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Scaling of the viscoelasticity of highly filled carbon black polyethylene composites above the melting point
Polymer 45 (2004) 7681–7692 www.elsevier.com/locate/polymer
Scaling of the viscoelasticity of highly filled carbon black polyethylene composites above the melting point Karl-Michael Ja¨gera,*, Svein Staal Eggenb a
Borealis AB, Marketing and Development Centre Wire and Cable, 444 86 Stenungsund, Sweden b Borealis A/S, Polyolefin Research, 3960 Stathelle, Norway Received 26 April 2004; received in revised form 10 August 2004; accepted 19 August 2004
Abstract Different types of linear low-density polyethylene and ethylene butylacrylate copolymers were mixed with various types of carbon black in amounts between 25 and 40% by weight. Viscoelastic properties were measured using dynamic mechanical analysis applying a frequency sweep. Typically, the complex modulus approaches asymptotically a constant value at small frequencies, which is referred to as ‘yield modulus’. These results were analysed using a scaling approach according to which the complex modulus and the frequency are normalised by the yield modulus and the quotient of the yield modulus and the polymer viscosity, respectively. Thus a master curve is achieved for nearly all samples independent of the polymer and carbon black type and loading. A similar scaling behaviour has been observed earlier for differently concentrated suspensions of carbon black in Newtonian liquids, but not for filled polymers and different carbon blacks. Thus, contributions from polymer and carbon black to the compounds’ viscoelastic properties are discussed. q 2004 Elsevier Ltd. All rights reserved. Keywords: Polymer; Carbon black; Rheology
1. Introduction Several industrial applications of carbon black polymer composites, such as in semi-conducting power cable shields, require relatively high filler loading up to 40% by weight. Certainly, this high filler load has a strong impact on the flow properties of these compounds during processing by, for instance, cable extrusion. Carbon black forms a solid-like network structure in the polymer resulting in a yield stress tc that depends on the concentration and the nature of the filler but is nearly independent of the type of polymer and test temperature [1]. However, the experimental determination of the yield stress must be treated with care. Leblanc [2] pointed out that one might argue about the yield stress as a true characteristic of filled materials. Frequently, the yield stress is essentially only an extrapolated value taken from a small experimental window that does not reveal the true progress of the flow * Corresponding author. Tel.: C46 303 86019; fax: C46 303 77 05 96. E-mail address: [email protected] (K.-M. Ja¨ger). 0032-3861/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymer.2004.08.073
curve towards low shear rates and that does not reveal whether a yield stress actually exists. In fact, measurements on carbon black rubber compounds at very low shear rates down to 10K5 sK1 showed a gradual increase of the viscosity [3]. Another problem arises from the fact that most (perhaps all) highly filled carbon black polymer composites have no linear viscoelastic region. The flow curves obtained are clearly strain dependent even at very low strains and shear rates. This means that the test conditions affect the measured material properties. Thus a tested sample characterised by a calculated yield stress does not correspond to an unloaded sample in terms of its microstructure. It is clear from above that a yield stress obtained from standard rheology tests is not a material property alone, but depends also on the test conditions. Nevertheless, it is an engineering reality. In this sense one may be restricted to constant test conditions and use the obtained data for comparison purpose. Recent developments in the evaluations of rheological data [4–6] allow further conclusions to be drawn from the
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shear thinning behaviour of these samples. Coussot [4] describes the flow of clay-water suspensions by a HerschelBukley model, t Z tc C k$g_ n
(1)
where t denotes the shear stress magnitude and the constants k and n are positive parameters and g_ is the shear rate. His analysis shows a geometrical similarity of the flow curves at different filler concentrations for a given material. In consequence, all flow curve data follow a master curve in a dimensionless G–T-diagram at low shear rates. In this data representation, the dimensionless shear stress is TZt/tc and the dimensionless shear rate is GZ _ c ; where m is the viscosity of the force-free suspenm$g=t sion, which is the viscosity that depends only on the volume fraction of non-interacting particles as defined by Chong et al. [7]. The existence of this master curve leads to the conclusions that, first, the rupture and restoration of particle links dissipate most of the energy leading to the shear thinning yield stress behaviour and, secondly, the networks’ microstructure is disordered and similar for a wide range of concentrations. On the basis of this similarity, Lin and Chen [5] elaborated a fractal scaling model (links–nodes–blobs model), while Trappe and Weitz [6] argued in terms of a percolation concept. The latter authors studied carbon black suspensions. It is worth pointing out that Trappe and Weitz [6] did not apply the exactly same scaling method as proposed by Coussot [4]. Instead of normalising the shear rate and the shear stress by the yield stress and the viscosity of the force free suspension, they used a frequency scale factor a and a modulus scale factor b. The latter has been applied to both, the real and the imaginary modulus. Both scale factors are linearly related. In order to allow temperature dependent variations in the fluid viscosity hf, they introduced the latter into this relationship, so that bfa/hf. Although the dynamic response and the steady shear-rate response is likely not identical, one recovers immediately a similar data treatment as in the G–T-diagram in terms of utilizing the geometrical similarity of the flow curves, recognizing the close relationship between the modulus and the shear stress on one side and the frequency and the shear rate on the other side. A difference, however, is that G takes the viscosity of the force-free suspension into account whereas the treatment by Trappe and Weitz considers the viscosity of the fluid only. Later on we will show that the latter method leads to better results for our systems. Coussot [4] argued that the rupture and restoration process is responsible for the strong shear thinning of studied systems and thus contributes substantially to the master curve discussed above. He thought this process as being due to the agglomeration of solid particles in flocs, which are ruptured during flow and subsequently restored [4], and the yield stress as number of links per surface unit (as function volume fraction, dispersion and geometry of the
