Open Access Article

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Muchao Qu^{ac},
Fritjof Nilsson*^{b},
Yijing Qin^{ac},
Guanda Yang^{ac},
Qun Gao^{d},
Wei Xu^{d},
Xianhu Liu^{e} and
Dirk W. Schubert^{ac}
^{a}Institute of Polymer Materials, Friedrich-Alexander-University Erlangen-Nuremberg, Martensstr. 7, 91058 Erlangen, Germany. E-mail: muchao.qu@fau.de
^{b}KTH Royal Institute of Technology, School of Chemical Science and Engineering, Fibre and Polymer Technology, SE-100 44 Stockholm, Sweden
^{c}Bavarian Polymer Institute (BPI), Key Lab ‘Advanced Fiber Technologies’, Dr-Mack-Str. 77, 90762 Fürth, Germany
^{d}School of Automobile and Transportation Engineering, Guangdong Polytechnic Normal University, Guangzhou 510450, China
^{e}Key Laboratory of Materials Processing and Mold, Ministry of Education, National Engineering Research Center for Advanced Polymer Processing Technology, Zhengzhou University, No. 97-1 Wenhua, Jinshui District, Zhengzhou 450002, China

Received
6th October 2019
, Accepted 16th December 2019

First published on 24th January 2020

In order to study the electrical conductivity of anisotropic PMMA/carbon fiber (CF) composites, cylindrical PMMA/CF filaments were extruded through a capillary rheometer, resulting in an induced CF orientation along the extrusion direction. The aspect ratios of the CFs in the filaments were accurately regulated using a two-step melt mixing process. By measuring the vertical and horizontal resistances of filaments where the outermost layer was successively peeled off, the anisotropic conductivities could be calculated. This was done using a novel analytical model where each cylindrical composite filament was defined as a structure consisting of three concentric cylinders with potentially different conductivities and CF orientations. The electrical conductivity increased with the degree of fiber orientation along the voltage direction and the effects of anisotropy and measurement direction were incorporated into the (isotropic) McLachlan equation. The required distance for electrical contact between the CFs was calculated to be 16 nm. Finite element (FEM) simulations were successfully utilized to confirm the data.

The electrical conductivity of a CPC is not only influenced by the filler volume fraction and the filler- and matrix conductivities, but also by the shape and geometrical arrangement of the fillers. For example, the aspect ratio (AR) of the carbon fibers must be considered when designing CF composites. It has been reported that a larger AR leads to a lower percolation threshold.^{10} Moreover, the orientation of the CFs inside the CPCs also influences the composite conductivity significantly. In our previous work an equation for predicting the percolation threshold of anisotropic fiber composites was presented, which accounts for the influence of the AR and orientation of the fibers.^{11} It was also proved that a greater orientation leads to a higher percolation threshold.^{11} The main challenge when studying CPCs is to predict the conductivity precisely. Due to the shear gradient across the channel section, the average angle between the CFs and the extrusion direction decreases from the outer part of the composite to the center such that the CFs in the middle part of the composite cylinder becomes less oriented. The composite conductivity for a section of an extruded filament is thus depending on the radial position of the material. In this study, the radial dependence of the CB orientation will be exploited for revealing a relationship between the CF orientation and the composite conductivity for PMMA/CF composites.

For anisotropic CPCs, Weber and Kamal proposed the “contact model”^{12} to predict the longitude conductivity (measuring voltage direction parallel to the fibers orientation, eqn (1)) and transverse conductivity (measuring voltage direction perpendicular to the fibers orientation, eqn (2)):

(1a) |

(2a) |

(3) |

X is a factor depending on the contact number of fibers:

(4) |

It can be noted that eqn (1a) and (2a) becomes zero when φ_{f} = 0. In a real composite the conductivity will rather approach the polymer conductivity σ_{m} when the filler fraction decreases towards zero. The equations thus become significantly more realistic by adding the contribution of σ_{m}, such that:

(1b) |

(2b) |

A two-step mixing procedure was applied in this study (Fig. 1). Prior to processing, all the materials were dried under vacuum at 80 °C for 24 hours. PMMA/CF composites were prepared by melt mixing in an internal kneader PolyDrive (Haake, 557-8310) at a temperature of 200 °C and a rotation speed of 60 rpm for 20 minutes. The composites (50 vol%) produced according to this process are referred as 1st-step mixing.^{11}

Fig. 1 Process flow chart of the two-step melt mixing method, in order to control the AR of the CFs.^{11} |

Composites from 1st-step mixing were treated as master batches (MB), and portions of these batches were further diluted with pure PMMA to the required concentration (2nd-step mixing): 10 vol%; 20 vol%; 30 vol%; 35 vol%; 40 vol%. After melt mixing, all the composites, with controlled aspect ratio of CFs, were ground into granules and dried under vacuum at 80 °C for 24 h. After drying, the composite granules were extruded at 200 °C in a capillary rheometer (Göttfert, Rheograph 2003) at a constant extrusion speed of 0.08 mm s^{−1}, with a die of 10 mm in length, 3 mm in diameter, to induce the orientation of CFs in the extruded composites filament.

Fig. 2 Polishing approach to assess the orientation of CFs in the CPC filament, in order to reveal the CFs orientation. |

Afterwards, a “peeling-off” procedure was applied for further investigation of the longitudinal resistivity of the cylindrical filaments. As described in Fig. 4, the original extruded cylindrical filament (with 1.5 mm radius) was approximated as a composite object consisting of a thin cylinder (yellow in Fig. 4(c), with radius 0.9 mm) covered by two cylindrical shells (green and blue) with thickness 0.3 mm, respectively. The conductivity inside each of the three geometrical sub-domains was assumed to be approximately constant. In order to reveal these local conductivities, the outermost radial layer of each sample was step-by-step peeled off. After each peeling step, the samples were ultrasonic washed with ethanol, dried at 80 °C for 12 hours, the end-sections were covered with silver and the longitudinal resistivity was measured. Consequently, for each CF volume fraction, four longitudinal resistances (R_{L,a} to R_{L,c}) were measured, from which the local longitudinal conductivities could be calculated.

(5) |

Fig. 5 Transverse volume resistivity measurement on the PMMA/CFs filament in the horizontal direction. |

The exposed rectangular areas of the cylindrical filament were then coated with silver paste and the resistivity between the faces was measured with a constant (1 V) voltage without heating the sample. The direction of the voltage was applied perpendicular to the extrusion direction of the filament, i.e. the direction of CFs orientation, also using a Keithley 6487 Pico ammeter.

During each polishing step, the sample was polished until the target width (1, 1.8 or 2.4 mm) of the exposed surfaces was reached, as shown in Fig. 6. The subdomain coloring of the filament in Fig. 6 corresponds to the coloring of Fig. 4. Between each polishing step, the sample was ultrasonic washed with ethanol and dried. Both sides of the sample were coated with silver and the resistance between the coated surfaces was measured. Four transversal resistances (R_{T,a} to R_{T,c}) were thus measured from each specimen. These resistances were in turn used to estimate the local transversal conductivity of each subdomain.

Measured CF-orientations 〈cos^{2}θ〉 were gathered for 6 different CF vol% and 3 different radial positions (layers A–C) in Table 1. The CF-orientation is generally reduced from the rim part (A) to the center part (C), indicating that these regions were subject to a corresponding decreasing shear stress during the extrusion process.

Region | cos^{2}θ |
|||||
---|---|---|---|---|---|---|

10 vol% | 20 vol% | 30 vol% | 35 vol% | 40 vol% | 50 vol% | |

A (blue) | 0.991 | 0.989 | 0.988 | 0.990 | 0.987 | 0.943 |

B (green) | 0.986 | 0.987 | 0.977 | 0.984 | 0.979 | 0.976 |

C (yellow) | 0.985 | 0.964 | 0.978 | 0.954 | 0.947 | 0.938 |

3.2.1 Analysis on the longitude conductivity σ_{∥}. For each CF-concentration, four longitudinal resistances (R_{L,a}–R_{L,d}) had been measured. The next objective was thus to convert the (global) resistance data to the (local) conductivities of the concentric filament layers. It was observed that the geometry of Fig. 4(c) can be considered as a parallel coupling (resistivity R_{L,b}) between the green cylindrical shell (resistivity R_{L,green}) and the yellow solid cylinder (resistivity R_{L,c}). Since the longitudinal resistances R_{L,b} and R_{L,c} are known, the unknown R_{L,green} of the green shell can be calculated from:

(6) |

The same procedure was used to calculate the resistance R_{L,blue} for the blue cylindrical shell. Since the dimensions of the three cylinders were known, the longitudinal electrical conductivity of each region σ_{∥,A} (blue region), σ_{∥,B} (green region) and σ_{∥,C} (yellow region) from the 3 mm filament could thus be calculated. If L is the length of the sample, S is the area of the sample, and R is the corresponding resistance, then the conductivity σ was calculated as follows:

(7) |

3.2.2 Analysis of the transverse conductivity σ_{⊥}. In order to determine the transversal volume resistance (R_{T}) in the horizontal direction, the samples were polished as presented in Fig. 6. Since the length of the investigated samples in Fig. 6 always remain the same, therefore, only the cross section of the samples are presented in this section. The three cross sections of the sample (with three different polished width W_{a}, W_{b}, W_{c}), as well as the three corresponding measured resistances R_{T,a}, R_{T,b}, R_{T,c} are presented in Fig. 9(a). Three concentric circles are assumed, with a constant resistivity ρ_{1}, ρ_{2}, and ρ_{3} in each region (blue, green and yellow, respectively). This section presents the strategy on calculating the resistivity ρ_{1}–ρ_{3} from the experimental resistances R_{T,a}–R_{T,c}.

where W(y) is the width of the sample at the position y, L is the length of the sample as presented in Fig. 6. Thus, the resistivity ρ_{1} can be obtained. Similarly, the resistance of structure R_{2} presented in Fig. 10(a) is given by:

which is further assumed as a parallel connection between R_{L} (left), R_{C} (center) and R_{R} (right), as presented in Fig. 10(b):

Fig. 9 (a) Collection of three cross sections of the sample from Fig. 6. (b) A rough assumption on the resistance, based on a series connection, to calculate R_{1}. (c) The schematic diagram of the resistors series connection based on (b). |

The applied voltage direction is given along the Y-axis, thus the measured resistance R_{T,a} is roughly considered as a series connection between the resistors in Fig. 9(b) and (c). Therefore, the resistance of the shadowed area R_{1}:

(8) |

(9) |

(10) |

Combining eqn (9) and (10), yields:

(11) |

Since the resistivity ρ_{1} (blue material, as presented in Fig. 10(c)) is known, the resistor R_{R} in Fig. 10(b) is considered as a series connection of infinite thin layers of material, which can be obtained by the area calculated by integral. Thus, the resistance of R_{C} can be calculated using eqn (11), as well as the resistivity ρ_{2} (green material) using similar strategy as eqn (8).

As all the resistivity ρ_{1}–ρ_{3} are obtained, the transverse conductivity σ_{⊥} of each four region can be calculated. All the logarithmic values of the longitudinal electrical conductivity σ_{∥} and the transversal electrical conductivity σ_{⊥} are presented in different color in Fig. 11. The 〈cos^{2}θ〉 values in the corresponding region are also shown.

It can be noted that the longitudinal electrical conductivity σ_{∥} is always higher than transverse σ_{⊥}. Presumably it is because that a parallel orientation of CFs to the voltage applied is a more effective in building an electrical pathway, which has also been reported in the open literature.^{15–18}

Fig. 12 (a) The mesh size in the finite element analysis of software Comsol Multiphysics; (b) the simulated model of the sample, the blue plate present electrode. |

As shown in Fig. 12, the geometry of each model consists of the mid-section of a horizontal cylinder comprising three concentric cylindrical layers. The conductivity of each cylindrical layer is taken from the theoretical approach in Chapter 3.2 while the conductivity of the gold-plated electrodes, shown as blue regions in Fig. 12(b), is set to 4.1 × 10^{7}. The electrode on the upper surface provides a voltage of 1 V and the electrode on the lower surface is grounded, i.e. set to 0 V. No-flux boundary conditions were used on remaining external boundaries and continuity boundary conditions were used for all internal boundaries. A sufficiently dense computational mesh was constructed and the static transverse DC resistivity of the composite objects was finally computed and compared with the corresponding experimental data.

