characterization and modeling of anisotropic electrically

Characterization and Modeling of

Anisotropic Electrically Conductive

Composite Filaments comprising PMMA

and Carbon Fillers

Charakterisierung und Modellierung von

anisotropen elektrisch leitfähigen

Komposit-Filamenten aus PMMA und

Carbon Füllstoffen

Der Technischen Fakultät

der Friedrich-Alexander-Universität Erlangen-Nürnberg

zur

Erlangung des Doktorgrades Doktor-Ingenieur

vorgelegt von

Muchao Qu

aus Heilongjiang, China

Als Dissertation genehmigt von der

Technischen Fakultät der

Friedrich-Alexander-Universität Erlangen Nürnberg

Tag der mündlichen Prüfung: 14/Dec/2018

Vorsitzender des

Promotionsorgans:

Prof. Dr.-Ing Reinhard Lerch

Gutachter: Prof. Dr. rer. nat. habil. Dirk W. Schubert

Prof. Dr. Fritjof Nilsson (KTH, Sweden)

List of publications

A. Peer reviewed Papers

1. Qu, M., & Schubert, D. W. (2016). Conductivity of melt spun PMMA composites with

aligned carbon fibers. Composites Science and Technology, 136, 111-118.

2. Qu, M., Nilsson, F., Qin, Y., Yang, G., Pan, Y., Liu, X., ... & Schubert, D. W. (2017).

Electrical conductivity and mechanical properties of melt-spun ternary composites comprising

PMMA, carbon fibers and carbon black. Composites Science and Technology, 150, 24-31.

3. Qu, M., Nilsson, F., & Schubert, D. W. (2018). Effect of Filler Orientation on the Electrical

Conductivity of Carbon Fiber/PMMA Composites. Fibers, 6(1), 3.

B. Conference contributions

1. Qu, M., Schubert, D.W. Poster presentation. Conductivity of Melt Spun PMMA Composites

with Aligned Carbon Fibers. The 24th Annual World Forum on Advanced Materials, in Poznań,

Poland, 2016.

2. Qu, M., Schubert, D.W. Poster presentation & Paper. Research on melt spun PMMA

composites with aligned carbon fibers. 17th European Conference on Composite materials

ECCM17, in Munich, Germany, 2016.

3. Qu, M., Schubert, D.W. Poster presentation. Conductivity of Melt-spun PMMA Composites

with Aligned Carbon Fibers. Polymer Networks Group conference, in Stockholm, Sweden,

2016.

II List of publications

4. Qu, M., Nilsson, F., Schubert, D. W. Poster presentation. Electrical conductivities of melt

spun PMMA/aligned CFs/CB composite filaments. Nordic Polymer Days 2017, in Stockholm,

Sweden, 2017.

5. Qu, M., Nilsson, F., Schubert, D. W. Poster presentation. Electrical conductivities of melt

spun PMMA/aligned CFs/CB composite filaments. International Polymer processing Society

2017, in Dresden, Germany, 2017.

6. Qu, M., Schubert, D.W. Oral presentation. Conductivity of Melt Spun PMMA Composites

with Aligned Carbon Fibers. 15th biennial international Bayreuth Polymer Symposium, in

Bayreuth, Germany, 2017.

7. Qu, M., Schubert, D.W. Poster presentation. Conductivity of Melt Spun PMMA Composites

with Aligned Carbon Fibers. 25th Annual World Forum on Advanced Materials, in Kuala

Lumpur, Malaysia, 2017.

C. Awards

International Union of Pure and Applied Chemistry (IUPAC) best Poster Prize, Kuala Lumpur,

Malaysia, 2017

Table of contents

List of publications ...................................................................................................................... I

Table of contents ...................................................................................................................... III

1 Introduction ............................................................................................................................. 1

2 Literature review ..................................................................................................................... 3

2.1 Percolation threshold models for conductive polymer composite ............................... 3

2.2 Electrical conductivity models for conductive polymer composite ............................. 6

2.3 Ternary conductive polymer composite ..................................................................... 12

2.4 Outline of the thesis .................................................................................................... 15

3 Experimental section ............................................................................................................. 17

3.1 Materials ..................................................................................................................... 17

3.2 Composites preparation .............................................................................................. 18

3.2.1 1st step melt mixing process of binary composites and master batches .......... 18

3.2.2 2nd step mixing process for AR control and ternary composites system ......... 19

3.3 Extrusion process ....................................................................................................... 21

3.4 Thermogravimetric analysis (TGA) ........................................................................... 22

3.5 Morphology ................................................................................................................ 22

3.5.1 Investigation on CFs length distribution ......................................................... 22

3.5.2 Investigation on the orientation and distribution of CFs ................................. 23

3.5.3 Morphological observation of the carbon fillers ............................................. 23

3.6 Electrical conductivity measurement ......................................................................... 24

3.6.1 Vertical conductivity measurement ................................................................. 24

3.6.2 Horizontal conductivity measurement ............................................................ 25

4 PMMA/CF binary system ..................................................................................................... 27

4.1 Introduction ................................................................................................................ 28

4.2 Thermogravimetric analysis (TGA) of the PMMA/CF Composites .......................... 29

4.3 Morphological study of the anisotropic PMMA/CF composite filament .................. 30

IV Table of contents

4.4 Investigation on the distribution of CFs aspect ratio in the CPCs ............................. 33

4.5 Investigation on the orientation of CFs in the CPCs .................................................. 35

4.6 Electrical conductivity of the PMMA/CF binary composites .................................... 41

4.6.1 Anisotropic electrical conductivity of the composite filament. ...................... 41

4.6.2 Influence of CFs AR on electrical conductivity .............................................. 47

4.6.3 Influence of CFs orientation on electrical conductivity .................................. 56

4.7 Conclusion .................................................................................................................. 63

5 PMMA/CB/CF ternary system .............................................................................................. 65

5.1 Introduction ................................................................................................................ 66

5.2 Morphological study on the PMMA/CB/CF ternary composites............................... 68

5.3 Conductivity of the composite filament ..................................................................... 69

5.3.1 Percolation threshold of binary PMMA/CF and PMMA/CB samples ............ 69

5.3.2 Contour plot of conductivity on PMMA/CB/CF ternary composite ............... 70

5.4 Novel equations for electrical behavior of ternary composite ................................... 74

5.5 Conclusion .................................................................................................................. 79

6 PMMA/CB/CNTs ternary system ......................................................................................... 81

6.1 Introduction ................................................................................................................ 82

6.2 Morphology of the ternary composite filament .......................................................... 84

6.3 Conductivity of the binary PMMA/CNTs and PMMA/CB composite ...................... 85

6.4 Experimental contour plot of conductivity on ternary composite .............................. 87

6.5 Reanalysis of the literature ......................................................................................... 90

6.6 Synergasm – a novel synergy definition for ternary composites ............................... 94

6.7 Conclusion .................................................................................................................. 97

7 Summary ............................................................................................................................... 99

8 Summary (in German) ......................................................................................................... 101

9 Appendix ............................................................................................................................. 103

Abbreviations and symbols .................................................................................................... 111

References .............................................................................................................................. 115

Acknowledgement .................................................................................................................. 123

1 Introduction

Poly(methyl methacrylate) (PMMA) is a thermoplastic with favorable processing conditions.

It has been widely applied in the industry because of its transparent properties, e. g. plastic

optical fibers (POF) [1-3]. Moreover, the amorphous thermoplastic PMMA is one of the most

optimal polymer matrix to produce composites, because the influence of the crystallization on

the distribution of the fillers could be ignored [4, 5]. Many fillers has been reported as an

enhancement for PMMA based composites, while the transparency of the PMMA could be

barely reduced by the addition of the fillers [6-12], which provides the possibility for producing

a conductive POF.

However, the conductivity of anisotropic PMMA composites filament with conductive fillers

has been rarely reported. Both mechanism of conductive fillers under shear field as well as

investigation on mathematical models still remain unclear. Therefore, the main purpose of this

thesis focuses on the electrical conductivity of PMMA composites doped with conductive fillers,

which belongs to the definition of conductive polymer composites (CPCs).

The CPCs have been widely used in many fields, such as anti-static materials, electromagnetic

interference (EMI) shielding, sensor and conductors [13, 14]. The most widely used conductive

2 1 Introduction

fillers are carbon fibers (CFs) [15-25], carbon black (CB) [4, 5, 26-29] and carbon nanotubes

(CNTs) [30-40]. During the extrusion process, an orientation of the inner fillers (especially CF

and CNTs) parallel to the extrusion direction could be induced. Therefore, the electrical

conductivities of the CPCs with anisotropic fillers can be thus different from that with isotropic

fillers.

In this thesis, the binary PMMA/CF composite filament (with up to 60 vol. % CF) have been

studied, in order to reveal the influence of (1) aspect ratio (AR) of fillers; (2) orientation of

fillers on the electrical conductivity of the CPCs. Besides the development of existing equation

and theories of electrical conductivities, a novel approach has been proposed to explain the

counterintuitive phenomenon.

In addition, binary PMMA composite filament doped with carbon black (CB) and carbon

nanotubes (CNTs) have also been presented. The ternary composite filament comprising

PMMA/CB/CF has been studied with a broad range of composite compositions (up to 50 vol. %

CF and 20 vol. % CB). Experimental conductivity contour plots for PMMA/CB/CF ternary

composites filament have been presented for the first time.

Furthermore, based on a model for predicting the percolation threshold of ternary composites

filament, a novel equation has been proposed to predict the conductivity of ternary composites

filament, showing results in agreement with corresponding experimental data.

Moreover, ternary composite filament comprising PMMA/CB/CNTs (up to 30 vol. % CNTs

and 20 vol. % CB) has also been studied. A novel quantified definition ‘Synergasm’ was

proposed, which is able to precisely describe the synergic effect between CB and CNTs for the

first time.

2 Literature review

2.1 Percolation threshold models for conductive polymer composite

CPCs have been extensively investigated in both academia and industry, because the

combination of promising material properties and comparatively simple manufacturing

processes often imply commercially interesting materials. Nowadays, the conductivity of CPCs

is generally explained by “conductive pathways” in the composites, which are formed by

conductive fillers [41-44]. As the fillers fraction increases, the number of “conductive pathways”

growth, and consequently, the conductivity of the composite also increases. An electrical

percolation threshold is defined as a certain critical value filler fraction when the conductivity

of the composite increases by several orders of magnitude [45-47].

Many theories have been suggested for describing the relationship between filler fraction and

electrical conductivity for composites consisting of polymers and conductivity fillers. The most

classical theory discussing percolation threshold of CPC is [48, 49]:

t

c )(0 (only when > c) (2.1)

4 2 Literature review

where σ and σ0 are the conductivities of the composite and the polymer matrix, respectively.

is the volume fraction of fillers and c is the percolation threshold. For composites with

filler fractions > c, the experimental results can be fitted by plotting log(σ) against log(

c) and regulating c until the best linear fit is obtained.

Another model for the percolation threshold of cylindrical fillers has been presented by

Balberg [65, 66], considering the angle between two cylinders (abstract model of carbon fibers,

Figure 2.1 (a)). In the simulation, the cylinder was assumed to have a hemispherical "cap" at

both ends, which simplifies the calculation of the contact cylinders. The length (L) as well as

the width (W) of the cylinder were considered.

Figure 2.1 Theoretical Models for percolation threshold description, presented by Balberg. (a)

Two carbon fibers were contacted assuming as two capped cylinders, the excluded volume can be

considered as sum of volume of: (b) a sphere; (c) a cylinder; (d) a parallelepiped.

Considering the orientation of the cylinder in a 3-D system, where it can just contact with the

other cylinder, and the excluded volume of the capped cylinder is a sum of volume from a

sphere with radius W (Figure 2.1 (b)), a cylinder with radius W and height L (Figure 2.1 (c))

and a parallelepiped with both length L (angle γ) and height 2W (Figure 2.1 (d)):

𝑉𝑒𝑥 =4

3𝜋𝑊3 + 2𝜋𝑊2𝐿 + 2𝑊𝐿2𝑠𝑖𝑛𝛾 (2.2)

2 Literature review 5

where γ is the angle between two cylinders in the system. The percolation threshold in this

model could be calculated as:

𝜑𝑐 = 𝐾 ∙𝑉𝑓

𝑉𝑒𝑥= 𝐾 ∙

1

6𝜋𝑊3+

1

4𝜋𝑊2𝐿

(4

3)𝜋𝑊3+2𝜋𝑊2𝐿+2𝑊𝐿2<𝑠𝑖𝑛𝛾>

(2.3)

where Vf is the volume of the capped cylinder, <sin γ> the average sinusoidal value of each

pair of cylinders in the system. Dividing the top and bottom by W3, an equation between φc and

the aspect ratio AR = L/W (aspect ratio), is obtained:

𝜑𝑐 = 𝐾 ∙1

6𝜋+

1

4𝜋∙𝐴𝑅

(4

3)𝜋+2𝜋∙𝐴𝑅+2∙𝐴𝑅2∙<𝑠𝑖𝑛𝛾>

(2.4)

From equation 2.4, it can be noted that: with a larger AR of the fibers, the percolation

threshold of the composites is shifted towards a lower concentration of fillers. It has been

reported that the internal orientation of the CFs also influences the threshold of the composite

[50]. A decreased particle size, when AR and all other variables are kept constant, generally

result in a decreased percolation threshold due to shorter average inter-particle distances [51]

An increased applied voltage (or temperature) can potentially result in a decreased percolation

threshold, because the maximum hopping/tunneling distances for the electrons will increase if

they have higher energy [52]. If the fillers are surface modified with dense insulating grafts, the

shortest possible interparticle distance will be limited, resulting in a higher and narrower

percolation threshold [53]. The dispersion of the fillers also effect the percolation threshold [54].

The main factors that influence the percolation threshold of composites (in general) are

summarized in Table 2.1.


Table 2.1 Factors which influence the percolation threshold of composites.

Orientation

↑

Dispersion

↑

E-

field ↑

Grafts

↑

Aspect

ratio ↑

Size

↑

Percolation

threshold c

If the CFs are considered to be perfect cylinders, it is possible to calculate the percolation

threshold from a simple mathematical model. Chippendale et al. [24] considered a thick layer

laminate with totally oriented CFs as a model of a carbon-fiber-reinforced polymer (CFRP).

Figure 2.2 shows a schematic 2D cross section. The percolation threshold for a CF composite

can be simplified by considering the black circles (CFs) inside the red square. With the

Chippendale model and the boundary conditions, the result is φc (fiber removal model) = 40

vol. %. The CFs in practical situations are not, however, perfectly parallel to each other, which

means that the true percolation threshold is shifted towards a lower concentration.

Figure 2.2 Theoretical Models for percolation threshold description, presented by Chippendale,

with black circles denote CFs.

2.2 Electrical conductivity models for conductive polymer composite

Many models have been proposed to describe the relationship between filler fraction and

electrical conductivity for composites consisting of polymers and conductivity fillers, because

the property of main interest is often the precise composite conductivity rather than the

percolation threshold [55-57].


Voet et al. [58] have proposed a simplified equation to describe the relationship between

conductivity of CPCs σ and the volume fraction of the fillers φ, with an adjustable parameter

K:

log 𝜎 = 𝐾 ∙ 𝜑1/3 (2.5)

Although this equation was simple and neat, the conductivity of the filler was not even

considered, which means all the adjustable elements in the CPCs was unified using the

parameter K. The equation is presented with K=1 in Figure 2.3.

Thongruang et al. [59] have further developed this model, where a linear relationship is

believed to describe the conductivity of CPC and the fillers. In equation 2.6, σmax refers to the

maximum value of the conductivity of CPC. φ and σf are the volume fraction of the fillers and

the conductivity of fillers, respectively.

σ𝑚𝑎𝑥 =2

3𝜋∙ 𝜑 ∙ 𝜎𝑓 (2.6)

It must be noted that the boundary conditions are questionable in this equation: when φ→0,

i.e. pure polymer system without any fillers, σmax = 0, which differs from the conductivity of

polymer itself. And when φ→1, i.e. pure fillers without any polymer, σmax = 0.21σf. The

equation from Thongruang is presented in Figure 2.3, with σf was taken as 589 S/cm from the

CFs [69].

Scarisbrick [60] has proposed a statistical model considering the influence the conductivity

of the fillers:

𝜎𝑐 = 𝜎𝑓 × 𝜑 × 𝜑[𝑒𝑥𝑝(𝜑−2/3)] × 𝐶2 (2.7)

where C2 is a geometric factor and varies from 1 to 0.003. φ, σf and σc are the volume fraction

of the fillers, the conductivity of fillers and the conductivity of CPCs, respectively. In this


equation, the conductivity of CPC can be more precisely predicted by adjusting the parameter

C2.However, the disadvantage of equation 2.7 is that the contribution of conductivity of

Polymer matrix was still not considered. The equation is presented with C2=1 and C2=0.003 in

Figure 2.3, with σf was taken as 589 S/cm from the CFs [69].

Sohi et al. [55] has proposed another model (equation 2.8) based on equation 2.7:

𝜎𝑐 = C × 𝜎𝑓 ×𝐴𝑅

10× 𝑆 × 𝜑 × 𝜑[𝑒𝑥𝑝(𝜑−2/3)] + (1 − 𝜑)𝜎𝑚 (2.8)

where fillers aspect ratio AR, and the conductivity of polymer matrix σm are considered. C is

the geometric factor and varies from 1 to 0.001. AR is the aspect ratio of the fillers and S is the

surface to volume ratio in μm-1. The equation 2.8 is presented with C=1 and C=0.001 in Figure

2.3, with σf =589 S/cm, and AR=100.

Bueche [61] has proposed a model, also considering the polymer matrix conductivity:

𝜎𝑐 = 𝜑 ∙ 𝜎𝑓 + (1 − 𝜑) ∙ 𝜎𝑚 (2.9)

where σc, σm and σf are the conductivities of the composite, the polymer matrix and the

conductive fillers, respectively. is the volume fraction of fillers. The conductivity of CPC is

thus considered as a sum of conductivity contribution from the polymer matrix and the

conductivity of fillers. The equation 2.9 is presented in Figure 2.3, with σf =589 S/cm.

Based on the model from Bueche (equation 2.9), McCullough [62] has added modified

components, which is more precise:

𝜎𝑐 = 𝜑𝜎𝑓 + (1 − 𝜑)𝜎𝑚 − [𝜑(1 − 𝜑)𝑆(𝜎𝑓 − 𝜎𝑚)2/(𝑉𝑓𝜎𝑓 + 𝑉𝑚𝜎𝑚)] (2.10)

V𝑓 = (1 − S) ∙ 𝜑 + S ∙ (1 − φ) (2.11)


V𝑝 = S ∙ 𝜑 + (1 − S) ∙ (1 − φ) (2.12)


conductive fillers, respectively. is the volume fraction of fillers, and S refers to a structure

factor, which varies from 0 to 1. The equation 2.10-2.12 is presented with S=0 and S=0.5 in

Figure 2.3, with σf =589 S/cm. However, the orientation as well as the geometry of the fillers

are not considered in this model.

Figure 2.3 Theoretical Models for prediction the conductivity of CPC as a function of filler volume

fraction (equation 2.5-2.12).

