus G00 as a function of Φ for the samples at Φ g 2.2% (open symbols) and fitting curves obtained using the relationship G00 = K(Φ Φc)ν for different values of Φc: Φc = 0% (- - -), 0.5% (---), 1% ( 3 3 3 ), 1.5% (;), 2% (- 3 -), and 2.2% ( 3 3 3 ). The solid point represents the estimation of G00 for the sample at Φ = 1.7% as deduced from the master curve of Figure 7.
spectroscopy to detect the sol-gel transition, Inoubli et al. estimated Φc ≈ 2.5% for polybutylacrylate-silica nanocomposites;26 similarly, Cassagnau found Φc = 3.3% for ethylene vinyl acetate copolymer-silica composites.27 Using light scattering methods, Piau et al. estimated Φc ≈ 1% for silica-silicone gels.28 On the other hand, percolation theories predict the elastic modulus of the particle network near the percolation threshold to grow in a critical fashion with Φ, G00 ∼ (Φ - Φc)ν, suggesting a simple way to estimate Φc.3 Following the percolation approach, we fit the relationship G00 = K(Φ - Φc)ν to the low-frequency elastic moduli of the samples at Φ g 2.2% setting K and ν as fitting parameters while keeping Φc constant. The procedure was repeated for different values of Φc, and the results are reported in Table 1 and plotted in Figure 10. A weak dependence of the regression R2 on the value of Φc can be noticed, suggesting that a mere best fitting approach is not a reliable method for the detection of the percolation threshold in our systems. Instead, we use the following procedure: once K and ν have been obtained with reference to the samples at Φ g 2.2, we are able to extrapolate a value of G00 also for the samples at lower Φ using the equation G00 = K(Φ - Φc)ν. The results of such an extrapolation are reported in the last column of Table 1 for the sample at Φ = 1.7%. On the other hand, from Figure 4a, we estimate that the maximum acceptable value of G00 for this sample (26) Inoubli, R.; Dagreou, S.; Lapp, A.; Billon, L.; Peyrelasse, J. Langmuir 2006, 22, 6683. (27) Cassagnau, P. Polymer 2003, 44, 2455. (28) Piau, J.-M.; Dorget, M.; Palierne, J.-F.; Pouchelon, A. J. Rheol. 1999, 43, 305.
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is ∼50 Pa, that is, the lowest value of G0 measured in the investigated frequency range. Thus, by comparing such a limit value with the extrapolated values of Table 1, we conclude that values of Φc lower than 1.5% must be discarded. On the other hand, we observe that R2 decreases for Φc greater than 1.5%. In the light of the previous observations, we can assert that Φc = 1.5% represents a reasonable, though approximate, estimate for the percolation threshold of our systems. This value is also in agreement with that found by Pouchelon and Vondracek29 through rheological analyses and the one inferred by Barthel30 from theoretical considerations. Further support to the reliability of our estimation of Φc comes from the analysis of the master curve. In fact, an important consequence of the scaling shown in Figure 7 is that the network elasticity can be estimated for all samples above Φc from the factor b required to vertically scale the data. This allows determining G00 even for very tenuous networks having a modulus too low to be directly measured in reasonable experimental times through a simple frequency scan.7 This is the case of the sample at Φ = 1.7%, which is above the estimated percolation threshold Φc = 1.5%, but whose plateau modulus cannot be directly inferred from the experimental data of Figure 4a. Hence, shifting the curve of this sample on the master curve (solid tilted squares in Figure 7), we can estimate its elasticity G00. Once added to the data of Figure 10, the extrapolated point (solid symbol) well follows the critical behavior for Φc = 1.5%, supporting the consistency of the adopted procedure. The value of the critical percolation exponent ν depends on the stress-bearing mechanism of the percolating filler network, and it ranges from ν ≈ 2.1 for systems in which the particles are free to rotate about each other to ν ≈ 3.75 in the case of networks in which the chains can resist stress either by stretching of bonds or by unbending the chain.31,32 For Φc = 1.5%, we find ν = 2.8, in line with the result by Grant and Russell, who found a universal response with ν = 3 ( 0.5 for suspensions of octadecyl silica particles irrespective of the strength of interparticle attraction.33 Looking at the data of Table 1, however, we stress the high sensitivity of the values of ν on Φc, which makes it difficult to draw unambiguous conclusions about the actual stress-bearing mechanism in the studied systems. Alternative