The effect of DBP of carbon black on the dynamic self-assembly in a polymer melt

Abstract

The dynamic percolation behavior of conductive fillers in conductive polymer composites (CPCs) has drawn wide interest due to its crucial influence on the final properties. It is thought that the viscosity of the neat polymer, filler–polymer interaction and entanglement in the filler network are crucial issues. Meanwhile, the structure and related characteristics of the filler is an important parameter for determining filler properties and various functionalities. However, their influence on the filler dynamic self-assembly process in a polymer matrix has been barely investigated. Herein, three types of carbon black (CB) with different dibutyl phthalate (DBP) absorption have been used to study the electrical percolation behavior in thermoplastic polyurethane with methods including in situ electrical measurement during isothermal annealing, scanning electron microscopy (SEM), and rheological study as well as theoretical analysis. It is observed that a higher DBP value leads to a lower percolation threshold. During dynamic percolation, the activation energy increases almost linearly with DBP absorption. It is thought that the more stable pre-formed conductive networks for CB with more DBP are responsible. Thus, the driving force for the self-assembly process is lower for CB with more DBP. This study provides new insight for the dynamic self-assembly process of a functional filler in a polymer matrix.

---

1. Introduction

Various functional polymer composites, including electrically conductive,¹ thermally conductive,² dielectric,³ electromagnetic interference (EMI) shielding⁴ and pressure/strain/damage sensing^5–7 composites etc., have been extensively investigated. These functionalities need to be delivered by various functional fillers and their network structure. The morphological control of these functional fillers or structures during the network formation process is shown to play a vital role in the final properties of these functional polymer composites. There are many methods that can be used to control the filler network morphology and resulting functionalities.⁸ For instance, electrically conductive polymer composites (CPCs) can be obtained by incorporating conductive fillers into insulating polymer matrices. Various conductive fillers, such as carbon black (CB),⁹ carbon nanotubes (CNTs),^10,11 carbon fiber (CF)¹² and graphene,^13,14 etc., can be used. Percolation theory has been used to describe the relationship between the composite conductivity and filler content.^15,16 With the filler content reaching a critical value (the percolation threshold), a drastic increase is observed in the composite conductivity due to the formation of conductive networks. Such a percolation threshold is shown to be largely influenced by various morphology controlling methods.^10,17–19

It is often thought that the filler morphology, including the orientation, dispersion, and network structure, and thus the related functionalities, are static for a given filler filled functional polymer composite system, especially without the influence of processing related shear field or chemical reaction. However, it has been reported that fillers can self-assemble into network structures in a polymer melt or solution.²⁰ And it could have a crucial effect on their functionalities. For instance, the construction of conductive networks in CPCs is a thermodynamically non-equilibrium dynamic process. There is an interfacial energy difference and chemical incompatibility between the filler and polymer matrix, which drives the aggregation and flocculation of filler particles or aggregates in the polymer melt to approach equilibrium.²¹ Thus, the formation of conductive networks is highly dependent on temperature (T) and time (t).^22–25 Thermal treatment at elevated temperatures could accelerate the structure evolution of percolating networks which is caused by self-assembly of the filler in the matrix melt.^26,27 Therefore, these fillers can be considered to have a certain degree of mobility, which allows the electrical conductivity to increase drastically at a certain annealing time (the percolation time) during such a dynamic percolation process. Wu et al.²⁸ studied the dynamic percolation of CB/PMMA, where CB and oxidized CB were compared. A quantitative relation between the percolation time, annealing temperature, CB concentration and CB–polymer interfacial energy is observed. the For CB/PMMA system, theoretical analysis as well as experimental results revealed that the activation energy for such a dynamic percolation of conductive network formation is close to the activation energy of η^*₀ (zero-shear-rate viscosity) for the neat polymer. It is thought that the filler mobility due to agglomeration could incisively reflect the mobility of the polymer layer surrounding the carbon filler surface, therefore, such dynamic percolation measurements can be potentially used to characterize polymer dynamics,²⁰ and particle–polymer interaction.²⁸ It is also reported that filler–matrix interaction has important influence on such an activation energy. The activation energy of the oxidized CB/PMMA system is higher than that of η^*₀ for the pure polymer, due to a decrease in the interfacial free energy and enhanced filler matrix interaction.²⁸ Meanwhile surface fluorinated CB can lead to reduced activation energy due to reduced interaction between the filler and matrix.²³ Regarding CPCs containing a higher aspect ratio filler, the dynamic process for multi-walled carbon nanotubes (MWNTs) and carboxyl-tethered MWNT (MWNT-COOH) filled PVDF systems was investigated by Zhang et al.²⁹ They observed a higher percolation time and activation energy when the interaction between the filler and matrix is enhanced. Similar results are also observed by Bao et al. under the influence of an electric field.^30,31 The same group demonstrated that graphene nanoplates have lower activation energy, hence form a conductive network more easily than CNTs do.³² However, the detailed mechanism responsible is not investigated.

