# Linear and Nonlinear Rheology Combined with Dielectric Spectroscopy of Hybrid Polymer Nanocomposites for Semiconductive Applications

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^{2}

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## Abstract

**:**

## 1. Introduction

#### Present Study

## 2. Materials and Methods

#### 2.1. Materials and Preparation

^{®}260G, TIMCAL Graphite and Carbon, Bodio, Switzerland, and the GnP used was xGnP Grade M5 from XG Sciences, Lansing, MI, USA. The CB was characterized by a surface area of 70 ${\mathrm{m}}^{2}$/g and a density of 1.8 g/${\mathrm{cm}}^{3}$, while the GnP had a thickness of 6–8 nm, a characteristic diameter of 5 $\mathsf{\mu}$m, a surface area of 120–150 ${\mathrm{m}}^{2}$/g and a density of 2.2 g/${\mathrm{cm}}^{3}$ (manufacturer data). The composition of the EBA-GnP-CB hybrid nanocomposites was optimized by Oxfall et al. [2], and Arino et al. [3] further improved the electrical properties of the nanocomposites by varying the processing conditions, such as screw geometry, temperature profile and screw speed. A Brabender, Brabender GmbH, Duisburg, Germany, 19/25D (barrel diameter of 19 mm and barrel length of $19\times 25$) single-screw extruder, equipped with a 1.5-mm radius circular die was used for sample preparation. More details on the compounding of the master batches can be found in [3].

