New insights into the elasticity and multi-level relaxation of filler network with studies on the rheology of isotactic polypropylene/carbon black nanocomposite

Abstract

The elastic properties and multi-level relaxation behavior of a filler network in isotactic polypropylene/carbon black (iPP/CB) nanocomposites were systematically investigated, which was instructive for the development and application of viscoelastic materials. Based on a two-phase model, master curves of the elastic modulus of composites with different CB concentrations were built to describe the elastic feature of CB networks in the composites. From the elasticity of the networks, it was found that the critical volume of particle for the formation of the elastic network is 2.4 vol% and that the value of critical exponents is 5.1 ± 0.3, indicating that the particle–particle interaction in the network is strong. Based on semi-dilute fractal theory, the value obtained for the fractal dimension of the filler network was d_f = 2.0 ± 0.1, which was in agreement with the reaction limited aggregation mechanism, namely that CB particles must overcome a great barrier to form a cluster. The relaxation behavior of the filler network was also studied. For composites with a CB content of 2.0 vol% (slightly lower than the elastic percolation threshold of 2.4 vol%), the relaxation behavior became slower with the extension of the annealing time. Furthermore, the CB particles aggregated to form a denser network or backbone and the distribution of relaxation units became narrower, leading to an increase of the relaxation modulus. For composites with a CB content of 13.6 vol%, 0.5 h annealing treatment brought a wider distribution of relaxation units, due to the formation of “short chains” of particles, while both relaxation time and relaxation modulus of the network increased. Further annealing treatment (>0.5 h) made no difference to the distribution of relaxation units and relaxation modulus of the network, but relaxation time of the networks kept increasing. The CB concentration dependence of the relaxation behavior of the network revealed that as CB content increased, the relaxation modulus of the filler network increased monotonously. However, both relaxation time and distribution of relaxation units decreased to a minimum value when CB content increased to 6.4 vol%, and then increased with CB content.

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1. Introduction

Polymer nanocomposites, which combine the flexibility and ease of processing of polymers with the excellent functional properties imparted by nanoparticles, have received ongoing interest.^1–5 Filler filled polymer nanocomposites, which can be depicted as a suspension of filler particles and/or agglomerates interspersed within the polymer medium, could significantly change the viscoelasticity of materials.^6–11 In well dispersed nanocomposites with fine tuning of the microscopic properties, the large interfacial area can be obtained due to the extremely high specific surface area of the nanoparticles, which leads to a strong interaction among the particles or between the particle and polymer.^12,13 It is generally believed that the interfacial interaction between a particle and polymer affects the viscoelasticity of the nanocomposites significantly. However, up until now, the interaction mechanism study is still under debate. Some researchers have pointed out that particle–polymer interaction can influence the viscoelastic properties of the system by a variety of different mechanisms. They further highlight that the effects on the dynamics of the polymer segments induced by particles modify the relaxation spectrum of polymers, and particle jamming may lead to slower relaxations.¹⁴ While other researchers considered that the segmental relaxation of a matrix within composites is not substantially altered by the adsorption of a polymer chain onto the filler surface.¹⁵ Jouault showed that only when the mixed network is formed for a typical rim-to-rim distance equal to twice the gyration radius of the polymer chains, is the transmission of stress realized in the polymer–particle network, and an additional elastic contribution could appear in the modulus.⁷ Filler concentration is another important factor in the viscoelasticity of composites. It is described in the literature that when the filler volume fraction increases to a critical value (rheological percolation threshold), three-dimensional networks of clusters,¹⁶ physical jamming,^17,18 transient networks or entrapped entanglements due to polymer adsorption on the filler surface, would be obtained in composites,¹⁹ resulting in qualitative changes of the relaxation spectra and liquid–solid transition. The second plateau, where the storage modulus (G′) of the composite becomes almost independent of the frequency as the filler loading increases at low frequencies,^6,9,20 is believed to be the result of the formation of a particle network. Furthermore, the effects caused by colloidal spherical particles (hydrodynamic effect) make the flow field uneven, which would lead to an enhancement of G′ for composite melt.^13,21 Generally speaking, in order to better understand the influence of nanoparticles on the viscoelasticity of composites, we should take the following factors into account: hydrodynamic effect, polymer–filler interface interaction, and filler network. These factors codetermine the viscoelasticity of the composites. However, the respective contribution cannot be determined in detail.