To conclude, there are still many issues that need to be investigated regarding the dynamic percolation process in CPCs, let alone the dynamic processes in other filler filled functional polymer composites. Among them, the influential issues and origin of the dynamic network formation process in CPCs are important and need further investigation. The structure and related characteristics of the filler is an important parameter for determining the filler properties and various functionalities. However, their influence on the dynamic self-assembly process in a polymer matrix has been barely investigated to the best of our knowledge. For instance, CB is widely used as a reinforcing and conductive filler for polymeric materials.³³ Their structure and network morphology play important roles in the final properties of these composites. It consists of nearly spherical primary particles, which are fused together in aggregates. The degree of aggregation for these CB particles is commonly known as the structure and is measured by a method based on dibutyl phthalate (DBP) absorption.³⁴ The percolation threshold of such composites is reported to be strongly influenced by DBP,³⁵ the specific surface area, and other properties of CB.^36–38 However, the effect of the CB structure, such as DBP absorption, on the dynamic percolation of CPCs has not been investigated to the best of our knowledge. And this might be able to reveal the mechanism of such a process from a different perspective. It could also shed some light on the dynamic network formation process of various functional fillers in a given polymer matrix.

Therefore, the dynamic percolation behavior of CPCs based on CB/thermoplastic polyurethane (TPU) is studied. Three types of CB with different DBP absorption are used as conductive fillers. A real-time measurement of resistivity and dynamic rheological studies are carried out to trace the dynamic percolation process of conductive network formation during annealing. The effect of DBP of CB on the static percolation threshold and dynamic percolation process of these CPCs is investigated. Characterizations on the rheological properties and the morphology of these CPCs are carried out to provide structural information during the dynamic percolation process. Meanwhile, a theoretical approach is used to investigate the activation energy of such a dynamic process for various CB filled CPCs. The correlation between the CB structure and the respective dynamic percolation process in the TPU melt is discussed.

2. Experimental

2.1 Materials and sample preparation

Polyester based TPU (Irogran PS 455-203) from Huntsman Corp. was used as the matrix in the composites. It consists of 9.9% 4,4′-methylenediphenylene isocyanate (MDI), 58.2% butyl diol, and 31.8% adipate segments with hydroxyl groups or carboxyl groups.⁷ Three types of CB with different DBP absorption were used as conductive fillers: Printex XE2B (diameter: 35 nm; N₂ surface area: 1000 m² g⁻¹; DBP adsorption number: 420 ml/100 g; Evonik Degussa); VXC72 (diameter: 30 nm; N₂ surface area: 254 m² g⁻¹; DBP adsorption number: 178 ml/100 g; Cabot); and Printex 35 (diameter: 31 nm; N₂ surface area: 65 m² g⁻¹; DBP adsorption number: 42 ml/100 g; Evonik Degussa). For convenience, Printex 35, Vulcan XC72, and Printex XE2B are labeled as P35, VXC72 and XE2B. Both the polymer and carbon particles were dried at 70 °C for 24 h under vacuum before being mixed in an internal mixer (XSS-300, Qingfeng Mold Factory, Shanghai, China) at 160 °C. These composites were then compression molded at 160 °C for 5 min under a pressure of 10 MPa followed by quenching.

2.2 Electrical resistivity measurements

Firstly, the resistivity of these compressed films was measured with a Keithley 6487 picoammeter under a constant voltage of 1 V to avoid a strong electric current. Silver paint was applied onto both ends of the sample to ensure good contact. Each data point represents the average of measurements on 4 different samples. Then, the sample resistivity was monitored during annealing at a given temperature. The resistivity was measured and recorded using a Keithley 6487 interfaced with a computer with an applied voltage of 1 V. The samples with a size of 5 mm × 20 mm are cut from the compressed film and fixed on a Teflon plate. Silver paint was applied onto both ends of the sample to ensure good contact between the sample and copper electrodes. These samples are placed in silicone oil equipped with a temperature-controlled apparatus (IKA) for annealing and to avoid oxidation.

2.3 Dynamic rheological tests

Rheological measurement was performed on a dynamic rheometer (Bohlin Gemini 2000, Malvern, British) with parallel plate geometry (diameter 25 mm). The parallel plate fixture with a diameter of 25 mm and a fixed space of 1.5 mm was used at angular frequencies from 0.01 to 100 Hz with the experimental temperature (155, 165 and 175 °C) to evaluate the influence of CB content and self-networking capability on the viscoelasticity of TPU. The dynamic time sweep was performed with a fixed frequency of 1 rad s⁻¹ under 1% strain and at a fixed temperature (155, 165 and 175 °C).

2.4 Scanning electron microscopy (SEM)

SEM observation was performed using FEI Inspect F SEM with an accelerating voltage of 20 kV. These samples were fractured in liquid nitrogen and the fracture surfaces were coated with gold before observation.