The simulation results based on the proposed transverse conductivity σ_{⊥} are presented in Fig. 13, together with the experimental resistance measurement. The simulation results are presented by solid circles. The simulated data are always a bit lower, but the deviation is less than 10% and thus negligible to the large scale. It can be seen that the simulation results demonstrating an agreement with the experimental measurement.

Fig. 14 (a) The logarithm longitudinal electrical conductivity σ_{L} and (b) the logarithm transverse electrical conductivity σ_{T} in different shells of the cylindrical filament. |

A distinct vertical shift (∼2 magnitudes) was observed between the longitudinal conductivity (solid conductivity curve, Fig. 14(a)) and the transverse conductivity (dashed conductivity curve, Fig. 14(b)). This shift was expected since the presence of highly conducting CFs primarily enhances the current flow (and thus the conductivity) along the fiber direction. A less pronounced vertical shift was also observed between the conductivities of the three radial layers in Fig. 14(a) and (b). The longitudinal conductivity was typically lowest for the innermost layer (C) while the transverse was typically lowest for the outermost shell (A). The explanation is that the CFs are highly anisotropic in the outer shells of the composite cylinder (layers A) but less oriented in the central part of the cylinder (layers B and C). The dominant vertical shift factor, which is increasing with increasing degree of CPC anisotropy, can be used to improve the McLachlan equation such that it (1) works better for anisotropic composites and (2) includes the effect of measurement direction. For typical composites, where the composite conductivity is dominated by the contribution from the fibers, the improved contact eqn (1b) and (2b) will coincide with the original contact eqn (1a) and (2a). A predicted ratio between the transverse and longitudinal conductivities can be achieved for such systems by combining eqn (1a) and (2a):

(12) |

In Fig. 15, experimentally measured values for the ratio between the transversal- and the longitudinal conductivities are plotted versus tan^{2}(θ). The experimental data points are shown as symbols in different colors, where each color corresponds to one of the shells A–C and each symbol corresponds to a specific CF filler concentration. The best linear fit (dashed line) leads to a slope of 1.03, which closely resembles eqn (12), i.e. a slope of 1.00.

Fig. 15 The ratio between σ_{trans}/σ_{long} as a function of tan^{2}θ according to eqn (12). The solid line shows eqn (12), while the dashed line indicates the best linear fit. |

Considering the large scattering at the experimental data points, the theoretical prediction is in reasonable agreement with the experimental data. It should also be noted that the CF orientations in this study were measured from 2D micrographs, resulting in slightly different angles than the correct 3D angles. Moreover, rewriting eqn (1)–(3) yields:

(13) |

The best fit line of eqn (13) is presented in Fig. 16 together with corresponding experimental data. A value of 18 vol% was obtained for φ_{c}. The φ_{t} became 0.84, indicating a saturated CF concentration of 84 vol%, after which all the CFs were participating in the conductive network. The diameter of the circle of contact between CFs was determined to 0.016 μm (illustrated as inset in Fig. 16), which is much higher than the reported value 2.1 × 10^{−4} μm,^{12} 1.4 × 10^{−7} μm (ref. 13) and 2.1 × 10^{−6} μm,^{14} indicating a much closer contact between highly oriented CFs in the anisotropic extruded filament. Since the theoretical predictions are promising, it is reasonable to calculate the difference Δf = σ_{long} − σ_{trans} from eqn (13) such that:

Δf = σ_{long} − σ_{trans} = σ_{long}(1 − tan^{2}(θ))
| (14) |

σ_{long} = σ*/tan^{2}(θ*)
| (15) |

Fig. 16 Fit curve in order to investigate the saturated concentration of CFs and the contact distance. |

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