#### 2.2. Rheological Characterization

#### 2.3. Extensional Rheology and the Molecular Stress Function Theory

#### 2.4. Combined Rheology: Dielectric Spectroscopy

## 3. Results and Discussion

#### 3.1. Linear and Nonlinear Oscillatory Shear

#### 3.2. Extensional Rheology

#### 3.3. Time Dependence and Electrical Conductivity

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

FT | Fourier-transform |

C | compression screw (2:1) |

CB | carbon black |

DES | dielectric spectroscopy |

EBA | ethylene-butyl acrylate |

GnP | graphite nanoplatelets |

LAOS | large amplitude oscillatory shear |

MSF | molecular stress function |

M | mixing screw (Maillefer + Saxton mixer) |

MWCNT | multi-walled carbon nanotubes |

OMMT | organomodified montmorillonite |

PCC | precipitated calcium carbonate |

PCL | polycaprolactone |

PE | polyethylene |

SAOS | small amplitude oscillatory shear |

SWCNT | single-walled carbon nanotubes |

## References

- Ferrari, A.C.; Bonaccorso, F.; Fal’ko, V.; Novoselov, K.S.; Roche, S.; Boggild, P.; Borini, S.; Koppens, F.H.L.; Palermo, V.; Pugno, N.; et al. Science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems. Nanoscale
**2015**, 7, 4598–4810. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Oxfall, H.; Ariu, G.; Gkourmpis, T.; Rychwalski, R.; Rigdahl, M. Effect of carbon black on electrical and rheological properties of graphite nanoplatelets/poly(ethylene-butyl acrylate) composites. Express Polym. Lett.
**2015**, 9, 66–76. [Google Scholar] [CrossRef] [Green Version] - Arino, R.; Diez, E.A.; Rigdahl, M. Enhancing the electrical conductivity of carbon black/graphite nanoplatelets: Poly(ethylene-butyl acrylate) composites by melt extrusion. J. Appl. Polym. Sci.
**2016**, 133. [Google Scholar] [CrossRef] - Induchoodan, G.; Kádár, R. Tailoring polymer nanocomposite microstructure by controlling orientation, dispersion and exfoliation of GnP in LDPE via extrusion flow. Trans. Nord. Rheol. Soc.
**2016**, 25, 187–191. [Google Scholar] - Song, Y.; Zheng, Q. Concepts and conflicts in nanoparticles reinforcement to polymers beyond hydrodynamics. Prog. Mater. Sci.
**2016**, 84, 1–58. [Google Scholar] [CrossRef] - Cassagnau, P. Linear viscoelasticity and dynamics of suspensions and molten polymers filled with nanoparticles of different aspect ratios. Polymer
**2013**, 54, 4762–4775. [Google Scholar] [CrossRef] - Kim, H.; Abdala, A.A.; Macosko, C.W. Graphene/Polymer Nanocomposites. Macromolecules
**2010**, 43, 6515–6530. [Google Scholar] [CrossRef] - Zhao, J.; Morgan, A.B.; Harris, J.D. Rheological characterization of polystyrene–clay nanocomposites to compare the degree of exfoliation and dispersion. Polymer
**2005**, 46, 8641–8660. [Google Scholar] [CrossRef] - Wagener, R.; Reisinger, T.J. A rheological method to compare the degree of exfoliation of nanocomposites. Polymer
**2003**, 44, 7513–7518. [Google Scholar] [CrossRef] - Aranguren, M.I.; Mora, E.; DeGroot, J.V.; Macosko, C.W. Effect of reinforcing fillers on the rheology of polymer melts. J. Rheol.
**1992**, 36, 1165–1182. [Google Scholar] [CrossRef] - Hassanabadi, H.M.; Wilhelm, M.; Rodrigue, D. A rheological criterion to determine the percolation threshold in polymer nano-composites. Rheol. Acta
**2014**, 53, 869–882. [Google Scholar] [CrossRef] - Hassanabadi, H.M.; Abbasi, M.; Wilhelm, M.; Rodrigue, D. Validity of the modified molecular stress function theory to predict the rheological properties of polymer nanocomposites. J. Rheol.
**2013**, 57, 881–899. [Google Scholar] [CrossRef] - Leblanc, J.L.; Jäger, K.M. Investigating Nonlinear Viscoelastic Properties of Molten Carbon Black/Poly(ethylene-co-butyl acrylate) Composites, Using Fourier Transform Rheometry and Other Test Techniques. J. Appl. Polym. Sci.
**2016**, 101, 4071–4082. [Google Scholar] [CrossRef] - Ahirwal, D.; Palza, H.; Schlatter, G.; Wilhelm, M. New way to characterize the percolation threshold of polyethylene and carbon nanotube polymer composites using Fourier transform (FT) rheology. Korea-Aust. Rheol. J.
**2014**, 26, 319–326. [Google Scholar] [CrossRef] - Lim, H.T.; Ahn, K.H.; Hong, J.S.; Hyun, K. Nonlinear viscoelasticity of polymer nanocomposites under large amplitude oscillatory shear flow. J. Rheol.
**2013**, 57, 767–789. [Google Scholar] [CrossRef] - Nelson, J.K.; Hu, Y. Nanocomposite dielectrics—Properties and implications. J. Phys. D
**2005**, 38, 213–222. [Google Scholar] [CrossRef] - Jeong, K.U.; Lim, J.Y.; Lee, J.Y.; Kang, S.L.; Nah, C. Polymer nanocomposites reinforced with multi-walled carbon nanotubes for semiconducting layers of high-voltage power cables. Polym. Int.
**2010**, 59, 100–106. [Google Scholar] [CrossRef] - Oxfall, H. Manufacturing and Characterization of Filled Polymeric Systems. Ph.D. Thesis, Chalmers University of Technology, Gothenburg, Sweden, 2013. [Google Scholar]
- Sinha Ray, S.; Okamoto, M. Polymer/layered silicate nanocomposites: A review from preparation to processing. Prog. Polym. Sci.
**2003**, 28, 1539–1641. [Google Scholar] [CrossRef] - Kremer, F.; Schönhals, A. (Eds.) Broadband Dielectric Spectroscopy; Springer: Berlin/Heidelberg, Germany, 2003.
- Potschke, P.; Abdel-Goad, M.; Alig, I.; Dudkin, S.; Lellinger, D. Rheological and dielectrical characterization of melt mixed polycarbonate-multiwalled carbon nanotube composites. Polymer