approaches to infer valuable information about the structure of aggregating particles start from the consideration that the filler network transmits stress through the chains of the elastic backbone. Modeling the colloidal gel as resulting from the aggregation of fractal flocs, Shih and co-workers developed a scaling theory relating the elastic modulus and the limit of linearity of a gel far from the gelation threshold to its fractal dimension.4 Provided that the flocs are large enough to be considered as weak springs, the model predicts G00 ∼ ΦR, with R = (3 þ x)/(3 - df) and γcr ∼ Φ-β, with β = (1 þ x)/(3 - df); x represents the fractal dimension of the backbone of the flocs, and its value must be greater than unity to provide a connected path. To estimate the limit of linearity of our samples, the viscoelastic moduli were recorded at increasing oscillation amplitude γ at ω = 0.063 rad/s, that is, a frequency low enough to consider the response of the materials governed by the particle network. We conventionally set γcr as the strain at which G0 deviates 5% from its constant low-strain value. G00 and γcr for the samples at Φ g 1.7% are plotted as a function of Φ in Figure 11. Hence, fitting a power-law to the data at Φ g 2.4% (solid ymbols of Figure 10) to (29) (30) (31) (32) (33)
Pouchelon, A.; Vondracek, P. Rubber Chem. Technol. 1989, 62, 788. Barthel, H. Colloids Surf. 1995, 101, 217. Arbabi, S.; Sahimi, M. Phys. Rev. B 1993, 47, 695. Sahimi, M.; Arbabi, S. Phys. Rev. Lett. 1993, 47, 703. Grant, M. C.; Russel, W. B. Phys. Rev. E 1993, 47, 2606.
2720 DOI: 10.1021/la902755r
Filippone et al.
Figure 11. Low-frequency plateau modulus G00 (circles, left axis) and critical strain amplitude γcr (triangles, right axis) as a function of Φ. Solid and dashed lines are the power-law fitting curves for G00 and γcr, respectively. Fittings are restricted to the data for samples at Φ g 2.4% (solid symbol).
fulfill the requirement of Shih’s model to be far from Φc, we get R = 4.7 ( 0.3 and β = 2.5 ( 0.2, which in turn lead to df = 2.1 ( 0.2 and x = 1.3 ( 0.2. Our value of df is in good agreement with those found through rheological techniques by Yzquiel et al.34 and Paquien et al.35 for silica-based suspensions, and its reliability is consistent with recent ultrasmall angle X-ray scattering analyses.14 Moreover, the value of x indicates a slightly tortuous connection path between the flocs, which appears to be in reasonable agreement with the TEM micrograph shown in Figure 3b.
Conclusions The linear viscoelastic behavior of PS/SiO2 nanocomposites at low particle volume fractions has been investigated as a function of filler content. We have found a strong propensity to flocculation of the pristine silica aggregates generated during the melt compounding process. As a consequence, the filled samples display aging phenomena and a complex viscoelastic response. Irrespective of filler content, the host polymer dominates the high frequency behavior and mere hydrodynamic effects simply cause a vertical shift of the moduli. Differently, clear alterations emerge over longer time scales due to mechanical interactions among aggregates. A whole space-spanning, stress-bearing network forms at filler loadings higher than a critical volume fraction. Once accounted for hydrodynamic effects related to the filler through the concept of shear stress equivalent deformation, the contribution of the particle network to the elasticity of the samples clearly emerges and can be singly studied using a twocomponent model. A master curve of the elastic modulus is obtained by properly normalizing the elastic modulus of samples at different composition. The scaling parameters have a precise physical meaning that clearly emerges once their inter-relationships are analyzed. In addition, the master curve allows estimating the modulus of networks too tenuous to be appreciated through a simple frequency scan. The network structure has been studied using both the percolation and fractal approaches, which led to the estimation of relevant physical parameters such as the percolation threshold and fractal dimensions of the filler network. Acknowledgment. The authors acknowledge Dr. Pellegrino Musto of the National Reaserch Council of Italy for FT-IR spectra and for useful discussions, and Dr. Dino Ferri from Polimeri Europa for kindly supplying the polystyrene matrix. (34) Yzquiel, F.; Carreau, P. G.; Tanguy, P. A. Rheol. Acta 1999, 38, 14. (35) Paquien, J. N.; Galy, J.; Gerard, J. F.; Pouchelon, A. Colloids Surf., A 2005, 260, 165.
Langmuir 2010, 26(4), 2714–2720