However, the limitation of the equations above (2.5-2.12) is not considering the orientation

as well as the geometry of the fillers. Furthermore, none of them has discussed about the

percolation threshold, which should be the most important concept in the study of CPCs.

Taipalus et al. [63] have presented a “contact model”, which is generally used for CPCs after

reaching the percolation threshold (Equation 2.13).


Xvd

dlff

cmc

)(

cos42

2

(2.13)


conductive fillers, respectively. d = diameter of the fibers, l = average length of the fibers, θ =

average angle between the inclination of fibers and the direction of the applied voltage, vf is the

volume fraction of the fillers, and X is a factor depending on the contact number of fibers, i.e.,

the vf. Finally, dc is the diameter of the circle of contact, which depends on the applied voltage

(Figure 2.4).

Figure 2.4 Schematic models of three different type of contact CFs in “contact model”: (a) a thin

layer of polymer is between two CFs ; (b) two CFs contact just on one single point; (c) a flat circle

of contact between CFs. [63]

In Equation 2.13, it must be noted that the diameter of the circle of contact dc has a direct

influence on the conductivity of CPCs, which can only be obtained by fitting the local data

points, and no comparative values can be found from literature. Therefore, Equation 2.13 is not

a suitable model to calculate the conductivity of composites, but is able to explain the

conductivity phenomena in a CPC.


Mamunya et al. [64] has proposed a model, including the filler and polymer surface energies.

According to this model, the electrical conductivity of CPC is presented as follows:

log 𝜎𝑐 = log 𝜎𝑝ℎ𝑖_𝑐 + (log 𝜎𝑚𝑎𝑥 − log 𝜎𝑝ℎ𝑖_𝑐) × (𝜑−𝜑𝑐

𝐹−𝜑𝑐)𝑘

(2.14)

k = (A − B ∙ γ𝑝𝑓) ×𝜑𝑐

(𝜑−𝜑𝑐)0.75 (2.15)

γ𝑝𝑓 = γ𝑝 + γ𝑓 − 2(γ𝑝γ𝑓)0.5 (2.16)

𝐹 =5

75

10+𝐴𝑅+𝐴𝑅

(2.17)

where σc, σphi_c and σmax are the electrical conductivity of the composite, of the composite at the

percolation threshold, of the composite at maximum volume fraction of fillers, respectively. F

is the maximum volume fraction of the conducting filler, corresponds to the filler geometry

aspect ratio (AR). And if the filler is spherical (AR=1) F = 0.64, which corresponds to the sphere

close packing. γp and γf surface energies of polymer and fillers, respectively. A and B are two

parameters with A=0.11 and B=0.03. The limitation of this model is that the data points with

filler volume fraction below percolation threshold can not be used. Therefore, in this thesis,

model from Mamunya has not been applied to obtain the percolation threshold.

The most effective model is the general effective medium (GEM) equation presented by

McLachlan [67, 68]:

( 1 − 𝜑)𝜎𝑚1 𝑠⁄

−𝜎𝑐1 𝑠⁄

𝜎𝑚1 𝑠⁄

+(1−𝜑𝑐)/𝜑𝑐∙𝜎𝑐1 𝑠⁄ + 𝜑

𝜎𝑓1 𝑡⁄

−𝜎𝑐1 𝑡⁄

𝜎𝑓1 𝑡⁄

+(1−𝜑𝑐)/𝜑𝑐∙𝜎𝑐1 𝑡⁄ = 0 (2.18)

where σm, σc, σf are the conductivities of the PMMA matrix, the composite and the CFs,

respectively. The volume fraction of CFs is denoted 𝜑 and the percolation threshold 𝜑𝑐. For

3-D systems, the exponential factors are generally set to s = 0.87 and t = 2. The advantage of


this model is that all the data points obtained from the experiment can be fitted using Mclachlan

model, and the percolation threshold can be thus obtained for further study.

2.3 Ternary conductive polymer composite

Conductive polymer composites (CPCs) have generated a great deal of interest due to its

high conductivity, low weight, and the ease of processing [70-73]. Besides CFs, Carbon

nanotubes (CNTs) and carbon black (CB) are two conductive fillers, which are also most

commonly used for the design of CPCs [74-79]. Recently, studies on the ternary CPCs

comprising CNTs and CB have been widely reported, because a combination of CB and CNTs

is a good way to improve the electrical properties while limiting the costs [80-88].

However, few theory has been developed for describing the conductivity of the ternary system.

A theoretical description [89] has been put forward to explain the synergistic effect. For a CPC

with a certain concentration of fillers i.e. aCFbCB, it can be assumed to be two resistors

(PMMA with a vol. % CF and PMMA with b vol. % CB) connected in parallel or series (Figure

2.5).

Figure 2.5 The resistance of PMMA/aligned CFs/CB composite filament considered as two

resistors - (a) parallel or (b) series connected.

(a) (b)


The theoretical conductivity magnitudes, both should be lower than the experimental

conductivity, assuming series (Equation 2.19) or parallel (Equation 2.20) connections of

conductive CB and CF networks:

𝜎𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 =1

2(𝜎𝑃𝑀𝑀𝐴/𝐶𝐵 + 𝜎𝑃𝑀𝑀𝐴/𝐶𝐹) (2.19)

𝜎𝑠𝑒𝑟𝑖𝑒𝑠 = 2(𝜎𝑃𝑀𝑀𝐴/𝐶𝐵∙𝜎𝑃𝑀𝑀𝐴/𝐶𝐹

𝜎𝑃𝑀𝑀𝐴/𝐶𝐵+𝜎𝑃𝑀𝑀𝐴/𝐶𝐹) (2.20)

where σPMMA/CB, σPMMA/CF are the conductivities of composites with certain concentration of CB

only and CF only, respectively. The corresponding conductivity of PMMA/aligned CF and

PMMA/CB can be revealed from experiments. The conductivity values estimated by applying

this method can be thus compared with the experimental values. However, the conductivity of

binary composites (i.e. only one filler is used alone) cannot be obtained from equation 2.20,

which leads to a wrong conclusion from the boundary conditions.

Sun et al. [81] have proposed a percolation threshold model based on the assumption that the

excluded volume of the two fillers can be added together linearly. The percolation threshold of

a conductive composite with two fillers can then be predicted as:

KBc

B

Ac

A ,,

(2.21)

where φA and φB are the volume fractions of filler A and filler B in ternary composites,

respectively, while φc,A and φc,B are the corresponding percolation concentrations when filler A

or B is used alone in binary composites. When K=1, the percolation threshold is just reached.

When K>1 the fillers in the composites connect each other (i.e. the composites are conductive)

and when K<1 the composites are insulating. Based on this model, the potential synergic effects

between CFs and CB have been discussed [89], defining “synergy” as a lower percolation

threshold of the mixture than predicted by Eq.1 with K=1. It should be noted that, according to


Eq. 1, a linear relationship between volume fractions φA and φB can be established once the

percolation thresholds φc,A and φc,B are determined.

In spite of this, for a ternary composites with CB and CNTs, a “synergistic effect” is still

widely used to indicate the enhancement of the conductivity [74-88] yet without unified clear

definition. Figure 2.5 is a typical schematic diagram for the “synergy”: the CNTs are considered

as 1-D filler with a large aspect ratio (AR), while CB are 0-D spherical particles. Therefore,

due to the geometry of both fillers, the electrical pathway was expected to be easier formed in

the CNTs/CB ternary CPCs than the binary CPCs with only one filler. Different morphological

structure was proposed to explain such phenomenon, such as “grape-like” [82, 85], or

“conductivity-bridge” [79, 83].

Figure 2.5 Schematic structure of ternary CPCs with CNTs and CB. (a) The well-known “synergy”

structure between CNTs and CB; (b) The well-dispersed CB particle in fact with the same volume

fraction; (c) A higher CB volume fraction is needed to achieve the “synergy” model in (a).

Unfortunately, there is still no clear definition on the “synergistic effect” on the conductivity

between CNT and CB. From the basic concept, it can only be assumed that this is an effect

shows 1+1˃2. Until now, there are three main perspectives to explain the “synergistic effect”

between CNT and CB:

1st synergy: A “small amount” of CNTs (or CB) (with either volume or weight percent) will

“dramatically” increase the conductivity of the CPCs [75, 76].

𝜎(𝜑𝐶𝑁𝑇𝑠)𝐶𝑁𝑇𝑠 < σ(𝜑𝐶𝑁𝑇𝑠, 𝜑𝐶𝐵)𝑡𝑒𝑟, with φCB<< φCNTs (2.22)

𝜎(𝜑𝐶𝐵)𝐶𝐵 < σ(𝜑𝐶𝑁𝑇𝑠, 𝜑𝐶𝐵)𝑡𝑒𝑟, with φCNTs<< φCB (2.23)


2nd synergy: The conductivity of the ternary CPCs is higher than that of either binary CPCs,

while the total filler fraction keeps constant. Sometimes an optimal ratio between CNTs and

CB was presented. [77, 80, 82-86, 88]

𝜎(𝜑𝐶𝐵)𝐶𝐵 < σ(𝜑𝐶𝑁𝑇𝑠, 𝜑𝐶𝐵)𝑡𝑒𝑟 > 𝜎(𝜑𝐶𝑁𝑇𝑠)𝐶𝑁𝑇𝑠, with φCNTs+ φCB=constant (2.24)

3rd synergy: The experimental percolation threshold of ternary CPCs is lower than the

predicted one according to Equation 2.21, which is proposed for the situation when the

percolation threshold of ternary CPCs is just reached, and the predicted percolation threshold

of the ternary CPCs is defined as (φCNT+φCB). Usually, a certain ratio between CNT and CB

were used to produce the ternary composite. Once the percolation thresholds φc,CNT and φc,CB

are determined, then the predicted percolation threshold of the ternary CPCs can be calculated.

Based on this model, the 3rd synergic effects between CFs and CB have been widely discussed

[77, 78, 82-88].

2.4 Outline of the thesis

In this thesis, efforts have been made on the morphological and electrical behavior of

anisotropic filament from PMMA based conductive composites. In Chapter 2, the background

information as well as the models describing electrical conductivity of CPCs were presented.

In Chapter 3, the materials, the processing and characterization methods involved in this thesis

were introduced. In Chapter 4, the binary PMMA/CF composite filament have been studied, in

order to reveal the influence of (1) aspect ratio (AR) of fillers; (2) orientation of fillers on the

electrical conductivity of the CPCs. Besides the development of existing equation and theories

of electrical conductivities, a novel approach has been proposed to explain the counterintuitive

phenomenon. In Chapter 5, the ternary composite PMMA/CB/CF has been studied. Based on

a model for predicting the percolation threshold of ternary composites, a novel equation has

been proposed to predict the conductivity of ternary composites. In Chapter 6 ternary composite


comprising PMMA/CB/CNTs has also been presented. A novel quantified definition

‘Synergasm’ was proposed, which is able to precisely describe the synergic effect between CB

and CNTs for the first time. The conclusion of this thesis is summarized in Chapter 7 and 8 (in

the language of German). Chapter 9 presents the additional information as supplementary

material for this work.

3 Experimental section

3.1 Materials

In this work, Poly(methyl methacylate) (PMMA) was chosen as the polymer matrix, due to

its high impact strength, lightweight and favorable processing conditions. Moreover, as an

amorphous thermoplastic polymer, the influence of the rare PMMA polymer crystallization on

filler distribution can be thus minimized and ignored. Commercial PMMA Plexiglas 7N was

obtained from Evonik Röhm GmbH, with a weight-average molar mass 𝑀𝑤 of 99 kg/mol, a

polydispersity index of 1.52, and a density 1.19 g/cm3.

The carbon fillers used in this work are carbon fibers (CF), carbon black (CB) and carbon

nanotubes (CNTs). The CF segments (Figure 3.1 (a)) were obtained from Tenax® - JHT C493

(Toho Tenax Europe GmbH, Wuppertal, Germany) with a diameter of 7 µm, an initial length

of 6 mm, a specific resistance of 1.7×10-3 Ω/cm, and a density 1.79 g/cm3. CB (Figure 3.1 (b))

was Printex XE2 from Evonik Degussa, with a specific surface area of 900 m2/g measured by

the BET-method. The mean diameter of the primary CB particles was around 35 nm and the

density at room temperature was 2.13 g/cm3. CNTs (Figure 3.1 (c)) were Bayertubes® C 150

18 3 Experimental section

P, with an outer diameter of 13 nm, an inner diameter of 4 nm, an average length of 1 µm and

a bulk density of 130-150 kg/m3.

Figure 3.1 Morphological photos of the carbon fillers used in this work: (a) carbon fibers; (b)

carbon black; (c) carbon nanotubes.

3.2 Composites preparation

3.2.1 1st step melt mixing process of binary composites and master batches

Prior to processing, all the materials in this study were dried 24 hours under vacuum at 80 ˚C.

The composites were produced by melt mixing procedure in an internal kneader PolyDrive

(Haake, 557-8310) (Schwerte, Germany) at a temperature of 200 ˚C and a rotation speed of 60

rpm. The melt mixing recipe is presented in Figure 3.2, where the binary composites system

PMMA/CF (Figure 3.2 (a), blue circle), PMMA/CB (Figure 3.2 (b), black circle) and

PMMA/CNTs (Figure 3.2 (c), red circle) were prepared.

The PMMA and the corresponding carbon fillers were simultaneously fed into the kneader

and mixed for 10 minutes. Parts of carbon fillers were found to be stuck in the gap of the

machine (especially with high concentration of CF). These fillers were not participated in the

melt mixing procedure and were therefore manually removed. The remaining composites were

mixed for another 10 min under the same conditions. The binary PMMA/CF composites were

produced with 10-60 vol. % CF, the binary PMMA/CB composites were produced with 1-20

3 Experimental section 19

vol. % CB, and the binary PMMA/CNTs composites were produced with 0.01-30 vol. % CNTs.

The composites, with 40 vol. % CFs, 50 vol. % CFs 60 vol. % CFs and 20 vol. % CB were

named as master batches (MB) and marked with bold circle. Therefore, this process was named

as 1st step melt mixing process, because only binary composites system and MB composites

were produced in this procedure. The following nomenclature is used for the samples: 50CF

means that the expected concentration of CF was 50 vol. %, 20CB means that the expected

concentration of CB was 20 vol. %, created by the 1st-step mixing process. After melt mixing,

all the composites were ground into granules and dried under vacuum at 80 ˚C for 24h.

Figure 3.2 Process flow chart of the 1st step melt mixing method. (a) PMMA/CF binary composites,

those which concentration were 40 vol.%, 50 vol.% and 60 vol.% CFs are used as master batches

for further dilution; (b) PMMA/CB binary composites, those which concentration were 20 vol.%

CB are used as master batches for further dilution; (c) PMMA/CNT binary composites.

3.2.2 2nd step mixing process for AR control and ternary composites system

Composites with concentrations of 40, 50 and 60 vol. % CFs 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), as illustrated in Figure 3.3 (a). The following

nomenclature is used for the samples, which will be discussed in Chapter 4: 10(-50)CF means


that the composite from 2nd-mixing with an expected CF concentration of 10 vol. % was

obtained by dilution of the master batches 50CF. Using this two-step mixing procedure, the

length or the aspect ratio (AR) of CFs can be controlled, as presented by Starý et al. [23].

The PMMA/CB/CF ternary composites were produced by diluting the mater batches 50CF

with pure PMMA and CB to the required concentration (Figure 3.3 (b)), which will be discussed

in Chapter 5. The following nomenclature is used for the samples: aCB and bCF denote the

composites with a % volume fraction of CB, and b % volume fraction of CFs, respectively.

Thus, the sample with the name aCBbCF presents the ternary composite filled with a vol. %

CB and b vol. % CFs and (100-a-b) vol. % PMMA. In this study, a ∈ [0, 20], and b ∈ [0,

50].

The PMMA/CB/CNTs ternary composites were obtained by diluting the mater batches 20CB

with pure PMMA and CNTs to the required concentration (Figure 3.3 (c)), which will be

discussed in Chapter 6. The following nomenclature is used for the samples: mCB and nCNTs

denote the composites with m % volume fraction of CB, and n % volume fraction of CNTs,

respectively. Thus, the sample with the name mCBnCNTs presents the ternary composite filled

with m vol. % CB and n vol. % CNTs and (100-m-n) vol. % PMMA. In this study, m ∈ [0,

20], and n ∈ [0, 30].

After melt mixing, all the composites were ground into granules and dried under vacuum at

80 ˚C for 24h.


Figure 3.3 Process flow chart of the 2nd step melt mixing method. (a) Using the MB from 1st step

mixing process, the length of CFs can be controlled in the produced composites; (b)

PMMA/CB/CF ternary composites were produced by diluting the MB (50CF) with CB and pure

PMMA; (c) PMMA/CB/CNTs ternary composites were produced by diluting the MB (20CB) with

CNT and pure PMMA.

3.3 Extrusion process

The composite granules after drying were anisotropic at 200 °C utilizing a capillary

rheometer (Göttfert, Rheograph 2003), (Göttfert, Buchen, Germany), using either a 10-mm

length die with a diameter D = 1 mm or a 10-mm length die with a diameter D = 3 mm. A

constant extrusion speed v = 0.08 mm/s of the pistol (with diameter D0 = 12 mm) was applied

using both dies. Therefore, the apparent shear rate of the anisotropic composite filaments can

be calculated from Equation 3.1 [107], resulting in (D = 1 mm) = 92.16 s−1, and (D = 3

mm) = 3.41 s−1.

=8∙𝑣∙𝐷0

2

𝐷3 (3.1)


Figure 3.4 The anisotropic PMMA composites using 3 mm die and 1 mm die.

The samples with 3 mm diameter will only be discussed in Chapter 4.6.1 and 4.6.3, while

the samples with 1 mm diameter will be discussed in all the chapters.

3.4 Thermogravimetric analysis (TGA)

The resulting CF concentrations of the composite filament were determined by

thermogravimetric analysis (TGA 2950, TA Instruments). Ca. 20-30 mg samples from the

corresponding CPCs were heated from room temperature to 600 ˚C with a heating rate of

10 °C/min under a nitrogen atmosphere.

3.5 Morphology

3.5.1 Investigation on CFs length distribution

In order to determine the CF length distributions in the composites, the PMMA/CF binary

composite filament were handled with acetone. PMMA matrix was dissolved and the CFs were

investigated using a light microscope (Leitz, Orthoplan P). The images obtained were then

analyzed using the JMicrovision image analysis software. At least 500 carbon fibers per

concentration were studied and measured.


3.5.2 Investigation on the orientation and distribution of CFs

In order to study the orientation of CFs inside the composite filaments, the composite

filaments were circle peeled (annular cut) (Figure 3.5, (a)), and ultrasonic washing with ethanol.

The samples were then observed with a light microscope, and the orientations of CFs on the

surface and inside were compared at the same time. The composite filament were also fixed

with epoxy resin. The epoxy resin as well the composite were then polished, until the width of

the exposed surface of the composite was equal to the original diameter of the composite

filament. (Figure 3.5, (b)). With the light microscope, 500 carbon fibers (for the 1 mm diameter

composite filament) or 1200 carbon fibers (for the 3 mm diameter composite filament) were

randomly chosen and the inclination between each carbon fibers and the extrusion direction can

be recorded and then analyzed with JMicrovision and Matlab software.