filler) times the mean interaction intensity. Indeed, Trappe and Weitz [6] altered the attractive forces between particles by adding dispersant to the carbon black suspended in oil. Interestingly, they were able to scale these data onto the same master curve as obtained from varying the carbon black concentration. They found that the critical onset of a solid-like network is not restricted to a change in the carbon black concentration, but can also occur as the interaction energy between particles is varied, which gives substantial support to the above arguments by Coussot. Focusing now on filled polymer systems the picture becomes somewhat more complicate, since the nature of the solid-like network structure may depend on whether interparticle attractions or interactions between the particles and the polymer chains are stronger. In either case when a strain rate field is applied to the system, the network structure breaks. Upon resting the network structure restores either partly or completely to its original state. The hydrodynamic resistance due to the viscoelastic polymer matrix is one factor governing the kinetics of the restoration process [8]. The network structure remains controversial. The cluster–cluster aggregation (CCA) model has been successfully applied to modelling e.g. the filler concentration dependence of mechanical properties of carbon black rubber compounds [9]. According to the CCA model, kinetic aggregation of filler particles in elastomers is based upon the assumption that the particles are allowed to fluctuate around their mean position in the matrix. Upon contact of neighbouring particles or clusters they stick together. Depending on the filler concentration this flocculation process leads to a network structure that can be considered as space-filling configuration of fractal CCA clusters. From this the elastic modulus, G, of the compound can be described as function of volume fraction carbon black, 4, involving the fractal dimension of the CCA clusters, df, the fractal dimension of the CCA cluster backbone, df,B, and the average elastic bending–twisting modulus of the different kinds of angular deformation of the cluster units, Gp, G zGp 4ð3Cdf;B Þ=ð3Kdf Þ ;
(2)
which assumes that the modulus of the filler network is higher than the modulus of the matrix (rigidity condition). Heinrich and Klu¨ppel [9] note that the temperature and frequency dependence of G is controlled by Gp that involves an immobilized, glassy bound rubber phase around the filler clusters. At temperatures above the glass–rubber transition the rigidity condition is no longer fulfilled. Cassagnau [10] studied the viscoelastic properties of silica filled ethylene vinyl acetate copolymer (EVA) in concentrated solutions and melts. He notes that the CCA model depicts well the dependency of the modulus on filler volume fraction. However, based on his results he proposed that Gp depends on the dynamic relaxation regime of the polymer chains. A further model classifies the polymer chains in a filled
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compound as either free or trapped to the particles [8]. Upon deformation, some chains get liberated from the particles and thus participate in free chain relaxation. Concurrently, some free chains are being trapped onto the particles. The dynamic balance between the number of free to trapped chains depends on the shear rate, the absolute strain on the chains and the relaxation times. Interactions between polymer chains and filler have been also considered by Liu et al. [11] analysing rheology data of nanotube polycarbonate composites. These authors described the network structure by both factors, the entanglement of the nanotubes themselves and polymer– filler interaction. It is clear from above, the nature of the network structure in highly filled melts, and thus the physical meaning of a yield stress (if it exists) and of a master curve in the G–Tdiagram, remains controversial, and depends largely on the investigated system. The exact theoretical description of the networks’ microstructure in terms of fractal or percolation theory is not targeted in this work. Rather we would like to better understand in what way the filler and the polymer contribute to the network structure and the viscoelastic properties of carbon black filled polyethylene copolymer systems. With this respect various types of carbon black and polyethylene copolymers at different filler concentrations and temperatures are compared in a G–T-diagram. This shows that the G–T-diagram results in a master curve not only for filled suspensions, but also for filled polymer systems. Based on these results, the viscosity of these compounds is discussed in terms of contributions of matrix polymer, carbon black network strength (yield stress) and time dependence of network rupture and restorations. Thus, this work focuses on a detailed description of the rheology of carbon black filled polyolefins.
All samples are compounds of a polyolefin (Table 1) mixed in a pilot scale Buss extruder (L/DZ11) with various types of carbon black (Table 2) in amounts between 25 and 40 wt% and a small amount of antioxidant. The pelletised Table 1 Base resins used MFR2,190 (g/10 min) EBA EBA EBA EBA LLDPE LLDPE LLDPE LLDPE
Table 2 Carbon black types used DBPA (ml/100 g) CB1 CB2 CB3 CB4 CB5 CB6
100–120 190–220 115–135 115–135 180–200 180–200
The DBPA number is a measure of the carbon blacks’ aggregated structure.
extrudate was compression moulded in 1.5 mm thick plates from which circular samples have been cut. The matrix polymers P1–P4 are polar ethylene butylacrylate copolymers (EBA) polymerised by the same highpressure process as low-density polyethylene (LDPE). In order to emphasise the role of the viscosity of the matrix polymer, some compounds are made with linear low-density polyethylene (LLDPE) polymerised using a single site catalyst in a low pressure process, P5–P8. The various types of carbon black used are characterised by their dibutylphthalat absorption (DBPA). This parameter is indicative of the aggregated structure of the carbon black type. Therefore, a high value can in first approximation be associated with a high contact probability between carbon black particles in a polymer melt at a given loading level. The experiments were carried out in a Rheometrics Dynamic Analyzer RDA II in plate–plate configuration at 190 8C applying a frequency sweep with a strain of 5%. All other test conditions than the mentioned above will be stated in the text. As it will be discussed later, the data obtained from these compounds lack of a linear viscoelastic region and are therefore depending on the set strain. A strain of 5% has been chosen since it appeared to be the smallest strain amplitude that resulted in reasonable measurement accuracy.
3. Results
2. Experimental
P1 P2 P3 P4 P5 P6 P7 P8
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6 18 1.1 15 18 3.8 6 20
3.1. Basic concepts Fig. 1 shows the complex viscosity of CB1/P1 composites with various carbon black concentrations in comparison to the base polymer. While the base polymer exhibits a Newtonian region, the filler in the composites causes a drastically increasing viscosity with decreasing shear rate. It is not clear from the limited experimental window whether the composites’ viscosity yields infinite values or may reach some plateau at very low shear rates. With increasing frequency the viscosity of the composites decreases, which is due to ruptured filler networks. Therefore, at high frequencies (short time intervals) most of the broken network links have not time enough to arrange in new links and the filler reinforces the melt mainly hydrodynamically, which is indicated by a flow curve progressing
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Fig. 1. The complex viscosity as function of frequency of the EBA base resin P1 unfilled and filled with various amounts of CB1.
about parallel to that of the matrix polymer. However, at small frequencies (longer time intervals) these links have time enough to restore and thus determine the viscosity. The complex shear modulus G* of the same data as in Fig. 1 plotted versus the angular frequency u is shown in Fig. 2. The experimental set-up (oscillating mode, plate– plate configuration) does not allow the precise determi-
nation of shear stress and shear rate mainly due to lack of a linear viscoelastic region. However, serving the discussion regarding the network structure, at constant test conditions a G*–u plot can be as first approximation associated with a tK g_ plot due to the close connection of the corresponding parameters. Typically the shear modulus (shear stress) of polymer
Fig. 2. The same data of the filled compounds shown in Fig. 1 are plotted as complex shear modulus. The symbols represent measured data. The lines are fit according to Eq. (5).
