A Unifying Approach for the Linear Viscoelasticity of Polymer

Oct 25, 2012 - We validate our approach through the building of master curves of the elastic modulus of samples at different composition. Possible gen...
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A Unifying Approach for the Linear Viscoelasticity of Polymer Nanocomposites Giovanni Filippone* and Martina Salzano de Luna Department of Materials and Production Engineering (INSTM Consortium−UdR Naples), University of Naples Federico II, P.le Tecchio 80, 80125 Naples, Italy S Supporting Information *

ABSTRACT: When the filler content is higher than a critical threshold, flocculation of nanoparticles in polymer melts eventually results in three-dimensional networks of clusters. The marked elastic feature of such structures combines with that of the polymer melt giving rise to a complex dependence of the linear viscoelastic moduli on frequency and filler content. We analyze a wide variety of polymer nanocomposite systems and show that a unifying description of their viscoelasticity is possible irrespective of the nature of pristine nanoparticles and the degree of polymer−filler affinity. We validate our approach through the building of master curves of the elastic modulus of samples at different composition. Possible general trends in the stress bearing mechanisms of the different kinds of network considered are also discussed. Given its generality, the proposed analysis is expected to be useful to describe a wide variety of complex fluids in which a superposition of the elasticity of the components is possible. In Newtonian suspensions increasing the filler volume fraction Φ eventually results in the arrest of the particle dynamics. This can happen in several disparate ways, and always leads to a disordered solid. The simplest case is that of hard sphere suspensions, in which the interparticle interaction is zero at all separations and infinitely repulsive at contact. The crowding of hard spheres results in a colloidal glass, whose solid-like features originate from the trapping of individual particles within the cages formed by the nearest neighbors.2 Actually, attractive colloidal suspensions better reflect the behavior of PNCs, in which van der Waals attractive forces between particles and aggregates are generally of major importance. In such systems, aggregation results in disordered clusters of particles, which may or may not span the whole space depending on filler content, interaction potential and applied stress.3 If a colloidal three-dimensional gel forms, then the suspension viscoelasticity can be rationalized with a model that combines the elasticity of the disordered particle network and the Newtonian viscosity of the suspending liquid.4,5 Despite the complexities stemming from the intrinsic nonNewtonian feature of polymer matrices, we have recently proved that a similar approach can be successfully extended to simple PNC systems.6−8 In the present paper, we deal with the generalization of our approach, showing that it can be easily used to capture the linear viscoelasticity of a wide variety of

1. INTRODUCTION Adding solid particles to polymer matrices is a common way to reduce costs and impart desired structural and functional features to the host material. Filled polymers can be described as suspensions of particles and particle aggregates dispersed in the polymer medium. Interactions between aggregates and matrix, as well as between particles, hinder the material deformability modifying both the solid- and melt-state behavior. In polymer microcomposites such effects become noticeable only at relatively high filler contents, that is when the filler particles are sufficiently close to each other to form a network that spans large sections of the material. Over the last 2 decades, the same reinforcing effects have been observed using very small amounts of inorganic nanoparticles. This has resulted in extensive researches in the field of polymer-based nanocomposites (PNCs). In order to fully understand the unique properties of PNCs, the morphological and structural implications which stem from the nanometric sizes of the filler have to be taken into account. Dealing with nanoparticles implies very large polymer−particle interfacial areas, very small wall-to-wall interparticle distances and relevant filler mobility. Indeed, nanoparticles can experience significant Brownian motions even in highly viscous mediums such as polymer melts.1 The result is that, unlike polymer microcomposites, PNCs behave as “living systems” evolving toward more favorable states in relatively short time scales. This makes such a class of materials reminiscent of colloidal suspensions, which can be taken as a starting point to which new complexities can be added step by step. © 2012 American Chemical Society

Received: July 29, 2012 Revised: October 9, 2012 Published: October 25, 2012 8853

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moduli were measured as a function of frequency in the linear regime, which had been previously evaluated for each sample through strain amplitude tests. The frequency sweep experiments were performed at T = 190 °C for the PS/O-clay samples and at T = 180 °C for the PS/ MWCNT ones. Thermogravimetric analyses (TGA mod. Q5000 by TA Instruments) were performed on each sample at the end of rheological tests to evaluate its actual filler content. The samples were heated in a dry nitrogen atmosphere from room temperature up to T = 700 °C at 10 °C min−1, and the residuals were recorded at T = 600 °C.