Most researchers studied the rheology properties, especially the relaxation behavior of the composite as a whole system. Tian Tang used a micromechanics model to calculate the effective stress relaxation stiffness of the linear viscoelasticity of the composite.²² Hanna J. Maria studied stress relaxation behavior of organically modified montmorillonite filled natural rubber/nitrile rubber nanocomposites to predict the performance of a material over long periods of time.²³ In their study, the research objects are both composites rather than filler networks, which mainly determine the function of the materials. In suspensions with soft particles, as the concentration of colloid particles increase, the particles packing together experience a jamming transition and a colloid network appears. This colloid network could relax in a shorter time range.^24–26 For filled polymer composites, filler networks belong to the scope of the colloid as well, so it can be predicted that filler networks in the nanocomposites are not the ideal Hooke solid, they possess the properties of viscoelasticity.^27,28 A frequency range of 10⁻² to 10² rad s⁻¹ has been accepted in the dynamic frequency sweep by extensive researches to investigate the viscoelasticity of nanocomposite melt.^29,30 However, in the conventional investigation range (10⁻² to 10² rad s⁻¹), the relaxation behavior of the colloid particle network in the polymer matrix can hardly be detected. Therefore, the test time is expanded to 10³ s, guaranteeing that most of the relaxation behavior of the filler network could be detected. (We used a cyclic stress relaxation test, and for each sample eight cycles were carried out. Thus, a longer test time for each circle may lead to an extremely long test time for a sample, which can even lead to degradation of the samples. We therefore think that 10³ s is the most suitable time scale for our test.) Compared to what has been reported in the literature, we try to describe the elasticity and multi-level relaxation of the filler network separately, rather than the whole system of the nanocomposite. For this purpose, a two-phase model put forward by Filippone^31–34 is proposed to account for the linear dynamic rheology behavior of nanofilled polymer melts and to describe the elastic properties of a filler network separated from the composite melt. Based on the fractal concept raised by Piau,³⁵ the micro-rheological structure of the filler network is disclosed. Furthermore, according to the relaxation behavior of the network, we tried to analyze the structure evolution of the filler or filler network during the whole test domain.

2. Materials and processing

A carbon black (VXC 68, Cabot) with a dibutyl phthalate (DBP) volume of 1.23 ml g⁻¹ and a primary particle size of 25 nm was used as received. A commercial isotactic PP (iPP, trade name T30S, melt flow rate of 2.3 g/10 min, at 230 °C and 2.16 kg load, PD is 3.46, supplied by Lanzhou petroleum Chemical Co, Ltd, China) was used as the matrix. Depending on the volume fraction of carbon black (CB), a series of iPP/CB composites were fabricated by melt processing using a twin-screw extruder with a temperature setting of 185 to 210 °C from hopper to die. The test samples were prepared by compression molding at a temperature of 200 °C and a pressure of 10 MPa for 5 min. The diameter of the samples is 25.0 mm and the thickness is 1.5 mm.

3. Characterization

The dynamic rheological measurements were performed on a stress-controlled rheometer (AR 2000, TA Instruments) equipped with parallel-plate geometry (diameter of 25 mm). The gap between the two plates was fixed at 1.35 mm to establish good contact between the geometry and test sample. The temperature was 200 °C and in order to obtain a stable inner structure, a time sweep for 2 h was carried out, with a fixed frequency of 0.5 rad s⁻¹ and a strain of 0.1% (it is in the linear viscoelastic region of the composite melts³⁶). The frequency sweep was then conducted in a frequency (ω) range from 0.05 to 500 rad s⁻¹ with the stain of 0.1% and six data points were collected in every decade (as the time to obtain a data point was 2π/ω). A stress relaxation analysis was taken out with the initial strain of 0.1%, and in the following 4 h, a strain of 0.1% was added every 0.5 h. Here it should be pointed out that before adding strain, the former strain in the samples was removed, so that an accumulation effect could be avoided.