Figure 3.5 Two approaches to assess the orientation of CFs inside the anisotropic composite; (a)

circle peeling, in order to compare the CFs on the surface and with those inside the composites;

(b) polishing, in order to obtain the statistical distribution of the CFs.

3.5.3 Morphological observation of the carbon fillers

The morphologies of the composites were studied by a scanning electron microscope (SEM).

The anisotropic samples were fractured in liquid nitrogen and the broken surfaces (cross section

or the middle section as presented in Figure 3.5 (b)) were sputtered with a thin layer of


palladium, were then analyzed using a SEM (Leica, LEO 435VP) equipped with a secondary

electron detector at an acceleration voltage of 10KV.

3.6 Electrical conductivity measurement

3.6.1 Vertical conductivity measurement

Since anisotropic the CPCs filaments were anisotropic, the electrical conductivity

measurement were carried out in two directions, in order to check the influence of the measuring

direction, i.e. the direction of the applied voltage on the conductivity of the CPCs. The vertical

measurement refers to a parallel relationship between the extrusion direction of the CPCs

samples and the applied voltage, and this measurement is used for discussion for all the chapters

in this study. The composite filament were cut into samples of 20 mm length and their end-

sections were polished in order to remove isolate polymer. Silver conductive paste was then

coated (exclusively) at the polished ends, to ensure enough contact between the samples and

the copper electrode, as shown in Figure 3.6.

Figure 3.6 Schematic of silver coated sample for the vertical conductivity measurement.

The electrical resistance R of the samples at room temperature was measured using a Keithley

6487 Pico ammeter (Tektronix, Beaverton, United States) at a constant voltage 1 V. The volume

conductivity σ was calculated as follows:

RD

L

2

4

(3.2)


where R is the electrical resistance of the composite, L is the distance between two silver-coated

ends of the sample, and D is the diameter of the composite filament. Because the diameters of

the extrudes samples were close to the diameters of the extrusion dies (1 mm and 3 mm), a

slight reduction of the fiber diameter of less than 5% was observed as a consequence of

stretching during fiber spinning by gravity. A Vernier caliper was used for the diameter

measurements on the samples. For each material composition, 20 composite samples were

manufactured and analyzed for the conductivity measurement. Each presented conductivity

data point in this study is thus the average of 20 individual measurements on different samples.

It must be noted that the room temperature, as well as the humidity in the air, has an influence

on the conductivity of the anisotropic sample and the measured values. Therefore, the

conductivity of pure PMMA have been determined for each different set of the experiment. The

pure PMMA fiber have been anisotropic and measured, using the same conditions as the other

composites.

3.6.2 Horizontal conductivity measurement

The horizontal measurement refers to a perpendicular relationship between the extrusion

direction of the CPCs samples and the applied voltage, and the results based on this

measurement is discussed in Chapter 4.6.1. The samples with 3 mm diameter were set into

epoxy resin, and polished on both side surface, until the exposed width of CPCs on the both

ends equal to required W. The surface of the sample were then coated with silver and measured.


Figure 3.6 Schematic of silver coated sample for the horizontal conductivity measurement.

The horizontal conductivity of the anisotropic filament can be thus determined using :

𝜎 =ℎ

𝐿∙𝑊∙𝑅 (3.3)

where R is the electrical resistance of the composite, L is the length of the sample, h is the

height of the epoxy resin, and W is the width of the sample. A rough estimation is proposed to

obtain the horizontal conductivity, which is presented in Chapter 9.1.

A Vernier caliper was used for the measurements on W and h. For each material composition,

3 composite samples were manufactured, polished to different W, and analyzed. Each presented

conductivity data point is thus the average of 3 individual measurements on different samples.

4 PMMA/CF binary system

28 4 PMMA/CF binary system

4.1 Introduction

Carbon fibers (CFs) are widely used as conductive fillers, and have gained increasing

attention in both academic and engineering aspects [15-25]. With plate-shaped [16], cuboid [15,

89] or even composite films [17], most of literature has discussed the effect of CFs with a

maximum concentration of 10 vol. %, because this is already above the percolation threshold

of CPCs with isotropic CFs, after which there is no significant change in the electrical properties

of the composites (Figure 2.3). In this study, the volume fraction of CFs in the CPCs were

achieved up to 60 vol.% (Chapter 4.2), in order to present a general investigation on the

electrical properties of the PMMA/CF system.

Besides the volume fraction, the aspect ratio (AR) of CFs has also a dramatically influence

on the percolation threshold of the CPCs [23, 97-99]. It has been reported that the percolation

threshold of the CPCs is approximately proportional to the reciprocal of the aspect ratio (AR-1)

of the fillers. In this study, CF aspect ratios were controlled by using a two-step melt mixing

method (Chapter 4.4), the influence of the carbon fibers’ AR on the electrical properties is

presented in Chapter 4.6.2.

Moreover, it has also been reported that the threshold of the composite also depends on the

orientation of the CFs [8, 9]. With a higher orientation of CFs, the percolation threshold is

shifted towards a higher volume fraction of fillers. Unlike CB (dot-like, 0-dimensional

particles), CFs with a high aspect ratio (AR) should be considered as 1-dimensional particles.

Due to the shear deformation during the extrusion process, the orientation of 1-D particles can

be induced. Another typical 1-D conductive filler, CNTs, however, due to the irregular shape

(fold, spiral) and much smaller scale compared to CFs, are not chosen and analyzed in this

chapter. In the past few decades, extensive efforts have been generally focused on polymer/CFs

composites with isotropic CFs. To our knowledge, polymer/CFs composites with highly

4 PMMA/CF binary system 29

anisotropic CFs has been rarely reported in the open literature. Therefore in this chapter,

composite with different CFs concentrations were anisotropic into composite filament with two

different diameters using a capillary rheometer, to induce CF orientation. The morphology on

the anisotropic PMMA/CF CPCs is shown as a “fibers in fiber” structure in Chapter 4.3. The

distribution of CF orientation in the CPCs is investigated and presented in Chapter 4.5 and the

influence of the fillers orientation on the electrical properties of PMMA/CF CPCs is discussed

in Chapter 4.6.3.

4.2 Thermogravimetric analysis (TGA) of the PMMA/CF composites

In order to determine the actual CF volume fractions in the PMMA/CF composites from the

designed CF volume fraction, the composites filament were tested by TGA measurements.

Composites from the 1st step melt mixing with five different designed CFs fractions (10; 20; 30;

40; 50; 60 vol. %) as presented in Figure 4.1. Since the actual CF weight fractions in the

composites were given as results from TGA, thus the corresponding actual CF volume fractions

can be calculated by Equation 4.1,

PMMACF

CF

wtwt

wtvol

/.%)1(/.%

/.%.%

(4.1)

where ρCF = 1.79 g/cm3 and ρPMMA = 1.19 g/cm3. The processing parameters (T = 200 °C,

extrusion speed of pistol = 0.08 mm/s) during the extrusion process has been optimized for

PMMA/CF composites, in order to get the smooth and uniform filament composite. Because of

the non-uniform distribution of CFs in the PMMA matrix, the TGA results could be difference

on samples even from the same extrusion process. Therefore, Figure 4.1 presents a certificate

of the difference between the actual CF concentration in the composites and the designed CF

concentration. This is because parts of CFs were lost and removed during the melt mixing


process. These differences are indicated with error bars on the volume fraction of CFs in this

study.

Figure 4.1 The actual weight fraction and actual CFs volume fraction in the samples from 1st step

melt mixing. A difference between actual and designed CFs volume fraction can be found.

4.3 Morphological study of the anisotropic PMMA/CF composite filament

The morphology of the anisotropic PMMA/CF composite filament with 1 mm diameter is

investigated. The cross sections of composite filament with different concentration of CFs are

shown in Figure 4.2 (a) and (b). The CFs can be seen inside the composite filament, as well as

lots of "black holes", which indicate the positions of pulled out CFs. It should be noted that

almost all the CFs and the "black holes" are basically perpendicular to the cross section, i.e.

parallel to the axis of the composite filament (the extrusion direction). The red circle in Figure

4.2 (a) presents an exception, which shows the situation when a single CF is not parallel to the

axis of the composite filament, but this situation was rarely observed in our work.


Figure 4.2. Cross sections of composite filament, different concentrations of CFs: (a) = 19.09

vol. %, (b) = 36.30 vol. %, one-step mixing. Red circles: a single CF is not parallel to the axis of

the composite filament.

The amount of PMMA matrix among the CFs is significant in Figure 4.2 (a). The CFs are

rarely connected, even when the concentration of CFs reaches almost 20 vol. %. As the

concentration of CFs increases, the agglomeration of CFs can be seen as in Figure 4.2 (b) with

the 36.3 vol. % CF. It is generally considered, the conductivity of the composites depends on

the amount of "conductive pathway", which are built up by the connection of CFs with each

other. From this point of view, it can be assumed that the composite in Figure 4.2 (b) is

conductive, while that in Figure 4.2 (a) is not. The conductive data are presented in Chapter

4.6.


Figure 4.3 Schematic view of the orientation of CFs inside the composites; (a) due to the different

shear deformation of the composite melt, CFs have different orientations in different regions of

the composites; (b) schematic view of the composite, with different orientations of CFs in different

regions.

The flow pattern of the composite melt inside the capillary rheometer is shown as Figure 4.3

(a). The composite melt near the wall of the capillary (rim area marked dark grey) is subject to

higher shear deformation than the peripheral composite melt in the middle capillary (center area

marked light grey). After extrusion, the isotropic CFs (red segments) at first are differently

oriented due to the shear gradient across the channel section of the composites cylinder. Based

on this, the 3-D schematic view of the anisotropic composite filament with CFs is presented in

Figure 4.3 (b). The small white cylinders denote carbon fibers, the light grey and dark grey

parts denote the PMMA matrix, with lower (center part) and higher (rim part) shear deformation,

respectively. The CFs in the outer part of the composite are highly anisotropic along the axis of

the composite, i.e. the extrusion direction, while those in interior of the composite are less

oriented.


Figure 4.4 Anisotropic PMMA/CF composite filament, after circle peeling, showing the isotropic

CFs in the center and the oriented CFs in the rim, Nr. 10(-60)CF.

Figure 4.4 presents confirmation of the hypothesis that the orientation of CF is greater in the

outer regions. After circle peeling and ultrasonic washing with ethanol, the inner and outer parts

of the composite filament can be observed at the same time. It is clear that the CFs on the surface

of the composite are well aligned, and the CFs inside are isotropic. Therefore in this study, a

“fibers in fiber” structure of all the PMMA/CF samples are produced by using the extrusion

procedure, containing an increasing orientation degree of CFs from the center of the composite

filaments to the rim.

4.4 Investigation on the distribution of CFs aspect ratio in the CPCs

The aspect ratios of CFs in the composite filament are controlled by using the 2-step melt

mixing procedure. After dissolving the PMMA matrix with acetone, the PMMA/CF composite

filament were made into casting films.


Figure 4.5 Length and distribution of CFs in the composites: (a) and (d)-Sample 10CF; (b) and

(e)-Sample 20CF; (c) and (f)-Sample 50CF.

In the cast films of the composites, the lengths of the CFs were able to be directly observed

under a microscope (Figure 4.5 (a)-(c)). The uncertainty "Δ" of the average "m" was estimated

as:

∆= ±1.96×𝑆𝐷

√𝑁 (4.2)

N is the number of units studied, equal to 500 in this work, and SD is the standard deviation,

which means that the true mean µ is with 95% confidence in the range m ± Δ [96]. Figure 4.5

(d-f) shows the frequency distributions of the lengths of CFs in Figure 4.5 (a-c), respectively.

The results show that: the length of CFs after melt mixing decreased when the concentration of

CFs was increased. Presumably because the probability of CF collision under the melt viscosity

of composites with a higher CF content is increased, i. e. a higher CFs concentration of the

composite melt presents a higher viscosity, which leads to shorter CFs in the composite melt.

The average length of CFs was not changed by dilution from the master batches with pure

PMMA in the second step, as has been shown by Starý et al. [23]. The lengths of CFs in all the


composites in this study, with different concentration of CFs, are present in Figure 4.6, which

shows that different aspect ratio of CFs (6.3, 9.2 and 12.0) were achieved. A possible model

describing the relationship between the average value of CFs and the volume fraction of CFs is

left for future study.

Figure 4.6 The average length of the CFs as a function of the volume fraction.

4.5 Investigation on the orientation of CFs in the CPCs

Johannsen et al. [100] has reported similar anisotropic PMMA/CNTs “fibers in fiber”

structure, where a lower concentration of CNTs in the boundary layer of the composite

filaments is found compared to the center, due to the extrusion swell effect. However in this

study, hardly any expansion of PMMA/CF composite after leaving the nozzle were observed.

Presumably because the properties of PMMA polymer matrix as well as the high concentration


of CFs contributes to a negative extrusion swell effect. Therefore, only the orientation of CFs

in different region of the composite filament are discussed.

In order to investigate the statistical distribution of the CF orientation, the composites were

set in epoxy resin and polished. Figure 4.7 shows a section through the middle axis of the sample

with 1mm diameter under a microscope. The inner and outer parts of the composite are roughly

divided by two red lines. The CFs in the outer region are relative highly anisotropic, whereas

these in the inner region have a significant inclination. 500 CFs were randomly chosen from

the picture and their inclinations were determined. The mean value of the two borderlines (i.e.

the axis of the composite, or the extrusion direction) was defined as 0°, and the orientation of

the CFs with respect to this line were in a range of [-90°, 90°]. The frequency distribution of

the inclination is shown as illustration at bottom left in Figure 4.7. It should be noted that these

angles of inclination are taken from a two-dimensional image, which differs from the real

circumstance in a 3-D system. This method in this work thus provides only a rough estimate of

the distribution.

Figure 4.7 The middle section of the 1mm diameter anisotropic PMMA/CF fiber, with the

frequency distribution of the angles between the CFs and the axis of the composite filament, with

Nr. 60CF.


Figure 4.8 (a) presents the investigation on the inclinations of 500 CFs in an anisotropic

composite filament with 1 mm diameter. The randomly chosen CFs are marked with red

segments. The inclination between each CF segment and the borderlines of the composite

filament is presented in Figure 4.8 (b), where 500 data points are presented in an X-axis range

of [-r, r] (r is the radius of the polished composite filament). Divide the data points into 20

intervals based on their X-values [-r, -0.9r); [-0.9r, -0.8r); …[0.9r, r], and calculate their mean

value of the inclinations. Thus the 500 data points can be concluded into 20 points (Figure 4.8

(c)) and a significant curve can be estimated. This is highly related to the stress distribution

across the section of pipe, where fluid flows through [101-104]. This section can be further

carried out and verified by Finite Element Analysis (FEM), which is left for future study, and

will not be discussed in this thesis.

Figure 4.8 Investigation on the distribution of CFs. (a) polished PMMA/CF sample and the 500

CFs marked as red segments; (b) the radical position of CFs and the inclination of each CF; (c)

divide X-axis into 20 interval, the mean value of the inclination and the fitted curve, consists with

the shear stress distribution diagram of a pipe in pipeline hydrodynamics.

In this study, <sin γ> is used to describe the orientation of CFs in the composite filament [65,

66]. <sin γ> is the average sinusoidal value of the angle γ between each pair of CFs in the

system:


< sin 𝛾 >=∑ ∑ 𝑠𝑖𝑛|𝜃𝑖−𝜃𝑗|

𝑖−1𝑗=1

n𝑖=1

∑ 𝑖n−1𝑖=0

(4.3)

where n=500 for the 1 mm diameter samples, and n=1200 for the 3 mm diameter samples. <sin

γ> was obtained with Matlab software, and is used in a discussion of excluded volume theory

in Chapter 4.6.2. The results of the samples with 1 mm diameter (plain segments) and 3 mm

diameter (shadowed segments) are shown in Figure 4.9, where CFs with AR≈12, 9 and 6 are

shown in black, red and blue respectively. The value <sin γ> of composite filament with each

volume fraction and each CFs AR is presented in three parts: the upper value refers to the <sin

γ> obtained from CFs in inner region of the composite filament (Figure 4.7), the lower refers

to data from outer region of the composite filament (Figure 4.7), and the middle value is the

average.

Figure 4.9 The orientation of CFs <sin γ> vs. volume fraction of CFs, considering different

CFs AR and different diameters of the composite filament.

It can been estimated that either the longer CFs or the higher concentration of CFs, contributes

to a smaller <sin γ>, i.e. to a greater orientation of CFs, presumably because either the longer

CFs or the higher concentration of CFs enhances the viscosity of the composite melt [23]. When


the composite melt flows in the capillary rheometer, the composite melt with higher viscosity

develops more shear stress, which leads to a greater orientation of the CFs, i.e. a smaller value

of <sin γ>.

Moreover, the shear rate of the composite filament during the extrusion process also

influences the orientation of the CFs. For a system with totally aligned cylinders, when the

shear rate tends to infinity, <sin γ>=0. For a system with isotropic cylinders, when the shear

rate equals to zero:

< 𝑠𝑖𝑛𝛾 >𝑖𝑠𝑜=

∫ 𝑠𝑖𝑛 𝛾 𝑑𝛾𝜋0

∫ 𝑑𝛾𝜋0

=2

𝜋≈ 0.64 (𝑓𝑜𝑟 2𝐷 𝑠𝑦𝑠𝑡𝑒𝑚)

∫ 𝑠𝑖𝑛 𝛾∙𝑠𝑖𝑛 𝛾 𝑑𝛾𝜋0

∫ 𝑠𝑖𝑛 𝛾 𝑑𝛾𝜋0

=𝜋

4≈ 0.785 (𝑓𝑜𝑟 3𝐷 𝑠𝑦𝑠𝑡𝑒𝑚)

(4.4)

Therefore, considering these two boundary conditions, a model from Prof. Schubert is

suggested to describe the relationship between the <sin γ> and the shear rate :

< sin γ >=< 𝑠𝑖𝑛𝛾 >𝑖𝑠𝑜× (1−𝐵

1+(/𝐴)2+ 𝐵) (4.5)

where A and B are two adjustable parameters. Discussion on the relationship and model in

details is left for further study in the future.

Fig. 4.10 (a) shows the middle section of a cylindrical 3 mm PMMA/CF filament with 35

vol. % CFs. The figure is intended as an illustration of how the CF-orientation is determined in

this study and the sample is polished as presented in Fig. 3.5. The CFs can be seen as bright

segments in the micrograph. 1200 CFs were randomly chosen from the picture and the

inclination between each CF and the extrusion direction were determined (Fig. 4.10 (b)),

resulting in an averaged inclination angle θ which can be inserted in Eq. 2.13. The marked areas

A (red), B (green), C (blue) and D (purple) in Fig. 4.10 (b) correspond to the assumed structure

in Fig. 4.11 and Fig. 4.12. It should be noted that these inclination angles are taken from a two-


dimensional (2-D) image, which differs from the real circumstance in a 3-D system, however

in the chosen plane this effect is negligible. 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. The CFs in the center part of the composite cylinder are thus less

oriented. Moreover, the scattering in the CF orientation (Fig. 4.10 (b)) is symmetric around the

center due to axisymmetric geometry (r=0), resulting in a uniform CF-orientation distribution

in the extruded filament samples. The 1200 data points in Fig. 4.10 (b) are subsequently divided

into 15 intervals according to the X-position, and the average value of absolute CFs inclination

as a function of distance from the center axis is presented is Fig. 4.10 (c), which consists with

the shear stress distribution in the capillary


Fig. 4.10 (a) The middle section of the 3mm diameter anisotropic PMMA/CF filament with 35

vol. % CF. (b) The 1200 CFs inclination vs. radical position of the sample. (c) Average value of

absolute CFs inclination as a function of the CF distance from the center axis.