PNCs which experience an arrest of the relaxation dynamics above a critical filler content Φc. To support the generality of our conclusions, we analyze experimental data both produced by ourselves and taken from the literature. In particular, we have deliberately selected a series of PNCs very different among them, showing that our analysis works irrespective of the nature of polymer and filler. Besides demonstrating that the complex Φ-dependent relaxation spectrum of PNCs can be simply described in terms of only two main families of dynamical species, our analysis allows for the accurate detection of the elasticity of very tenuous networks, which would be difficult to be detected through conventional linear viscoelastic analyses. In addition, we show that our approach leads to a confident identification of the filler percolation threshold. This allows us to compare the stress bearing mechanisms of the various kinds of elastic networks which form in the considered PNCs, looking for possible common trends. Finally, the possibility to extend our analysis to PNCs with glassy dynamics is briefly discussed.

3. RESULTS AND DISCUSSION 3.1. Network Formation and Two-Phase Model. Unless specific polymer−filler interactions establish, nanoparticles are generally difficult to be dispersed within polymer matrices. The hydrodynamic forces developed during intense melt mixing break up the initial aggregates down to small clusters of primary particles. Above the melting or glass transition temperature of the polymer matrix, however, the clusters are inclined to reassemble into bigger structures because of interparticle attraction. Such a flocculation process can be monitored by measuring the linear viscoelastic moduli at frequency ω low enough to consider negligible the elastic contribution of the matrix.1 The time evolution of G′ and G″ is reported in Figure 1 for four PNCs with negligible polymer−particle energetic

2. EXPERIMENTAL SECTION Details on the materials and on the preparation and characterization techniques of the PNCs taken from the literature are reported in the corresponding reference papers. The following experimental section refers to two series of PNCs based on polystyrene (PS) filled with either organo-modified montmorillonite (O-clay) or multiwalled carbon nanotubes (MWCNTs), which have not been reported in previous papers. 2.1. Raw Materials and PNC Preparation. The polymer matrix is atactic polystyrene (PS, kindly supplied by Polimeri Europa) with average molecular weight Mw = 125 KDa, polydispersity index Mw/Mn = 2, zero-shear rate viscosity η0 = 1.7 × 103 Pa s at temperature T = 200 °C and glass transition temperature Tg = 100 °C. The fillers are organo-modified clay and multiwalled carbon nanotubes. Specifically, the O-clay is Cloisite 15A by Southern Clay Products, a montmorillonite organo-modified with dimethyl dihydrogenatedtallow quaternary ammonium salt with organic content of ∼43% and a mass density ρ = 1.66 g cm−3. The MWCNTs were synthesized by fluidized bed chemical vapor deposition (FBCVD) using a γ-alumina substrate impregnated with iron as bed material, ethylene as carbon source and nitrogen as fluidizing agent;9 the purity attained is >99.5%, with residual iron and aluminum contents lower than 0.5 wt %. PNC samples at different content of O-clay (Φ up to 0.018) were prepared by melt mixing the polymer and the filler using a twin-screw microcompounder (Xplore by DSM). The extrusions were performed at T = 200 °C and screw speed ∼150 rpm, corresponding to average shear rates of ∼50 s−1. PNC samples at different content of MWCNTs (up to Φ = 0.0086, assuming the same density of pure graphite, ρ = 2.1 g cm−3) were prepared by a masterbatch melt mixing technique. First, a masterbatch with relatively high nanotube loading (Φ≈0.03) was prepared by melt compounding the constituents at T = 200 °C and screw speed ∼200 rpm; the resulting masterbatch was then melt compounded with neat PS in a second extrusion step for adjusting the composition. To minimize thermo-oxidative degradation phenomena, the extrusions were all performed under nitrogen atmosphere, and the polymer and the fillers were dried for ∼16 h at T = 95 °C prior to melt mixing. Finally, the extruded samples were granulated and compression-molded in the form of disks (diameter 40 mm, thickness ∼1.5 mm) suitable for rheological analyses. 2.2. Characterization. Rheological tests were carried out using a stress-controlled rotational rheometer (mod. ARG2 by TA Instruments) in parallel plate geometry (plate diameter 40 mm). Lowfrequency (ω = 0.1 rad s−1) time sweep experiments were performed on each sample to investigate the temporal evolution of the linear viscoelastic properties. The tests were carried out at T = 190 °C for the PS/O-clay samples and at T = 220 °C for the PS/MWCNT ones. Once reached the steady state, the elastic (G′) and viscous (G″) shear