Strain sweep was carried out at a fixed frequency of 0.5 rad s⁻¹ and the strain from 0.01% to 50% was used to investigate the nonlinear viscoelastic behavior of the annealed composites. At the same time, a Keithley 6517B was used to synchronously monitor the electrical resistivity of the samples with the copper electrodes making contact with the geometry, and a voltage of 0.1 V. To prevent ] thermal degradation of the matrix, all the rheological experiments were conducted in a nitrogen atmosphere.

4. Results and discussion

4.1 Elasticity of the filler network

A model of a solid network interspersed in a background fluid was put forward by Trappe and his coworkers,³⁷ where they accounted for the scaling by combining the elasticity of the solid network formed by the particles and the viscosity of the suspending fluid. Later, based on this model, Filippone and his coworkers put forward a two-phase model, which made it possible to separate the contributions of each phases in the nanocomposite and proved its application with the hydrodynamic effect of the filler particles being taken into account.^31,33,34 In this model, at a low frequency the stress response of the composites is dominated by the cluster network, which exhibits a ω-independent elastic modulus, while the polymer dynamics prevail at a high frequency.

In the iPP/CB composite melt, the CB clusters are inclined to reassemble into bigger structures due to inter-particle attraction.^36,38,39 From Fig. 1a, we can see that the elastic modulus of the composite increases during the earlier stage, then it reaches a steady value, meaning that the flocculation of the filler particles leads to an enhancement effect on the composite melt. It is assumed that an equilibrium structure of the composite melt can be obtained after a 2 h melt annealing treatment.³⁹ The frequency-dependent storage modulus of the iPP/CB composite melt based on the equilibrium state is shown in Fig. 1b. It is obvious that CB nanospheres have a dramatic effect on the relaxation behavior of the composite. As the particle loading increases, the elastic modulus G′ increases and the linear viscoelastic data indicates a transition to a solid-like response (a second plateau) at low oscillation frequencies for the composite with particle volume fractions of 8.7 vol% (Fig. 1b). The two-phase model holds the concept that the appearance of the low frequency plateau was contributed to the formation of the filler network, and the value of the plateau modulus reflects the strength of filler network elasticity.^31,33,34,37

Fig. 1 (a) Evolution of storage modulus (G′) for the iPP/CB composite melts during melt annealing for 2 h. (b) Frequency dependence of G′ for the iPP/CB composite melts. (c) Determination of the two shift factors, a and b. (d and e) Fabrication of the master curve of G′ for the iPP/CB composite melts with different CB concentrations. (f) Comparison between the plots of G′′ (iPP) versus ω and b/B (c) versus a.

More details are illuminated by the two-phase model proposed by Filippone.^31,33,34 An empirical amplifying factor B(c) (B(c) = G*(c)/G*(c = 0), where G*(c) and G*(c = 0) are the complex shear modulus of the filled sample and neat matrix in the high-frequency region) is applied to account for the increased gap available to the fluid because of the presence of the particles. Once amplified by B(c), the loss modulus G′′(ω) of the neat polymer crosses the cluster network elasticity, b(c) = G′_n(c), where G′_n(c) is a low-frequency plateau of the storage modulus, and b(c) and G′_n(c) are both concentration dependent. A characteristic frequency a(c) is obtained simultaneously, which sets the transition from the domain governed by the filler (ω < a(c)) to that dominated by the matrix (ω > a(c)) (Fig. 1c).³² However, for samples without storage plateau scaling, the G′ curves onto the master curve is not allowed in obedience to the two-phase model which facilitates the identification of the percolation threshold as well. Thus, for these samples we assume that the plateau would occur at a lower frequency, and set a series of the b(c)and a(c) with different values, choosing the appropriate value to obtain the master curve. (The correctness of the choosing of b(c) and a(c) is shown in Fig. 1f, it is obvious that data b/B vs. a(c) is right in the curve of G′′ vs. ω.) Then a precise track in the plane of G′(c)/b(c)–ω/a(c) is established,³² and it is not difficult to find that when excluding the high-frequency domain, the collapse of the G′ data sets on the corresponding master curves is perfect for all samples shown in Fig. 1d. Filippone pointed out that viscosity of composites is dominated by the polymer matrix without any contribution from filler particles at a high frequency, so therefore the data of G′(c)/b(c) vs. ω/a(c) in the high frequency domain can be ignored (Fig. 1e).