**2004**, 45, 8863–8870. [Google Scholar] [CrossRef] - Alig, I.; Skipa, T.; Lellinger, D.; Potschke, P. Destruction and formation of a carbon nanotube network in polymer melts: Rheology and conductivity spectroscopy. Polymer
**2008**, 49, 3524–3532. [Google Scholar] [CrossRef] - Moreira, L.; Fulchiron, R.; Seytre, G.; Dubois, P.; Cassagnau, P. Aggregation of Carbon Nanotubes in Semidilute Suspension. Macromolecules
**2010**, 43, 1467–1472. [Google Scholar] [CrossRef] - Bharati, A.; Wubbenhorst, M.; Moldenaers, P.; Cardinaels, R. Effect of Compatibilization on Interfacial Polarization and Intrinsic Length Scales in Biphasic Polymer Blends of PαMSAN and PMMA: A Combined Experimental and Modeling Dielectric Study. Macromolecules
**2016**, 49, 1464–1478. [Google Scholar] [CrossRef] - Figuli, R.; Schwab, L.; Lacayo-Pineda, J.; Deckmann, H.; Wilhelm, M. Combined Dielectric (DEA) and Dynamic Mechanical Thermal Analysis (DMTA) in Compression Mode; Kautschuk Gummi Kunststoffe (KGK): Leverkusen, Germany, 2016; pp. 30–35. [Google Scholar]
- Costa, L.; Achour, M.; Graça, M.; Hasnaoui, M.E.; Outzourhit, A.; Oueriagli, A. Dielectric properties of the ethylene butylacrylate/carbon black nanocomposites. J. Non-Cryst. Solids
**2010**, 356, 270–274. [Google Scholar] [CrossRef] - Kádár, R.; Naue, I.F.; Wilhelm, M. First normal stress difference and in-situ spectral dynamics in a high sensitivity extrusion die for capillary rheometry via the ‘hole effect’. Polymer
**2016**, 104, 193–203. [Google Scholar] [CrossRef] - Merger, D.; Wilhelm, M. Intrinsic nonlinearity from LAOStrain—Experiments on various strain- and stress-controlled rheometers: a quantitative comparison. Rheol. Acta
**2014**, 53, 621–634. [Google Scholar] [CrossRef] - Hyun, K.; Wilhelm, M.; Klein, C.; Cho, K.S.; Nam, J.; Ahn, K.; Lee, S.; Ewoldt, R.; McKinley, G. A review of nonlinear oscillatory shear tests: Analysis and application of large amplitude oscillatory shear (LAOS). Prog. Polym. Sci.
**2011**, 36, 1697–1753. [Google Scholar] [CrossRef] - Walters, K. Rheometry; Chapman and Hall: London, UK, 1975. [Google Scholar]
- Calin, A.; Wilhelm, M.; Balan, C. Determination of the non-linear parameter (mobility factor) of the Giesekus constitutive model using {LAOS} procedure. J. Non-Newton. Fluid Mech.
**2010**, 165, 1564–1577. [Google Scholar] [CrossRef] - Cziep, M.A.; Abbasi, M.; Heck, M.; Arens, L.; Wilhelm, M. Effect of Molecular Weight, Polydispersity, and Monomer of Linear Homopolymer Melts on the Intrinsic Mechanical Nonlinearity ${3}_{0}^{Q}$(ω) in MAOS. Macromolecules
**2016**, 49, 3566–3579. [Google Scholar] [CrossRef] - Hyun, K.; Wilhelm, M. Establishing a New Mechanical Nonlinear Coefficient Q from FT-Rheology: First Investigation of Entangled Linear and Comb Polymer Model Systems. Macromolecules
**2009**, 42, 411–422. [Google Scholar] [CrossRef] - Ewoldt, R.H.; Hosoi, A.E.; McKinley, G.H. New measures for characterizing nonlinear viscoelasticity in large amplitude oscillatory shear. J. Rheol.
**2008**, 52, 1427–1458. [Google Scholar] [CrossRef] - Rolón-Garrido, V.H. The molecular stress function (MSF) model in rheology. Rheol. Acta
**2014**, 53, 663–700. [Google Scholar] [CrossRef] - Abbasi, M.; Golshan Ebrahimi, N.; Nadali, M.; Khabazian Esfahani, M. Elongational viscosity of LDPE with various structures: Employing a new evolution equation in MSF theory. Rheol. Acta
**2012**, 51, 163–177. [Google Scholar] [CrossRef] - Abbasi, M.; Golshan Ebrahimi, N.; Wilhelm, M. Investigation of the rheological behavior of industrial tubular and autoclave LDPEs under SAOS, LAOS, transient shear, and elongational flows compared with predictions from the MSF theory. J. Rheol.
**2013**, 57, 1693–1714. [Google Scholar] [CrossRef] - Polizos, G.; Tuncer, E.; Tomer, V.; Sauers, I.; Randall, C.; Manias, E. Dielectric Spectroscopy of Polymer-Based Nanocomposite Dielectrics with Tailored Interfaces and Structured Spatial Distribution of Fillers. In Nanoscale Spectroscopy with Applications; CRC Press: Boca Raton, FL, USA, 2013. [Google Scholar]
- Meins, T.; Hyun, K.; Ratzsch, K.; Friedrich, C.; Struth, B.; Wilhelm, M. Combined methods in Rheology: Rheo-SAXS, Rheo-NMR and Rheo-Dielectric to bridge length and time scales. Annu. Trans. Nord. Rheol. Soc.
**2011**, 19, 201–206. [Google Scholar] - Meins, T.; Dingenouts, N.; Kübel, J.; Wilhelm, M. In Situ Rheodielectric, ex Situ 2D-SAXS, and Fourier Transform Rheology Investigations of the Shear-Induced Alignment of Poly(styrene-b-1,4-isoprene) Diblock Copolymer Melts. Macromolecules
**2012**, 45, 7206–7219. [Google Scholar] [CrossRef] - Hyun, K.; Höfl, S.; Kahle, S.; Wilhelm, M. Polymer motion as detected via dielectric spectra of 1,4-cis-polyisoprene under large amplitude oscillatory shear (LAOS). J. Non-Newton. Fluid Mech.
**2009**, 160, 93–103. [Google Scholar] [CrossRef] - Rolón-Garrido, V.; Wagner, M. The MSF model: Relation of nonlinear parameters to molecular structure of long-chain branched polymer melts. Rheol. Acta
**2007**, 46, 583–593. [Google Scholar] [CrossRef] - Chan, Y.; White, J.L.; Oyanagi, Y. A Fundamental Study of the Rheological Properties of Glass-Fiber-Reinforced Polyethylene and Polystyrene Melts. J. Rheol.
**1978**, 22, 507–524. [Google Scholar] [CrossRef] - Takahashi, T.; Wu, W.; Toda, H.; Takimoto, J.I.; Akatsuka, T.; Koyama, K. Elongational viscosity of ABS polymer melts with soft or hard butadiene particles. J. Non-Newton. Fluid Mech.
**1997**, 68, 259–269. [Google Scholar] [CrossRef] - Li, L.; Masuda, T.; Takahashi, M. Elongational flow behavior of ABS polymer melts. J. Rheol.
**1990**, 34, 103–116. [Google Scholar] [CrossRef] - Schopp, S.; Thomann, R.; Ratzsch, K.F.; Kerling, S.; Altstädt, V.; Mülhaupt, R. Functionalized Graphene and Carbon Materials as Components of Styrene-Butadiene Rubber Nanocomposites Prepared by Aqueous Dispersion Blending. Macromol. Mater. Eng.
**2014**, 299, 319–329. [Google Scholar] [CrossRef] - Heaney, M.B. Electrical Measurement, Signal Processing, and Displays; CRC Press: Boca Raton, FL, USA, 2003; Chapter 7. [Google Scholar]