Figure 1. Time evolution of G′ (full symbols) and G″ (empty symbols) at ω = 0.1 rad s−1 normalized over the corresponding values of the neat matrix for PNCs based on PS filled with different kinds of nanoparticles: fumed silica (SiO2) at Φ = 0.022 (diamonds), MWCNTs at Φ = 0.0033 (squares), graphite nanoplatelets (GNPs) at Φ = 0.036 (triangles), and O-clay (circles) at Φ = 0.012. The filler content of each sample is above the filler percolation threshold.

interactions. The samples are based on the same polystyrene matrix filled with different kinds of nanoparticles. The filler content is above the percolation threshold, which was estimated for each system as described in paragraph 3.4. The samples share the same qualitative behavior: the elastic modulus increases during the earlier stage, then it reaches a steady state value; a similar trend is noticed for the loss modulus, but the changes are much less pronounced. The differences in the growth kinetics reflect the different interparticle attractive forces and mobility of the flocculating clusters, ultimately related to their initial size and shape and melt temperature. Irrespective of the nature of the primary particles, however, once reached the steady state all the systems can be depicted as three-dimensional networks of nanoparticle clusters interspersed with the host polymer. We argue that such 8854

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simply causes an apparent increase of the viscoelastic moduli due to hydrodynamic interactions among the clusters. An empirical amplifying factor B(Φ) allows to account for the reduced gap available to the fluid because of the presence of the particles.11 In the case of PNCs, B(Φ) can be estimated as the ratio between the complex shear moduli, G*(ω), of filled sample and neat matrix in the high-frequency region.6 Once amplified by B(Φ), the G″(ω) of the neat polymer crosses the cluster network elasticity, bΦ = G′0(Φ), at a characteristic Φdependent frequency, aΦ, which sets the transition from the regime governed by the filler (ω < aΦ) to that dominated by the matrix (ω > aΦ) (Figure 3). Because of the independency of the responses of polymer and network, the higher the filler content, the stronger the network and, as a result, the wider the frequency range in which the latter prevails over the polymer. As a consequence, aΦ and bΦ can be respectively set as the horizontal and vertical shift factors for the building of a master curve of the elasticity of samples at different composition. Such a procedure has been successfully applied to two distinct series of PNCs without noticeable polymer−particle interactions.6,8 The two master curves taken from the corresponding reference papers are reported in Figure 4. The good quality of the scaling for both systems confirms the validity of the basic assumptions of the two-phase model. We stress that the overlay of the G′ curves is

a common structure of noninteracting PNCs may reflect in a similar macroscopic response. Specifically, the viscoelasticity of such systems arises from the combination of the response of the space-spanning network made of clusters of nanoparticles and that of the polymer matrix. Although essentially elastic, the network may exhibit slow dissipative relaxation dynamics because of space rearrangements of the clusters.10 Therefore, both the filler and matrix phases of PNCs are viscoelastic by nature. As a first approximation, however, we neglect the viscous connotation of the cluster network and the elastic one of the polymer. Under this assumption, the entire elasticity of the PNC derives from the filler and scales with its content, whereas the viscous connotation is totally encompassed in the unfilled matrix. A relevant consequence of such a physical picture, illustrated in Figure 2 and hereafter called “two-phase model”, is the

Figure 2. Schematic representation of the origin of viscoelasticity in PNC samples above the percolation threshold. The nature of the pristine nanoparticles is unessential for the underlying physics of the two-phase model.

possibility of separating the contributions of each one of the phases. As shown in Figure 3 for a representative PS-based sample filled with MWCNTs, at low frequency the PNC response is dominated by the cluster network, which exhibits a ω-independent elastic modulus G′0. On the other hand, the polymer dynamics prevail at high frequency, where the filler

Figure 3. Example of evaluation of the scaling factors to build the master curve of G′. Full triangles represent the elastic modulus of the PNC (PS/MWCNT at Φ = 0.0086), the dashed line is the viscous modulus of the unfilled polymer, the solid line is the viscous modulus of the polymer amplified by B(Φ) to account for hydrodynamic effects. The horizontal and vertical shift factors to build the master curve of G′ are pointed out. The elastic modulus of the unfilled polymer (dotted gray line) and the viscous modulus of the PNC (empty gray triangles) are shown as well.