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melts with high carbon black loading approaches asymptotically at constant values (yield stress) with decreasing frequency. Increasing the filler concentration leads to a higher ‘yield modulus’, while the flow curves change hardly their geometrical shape. Differently, when the base polymer is changed at constant carbon black concentration, the flow curves differ distinctly, but arrive at about the same ‘yield modulus’ (Fig. 3). Fig. 3 shows also data of a measurement at 130 8C, which agree with the same observation: the yield modulus is nearly independent of the base polymer and test temperature. Only at higher shear rates the influence of the base polymer manifest itself in a characteristic flow curve. Focusing on the LLDPE compounds shows generally that a stronger up-turn of G* at increasing frequencies corresponds to less shear-thinning of the complex viscosity (Fig. 4). The previously discussed strain dependence is clearly demonstrated in Fig. 5. The complex modulus decreases with increasing strain amplitude applied. Note that the data have been obtained by tests with plate–plate configuration. Therefore, the data correspond to a deformation distribution rather than to a single value. It is clear from above that this type of experiment has to be treated with care. Nevertheless, its results can be used for comparison purpose when the test parameters are strictly kept constant. 3.2. The G–T-diagram In the following, we treat our data based on methods described in Section 1. For this evaluation the dimensionless
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parameters G and T are adapted to the experimental parameters G* and u, T Z G =Gc
(3)
and G Z hf $u=Gc :
(4) Gc
represents the yield modulus In these expressions, extrapolated from data as presented in Figs. 2 and 3. The complex viscosity of the unfilled polymer is denoted by hf. This approach deviates from the method by Coussot [4] due to the use of the polymer viscosity instead of the viscosity of the force-free composite. It is with this respect more similar to the method by Trappe and Weitz [6]. The extrapolation is performed by fitting the experimental data to a function according to Eq. (1), G Z Gc C K$un ;
(5)
where K is a constant. Since this expression can be written as Gc;ðuZ1Þ Z Gc C K at the frequency uZ1, the constant K can be replaced by the corresponding measured value of GðuZ1Þ minus the yield modulus. Thus a curve-fit can be performed by choosing the best fitting pair of Gc and n values. The curves obtained are plotted in Figs. 2, 3 and 5 together with the corresponding measured values shown as data points. In all cases a good fit is achieved. The Gc values from the curve fit and the separately measured polymer viscosity shown in Fig. 4 can then be used to scale the flow curves using Eqs. (3) and (4). The
Fig. 3. The influence of the matrix polymer on the complex shear modulus. At given carbon black content the ‘yield modulus’ hardly changes while the compounds show distinct differences at higher frequencies when the base resin or test temperature is changed. The symbols represent measured data. The lines are fit according to Eq. (5).
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Fig. 4. The complex viscosity of some base resins.
same data as in Fig. 2 are shown in Fig. 6(a) in a G–Tdiagram. In fact, all curves coincide in a master curve. Experimental data of compounds of other carbon black types mixed into the same base resin result in even more convincing master curves (Fig. 6(b)).
In order to demonstrate the role of the polymer viscosity in the G–T-diagram, a similar scaling without hf has been performed. Accordingly, Fig. 7 shows G =Gc plotted versus u=Gc : Obviously, the shear thinning behaviour of the polymer at high shear rates has an impact on the geometrical
Fig. 5. The complex shear modulus of the EBA base resin P1 filled with about 39%-wt CB1 measured using different strain amplitudes. The symbols represent measured data. The lines are fit according to Eq. (5).
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Fig. 6. (a) The data shown in Fig. 2 have been normalised according to Eqs. (3) and (4). This procedure results in a master curve for all carbon black concentrations. (b) The normalisation is tested with results obtained from P1 mixed with CB3.
shape of the flow curves. However, if the base polymer is taken into account the master curve in the G–T-diagram depends on the characteristic times of the rupture and restoration of the carbon black network only as it will be discussed later. Obviously, replacing hf by the concentration dependent viscosity of the force-free composite m would lead to a
separation of the individual curves, which justifies our modification of Coussot’s model. The success of this modification can be further demonstrated by plotting data of composites of the same carbon black mixed into different polymers (Fig. 8). Again a master curve is obtained. Fig. 8 includes also a curve measured at 130 8C. This curve does not coincide with the master curve as well as the others,
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Fig. 7. The normalisation as in Fig. 6 is performed without considering the viscosity of the base resin. Clearly, including the polymer viscosity is required in order to achieve a master curve.
which may be explained by a time and temperature dependent re-arrangement of the carbon black network that typically occurs in such materials [12,13]. Experimental results of various carbon blacks mixed into
the EBA P1 in various concentrations between 24 and 42% by weight are presented in a G–T-diagram (Fig. 9). Again nearly all individual flow curves follow the same master curve. Only an acetylene black CB2 does not follow this
Fig. 8. The concept of the normalisation in a G–T-diagram leads also to a master curve when the base resin is changed. The displayed data refer to compounds shown in Fig. 3.
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Fig. 9. Data of all carbon black types listed in Table 2 mixed into the EBA P1 in amounts between 24 and 42%-wt are presented in the G–T-diagram (24 samples in total). Only data of the acetylene black CB2 (four different concentrations) does not coincide with the general master curve.
trend. However, the deviation of CB2 from the master curve cannot be sufficiently explained within this study. Finally data obtained using different strain amplitudes (Fig. 5) are treated in a G–T-diagram (Fig. 10). Obviously,
the strain amplitude has an effect on the time dependencies of the rupture and restoration processes of network links. This section has shown that not only composites with different filler loading follow a master curve, but also
Fig. 10. Rheology data obtained using different strain amplitudes do not coincide in a single master curve. The data refer to Fig. 5. The compound is P1 filled with about 39%-wt CB1.
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composites of various carbon black types and polymer types follow the same trend. (Only an acetylene black CB2 deviates from this behaviour.) From this an underlying similarity of all tested compounds regardless the polymer type, carbon black type/loading and test temperature has been demonstrated. It is possible to distinguish between the influence of the matrix polymer (and temperature) and of the time dependent network arrangement on the composites’ viscosity. 3.3. The yield modulus The G–T-diagram is insensitive to the network strength of carbon black. In order to study the latter, one could plot the yield modulus obtained from the curve-fit versus the filler concentration. However, since the calculated value of the modulus is rather sensitive to the exponent n, this plot would show some data scatter. It appears more suitable and certainly more convenient to apply a simple curve shift of viscosity data along the frequency axis so that the flow curves overlap with an arbitrarily chosen reference curve. It can be derived that two curves that coincide in a master curve in a G–T-diagram according to Eqs. (3) and (4), follow the same curve also in a presentation of h/hf versus hf $u= Gc : Therefore, one can plot href/hf,ref of any arbitrarily chosen reference curve versus hf,ref$u and shift the hi/hf,idata of another curve along the frequency axis by adjusting a shift factor g according to hf,i$u/gi until it matches the reference curve (Fig. 11). The determined shift factor g is then defined by,
gi Z
Gc;i f Gc;i : Gc;ref
(6)
This shows that the simple curve shift of measured data allows obtaining a parameter g that is proportional to the yield modulus without the need of any extrapolation towards low shear rates and without any curve fit. A shift is exemplified in Fig. 11 using the same data as in Fig. 1, where the reference curve corresponds to the composite of 39% by weight CB2 mixed into the EBA P1. Performing such a shift with all available flow curves and plotting the shift factor versus the filler concentration results in Fig. 12. The curves obtained in this way display a distinct difference between the carbon black types regardless the polymer they were mixed with.
4. Discussion The Figs. 2 and 3 exemplify that rheology data show distinct differences when the measured compounds are based on different polymers, when the carbon black type or content is changed, or when the test temperature is altered. However, the data treatment in a G–T-diagram can clearly show a correlation between those different curves and thus allows conclusions regarding three basic contributions affecting the viscosity of the highly filled carbon black polymer composites. First, the strength of the carbon network manifests itself in a yield modulus (yield stress). Secondly, when shear is applied the carbon black network links begin to rupture and
Fig. 11. The same data as in Fig. 1 are normalised by the viscosity of the matrix polymer P1. Then the curves are shifted along the frequency axis using a factor g until they coincide with a reference curve. The reference curve is the compound of about 39%-wt CB2 mixed into P1.
