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# Influence of thermo-rheological history on electrical and rheological properties of polymer–carbon nanotube composites

I. Alig, D. Lellinger and T. Skipa,     German Polymer Institute, Germany

## Abstract:

Electrical conductivity and mechanical properties of carbon nanotube (CNT)–polymer composites depend greatly on the structure of the filler network. Therefore, knowledge of the relation between thermo-rheological history and the structure of the filler network in the polymer matrix is of great scientific and industrial interest. This chapter reviews the knowledge on the relation between filler history created by the processing conditions, the structure of the nanotube network and the resulting composite properties. An overview of model approaches for the description of the electrical conductivity in composites filled with interacting fillers is given. Rheo-electrical laboratory experiments with well-defined shear flow conditions as well as in-line conductivity measurements during processing of composite melts are presented and discussed.

Key words

carbon nanotube–polymer composites

electrical conductivity

filler network

rheology

thermo-rheological history

## 10.1 Introduction

Adding small amounts of carbon nanotube (CNT) additive turns an electrically insulating polymer into a conductive composite and can lead to reinforcement of its mechanical properties, which is usually not possible with small amounts of conventional conductive fillers (Szleifer and Yerushalmi-Rozen, 2005). Apart from the traditional application of conductive or reinforced plastics, new specific applications of nanotube-filled polymers have been proposed in the past few years, e.g. for gas-sensing (An et al., 2004), liquid crystal displays (Russell et al., 2006), optical transparent films (Carroll et al., 2005), or smart materials or actuators (Kang et al., 2006; Li et al., 2008). Electrical conductivity and reinforcement in such composite materials are usually explained by the formation of a percolating network of interconnected filler particles, which transfer the electrical current and/or the mechanical stress (Klüppel, 2003). In this picture, the high aspect ratio (length to diameter ratio) of CNT is considered to be one of the main advantages of carbon nanotubes with respect to other fillers. The high aspect ratio of CNT of up to 1000 theoretically enables the formation of a nanotube network at extremely low critical volume fractions ~ 1/(aspect ratio) (Balberg et al., 1984). However, in reality, the CNT are bent objects, which tend to form agglomerates, so that the experimental percolation concentrations usually exceed the theoretical expected values for fully elongated objects.

In Fig. 10.1 the theoretically predicted (Balberg et al., 1984) dependence of the critical volume concentration of fillers at percolation threshold is plotted versus the aspect ratio of the fillers. The grey area indicates the typical range of experimentally determined percolation concentrations for melt mixed thermoplastics containing CNT, which has been found between about 0.1 to 5 vol.% (Alig et al., 2008b). These values are one to two orders of magnitude above the theoretically expected values. At first glance, this difference can be explained by the worm-like structure of the CNT and/or by remaining (primary) CNT agglomerates. However, the nanotube fraction for the insulator–conductor transition in thermoplastic polymers is still low, so that – in contrast to classical conductive fillers – the desired properties of the polymer matrix (for example, its elongation to break) are almost unchanged.

10.1 Critical volume fractions of fillers at percolation threshold versus aspect ratio (after Balberg et al., 1984).

Although different types of multi-walled carbon nanotubes (MWNTs) are commercially available, the industrial acceptance of the nanotube–polymer composites is still limited. Apart from other reasons, this is due to the broad variation of their physical properties depending on the processing conditions. During compounding, extrusion and injection moulding of the composites, polymer melts undergo moderate to very strong mechanical deformations which affect the state of filler network. It was shown previously (Alig et al., 2007a; Alig et al., 2008a; Pegel et al., 2008; Lellinger et al., 2008; Villmow et al., 2008a; Villmow et al., 2008b), that the state of the nanotube network – which is induced by processing of the polymer melt and frozen (‘frustrated’) in the short period prior to vitrification or crystallization – mainly defines the physical properties of the final plastic part. An example of conductivity changes during isothermal annealing and crystallization during cooling is shown in Fig. 10.2 for a pressed plate of polypropylene with 2 wt% CNT (Alig et al., 2008b). The temperature program is shown in the upper part of Fig. 10.2: after very fast heating from 100 °C up to 200 °C (which is above the melting temperature of Tm ≈ 145 °C), the sample was annealed for 5 hours at 200 °C. The annealed sample was then cooled below the crystallization temperature with a cooling rate of 0.2 K/min. The initial conductivity at 100 °C represents a situation below the electrical percolation threshold, since the conductive CNT network was at least partially destroyed by the ‘squeeze flow’ during pressing. During fast melting and isothermal annealing, the composite undergoes a thermally induced insulator–conductor transition and tends to approach an equilibrium value of the conductivity for long annealing times. During cooling, the polypropylene crystallizes and the decrease of the conductivity is associated with a reduction of the matrix conductivity due to immobilization of the charge carriers in the crystalline (stepwise decrease) and in the amorphous phase (continuous decrease). The conductive filler network formed during annealing becomes frustrated in the semicrystalline structure.

10.2 Influence of thermo-rheological history on the electrical conductivity σ′ of a CNT–polypropylene composite. The upper graph shows the temperature program applied to a pressed sample (after squeeze flow).

As will be shown below (see Sections 10.3.1 and 10.4), rheo-electrical experiments combining measurements of electrical conductivity and rheological properties under well-defined shear conditions (Kharchenko et al., 2004; Alig et al., 2007b; Alig et al., 2008d; Obrzut et al., 2007; Lellinger et al., 2009; Skipa et al., 2009; Skipa et al., 2010) can provide a deeper understanding of the underlying physical mechanisms of the CNT network formation and destruction. The first indication of destruction of electrical and rheological nanotube networks under steady shear flow was given by Kharchenko et al. (Kharchenko et al., 2004) for polypropylene (PP) composites filled with MWNT. The breakdown of the conductive nanotube network for a wide variation of shear rates and MWNT concentrations was somewhat later observed by Obrzut et al. (Obrzut et al., 2007) for PP–MWNT. A tremendous decrease in electrical conductivity during transient shear deformation was monitored in similar laboratory experiments for polycarbonate (PC)–MWNT composite melts (Alig et al., 2007b; Alig et al., 2008d). The time-dependent recovery of the conductivity in the quiescent state (following the transient shear) was attributed to the reformation of the conductive network in the melt.