Figure 4. Master curves of G′ of (a) PS/SiO2 fumed and (b) PS/GNP samples. Data are taken from refs 6 and 8, respectively. The insets show representative TEM images of clusters of nanoparticles embedded in the PS matrix. 8855

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3.2. Generalizing the Two-Phase Model. Nanoparticles Nature. According to Figure 2 we assume the existence of two main populations of dynamical species: the unfilled polymer, whose dynamics, hydrodynamic effects apart, are not affected by the filler, and the elastic three-dimensional network made by clusters of nanoparticles, which is unable to relax in the investigated time window. As a consequence, we should be able to scale the elastic moduli of different kinds of PNCs irrespective of the size and shape of the pristine particles, which simply represent the elementary building blocks of the clusters of which the network is made. We also observe that, if negligible polymer−particle interactions exist, then the network elasticity originates from mechanical interactions among bare clusters. In such a case the temperature is expected not to affect the network modulus appreciably, whereas it clearly influences the PNC high-frequency behavior dominated by the polymer. This is shown in Figure 6a for a representative PS-based sample at Φ = 0.0086 of MWCNTs: the low-frequency plateau does not change with temperature, so shifting the curves along the log frequency axis by a factor aT causes their overlap (Figure 6b); in addition, using the same empirical shift factors we generate a master curve of the linear viscoelastic moduli of the neat polymer according to the time−temperature superposition principle (Figure 6c), which confirms the independence of the responses of cluster network and unfilled polymer. Therefore, increasing the temperature in PNCs characterized by weak polymer−particle interactions has the same effect as increasing the filler content: it results in a widening of the frequency range in which the cluster network prevails over the matrix. As a result, the master curves of G′ produced at different temperatures can be correctly compared. This is done in Figure 7 for four PNC systems based on the same polystyrene matrix filled with different types of noninteracting nanoparticles. The data sets concerning the samples containing fumed silica (SiO2) and graphite nanoplatelets (GNPs) have been already presented and discussed in previous works;6,8 the master curves for the samples based on O-clay and MWCNTs have been built using the procedure described above. When the network elasticity was to low to intercept the amplified G″(ω) of the unfilled polymer within the investigated frequency range, terminal Maxwellian behavior was assumed for the neat matrix, whose loss modulus was prolonged at low frequency according to the scaling law G″ ∼ ω1. The numerical values of B(Φ) and the pairs (aΦ; bΦ) used to build the master curves are listed in the Supporting Information (Table S1). The G″(ω) curves of the pure PS matrix at the various temperatures are shown as well together with the shift factors lying on them (Figure S2, Supporting Information). All the data collapse into a single master curve, which means that common dynamics characterize the considered PNCs irrespective of the nature of the primary particles. Specifically, a frequency-independent elasticity emerges at ω/aΦ ≪ 1, the only contribution to the PNC elasticity stemming from the cluster network. In contrast, at ω/aΦ ≫ 1 the polymer elasticity dominates the response, and all the samples share the relaxation dynamics of the common PS matrix. As a consequence, the nonscalable tails at high scaled frequency reflect the polymer relaxation processes and do not invalidate the good overall quality of the scaling. The transition between the two regimes occurs as a gradual slowing down of the relaxation processes. According to the two-phase model, at ω/aΦ ∼ 1 cluster network and polymer matrix equally contribute to the PNC elasticity. Actually, the nanoparticles slow down the polymer

not obtained by looking for their partial superposition, being instead a direct consequence of the precise physical meaning of the shift factors. Vice versa, once built by referring to the samples at Φ high enough to clearly identify the G′0 of the corresponding networks, the master curve can be exploited to infer the elasticity of tenuous networks, whose strength is instead too low to be easily detected through usual viscoelastic analyses. Such a predictive feature of the master curve is shown in Figure 5a for a representative PS/MWCNT sample. The