3. Results and discussion

3.1 Electrical percolation of composites

The electrical resistivity measurement was carried out at room temperature to investigate the effect of DBP of CB on the static percolation behavior. Fig. 1 shows the dependence of electrical resistivity on the CB content for different CPCs. As the filler content increases, all three systems (P35/TPU, VXC72/TPU and XE2B/TPU) exhibit a drastic transition from electrical insulator to conductor due to the formation of conductive networks. The percolation threshold φ_c can be calculated based on the power law equation:³⁹

σ_c = σ_f(φ − φ_c)^t (1)

where σ_c is the conductivity of the CPCs, σ_f is the filler conductivity, and φ is the filler content. It was noted that the φ_c is near 13 vol% for P35/TPU, 6.1 vol% for CPCs with VXC72, and 3.1 vol% for CPCs with XE2B, indicating a decreasing φ_c with increasing CB DBP absorption (υ). The DBP absorption number is a CB structure parameter characterizing the tendency to form CB networks.³⁶ It has been reported that the percolation threshold in CB filled polymer composites depends on υ in a single polymer matrix. The Janzen equation³⁵ has been proposed to predict the percolation threshold φ_c:

φ_c = 1/(1 + 4ρυ) (2)

where φ_c is the volume fraction at percolation, ρ is the density of CB, and υ is the DBP value in cm³ g⁻¹. Eqn (2) also shows a decrease in φ_c with increasing υ. This is reasonable because CB with a high υ exhibits a strong self-agglomeration capability to form CB networks in the polymer matrix. The results of the experimental (φ_c) and theoretical (φ^*_c) percolation threshold of different CB/TPU composites are listed in Table 1. As the DBP of CB follows XE2B > VXC72 > P35, the percolation of the composites is XE2B < VXC72 < P35. Comparing the current experimental (φ_c) with the theoretical (φ^*_c) percolation threshold, it is noted that the experimental (φ_c) results agree with the theoretical (φ^*_c) prediction quite well for VXC72 and XE2B, but not for the ones containing P35. Other reports from the literature are also listed. For relatively low DBP CB, Janzen et al. showed that the Janzen equation agrees with experimental results. However, Balberg et al.^40,41 demonstrated that two CBs with different DBP based CPCs disagree with the Janzen equation. Therefore, it is thought that the percolation behavior of a given CPC is significantly influenced by several issues: the filler type and properties, matrix and processing related parameters. And it can hardly be determined by the filler properties alone (such as DBP). Therefore, the dynamic percolation behavior of these systems should be studied to understand the influence of DBP on the network formation process in these systems.

Fig. 1 Volume resistivity as a function of CB weight fraction for P35, VXC72, and XE2B filled TPU nanocomposites at room temperature.

Table 1 Experimental (φ_c) and theoretical (φ^*_c) percolation threshold, and conductivity exponent (t) for different CB/TPU composites

Sample name	φc (vol)	t	υ (cm^3/100 g)	φ^*c (vol)	Source
P35/TPU	13.0%	7.2	42	24.3%	Current study
VXC72/TPU	6.1%	6.8	178	7.1%	Current study
XE2B/TPU	3.1%	4.4	420	3.3%	Current study
N880 CB/SBR	31.0%	—	37	26.8%	Janzen^35
Acetylenic CB/natural rubber	8.0%	—	125	9.8%	Janzen^35
CB/PE	39%	6.4	43	23.9%	Balberg^41
CB/PVC	10%	2	350	3.7%	Balberg^40

---

As for the exponent t, it is often thought to be related to the system dimensionality of the composite, with a value of 1 to 1.3 for 2D and 1.6 to 2.0 for 3D distribution of fillers. However, it is noted that the value of t is related to DBP of CB; the less DBP of CB, the higher the t values. Balberg et al.⁴¹ observed similar results and it is proposed that the inter-particle distance distribution has a large width to enable the non-universal high t values in the composites with low DBP CB, while the system is well approximated by an almost equal small inter-particle distance in the composites of high structure CB, and this yields a universal percolation-like behavior.

3.2 Dynamic percolation of composites

To study the conductive network formation by dynamic percolation measurement, the selected filler concentration should be lower than the percolation threshold of the composites. Due to the different percolation of P35, VXC72 and XE2B filled composites, the CB content near or below the percolation threshold is selected. Fig. 2(a)–(c) show the time dependency of resistivity at 165 °C for the P35, VXC72 and XE2B filled TPU composites that have various CB concentrations. The electrical resistivity of these composites decreases slightly as the annealing time increases in the first stage, and then drastically decreases when a critical time is reached, i.e., the transition from insulator to conductor. The critical annealing time is defined as the percolation time, t_p (as shown in Fig. 2(a)), which can be used to determine the timing of conductive network formation. It is observed that: (1) with an increasing CB content, given that the conductive CB network is constructed with a slight migration of the dispersed CB particles, the dynamic percolation curves are shifted to a shorter percolation time; (2) the shape of these dynamic percolation curves for the P35, VXC72 and XE2B filled composites does not change for different CB concentrations, which suggests that the primary characteristic of the conductive network structure does not change with CB concentration; and (3) under the same percolation time (t_p), the content of CB follows the order XE2B < VXC72 < P35 due to the difference in their DBP absorption values.

Fig. 2 Dynamic percolation and dynamic storage modulus as a function of time at 165 °C for the different composites: (a and d) P35/TPU, (b and e) VXC72/TP, and (c and f) XE2B/TPU with various filler loading. (g) The variation of dynamic percolation and rheology in percolation time as a function of the CB content for the different composites at 165 °C.