Parallel to the laboratory experiments, the influence of shear and elongation flow on the electrical conductivity of CNT polymer composites has also been found in processing-related studies (Alig et al., 2007a; Alig et al., 2008a; Pegel et al., 2008; Lellinger et al., 2008; Villmow et al., 2008a; Villmow et al., 2008b) during extrusion, pressing and injection moulding. The destruction of the conductive filler network by mechanical deformation (nanotube de-agglomeration and/or orientation) was observed for the first time in in-line conductivity experiments during extrusion of polypropylene (Alig et al., 2007a), PC and polyamide 6 (PA6) (Alig et al., 2008a) containing different amounts of MWNT. (Pegel et al., 2008) showed examples of deformation-induced agglomeration and de-agglomeration of CNT in PC composites in the squeeze flow during pressing. Villmow et al. (Villmow et al., 2008a; Villmow et al., 2008b) demonstrated the destructive effect of mechanical deformation on the electrical conductivity of injection-moulded plastic parts. In (Villmow et al., 2008a), a conductivity variation up to 5 orders of magnitude within the same injection-moulded PC–MWNT plastic part was found, which they attributed to different flow conditions. Similar results were obtained by in-line and off-line conductivity measurements during and after injection moulding for the plastic parts of PC–MWNT, polyamide–MWNT and an acrylonitrile-butadiene-styrene–polycarbonate blend (Bayblend™) with MWNT for a wide variation of the processing parameters (Lellinger et al., 2008). The strong dependence of the conductivity on injection velocity, melt temperature and viscosity was explained by the competition between deformation-induced formation and destruction of the conductive CNT network.

In addition to nanotube-filled (highly viscous) thermoplastic polymers, there are several works on low viscous colloidal suspensions (Lin-Gibson et al., 2004; Hobbie and Fry, 2006, 2007) and epoxy resins filled with MWNT (Martin et al., 2004; Sandler et al., 2004; Kovacs et al., 2007; Ma et al., 2007). In those systems, shear can also induce agglomeration of nanotubes. The process of nanotube agglomeration can be considered similar to flow-induced flocculation of particles in colloidal suspensions (Switzer and Klingenberg, 2004; Vermant and Solomon, 2005; Guery et al., 2006) or shear-induced phase separation in polymer blends (see e.g. Hobbie et al., 2002; Han et al., 2006). The structures of the filler network which appear under shear can also lead to a notable change in the composites’ viscosity and conductivity (Lin-Gibson et al., 2004; Martin et al., 2004; Sandler et al., 2004; Switzer and Klingenberg, 2004;Vermant and Solomon, 2005; Guery et al., 2006; Hobbie and Fry, 2007; Kovacs et al., 2007; Ma et al., 2007). These findings correspond to earlier experiments by Schueler et al. (Schueler et al., 1996; Schueler et al., 1997) on epoxy systems containing carbon black (CB). The authors attributed the increase of electric conductivity to the agglomeration of CB induced by external shear forces. The agglomeration of particles in sheared carbon black-filled epoxies was explained analogically to colloidal dispersions by the interplay between attractive London–van der Waals forces and repulsive Coulomb forces causing a potential energy barrier between filler particles. Since it is difficult to overcome such barrier by thermal energy alone, shear deformation can accelerate the agglomeration process. Furthermore, it was theoretically shown (Switzer and Klingenberg, 2004) that rigid elongated objects like carbon nanotubes in the presence of shear will always aggregate due to friction forces (even without attraction between them).

In order to demonstrate the influence of thermo-rheological history on the electrical and rheological properties on CNT–polymer composites, the results of rheo-electrical laboratory experiments, which allow simultaneous measurements of the electrical conductivity and the complex dynamic shear modulus (G* = G′ + i G″) under well-defined shear conditions will be shown in Section 10.4 (Lellinger et al., 2009; Skipa et al., 2010; Skipa et al., 2009). The results of rheo-electrical experiments are discussed together with in-line conductivity measurements during extrusion (Section 10.4) and injection moulding (Section 10.5).

## 10.2 Background

### 10.2.1 Percolation and insulator–conductor transition

In the framework of the percolation theory, a mixture of two materials with different conductivities (here: CNT in a polymer matrix) can be modelled by the lattice built up by bonds chosen randomly to be a conductor or an insulator (Clerc et al., 1990; Sahimi, 1994; Stauffer and Aharony, 1994; Bunde and Havlin, 1996). The percolation threshold pc is defined as a critical value of volume fraction p of the conductive component, separating the states, where only finite clusters of the conductor exist p < pc and, where conduction paths between opposite edges of the lattice (infinite cluster) appear p > pc. The macroscopic frequency dependent complex conductivity σ* = σ' + " and complex permittivity ε* = ε'–" of percolating systems have been described by two different physical models: the equivalent circuit (or intercluster polarization) model (Kirkpatrick, 1973; De Gennes, 1976b; Efros and Shklovskii, 1976; Straley, 1976; Bergman and Imry, 1977; Webman et al., 1977; Stephen, 1978; Stroud and Bergman, 1982; Wilkinson et al., 1983) and the anomalous diffusion model (Scher and Lax, 1973; De Gennes, 1976a; Stauffer, 1979; Straley, 1980; Gefen et al., 1981; Gefen et al., 1983; Laibowitz and Gefen, 1984; Weiss and Rubin, 1983; Hong et al., 1986).

It has been established theoretically and experimentally (Kirkpatrick, 1973; De Gennes, 1976b; Efros and Shklovskii, 1976; Straley, 1976; Bergman and Imry, 1977;Webman et al., 1977), that near the critical concentration pc the DC conductivity and the static permittivity follow power laws:

[10.1]

[10.2]

[10.3]

where ω = 2πf is the angular frequency. The critical exponents s and t are assumed to be universal and depend only on the dimension of the percolation system and not on the details of cluster geometry (Kirkpatrick, 1973; De Gennes, 1976b; Efros and Shklovskii, 1976; Straley, 1976; Bergman and Imry, 1977; Stephen, 1978; Webman et al., 1977; Clerc et al., 1990). The currently accepted values of these exponents (Clerc et al., 1990; Sahimi, 1994; Stauffer and Aharony, 1994; Bunde and Havlin, 1996) are s = t ≈ 1.3 for two dimensions and s ≈ 0.73, t ≈ 2.0 for three dimensions.

For example, in Fig. 10.3, the concentration dependencies of the DC conductivity at room temperature for carbon black (CB) in natural rubber (left graph) and MWNT in a PC matrix (right graph) are compared. For the MWNT–PC composite the percolation threshold (pc) is observed at a concentration between 1 and 1.5 wt% (Pötschke et al., 2003). In the case of CB, between 8 and 20 wt% are needed to reach percolation threshold. The solid lines fit with Equations 10.1 and 10.2 with exponents s = 0.73, t = 3.5 (CB) and t = 2.1 (MWCNT). The values for pc were 11.9 wt% and 1.44 wt% for CB and MWNT, respectively. The power law behaviour for the DC conductivity values above pc is visualized in a log-log plot of conductivity versus reduced filler content (ppc) in Fig. 10.4. Further details are given in (Pötschke et al., 2003; Alig et al., 2007a).

10.3 DC conductivity versus volume content of carbon black in natural rubber (a) and for MWNT in PC (b). The percolation thresholds (pc) are indicated by a dotted line. The solid lines are fits with Equations 10.1 and 10.2.