Figure 5. Possible scaled curves of G′ in accordance to the two-phase model for PS/MWCNT samples at Φ = 0.0019 (a) and Φ = 0.00035 (b). The data are superimposed on the master curve derived from samples at higher filler contents (full circles). The pairs of scaling factors (aΦ; bΦ) are shown in the insets (coordinates of the symbols) together with the amplified G′′ of the neat polymer (solid and dashed line). Dashed lines in part a and b represent the track on which the point at the lowest frequency of the curve to be scaled must move with changing the pair (aΦ; bΦ). Full diamonds in part a partially lie on the master curve. This univocally sets the elasticity bΦ of the cluster network.

positioning of each G′ curve on the master curve is unambiguous due to interrelationship between the shift factors, which establishes a precise track in the plane G′/bΦ − ω/aΦ on which the curve to be scaled can move. This precisely sets the network elasticity bΦ=G′0. On the other hand, as clarified in Figure 5.b, scaling the G′ curves of samples below Φc onto the master curve is not allowed in obedience to the two-phase model. This facilitates the identification of the percolation threshold, which must be sought in a range of compositions which is inferiorly limited by the highest Φ of the nonscalable G′ curves. 8856

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Figure 6. (a) G′(ω) curves of a PS/MWCNT sample at Φ = 0.0086 obtained at three different temperatures: T = 180 °C (squares), 190 °C (circles), and 200 °C (triangles). (b) Overlay of the curves of part a obtained by shifting them along the log frequency axis by an empirical factor aT (reported in the inset). (c) Master curves of G′ (full symbols) and G″ (empty symbols) of the neat PS at the reference temperature T = 180 °C obtained using the same horizontal shift factors aT shown in the inset of part b; symbols refer to the same temperatures as in part a.

reflects the interplay between the filler in its aggregated state and the host polymer. 3.3. Generalizing the Two-Phase Model. Polymer Nature and Polymer−Particle Interactions. The key assumption of the two-phase model is the coexistence of two main dynamical species with distinct relaxation time scales: the free polymer and some three-dimensional structure which is unable to relax, at least within the experimentally accessed time scales. This confers wide generality to the two-phase model, as the nature of latter phase is not strictly defined. To support such a conclusion, the master curves of G′ of several PNCs taken from the literature are reported in Figure 8a. We intentionally selected polymer−nanoparticle pairs which differ among them in terms of filler size, shape and surface chemistry, matrix viscosity, presence of compatibilizing agents and degree of polymer−filler affinity. The main features of each system are summarized in Table 1. The numerical values of B(Φ) and the pairs (aΦ; bΦ) used to build the master curves are listed in the Supporting Information (Table S2). The G″(ω) curves of the pure matrices on which the shift factors lie are shown as well (Figure S3, Supporting Information). Excluding the high-frequency regime, the collapse of the G′ data sets on the corresponding master curves is good for each system. Particularly noticeable is the result for PNCs in which good polymer particle affinity is expected due to either the filler surface chemistry or the presence of a compatibilizing agent. This corroborates the conclusions of a recent paper by our group, in which we proposed that the two-phase model should work even in the case of polymer-mediated filler networks.7 Regardless of the presence of a fraction of chains linked to the particle, in fact, most of the polymer retains its own characteristic dynamics, which remain much faster than those of the hybrid network thus restoring the underlying physics of the two phase model. The overlay of the master curves of the various systems is shown in Figure 8b. Each data set approaches the common horizontal asymptote G′/bΦ = 1 with its own characteristic dynamics. Keeping in mind the negligible effect of nanoparticle size and shape on the PNC relaxation spectrum, we primarily ascribe the scattering of the data at ω/aΦ ∼ 1 to the peculiar dynamics of the polymer in each of the selected systems. In particular, the different relaxation spectra of the various matrices bring about different shapes of the G′ curves at

Figure 7. Master curve of G′ of four different PNC systems based on the same PS matrix tested at different temperatures: PS/SiO2 fumed at T = 200 °C (red), PS/O-clay at T = 190 °C (green), PS/GNP at T = 180 °C (blue), and PS/MWCNT at T = 180 °C (black). The filler volume fraction are listed in the inset for each system. The continuous line represents the elastic modulus of an ideal PNC sample in which the ω-independent network elasticity simply adds up to the storage modulus of the neat PS matrix.