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Fig. 12. The shift factor g, a measure of the ‘yield modulus’, is plotted versus the carbon black content for various carbon black types mixed into different base resins. (Base resins cannot be distinguished in this plot).
restore, which causes a strong shear thinning. Thirdly, the viscosity of the unfilled matrix polymer affects the viscosity of the filled polymer melt. It has been shown in Fig. 12 that the network strength characterised by the yield modulus does not depend on the viscosity of the (unfilled) polymer matrix, the polymer polarity (comparing acrylate copolymers with LLDPE), or the molecular size distribution (comparing materials from the high pressure process with single-site materials from the low pressure process), and varies only little with temperature. Since those parameters would be expected to change, for instance, the ratio of polymer chains trapped to particles to free chains as described in [8], we may conclude that a polymer network or a polymer–filler network does not significantly affect the network strength in the systems studied within this work. Thus a striking similarity between carbon black polyolefin melts and carbon black oil suspensions is recognized. Therefore, it appears valid to adapt Coussot’s [4] description of a filler network and of the rupture and restoration process as being due to opening and closing of filler links. Accordingly, the network strength results theoretically from the number of links per surface unit times the mean interaction intensity [4]. The number of links correlates with the contact probability of carbon black aggregates as modelled by using, for instance, percolation or fractal theories. Accordingly, the network is formed at a critical volume fraction of carbon black. Increasing the carbon black loading above that critical concentration causes the
elastic network to become rapidly stronger (Fig. 12). Further, the aggregated structure measured as DBPA increases the contact probability. Indeed, the higher structured carbon black types CB2, CB5 and CB6 rank on higher yield stress levels than the lower structured types CB1, CB3 and CB4. However, the order according to the DBPA is not strictly retained comparing for instance CB5 and CB6, or CB1 and CB3 in Fig. 12. Possible explanations for this observation are numerous. For instance, the DBPA number is only a rough description of the aggregate structure, but does not reveal details about the aggregate geometry, such as its bulkiness. Further, it does not necessarily describe the aggregate structure, after the carbon black is mixed into the polymer; i.e. the measured DBPA value is sensitive to carbon black agglomerates that may be separated during mixing. Finally, the mean interaction intensity between carbon black aggregates, often discussed in terms of van der Waals forces and the ability to form corresponding bonds [14], may account for differences in network strength between various carbon black types as it has been demonstrated by Trappe and Weitz [6]. Interestingly, Klu¨ppel [14] argues that the stiffness of a particle–particle bond depends on the ability to squeeze out the bound polymer chains from the contact area under the attractive action of van der Waals force between the filler particles. Heinrich and Klu¨ppel [9] also mentioned that glassy bound rubber undergoes a transition above about 70 8C. In fact, Amari et al. [15] studied acetylene black EVA compounds, which are similar to the systems
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investigated within this work, and concluded that the viscoelastic properties in the solid state are mainly affected by the viscoelasticity of the disperse medium, but in the molten state by the flocculated filler structure. The above indeed suggests that generally polymer–filler interaction do affect the network strength. However, this further suggests that these interactions between polyethylene copolymers and carbon black become negligible in the melt. This again supports our assumption of a dominating filler network that is not affected by the viscosity of the polymer matrix. When shear is applied to the test sample, the network links begin to rupture and restore in a characteristic time dependent process. The existence of a master curve in the G–T-diagram indicates that corresponding network structures are disordered and similar in the sense of percolation or fractal theories [4–6]. Further, the characteristic time dependence is applicable to nearly all studied carbon black types and concentrations. These few differences between carbon black types are surprising. Obviously, typical colloidal carbon black properties, such as DBPA, specific surface area, primary particle size etc. are not affecting the rupture and restoration process. Interestingly, however, the acetylene black CB2 differs from the furnace black types. Therefore, possibly the carbon black surface characteristics, such as the level of graphitisation being generally higher for acetylene blacks than for furnace blacks, may have an unexpected contribution. However, due to lack of experimental evidence this is highly speculative at this stage. Since the master curve in the G–T-diagram is independent of the matrix polymer, it seems that the polymer chains do not involve in the characteristic time dependence of the rupture and restoration process of network links, which is consistent with the model of a network of particle–particle bonds. This behaviour is again similar to that of carbon black suspensions. However, the polymer does play a role in the overall viscoelastic behaviour and shear thinning behaviour of the compounds. This is evident from the empirically found relationship between the polymer viscosity, hf, and the normalised frequency scale, G, in Eq. (4). Possibly this reflects the hydrodynamic resistance to the restoration process due to the viscoelastic polymer matrix surrounding the filler particles as mentioned by Joshi and Leonov [8].
5. Conclusions Dynamic spectroscopy has been performed on a series of highly filled carbon black polymer composites based on various types of carbon blacks mixed into different EBA and LLDPE polymers. These melt rheology data show distinct differences when the test temperature or any of the
components are changed. However, a data treatment in a G–T-diagram can clearly show a correlation between those different curves and thus allows conclusions regarding influence of the filler and the polymer on the viscoelastic behaviour of the molten compounds. First, the strength of the network structure manifests itself in a yield modulus (yield stress). This yield modulus depends on the carbon black type and loading only indicating that the network structure is determined by particle–particle bonds. Second, when shear is applied the carbon black network links begin to rupture and restore, which causes a strong shear thinning. It is shown that this process of rupture and restoration of network links follows a characteristic time dependence that does neither depend largely on carbon black type and loading nor on type of polyethylene copolymer. The systems studied within this work exhibit therefore a similar scaling behaviour and a similar network structure than particle filled suspensions that has been investigated earlier [4,6]. However, differently to suspensions the viscoelasticity of the matrix polymer affects the viscosity of the filled polymer melt particularly at higher shear rates. Although the polymer seems not to take part in the characteristic time dependence of the rupture and restoration process of the carbon black network links, it does play a role in the overall viscoelastic properties and shear thinning of the materials. Thus, in view of the controversially discussed role of the polymer in the nature of the network structure and in the viscoelastic behaviour of filled systems, this work has contributed to an overall better understanding for a relatively broad series of carbon black filled polyethylene copolymer melts.