10.4 DC conductivities above the percolation threshold versus reduced MWNT content in PC. The solid line represents a fit with Equation 10.2.

The frequency dependencies of the real part σ' (ω) of the conductivity and the real part of the permittivity ε' (ω) at the percolation threshold also show a power-law behaviour (Straley, 1976; Bergman and Imry, 1977; Webman et al., 1977; Laibowitz and Gefen, 1984):

[10.4]

[10.5]

where critical indices x and y should satisfy the relation x + y = 1 (Bergman and Imry, 1977; Webman et al., 1977; Laibowitz and Gefen, 1984).

In the anomalous diffusion model the transport properties of the percolation system are formulated terms of diffusion (random walk) within the clusters. The correlation time τξ, which a charge carrier needs to traverse a cluster of correlation length ξ is inversely proportional to a critical frequency ωξ. This frequency shows a critical slowing down at percolation concentration: , where dw is the effective fractal dimensionality of the random walk (‘diffusion exponent’) and v is the critical exponent of the correlation length. For frequencies ω < ωξ the charge carriers can explore different clusters within one period (normal diffusion), whereas for frequencies above ωξ, the charge carriers visit only parts of the percolation cluster within one period and anomalous charge carrier diffusion in the fractal percolation clusters takes place. The case ω < ωξ is expressed by a frequency independent plateau in σ' = σ'(ω).

As an example, in Fig. 10.5, the frequency dependences of the AC conductivity at room temperature are shown for MWNT–PC composites with different CNT content below and above the percolation threshold. The frequency dependence of the conductivity can be described in the frame of charge carrier diffusion in percolation structures or alternatively by resistor–capacitor composites (for details, see references in Pötschke et al., 2003; Alig et al., 2007a). Below the percolation threshold, the expected slowing down of the critical frequency with increasing MWNT content can be seen. The critical frequency is indicated in the experimental σ'(ω) curves for p < pc by the cross-over from DC plateau to power law behaviour.

10.5 AC conductivity as a function of frequency for PC with different MWNT content.

An alternative description of the insulator–conductor transition is the generalized effective medium (GEM) theory (Equation 10.6) (see McLachlan et al., 1990; Andrews et al., 2001; Almond and Bowen, 2004):

[10.6]

where σDC, σmatrix and σfiller are the electrical conductivities of the composite, the polymer matrix and the filler. It will be shown in the next section, that the ‘fillers’ are not necessarily the individual nanotubes. The assumption that conductive (sphere-like) agglomerates form the conductive filler network has been shown to be rather successful to describe the time-evolution of the electrical conductivity under shear and in the quiescent melt (Alig et al., 2007b; Alig et al., 2008a; Alig et al., 2008c; Lellinger et al., 2009; Skipa et al., 2009; Skipa et al., 2010). In this model, the filler content p has to be replaced by a time-dependent agglomerate concentration pA(t) and σfiller becomes the conductivity of agglomerates σ0,A.

An empirical equation to describe the insulator–conductor transition has been proposed by (Fournier et al., 1997) and was successfully applied to polymers containing CNT by Coleman et al. (Coleman et al., 1998), Curran et al. (Curran et al., 2006) and McCullen et al. (McCullen et al., 2007). A modified Fournier equation has been proposed more recently (Alig et al., 2008c):

[10.7]

where f(p) = l/l + exp[b(ppc and b is an empirical parameter. This equation was also extended to the assumption of time-dependent agglomerate concentration pA(t) (Alig et al., 2008c; Deng et al., 2009; Skipa et al., 2009).

All three approaches (percolation, GEM theory and empirical Fournier equation) were compared for the same set of MWNT concentrations in PC (Alig et al., 2008c) and in polypropylene (Deng et al., 2009). It turns out that all three equations can fit the conductivity data with similar accuracy. Since the classical percolation theory assumes a random distribution of conductive fillers, it does not describe the distribution of fillers with (attractive) interactions. In order to keep the advantages of the percolation approach, we had to assume that the agglomerates (forming the percolation network) are randomly distributed (Alig et al., 2007b).

### 10.2.2 Primary and secondary agglomeration

#### Primary agglomeration

A first indication of the influence of melt mixing on electrical conductivity was found by measurements of the frequency dependent conductivity on samples prepared by a twin-screw microcompounder (DACA) at different mixing conditions (Pötschke et al., 2003). For a MWNT concentration close to the percolation threshold (1.0 wt% in PC), the mixing time and the screw speed of the microcompounder were varied at a constant processing temperature of 260 °C.

The measurements of AC conductivity (Fig. 10.6) were performed at room temperature on pressed plates prepared under controlled (identical) conditions from extruded strands. The differences in dispersion of the MWNT in the PC matrix are reflected in the AC conductivity spectra. The increase of the low frequency plateau values of the conductivity and in the cross-over frequencies in Fig. 10.6 indicate a transition from an insulator (150 rpm for 5 min.) to a conducting filler network with increasing mixing time or screw speed. These changes cannot be explained in the frame of classical percolation theory alone. One has to assume an increase in the effective amount of nanotubes contributing to the conductive network by destruction of the primary agglomerates during melt mixing. An example of remaining primary agglomerates is shown in Fig. 10.7 for a PC–MWNT composite (0.6 vol.% MWNT) after insufficient melt mixing (for 5 min. with a screw speed of 50 rpm at 300 °C) and gently pressing at 300 °C. The conductivity of this sample is 5 × 10–5 S/cm and below the value of 5 × 10–4 S/cm for the same material mixed for 15 min. with the same parameters and pressed under identical conditions (see Fig. 10.8(b)).

10.6 Influence of melt mixing: AC conductivity as a function of frequency for different mixing conditions (screw speed of the microcompounder and mixing time) for PC with 1.0 wt.% MWNT (measurements on pressed plates at room temperature).

10.7 Example of remaining primary agglomerates: transmission electron micrograph of PC with 0.6 vol.% MWNT (mixing time: 5 min., screw speed: 50 rpm, mixing temperature: 300 °C and pressing at T = 300 °C).

10.8 Transmission electron micrographs from PC–MWNT plates with 0.6 vol.% MWNT: (a) non-conductive state (σDC < 10–16 S/cm) with well-dispersed nanotubes (mixing time of 15 min. at 250 °C and pressed at 265 °C); (b) conductive state (σDC = 5 × 10–4 S/cm) with nanotube agglomerates mixed for 15 min. at 30o°C and pressed at 300 °C.

#### Secondary agglomeration

It is usually assumed that nanofillers have to be well dispersed in the polymer matrix to achieve the desired properties. Therefore, a lot of effort is spent during sample preparation or polymer melt processing (especially during compounding) to avoid agglomeration of nanofillers. The left transmission electron microscope (TEM) image in Fig. 10.8 shows a sample with well-dispersed nanotubes, where the primary agglomerates are assumed to be almost destroyed by the pre-treatment: melt mixing for 15 min. with 50 rpm at 250 °C and pressing at 250 °C with 0.5 mm/min. However, for this sample, the DC conductivity is below 10–16 S/cm, which is in the order of magnitude of the conductivity of the net polymer.