dynamics even in case of weak polymer−particle interactions by way of confinement effects and topological constraints. As a consequence, the PNC relaxation spectrum in the intermediate regime reflects the relaxation of the PS matrix in the nanocomposite, which is different from that of the unfilled polymer to which we refer in the two-phase model. This is clarified in Figure 7, where the continuous line represents the elastic modulus of an ideal sample in which the ω-independent network elasticity simply adds up to the G′(ω) of the unfilled PS according to our simplified picture. In such a case the transition would be evidently much sharper than that observed in the real PNCs. This proves the retardation effect of the filler on the polymer dynamics. Surprisingly, however, neither the size nor the shape of the pristine nanoparticles noticeably affect the shape of the curves in the intermediate regime. This means that the single nanoparticles play a secondary role in determining the PNC relaxation spectrum, which essentially 8857

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Figure 8. (a) Master curves of G′ for the systems taken from the literature. From bottom to top: refs 12, 7, 15, 13, 16, and 14. The curves have been vertically shifted for sake of clarity. Different symbols are used for each composition. Increasing the Φ: squares, circles, triangles, inverted triangles, diamonds. (b) Detail of the master curves shown in (a) without vertical shifting. Only one out of two experimental points is reported. A single symbol has been used for each master curve: squares,12 circles,7 inverted triangles,13 triangles,14 diamonds,15 and half-squares.16 Colors are the same as in part a.

high and intermediate values of ω/aΦ. In addition, in the systems with non-negligible polymer−particle interactions, the specific dynamics of the polymer chains adsorbed onto the nanoparticle surface also affect the PNC relaxation modes.17 Moreover, an influence of the testing temperature on the network elasticity is expected in polymer-mediated cluster networks, and this could imply additional changes in the relaxation spectra. As a consequence, despite the common elementary principles on the basis of the melt state relaxation processes, and unless comparing systems with similar polymer matrices and typology of network, the building of a universal master curve of the elasticity of PNCs is precluded by the specificity of the each system. 3.4. Network Elasticity and Stress-Bearing Mechanisms. The two-phase model applies for samples with filler volume fractions greater than a critical value Φc. This threshold represents the minimum particle volume fraction necessary for the formation of a space-spanning network, which can be either formed by bare clusters or mediated by a fraction of adsorbed polymer. According to the percolation theory, the network is expected to exhibit critical behavior just above Φc, its elasticity growing with Φ as G′0 = k(Φ − Φc)ν.18 We estimate the values of Φc by fitting the previous law to the vertical shift factors for the building of the master curves setting k and ν as fitting parameters while keeping Φc constant. The procedure is repeated for different Φc in a range of composition inferiorly limited by the highest Φ of the G′ curves which cannot be scaled on the corresponding master curve (see Figure 5b). Hence the percolation threshold is identified as the value of Φc which returns the maximum regression coefficient R2. The power-law dependences of the network elasticity are shown in Figure 9 for all the investigated systems; the numerical values of the coefficients are summarized in Table 2, where the available Φc declared by the authors of the reference papers are also reported for comparison. The agreement between the deduced and declared Φc is generally good. For each system the elasticity well follow the predicted growth with filler content. The concentric symbols

have been deduced by exploiting the predictive feature of the master curve as shown in Figure 5a. The extrapolated points result well aligned to those at higher Φ, supporting the reliability of our procedure. The value of the critical exponent ν depends on the specific stress-bearing mechanism. For networks with energetic particle−particle interactions, Arbabi and Sahimi distinguished between systems with purely central forces, in which the particles are free to rotate about each other and ν ≈ 2.1, and networks with bond-bending forces, which can bear stresses also by the unbending of their branches and ν ≈ 3.75.19,20 On the other hand, if good polymer−filler affinity exists, polymer bridging can result in long-lived bonds between the particles which contribute to the stress transfer. Comparing different kinds of polymer−nanoparticle gels, Surve et al. suggested a universal trend with ν≈1.88 for polymer-mediated particle networks, in contrast to systems with strong particle− particle interactions which may exhibit elasticity exponents as high as ν = 5.3.21 The investigated systems exhibit values of ν ranging between ∼0.92 and ∼4.39. Trends which clearly relate the size and shape of the primary nanoparticles to the elasticity coefficient cannot be identified. Nonetheless, in line with the conclusions by Surve et al. the data of Figure 9 can be reasonably divided in two groups depending on the degree of polymer−filler interaction, and the systems in which good affinity is declared or presumed have lower values of ν. On the other hand, in the immediate vicinity of Φc the elasticity of the networks which form in the absence of noticeable polymer− particle interactions is generally lower. The strength of percolating networks reflects a complex interplay between energetic and structural features. The steepness of the interaction potential sets the attractive force between the network elements, but the mechanical strength ultimately depends on the way in which they are arranged in the space. It is possible that the better dispersion of the clusters in case of good polymer−particle affinity may result in finer networks, which are more effective in bearing the stress than the tenuous fractal structures forming just above Φc in noninteracting PNCs. Targeted analyses, here prevented by the specificity of 8858