Measured CF-orientations <cos2θ˃ were gathered for 6 different CF vol. % and 4 different

radial positions (layers A-D) in Tab. 4.1. The CF-orientation is generally reduced from the rim

part (A) to the center part (D), indicating that these regions were subject to a corresponding

decreasing shear stress during the extrusion process.

Tab. 4.1 The average orientation of CFs <cos2θ˃ in the different position of the 3 mm diameter anisotropic

PMMA/CF filament with different CFs concentration.

Region 𝒄𝒐𝒔𝟐𝜽

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

A (red) 0.991 0.989 0.988 0.990 0.987 0.943

B (green) 0.984 0.988 0.982 0.983 0.980 0.979

C (blue) 0.988 0.986 0.966 0.983 0.976 0.967

D (purple) 0.985 0.964 0.978 0.954 0.947 0.938

4.6 Electrical conductivity of the PMMA/CF binary composites

4.6.1 Anisotropic electrical conductivity of the 3mm diameter composite filament.

As known, the fillers orientation has a dramatic influence on the electrical conductivity of

CPC. Therefore, the electrical conductivity of the composite filament with aligned CFs should

be divided as vertical conductivity σV or σ∥ (parallel relationship between applied voltage and

extrusion direction) or σL (longitudinal conductivity) and horizontal conductivity σH or σ⊥

(perpendicular relationship between applied voltage and extrusion direction) or σT (transverse

conductivity). Moreover, due to the shear gradient across the channel section, the CFs


orientation were decreased from the outer part of the composite to the center, which leads to

different conductivity in the radial direction of the composites filament, i.e. the value of <sin γ>

increases from the outer region of the composite filament to the inner region of that. Thus, in

order to investigate the vertical conductivity σV and the σH, the different composition, i.e. the

non-uniform orientation of CFs in the CPCs must be taken into consideration.

A CPC filament with a radially uniform filler fraction, which is assumed in this study, can be

modeled as a resistor network (Fig. 4.11 (a)). A filament for example consisting of two radially

separated material layers (i.e. one inner region and one outer region) can be represented by the

resistor network of Fig. 4.11 (b), where the colors of the resistors (red, blue, green) indicate

different resistance values. The radial (green) resistors are neglectable for the conductivity,

which means that the CPC filaments of this study can be modeled as resistive networks with

four (non-connected) concentric cylinders (Fig. 4.11 (c)). The electrical properties of the CPCs

were (among others) analyzed using this model.

Fig. 4.11 The assumed structure of a CPC filament in this study. (a) A simple resistor network

mimicking a filament with radially independent conductivity; (b) A resistor network with two

concentric layers, connected radially with green resistors; (c) a resistor network with four

concentric layers, where the radial resistors were omitted.

4.6.1.1 Analysis on the longitude conductivity σ∥


In order to investigate the conductivity of each layer, the filament was considered as four

concentric circles at first, with same resistivity in each region (Figure 4.12 (a)). A “peeling-off”

procedure was applied for the investigation of the vertical conductivity σV or σ∥ of filaments.

At first, the conductivity of samples with certain length was measured at the original status

(Figure 4.12 (a)). Afterward, the sample was circle peeled to the expected radius (annular cut,

Figure 4.12 (b) to (d)), to remove the outer composites and then ultrasonic washed with ethanol

and dried for 24 hours. Both ends of samples were coated with silver for the conductivity

measurement (Chapter 3.6.1). Therefore, for each concentration of composites, four vertical

resistance (Rv,a- Rv,d) were obtained.

Figure 4.12 “peeling-off” procedure to obtain the longitudinal resistivity of the different regions

from the CPC filament. (a) Original extruded filament with 1.5 mm radius; (b) peeled-off filament

with 1.2 mm radius; (c) peeled-off filament with 1.0 mm radius; (d) peeled-off filament with 0.9

mm radius. The light-colored discs at the bottom only indicate the original geometry of the

filament.

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. 4c can be considered

as a parallel coupling (resistivity RL,c) between the blue cylindrical shell (resistivity RL,blue) and

the purple solid cylinder (resistivity RL,d). Since the longitudinal resistances RL,c and RL,d are

known, the unknown RL,blue of the blue shell can be calculated from:

1

𝑅𝐿,𝑐=

1

𝑅𝐿,𝑏𝑙𝑢𝑒+

1

𝑅𝐿,𝑑 (4.6)


The same procedure was used to calculate the resistances RL,green and RL,red for the green and

red cylindrical shells. Since the dimensions of the four cylinders were known, the longitudinal

electrical conductivity of each region σ∥, A (red region), σ∥, B (green region) σ∥, C (blue region)

and σ∥, D (purple region) from the 3 mm filament could thus be calculated. The conductivity σ

was calculated as follows:

𝜎 =𝐿

𝑆∙𝑅 (4.7)

where L is the length of the sample, S is the area of the sample, and R is the corresponding

resistance.

4.6.1.2 Analysis on the longitude conductivity σ⊥

In addition, a “polishing” procedure was applied for the investigation of the horizontal

conductivity σH or σ⊥ of filaments. The samples were set into epoxy resin and polished till the

targeted width is exposed (Figure 4.13, from (a) to (d)). The sample was then ultrasonic washed

with ethanol and dried for 24 hours. Both sides of samples were coated with silver for the

conductivity measurement (Chapter 3.6.2). Therefore, for each concentration of composites,

four horizontal resistances (RH,a- RH,d) were obtained for further calculation.


Figure 4.13 A four-step polishing procedure for obtaining the local transversal conductivites of a

CPC filament. Each filament is polished until the desired width (W) of the exposed sample is

reached. (a) Wa=1 mm, (b) Wb=1.8 mm, (c) Wc=2.2 mm, (d) Wd=2.4 mm.

In order to determine the transversal volume resistance (RT) in the horizontal direction, the

samples have been polished as presented in Fig. 4.13. Since the length of the investigated

samples in Fig. 4.13 always remain the same, therefore, only the cross section of the samples

are presented in this section for a better understanding. The four cross sections of the sample

(with four different polished width WA, WB, WC, WD), as well as the four corresponding

measured resistances RT,a, RT,b, RT,c, RT,d are presented in Fig. 4.14 (a). Four concentric circles

are assumed, with a constant resistivity ρ1, ρ2, ρ3 and ρ4 in each region (red, green, blue and

purple, respectively). This section presents the strategy on calculating the resistivity ρ1-ρ4 from

the experimental resistances RT,a -RT,d.

Fig. 4.14 (a) Collection of four cross sections of the sample from Fig. 4.13. (b) A rough assumption

on the resistance, based on a series connection, to calculate R1.

The applied voltage direction is given along the Y-axis, thus the measured resistance RT,a is

roughly considered as a series connection between the resistors in Fig. 4.14(b) -(c). Therefore,

the resistance of the shadowed area R1:


𝑅1 =𝑅𝑇,𝑎−𝑅𝑇,𝑏

2≅ ∫

𝜌1

𝑊(𝑦)∙𝐿

𝑦1

𝑦2𝑑𝑦 (4.8)

where W(y) is the width of the sample at the position y, L is the length of the sample as presented

in Fig. 4.14. Thus, the resistivity ρ1 can be obtained.

Similarly, the resistance of structure R2 presented in Fig. 4.15 (a) is given by:

𝑅2 ≅𝑅𝑇,𝑏−𝑅𝑇,𝑐

2 (4.9)

Fig. 4.15 (a) The resistor R2 and the corresponding structure. (b) A rough assumption of the

resistor R2 based on a parallel connection. (c) A rough estimation on the resistance of RR, based

on a series connection.

which is further assumed as a parallel connection between RL (left), RC (center) and RR (right),

as presented in Fig. 4.15 (b):

𝑅2 ≅ 1/(1

𝑅𝐿+

1

𝑅𝐶+

1

𝑅𝑅)) (4.10)

Combing Eq. (4.9) and Eq. (4.10), yields:

𝑅𝐶 ≅ 1/(2

𝑅𝑇,𝑏−𝑅𝑇,𝑐− 2/𝑅𝑅) (4.11)

Since the resistivity ρ1 (red material, as presented in Fig. 4.15 (c)) is known, the resistor RR in

Fig. 4.15 (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 RC can be calculated using

Eq. 4.11, as well as the resistivity ρ2 (green material) using similar strategy as Eq. 4.8.


As all the resistivity ρ1-ρ4 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. 4.16. The <cos2θ˃

values in the corresponding region are also shown.

Fig. 4.16 The logarithm longitude electrical conductivity σ∥ and transverse electrical conductivity

σ⊥ are presented in different colors, as well as the <cos2 θ> value in each corresponding region.

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].

4.6.1.3 Analysis on the longitudinal and transverse conductivities

The longitudinal and transverse conductivities of the different CPC layers are presented in

Fig. 4.17. Balberg has predicted that the percolation threshold of a CPC depends on the filler


aspect ratio (AR) and the orientation of the fillers. Therefore, in this study, the percolation

threshold of the transverse samples and longitude samples should be equal. According to the

McLachlan equation, the conductivity could then be precisely calculated, using the percolation

threshold and the PMMA and CF conductivities as input. Usage of the McLachlan equation

would thus result in identical transversal and longitudinal conductivities also for anisotropic

composites, which is unrealistic and in contrast to the results of this study. The discrepancy is

most probably explained by the fact that the McLachlan equation assumes isotropic composites

and that neither McLachlan nor Balberg has not considered the measuring direction, i.e. the

direction of applied voltage on the samples.

Fig. 4.17 The logarithm longitudinal electrical conductivity σL and the logarithm transverse

electrical conductivity σT in different shells of the cylindrical filament. (a) Comparing between σL

and σT in layer A and B; (b) comparing between σL and σT in layer C and D.

For anisotropic CPCs, Weber and Kamal proposed the “contact model” [121] to predict the

longitude conductivity (measuring voltage direction parallel to the fibers orientation, Eq. (4.12))

and transverse conductivity (measuring voltage direction perpendicular to the fibers orientation,

Eq. (4.13)):

𝜎𝑐,𝑙𝑜𝑛𝑔 =4∙𝑑𝑐∙𝐿∙𝑐𝑜𝑠

2𝜃

𝜋∙𝑑2∙ 𝛽 ∙ 𝜑𝑓 ∙ 𝜎𝑓 ∙ 𝑋 (4.12)

𝜎𝑐,𝑡𝑟𝑎𝑛𝑠 =4∙𝑑𝑐∙𝐿∙𝑠𝑖𝑛

2𝜃

𝜋∙𝑑2∙ 𝛽 ∙ 𝜑𝑓 ∙ 𝜎𝑓 ∙ 𝑋 (4.13)


where 𝜎𝑐,𝑙𝑜𝑛𝑔and 𝜎𝑐,𝑡𝑟𝑎𝑛𝑠 are the longitude and transverse conductivities of the composite,

respectively. d = diameter of the fibers, L = average length of the fibers, θ = average angle

between the inclination of fibers and the direction of the applied voltage. 𝜑𝑓 and 𝜎𝑓 are the

volume fraction of the fillers and the conductivity of the fillers, respectively. 𝑑𝑐 is the diameter

of the circle of contact, which depends on the applied voltage [121]. 𝛽 is a coefficient to

describe the participating fillers in the conductive network. For fillers with volume fraction 𝜑𝑓

below percolation threshold 𝜑𝑐, 𝛽=0; When the filler volume fraction 𝜑𝑓 reaches a saturated

volume fraction 𝜑𝑡, 𝛽=1; For fillers volume fraction in the range 𝜑𝑐 < 𝜑𝑓 < 𝜑𝑡, 𝛽 follows

𝛽 =𝜑𝑓−𝜑𝑐

𝜑𝑡−𝜑𝑐 (4.14)

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

𝑋 = 0.59 + 2.25 ∙𝛽∙𝜑𝑓

𝜑𝑡 (4.15)

(It can be noted that Eqs 4.12-4.13 becomes zero when 𝜑𝑓 =0. In a real composite the

conductivity will rather approach the polymer conductivity 𝜎𝑎, which is small but larger than

zero, when the filler fraction decreases. The equations could therefore become more realistic

by adding 𝜎𝑎.)

A distinct vertical shift between the longitudinal conductivity (solid conductivity curve) and

the transverse conductivity (dashed conductivity curve) is clearly observed in Fig. 4.17 (a), but

only a slight vertical shift is observed in Fig. 4.17 (b). The explanation is that the CFs are highly

anisotropic in the outer shells of the composite cylinder (layers A and B) but less oriented in

the central part of the cylinder (layers C and D). This 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, which is obtained by a combination with contact model. Combing Eq.1

and Eq.2 yields:


𝜎𝑡𝑟𝑎𝑛𝑠

𝜎𝑙𝑜𝑛𝑔=< tan2𝜃 >=

1−<𝑐𝑜𝑠2𝜃>

<𝑐𝑜𝑠2𝜃> (4.16)

which is presented in Fig. 4.18 as a solid line. The experimental data points in this study are

shown as points in different colors corresponds with different layers and the data from samples

with different concentration of CFs are marked with different symbols. The best linear fit (blue

dashed line) leads to a slope of 1.05, consisting with the Eq. 4.16.

Fig. 4.18 The ratio between 𝝈𝒕𝒓𝒂𝒏𝒔/𝝈𝒍𝒐𝒏𝒈 as a function of 𝐭𝐚𝐧𝟐𝜽 according to equation 4.16. The

red line shows Eq. 4.16, while the dashed line indicates the best linear fit.

Considering the relatively 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 Eq.1-3 yields:

𝜎𝑙𝑜𝑛𝑔

cos2𝜃=

𝜎𝑡𝑟𝑎𝑛𝑠

sin2𝜃=

4∙𝑑𝑐

𝜋∙𝑑2∙𝜑𝑓−𝜑𝑐

𝜑𝑡−𝜑𝑐∙ 𝜑𝑓 ∙ 𝜎𝑓 ∙ 𝑋 (4.17)


Fig. 4.19 Fit curve in order to investigate the saturated concentration of CFs and the contact

distance.

The best fit line of Eq. 4.17 is presented in Fig. 4.19 together with corresponding experimental

data. A value of 18 vol. % was obtained for 𝜑𝑐. The 𝜑𝑡 became 0.58, indicating a saturated

CF concentration of 58 vol. %, after which all the CFs were participating in the conductive

network. The diameter of the circle of contact between CFs was obtained as 0.023 μm

(illustrated as inset in Fig. 4.19), which is much higher than the reported value 2.1E-4 μm [121],

1.4E-7 μm [122] and 2.1E-6 μm [63], 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=σlong-σtrans from Eq. 4.17 such that:

Δf= σlong – σtrans =σlong(1-tan2(θ)) (4.18)

σlong=σ*/ tan2(θ*)) (4.19)

where σ* is the predicted conductivity from the unmodified (isotropic) McLachlan equation

and θ* is the average angle in a perfectly isotropic composite. With this modification the

McLachlan equation can most probably predict both longitudinal and longitudinal composite

conductivities for anisotropic composites with average angle θ of filler against the electrical


field. Additional experimental measurements will however be needed to further strengthen the

revealed effect and hypotheses for explanation.

4.6.2 Influence of CFs AR on electrical conductivity

Figure 4.20 shows the logarithm of vertical conductivity of the samples plotted versus the CF

volume fraction. The limit of concentration of CFs in the composites which can be anisotropic

in the capillary rheometer is 56.44 vol. % (60CF), because composites with a higher

concentration have too high a viscosity. Thus, the maximum electrical conductivity found in

this work was 3.57 S·cm-1.

Figure 4.20 Logarithm of conductivity vs CF volume fraction for the composites from different

master batches and one-step mixing.

The GEM equation (Equation 2.18) is used to obtain the percolation threshold. The

conductivity of the CFs, σf was stated by the manufacturer to be 5.89×102 S·cm-1, and the

conductivity of the PMMA matrix, σm (3.64×10-9 S·cm-1) was measured on a pure PMMA fiber

anisotropic under the same conditions as the other samples. The empirical exponent s was taken


to be 7. Due to the non-uniformed aspect ratio of the CFs from the 1st step mixing, these data

(red circles) are not further discussed.

The fitted theoretical curves in Figure 4.20 are shown with dotted lines, and the percolation

threshold φc at three different volume fraction of CFs were obtained: The percolation threshold

of composites from MB40 (green triangles) is 21.34±8.16 vol. %; of composites from MB50

(blue squares) is 26.39±4.81 vol. %; and of composites from MB60 (black pentagons) is

30.51±4.53 vol. %. From each fitted curve, the concentration with a second derivative equal to

zero (where the conductivity increases most rapidly) was also calculated as another description

of the percolation threshold: φMB40 = 25.51 vol. %, φMB50 = 30.43 vol. %, φMB60 = 34.38 vol. %,

and these values do not differ significantly from the fitted results. Composite filaments with a

higher CF AR facilitate the construction of a conductive network, and therefore reduce the

percolation threshold.

The GEM equation from McLachlan describes system with isotropic fillers. In this study, the

CFs were well oriented inside the composites and these are only small inclinations between the

CFs, which decrease the possibility for CFs to contact each other. A possible modification of

the GEM equation taking into consideration the orientation of the fillers to cover the effects

described above is left for future study.

4.6.2.1 Modification on the excluded volume theory from Balberg

In the study of Starý et al. [23], the plate-formed PMMA/CF samples were affected by

centrifugal force at 200 °C. The dispersion of CFs was not totally random, but the distribution

of orientation of the CFs was still uniform in the plane of the plate, i.e. isotropic in the range of

360°. Therefore, the value of <sin γ> (the average angle between each pair of random fibers in

the system) according to Starý was also considered as 0.64.


For cylinders with L>>W in Eq. 2.3, the AR2 term in the denominator dominates, so that φc is

approximately proportional to the reciprocal of the aspect ratio (AR-1). Several reports based

on this approximation were found [23, 97-99]. In Figure 4.21 the percolation threshold φc from

literature and this study are presented with hollow symbols, which are plotted versus AR-1

(dashed horizontal axis). In the study of Yi & Choi [97] (marked with crossed inverted triangles),

extremely low percolation was reported. Because the composites were prepared below the

melting temperature, which leads to an agglomerate of CFs only on the surface of the polymer

granules. Consider following preconditions: 1. the polymer granules as perfect spheroids with

regular size; 2. the CFs exist only between them; 3. the efficiency of close packing is

approximately 0.74. In this case, the data of Yi & Choi has been divided by φc with (1 - 0.74 =

0.26) (marked with inverted triangles), in order to compare their data with the other literature.

Figure 4.21 Percolation threshold φc vs (a) dashed horizontal axis: Simplified formula from

Equation 2.4, with large AR, and regardless of <sin γ>; and (b) solid horizontal axis: complete

formula from Equation 2.4, considering <sin γ>.