References [1] Wu G, Asai S, Sumita M, Hattori T, Higuchi R, Washiyama J. Colloid Polym Sci 2000;278:220 and references therein. [2] Leblanc JL. Prog Polym Sci 2002;27:627. [3] Barre`s C, Leblanc JL, Guilet S. Polym Test 2000;20:329. [4] Coussot P. Phy Rev Lett 1995;72(20):3971 and references therein. [5] Lin C-R, Chen W-J. Colloid Poly Sci 1999;277:1019. [6] Trappe V, Weitz DA. Phys Rev Lett 2000;85(2):449. [7] Chong JS, Christiansen EB, Baer AD. J Appl Polym Sci 1971;15: 2007–21. [8] Joshi PG, Leonov AI. Rheol Acta 2001;40:350–65. [9] Heinrich G, Klu¨ppel M. Adv Polym Sci 2002;160:1–44. [10] Cassagnau P. Polymer 2003;44:2455–62. [11] Liu C, Zhang J, He J, Hu G. Polymer 2003;44:7529–32. [12] Ja¨ger KM, McQueen DH. Kautsch Gummi Kunstst 1999;52(11):734. [13] Ja¨ger KM, McQueen DH, Vilcakova J. J Phys D: Appl Phys 2002;35: 1068. [14] Klu¨ppel M. Adv Polym Sci 2003;164:1–86. [15] Amari T, Uesugi K, Suzuki H. Prog Org Coat 1997;31:171–8.
Scaling of the viscoelasticity of highly filled carbon black polyethylene composites above the melting point Karl-Michael Ja¨gera,*, Svein Staal Eggenb a
Borealis AB, Marketing and Development Centre Wire and Cable, 444 86 Stenungsund, Sweden b Borealis A/S, Polyolefin Research, 3960 Stathelle, Norway Received 26 April 2004; received in revised form 10 August 2004; accepted 19 August 2004
Abstract Different types of linear low-density polyethylene and ethylene butylacrylate copolymers were mixed with various types of carbon black in amounts between 25 and 40% by weight. Viscoelastic properties were measured using dynamic mechanical analysis applying a frequency sweep. Typically, the complex modulus approaches asymptotically a constant value at small frequencies, which is referred to as ‘yield modulus’. These results were analysed using a scaling approach according to which the complex modulus and the frequency are normalised by the yield modulus and the quotient of the yield modulus and the polymer viscosity, respectively. Thus a master curve is achieved for nearly all samples independent of the polymer and carbon black type and loading. A similar scaling behaviour has been observed earlier for differently concentrated suspensions of carbon black in Newtonian liquids, but not for filled polymers and different carbon blacks. Thus, contributions from polymer and carbon black to the compounds’ viscoelastic properties are discussed. q 2004 Elsevier Ltd. All rights reserved. Keywords: Polymer; Carbon black; Rheology
1. Introduction Several industrial applications of carbon black polymer composites, such as in semi-conducting power cable shields, require relatively high filler loading up to 40% by weight. Certainly, this high filler load has a strong impact on the flow properties of these compounds during processing by, for instance, cable extrusion. Carbon black forms a solid-like network structure in the polymer resulting in a yield stress tc that depends on the concentration and the nature of the filler but is nearly independent of the type of polymer and test temperature [1]. However, the experimental determination of the yield stress must be treated with care. Leblanc [2] pointed out that one might argue about the yield stress as a true characteristic of filled materials. Frequently, the yield stress is essentially only an extrapolated value taken from a small experimental window that does not reveal the true progress of the flow * Corresponding author. Tel.: C46 303 86019; fax: C46 303 77 05 96. E-mail address: [email protected] (K.-M. Ja¨ger). 0032-3861/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymer.2004.08.073
curve towards low shear rates and that does not reveal whether a yield stress actually exists. In fact, measurements on carbon black rubber compounds at very low shear rates down to 10K5 sK1 showed a gradual increase of the viscosity [3]. Another problem arises from the fact that most (perhaps all) highly filled carbon black polymer composites have no linear viscoelastic region. The flow curves obtained are clearly strain dependent even at very low strains and shear rates. This means that the test conditions affect the measured material properties. Thus a tested sample characterised by a calculated yield stress does not correspond to an unloaded sample in terms of its microstructure. It is clear from above that a yield stress obtained from standard rheology tests is not a material property alone, but depends also on the test conditions. Nevertheless, it is an engineering reality. In this sense one may be restricted to constant test conditions and use the obtained data for comparison purpose. Recent developments in the evaluations of rheological data [4–6] allow further conclusions to be drawn from the
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shear thinning behaviour of these samples. Coussot [4] describes the flow of clay-water suspensions by a HerschelBukley model, t Z tc C k$g_ n
(1)
where t denotes the shear stress magnitude and the constants k and n are positive parameters and g_ is the shear rate. His analysis shows a geometrical similarity of the flow curves at different filler concentrations for a given material. In consequence, all flow curve data follow a master curve in a dimensionless G–T-diagram at low shear rates. In this data representation, the dimensionless shear stress is TZt/tc and the dimensionless shear rate is GZ _ c ; where m is the viscosity of the force-free suspenm$g=t sion, which is the viscosity that depends only on the volume fraction of non-interacting particles as defined by Chong et al. [7]. The existence of this master curve leads to the conclusions that, first, the rupture and restoration of particle links dissipate most of the energy leading to the shear thinning yield stress behaviour and, secondly, the networks’ microstructure is disordered and similar for a wide range of concentrations. On the basis of this similarity, Lin and Chen [5] elaborated a fractal scaling model (links–nodes–blobs model), while Trappe and Weitz [6] argued in terms of a percolation concept. The latter authors studied carbon black suspensions. It is worth pointing out that Trappe and Weitz [6] did not apply the exactly same scaling method as proposed by Coussot [4]. Instead of normalising the shear rate and the shear stress by the yield stress and the viscosity of the force free suspension, they used a frequency scale factor a and a modulus scale factor b. The latter has been applied to both, the real and the imaginary modulus. Both scale factors are linearly related. In order to allow temperature dependent variations in the fluid viscosity hf, they introduced the latter into this relationship, so that bfa/hf. Although the dynamic response and the steady shear-rate response is likely not identical, one recovers immediately a similar data treatment as in the G–T-diagram in terms of utilizing the geometrical similarity of the flow curves, recognizing the close relationship between the modulus and the shear stress on one side and the frequency and the shear rate on the other side. A difference, however, is that G takes the viscosity of the force-free suspension into account whereas the treatment by Trappe and Weitz considers the viscosity of the fluid only. Later on we will show that the latter method leads to better results for our systems. Coussot [4] argued that the rupture and restoration process is responsible for the strong shear thinning of studied systems and thus contributes substantially to the master curve discussed above. He thought this process as being due to the agglomeration of solid particles in flocs, which are ruptured during flow and subsequently restored [4], and the yield stress as number of links per surface unit (as function volume fraction, dispersion and geometry of the