A sample prepared by mixing for 15 min. with 15 rpm at 300 °C and pressed at T = 300 °C shows a considerable nanotube agglomeration (see Fig. 10.8 (b)). Surprisingly, the electrical conductivity of this sample is about twelve orders of magnitude higher (5 × 10–4 S/cm) than that for the sample pressed at lower temperature and showing well-dispersed nanotubes. It has been assumed that the higher viscosity at 250 °C leads to a higher shear stress during compounding and in the squeeze flow during pressing, which destroys the conductive agglomerates. On the other hand, the lower viscosity of the polymer matrix at 300 °C may accelerate nanotube agglomeration during pressing. The considerably higher conductivity of the sample with secondary agglomerates (Fig. 10.8 (b)), which are assumed to be formed by attractive interactions between nanotubes, compared to the sample (Fig. 10.8 (a)) with well-dispersed CNT is remarkable. A more detailed study of the influence of extrusion and pressing conditions on the secondary agglomeration (TEM and conductivity) can be found in (Pegel et al., 2008).

A similar increase in conductivity was found for PC–MWNT melts during isothermal annealing in the quiescent melt (absence of shear) and has also been attributed to secondary agglomeration of nanotubes and formation of a conductive network of interconnected agglomerates (Alig et al., 2007b; Alig et al., 2008a; Alig et al., 2008c; Alig et al., 2008d). Fig. 10.9 shows the DC conductivities for a series of injection-moulded samples before and after annealing for one hour at 230 °C. The samples after injection moulding represent a state with almost well-dispersed nanotubes (Alig et al., 2008c). The increase of conductivity with annealing time is discussed in terms of a time-dependent insulator–conductor transition due to nanotube agglomeration or by ‘dynamic percolation’ (Alig et al., 2008c).

10.9 Insulator–conductor transition for injection-moulded PC plates with MWNT. The open circles represent conductivity after injection moulding (dispersed nanotubes) and the closed circles represent the conductivity after annealing for about 5 hours at 230 °C.

Based on these results, secondary agglomeration is considered to be a key process in understanding the dependence of electrical conductivity on thermal and rheological prehistory. Therefore, it was discussed for PC–MWNT that the dispersed state of nanotubes leads to a low conductivity due to the insulating polymer chains in the contact regions between the nanotubes (Pötschke et al., 2004; Alig et al., 2007b; Alig et al., 2008d). In this picture, the nanotube agglomeration is assumed to result in a denser packing of nanotubes inside the agglomerates and correspondingly in a smaller distance between MWNT. Very small inter-particle distances (~ few nm, almost physical contacts) are necessary for low contact resistance (Ruschau et al., 1992; Gojny et al., 2005; Gojny et al., 2006; Meier and Klüppel, 2008) and efficient electron transport through the conductive filler network. For the electron transport between CB, tunnelling of electrons was proposed (Meier and Klüppel, 2008). This explanation can be extended to CNT contact regions. We assume that sufficiently small MWNT distances can only be achieved inside the densely packed filler agglomerates.

### 10.2.3 Modelling of build-up and destruction of agglomerates

The growth of the conductive network in a polymer melt can be considered in the simplest case as a clustering process in which two non-conductive particles (e.g. well-dispersed nanotubes) interact and create a conductive agglomerate A. For attractive particles in a quiescent melt, this leads to a time-dependent increase of the volume concentration of agglomerates pA(t). These agglomerates are assumed to form the conductive filler network. For a description of the agglomeration of nanotubes, a kinetic equation of n-th order can be written.

In the presence of steady shear, the kinetic equation has to be extended by two additional shear-dependent terms for shear-induced destruction and agglomeration (Alig et al., 2007b; Alig et al., 2008a; Alig et al., 2008c; Alig et al., 2008d). For the assumption of additivity and first order kinetics one can write:

[10.8]

where pA is the volume concentration of conductive agglomerates, pA0 is the starting agglomerate concentration, pA∞ is the value for and t → ∞, k0 is the kinetic coefficient for quiescent agglomeration, for shear-stimulated agglomeration and for the shear-stimulated destruction process. The coefficients k1 and k2 are assumed to depend on shear rate. Equation 10.8 can be rewritten as:

[10.9]

with k = (k0 + k1).

For steady shear conditions, the concentration of agglomerates approaches a steady state value, which is determined by the interplay of destruction and build-up effects on filler network in flowing matrices (Skipa et al., 2009, 2010).

The solution of Equation 10.9 pA(t) can be set alternatively into the percolation equations (Equations 10.1 and 10.2), the effective medium approach (Equation 10.6) or the modified Fournier expression (Equation 10.7) to describe the conductivity in the quiescent melts or melts under shear (Alig et al., 2007b; Alig et al., 2008a; Alig et al., 2008c; Alig et al., 2008d). The filler content pfiller in Equations 10.1 and 10.2, 10.6 and 10.7 has to be replaced by the time-dependent content of conductive agglomerates pA (t). For σfiller and σmatrix the conductivity values of the nanotube agglomerates σA and polymer matrix have to be taken. The value pc is now the percolation threshold of the percolating conductive agglomerates. In the following sections, example fits to the time-dependent conductivity data during quiescent agglomeration and shear-stimulated insulator–conductor transition will be shown. For a more detailed description, the shape and size distribution of conductive agglomerates/clusters has to be considered. A first attempt to consider different sizes of agglomerates has been discussed by Skipa et al. (2010).

## 10.3 Measuring techniques and materials

### 10.3.1 Rheo-electrical experiments

The time-dependent conductivity measurements were performed using a Novocontrol impedance analyser coupled with a laboratory rheometer (Ares, from Rheometric Scientific) in which the rheometer tools (plate-plate geometry) act as electrodes (Alig et al., 2007b; Alig et al., 2008c; Alig et al., 2008d). The set-up is schematically shown in Fig. 10.10.

10.10 Schematic representation of the rheometer combined with conductivity (dielectric) spectroscopy.

For most of the experiments, the plates were equipped with ring electrodes (inner diameter 19 mm, outer diameter 25 mm). By using ring electrodes, it is possible to ensure a relatively narrow distribution of the shear rates in the relevant region of electrical field during rotation. Tangential shear is applied to the melt by rotating the lower rheometer plate, and the electrical conductivity is measured perpendicular to the direction of applied shear. This monitors the electrical conductivity in the volume between the two ring electrodes. The set-up covers a broad frequency range from 10–3 to 107 Hz for the dielectric measurements (Alig et al., 2008d; Lellinger, 2009). Since recording of a conductivity (or permittivity) spectrum over a wide range of frequencies needs at least several minutes, the measured (properties) can change during the measurement. Thus, for the monitoring of fast (time-dependent) processes, the conductivity measurements were performed at a single frequency. In most cases the frequency of f = 1 kHz could be taken as representative of the DC conductivity as discussed in (Alig et al., 2008d).