It is important to note that the precondition for this approximation (dashed coordinate system)

is large values of the aspect ratio (AR). If the aspect ratio is much smaller, the prediction of

percolation threshold must still follow Equation 2.4. The data calculated from the complete

formula are marked with solid symbols in the coordinate system with solid horizontal axis in

Figure 4.16. All these data fall on the same master line. The fitted result leads to K=3.4.

In a composite system with totally aligned CFs, the orientation of CFs <sin γ> is defined as

0. According to Equation 2.4, the value of φc is therefore always equal to 0.125·K, i.e. φc ≈ 42.5

vol. %, which is consistent with the totally aligned CFRP model (φc = 40 vol. %,) from

Chippendale et al. [24]. In practice, however, the CFs cannot be totally aligned. The average

<sin γ> of all the anisotropic composite filament in this study was obtained by fitting equation

2.4 (with K=3.4) to give : <sin γ> = 0.21, which is consistent with the calculated <sin γ>=0.17

of sample Nr. 60CF, and is consistent with the rough average value in Figure 4.9 . With K=3.4,

the equation from Balberg is presented in Figure 4.22, the relationship between percolation

threshold and the orientation of CFs average <sin γ> is presented.


Figure 4.22 Modified equation from Balberg (Equation 2.4), with K=3.4. Fillers with different AR

in this work are presented in different color, and fillers with extreme large AR (e.g. CNTs) are

presented in dashed lines.

4.6.3 Influence of CFs orientation on electrical conductivity

The precise relationship between electrical conductivity and orientation of fillers are still

unclear. However, it is possible to compare the electrical conductivity of composites filaments

produced utilizing extrusion dies with 1-mm and 3-mm diameter, respectively, will be

compared and discussed. Since the CFs become more orientated along the extrusion direction

when using the die a smaller diameter (i.e. the D=1 mm sample contains higher oriented CFs

than that in D=3 mm sample), the relationship between electrical conductivity of the specimen

and average CFs orientation could then be investigated.

Figure 4.23 shows the logarithm of the electrical conductivity of samples plotted versus the

CF concentration. The conductivities of anisotropic composites filaments with 1-mm and 3-

0.2 0.4 0.6 0.8 1.00.0

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

perc

ola

tin

o t

hre

sh

old

sin gama average

AR6

AR9

AR12

AR100

AR1000

0.64


mm diameter, are shown with red circles and black squares, respectively. Error bars in the

horizontal direction indicate the uncertainty in filler fraction (i.e. the difference between actual

and designed CFs volume fraction). The conductivity of pure PMMA anisotropic filament was

determined as 1.16 × 10−9 S/cm. The percolation thresholds of the composites filament were

obtained by fitting the GEM equation from McLachlan (Equation 2.18), which is presented in

Figure 4.23. The percolation thresholds φc of composites filament with 1 mm and 3 mm

diameter are: e 𝜑𝑐 (D = 1 mm) = 32.0 ± 5.9 vol. %, and 𝜑𝑐 (D = 3 mm) = 20.0 ± 2.5 vol. %. The

percolation thresholds determined by using the classical percolation theory (Equation 2.1): 𝜑𝑐(D

= 1 mm) = 31.9 vol. % and 𝜑𝑐(D = 3 mm) = 20.3 vol. %, which are consistent with the data obtained

from the GEM equation. The corresponding illustrations of the fit results utilizing Equation 2.1

are presented as insets in Figure 4.23.

During the extrusion process with the same speed of the pistol (0.08 mm/s), the composites

melt through 1-mm diameter die encountered higher shear rates ((D = 1 mm) = 92.16 s−1) and

therefore, larger shear deformations occurred for the samples with 1-mm diameter. As a

consequence, the CFs in the anisotropic filament with 1 mm diameter present a higher

orientation. Since the AR of CFs were controlled in this chapter, the average CF orientation

was the only difference between samples with 1 mm diameter and 3 mm diameter. The results

presents that a higher orientation of the CFs shift the percolation threshold to a higher CFs

volume fraction, as proved in Chapter 4.6.2. Therefore, the electrical conductivity of the

composites with 1 mm diameter is significantly lower than that of the samples with 3 mm

diameter in the region of CFs between the two percolation thresholds (20–32 vol. %).


Figure 4.23 The logarithm of the electrical conductivity vs. CF volume fraction for anisotropic

filaments with 1-mm diameter and 3-mm diameter, respectively. The solid lines show the best fit

with the McLachlan equation (Equation 2.18). The two insets illustrate the linear fit using

(Equation 2.1) for the 1-mm diameter and 3-mm diameter, respectively.

Moreover, it must be noted in the Figure 4.23 that the conductivity of the specimen with 1

mm diameter (red circles) rapidly increased ca. 6 magnitudes around the percolation threshold

and then kept almost constant when the CFs volume fraction are above the percolation

threshold. However, the conductivity of the specimen (black squares) with 3 mm diameter has

only increased 3 magnitudes near the percolation threshold, but kept increasing slowly after the

percolation threshold.


At CF volume fraction with approximately 50 vol. %, the specimen with 1 mm and 3 mm

diameter approached a similar value. This phenomenon can be explained by the “contact

model” [63], which is typically used for predicting conductivity CPCs with a filler volume

fraction beyond the percolation threshold (Equation 2.13).

In this chapter, the vertical conductivity is discussed, i.e. the measuring voltage was applied

along the extrusion direction of the filament and thus, the exponent cos2θ (the orientation of

CFs) increases when the tiny CFs in the CPCs become more oriented to the extrusion direction.

However, the CFs in the specimen with 1 mm diameter, were already highly anisotropic to the

extrusion direction, even at a low filler fractions (θ ≈ 0 => cos2θ ≈ 1), therefore, the cos2θ-term

from samples with 1 mm diameter almost keeps constant. Comparing to that, the CFs in the

specimen with 3 mm diameter, were firstly more isotropic at a low filler fraction, and got more

and more oriented to the extrusion direction as the concentration of CFs increases. Therefore,

the cos2θ value as well as the conductivity (which only depends on cos2θ value in Equation 2.13

in this case) of the samples with 3 mm diameter increased gradually with an increasing filler

concentration. When the CFs volume fraction in the composites reach at very high filler

fractions (beyond the percolation threshold), the CFs in both samples with 1 mm and 3 mm

diameter were highly anisotropic. Therefore, the final electrical conductivity values get similar

to each other.

It can be also noted that when the CFs volume fraction is far beyond the percolation threshold

(i.e., higher than 35%), the conductivity of the anisotropic filaments with 3 mm diameter is

lower than that with 1mm diameter. This means, an increased CFs orientation leads to a higher

electrical conductivity along the orientation direction, which is consistent with reported

literature [105-107]. Taking the morphological study into consideration, a novel approach was

proposed to explain the phenomena.


4.6.3.1 A novel approach supported by the corresponding morphological study

The explanation of the different conductivities between samples with 1 mm and 3 mm

diameter lies in the volume fraction of CFs. Figure 4.24 presents a schematic interpretation of

the CFs orientation and CF volume fraction.

The first precondition of this approach is that the midpoints of CFs are fixe and the

independent CFs can only rotate around their midpoints in a 2-D system. Therefore, the CF

volume fraction depends on the midpoints of the CFs and the rotation of each CF leads to a

different CFs orientation in the CPCs. Thus, a circle is formed, considering the rotation range

of each single CF around the middle point, which is consistent with the excluded volume

concept from Balberg [65, 66]. The measuring direction of the CPCs is applied from top to

bottom of the figures, which is shown with the two-way arrows.

Based on this hypothesis, the close packing of the circles refers to the CFs volume fraction,

which just reaches the percolation threshold, i.e. the CFs just can not reach or contact each other

(Figure 4.24 (a) and (b)). In this case, there is no difference between samples with anisotropic

CFs (sample with 1 mm diameter in this case, Figure 4.24 (a)) and samples with isotropic CFs

(sample with 3 mm diameter in this case, Figure 4.24 (b)). This is the reason why the composite

filament with 1 mm diameter present a lower conductivity in the range of CFs below 15 vol. %

in Figure 4.25.


Figure 4.24 Schematic interpretation of CPCs with anisotropic CFs (a,c,e) and CPCs with

isotropic CFs (b,d,f). The structure of CFs are presented as below percolation threshold (a,b); just

reach percolation threshold (c,d); and far beyond percolation threshold (e,f).

When the volume fraction of the CFs increases, it can be considered as an increase of the

middle points of the CFs in the same volume, which can be further simplified as a proportional

reduction of the hexagon size of the CFs middle points (Figure 4.24 (c) and (d)). In this case,

the overlapping are be CFs in Figure 4.24 corresponds with the “soft-shell/hard-core” structure

of the CFs [14]. It can be seen from Figure 4.23 (c) that the CFs were parallel to each other and

oriented along the measuring direction. Therefore, the possibility is reduced for CFs to reach

each other and build a conductive pathway. However, by just adjusting the orientation of CFs

(a rotation of CFs in Figure 4.24 (d)), the percolation threshold of the same volume fraction of

the CFs are reached. With the successful build of the electrical pathway, the conductivity of the

CPCs dramatically increase. This is the reason why the composite filament with 1 mm diameter

present a lower conductivity in the range of CFs 20-35 vol. % in Figure 4.23.


When the CF volume filler fractions increases beyond the percolation thresholds (Figure 4.24

(e) and (f)), i.e. conductive network are formed in both CPCs, and filament with 1 mm and 3

mm diameter turn to be both conductive. However, the difference between two kinds of

conductive pathways must be noted: (1) along one single CF; (2) transfer between the contact

points between two connected CFs. It can be logical that the first pathway is more efficient with

a higher conductivity, which appears more in the system with oriented CFs, as presented in

Figure 4.24 (e). Therefore, the CPCs with oriented CFs along the measuring direction presents

a higher conductivity than that with isotropic CFs. This is the reason why the composite

filament with 1 mm diameter present a higher conductivity in the range of CFs higher than 35

vol. % in Figure 4.23.

Figure 4.25 (g) presents a general morphological picture of the PMMA/CF CPC filament,

where a single carbon fiber and the extrusion direction were shown. Figure 4.25 (a)-(f) present

the fiber orientation in anisotropic composite filament with 10 vol. % (Figure 4.25, (a)(b)), 30

vol. % (Figure 4.25, (c)(d)) and 50 vol. % CFs (Figure 4.25, (e)(f)) respectively. The upper row

of figures (Figure 4.25, (a)(c)(e)) refers to the filament with 1 mm diameter, containing more

oriented CFs; the lower (Figure 4.25, (b)(d)(f)) refers to the filament with 3 mm diameter,

containing less oriented CFs. It can be noted that the CFs in the filament with 3 mm diameter

are more isotropic than that in the 1 mm diameter filament. Moreover, it can be seen that the

orientation of the CFs increases, as the filler fraction increases. This is the certificate of the

novel approach in Figure 4.24.


Figure 4.25 Morphological study on the orientation of inner CFs with 1-mm diameter (g,a,c,e) and

with 3-mm diameter (b,d,f). (a)(b) show filament with 10 vo.% CFs, which is below percolation

threshold; (c)(d) show filament with 30 vo.% CFs, which just reach percolation threshold; (e)(f)

show filament with 50 vo.% CFs, which far beyond percolation threshold.

4.7 Conclusion

In Chapter 4, PMMA/CF anisotropic filament are presented. The CPCs were produced under

high concentration (Chapter 4.2). Different lengths of CFs (with aspect ratios 6.3, 9.2, and 12.0,

respectively) were obtained with a two-step melt mixing process (Chapter 4.4). Due to the

extrusion process, highly anisotropic CFs were obtained in the composites, presenting a “fibers

in fiber” structure (Chapter 4.3). Moreover, due to the shear gradient across the channel section,

the CFs orientation were decreased from the outer part of the composite to the center (Chapter

4.5)


In Chapter 4.6.1, the vertical and horizontal electrical conductivity of the anisotropic

anisotropic PMMA/CF composite with 3 mm diameter were discussed. The “peeling-off” and

“polishing” methods were developed, in order to reveal different conductivity in the radial

direction of the composites filament.

In Chapter 4.6.2, the vertical electrical conductivity of PMMA/CF filaments with 1 mm

diameter were revealed. The conductivity of composite filaments at room temperature was

compared with the McLachlan GEM equation and Balberg excluded volume theory. An

increase the filler aspect ratio led to a decrease in the percolation threshold. Taking into

consideration the filler orientation, the slope K=3.4 in excluded volume theory was estimated

for the first time.

In Chapter 4.6.3, the conductivity of PMMA/CF filaments (with controlled aspect ratio of

CFs) with 1-mm and 3-mm diameter were compared. The results show that a higher orientation

of the CFs can shift the percolation threshold to a higher volume fraction. However, when the

CFs volume fraction beyond the percolation threshold, a higher orientation of the CFs along the

measuring direction leads to a higher conductivity of the CPCs. A novel explanation based on

the morphological study was proposed to explain the phenomenon.

5 PMMA/CB/CF ternary system

66 6 PMMA/CB/CF ternary system

5.1 Introduction

Besides carbon fibers (CFs), carbon black (CB) is also the most commonly used conductive

fillers, which can be incorporated into a polymer matrix for facilitating the formation of a CPCs.

In this chapter, the electrical conductivity of anisotropic composites filament consisting of

PMMA containing both aligned carbon fibers (CF) and carbon black (CB) will be presented. A

broad range of composite compositions (up to 50 vol. % CF and 20 vol. % CB) has been studied.

The CB (XE2) used in this chapter is highly structured and the primary particles are fused to

small aggregates which in turn form agglomerates [14], resulting in a significantly reduced

percolation threshold. As discussed in Chapter 4, the PMMA/aligned CFs composites filament

presents a higher percolation threshold than those with isotropic CFs. Therefore, the study on

the conductivity of PMMA/aligned CFs/CB has arisen the interests.

Literature exist reporting a synergistic effects of CPCs containing both CFs and CB [108-

112]. In a system with isotropic CFs and CB particles, clusters of CB may form short

“conductive pathways” between the CFs. Therefore, the conductivity of composites containing

both fillers can be enhanced, as compared to those with only CFs or CB at the same

concentration of fillers.

Figure 5.1 Schematic of the conductive pathway formed by CB and CFs in the CPCs. (a) CPCs

with isotropic CFs, conductive pathway are contributed also by CFs themselves; (b) CPCs with

highly orientated CFs, the contacts between CFs are reduced. With the addition of CB particles,

a fence-like structure can be formed; (c) cross section view of composite filament with fence-like

structure.

5 PMMA/CB/CF ternary system 67

However, in a CPC system with isotropic fillers (Figure 5.1(a)), the contribution of

conductive pathways formed between CFs themselves to the conductive networks must also be

considered [108-112]. In order to evaluate the real synergic effect between CFs and CB, the

effect of the CF-pathways in a PMMA/CB/CF system should be minimized. Therefore, the

extrusion process is applied, in order to induce a maximum alignment of the CFs in the CPCs

and consequently, to reduce the contacts between the CFs. With the addition of CB particles,

“conductive pathways” can also be formed between the parallel CFs, such that fence-like

structure is formed. (A side view and a cross section view of an idealized fence-like geometry

is shown in Figure 5.1 (b) and Figure 5.1 (c), respectively). Thus, the real synergistic effect

between CB and CFs in an anisotropic composite filament can be then revealed.

Synergistic effects were also reported between CB and carbon nanotubes (CNT) in ternary

composites [113-115]. CNT and CF are both suitable conductive 1-D fillers. In this chapter,

CFs were chosen at first due to their (1) improved melt mixing dispersion (2) more distinct

cylindrical shape, (3) larger size, which simplifies microscopy analysis and (4) higher

percolation threshold, which enabled a more genuine validation of our analytical conductivity

model.

Several theories [116, 117] have been suggested to describe the conductivity of binary

composites, but few models consider the conductivity of ternary composites with two

conductive fillers. Sun et al. [81] have proposed a percolation threshold model based on the

assumption that the excluded volume of the two fillers can be added together linearly. The

percolation threshold of a conductive composite with two fillers can then be predicted as

Equation 2.21. Therefore, a conductivity contour plot versus φCFs and φCB was presented in this

chapter, such that the linear Equation 2.21 could be evaluated.

Since the experimental conductivity measurements did not fully support Equation 2.21, an

improved original equation was proposed in this work for predicting the electrical percolation


threshold of ternary composites containing two different conductive fillers. In addition, based

on the improved percolation threshold equation, a novel equation for predicting the electrical

conductivity of ternary composite was also developed.

5.2 Morphological study on the PMMA/CB/CF ternary composites

Figure 5.2 shows a cross section of a PMMA/CF/CB composite filament (with parallel CF

fibers), which proves the assumed fence-like structure of Fig. 1b. In Figure 5.2 (a) and (b), the

cross section of the complete sample is presented. CFs as well as “black holes" (positions of

pulled out CFs) can be observed inside the composite. It can be seen that almost all the CFs and

the “black holes” are parallel to the axis of the composite filament, which indicates that the CFs

are highly anisotropic in the composite filament (more discussion about CF orientation can be

found in Chapter 4.2). Due to the alignment of the CFs along the extrusion direction of the

composites filament, almost no conductive pathways are formed directly between the CFs.

Figure 5.2 (c), shows a single CF surrounded by CB particles, revealing that the parallel CFs

were connected by the small CB particle. Therefore, the conductive pathways are formed

through the PMMA matrix. Typical conductive pathways in PMMA/CF/CB systems can thus

be visualized with images like Figure 5.2 (c). Moreover, the CB particles are well dispersed and

no significant cluster or agglomerates are found (Figure 5.2 (d)).


Figure 5.2 SEM micrographs of cross sections of composite filament with different magnification,

sample: 2CB3CF.

5.3 Conductivity of the composite filament

5.3.1 Percolation threshold of binary PMMA/CF and PMMA/CB samples

For PMMA/CB and PMMA/CF binary composites filament, the electrical conductivity, as a

function of filler fraction, is presented in Figure 5.3. The X- and Y-axis present the CB and CF

volume fractions in the composites filament, respectively. Until now, several models have been

proposed to describe percolation phenomena at critical levels of conductive fillers and to predict

the electrical conductivity behavior of CPCs. The most classical percolation theory equation is

Equation 2.1 [48-49]. For composites with filler concentrations φ>φc, the experimental results

(a) (b)

(c) (d)


are fitted by plotting log σ against log (φ-φc) and regulating φc until the best linear fit is obtained.

Thus, the estimated percolation thresholds for the anisotropic PMMA matrix composite

filament become: φc, CB =3.9 vol. %, and φc, CF =31.8 vol. % (Figure 5.3), which is consistent

with the percolation threshold of PMMA/CF 30.43 vol. % as reported in Chapter 4.6.2. A strong

alignment of the anisotropic CFs was induced by the extrusion process, and consequently, the

percolation threshold of the PMMA/CFs composite filament becomes much higher than of the

reported PMMA/CFs binary composites filament with isotropic CFs [108-112].

Figure 5.3 Conductivity of PMMA/CF and PMMA/CB composites vs. filler volume fraction in

three-dimensional coordinates. Standard deviations, based on 10 measurements/data points, are

shown with error bars. The two insets illustrate the linear fit of the t-value for PMMA/CF (red

square) and PMMA/CB (blue spheroids), respectively.