From the master curve, the elasticity modulus of the filler network (G′_n = b) in the composite melt can be obtained, as shown in Fig. 2d. Furthermore, the storage modulus of the CB network as a function of CB concentration follows the power law:³⁴

G′_n ∝ (c − c_p)^t (1)

here, c_p is the elastic percolation threshold; c is filler concentration; t is the critical exponent, which relates to the ability of the networks to bear stress in microscopic force laws.

Fig. 2 Storage modulus of the CB network as a function of CB concentration. The solid line and the inset represent the result of the fit according to eqn (1).

By plotting G′_n vs. CB loading and fitting it to function (1), the critical particle volume for the formation of the elastic network in composite (c_p) is 2.4 vol% and the value of critical exponents is 5.1 ± 0.3 (Fig. 2 and the inset in Fig. 2). According to the percolation power law theory, as long as the filler concentration is above critical threshold, the particle network could build. (In this study, the critical threshold is 2.4 vol%, meaning that when the CB content was above 2.4 vol%, a CB network could build.) For networks with energetic particle–particle interactions, Arbabi and Sahimi distinguished the systems with central forces dominating when t is about 2.1, in which the particles are free to rotate. Furthermore, in networks with bond-bending forces dominating when t is about 3.75, particles can bear stresses by the unbending of their branches.^32,40–42 Surve et al. suggested that for polymer-mediated particle networks a universal trend with t ≈ 1.88 was obtained,⁴³ in contrast to systems with strong particle–particle interactions which exhibited elasticity exponents as high as t ≈ 5.3.³² In our study, the value of critical exponents is 5.1 ± 0.3, indicating that the particle–particle interaction is strong in our system.

4.2 Nonlinear viscoelasticity and fractal dimension of the filler network

The Payne effect was first put forward by Payne in 1962 in the study of the dynamic modulus of carbon black-loaded natural rubber vulcanizates.^44,45 The effect of amplitude-dependence of the dynamic viscoelastic properties of filled-rubbers, is often referred to as the Payne effect (non-linear behavior), i.e., the decrease of the modulus with increasing deformation ratio. For pure polymer, the non-linear behavior can be imagined to be associated with the mechanism of chain disentanglements. For nano-filled polymer composites, the mechanisms of non-linear behavior become more complex. It is generally explained in terms of the breakdown process occurring in the filler network, the disentanglement of polymer chains and the desorption of the polymer chains from the particle surface.^46,47

Fig. 3 shows the nonlinear viscoelastic behavior of the composite melts with different CB concentrations. For pure iPP matrix, the storage modulus almost remains constant during the whole test domain (γ < 50%). However, for the iPP/CB composite melt, as the strain reaches a critical value (γ_c), the modulus of the composite melt decreases with the increasing of the strain. Furthermore, the degree of non-linearity increases with filler concentration (as CB concentration increases, the critical strain value reduced from ∼2% to ∼0.2%). There is no doubt that the filler network existing in the annealed iPP/CB composite melt plays the role of backbones, so at the critical value (γ_c) the decreasing of the storage modulus can be contributed to the breakdown of the filler network.