For simultaneous electrical and rheological measurements during steady shear, the rheometer was connected to a data acquisition system (NI PXI-10042) with an analogue output for the motor control (Lellinger et al., 2009; Skipa et al., 2009; Skipa et al., 2010). Using the analogue outputs of the rheometer and the electrometer, it is possible to measure strain, torque and conductivity with high acquisition rate, and this is perfectly synchronized with the motor control signal. With the modified rheometer it became possible to apply a combined deformation program overlaying steady and oscillatory motions. The motor thus can simultaneously perform steady rotation γsteady (shear flow) and short oscillations. . From the torque data of the oscillations the complex shear modulus of the composite melt can be deduced. The oscillation frequency and the strain rate amplitude were typically taken to be f = 1 Hz and rad/s (corresponding to γ0 = 1% of sample strain amplitude). For calculation of the transient shear viscosity the torque averaged over one period of the overlaid oscillation 〈τ(t)〉 was divided by the averaged shear rate . The set-up allowed simultaneous (time-resolved) monitoring of the DC conductivity (σDC), the components of the complex shear modulus (G′ and G″) and the transient viscosity (η+) under quasi-stationary shear conditions. To measure G′ and G″ in quiescent melts, a small oscillatory shear of 1% strain was applied. The influence of such small oscillations on the electrical conductivity can almost be ignored in the ‘quasi-quiescent’ experiment (Skipa et al., 2010).

### 10.3.2 In-line monitoring during extrusion

For in-line measurements during extrusion a measurement slit die is flanged to the outlet of a twin-screw extruder. A typical set-up for laboratory-scale experiments is shown in Fig. 10.11. Figure 10.11 (b) shows the two half-shells of a slit die with two rectangular electrodes in capacitor geometry (Alig et al., 2008a). For insulation, the electrodes are embedded in a ceramic inlay. After mounting, the two electrode areas are face to face and the assembly was attached to the extruder. An LCR meter (HP4284A, from Agilent) was used to measure parallel capacitance Cp and tanδ in a frequency range from 21.5 Hz to 1 MHz. For experimental details, see (Alig et al., 2007a; Alig et al., 2008a; Alig et al., 2008b). The real parts of the complex conductivity σ′ and permittivity ε′ spectra were calculated from the air capacitance and the measured values of Cp and tanδ, respectively. For measurements of the DC conductivity another electrode was connected to an electrometer. In order to investigate the time-dependent changes of AC or DC conductivity under quiescent melt conditions, the extruder was stopped for a few minutes.

10.11 Measurement slit die for in-line conductivity measurements flanged to the outlet of a twin-screw laboratory extruder (a) and opened slit die in capacitor geometry (b).

### 10.3.3 In-line measurements during injection moulding

For time-resolved in-line measurements of the DC conductivity an injection moulding machine (Klöckner Ferromatik Desma FX75-2 F, Germany) was equipped with a mould (Fig. 10.12) containing three borings on every side to accommodate conductivity and/or pressure sensors in different arrangements (Lellinger et al., 2008). The dimensions of the produced model parts were 220 × 30 × 6 mm3. The electrical conductivity sensor (Fig. 10.12) consists of an inner electrode (diameter 9 mm) electrically insulated by 2 mm thick ceramic material from the ground electrode (outer shell of the sensor and the whole mould). In this study the conductivity sensor was mounted in the central position between two pressure sensors. An electrometer (Keithley 6514) was used to measure the resistance R at constant current. By FEM simulations of the electrode configuration, the geometrical factor A/d was estimated to determine the specific electrical conductivity σ = (R A/d)–1 (Lellinger et al., 2008).

10.12 Scheme of the mould, the plastic part and the conductivity sensor used for in-line conductivity measurements during injection moulding.

The corresponding off-line measurements of the conductivities at the same location of the final parts were conducted using two opposing electrodes with 17 mm diameter. The geometrical factor to calculate the specific electrical conductivities for this electrode configuration was calculated by FEM as well.

### 10.3.4 Materials and samples

The nanotubes used in this study were MWNTs supplied as PC masterbatches from Hyperion Catalysis International, Inc. (Cambridge, MA, USA) or as masterbatches and premixed compounds from Bayer Technology Services (Leverkusen, Germany). According to the supplier, the diameter of Hyperion nanotubes is approximately 10 nm and its length is above 10 μm. The Baytubes® are high purity multi-walled carbon nanotubes with an outer diameter of 5 to 20 nm, a length of 1 to 10 μm and an electrical conductivity of above 104 S/cm (specification, Baytubes®).

Using a DACA microcompounder or a Haake PTW16/25 twin screw laboratory extruder, the masterbatches from Hyperion Catalysis International, Inc. were melt mixed into PC of the same type as used for the masterbatches. Details of the mixing conditions are given in (Pötschke et al., 2003; Alig et al., 2007a; Alig et al., 2008a). Injection-moulded plates were prepared by Bayer Technology Services from PC (Makrolon 2600, Bayer MaterialScience AG) containing different amounts of high purity Baytubes®. The process parameters for injection moulding are given in (Alig et al., 2008c; Skipa et al., 2009; Skipa et al., 2010). For the rheo-electrical measurements, round samples of 25 mm and 2 mm thickness diameter were cut from injection-moulded plates. Transmission electron microscopy (TEM) investigations showed that the injection-moulded samples contain dispersed MWNT with only a small amount of agglomerates (Alig et al., 2008c; Skipa et al., 2009). It was also shown by X-ray experiments that these composites do not contain significant amounts of oriented MWNT (Richter et al., 2009). The electrical conductivity of the injection-moulded plates at room temperature was found to be quite low, having a percolation threshold well above 5 wt% MWNT (Alig et al., 2008c; Skipa et al., 2009).

The in-line investigations during injection moulding (Lellinger et al., 2008) were performed for a wide variety of the processing parameters on PC, an acrylonitrile-butadiene-styrene/polycarbonate blend (Bayblend™) and polyamide-12 (PA12) containing different amounts of high purity multi Baytubes®. The PC and Bayblend™ compounds were provided by Bayer Technology Services and the polyamide-12/MWNT compounds were provided by Evonik Industries AG (Marl, Germany).