filler) times the mean interaction intensity. Indeed, Trappe and Weitz [6] altered the attractive forces between particles by adding dispersant to the carbon black suspended in oil. Interestingly, they were able to scale these data onto the same master curve as obtained from varying the carbon black concentration. They found that the critical onset of a solid-like network is not restricted to a change in the carbon black concentration, but can also occur as the interaction energy between particles is varied, which gives substantial support to the above arguments by Coussot. Focusing now on filled polymer systems the picture becomes somewhat more complicate, since the nature of the solid-like network structure may depend on whether interparticle attractions or interactions between the particles and the polymer chains are stronger. In either case when a strain rate field is applied to the system, the network structure breaks. Upon resting the network structure restores either partly or completely to its original state. The hydrodynamic resistance due to the viscoelastic polymer matrix is one factor governing the kinetics of the restoration process [8]. The network structure remains controversial. The cluster–cluster aggregation (CCA) model has been successfully applied to modelling e.g. the filler concentration dependence of mechanical properties of carbon black rubber compounds [9]. According to the CCA model, kinetic aggregation of filler particles in elastomers is based upon the assumption that the particles are allowed to fluctuate around their mean position in the matrix. Upon contact of neighbouring particles or clusters they stick together. Depending on the filler concentration this flocculation process leads to a network structure that can be considered as space-filling configuration of fractal CCA clusters. From this the elastic modulus, G, of the compound can be described as function of volume fraction carbon black, 4, involving the fractal dimension of the CCA clusters, df, the fractal dimension of the CCA cluster backbone, df,B, and the average elastic bending–twisting modulus of the different kinds of angular deformation of the cluster units, Gp, G zGp 4ð3Cdf;B Þ=ð3Kdf Þ ;
(2)
which assumes that the modulus of the filler network is higher than the modulus of the matrix (rigidity condition). Heinrich and Klu¨ppel [9] note that the temperature and frequency dependence of G is controlled by Gp that involves an immobilized, glassy bound rubber phase around the filler clusters. At temperatures above the glass–rubber transition the rigidity condition is no longer fulfilled. Cassagnau [10] studied the viscoelastic properties of silica filled ethylene vinyl acetate copolymer (EVA) in concentrated solutions and melts. He notes that the CCA model depicts well the dependency of the modulus on filler volume fraction. However, based on his results he proposed that Gp depends on the dynamic relaxation regime of the polymer chains. A further model classifies the polymer chains in a filled
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compound as either free or trapped to the particles [8]. Upon deformation, some chains get liberated from the particles and thus participate in free chain relaxation. Concurrently, some free chains are being trapped onto the particles. The dynamic balance between the number of free to trapped chains depends on the shear rate, the absolute strain on the chains and the relaxation times. Interactions between polymer chains and filler have been also considered by Liu et al. [11] analysing rheology data of nanotube polycarbonate composites. These authors described the network structure by both factors, the entanglement of the nanotubes themselves and polymer– filler interaction. It is clear from above, the nature of the network structure in highly filled melts, and thus the physical meaning of a yield stress (if it exists) and of a master curve in the G–Tdiagram, remains controversial, and depends largely on the investigated system. The exact theoretical description of the networks’ microstructure in terms of fractal or percolation theory is not targeted in this work. Rather we would like to better understand in what way the filler and the polymer contribute to the network structure and the viscoelastic properties of carbon black filled polyethylene copolymer systems. With this respect various types of carbon black and polyethylene copolymers at different filler concentrations and temperatures are compared in a G–T-diagram. This shows that the G–T-diagram results in a master curve not only for filled suspensions, but also for filled polymer systems. Based on these results, the viscosity of these compounds is discussed in terms of contributions of matrix polymer, carbon black network strength (yield stress) and time dependence of network rupture and restorations. Thus, this work focuses on a detailed description of the rheology of carbon black filled polyolefins.
All samples are compounds of a polyolefin (Table 1) mixed in a pilot scale Buss extruder (L/DZ11) with various types of carbon black (Table 2) in amounts between 25 and 40 wt% and a small amount of antioxidant. The pelletised Table 1 Base resins used MFR2,190 (g/10 min) EBA EBA EBA EBA LLDPE LLDPE LLDPE LLDPE
Table 2 Carbon black types used DBPA (ml/100 g) CB1 CB2 CB3 CB4 CB5 CB6
100–120 190–220 115–135 115–135 180–200 180–200
The DBPA number is a measure of the carbon blacks’ aggregated structure.
extrudate was compression moulded in 1.5 mm thick plates from which circular samples have been cut. The matrix polymers P1–P4 are polar ethylene butylacrylate copolymers (EBA) polymerised by the same highpressure process as low-density polyethylene (LDPE). In order to emphasise the role of the viscosity of the matrix polymer, some compounds are made with linear low-density polyethylene (LLDPE) polymerised using a single site catalyst in a low pressure process, P5–P8. The various types of carbon black used are characterised by their dibutylphthalat absorption (DBPA). This parameter is indicative of the aggregated structure of the carbon black type. Therefore, a high value can in first approximation be associated with a high contact probability between carbon black particles in a polymer melt at a given loading level. The experiments were carried out in a Rheometrics Dynamic Analyzer RDA II in plate–plate configuration at 190 8C applying a frequency sweep with a strain of 5%. All other test conditions than the mentioned above will be stated in the text. As it will be discussed later, the data obtained from these compounds lack of a linear viscoelastic region and are therefore depending on the set strain. A strain of 5% has been chosen since it appeared to be the smallest strain amplitude that resulted in reasonable measurement accuracy.
3. Results
2. Experimental
P1 P2 P3 P4 P5 P6 P7 P8
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6 18 1.1 15 18 3.8 6 20
3.1. Basic concepts Fig. 1 shows the complex viscosity of CB1/P1 composites with various carbon black concentrations in comparison to the base polymer. While the base polymer exhibits a Newtonian region, the filler in the composites causes a drastically increasing viscosity with decreasing shear rate. It is not clear from the limited experimental window whether the composites’ viscosity yields infinite values or may reach some plateau at very low shear rates. With increasing frequency the viscosity of the composites decreases, which is due to ruptured filler networks. Therefore, at high frequencies (short time intervals) most of the broken network links have not time enough to arrange in new links and the filler reinforces the melt mainly hydrodynamically, which is indicated by a flow curve progressing
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Fig. 1. The complex viscosity as function of frequency of the EBA base resin P1 unfilled and filled with various amounts of CB1.
about parallel to that of the matrix polymer. However, at small frequencies (longer time intervals) these links have time enough to restore and thus determine the viscosity. The complex shear modulus G* of the same data as in Fig. 1 plotted versus the angular frequency u is shown in Fig. 2. The experimental set-up (oscillating mode, plate– plate configuration) does not allow the precise determi-
nation of shear stress and shear rate mainly due to lack of a linear viscoelastic region. However, serving the discussion regarding the network structure, at constant test conditions a G*–u plot can be as first approximation associated with a tK g_ plot due to the close connection of the corresponding parameters. Typically the shear modulus (shear stress) of polymer
Fig. 2. The same data of the filled compounds shown in Fig. 1 are plotted as complex shear modulus. The symbols represent measured data. The lines are fit according to Eq. (5).