Fig. 3 Nonlinear viscoelastic behavior of the composite melts with different CB concentrations.

More details about the breakdown of the filler network are obtained by the changing of the resistivity as a function of strain amplitude, as shown in Fig. 4. At a fixed frequency, the storage modulus of the iPP/CB (8.7 vol%) composite melt decreases dramatically with the deformation increasing to a critical strain (∼0.4%). However the resistivity of the sample does not show any synchronous changing, replaced by a lagging increasing of resistivity when the deformation reaches a higher strain of ∼3%, which is much larger than that of ∼0.4%. This indicates that at the beginning of the network deformation, CB particles do not separate immediately with the strain but keep close to each other, which guarantees the existence of the electronic conductive path. Therefore the decrease of storage modulus at the critical strain γ_c only reflects the yield behavior of the filler network under an external force effect. However, the particle clusters are not separated until a much larger strain (here the strain is ∼3%) is applied. This deformation may break the touching point of the filler network and separate the particle clusters, which leads to the increase of resistivity. Fig. 4b shows the relative resistivity (ρ/ρ₀, ρ₀ is the resistivity at the lowest strain amplitude) for the composite melts with different CB concentrations as a function of strain amplitude. Obviously, the resistivity of samples with higher CB concentrations is less sensitive to the deformation, indicating that at higher CB concentrations, the separation of a network or cluster is more difficult. However, this network is much easier to yield, as shown in Fig. 3. In order to clarify the structural information of the filler network in detail, we introduce the concept of semi-dilute fractal theory.³⁵

Fig. 4 (a) Storage modulus (G′) and resistivity (ρ) of the composite melt with 8.7 vol% CB as functions of strain amplitude. (b) Relative resistivity (ρ/ρ₀, ρ₀ represents the resistivity at the lowest strain amplitude) for the composite melts with different CB concentrations as a function of strain amplitude (γ).

The formulation of the non-fluctuating semi-dilute fractal concept relates fractal dimension to the rheological scaling laws:³⁵

G′_c ∝ c^5/(3−d_f) (2)

Based on this scaling law, we realised that the power law of c is 4.9 ± 0.6 and the fractal dimension is 2.0 ± 0.1 for iPP/CB composites with a filler network (Fig. 5). It is reported that in dilute suspensions, a diffusion limited, cluster–cluster aggregation process yields d_f ≈ 1.75, while the reaction limited aggregation yields d_f = 2.1 ± 0.1, suggesting that suspensions aggregating by these mechanisms should form gels.⁴⁸ The exponent reflects the topology of the samples and the motion ability of each part. If the fractal dimension of the cluster is relatively large and the cluster is dense, the exponent would be large. Scaling law power values for the elastic modulus found in the literature for similar compounds are:⁴⁹ (1) G′_c ∼ c^3.5±0.2 for the mechanism of diffusion limited cluster–cluster aggregation; (2) G′_c ∼ c^4.5±0.5 for the mechanism of reaction limited cluster–cluster aggregation. Our result shows that G′_c ∼ c^4.9±0.6, indicating that when CB particles collide with each other, they cannot aggregate into a cluster directly, only those who overcome the strong barrier could turn into a cluster. Thus in our study, the formation of the CB network meets the reaction limited aggregation mechanism. In order to discuss the kinetics of the aggregation process, the mobility of the polymer matrix was investigated by dynamic rheological measurements. Fig. 6 shows the temperature dependence of the gel time for samples with 6.4 vol% CB, and the inset shows the Arrhenius plot of the zero shear viscosity of the neat iPP as a function of temperature. It is observed that the activation energy for the formation of a gel is larger than that of the polymer viscosity (48.8 kJ mol⁻¹ > 39.7 kJ mol⁻¹), indicating that in the iPP/CB composite melt the aggregation of CB particles does not comply with the mechanism of diffusion limited cluster–cluster aggregation, and it has to overcome a great barrier to form a cluster. This meets the mechanism of reaction limited cluster–cluster aggregation, as our previous study showed.³⁹ Similar reports have also been given by Wu.^50,51

Fig. 5 Concentration dependence of the critical storage modulus (G′_c) at the critical strain γ_c, where the storage modulus of the composite is about 0.95 times that of the storage modulus in the linear domain. Note that the concentration should be well above the percolation threshold in order to obtain the power-law relationship. Solid lines are the power-law fitting curves for G′_c.