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Macromolecules a When not explicitly specified in the reference paper, η0 has been estimated as the dynamic viscosity η′=G″·ω in the limit of low frequency. bWhen not expressed in volume fraction in the reference paper, the Φ was calculated using the mass fraction and the density value indicated by the authors or in the technical datasheet. cSame kind of fumed silica as in the sample PS/SiO2 of ref6. dSamples prepared by melt compounding. eSamples prepared by freeze-drying. fThe systems taken from refs 15 and 16 are based on SWCNTs synthesized using the same technique.

0.0014; 0.0034; 0.0069; 0.014. supposed to be good by the authors poly(methyl methacrylate) (PMMA)

η0≈7.7 × 103 Pa·s T = 200 °C

16

15 0.00147; 0.00183; 0.002; 0.0025. η0 ≈ 2 × 10° Pa·s T = 20 °C

14 0.011; 0.016; 0.026; 0.042; 0.053.

unspecified; supposed to be good due to the compatibilizer supposed to be good by the authors η0 ≈ 6.1 × 103 Pa·s T = 160 °C

linear low density polyethylene (LLDPE) plus maleic anhydride grafted polyethylene (PE-g-MA) unsaturated polyester resin (UPR)

Unspecified; supposed to be good Poly(ε-caprolactone) (PCL)

η0≈1.5 × 104 Pa·s T = 80 °C

13

7

MCd: 0.048; 0.062; 0.071; 0.075. FDe: 0.034; 0.044; 0.047; 0.062. 0.015; 0.025; 0.051. Poly(ethylene oxide) (PEO)

η0≈2.1 × 104 Pa·s T = 110 °C

Good

12 0.02; 0.03; 0.04. Strong

Colloidal silica (SiO2 Ludox AS30) Fumed Silica (SiO2 Aerosil A150)c Organo-modified clay (Oclay Cloisite 30B) Organo-modified clay (Oclay Cloisite 20A) Single-walled carbon nanotubes (SWCNTs)f Single-walled carbon nanotubes (SWCNTs)f

poly(ethylene oxide) (PEO)

5

η0≈2 × 10 Pa·s T = 75 °C

ref Φ of the samples which G′ curves are scaled onto master curvesb degree of polymer−particle interaction zero-shear rate viscosity of the neat polymera and testing temperature polymer matrix filler

Table 1. Main Features of the Investigated Systems Taken from the Literature

Article

Figure 9. Power law dependence of the network elasticity on the reduced filler content Φ − Φc for the investigated systems. Full and empty symbols refer to PNCs with good and weak polymer−particles interactions, respectively. Concentric symbols represent the values of network elasticity which have been inferred from the master curve as described in Figure 5a.

Table 2. Percolation Thresholds Φc and Parameters k and ν of the Percolation Law G′0 = k*(Φ − Φc)ν Estimated As Described in the Main Texta system (reference) PCL/O-clay13 LLDPE+PE-gMA/O-clay14 UPR/ SWCNTs15 PMMA/ SWCNTs16 PEO/SiO2 colloidal12 PEO/SiO2 fumedb,7 PEO/SiO2 fumedc,7 PS/GNPs8 PS/SiO2 fumed6 PS/O-clay PS/MWNTs

Φc (estimated)

ν

k

Φc (reported in the reference paper)