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melts with high carbon black loading approaches asymptotically at constant values (yield stress) with decreasing frequency. Increasing the filler concentration leads to a higher ‘yield modulus’, while the flow curves change hardly their geometrical shape. Differently, when the base polymer is changed at constant carbon black concentration, the flow curves differ distinctly, but arrive at about the same ‘yield modulus’ (Fig. 3). Fig. 3 shows also data of a measurement at 130 8C, which agree with the same observation: the yield modulus is nearly independent of the base polymer and test temperature. Only at higher shear rates the influence of the base polymer manifest itself in a characteristic flow curve. Focusing on the LLDPE compounds shows generally that a stronger up-turn of G* at increasing frequencies corresponds to less shear-thinning of the complex viscosity (Fig. 4). The previously discussed strain dependence is clearly demonstrated in Fig. 5. The complex modulus decreases with increasing strain amplitude applied. Note that the data have been obtained by tests with plate–plate configuration. Therefore, the data correspond to a deformation distribution rather than to a single value. It is clear from above that this type of experiment has to be treated with care. Nevertheless, its results can be used for comparison purpose when the test parameters are strictly kept constant. 3.2. The G–T-diagram In the following, we treat our data based on methods described in Section 1. For this evaluation the dimensionless
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parameters G and T are adapted to the experimental parameters G* and u, T Z G =Gc
(3)
and G Z hf $u=Gc :
(4) Gc
represents the yield modulus In these expressions, extrapolated from data as presented in Figs. 2 and 3. The complex viscosity of the unfilled polymer is denoted by hf. This approach deviates from the method by Coussot [4] due to the use of the polymer viscosity instead of the viscosity of the force-free composite. It is with this respect more similar to the method by Trappe and Weitz [6]. The extrapolation is performed by fitting the experimental data to a function according to Eq. (1), G Z Gc C K$un ;
(5)
where K is a constant. Since this expression can be written as Gc;ðuZ1Þ Z Gc C K at the frequency uZ1, the constant K can be replaced by the corresponding measured value of GðuZ1Þ minus the yield modulus. Thus a curve-fit can be performed by choosing the best fitting pair of Gc and n values. The curves obtained are plotted in Figs. 2, 3 and 5 together with the corresponding measured values shown as data points. In all cases a good fit is achieved. The Gc values from the curve fit and the separately measured polymer viscosity shown in Fig. 4 can then be used to scale the flow curves using Eqs. (3) and (4). The
Fig. 3. The influence of the matrix polymer on the complex shear modulus. At given carbon black content the ‘yield modulus’ hardly changes while the compounds show distinct differences at higher frequencies when the base resin or test temperature is changed. The symbols represent measured data. The lines are fit according to Eq. (5).
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Fig. 4. The complex viscosity of some base resins.
same data as in Fig. 2 are shown in Fig. 6(a) in a G–Tdiagram. In fact, all curves coincide in a master curve. Experimental data of compounds of other carbon black types mixed into the same base resin result in even more convincing master curves (Fig. 6(b)).
In order to demonstrate the role of the polymer viscosity in the G–T-diagram, a similar scaling without hf has been performed. Accordingly, Fig. 7 shows G =Gc plotted versus u=Gc : Obviously, the shear thinning behaviour of the polymer at high shear rates has an impact on the geometrical
Fig. 5. The complex shear modulus of the EBA base resin P1 filled with about 39%-wt CB1 measured using different strain amplitudes. The symbols represent measured data. The lines are fit according to Eq. (5).
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Fig. 6. (a) The data shown in Fig. 2 have been normalised according to Eqs. (3) and (4). This procedure results in a master curve for all carbon black concentrations. (b) The normalisation is tested with results obtained from P1 mixed with CB3.
shape of the flow curves. However, if the base polymer is taken into account the master curve in the G–T-diagram depends on the characteristic times of the rupture and restoration of the carbon black network only as it will be discussed later. Obviously, replacing hf by the concentration dependent viscosity of the force-free composite m would lead to a
separation of the individual curves, which justifies our modification of Coussot’s model. The success of this modification can be further demonstrated by plotting data of composites of the same carbon black mixed into different polymers (Fig. 8). Again a master curve is obtained. Fig. 8 includes also a curve measured at 130 8C. This curve does not coincide with the master curve as well as the others,
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Fig. 7. The normalisation as in Fig. 6 is performed without considering the viscosity of the base resin. Clearly, including the polymer viscosity is required in order to achieve a master curve.
which may be explained by a time and temperature dependent re-arrangement of the carbon black network that typically occurs in such materials [12,13]. Experimental results of various carbon blacks mixed into
the EBA P1 in various concentrations between 24 and 42% by weight are presented in a G–T-diagram (Fig. 9). Again nearly all individual flow curves follow the same master curve. Only an acetylene black CB2 does not follow this
Fig. 8. The concept of the normalisation in a G–T-diagram leads also to a master curve when the base resin is changed. The displayed data refer to compounds shown in Fig. 3.
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Fig. 9. Data of all carbon black types listed in Table 2 mixed into the EBA P1 in amounts between 24 and 42%-wt are presented in the G–T-diagram (24 samples in total). Only data of the acetylene black CB2 (four different concentrations) does not coincide with the general master curve.
trend. However, the deviation of CB2 from the master curve cannot be sufficiently explained within this study. Finally data obtained using different strain amplitudes (Fig. 5) are treated in a G–T-diagram (Fig. 10). Obviously,
the strain amplitude has an effect on the time dependencies of the rupture and restoration processes of network links. This section has shown that not only composites with different filler loading follow a master curve, but also
Fig. 10. Rheology data obtained using different strain amplitudes do not coincide in a single master curve. The data refer to Fig. 5. The compound is P1 filled with about 39%-wt CB1.
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composites of various carbon black types and polymer types follow the same trend. (Only an acetylene black CB2 deviates from this behaviour.) From this an underlying similarity of all tested compounds regardless the polymer type, carbon black type/loading and test temperature has been demonstrated. It is possible to distinguish between the influence of the matrix polymer (and temperature) and of the time dependent network arrangement on the composites’ viscosity. 3.3. The yield modulus The G–T-diagram is insensitive to the network strength of carbon black. In order to study the latter, one could plot the yield modulus obtained from the curve-fit versus the filler concentration. However, since the calculated value of the modulus is rather sensitive to the exponent n, this plot would show some data scatter. It appears more suitable and certainly more convenient to apply a simple curve shift of viscosity data along the frequency axis so that the flow curves overlap with an arbitrarily chosen reference curve. It can be derived that two curves that coincide in a master curve in a G–T-diagram according to Eqs. (3) and (4), follow the same curve also in a presentation of h/hf versus hf $u= Gc : Therefore, one can plot href/hf,ref of any arbitrarily chosen reference curve versus hf,ref$u and shift the hi/hf,idata of another curve along the frequency axis by adjusting a shift factor g according to hf,i$u/gi until it matches the reference curve (Fig. 11). The determined shift factor g is then defined by,
gi Z
Gc;i f Gc;i : Gc;ref
(6)
This shows that the simple curve shift of measured data allows obtaining a parameter g that is proportional to the yield modulus without the need of any extrapolation towards low shear rates and without any curve fit. A shift is exemplified in Fig. 11 using the same data as in Fig. 1, where the reference curve corresponds to the composite of 39% by weight CB2 mixed into the EBA P1. Performing such a shift with all available flow curves and plotting the shift factor versus the filler concentration results in Fig. 12. The curves obtained in this way display a distinct difference between the carbon black types regardless the polymer they were mixed with.