Then, the effect of temperature on the percolation time is investigated to study this issue further. In order to facilitate comparison, samples with the same initial resistivity at room temperature with their CB content lower than the percolation threshold were selected. Therefore, 20 wt%, 7.5 wt% and 5 wt% for the P35, VXC72 and XE2B filled composites are selected according to Fig. 1, respectively. Fig. 3 shows the dynamic percolation curves for different CB filled TPU composites at various annealing temperatures. Similarly, the dynamic percolation curves are shifted to a lower percolation time with the increasing annealing temperature; this is because the increase in the temperature accelerates both the relaxation of TPU molecules and the Brownian motion of CB, resulting in a decrease in the viscosity resistance of the TPU melt and an increase in the self-assembly rate of CB. The difference between the P35, VXC72 and XE2B filled composites is their final resistivity: XE2B > VXC72 > P35, as shown in Fig. 3. This may be due to the highest DBP value of XE2B. Under annealing, XE2B tends to form more thorough conductive networks. Fig. 4(a)–(c) show the electrical resistivity as a function of φ for the P35, VXC72 and XE2B filled composites at room temperature, as well as annealing at 155, 165, and 175 °C for 30 min. These curves at elevated temperatures show a similar tendency: the percolation threshold of the composites decreases with the increasing annealing temperature. Meanwhile, the percolation threshold of these CPCs after annealing still obeys the Janzen equation as shown in Fig. 4(d). It also suggests that the primary characteristic of the conductive network structure does not change with annealing. This conclusion is consistent with the dynamic percolation measurement.

Fig. 3 Temperature dependence of dynamic percolation curves for (a) P35/TPU, (b) VXC72/TPU, and (c) XE2B/TPU composites filled with a different content of CB; the variation in percolation time as a function of CB content for the different composites at (d) 155 °C, (e) 165 °C and (f) 175 °C.

Fig. 4 Volume resistivity as a function of CB weight fraction for (a) P35/TPU, (b) VXC72/TPU, and (c) XE2B/TPU composites at room temperature and at elevated temperature after being annealed for 30 min. And the percolation as a function of DBP (d).

As is well known, a filler mixed with a polymer can remarkably change the viscoelasticity of the composites due to interaction between the polymer and filler.^42,43 Therefore, viscoelasticity is widely used to characterize polymer dynamics and microstructural evolution of the composites.⁴⁴ Huang et al.⁴⁵ investigated the linear viscoelasticity of a CB filled isotactic polypropylene (PP) melt in which the CB particles can aggregate to form a percolating network accompanied by an increasing storage modulus (G′) with time. Cao et al.⁴⁶ measured the conductive and rheological properties simultaneously in the PE/CB system to investigate the influence of temperature and filler content on percolation time to R and G′. It is thought that three types of network in the CB filled composites are expected: a temporary filler network, a temporary polymer network formed by entanglement, and a combined filler–polymer network.⁴⁷ The combined filler–polymer network has been proposed to explain the rheological behaviors of CB filled polymers. Incorporation of CB into polymers would reduce the mobility of polymer chains between CB aggregates,⁴⁸ which leads to the increment in the modulus. As shown in Fig. 2(e) and (f), the storage modulus (G′) of the composites remains constant with the increasing annealing time at the beginning, however, after a critical time the storage modulus abruptly increases. It is thought that such an increase is caused by the gradual formation of the combined networks described above. By comparing the dynamic percolation and dynamic storage modulus in Fig. 2(g), it can be noted that the percolation time obtained by electrical measurement as well as the onset time obtained by rheological measurement is similar, which confirms the presence of a dynamic network formation process during annealing.

To investigate the self-assembly process of the filler network in these CB/TPU composites, the morphology of P35/TPU before and after annealing is studied. The XE2B/TPU and VXC72/TPU composites were not shown because of their rather low CB content, which makes it difficult to observe the morphology of the CB networks.

Fig. 5(a) shows the composites containing 25 wt% P35 before annealing. It can be observed that CB is rather homogeneously dispersed in the matrix, with particles separated from each other. These particles coagulate together to form networks after annealing for 30 min as shown in Fig. 5(b). The self-networking of the CB particles implies that this CB was in an unstable state in the composite melt. According to Sumita et al.⁴⁹ and Wessling,⁵⁰ the dispersion process of particles in a polymer matrix does not result in a thermodynamic equilibrium system. There is an interfacial energy difference and chemical incompatibility between the filler and polymer matrix that results in the aggregation and flocculation of the filler particles or aggregates in the polymer melt. Therefore, the above rheological and morphological studies confirm the presence of dynamic self-assembly of the CB particle networks in the TPU melt.

Fig. 5 SEM of the P35/TPU composites containing 25 wt% P35 before annealing (a) and after annealing (b).

3.3 Theoretical analysis

Fig. 6(a)–(c) present the Arrhenius plots of t_p versus the inverse of annealing temperature for the P35, VXC72 and XE2B filled composites, respectively. A parallel linear relationship is observed for different CB concentrations. From these plots, the activation energy of conductive network formation can be calculated according to eqn (3):²³

ln(t_p) = ln A − E_a/RT (3)