0.006 0.006

1.78 3.07

5.0 × 107 1.8 × 108

>0.005 ∼0.01

0.000 55

2.07

9.3 × 107

∼0.001

0.001 33

0.92

6.2 × 105

∼0.0008

0.012

1.11

1.5 × 107

>0.01

0.041

1.74

3.8 × 107

0.041

0.029

1.59

2.7 × 107

0.029

0.0181 0.015 0.0067 0.0015

4.39 2.8 2.68 1.97

9.0 5.9 1.8 4.0

× × × ×

0.0181 0.015 − −

1010 108 108 107

The Φc values indicated in the reference papers are reported for comparison. Note that the nanoparticles used in refs 6 and 7 are exactly the same; those of refs 15 and 16 are synthesized using the same technique. bSamples prepared by melt compounding. cSamples prepared by freeze-drying. a

the considered systems, should be performed to assess the correctness of the previous conjecture. 3.5. Considerations on PNCs with Glassy Relaxation Dynamics. The two-phase model foresees that the network is unable to relax within the experimentally accessed time scales. This allows for the unambiguous identification of its elasticity, from which the scaling factors for the building of the master curve of G′ can be univocally deduced. PNCs, however, may exhibit slow glassy dynamics reminiscent of many soft materials.22 As a result of internal rearrangements, the elastic modulus of such kind of systems progressively relaxes, possibly exhibiting an ultimate relaxation over long time scales. This makes it impossible to rigorously apply the two-phase model, unless the coordinates of the points at which the transition 8859

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between the regimes dominated by the two main phases takes place are redefined. The most rational choice in such cases is the additional crossover point between the ω-dependent linear viscoelastic moduli that appears at low frequencies for filler contents high enough. In this way the viscous connotation of the neat matrix is not directly implicated, which makes less rigorous the procedure to separate the contributions of the two phases. However, following this route, we were able to scale both the moduli of a PNC system with glassy dynamics on a single pair of master curves.23 It is possible that referring to the G″ of the PNC partially accounts for the needing to amplify the viscous modulus of the neat matrix due to the hydrodynamic effects, restoring in part the physical meaning of the two-phase model. The possibility to extend such an approach to other systems with glassy dynamics remains to be proved. Alternatively, the network could be modeled as a viscoelastic phase with its own relaxation spectrum. This would enable the usage of rheological models employed for cocontinuous polymer blends, in which the network could be assumed as the more elastic of the two phases.24

Article

ASSOCIATED CONTENT

S Supporting Information *

Amplification factors to accounts for hydrodynamic effects, B(Φ), and shift factors for the building of the master curves of Figures 6 and 7. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: gfi[email protected]. Telephone: +39 081 7682104. Fax: +39 081 7682404. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been partially supported by the Italian Ministry of University and Research (MIUR) through the PRIN 2009 prot. 2009WXXLY2_002. The authors kindly thank Dr. Andrea Causa for his support and useful discussions.

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4. CONCLUSIONS PNCs exhibit a series of peculiar behaviors in the melt state which originate from the nanometric sizes of the filler. Namely, the particles move and rearrange in the host polymer medium in relatively short time scales, forming aggregates which eventually assemble into three-dimensional elastic networks of clusters. This allows for a general description of their linear viscoelasticity irrespective of the nature of pristine nanoparticles and polymer matrix. We have demonstrated that the complex relaxation spectrum of PNCs can be satisfactorily described in terms of only two main populations of dynamical species: the free polymer matrix, whose dynamics, hydrodynamic effects apart, are not affected by the filler, and an elastic threedimensional network, which can be either formed by bare clusters or mediated by a fraction of adsorbed polymer. Because of the marked differences in the relaxation time scales, the contributions of the two families are independent and separable, which allows for the building of a master curve of G′ of samples at different composition. The master curve can be used to univocally infer the elasticity of networks which are too tenuous to be detected through usual viscoelastic analyses. In addition, the physical constraints invoked when building the master curve allow to identify the samples whose filler content is below the percolation threshold, which can be more accurately estimated. The generality of our approach has been validated by analyzing a series of experimental data on PNCs which differ among them in terms of filler size and shape, polymer nature, and degree of polymer−particle affinity. The results are satisfactory, each experimental data set being scalable to build a master curve of G′ of samples at different filler content. Moreover, the deduced percolation thresholds are in line with those reported by the authors. Finally, the comparison between the elasticity exponents suggests a general trend for the structure of the networks, whose Φ-dependent absolute strength and stiffening seem to primarily depend on the degree of polymer−particle interaction. The simple underlying physics of the two-phase model bestows wide generality on it. Besides capturing the Φdependent viscoelasticity of PNCs, the proposed analysis is expected to be useful to describe a wide variety of other kinds of complex fluids in which a superposition of the elasticity of the components is possible.

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dx.doi.org/10.1021/ma301594g | Macromolecules 2012, 45, 8853−8860