4. Discussion The Figs. 2 and 3 exemplify that rheology data show distinct differences when the measured compounds are based on different polymers, when the carbon black type or content is changed, or when the test temperature is altered. However, the data treatment in a G–T-diagram can clearly show a correlation between those different curves and thus allows conclusions regarding three basic contributions affecting the viscosity of the highly filled carbon black polymer composites. First, the strength of the carbon network manifests itself in a yield modulus (yield stress). Secondly, when shear is applied the carbon black network links begin to rupture and
Fig. 11. The same data as in Fig. 1 are normalised by the viscosity of the matrix polymer P1. Then the curves are shifted along the frequency axis using a factor g until they coincide with a reference curve. The reference curve is the compound of about 39%-wt CB2 mixed into P1.
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Fig. 12. The shift factor g, a measure of the ‘yield modulus’, is plotted versus the carbon black content for various carbon black types mixed into different base resins. (Base resins cannot be distinguished in this plot).
restore, which causes a strong shear thinning. Thirdly, the viscosity of the unfilled matrix polymer affects the viscosity of the filled polymer melt. It has been shown in Fig. 12 that the network strength characterised by the yield modulus does not depend on the viscosity of the (unfilled) polymer matrix, the polymer polarity (comparing acrylate copolymers with LLDPE), or the molecular size distribution (comparing materials from the high pressure process with single-site materials from the low pressure process), and varies only little with temperature. Since those parameters would be expected to change, for instance, the ratio of polymer chains trapped to particles to free chains as described in [8], we may conclude that a polymer network or a polymer–filler network does not significantly affect the network strength in the systems studied within this work. Thus a striking similarity between carbon black polyolefin melts and carbon black oil suspensions is recognized. Therefore, it appears valid to adapt Coussot’s [4] description of a filler network and of the rupture and restoration process as being due to opening and closing of filler links. Accordingly, the network strength results theoretically from the number of links per surface unit times the mean interaction intensity [4]. The number of links correlates with the contact probability of carbon black aggregates as modelled by using, for instance, percolation or fractal theories. Accordingly, the network is formed at a critical volume fraction of carbon black. Increasing the carbon black loading above that critical concentration causes the
elastic network to become rapidly stronger (Fig. 12). Further, the aggregated structure measured as DBPA increases the contact probability. Indeed, the higher structured carbon black types CB2, CB5 and CB6 rank on higher yield stress levels than the lower structured types CB1, CB3 and CB4. However, the order according to the DBPA is not strictly retained comparing for instance CB5 and CB6, or CB1 and CB3 in Fig. 12. Possible explanations for this observation are numerous. For instance, the DBPA number is only a rough description of the aggregate structure, but does not reveal details about the aggregate geometry, such as its bulkiness. Further, it does not necessarily describe the aggregate structure, after the carbon black is mixed into the polymer; i.e. the measured DBPA value is sensitive to carbon black agglomerates that may be separated during mixing. Finally, the mean interaction intensity between carbon black aggregates, often discussed in terms of van der Waals forces and the ability to form corresponding bonds [14], may account for differences in network strength between various carbon black types as it has been demonstrated by Trappe and Weitz [6]. Interestingly, Klu¨ppel [14] argues that the stiffness of a particle–particle bond depends on the ability to squeeze out the bound polymer chains from the contact area under the attractive action of van der Waals force between the filler particles. Heinrich and Klu¨ppel [9] also mentioned that glassy bound rubber undergoes a transition above about 70 8C. In fact, Amari et al. [15] studied acetylene black EVA compounds, which are similar to the systems
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investigated within this work, and concluded that the viscoelastic properties in the solid state are mainly affected by the viscoelasticity of the disperse medium, but in the molten state by the flocculated filler structure. The above indeed suggests that generally polymer–filler interaction do affect the network strength. However, this further suggests that these interactions between polyethylene copolymers and carbon black become negligible in the melt. This again supports our assumption of a dominating filler network that is not affected by the viscosity of the polymer matrix. When shear is applied to the test sample, the network links begin to rupture and restore in a characteristic time dependent process. The existence of a master curve in the G–T-diagram indicates that corresponding network structures are disordered and similar in the sense of percolation or fractal theories [4–6]. Further, the characteristic time dependence is applicable to nearly all studied carbon black types and concentrations. These few differences between carbon black types are surprising. Obviously, typical colloidal carbon black properties, such as DBPA, specific surface area, primary particle size etc. are not affecting the rupture and restoration process. Interestingly, however, the acetylene black CB2 differs from the furnace black types. Therefore, possibly the carbon black surface characteristics, such as the level of graphitisation being generally higher for acetylene blacks than for furnace blacks, may have an unexpected contribution. However, due to lack of experimental evidence this is highly speculative at this stage. Since the master curve in the G–T-diagram is independent of the matrix polymer, it seems that the polymer chains do not involve in the characteristic time dependence of the rupture and restoration process of network links, which is consistent with the model of a network of particle–particle bonds. This behaviour is again similar to that of carbon black suspensions. However, the polymer does play a role in the overall viscoelastic behaviour and shear thinning behaviour of the compounds. This is evident from the empirically found relationship between the polymer viscosity, hf, and the normalised frequency scale, G, in Eq. (4). Possibly this reflects the hydrodynamic resistance to the restoration process due to the viscoelastic polymer matrix surrounding the filler particles as mentioned by Joshi and Leonov [8].
5. Conclusions Dynamic spectroscopy has been performed on a series of highly filled carbon black polymer composites based on various types of carbon blacks mixed into different EBA and LLDPE polymers. These melt rheology data show distinct differences when the test temperature or any of the
components are changed. However, a data treatment in a G–T-diagram can clearly show a correlation between those different curves and thus allows conclusions regarding influence of the filler and the polymer on the viscoelastic behaviour of the molten compounds. First, the strength of the network structure manifests itself in a yield modulus (yield stress). This yield modulus depends on the carbon black type and loading only indicating that the network structure is determined by particle–particle bonds. Second, when shear is applied the carbon black network links begin to rupture and restore, which causes a strong shear thinning. It is shown that this process of rupture and restoration of network links follows a characteristic time dependence that does neither depend largely on carbon black type and loading nor on type of polyethylene copolymer. The systems studied within this work exhibit therefore a similar scaling behaviour and a similar network structure than particle filled suspensions that has been investigated earlier [4,6]. However, differently to suspensions the viscoelasticity of the matrix polymer affects the viscosity of the filled polymer melt particularly at higher shear rates. Although the polymer seems not to take part in the characteristic time dependence of the rupture and restoration process of the carbon black network links, it does play a role in the overall viscoelastic properties and shear thinning of the materials. Thus, in view of the controversially discussed role of the polymer in the nature of the network structure and in the viscoelastic behaviour of filled systems, this work has contributed to an overall better understanding for a relatively broad series of carbon black filled polyethylene copolymer melts.
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