## 10.4 Destruction and formation of electrical and rheological networks

### 10.4.1 Conductive filler network

#### Shear induced destruction and conductivity recovery

In Fig. 10.13 time-dependent conductivity measurements for the same PC–MWNT composite with 0.6 vol.% MWNT using the rheo-electrical set-up (Fig.10.13 (a)) and the in-line set-up (Fig. 10.13 (b)) are arranged together to illustrate the effect of shear-induced destruction of the conductive filler network and conductivity recovery in the quiescent melt. The first time interval in Fig. 10.13 (a) shows a conductivity increase by about three orders of magnitude during isothermal annealing of the as-prepared sample measured in the rheo-electrical set-up at 230 °C for 2 hours. This increase can be explained by secondary agglomeration after the sample is heated above its glass transition temperature (Alig et al., 2007b; Alig et al., 2008b). During the short transient shear (ts = 10 s, dγ/dt = 1 s–1), the electrical conductivity decreases tremendously, which can be explained by the (at least partial) destruction of the conductive filler network in the shear field. Due to the reformation of the conductive filler network in the quiescent melt (t > 120 min), the conductivity increases again (‘conductivity recovery’). Parallel measurements of the real part of the shear modulus G′ in the quiescent melt (not shown) also exhibit an increase of G′ in this time interval. However, the shear modulus increases only from about 60 Pa to about 1000 Pa, whereas the conductivity increases by about 6 orders of magnitude in the same time interval. This ‘modulus recovery’ indicates that the ‘healing’ of the viscoelastic filler network after mechanical deformation is somehow related to the reformation of the conductive filler network, although the stress transfer and the charge transport mechanisms are different (see below).

10.13 Indication of shear-induced destruction of the conductive filler network and conductivity recovery in the quiescent melt. Time-dependent conductivity measurements of a PC–CNT composite with 0.6 vol.% MWNT: (a) in a rheometer during isothermal annealing after short transient shear (ts = 10 s, dγ/dt = 1 s–1); (b) in a measurement slit die during melt extrusion at different melt temperatures (screw speed: 175 rpm) and after the extruder has stopped.

In Fig. 10.13 (b), an example (Alig et al., 2008a) of in-line conductivity measurements during extrusion is shown for the same material. The dependence of the conductivity on processing time was studied using the slit die shown in Fig. 10.11. During extrusion of the PC–MWNT melt (screw speed: 175 rpm), the conductivity is very low, since the conductive filler network is almost destroyed by the shear and elongation flow in the extruder and/or the slit die. The conductivity recovers by reformation of the conductive filler network in the quiescent melt (after stopping the extruder).

In Fig. 10.14, the temperature dependence of the conductivity recovery is shown for the same PC–MWNT as in Fig. 10.13 in the quiescent melt after a short shear deformation (shear rate dγ/dt = 1 s–1 for 10 s) in the rheometer (a) and after the extruder was stopped (b) (Alig et al., 2008a). The data in Fig. 10.14 (b) are taken from Fig. 10.13 (b). As expected, the conductivity recovery becomes faster with increasing melt temperature. This temperature dependence is typical of a thermal activated process and can be related to the lower melt viscosity and the faster nanotube agglomeration.

10.14 Temperature dependence of the conductivity recovery in PC containing 0.6 vol.% MWNT in a rheometer with ring electrodes after a short shear deformation (1 rad/s for 10 s) (a) and in a measurement slit die after the extruder was stopped (b). The data after stopping the extruder are taken from Fig. 10.13 (b).

#### Steady shear conditions

To study the shear-induced destruction of a conductive filler network in more detail, time-dependent conductivity measurements were performed in the rheo-electrical set-up for a PC–MWNT composite with initially well-agglomerated MWNT (Fig. 10.15). The initial state was prepared by one hour of isothermal annealing (not shown). First, a steady shear deformation (shear rate of 0.02 rad/s) was applied for one hour, followed by isothermal annealing without shear. During one hour of steady shear the conductivity tends to approach an equilibrium value which indicates a dynamic equilibrium state of the conductive filler network. In the quiescent melt (after shear deformation), the conductivity increases due to the reformation of the conductive CNT network (Skipa et al., 2009; Skipa et al., 2010). Interestingly, the kinetic constant for destruction (assuming first order kinetics) is faster than the kinetic constant for quiescent melt recovery.

10.15 Time-dependent conductivity for an initially agglomerated PC–CNT composite with 0.6 vol.% MWNT in a rheometer under steady shear (dγ/dt = 0.02 rad/s for 1 hour) and during isothermal annealing after shear at 230 °C.

In Fig. 10.16, the formation of a conductive nanotube network by agglomeration during quiescent annealing (triangles) and by shear-stimulated coalescence of nanotubes under steady shear (squares) are compared for an MWNT–PC melt with 1 wt% MWNT at 230 ° C. Both samples represent initially well-dispersed nanotubes (Skipa et al., 2009). It is obvious that a small shear deformation (0.02 rad/s) can already induce an insulator–conductor transition with a conductivity increase by about 6 orders of magnitude. In contrast, the composite melt annealed in the quiescent melt without shear shows a much slower increase of the DC conductivity, which can be attributed to a diffusion controlled agglomeration of attractively interacting nanotubes. Details of the fits are given by (Skipa et al., 2010).

10.16 Conductivity of MWNT–PC melts measured during steady shear (upper curve) and quiescent annealing (lower curve). The solid lines represent fits. The inset schematically shows the measuring cell with the sample. The direction of the applied shear is perpendicular to the electrical field for conductivity measurement.

In contrast to quiescent polymer melts where the nanotube network formation is driven by Brownian diffusion of attractively interacting nanofillers, the formation of the network in external shear fields can be explained by a ‘picking-up’ mechanism under steady shear (Skipa et al., 2009, 2010), where the nanotubes in a shear gradient are collecting other nanotubes on their way and sticking to each other.

#### Steady state network

In Fig. 10.17, the time development of the DC conductivity under steady shear (shear rate: 0.02 rad/s) is compared for two samples (same material with 1 wt% MWNT in PC), one with initially ‘dispersed’ (squares) and the other with ‘agglomerated’ nanotubes (circles). Interestingly, both samples approach for sufficient duration of steady shear flow the same value of the electrical conductivity. This steady conductivity value is assumed to represent a dynamic equilibrium state of the conductive filler network. This supports the assumption of a stationary equilibrium between destruction and build-up of the filler network under steady shear (Skipa et al., 2009, 2010).

10.17 Time dependence of the DC conductivity during 1 hour of steady shear (shear rate: 0.02 rad/s) for two samples (1 wt.% MWNT) with initially ‘dispersed’ (squares) and ‘agglomerated’ nanotubes (circles).

### 10.4.2 Conductive and viscoelastic filler network

#### Relation between electrical and rheological properties

In Fig. 10.18, the conductivity (upper part) and the components of the complex shear modulus, G′ (squares) and G″ (triangles), for a MWNT–PC melt with 1 wt% MWNT measured during steady shear (0.02 rad/s) at 230 °C are plotted versus time. The initial sample (t = 0), prepared by injection moulding (Skipa et al., 2009, 2010), represents a state with almost well-dispersed nanotubes.