**Figure 1.**The screw types used to control the deformation history of the nanocomposites analyzed using a Brabender 19/25D single-screw extruder: (

**a**) C-screw, conventional geometry with 2:1 compression ratio; and (

**b**) M-screw, distributive mixing screw composed of a Maillefer region a Saxton mixing element.

**Figure 2.**Generic principle of Fourier transform rheology showing time-dependent linear and nonlinear stress material response to a sinusoidal strain input and corresponding Fourier transform of the output signal showing the presence of higher harmonics in the nonlinear case.

**Figure 3.**Dynamic strain sweep ($\mathsf{\omega}=5$ rad/s) comparing the dynamic moduli, ${G}^{\prime}$ and ${G}^{\prime \prime}$, and the relative first higher harmonic, ${I}_{3/1}$, for poly(ethylene-butyl acrylate) (EBA) at 160 °C.

**Figure 4.**Simultaneous rheo-dielectric measurements performed to characterize the EBA nanocomposites. (

**a**) Diagram of the combined rheometer and dielectric analyzer for determining the raw stress data and the dielectric spectra in situ. (

**b**) Image of the rheo-dielectric geometries.

**Figure 5.**Dynamic frequency sweep measurements in the linear viscoelastic regime of the samples investigated, prepared using (

**a**) the conventional screw, C, and (

**b**) the distributive mixing screw, M. In both diagrams, the EBA matrix and EBA-GnP are plotted as references.

**Figure 6.**Strain amplitude dependence of the third relative higher harmonic, ${I}_{3/1}$, at various angular frequencies for: (

**a**) EBA, (

**b**) EBA-GnP, (

**c**) 160C100 and (

**d**) 160M100.

**Figure 7.**Extensional measurements (symbols) and the MSF theory predictions (solid lines) for (

**a**) EBA, (

**b**) EBA-GnP(1), (

**c**) EBA-GnP(2), (

**d**) 180C50 (

**e**) 160M100 and (

**f**) 160C100. The EBA-GnP(2) contains 9.5 vol% of GnP and was analyzed in order to test the limits of the strain hardening behavior. The dashed line, LVE, represents the linear viscoelastic envelope.