where E_a is the activation energy, R is the gas constant, and T is the absolute temperature. The activation energy is calculated to be 78.2, 108.7, and 147.6 kJ mol⁻¹ for the P35, VXC72 and XE2B filled composites, respectively. This suggests that the activation energy of the conductive network formation is independent of the CB content. Accordingly, it provides an advantageous method to predict the percolation time for a given system at a certain temperature because of the linear relationship between the percolation time and annealing temperature. The activation energy of the P35, VXC72 and XE2B filled composites follows the order XE2B > VXC72 > P35. Regarding the driving force of such a dynamic process, the excess interfacial energy of the conductive composites is considerably high, therefore, CB aggregates in the melt tends to agglomerate in order to reduce such energy.⁵¹ The driving force for the agglomeration might be the strong dispersive interaction between adjacent CB and the matrix as well as the depletion interaction between adjacent CB aggregates.⁵² As the high DBP XE2B more easily agglomerates than P35 and VXC72 in the process of mixing and molding, it can form more stable conductive networks in the CPCs. Similarly, the conductive network of VXC72 is more stable than P35. As a result, the excess interfacial energy of the conductive composites follows the order XE2B < VXC72 < P35, so the driving force for the agglomeration is XE2B < VXC72 < P35, which indicates that P35 more easily forms a conductive network under annealing. Therefore, the activation energy of the P35, VXC72 and XE2B filled composites is XE2B > VXC72 > P35. At the same time, a linear relationship between the activation energy and DBP absorption is observed in Fig. 6(d), which shows that the activation energy is in direct proportion to the DBP absorption. Meanwhile, the relationship between DBP of CB and the activation energy is also plotted for several other systems in the literature^20,46 as shown in Fig. 6(d). It is noted that the polymer matrix has quite an important influence on the activation energy. For a HDPE based system (even though the HDPE is neither identical, nor from the same study), the activation energy increases with increasing DBP, which agrees with our findings. Overall, it is clear that more study is needed to construct a clearer relationship between DBP and the activation energy for different systems.

Fig. 6 Arrhenius plot of the shift factor for (a) P35/TPU, (b) VXC72/TPU and (c) XE2B/TPU containing various concentrations of carbon black. (d) exhibits the relationship of E_a and DBP.^20,46

Moreover, the activation energy of conductive network formation for the P35/TPU, VXC72/TPU and XE2B/TPU composites are compared with the activation energy calculated from the zero-shear-rate viscosity (η^*₀) for these composites. Fig. 7(a), (c) and (e) give complex viscosity as a function of ω for TPU, TPU/VXC72-3% and TPU/XE2B-2%, respectively. The zero-shear-rate viscosity (η^*₀) was obtained at the terminal frequency zone (ω → 0). η^*₀ is related to reciprocal temperature (1/T) by the Arrhenius equation:

where is the activation energy of the zero-shear-rate viscosity (η^*₀). According to the Arrhenius plot of η^*₀ against (1/T), as shown in Fig. 7(b), (d) and (f), is determined as 86.8 kJ mol⁻¹ (TPU), 123.8 kJ mol⁻¹ (TPU/VXC72-3%) and 138.4 kJ mol⁻¹ (TPU/XE2B-2%), respectively. The values for composites with a different filler content can be found in Fig. 8.

Fig. 7 Angular frequency dependence of the complex viscosity at various temperatures for pure TPU (a), 3 wt% VXC72/TPU (c) and 2 wt% XE2B/TPU (e); and the Arrhenius plot of the zero-shear-rate viscosity for pure TPU (b), 3 wt% VXC72/TPU (d) and 2 wt% XE2B/TPU (f).

Fig. 8 Activation energy calculated from percolation time (t_p) and polymer viscosity (η₀) as a function of CB weight fraction for the P35/TPU, VXC72/TPU and XE2B/TPU composites.

Fig. 8 shows the activation energy of t_p and η^*₀. The zigzag lines are the activation energy of η^*₀. The activation energy for the neat polymer matrix is lower than that for the filled composites with the lowest filler content in this system. This is thought as reasonable because the incorporation of CB into the polymer matrix generally results in mobility restriction of the polymer between the CB particles. Furthermore, the network formation ability of the CB is low due to the low content. Then, it decreases with a further increase of the filler content. The possible reason is that the network formation ability increases with increasing filler content, which leads to a decreased activation energy. For the P35 filled TPU composites, the activation energy increases slightly with a further increase in the filler content above 15 wt%. The mechanism responsible for this is not clear and needs further study. It is found that the activation energy value of conductive network formation for P35/TPU (78.2 kJ mol⁻¹) is very close to the activation energy of η^*₀ for pure TPU (86.78 kJ mol⁻¹, see Fig. 7(a) and (b)).

It seems that the contact process between two carbon particles can be equivalent to the excluding process of polymer molecules between two particles. Since the particle–particle interaction force is very weak, the shear stress and shear rate in the polymer melts are so weak that the movement of the polymer melts between CB particles might be regarded as a state of zero-shear-rate. This result agrees with Wu et al. in the PMMA/CB0 system (as shown in Table 2). However, the activation energy value of conductive network formation for the VXC72/TPU (108.7 kJ mol⁻¹) and XE2B/TPU (147.6 kJ mol⁻¹) composites is much higher than that of η^*₀ for pure TPU (86.8 kJ mol⁻¹), but close to the activation energy of the 3% VXC72/TPU (123.8 kJ mol⁻¹, Fig. 7(c) and (d)) and 2% XE2B/TPU (138.4 kJ mol⁻¹, Fig. 7(e) and (f)) composites, respectively. For the VXC72 and XE2B filler TPU composites, the high structure CB is well represented by elongated particles,⁴¹ which can increase the viscosity of the mixture more considerably as compared to the low structure CB, resulting in an increase in the activation energy value of η^*₀. A similar result was reported by Zhang et al. in VGCF/PVDF composites and Bao et al. in the CNT/EVA composites (as shown in Table 2).¹² Different from various reports and opinions in the literature as discussed in the introduction, the current study demonstrates that the activation energy of the electrical dynamic percolation process is heavily dependant on the filler structure. Moreover, such an activation energy obtained from rheological measurement shows first an increase, then an decrease with the increasing filler content. The filler structure is also shown to have a significant effect on such an activation energy.