10.18 Shear-induced insulator–conductor (upper graph) transition in a PC–MWNT melt (230 °C, 1 wt.% MWNT) during steady shear (shear rate: 0.02 rad/s) and simultaneously measured storage (G′) and loss modulus (G″) (lower graph).

Both conductivity and rheological measurements were performed in parallel and reflect the same phenomenon which takes place in the shear flow. The initial melt had a very low conductivity of about 10–10 S/cm. This value is similar to that of the pure PC melt. At the same time, the initial sample possesses a quite high storage modulus of G′ = 13 kPa and a loss modulus of G″ = 25 kPa, measured with a frequency of 1 Hz and 6.28% strain amplitude. For comparison, the pure PC has G′ = 4 kPa and G″ = 25 kPa.

When steady shear is applied, a tremendous increase of the DC-conductivity (as shown previously in Figs 10.16 and 10.17) and a simultaneous decrease of G′ and G″ are observed. As discussed in Section 10.4.1, both processes can be related to the formation of a filler network formed by agglomerates (Skipa et al., 2009, 2010). The nanotube agglomeration and network formation are schematically shown in Fig. 10.18 as insets. However, for the electrical conductivity, the network of agglomerated CNT is preferred, whereas it is not optimal for the mechanical reinforcement. In general, the mechanisms for charge transport and mechanical reinforcement in composite materials are expected to be different (Rothon, 2003). In order to obtain a conducting pathway in the matrix, an electrical percolation network of interconnected filler conductive particles is necessary. Very small inter-particle distances (~ 1 nm, almost physical contact) are needed for low contact resistance and efficient electron transport through the conductive network. Since the nanotubes in thermoplastic composites are considered to be surrounded or bridged by polymer chains (Pötschke et al., 2004), very small intertube distances are assumed to be achieved only by dense packing of nanotubes inside the agglomerates. In contrast, for efficient reinforcement, a strong interfacial interaction of nanotubes with the matrix and a homogeneous space distribution is needed. The optimal mechanical reinforcement is thus expected for well-mixed MWNTs which strongly interact with the polymer chains. Direct contact between the nanotubes is not necessary in this case. During agglomeration of the nanotubes, a new distribution of the nanotubes is created which results in a new (less efficient) type of the reinforcement: macro-fillers with a low aspect ratio (spherical-like agglomerates) replace nanofillers with high aspect ratio (individual MWNTs).

#### Shear rate dependence and dynamic filler network

In order to investigate the shear-rate dependence of the electrical and mechanical properties under steady flow condition, six similar samples (PC–MWNT, 1 wt% MWNT) were cut out of one injection-moulded plate (Alig et al., 2008c) and thermo-rheologically pre-treated in order to obtain samples with ‘well-agglomerated’ nanotubes and a high level of electrical conductivity. For details of the pre-treatment, see (Skipa et al., 2010). Different shear rates varying from 0.02 rad/s to 0.5 rad/s were applied to the thus prepared samples. In Fig. 10.19, the electrical conductivity (a), the storage modulus G′ (b), and the transient shear viscosity (c) are shown as a function of time after the shear deformation started. The transient viscosity shown in Fig. 10.19 (c) shows a viscosity overshoot, which shifts toward shorter times with increase of the shear rate. The pure PC does not show shear-thinning for the shear rates and temperatures used in this experiment and does not exhibit such a shear overshoot. Furthermore, no indication of (re)-orientation of MWNTs has been found for this system by X-ray measurements (Richter, 2009). Therefore, the appearance of maxima in the transient viscosity curves and the shear thinning can be ascribed to the shear-induced destruction of the MWNT network. For a more detailed discussion and further references, see (Skipa et al., 2010).

10.19 Time evolution of electrical conductivity (a); real part of the storage modulus G′ (b); and the transient shear viscosity (c) of a PC–MWNT melt (230 °C, 1 wt.% MWNT) with initially well-agglomerated nanotubes during 1 hour of steady shear for different shear rates.

After 1 hour of shear, all three quantities achieved for every shear rate almost constant values representing a stationary state of the dynamic filler network. Correspondingly, the values of, σDC, G′, G″ and η+ at t = 600 s can be taken for the stationary (subscript: stat) values for t → ∞. In Fig. 10.19 σstat = σDC (t → ∞) and ηstat = η+ (t → ∞) are plotted versus shear rate. The decrease of σstat and ηstat with the increase of the shear rate is due to the dominance of destruction of the ‘conducting’ and ‘viscoelastic’ networks, respectively. However, the steady state values of the conductivity, the shear modulus G′ and G″ (not shown) and the viscosity represent a dynamic equilibrium between destruction and build-up of the filler network in the steady shear flow. The assumption of a steady state of the ‘dynamic filler network’ seems to be of general importance to attractive interacting fillers in a liquid matrix. A similar shear rate dependence of electrical conductivity and viscosity was reported by Kharchenko et al. (2004) and Obrzut et al. (2007). In these papers, the time-evolution of the conductivity and the rheological properties are not investigated. Kharchenko et al. (2004) described the shear thinning of a PP–MWNT composite by the empirical Carreau equation (Tanner, 2000), where τη and are the characteristic time and the reduced shear rate, respectively. The decrease of the electrical conductivity was described by an analogous empirical function . The decrease of both quantities with was explained by the reduction in the number of nanotube contacts by orientation or by disruption in the flow.

The steady state conductivity data extracted from Fig. 10.19, we assumed as a simple semi-empirical equation (Skipa et al., 2010):

[10.10]

The solid line in Fig. 10.20 (a) represents a fit with Equation 10.10. The fit parameters are: t1 = 64.5 s and σ0 = 8.9 × 10–4 S/cm. The exponent t was taken to be 2. For the fit of the shear-dependent viscosity, the empirical Carreau equation (see above) has been assumed. Further details are discussed in (Skipa et al., 2010).

10.20 Stationary values for the DC conductivity (a) and viscosity (b) after 600 s of steady shear vs. shear rate (data extracted from Fig. 10.19). The lines indicate fits as discussed in the text.

## 10.5 Influence of processing history

Examples of the influence of pressing conditions, thermal annealing and extrusion conditions have already been shown in Sections 10.2 and 10.4. These examples are representative of the melt stage and concentrated on the influence of processing conditions on the CNT dispersion: destruction of primary agglomerates and the formation and destruction conductive (or viscoelastic) network by secondary agglomeration. However, the last step of melt processing is the ‘frustration’ of the filler network by solidification of the plastic part (crystallization or passing the glass transition). Here, we will consider the influence of processing conditions on the electrical conductivity during injection moulding, which is the final process step for most plastic parts. Using the in-line set-up for conductivity measurements shown in Fig. 10.12, different polymer–CNT compounds (MWNT in PC, polyamide and blends) were therefore investigated under different injection moulding conditions (Lellinger et al., 2008).