**Figure 8.**Transient development of the dynamic moduli and the Q-parameter in SAOS, $\mathsf{\gamma}=1$% and $\mathsf{\omega}=0.5$ rad/s: (

**a**) 160C100 and (

**b**) 160M100. All measurements were performed at 160 °C.

**Figure 9.**Transient development of the dynamic moduli and the Q-parameter in LAOS, $\mathsf{\gamma}=100$% and $\mathsf{\omega}=1$ rad/s, for (

**a**) EBA-GnP, (

**b**) 160C100, (

**c**) 180C50 and (

**d**) 180M100. All measurements were performed at 160 °C.

**Figure 10.**Transient development of the dielectric loss, ${\mathsf{\u03f5}}^{\prime \prime}$, spectra for: (

**a**) EBA-GnP, (

**b**) 160C50, (

**c**) 180C50 and (

**d**) 180M100.

**Figure 11.**Interpretation of the dynamic behavior of the hybrid nanocomposites studied in the nonlinear viscoelastic regime, based on the model of Leblanc and Jäger [13].

**Figure 12.**Electrical conductivity as the plateau at low frequencies of the real part of the complex dielectric conductivity [46], ${\mathsf{\sigma}}^{\prime}$, for the samples under investigation: (

**a**) examples of the transient dependence and (

**b**) conductivity at the onset of ${I}_{3/1}$ steady variation as a function of the processing (die) Weissenberg number, $Wi$.

**Figure 13.**Conductivity in the circuit as a function of the Weissenberg number inside the extrusion die, $Wi=\lambda {\dot{\gamma}}_{a}$, of extruded strands before pelletizing and rheological characterization. Data processed from Arino et al. [3].

**Table 1.**Polymer nanocomposite samples characterized: composition and processing method. The following notations are used: CB, carbon black; GnP, graphite nanoplatelets; ${T}_{p}$, processing temperature (die); ${\dot{\mathsf{\gamma}}}_{a}$, apparent processing shear rate (in the extrusion die) defined in Equation (1); $\mathsf{\lambda}$, polymer characteristic relaxation time, defined as the inverse of the angular frequency at the crossing between the dynamic moduli; and $Wi$, the Weissenberg number defined as $Wi={\dot{\mathsf{\gamma}}}_{a}\mathsf{\lambda}$.

Sample | CB/wt % | GnP/wt % | Screw Type | ${\mathit{T}}_{\mathit{p}}$/°C | n/${\mathbf{s}}^{-1}$ | ${\dot{\mathsf{\gamma}}}_{\mathit{a}}$/${\mathbf{s}}^{-1}$ | $\mathsf{\lambda}$/${\mathbf{s}}^{-1}$ | $\mathit{Wi}$ |
---|---|---|---|---|---|---|---|---|

EBA | - | - | C | 160 | 100 | 1080 | 0.02 | 22 |

GnP(1) | - | 15 (7 vol %) | C | 160 | 50 | 540 | 0.04 | 22 |

CB + GnP = 5 vol % | ||||||||

160C50 | 20 | 80 | C | 160 | 50 | 540 | 0.03 | 16 |

160C100 | 20 | 80 | C | 160 | 100 | 1080 | 0.03 | 32 |

180C50 | 20 | 80 | C | 180 | 50 | 540 | 0.03 | 16 |

180C100 | 20 | 80 | C | 180 | 100 | 1080 | 0.03 | 32 |

160M100 | 20 | 80 | M | 160 | 100 | 480 | 0.03 | 14 |

180M100 | 20 | 80 | M | 180 | 100 | 480 | 0.03 | 14 |

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**MDPI and ACS Style**

Kádár, R.; Abbasi, M.; Figuli, R.; Rigdahl, M.; Wilhelm, M.
Linear and Nonlinear Rheology Combined with Dielectric Spectroscopy of Hybrid Polymer Nanocomposites for Semiconductive Applications. *Nanomaterials* **2017**, *7*, 23.
https://doi.org/10.3390/nano7020023

**AMA Style**

Kádár R, Abbasi M, Figuli R, Rigdahl M, Wilhelm M.
Linear and Nonlinear Rheology Combined with Dielectric Spectroscopy of Hybrid Polymer Nanocomposites for Semiconductive Applications. *Nanomaterials*. 2017; 7(2):23.
https://doi.org/10.3390/nano7020023

**Chicago/Turabian Style**

Kádár, Roland, Mahdi Abbasi, Roxana Figuli, Mikael Rigdahl, and Manfred Wilhelm.
2017. "Linear and Nonlinear Rheology Combined with Dielectric Spectroscopy of Hybrid Polymer Nanocomposites for Semiconductive Applications" *Nanomaterials* 7, no. 2: 23.
https://doi.org/10.3390/nano7020023