Fig. 6 Temperature dependence of the gel time for the sample with a CB concentration of 6.4 vol%. The inset shows the Arrhenius plot of the zero shear viscosity of the neat iPP as a function of temperature.³⁹

4.3 Long time relaxation behavior of the filler network

4.3.1 Relaxation phenomenon of the filler network. The two-phase model proposed by Filippone holds the view that the elasticity of the network can maintain a constant at the low frequency domain, because the filler networks do not relax with the relaxation of the polymer matrix. In reality, filler networks in the iPP/CB composite are not ideal Hooke solids, and only when the time scale is long enough, would the relaxation behavior occur. We compare the relaxation behavior of neat iPP with that of the iPP/CB composite (CB content is 2.0 vol%, which is slightly lower than the elastic percolation threshold 2.4 vol%) after melt annealing at 200 °C for 2 h, as shown in Fig. 7a. It is obvious that stress can relax quickly in the pure iPP matrix. We find that the relaxation modulus (G_t) of pure iPP is about 3 × 10⁴ Pa when the time is as short as 0.01 s, and it decreases to 0.8 Pa when the time reaches 50 s. The reason for this phenomenon is that at 200 °C, the iPP molecular chains can disentangle from each other and can relax fully. However, the stress relaxation behavior of the iPP/CB composite becomes more complex. Within 1 s, the stress relaxation behavior of iPP/CB composite is similar to that of pure iPP. Beyond 1 s, a slow relaxation behavior appears, which gradually evolves in the process of the annealing, as shown in Fig. 7b. As for the composite sample (2.0 vol%) without annealing treatment, only short time relaxation behavior can be observed. However, after melt annealing for 0.5 h, a transition of the relaxation modulus (G_t) of the composite arises, namely that a slow relaxation behavior emerges. As annealing time increases, relaxation behavior becomes slower and slower, indicating that the appearance of slow relaxation behavior is caused by constantly changing the internal structure during annealing treatment. The aggregation of the CB particles during the melt annealing process may significantly affect the viscoelasticity of the composite, leading to the appearance of the slow relaxation behavior. At the same time, it should be pointed out that beyond the time scale of 100 s the relaxation modulus decreases with the extension of time, indicating that the CB network still can relax, although it is very slow. Therefore, the filler networks should not be regarded as the ideal Hook solid, and they should belong to a viscoelastic solid. Furthermore, we find that annealing time only affects the long time relaxation behavior and makes no difference to the short time relaxation behavior (within 1 s), as shown in Fig. 7b. This can be explained by the fact that the short time relaxation behavior is dominated by the relaxation of the polymer matrix, and it is independent of the annealing treatment.

Fig. 7 (a) Relaxation behavior of neat iPP and the iPP/CB composite (2.0 vol% CB) after melt annealing at 200 °C for 2 h. (b) Influence of the melt annealing time on the slow relaxation process of the iPP/CB composite (2.0 vol% CB). (c) The fitting parameters of the KWW function (eqn (3)) as functions of annealing time.

The relaxation process of the filler network can be described by the Kohlrausch–Williams–Watts (KWW) relaxation function:^52,53

G_n(t) = G₀ × e^−(t/τ)m (3)

here, G_n(t) is the relaxation modulus of the filler network at the time of t; G₀ is the relaxation modulus of the filler network at the time of 0 (initial modulus) (as shown in Fig. 7b, G₀ is the intersection point of the extension of the dashed line and the vertical coordinate); τ is the characteristic relaxation time of the network, reflecting an “average” relaxation time; m (0 < m ≤ 1) is the stretched exponential constant, which is a measure of the distribution of relaxation times (m → 1 represents a narrow distribution of the relaxation time).