5.3.2 Contour plot of conductivity on PMMA/CB/CF ternary composite

A conductivity contour plot diagram based on the experimental data (binary composites

and ternary composites) is presented in Figure 5.4, where the X-axis and Y-axis correspond to


CB volume fraction and CF volume fraction, respectively. The black squares on the contour

plot show the composition of CF and CB from actual tested samples. The 10-logarithm of the

electrical conductivity of CPCs is presented in colors while iso-contour lines equating 1

magnitude are drawn in black. The composite conductivity obviously increases steeply with

growing filler concentration.

Figure 5.4 Logarithm value of conductivity of composite filament as a function of filler volume

fraction of CFs and CB in a contour plot diagram presented in this work.

The percolation concentrations of CB and CF are marked on the X-axis (3.9, 0) and Y-axis

(0, 31.8), respectively. The white dashed diagonal line a (Y=-8.17X+31.8), which connect the

two percolation points, correspond to Equation 2.21 with K=1, showing the predicted

percolation threshold of the ternary composites. The percolation threshold should follow this

line if the excluded particle volumes are linearly additive, i.e. if Equation 2.21 is valid.

The dramatic changes in electrical conductivity must be considered, when the volume fraction

of fillers is near the percolation concentration, which can result in a large standard deviation for


the experimental measurements (See Figure 5.3). Therefore, two additional parallel lines (line

b and c) were also added on the contour plot, with fillers volume fraction are above the

percolation threshold. Line c was created by passing the maximum investigated CF volume

fraction (0, 50), with the same slope as line a. The intermediate value of y-intercepts from line

a and line c, was applied as the y-intercept of line b. These lines correspond to K=1.28 and

K=1.56, i.e. filler fractions 1.28 respective 1.56 times larger than the presented percolation

threshold filler fractions.

Conductivities of composite filament with CF/CB compositions following the three diagonal

lines were investigated and presented in Figure 5.5. For all three diagonal lines (on X-Y plane),

the conductivities (Z value) at the ends of the line were higher than in the middle, indicating a

negative collective effect between CB and aligned CFs. This result is in agreement with the

right-pointed protrusion (positive curvature) of contour lines in Figure 5.4, which is in contrast

to the slight “synergetic effect” [108-112] previously observed. However, small protrusions to

the left of the contour lines can be still observed for CF filler fractions below 10 vol. %,

indicating local synergetic effects similar to those previously reported [108-112].

The conductivity contour-lines of the composites filament are clearly concave and materials

showing synergetic effects must by definition also show non-linear behavior. In order to

describe such materials, the linear assumption of Equation 2.21 is thus clearly not valid,

although the equation still can be used for determining synergy. Therefore, an extended and

more general equation is required for predicting the conductivities of CPCs containing two (or

more) fillers.


Figure 5.5 Logarithm value of conductivity of composites with composition of CFs/CB on the three

diagonal lines (from Figure 5.4) vs. filler volume fraction of aligned CFs and CB in this work.

As a complement to the measurements, the literature on conducting polymer composites with

isotropic CF and CB in polymer matrix (thermosetting based shape-memory polymer [29],

acrylonitrile butadiene styrene [108] and polypropylene [109-111]) have been reanalyzed and a

conductivity contour plot of the results is presented in Figure 5.6. The literature data present

higher composite conductivities than the data from the anisotropic composite filament in this

work (Figure 5.3), probably mainly because the axial CF-orientations in this work obstruct the

construction of conductive networks and thus reduce the composite conductivities. It should

thus also be noted that the polymer matrix in the literature review is other polymer matrix

instead of PMMA in this study.


Figure 5.6 logarithm value of conductivity of composite filament vs. filler volume fraction of CFs

and CB in a contour plot diagram on the basis of literature reanalysis.

5.4 Novel equations for electrical behavior of ternary composite

If Equation 2.21 would hold, the iso-curve between φc,CB and φc,CF in Figure 5.4 would be a

straight line. Since the measured electrical conductivities showed distinctly curved iso-curves,

a new equation was required for describing the percolation behavior of CPC’s containing two

kinds of fillers more accurately. This conclusion was also strengthened by literature data [108-

110], showing synergetic (i.e. non-linear) effects by combining certain kinds of conductive

fillers. In order to simplify the mathematical expressions in Equation 2.21, the relative filler

fractions α and β were introduced, which are defined as the ratios between present volume

fraction φ and percolation threshold φC, for CFs and CB, respectively:

CFcCF ,/ (5.1)


CBcCB ,/ (5.2)

Equation 2.21 could thus be reformulated as:

K (5.3)

A corresponding improved non-linear equation should be able to describe both linear,

concave and convex curves and give correct values at α =0 or β =0. The least complicated

function fulfilling those criteria is probably:

𝑍 = (𝛼𝑁 + 𝛽𝑁)𝑁 (5.4)

where the physical meaning of Z is the same as K in Equation 2.21 and Equation 5.3, i.e. when

Z=1, the percolation threshold just reached. κ is an adjustable interaction parameter. If N =1 the

equation becomes linear, if N <1 the two fillers are synergetic and if N >1 the fillers are anti-

synergetic. Equation 5.4 can thus be used as an improved alternative to Equation 2.21 for

describing the percolation threshold of CPCs containing two fillers. The ‘synergy’ can also be

defined as: when the curvature parameter N becomes smaller than 1 when the experimental data

is fitted to Equation 5.4.

However, the property of main interest is often the composite conductivity rather than the

percolation threshold. Therefore, an equation for predicting the conductivity of three-

component composites was also developed. The conductivity of a composite containing only

one filler can often be accurately described, as function of filler fraction φ, with the McLachlan

Equation 2.18 [67, 68]. The hypothesis for the novel equation was that a linear combination of

the conductivities of two binary composites, each with one single type of filler, could be used

to describe the conductivity of a ternary composite, containing both types of fillers. This

hypothesis correspond to coupling the binary composites in parallel. According to this

hypothesis, the new equation (assuming CB and CF fillers) could be written as:


CBCBcCFCFcCBCFtot ZZ ,, ,1min,1min),(

(5.5)

The single-filler percolation thresholds φc,CF and φc,CB as well as the φ-dependent single-

filler conductivities σ(φCF)CF and σ(φCB)CB are determined with Equation 2.18. As a safety

routine, a restriction was added such that the functions could not exceed the values of their

corresponding filler conductivities.

The prediction capability of Equation 5.5 was evaluated using the experimental data

presented in Figures 5.3-5.5. As a first step, the single filler conductivities of PMMA/CB and

(un-axial) PMMA/CF composites filament were fitted with the McLachan equation (Equation

2.18), resulting in two log(σ) versus φ curves. The estimated filler conductivities were σCB

=1.34*102 S/cm for CB and σCF=4.06 S/cm for CF. The fitted percolation thresholds with

McLachan equation were φc,CF =33.5 vol. %. and φc,CB =3.9 vol. %, which are consistent with

the value obtained from Eq. 3: φc, CF =31.8 vol. % and φc, CB =3.9 vol. %. The result, as presented

in Figure 5.7, shows a good agreement between the experimental conductivities and the best fit

of Equation 2.18.


Figure 5.7 Conductivity of composite filament vs. filler volume fraction of aligned CFs and CB,

fitting with McLachlan equation to determine the electrical conductivity of fillers in the

anisotropic fiber consisting of ternary composites.

Equation 5.5, as implemented in the Matlab(R) software, was then used for creating a two-

dimensional conductivity contour plot. Equation 2.18 (with parameters as in Fig. 5.7: σCF, σCB,

φc,CF, φc,CB), and Equation 5.4 (with N = 1.47) were used as input. The resulting model waterfall

plot (Figure 5.8) and contour plot (Figure 5.9) show a trend which fits the corresponding the

experimental contour plot (Figure 5.4) well.


Figure 5.8 Logarithm value of conductivity of composite filament vs. filler volume fraction of CFs

and CB in a waterfall plot diagram, as generated with equation 5.8.

Figure 5.9 Logarithm value of conductivity of composite filament vs. filler volume fraction of CFs

and CB in a contour plot diagram, as generated with equation 5.8.


Finally, Equation 5.5, with settings as above, was compared to the experimental

conductivity data along the three lines of Figure 5.6. The result was presented in Figure 5.10.

Since the experimental non-linear trend was approximately in agreement with Equation 5.5, the

new equations are clearly more accurate than previously existing models for describing the

electrical conductivity and the percolation threshold for composites containing two different

conductive fillers.

Figure 5.10 Logarithm value of conductivity of composite filament vs. filler volume fraction of CB

in a 2-D coordinates system. Filled symbols represent experimental data while solid lines represent

the simulation curves with predicted data from Equation 5.8, where line a-c are from Figure 5.4.

5.5 Conclusion

In this study, anisotropic PMMA/CB/CF ternary composites filament, with a broad range of

composites compositions (up to 50 vol. % CF and 20 vol. % CB) were studied. SEM

micrographs exposed a “fence-like” morphological structure between the CB and the CF.


Electrical conductivity measurements were used to determine the electrical percolation

thresholds of binary PMMA/CF and PMMA/CB composites filament and to reveal the relation

between filler fraction and the ternary composite filament conductivity for PMMA/CB/CF.

Conductivity contour plots, showing conductivity as a function of filler fraction, were presented

for PMMA/CF/CB composites filament for the first time; one extensive plot based on data from

this study and one based on already existing literature data. Distinctly curved iso-lines were

observed. Synergetic effects between CFs and CB were found in the regions below 10 vol. %

CF, observed as left-pointed protrusions of the contour-lines. Moreover, based on a model for

predicting the percolation threshold of ternary composites, a novel equation for predicting the

electrical volume conductivity of ternary composites was proposed, showing results close to

corresponding experimental data. Since the study in this chapter comprises highly systematic

measurements of electric conductivity for PMMA/CB/CF composites as well as the novel and

significantly improved effective-media equations for conductive ternary composites, it provides

a general guidance for the design and analysis of CPC.

6 PMMA/CB/CNTs ternary system

82 6 PMMA/CB/CNTs ternary system

6.1 Introduction

Conductive polymer composites (CPCs) are promising materials for many electrical

applications due to their high conductivity, low weight and simple processing [70-73]. Carbon

nanotubes (CNTs) and carbon black (CB) are two of the most commonly used conductive fillers

in CPCs [74-79]. Several studies on ternary CPCs comprising CNTs and CB have recently been

reported, indicating that the electrical properties of ternary composites comprising CB and

CNTs can potentially be superior to corresponding binary CF- or CB-composites [80-88]. The

conductivity of binary CPCs can be modeled, as a function of filler fraction, with several

existing analytical models from the literature [57-64]. McLachlan's general effective medium

(GEM) equation [67, 68] (Equation 2.18) is probably the most reliable equation that can model

the composite conductivity both below and above the electrical percolation threshold.

The conductivities of ternary composites can also be modeled with analytical expressions,

but the number of existing models is far more limited [81]. Sun et al [81] suggested that the

percolation threshold of a ternary composite can be approximated by a straight line between the

percolation thresholds of the two different fillers. If φCNT and φCB are the (volume or weight)

fractions of CNTs and CB while φc,CNT and φc,CB are the percolation thresholds of CNT and CB

in binary composites (Equation 2.21), then Suns expression becomes:

1,,

CNTc

CNT

CNTc

CNT

(6.1)

The experimentally observed percolation threshold for ternary carbon fiber (CF) / CB

composites was however rather a convex curve than a straight line between φc,CNT and φc,CB.

Therefore an improved equation was suggested in Chapter 5.4.

Equation 5.4, which describes the curvature of the percolation threshold for ternary

composites, was used to derive a corresponding conductivity equation. The conductivities for

the binary composites (denoted σCB for the CB and σCNT for the CNT) can be calculated with

6 PMMA/CB/CNTs ternary system 83

Equation 2.18 and the expression for the ternary composites, which was successfully applied

on CF/CB composites (Equation 5.5).

A “synergistic effect” is often reported for ternary composites with CB and CNTs [72, 74-

88]. One possible explanation for an eventual “synergistic effect” is that electrical pathways

could potentially form more easily in ternary CNT/CB composites than in binary CNT- or CB

composites since the small, spherical CB fillers bridges the long CNT-fillers and thus improves

the contacts in the resulting electrical network [79, 83], [82, 85]. However, several definitions

of “synergy” exist, and the choice of definition will strongly influence the conclusion on

whether a ternary composite is subject to “synergy” or not. The current classifications of

“synergy” are:

1st synergy: When a “small amount” of a second filler “dramatically” increases the

conductivity of the CPC.

2nd synergy: When the conductivity of the ternary CPC is higher than the conductivity

of either of the two binary CPCs, while keeping the total filler fraction constant.

3rd synergy: When the experimental percolation threshold of the ternary CPC is lower

than the predicted straight line between the percolation thresholds of the two fillers

(Eq.2).

4th synergy: When the curvature parameter N becomes smaller than 1 when the

experimental data is fitted to Equation 5.5.

The 1st synergy criterion has been fulfilled for a few ternary CNT/CB composites [74, 75].

The 2nd synergy criteria and a suggested optimal ratio between CNTs and CB has been

discussed in several studies [77, 80, 82-86, 88] and also the 3rd synergic effect between CFs and

CB have often been mentioned [77, 78, 82-88]. The definition of the 4th synergy, which is

closely related to the definition of the 3rd synergy, was presented in Chapter 5. It should be


noted that the previously published conductivity data for ternary CNT/CB composites is overall

rather limited.

The true “synergistic effect” between the two fillers of a ternary composite can only be

revealed with a reliable conductivity contour plot covering a broad volume fraction range of

each filler. Such a contour plot has previously been presented for ternary carbon fiber (CF)/CB

composites, but never for CNT/CB mixtures. The aim of this study is to make and analyze such

a conductivity contour plot for ternary CNT/CB composites.

In order to minimize the interaction between the CNTs themselves and thus highlight the

interaction between the CNTs and the CB, the ternary CPCs specimen of this study were

anisotropic from a capillary rheometer. The aim was to induce an orientation of the CNTs and

reduce the contacts between the CNTs. The resulting composites become anisotropic, while the

existing literature mainly discusses CPCs with isotropic orientation of CNTs [75-80, 82-84, 86-

88].

6.2 Morphology of the ternary composite filament

Figure 6.1 shows (a) the cross section (b) the middle section of a ternary composite filament

with 5.44 vol. % CNTs and 6 vol. % CB. In Figure 6.1(a), the CNTs (brighter dots) were found

to be perpendicular to the flat page (or the plane of the paper). Because the diameter of the

CNTs were similar to the diameter of the primary CB particle, it is difficult distinguish the

CNTs and CB. In Figure 6.1(b), the CNTs were oriented along the extrusion direction, and the

CB aggregates were found to be located between them. This “grape-like” structure is the

theoretical basis of the reported “synergy” between CB and CNTs as reported before [82, 86].


Figure 6.1 Morphology of the anisotropic ternary CPCs filament with 5.44 vol. % CNTs and 6

vol. % CB: (a) the cross section of the specimen; (b) the middle section of the specimen.

6.3 Conductivity of the binary PMMA/CNTs and PMMA/CB composite

Figure 6.2 shows the logarithm of the conductivity plotted versus the CB (black squares) and

CNT (red circles) volume fraction, respectively. Each data point represents the average of 10

measurements, with standard deviations shown with vertical error bars. Horizontal error bars

indicate the uncertainty in filler fraction. The data was fitted with the McLachlan equation,

(Equation 6.1) and the percolation thresholds were determined to φc,CB=3.9 vol.%, φc,CNTs=5.0

vol.%. The reliability of the McLachlan percolation thresholds was confirmed by comparing

the results with another classical percolation theory equation (Equation 2.1). For composites

with filler concentrations φ>φc, the experimental results were fitted by plotting log σ versus φ,

as presented in the insets of Figure 6.2, also resulting in φc,CB=3.9 vol.%, φc,CNTs=5.0 vol.%.


Figure 6.2 The logarithm of the conductivity vs. filler volume fraction of CNTs and CB,

respectively. The solid lines show the best fit with the GEM equation (Equation 2.21). The two

illustrate the linear fit using (Equation 2.1).

The conductivity curves for CB and CNT were surprisingly similar, considering that the

electrical percolation threshold typically decreases when the aspect ratio (i.e. length/width) of

the particles increases [23]. A lower φc would thus have been expected for the strongly

elongated CNTs than for the almost spherical CB. The explanations for the non-intuitive

experimental results are that (1) CB aggregates in a structured way that increases the effective

aspect ratio and conductivity [90] and (2) the shear-induced fiber orientations increases the

percolation threshold for the oriented CNT composites as compared to more isotropic CNT-

composites. Kim has reported a similar electrical behavior for binary CPCs containing CB and

oriented CNTs [73]. Therefore, the “synergy” between CB and oriented CNTs could be


different in this study than that between CB and isotropic CNTs as reported before [75-80, 82-

84, 86-88]. Schematic electrical networks in anisotropic composites, comprising either CNT or

CB, are presented in Figure 6.3.

Figure 6.3 Schematic structure of electrical pathway formed by (a) CNTs and (b) CB particles in

a binary CPC system.

6.4 Experimental contour plot of conductivity on ternary composite

A logarithmic contour plot based on the experimental conductivity data is presented in Figure

6.4, where the X- and Y-axis correspond to CB- and CNT-volume fraction, respectively. The

filler fraction range was maximized with respect to the capability of the capillary rheometer. At

higher filler fractions the viscosity grew so high that the composites could not be extruded out

of the die. The black squares on the contour plot are the actual tested samples while the black

iso-contour lines correspond to 1 magnitude conductivity differences. The conductivity

increases steeply with growing filler concentration, especially near the percolation threshold

curve (red solid line). The results of Figure 6.4 will be systematically analyzed using the four

definitions of synergy.


Figure 6.4 Logarithm value of conductivity of composite filament vs. filler volume fraction of

CNTs and CB in a contour plot diagram, with definition on traditional 1st-3rd synergy.

The 1st synergy, which is fulfilled when a small addition of a second filler do not radically

increase the conductivity, was not clearly observed for any of the experimental data points.

Anyway, according to the shape of the contour-lines, two regions with potential 1st synergy

could eventually be present at high filler contents along the X- and Y-axis, as marked with

green solid lines. This possibility is however not confirmed experimentally and the eventual

synergetic conductivity increase must anyhow be weak since the material is already percolated.

It could also be argued that some ternary composites just above the percolation could fulfill the

1st synergy criterium, but since a similar conductivity increase would occur also for

corresponding binary composites, this argument is rather weak. Overall, the 1st synergy

criterion is undesirably inexact and should generally be avoided when evaluating synergy of

ternary composites.


The 2nd synergy definition is fulfilled when the conductivity of the ternary CPCs is higher

than the conductivity of either binary CPC, while keeping the total filler fraction m=(φCNT + φCB)

constant. This criterion is satisfied for any point C (along the magenta-colored line y=m-x) with

higher conductivity that the conductivities of points A=(0,m) and B=(m,0). From the contour

lines of Figure 6.4, it can be observed that the second synergy criterion is probably fulfilled at

high filler fractions, but at low fillers contents the composites are rather anti synergetic.