Table 2 Activation energy of percolation time (t_p) and polymer viscosity (η₀) for different composites

Sample name	Activation energy of tp (kJ mol^−1)	Activation energy of η^*0 (kJ mol^−1)	Reference
PMMA		158	28
PMMA/CB0	168
PMMA/CB50	194
PVDF		62	12
PVDF/VGCF (100/5)		135
PVDF/VGCF	144
EVA/CNT (99/1)		46.3	30
EVA/CNT-COOH (99/1)		51.5
EVA/CNT	79.4
EVA/CNT-COOH	92.7
TPU		86.8	Current study
TPU/VXC72 (97/3)		123.8
TPU/XE2B (98/2)		138.4
TPU/P35	78.2
TPU/VXC72	108.7
TPU/XE2B	147.6

---

In theory, there should be a critical value for the conductive filler content, below which the conductive network can not be formed under any conditions. Zhang et al.²² proposed a thermodynamic percolation model to predict the percolation time of the HDPE/IPP/VGCF system. The relationship between t_p and ϕ is given as follows:

where η is the viscosity of the matrix at a given annealing temperature, c is a constant, and ϕ is the volume fraction of filler in the matrix. ϕ* is the volume fraction of filler in the equilibrium state, and P(0) and P(∞) are the fractions of the conductive filler that join the conduction networks at t = 0 and at the equilibrium state (t = ∞), respectively. Considering the similarity in the network formation process, the thermodynamic percolation model was applied to the present system to analyze the relationship between t_p and ϕ.

Fig. 9(a)–(c) show the relationship between the volume fraction of different CB and 1/t_p for the P35, VXC72 and XE2B filled composites annealed at 155, 165 and 175 °C, respectively. When 1/t_p is extrapolated to 0, the value of ϕ* can be obtained as listed in Table 2. It is found that ϕ* of the P35, VXC72 and XE2B filled composites follows the order P35 > VXC72 > XE2B, demonstrating that the network formation of XE2B is easier than that of VXC72 and P35. This is consistent with the percolation threshold results at room temperature due to the difference in their DBP value. Moreover, the parameters P(0)/P(∞) and c/η are estimated from eqn (6):

where P(t_p) is the fraction of CB in the matrix which joins the conduction networks at t = t_p. P(t_p)/P(∞) is given as follows:

Fig. 9 The plots of filler volume fraction in the matrix versus 1/t_p for the (a) P35/TPU, (b) VXC72/TPU and (c) XE2B/TPU composites annealed at 155, 165, 175 °C; plots of ln[1 − P(t_p)/P(∞)] versus t_p for the (d) P35/TPU, (e) VXC72/TPU and (f) XE2B/TPU composites annealed at 155, 165 and 175 °C.

Fig. 9(d)–(f) show the plots of versus t_p for the P35, VXC72 and XE2B filled composites at different temperatures, respectively. A linear relationship between and t_p can be observed. The parameter c/η and P(0)/P(∞) can be calculated from the slope and the intercept of the line (Table 3). Finally, the relationship between t_p and ϕ can be calculated from eqn (5) using the value listed in Table 3. It is noted that eqn (5) fits the experimental results very well, indicating that the thermodynamic percolation model can be applied for the current system.

Table 3 Parameters of the percolation model for the P35, VXC72, and XE2B filled TPU systems

Sample	Temperature (°C)	φ* (vol%)	c/η	ln[(1 − P(0))/P(∞)]
P35/TPU	155	4.7	5.9 × 10^−3	−0.136
	165	4.7	8.8 × 10^−3	−0.166
	175	4.7	1.6 × 10^−2	−0.135
VXC72/TPU	155	1.22	1.9 × 10^−3	−0.120
	165	1.22	5.3 × 10^−3	−0.110
	175	1.22	9.2 × 10^−3	−0.098
XE2B/TPU	155	0.61	1.5 × 10^−3	−0.168
	165	0.61	3.9 × 10^−3	−0.156
	175	0.61	8.2 × 10^−3	−0.198

---

Fig. 10 shows the linear relationship between and the inverse of the different annealing temperatures for the P35, VXC72 and XE2B filled systems, and the number is the slope of the line. The correlative equation can be obtained as follows:

Fig. 10 A linear relationship of ln(c/η) versus the inverse of annealing temperature for the P35/TPU, VXC72/TPU and XE2B/TPU composites. The inner number is the slope of the line.

The viscosity of the system as a function of annealing temperature is as follows:

where η is the viscosity of the system, and k is the slope of the line. As is well known, the relationship between the zero-shear-rate viscosity, η₀, and the testing temperature is characterized as follows:⁵³where A is a constant, and ΔE_η0 is the activation energy of η₀.