Parameters τ and m are estimated by curve fitting according to the KWW equation, and the fitting results are shown in Fig. 7c. It is obvious that with an increase of annealing time, both the initial modulus G₀ and the characteristic relaxation time τ increase, which indicates that more CB particles take part in the formation of the filler network with increasing time, leading to a denser network structure. Thus the motion of the CB “chains” would be restricted by the surrounding particles. Similar researches on the relaxation behavior have been widely reported.^24,26 After 4 h annealing treatment, G₀ and τ keep increasing, indicating that the inner structure of the network does not reach equilibrium state yet. At the same time, the value of the stretched exponential constant m is within the range of 0.3–0.4. During the annealing treatment, the stretched exponential constant m increases with time, from which we can speculate that the distribution of the relaxation unit becomes narrower after the annealing treatment. Moreover, in addition to the network backbone, there are also many branched structures of CB “chains” in the CB network, whose motion ability is better than that of the backbone. As the annealing is performed, the surrounding CB particles may take part in the growth of the “chains”, thus these “chains” would become longer and denser, and eventually connect to a new backbone, as shown in Fig. 8a. In this way, the motion of the whole CB network would be restricted, leading to a narrower distribution of the relaxation units.

Fig. 8 Envolution of the network structure: (a) in composites with low CB concentrations (2.0 vol%), annealing treatment may give rise to the growth of CB branch chains that finally turn into backbones; (b) in composites with high CB concentrations (13.6 vol%), filler networks can be very dense and during early annealing treatment (0.5 h) may lead to the touching of longer CB chains, further touching between short CB chains can be very hard in long time annealing, however these chains can restrict the motion of backbones making the relaxation behavior of the backbone or network more difficult.

As for the composite with a higher CB concentration (13.6 vol%), the development of the relaxation process during melt annealing is shown in Fig. 9. Slow relaxation behavior of the samples without melt annealing is observed, indicating the filler network formed before annealing treatment. With increasing annealing time, a significant change in slow relaxation of the filler network takes place. The KWW function is used to fit the slow relaxation behavior, shown in Fig. 9a (dashed line), and the fitting results are shown in Fig. 9b. G₀ increases with annealing time in the first 0.5 h, however, it shows no increase in further annealing treatment (>0.5 h), indicating that the network in the composites with high filler concentrations would reach the state of equilibrium much more easily. During the whole annealing treatment, the relaxation time of the networks keeps increasing and finally reaches a state of stability after 3 h of annealing treatment. This kind of phenomenon suggests that some changes emerging within the network structure makes the network difficult to relax. Before annealing, m ≈ 0.41, and after 0.5 h of annealing treatment m decreases to 0.33 and further annealing only slightly decreases the stretched exponential constant m. It is believed that in a high concentration filler filled composite, the filler network would be very dense and that CB particles joined later would only form short CB “chains”, leading to a wide distribution of the relaxation unit, so that the value of m decreases. After annealing for 0.5 h, it is hard for short CB “chains” to make contact with each other, thus these short “chains” could not make a significant contribution to the modulus of the filler network. However, the existence of short CB “chains” would restrict the motion of the backbone, resulting in a longer relaxation time of the filler network, details are shown in Fig. 8b.

Fig. 9 (a) Development of the slow relaxation process in the composite melt with 13.6 vol% CB during melt annealing. (b) The fitting parameters of the KWW function (eqn (3)) as functions of annealing time.