The 3rd synergy definition is fulfilled if the percolation threshold contour curve (red solid

curve in the inset of Figure 6.4) is primarily situated below the line of Equation 6.1, which is

the dashed, red, diagonal, straight line (Y=-1.28X+5) going between the CNT percolation

threshold (0, 5) and the CB percolation threshold (3.9, 0). This synergy criterion is obviously

not fulfilled for our ternary CNT/CB composites, which is in agreement with several previous

studies [77, 83, 84, 88].

The 4th synergy, which is a more precise extension of the third definition, confirms that the

composite is dominantly anti-synergetic, because fitting with Equation 5.4 gives a curvature N

=1.3, which is clearly higher than unity. Figure 6.5 visualizes the conductivities calculated with

Equation 5.5 and the inset figure shows Equation 5.5 (solid lines) together with experimental

data (symbols) along four diagonal lines parallel to Equation 6.1 (dashed lines in Figure 6.5).

The modeling results are in reasonable agreement with the experimental measurements. It can

be noted that the curvature of the CNT/CB composites was qualitatively similar to the curvature

of ternary CF/CB composites.


Figure 6.5. Logarithm of the composite conductivity versus volume filler fraction of CNTs and CB

in a contour plot, as generated with Equation 5.5. The inset show experimental- and modeled data

along the four dotted lines.

6.5 Reanalysis of the literature

As a complement to our measurements and modeling results, the literature on ternary

composites with CNTs and CB in a polymer matrix (HDPE [91], TPU [92], epoxy resin[93],

UHMWPE [94], PVDF [75], PA6/ABS blend [78], PLA [79], PBT [80], Silicone Rubber [82],

TPE [83], PA12 [84], PP [85], PP-g-MA [86], PC [88]) was reanalyzed. Among these studies,

Chen et al. [78], Wu et al. [79], Dorigato et al. [80], Wen et al. [85] and Zhang et al. [86]

reported synergetic effects according to the 2nd or 3rd definition. The results are summarized in

Figure 6.6, showing the filler fractions where synergy has been reported for ternary CB/CNT

composites.


Figure 6.6 The composition between CNTs and CB reported in the literature, with different results

on ‘synergy’.

In contrast to the results of this paper, several studies from the literature have reported synergy

effects for ternary CNTs/CB composites. One possible reason for this discrepancy could be that

this study concerns oriented CNTs while most other studies have isotropic orientation of the

fillers. The choice of matrix material could also influence, due to eventual segregation effects.

Another explanation could simply be the difficulty of measuring and interpreting conductivity

results. For example, since the standard deviation is typically very high when measuring the

conductivity of composites close to the percolation threshold, it is easy to draw erroneous

conclusions based on statistical variations and random errors. In several of the mentioned

studies, the number of replicates is low or unknown, which accentuates the problem, because

the impact of single outliers increases. Several of the reported synergies are also based on

extrapolated data rather than comparisons between actual samples. Overall, even if synergies

between CNTs and CB could potentially exist, the hitherto presented pieces of evidence of


synergetic effects are surprisingly limited and could in worst case simply be the effect of

statistical variations.

As for the 1st synergy, can be described as the increase of the conductivity from the green

dashed line along either the X-axis (PMMA/CB binary system) or the Y-axis (PMMA/CNTs

binary system) to the green solid line. Figure 6.7 presents a schematic definition on the 1st

synergy, where some CNTs were added into the PMMA/CB binary system. Although this

synergy is found in our study accordingly, 1st synergy is believed to be an improper definition

due to the unfixed variable filler volume fraction.

Figure 6.7 Schematic definition on the 1st synergy, corresponds to Equation 2.23.

Additionally, the limit of 2nd synergy is only a comparison between the conductivity of

ternary CPCs with the binary CPCs, always expanded proportionally (Figure 6.8). Dorigato et

al. [80] presented the 2nd synergy with different m values as described in Figure 6.8. However,

the total filler fraction m were kept always below the percolation threshold of binary CPCs with

CB, which leads to a higher conductivity of ternary CPCs than the binary CPCs with CB below

percolation threshold. Wu et al. [79] has reported the conductivity of pre-foamed and post-

foamed CPCs (without any deviation on the conductivity), the 2nd synergy was found to exist

only after foaming (Anti synergy was reported before foaming). Nevertheless, Wen et al. [85]

reported a 2nd synergy between CNTs and CB after CPCs prepared by stretching extrusion only,

and an anti-synergy was presented without stretching.


Figure 6.8 Schematic definition on the 2nd synergy, corresponds to Eq. 2.24

Moreover, the 3rd synergy focus on the percolation region, where a large standard deviation

for the experimental measurements exist. Chen et al. [78] has produced binary CPCs with max.

4 wt. % CB, and chose the point 4 wt. % as the percolation threshold. This value is used in

Equation 6.5 to calculate the ternary percolation threshold as 0.97 wt. %, slightly higher than

0.8wt. %, as a prove of 3rd synergy. Besides, the reproducibility of the specimen could be also

questioned, due to no error bars on the presented conductivities. Zhang et al. [86] reported 10

volume fractions of the total filler content above the percolation threshold with CNT wt. %/CB

wt. %=1, and only 9 volume fractions were used to obtain the extremely low experimental

ternary percolation threshold as 0.21wt. % using Equation 6.5. Comparing with the calculated

ternary percolation threshold 3.0 wt. % using Equation 2.21, the most obvious 3rd synergy is

presented.

Therefore, to our best of knowledge, no convincing results were found according to the

common defined three “synergy”. Since the ternary CPCs has one more dimension (σ, φ1, φ2)

than the traditional binary CPCs (σ, φ), a new definition on the electrical interaction between

CNTs and CB is thus required.


6.6 Synergasm – a novel synergy definition for ternary composites

Since none of the existing synergy definitions can capture all relevant aspects of synergy,

thus a new definition “synergasm” was proposed to describe the electrical interactions between

CNTs and CB, as described in Fig. 6.7. A contour plot with a high number of isolines was

initially constructed from the experimental conductivity measurements. For pedagogical

reasons, one of these isolines is highlighted as a bold, black curve.

Fig. 6.7. Logarithm of the composite conductivity vs. volume filler fraction of CNTs and CB in a

contour plot diagram, defining a quantifiable “synergasm” between CNTs and CB. The inset

synergism map illustrates the synergasm effect of the different compositions of CNTs and CB from

this study.

The subsequent steps to define the “synergasm” are the following [123]:

1. Check the volume fractions of the fillers needed to achieve the conductivity of the

selected iso-line (the line of constant conductivity, denoted as “CC-lines”) if only one

type of filler would be used, here CB (Point D) and CNT (Point A) volume fraction,

respectively.


2. The filler with the lower required volume fraction to achieve the selected conductivity

is the benchmark. Example in Fig. 6.7 yields that point D with a CB volume fraction of

11% is the benchmark and not point A with a CNT volume fraction of 18%.

3. Consider lines of constant volume fraction (denoted as “CV-lines”), like the white lines

in Fig. 6.7, e.g. line𝐴𝐵, 𝐶𝐷 or 𝐸𝐹 easily constructed by

𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 = 𝑥 + 𝑦 (6.2)

with x = φ𝐶𝐵 and y = φ𝐶𝑁𝑇 as indicated by Fig. 6.7.

4. Moving these CV-lines so far towards the lower left (origin) parallel as long as it

becomes a tangent (line 𝐸𝐹) to the selected CC-line, having also no other intersection

point with the selected CC-line. Thus the contact point reveals a ternary recipe with the

most possible reduction of total volume fraction as compared to the previous determined

benchmark in step 2, therefore this is the point of strongest synergy of the two fillers

which is called point of synergasm, point G. All other cases, where such a construction

is not possible, do not have a synergy, only anti-synergy.

The strength of synergism can be evaluated by comparing the benchmark volume fraction (point

D) to the total volume fraction and the point of synergism point G, nevertheless the total volume

fraction in point G can be easily read at point E or F because point E, G and F laying on the

same line of constant volume fraction due to the construction scheme as described in step 4.

Therefore, for a selected constant conductivity line (the bold line here as an example), the

synergasm point can be revealed using following equation [123]:

S = 𝜑𝑅− 𝜑𝑇

𝜑𝑅× 100% (6.3)

where S denotes the strength of synergasm effect at a selected conductivity. 𝜑𝑅 and 𝜑𝑇 are

for the volume fraction of reference point (benchmark, in this case point D), and the volume

fraction of tangent point (the minimal CV-line, in this case point F)


It is remarkable that the points of synergasm on each CC-line are always on a line following

the relation:

φ𝐶𝑁𝑇 = 𝐴 × (φ𝐶𝐵 − φ𝐶𝐵,0) (6.4)

where φ𝐶𝐵,0 is a threshold value indicating that a certain amount of CB is needed to generate

a synergasm, and A is a coefficient between volume fraction of CNTs and CB in the CPC

system. From a microscopic point of view this is logical. Because the CNT are almost infinite

thin fibers thus the probability for a fiber-fiber contact becomes negligible small. Therefore

only the addition of the more spherical CB particles enables a significant bridging of the CNT

fibers and thus form conductive pathways. In this work focusing on the conductivity of the

anisotropic CPCs, the φ𝐶𝐵,0 and A are found to be 2 and 1.3, respectively, indicating the

minimal amount 2 vol. % of CB, as well as a volume proportion between CNTs and CB as 1.3

for further addition are required to achieve the synergasm.

Consider a reference volume VR, containing one single CNT (perfect cylinder with radius RCNT

and length LCNT) and corresponding CB particles (perfect sphere with radius RCB, with the

number n) to bridge the single CNT with the others: the volume fraction of CNT and CB can

be thus written as Eq.6.5 and Eq.6.6, respectively [123]:

𝜑𝐶𝑁𝑇𝑠 =𝜋∙𝑅𝐶𝑁𝑇

2∙𝐿𝐶𝑁𝑇

𝑉𝑅 (6.5)

𝜑𝐶𝐵 = 𝑛 ∙4

3∙𝜋∙𝑅𝐶𝐵

3

𝑉𝑅 (6.6)

which yields Eq. 6.7:

𝜑𝐶𝐵

𝜑𝐶𝑁𝑇𝑠=

4

3𝑛∙𝜋∙𝑅𝐶𝐵

3

𝜋∙𝑅𝐶𝑁𝑇2∙𝐿𝐶𝑁𝑇

= 1.3 (6.7)

Since the geometry of the CB and CNTs particle are known according to data sheet (RCB=17.5

nm, RCNT=6.5 nm, LCNT=1000 nm), the numbers of CB particles to bridge two CNTs involved


in the synergasm can be thus roughly estimated: n=7.7, indicating essential 7.7 CB particles in

average correspond to each single CNT in the CPC system. More geometric analysis on the

ratio between CB and CNTs is left for future study.

Fig. 6.8. Schematic description between the CNTs and CB to achieve the synergasm in the CPC

system. (a) a minimal amount of 2 vol. % CB; (b) besides the 2 vol. % CB, 7.7 CB particles in

average are required for each single CNT particle in the system [123].

6.7 Conclusion

In this study, the electrical conductivities of anisotropic ternary CPCs comprising PMMA,

CB and CNTs were measured in order to reveal an eventual “synergy” between oriented CNTs

and CB. The conductivities and the morphologies of the binary composites (PMMA/CNT and

PMMA/CB) were, as function of volume filler fraction, very similar to each other. The

experimental conductivities of the ternary CPCs were presented in a contour plot with isolines

and the results were evaluated using the four traditional “synergy” definitions. No strong

synergy effects were observed between the CB and the oriented CNT in this study and a careful

reexamination of CNT/CB conductivity data from the literature indicate that some of the

previously reported synergies could potentially be the result of random variations. Due to the

limitations of the traditional “synergy” definitions, a novel definition “synergasm” was

proposed, which is able to quantify the synergy more systematically. Accordingly, a


“synergasm” map is presented, showing the most obvious synergasm effects between CNTs

and CB.

7 Summary

In this thesis, binary composites PMMA/CF, PMMA/CB, PMMA/CNTs as well as ternary

composites PMMA/CB/CF and PMMA/CB/CNTs were produced. The morphological and

electrical behavior of the anisotropic filament CPCs were investigated. Based on this study,

theory from Balberg was further developed, and novel equation as well as theory was proposed.

The main results are summarized as below:

(a) The binary PMMA/CF CPCs were produced under high concentration (up to 60 vol. %

CF). Different lengths of CFs were obtained with a two-step melt mixing process. Due to the

extrusion process, highly anisotropic CFs were obtained in the composites, presenting a “fibers

in fiber” structure. Moreover, due to the shear gradient across the channel section, the CFs

orientation were decreased from the outer part of the composite to the center. The “peeling-off”

and “polishing” methods were proposed, presenting a rough estimation on the different

conductivities of composites filament. The results show that the measuring direction on the

CPCs containing anisotropic CFs could be the missing link between McLachlan equation and

the Blaberg theory. It has also been found that an increase the filler aspect ratio led to a decrease

in the percolation threshold. Taking into consideration the filler orientation, all the reanalyzed

data from literature were found resulting into a master line with a slope K=3.4, which was

100 7 Summary

evaluated for the first time. Moreover, comparing samples with 1mm and 3mm diameter shows

that a higher orientation of CFs along the measuring direction leads to a higher percolation

threshold, as well as a higher conductivity of CPCs beyond the percolation threshold. A novel

approach based on the morphological study was proposed to explain this phenomenon.

(b) The anisotropic PMMA/CB/CF ternary composites filament, with a broad range of

composites compositions (up to 50 vol. % CF and 20 vol. % CB) were studied. SEM

micrographs exposed a “fence-like” morphological structure between the CB and the CF.

Conductivity contour plots, showing conductivity as function of filler fraction, were presented

for PMMA/CF/CB composites for the first time; one extensive plot based on data from this

study and one based on already existing literature data. Distinctly curved iso-lines were

observed. Synergetic effects between CFs and CB were found in the regions below 10 vol. %

CF, observed as left-pointed protrusions of the contour-lines. Moreover, based on a model for

predicting the percolation threshold of ternary composites, a novel equation for predicting the

electrical volume conductivity of ternary composites was proposed, showing results close to

corresponding experimental data.

(c) The conductivities and the morphologies of the binary composites (PMMA/CNT and

PMMA/CB) filament were, as function of volume filler fraction, very similar to each other. The

experimental conductivities of the ternary CPCs filament PMMA/CB/CNTs (up to 30 vol. %

CNTs and 20 vol. % CB) were presented in a contour plot with isolines and the results were

evaluated using the four traditional “synergy” definitions. Due to the limitations of the

traditional “synergy” definitions, a novel definition “synergasm” was proposed, which is able

to quantify the synergy more systematically. Accordingly, a “synergasm” map is presented,

showing the most obvious synergasm effects between CNTs and CB.

8 Summary (in German)

In dieser Arbeit wurden binäre Komposite PMMA/CF, PMMA/CB, PMMA/CNTs sowie ternäre

Komposite PMMA/CB/CF und PMMA/CB/CNTs hergestellt. Das morphologische und elektrische

Verhalten der extrudierten Zylinder-Filament-Kompositen wurde untersucht. Die Theorie von Balberg

wurde weiterentwickelt und sowohl eine neue Gleichung als auch eine neue Theorie wurde

vorgeschlagen. Die Hauptergebnisse sind wie folgt zusammengefasst:

(a) Die binären PMMA/CF CPCs wurden in hoher Konzentration (bis zu 60 Vol. % CF) hergestellt.

Verschiedene Längen von CFs wurden mit einem zweistufigen Schmelzmischverfahren erhalten.

Aufgrund des Extrusionsverfahrens wurden hochorientierte CF in den Verbundwerkstoffen erhalten, die

eine "Fasern in Faser" Struktur aufwiesen. Aufgrund des Schergradienten über den Kanalabschnitt war

die Orientierung der CFs vom äußeren Teil des Kompostes zum Zentrum hin verringert. Es wurden die

"Abzieht" und "Polier" Methoden vorgeschlagen, die eine unterschiedliche Leitfähigkeit in der radialen

Richtung des Filament-Filament-Verbunds aufweisen. Eine Erhöhung des Aspekts Verhältnissen führte

zu einer Abnahme der Perkolationsschwelle. Unter Berücksichtigung der Füllstofforientierung wurde

die Steigung K = 3,4 in der Theorie des ausgeschlossenen Volumens zum ersten Mal geschätzt. Die

orientierten CFs im Komposite verschieben die Perkolationsschwelle auf einen höheren Wert, jedoch

ist die Leitfähigkeit oberhalb der Perkolationsschwelle für Komposite mit orientierten CFs höher. Dieses

Phänomen wurde durch den Unterschied in der durchschnittlichen Faserorientierung erklärt, der aus

102 8 Summary in German

Mikroaufnahmen beobachtet wurde. Es wurde eine neuartige Erklärung entwickelt, die befriedigend

zusammenfassen konnte, wie die Leitfähigkeit von Faserverbundwerkstoffen durch Faserorientierung

und Füllstoffanteil beeinflusst wird, und somit die experimentellen Leitfähigkeitsergebnisse erklären.

(b) Die extrudierten ternären PMMA/CB/CF Verbundwerkstoffe mit einer breiten Palette von

Verbundstoffzusammensetzungen (bis zu 50 Vol. % CF und 20 Vol. % CB) wurden untersucht. REM-

Aufnahmen zeigten eine "zaunähnliche" morphologische Struktur zwischen CB und CF.

Leitfähigkeitskonturen, die Leitfähigkeit als Funktion der Füllstofffraktion zeigen, wurden erstmals für

PMMA/CF/CB Komposite vorgestellt; ein umfangreicher Plot basiert auf Daten aus dieser Studie und

einer basierend auf bereits vorhandenen Literaturdaten. Deutlich gekrümmte Iso-Linien wurden

beobachtet. Synergieeffekte zwischen CF und CB wurden in den Regionen unter 10 Vol. % CF,

beobachtet als linksseitige Vorsprünge der Konturlinien. Darüber hinaus wurde basierend auf einem

Modell zur Vorhersage der Perkolationsschwelle ternärer Komposite eine neue Gleichung zur

Vorhersage der elektrischen Volumenleitfähigkeit von ternären Kompositen vorgeschlagen, die

Ergebnisse nahe an entsprechenden experimentellen Daten zeigt.

(c) Die Leitfähigkeiten und die Morphologien der binären Komposite (PMMA/CNT und PMMA/CB)

waren als Funktion der Volumen-Füllstoff-Fraktion sehr ähnlich. Die experimentellen Leitfähigkeiten

der ternären Komposite PMMA/CB/CNTs (bis zu 30 Vol. % CNTs und 20 Vol. % CB) wurden in einem

Konturdiagramm mit Isolinien dargestellt und die Ergebnisse wurden mit den vier traditionellen

"Synergie" -Definitionen ausgewertet. Aufgrund der Beschränkungen der traditionellen "Synergie"

Definitionen wurde eine neue Definition "Synergasm" vorgeschlagen, die in der Lage ist, die Synergie

systematisch zu quantifizieren. Dementsprechend wird eine "Synergie" Karte dargestellt, die die

offensichtlichsten Synergieeffekte zwischen CNTs und CB zeigt.