Comparing eqn (9) and (10), it is noted that:

From Fig. 10, the slope of the P35, VXC72 and XE2B filled composites is −9.5, −15.1 and −16.2, respectively. Then, ΔE_η0 can be calculated through eqn (11). ΔE_η0 of the P35, VXC72 and XE2B filled composites follows the order P35 < VXC72 < XE2B, which indicates a stronger interaction between XE2B and TPU. This result is consistent with the activation energy of conductive network formation discussed above.

Finally, the DBP adsorption number is a CB structure parameter characterizing the tendency to form the CB network. A high DBP absorption number is related to a high capability of the CB particles to aggregate to form a 3D network. It is widely used as an index to characterize the structure and aggregation or self-assembly ability of different CB in the literature. Quite often, different CB itself illustrates a similar morphology under SEM or TEM, but demonstrates a distinct different self-assembly behavior in a polymer matrix or in solution. Meanwhile, the intermolecular force between different CB and TPU, or among CB, or the relationship between the DBP value and the intermolecular force between CB and TPU, are indeed thought of as interesting and important topics for research. It might provide a detailed mechanism for the observed behavior and provide guidelines for future research. Nevertheless, tools such as AFM, polarized Raman spectroscopy, etc., can hardly provide any quantified value for the current system. In this perspective, more study is needed.

4. Conclusions

TPU based CPCs are prepared using three types of CB with different DBP absorption as fillers: P35, VXC72 and XE2B. It is found that the percolation threshold for the different CB filled TPU composites decreases with increasing DBP absorption, which partly agrees with the Janzen equation. This indicates that CB with a higher DBP value exhibits a stronger self-agglomeration capability to form a CB network in a polymer matrix. The dynamic process of CB network formation for various CB filled polymer composites was investigated by in situ electrical measurement during isothermal treatments, and SEM, as well as rheological study. It was found that the dynamic percolation is a function of isothermal treatment time, temperature and filler content. The percolation time, t_p, decreases with increasing temperature, and increases with decreasing filler content at a given temperature. The activation energy increases almost linearly with the increasing DBP value. It is thought that the aggregation of the fillers is caused by the interfacial energy difference between the filler and polymer matrix. As CB with more DBP can agglomerate more easily during mixing and molding, a more stable conductive network can be pre-formed more easily in the composites before isothermal annealing. As a result, the excess interfacial energy of the conductive composites follows the order XE2B < VXC72 < P35. Therefore, the driving force for agglomeration during isothermal annealing follows XE2B < VXC72 < P35, which indicates that P35 has a greater tendency to form a conductive network under annealing. Different from various opinions in the literature, it is illustrated that the activation energy of the electrical dynamic percolation process depends on the filler structure in the current study. Moreover, the activation energy obtained from rheological measurements shows first an increase, then a decrease with the increasing filler content. This study provides new insight into the dynamic percolation process of a functional filler in a polymer matrix.

Acknowledgements

We express our sincere thanks to the National Natural Science Foundation of China for financial support (51273117). H. Deng would like to thank the Ministry of Education (Program for New Century Excellent Talents in University, NCET-13-0383), the Innovation Team Program of Science & Technology Department of Sichuan Province (2014TD0002) and Sichuan Province for financial support (2013JQ0008). We also would like to thank one of the reviewers for fruitful advice regarding intermolecular forces in current system.