4.3.2 The influence of filler concentration on the relaxation behavior of the network. In this part, the effect of filler concentration on the relaxation behavior of the filler network is studied. Fig. 10a shows the relaxation behavior of the iPP/CB composite with different CB concentrations after annealing at 200 °C for 2 h. It is obvious that as the CB concentration increases, the relaxation modulus of the iPP/CB composite melt increases and within the entire test the amount of relaxation modulus reduction decreases. By fitting the slow relaxation process to the KWW equation, the value of G₀, τ and m are obtained, as shown in Fig. 10b. For all samples, G₀ monotonically increases with increasing CB concentration. However, it is unexpected to find that when the concentration of CB increases from 2.1 vol% to 6.4 vol%, the relaxation time of the samples decreases from 662 s to 0.08 s and at the same time m decreases, which indicates that the dispersion units of the networks become wider. It is uncommon to find that relaxation time decreases with an increase of colloidal particles. It is generally believed that in a nanocomposite with an increase of particles, mutual restraint between the particles would become stronger, resulting in a longer relaxation time of the filler network. This is in contrast with our results. It is speculated that when the CB concentration is low (2.1 vol%), CB particles aggregate to form a network throughout the sample and this network can be very loose. The movement of the network is the overall movement of the CB particles. Thus the relaxation time of the CB network would be relatively longer and the distribution of relaxation units would be narrower. As CB concentration increases, the branches of the network increase too. In this case not only the network backbone (long time relaxation) but also branches of the network (short time relaxation) exist in the filler network, so the distribution of relaxation units become wider, thus the value of m decreases and the average relaxation time of the overall network decreases. However, with further increasing of the CB particles (6.4 vol%), both relaxation time and stretched exponential constant m increase (relaxation units become narrower). Higher filler concentration would help to form a denser network and make the branches come into contact with each other, which finally forms the backbone of the network. In this case, the length of the relaxation units would become homogeneous, leading to an increase in the stretched exponential constant m. On the other hand, the restriction effect among each relaxation unit (especially branches) would lead to a longer relaxation time, resulting in the increasing of τ.

Fig. 10 (a) Relaxation behaviors of the iPP/CB composite melts (after annealing at 200 °C for 2 h) with different CB concentrations. (b) The fitting parameters of the KWW function (eqn (3)) for the slow relaxation process in the composite melts.

5. Conclusion

In summary, the elastic properties and multi-level relaxation behavior of the filler network in the iPP/CB composite were systematically studied. Firstly, a master curve of G′ of the composite with different CB concentrations was built based on the two-phase model proposed by Filippone. The rheological threshold, 2.4 vol%, was determined on the basis of a power law relation, and the critical exponent was in the vicinity of 5.1 suggesting strong interactions between the CB particles. In a non-linear viscoelasticity study, the degree of non-linearity of the composite increased with filler concentration. A formulation of the non-fluctuating semi-dilute fractal concept proposed by Piau allowed us to relate the fractal dimension to the rheological scaling laws (G′_c ∝ c^5/(3−d_f)), and a fractal dimension of d_f ≈ 2.0 was obtained, which is in good agreement with reaction limited cluster–cluster aggregation. The decreasing of the storage modulus at the critical strain γ_c reflected the yield behavior of the filler network.

The KWW equation was used to systematically study the relaxation behavior of the filler network. The results showed that in the iPP melt, the CB network was not the ideal Hooke solid and in long time scale it could relax as well. This relaxation behavior of the filler network was closely related to annealing time and filler concentration. For composites with low CB concentrations, the relaxation behavior became slower during annealing treatment, the CB particles or networks became denser and the distribution of relaxation units became narrower, leading to the increase in relaxation modulus. For composites with high CB concentrations, short time annealing treatment brought a wider distribution of relaxation units, as the formation of short chains, while both relaxation time and relaxation modulus of the network increased. However, in further annealing treatments the relaxation time of the network kept increasing, but the distribution of the relaxation units and relaxation modulus of the filler network did not show any difference. The CB concentration dependence of the relaxation behavior revealed that as CB content increased, the relaxation modulus of the filler network increased monotonously. However, both the relaxation time and distribution of relaxation units decreased to a minimum value when the CB content increased to 6.4 vol%, and then increased with CB content.

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grant No. 51103087, 51421061).

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