The electric conductivity during a typical injection cycle is shown in Fig. 10.21 for a polycarbonate–styrene-co-acrylnitrile blend (Bayblend®) containing 4 wt% MWNTs. At the start of the cycle, the mould is empty and the measured conductivity is below the instrument limit. When the melt approaches the conductivity sensor (a), the measured conductivity value increases with the degree of covering of the sensor until it reflects the specific electric conductivity of the part. After the sensor is fully covered by melt, the conductivity immediately decreases, which can be related to the cooling-down and vitrification of the melt at the sensor/mould surface. A few seconds later, the mould is completely filled, which induces a pressure increase up to the holding pressure (here: 400 bar). The pressure increases the density of the sample and thus reduces the distances between the nanotubes, which causes an increase of the conductivity (b). The conductivity in the following time period (c) is nearly constant, until the part breaks off the sensor (not shown).

10.21 Electric conductivity in the first few seconds of an injection cycle for a polycarbonate–styrene-co-acrylonitrile blend (Bayblend®) containing 4 wt.% MWNT (melt temperature 320 °C, mould temperature 90 °C, injection velocity 60 mm/s). For explanations, see text.

In Fig. 10.22, the in-line conductivity data of injection moulding of Bayblend® with 4 wt% MWNT are shown together with off-line measured values for different injection velocity, melt and mould temperature. For the in-line conductivity data, the values recorded 9 secs after the start of injection were taken. The offline data are measured at the same position on the vitrified sample. The good correlation between in-line and off-line data supports the assumption that the in-line values after 9 secs already represent a vitrified (glassy frozen) sample. The same was found for semicrystalline polyamide (Lellinger et al., 2008) where the measured values after about 10 secs represent a semicrystalline state. The tremendous decrease in the conductivity with increasing injection speed and decreasing temperature can be explained by the dominance of destruction of the conductive filler network due to increasing shear stress either because of an increasing shear rate (with injection velocity) or because of increasing viscosity (with temperature decrease). The relative high conductivity values at low injection velocities and high melt temperatures cannot be explained by the destructive effect of shear deformation alone.

10.22 Influence of injection speed v (a), melt temperature (b) and mould temperature Tmould (c) on the in-line and off-line measured electrical conductivity of a polycarbonate–styrene-co-acrylnitrile blend (Bayblend®) containing 4 wt.% MWNT.

Therefore, we assume that the measured conductivity values in the solidified state represent the ‘frustrated’ conductivity values resulting from the competition of deformation-induced build-up and destruction along the injection track. No significant effect of the mould temperature is seen. Since the mould temperature mainly determines the average cooling rate for the ‘frustration’ of the filler network by vitrification or crystallization, this finding supports our assumption that the conductivity values are mainly determined by the flow conditions and that the time for conductivity recovery is too short for typical injection cycles.

## 10.6 Conclusion

The influence of thermo-rheological prehistory on the electrical conductivity and the viscoelastic properties of polymer melts filled with carbon nanotubes is reviewed. For an illustration of shear-induced destruction and the build-up of the filler network and its reformation in the quiescent melt, time-resolved rheo-electrical laboratory experiments and in-line measurements during extrusion and injection moulding are presented and discussed. The experimental set-ups for simultaneous (time-resolved) measurements of the electrical conductivity and the storage (G′) and loss components (G″) of the complex shear modulus and for in-line conductivity measurements during extrusion and injection moulding are described.

The main results can be summarized as follows:

1. To achieve a polymer–carbon nanotube composite with high electrical conductivity, the primary agglomerates have first to be ‘dissolved’ in the polymer matrix to get well-separated nanotubes, which can then rearrange into secondary agglomerates forming the conductive filler network. For well-dispersed nanotubes, a high contact resistance between the nanotubes is expected due to the surrounding polymer chains, whereas nanotube agglomeration is assumed to result in denser packing, small inter-particle distances (which possibly allow tunnelling of electrons) and a low contact resistance. The latter allows efficient charge carrier transport through the conductive filler network.

2. Secondary agglomeration of attractive filler particles is assumed to be thermally activated (Brownian filler diffusion in the melt) and can be accelerated by external mechanical energy (shear-induced agglomeration).

3. Steady state values of electrical conductivity, storage (G′) and loss modulus (G″) for steady shear conditions indicate an equilibrium between build-up and destruction of the filler network (‘dynamic equilibrium’). The competition between formation and destruction of agglomerates under shear can be described by kinetic models.

4. A model combining agglomeration kinetics and an equation for the insulator–conductor transition was developed to describe the electrical conductivity in composites containing conductive fillers. Different approaches to the insulator–conductor transition have been tested: the classical percolation model, the modified Fournier equation, and the generalized effective medium (GEM) theory.

5. The mechanisms of charge carrier transport in a nanotube network and the effect of mechanical reinforcement are found to be different. From simultaneous measurements of electrical conductivities and shear modulus (G′ and G″), it became evident that a network formed by nanotube agglomerates increases the electrical conductivity, whereas the shear modulus is reduced. The optimal mechanical reinforcement is expected for randomly distributed nanotubes forming a combined (viscoelastic) nanotube–polymer network.

The results found for carbon nanotubes in a polymer matrix seem to have general importance for filler networks in viscoelastic or viscous matrices. Apart from its scientific interest, the results can help optimize industrial production of polymer–nanotube composites.

## 10.7 Acknowledgements

Part of the results was funded by the German Federal Ministry of Education and Research (BMBF) in the WING Program ‘Virtual Material Development’ (CarboNet, funding No. 03X0504E) and in the Framework Concept ‘Research for Tomorrow’s Production’ (CompoMel, funding No. 02PU2394). Furthermore, we thank the Bundesministerium für Wirtschaft (German Federal Ministry for Economic Affairs) via the Arbeitsgemeinschaft Industrieller Forschungsgesellschaften (AiF) for financial support of the AiF projects No. 122Z and 14454 N. We would like to acknowledge W. Böhm (DKI) and Dr. M. Engel (DKI, TU Darmstadt) for their help with dielectric and rheological measurements and Dr. M. Bierdel (Bayer Technology Services), Dr. H. Meyer (Bayer MaterialScience AG) and Dr. S. Hermasch (Evonik Industries AG, Marl, Germany) for providing part of the samples. Furthermore, we would like to thank Hyperion Catalysis International, Inc. (Cambridge, MA, USA) for providing MWNT masterbatches. Dr. M. Engel (DKI, TU Darmstadt), Dr. D. Xu (DKI), A. Ohneiser (DKI), H. Dörr (DKI), F. Pfleger (DKI) and G. Vulpius (DKI), we would like to acknowledge for their contribution to the in-line experiments. Prof. G. Heinrich and Dr. M. Grenzer (IPF, Dresden) we thank for helpful discussions. Last but not least, we would like to thank Dr. P. Pötschke (IPF, Dresden) for the very fruitful cooperation over the past few years.

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