Notes and references

1. E. T. Thostenson, Z. Ren and T.-W. Chou, Compos. Sci. Technol., 2001, 61, 1899–1912.
2. Z. Han and A. Fina, Prog. Polym. Sci., 2011, 36, 914–944.
3. R. Popielarz, C. Chiang, R. Nozaki and J. Obrzut, Macromolecules, 2001, 34, 5910–5915.
4. S. R. Dhakate, K. M. Subhedar and B. P. Singh, RSC Adv., 2015, 5, 43036–43057.
5. C.-Y. Li and T.-W. Chou, Nanotechnology, 2004, 15, 1493.
6. G. Zhou and L. Sim, Smart Mater. Struct., 2002, 11, 925.
7. L. Lin, S. Liu, Q. Zhang, X. Li, M. Ji, H. Deng and Q. Fu, ACS Appl. Mater. Interfaces, 2013, 5, 5815–5824.
8. H. Deng, L. Lin, M. Ji, S. Zhang, M. Yang and Q. Fu, Prog. Polym. Sci., 2014, 39, 627–655.
9. S. Zhang, L. Lin, H. Deng, X. Gao, E. Bilotti, T. Peijs, Q. Zhang and Q. Fu, Colloid Polym. Sci., 2012, 290, 1393–1401.
10. H. Deng, T. Skipa, E. Bilotti, R. Zhang, D. Lellinger, L. Mezzo, Q. Fu, I. Alig and T. Peijs, Adv. Funct. Mater., 2010, 20, 1424–1432.
11. S. Zhang, H. Deng, Q. Zhang and Q. Fu, ACS Appl. Mater. Interfaces, 2014, 6, 6835–6844.
12. C. Zhang, L. Wang, J. Wang and C.-A. Ma, Carbon, 2008, 46, 2053–2058.
13. X. Li, H. Deng, Z. Li, H. Xiu, X. Qi, Q. Zhang, K. Wang, F. Chen and Q. Fu, Composites, Part A, 2015, 68, 264–275.
14. X. An and C. Y. Jimmy, RSC Adv., 2011, 1, 1426–1434.
15. S. Kirkpatrick, Rev. Mod. Phys., 1973, 45, 574.
16. C. Zhang, J. Sheng, C. Ma and M. Sumita, Mater. Lett., 2005, 59, 3648–3651.
17. X. Gao, S. Zhang, F. Mai, L. Lin, Y. Deng, H. Deng and Q. Fu, J. Mater. Chem., 2011, 21, 6401–6408.
18. S. M. Zhang, eXPRESS Polym. Lett., 2011, 6, 159–168.
19. H. Xiu, Y. Zhou, J. Dai, C. Huang, H. Bai, Q. Zhang and Q. Fu, RSC Adv., 2014, 4, 37193–37196.
20. G. Wu, S. Asai and M. Sumita, Macromolecules, 2002, 35, 1708–1713.
21. V. Bravo, A. Hrymak and J. Wright, Polym. Eng. Sci., 2004, 44, 779–793.
22. C. Zhang, P. Wang, C.-a. Ma, G. Wu and M. Sumita, Polymer, 2006, 47, 466–473.
23. A. Katada, Y. Konishi, T. Isogai, Y. Tominaga, S. Asai and M. Sumita, J. Appl. Polym. Sci., 2003, 89, 1151–1155.
24. M. Traina, A. Pegoretti and A. Penati, J. Appl. Polym. Sci., 2007, 106, 2065–2074.
25. X. Yi, Function principle of filled conductive polymer composites, National Defence Industry Press, Beijing, 2004, p. 39.
26. G. G. Böhm and M. N. Nguyen, J. Appl. Polym. Sci., 1995, 55, 1041–1050.
27. B. H. Cipriano, A. K. Kota, A. L. Gershon, C. J. Laskowski, T. Kashiwagi, H. A. Bruck and S. R. Raghavan, Polymer, 2008, 49, 4846–4851.
28. G. Wu, S. Asai, C. Zhang, T. Miura and M. Sumita, J. Appl. Phys., 2000, 88, 1480.
29. C. Zhang, J. Zhu, M. Ouyang, C. Ma and M. Sumita, J. Appl. Polym. Sci., 2009, 114, 1405–1411.
30. Y. Bao, H. Pang, L. Xu, C.-H. Cui, X. Jiang, D.-X. Yan and Z.-M. Li, RSC Adv., 2013, 3, 24185–24192.
31. Y.-C. Zhang, D. Zheng, H. Pang, J.-H. Tang and Z.-M. Li, Compos. Sci. Technol., 2012, 72, 1875–1881.
32. H. Pang, C. Chen, Y.-C. Zhang, P.-G. Ren, D.-X. Yan and Z.-M. Li, Carbon, 2011, 49, 1980–1988.
33. J.-B. Donnet, Carbon black: science and technology, CRC Press, 1993.
34. W. Hess, C. Herd, J. Donnet, R. Bansal and M. Wang, Carbon Black Science and Technology, Marcel Dekker Inc., New York, 1993, p. 106.
35. J. Janzen, J. Appl. Phys., 1975, 46, 966.
36. A. I. Medalia, Rubber Chem. Technol., 1986, 59, 432–454.
37. T. Noda, H. Kato, T. Takasu, A. Okura and M. Inagaki, Bull. Chem. Soc. Jpn., 1966, 39, 829–833.
38. F. Ehrburger-Dolle, J. Lahaye and S. Misono, Carbon, 1994, 32, 1363–1368.
39. D. Stauffer and A. Aharony, Introduction to percolation theory, CRC Press, 1994.
40. I. Balberg, Phys. Rev. Lett., 1987, 59, 1305.
41. I. Balberg, Carbon, 2002, 40, 139–143.
42. Y. S. Lipatov, Relaxation and viscoelastic properties of heterogeneous polymeric compositions, Springer, 1977.
43. S. J. Chin, S. Vempati, P. Dawson, M. Knite, A. Linarts, K. Ozols and T. McNally, Polymer, 2015, 58, 209–221.
44. J. D. Ferry, Viscoelastic properties of polymers, John Wiley & Sons, 1980.
45. S. Huang, Z. Liu, C. Yin, Y. Wang, Y. Gao, C. Chen and M. Yang, Colloid Polym. Sci., 2011, 289, 1673–1681.
46. Q. Cao, Y. Song, Y. Tan and Q. Zheng, Polymer, 2009, 50, 6350–6356.
47. P. Pötschke, M. Abdel-Goad, I. Alig, S. Dudkin and D. Lellinger, Polymer, 2004, 45, 8863–8870.
48. L. Karasek and M. Sumita, J. Mater. Sci., 1996, 31, 281–289.
49. M. Sumita, K. Sakata, S. Asai, K. Miyasaka and H. Nakagawa, Polym. Bull., 1991, 25, 265–271.
50. B. Wessling, Adv. Mater., 1993, 5, 300–305.
51. K. Miyasaka, K. Watanabe, E. Jojima, H. Aida, M. Sumita and K. Ishikawa, J. Mater. Sci., 1982, 17, 1610–1616.
52. J. G. Meier, J. W. Mani and M. Klüppel, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 75, 054202.
53. C. Su, L. Xu, R.-J. Yan, M.-Q. Chen and C. Zhan, Mater. Chem. Phys., 2012, 133, 1034–1039.

---