9 Appendix

9.1 Rough estimation on the horizontal conductivity of CPCs filament

The proposed calculation is presented in Figure 9.1, where four concentric circles are assumed,

with same horizontal resistivity ρ1, ρ2, ρ3 and ρ4 in each region (red, green, blue and purple,

respectively). The center of the circles is set as the coordinate origin, with the radius of four

circles are r1, r2, r3 and r4, respectively.

Figure 9.1 Calculation horizontal resistance of the PMMA/CF filaments with 3 mm diameter using

the “Polishing” procedure.

Therefore, following equaions were obtained accodingly:

104 9 Appendix

𝑦12 + 𝑥2 = 𝑟1

2 = 𝑂𝑂02 = (

𝐷

2)2

(9.1)

𝑦22 + 𝑥2 = 𝑟2

2 = 𝑂𝐵02 = 𝐷2 − (𝐵1𝐵2/2)

2 (9.2)

𝑦32 + 𝑥2 = 𝑟3

2 = 𝑂𝐶02 = 𝐷2 − (𝐶1𝐶4/2)

2 (9.3)

𝑦42 + 𝑥2 = 𝑟4

2 = 𝑂𝐷02 = 𝐷2 − (𝐷1𝐷6/2)

2 (9.4)

𝑂𝐴0 = √𝑂𝐴22 − 𝐴0𝐴2

2 = √𝑟12 − (𝐴1𝐴2/2)2 (9.5)

where D=3 mm. Four resistances Ra, Rb, Rc and Rd, were measured, corresponding with

polished width WA (=1 mm), WB (=1.8 mm), WC (=2.2 mm) and WD (=2.4 mm), respectively.

The applied voltage direction is given along the Y-axis, thus the measured resistance Ra is

considered as series connection between R(α1), Rb and R(α1):

𝑅(𝛼1) =𝑅𝑎−𝑅𝑏

2 (9.6)

For the region α1, the resistance is considered as infinite sheet resistance connected in series.

The resistivity ρ1 for the red region can be then obtained from R(α1) as follows:

𝑅(𝛼1) = ∫ 𝜌1𝑑𝑦

𝐿∗2∗√𝑟12−𝑦2

𝑦=𝑂𝐴𝑜

𝑦=𝑂𝐵𝑜=

𝜌1

2∗𝐿(𝑎𝑟𝑐 𝑠𝑖𝑛

𝑂𝐴𝑜

𝑟1− 𝑎𝑟𝑐 𝑠𝑖𝑛

𝑟2

𝑟1) (9.7)

where L is the length of the filament along the extrusion direction. Similarly, The measured

resistance Rb can be considered as double resistances of parallel connection between R(β1),

R(β2), R(β1), then series connect with measured resistance Rc. The R(β1) (B0B2C4C3) can be

thus presented as following:

9 Appendix 105

𝑅(𝛽1, 𝐵0𝐵2𝐶4𝐶3) = 𝜌1 ∗𝑑𝑦

𝐿 ∗ (𝑋𝑚𝑎𝑥 − 𝑋𝑚𝑖𝑛)=𝜌1𝐿∗ ∫

𝑑𝑦

√𝑟12 − 𝑦2 −√𝑟22 − 𝑦2

𝑦=𝑂𝐵0

𝑦=𝑂𝐶0

=𝜌1𝐿∗ ∫

√𝑟12 − 𝑦2 +√𝑟22 − 𝑦2

𝑟12 − 𝑟22∗ 𝑑𝑦

𝑦=𝑂𝐵0

𝑦=𝑂𝐶0

=𝜌1

𝐿∗

1

𝑟12−𝑟22∗ (∫ √𝑟12 − 𝑦12 ∗ 𝑑𝑦

𝑦=𝑂𝐵0

𝑦=𝑂𝐶0+ ∫ √𝑟22 − 𝑦12 ∗ 𝑑𝑦

𝑦=𝑂𝐵0

𝑦=𝑂𝐶0) (9.8)

which can be further simplified as equation 9.8:

𝑅(𝛽1,𝐵0𝐵2𝐶4𝐶3) =𝜌1

𝐿∗

1

𝑟12−𝑟22∗1

2∗ (𝑟2 ∗ √𝑟12 − 𝑟22 + 𝑟1

2 ∗ 𝑎𝑟𝑐𝑠𝑖𝑛𝑟2

𝑟1− 𝑟3 ∗ √𝑟12 − 𝑟32 −

𝑟12 ∗ 𝑎𝑟𝑐𝑠𝑖𝑛

𝑟3

𝑟1+ 𝑟2 ∗ √𝑟22 − 𝑟22 + 𝑟2


𝑟2− 𝑟3 ∗ √𝑟22 − 𝑟32 − 𝑟2


𝑟2)

(9.9)

Therefore, the resistance R(β2) can be obtained as follows:

𝑅(𝛽2) = ∫ 𝜌2𝑑𝑦

𝐿∗2∗√𝑟22−𝑦2

𝑦=𝑂𝐵𝑜=𝑟2

𝑦=𝑂𝐶𝑜=𝑟3=

𝜌2

2∗𝐿(𝑎𝑟𝑐 𝑠𝑖𝑛

𝑟2

𝑟2− 𝑎𝑟𝑐 𝑠𝑖𝑛

𝑟3

𝑟2) (9.10)

𝑅(𝛽2) = 1/(2

𝑅𝑏−𝑅𝑐− 2/𝑅(𝛽1)) (9.11)

9.2 MATLAB code for conductivity of ternary composites (Equation 5.5)

clear,clc,clf,hold on, format compact

sigma_m=3.64E-10; %[S/cm] electrical conductivity of matrix sigma_fA=132.5; %[S/cm] electrical conductivity of filler A sigma_fB=63.05; %[S/cm] electrical conductivity of filler B t=2; %[] Constant t=2 in McLachan formula s=0.87; %[] Constant s=0.87 in McLachan formula k0=0; %[] Efficiency coefficient

kk=1.47;

phiA_C=0.039; %[] Percolation threshold with pure filler B (CB) phiB_C=0.02997; %[] Percolation threshold with pure filler A (CF) phiVA=[0:0.0025:0.2]'; %Examined filler fractions, filler A phiVB=[0:0.01:0.5]'; %Examined filler fractions, filler B

%phiVA=[0:0.005:1]'; %Examined filler fractions, filler A %phiVB=[0:0.01:1]'; %Examined filler fractions, filler B

106 9 Appendix

sInv=1/s; tInv=1/t; sigmaIntervalA=[sigma_m, sigma_fA]; sigmaIntervalB=[sigma_m, sigma_fB]; [XX,YY]=meshgrid(phiVA,phiVB); XX=XX'; YY=YY';

funk1=@(sigma,phiV,sigma_m,sigma_f,tInv,sInv,phi_c) (1-

phiV).*(sigma_m^sInv-sigma.^sInv)./(sigma_m^sInv+sigma.^sInv*(1-

phi_c)/phi_c)+... phiV.*(sigma_f^tInv-sigma.^tInv)./(sigma_f^tInv+sigma.^tInv*(1-

phi_c)/phi_c);

sigmaTotM=zeros(length(phiVA),length(phiVB));

for i=1:length(phiVA)

phiA_now=phiVA(i); alpha=phiA_now/phiA_C;

for j=1:length(phiVB)

phiB_now=phiVB(j); beta=phiB_now/phiB_C;

gamma=(alpha^kk+beta^kk)^(1/kk); phiA_Effective=min(1,gamma*phiA_C); phiB_Effective=min(1,gamma*phiB_C);

sigmaANow=fzero(@(sigma)

funk1(sigma,phiA_Effective,sigma_m,sigma_fA,tInv,sInv,phiA_C),sigmaInterval

A); sigmaBNow=fzero(@(sigma)

funk1(sigma,phiB_Effective,sigma_m,sigma_fB,tInv,sInv,phiB_C),sigmaInterval

B);

sigmaTotNow=( (alpha/(alpha+beta))*sigmaANow +

(beta/(alpha+beta))*sigmaBNow );

sigmaTotM(i,j)=sigmaTotNow;

end end

sigmaTotM(1,1)=sigma_m; Log10_sigmaTotM=log10(sigmaTotM);

figure(1) %surf(XX,YY,Log10_sigmaTotM) %alternative 1 surf(XX*100,YY*100,Log10_sigmaTotM),hold on %alternative 2 contour(XX*100,YY*100,Log10_sigmaTotM),hold on %alternative 2 XL=xlabel('\phi_ CB') YL=ylabel('\phi_ CF') ZL=zlabel('log10(\sigma)') TL=title('log10(\sigma) vs \phi')

9 Appendix 107

set(gcf,'Color',[1.0 1.0 1.0]) set([XL,YL,ZL,TL],'FontSize',16) set(gca,'LineWidth',2) grid on axis([0 20 0 20 -10 1])

lighting gouraud

camlight LEFT colormap jet set(light) view(3)

colorbar

get(gca) get(gcf)

9.3 MATLAB code for certain composition of CPCs (Figure 5.10 and 6.5)

clear,clc,clf,hold on, format compact

sigma_m=3.64E-9; %[S/cm] electrical conductivity of matrix sigma_fA=134; %[S/cm] electrical conductivity of of filler A sigma_fB=4.06; %[S/cm] electrical conductivity of of filler B

t=2; %[] Constant t=2 in McLachan formula s=0.87; %[] Constant s=0.87 in McLachan formula kk=1.40; %new line

phiA_C=0.039; %[] Percolation threshold with pure filler B (CB) %phiB_C=0.3185; %[] Percolation threshold with pure filler A (CF) phiB_C=0.335; %[] Percolation threshold with pure filler A (CF)

phiVA=[0:0.0001:0.061]; %Examined filler fractions, filler A %phiVB=[0:0.01:0.5]'; %Examined filler fractions, filler B phiVB=0; %Examined filler fractions, filler B startYV=[0.3185, 0.409, 0.500];

phiA_stop=startYV/8.17

A1=[0 -7.05773164 1 -6.96288 2 -7.10774 3 -7.65691 3.9 -6.485421808 ];

A2=[0 -1.18997 1 -2.39058 2 -2.97675 3 -2.77228 4 -2.65994 5 -2.32698 ];

A3=[0 -0.55

108 9 Appendix

1 -1.13798 2 -1.05686 3 -1.66065 4 -1.94081 5 -1.84822 6.12 -1.14 ];

expPHI1=A1(:,1); expPHI2=A2(:,1); expPHI3=A3(:,1);

expSIGMA1=A1(:,2); expSIGMA2=A2(:,2); expSIGMA3=A3(:,2);

sInv=1/s; tInv=1/t; sigmaIntervalA=[sigma_m, sigma_fA]; sigmaIntervalB=[sigma_m, sigma_fB]; [XX,YY]=meshgrid(phiVA,phiVB); XX=XX'; YY=YY';

funk1=@(sigma,phiV,sigma_m,sigma_f,tInv,sInv,phi_c) (1-

phiV).*(sigma_m^sInv-sigma.^sInv)./(sigma_m^sInv+sigma.^sInv*(1-

phi_c)/phi_c)+... phiV.*(sigma_f^tInv-sigma.^tInv)./(sigma_f^tInv+sigma.^tInv*(1-

phi_c)/phi_c);

for j=1:length(startYV)

sigmaTotM=zeros(length(phiVA),length(phiVB));

for i=1:length(phiVA)

phiA_now=phiVA(i); alpha=phiA_now/phiA_C;

phiB_now=0.409-8.17*phiA_now;

beta=phiB_now/phiB_C; if beta<0 break end

gamma=(alpha^kk+beta^kk)^(1/kk); phiA_Effective=min(1,gamma*phiA_C); phiB_Effective=min(1,gamma*phiB_C);

sigmaANow=fzero(@(sigma)

funk1(sigma,phiA_Effective,sigma_m,sigma_fA,tInv,sInv,phiA_C),sigmaInterval

A); sigmaBNow=fzero(@(sigma)

funk1(sigma,phiB_Effective,sigma_m,sigma_fB,tInv,sInv,phiB_C),sigmaInterval

B);

9 Appendix 109

sigmaTotNow=( (alpha/(alpha+beta))*sigmaANow +

(beta/(alpha+beta))*sigmaBNow );

sigmaTotM(i)=sigmaTotNow;

end

Log10_sigmaTotM=log10(sigmaTotM);

figure(1) if length(phiVB)>1 surf(XX*100,YY*100,Log10_sigmaTotM),hold on %alternative 2 contour(XX*100,YY*100,Log10_sigmaTotM),hold on %alternative 2 ZL=zlabel('log10(\sigma)') else P1=plot(XX*100,Log10_sigmaTotM); hold on

P2=plot(expPHI1,expSIGMA1,'bd',expPHI2,expSIGMA2,'ro',expPHI3,expSIGMA3,'gs

'); set([P1;P2],'LineWidth',3); set([P1;P2],'MarkerSize',12, 'MarkerFaceColor','k') grid on end

9.4 MATLAB code for setting 20 intervals of inclination of CFs (Figure 4.8)

clear; format long; load abs.txt %: write the data like 0.15, not 0,15 otherwise cannot load

ndata=length(abs); shux=abs(:,1); shuy=abs(:,2); gap=0.1; % set the gap, like 0-0.1, 0.1-0.2................. ngroup=2/gap; for x=1:ngroup % x is used to set up the groups xiaxian=(x-1)*gap+(-1); shangxian=x*gap+(-1); jishuqi=1;

Xvalue=[]; Yvalue=[]; for y=1:ndata % y is used to scan the datas if shux(y)>=xiaxian&shux(y)<shangxian Xmatrix(x,jishuqi)=shux(y); Ymatrix(x,jishuqi)=shuy(y); Xvalue(jishuqi)=shux(y); Yvalue(jishuqi)=shuy(y); jishuqi=jishuqi+1;

end Xaverage(x)=mean(Xvalue); Yaverage(x)=mean(Yvalue); Xstdqop(x)=std(Xvalue); Ystdqop(x)=std(Yvalue); end

end X=rot90(Xaverage); Y=rot90(Yaverage);

110 9 Appendix

Xstd=rot90(Xstdqop); Ystd=rot90(Ystdqop); Result(:,1)=X; Result(:,2)=Xstd; Result(:,3)=Y; Result(:,4)=Ystd;

9.5 MATLAB code for average value of sin gama between CFs (Equation 4.3)

clear all; format long; load all.txt

ndata=length(all); shu=all(:,1); paixudeshu=sort(shu,'descend'); k0=0; j0=0; for x=1:ndata k0=0; j0=0;% set the original Wert of k0 and j0 for n=x:ndata j=paixudeshu(x)-paixudeshu(n); k=sin(j*pi/180); j0=j0+j; k0=k0+k; end a(:,x)=j0; b(:,x)=k0; end z0=0; for y=1:(ndata-1) z0=z0+y; end a0=sum(a)/z0; b0=sum(b)/z0;

Abbreviations and symbols

Abbreviations

ABS Acrylonitrile Butadiene Styrene

AC Alternating Current

BET Brunauer-Emmett-Teller

CB Carbon Black

CFs Carbon Fibers

CNTs Carbon Nanotubes

CPCs Conductive Polymer Composites

DC Direct Current

GE Graphene

HDPE High-Density PolyEthylene

MWCNTs Multi-Walled Carbon NanoTubes

PA6 PolyAmide 6

PBT PolyButylene Terephthalate

PC PolyCarbonate

PCL Poly-ε-CaproLactone

PE PolyEthylene

PLA Poly(Lactic Acid)

PMMA Poly(Methy-MethAcrylate)

PP PolyPropylene

PS PolyStyrene

112 Abbreviations and symbols

PVDF Poly(VinyliDene Fluoride)

SR Silicone Rubber

SEM Scanning Electron Microscope

SWNT Single-Wall NanoTube

TEM Transmission Electron Microscopy

TGA ThermoGravimetric Analysis

TPE ThermoPlastic Elastomer

TPU Thermoplastic PolyUrethanes

UHMWPE Ultra-High-Molecular-Weight PolyEthylene

Greek letters

γ the angle between two cylinders in Balberg theory

γp the surface tension of polymer

γ𝑓 the surface tension of fillers

γpf the interfacial tensions between polymer and fillers

γ shear rate

ρ electrical resistivity

σ electrical conductivity

σm the conductivity of polymer matrix

σc the conductivity of CPCs

𝜃 Average angle between two fibers in contact model

σf the intrinsic conductivity of the filler

φi the volume fraction of component i

φ the volume fraction of the filler

Abbreviations and symbols 113

φc the critical percolation threshold

Latin letters

𝐴 fitting parameter in equation 2.15

𝐴𝑅 Aspect Ratio of fillers

𝐵 fitting parameter in equation 2.15

C fitting parameter in equation 2.7 and 2.8

𝑑𝑐 the diameter of the circle contact in equation 2.13

𝐹 Maximum volume fraction in equation 2.17

I current

h the height of the sample in the 'peeling-off' procedure

𝐾 the coefficient in Balberg theory

𝐿 the length of cylinder in Balberg theory

𝑙 Average length of fibers in contact model

Mn number-average molar mass

Mw weight average molar mass

r the radius of the sample

R the resistance of the sample

𝑠 fit parameter in equation 2.18

S fit parameter in equation 2.8

t fit parameter in equation 2.18

T temperature

Tg glass-transition temperature

μ fit parameter in equation 2.6

U constant voltage

114 Abbreviations and symbols

Vf the volume of filler in Balberg theory

Vex the excluded volume of filler in Balberg theory

𝑊 the width of cylinder in Balberg theory

𝑋 volume factor in equation 2.13

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[123] A model explanation from Prof. Dirk Schubert.

Acknowledgement

• To Prof. Dr. rer. nat. habil. Dirk W. Schubert:

I have to admit that this is the most time-consuming part of my Ph.D. thesis, because there

are so many things I would like to thank you. I have researched my memories and finally, I

decided to write it as a letter.

One of the most grateful things in my study life is that you took me as your Ph.D. student

three years ago. When I looked back on myself in 2015, I was really simple and naive at that

time. All my grow-up, my progress and my achievement in the last three years, started from

when you gave me the key at the very first beginning, which opened the door to my entire new

life.

After 3 months at LSP, I gave my 1st presentation; after 6 months, I finished my 1st manuscript.

“I will pay you for conferences!” You said so. And starting from 2nd year, I was paid to seven

different conferences and met guys all over the world. I talked with lots of other Chinese

students, and none of them has a better Ph.D. environment than I do.

When I was trying to apply a scholarship offered from China, you refused to sign your name

on the prepared recommendation letter. “I have a better version than yours!” You said, then you

started writing a whole paragraph personally. One year later, there was another scholarship

offered from FAU, and you even wrote the whole recommendation letter in German for me.

I made mistakes (Facebook issue), for which I have blamed myself deeply. “Let it go, I trust

you.” You were smiling, “Everyone makes mistakes. Next time you’ll get better.” Yes, I have

really learned a lot and getting better. I am trying always less than four lines in an email; I am

124 Acknowledgement

considering stuff which only matters or value or contribute; I am successful in controlling my

temper, because I know I will never have enough bullets.

There were really countless memories with you, e.g. Sun Tzu, Magical Pilze, Chinese chess,

Asian culture and so on. And the most vivid moments of you floating before my eyes, was when

you were trying to solve a mathematics problem. The equations and symbols were under your

control and flowing smoothly through the pencil.

You are a unique Professor. I have never imagined that a professor of polymer who started

from physics, takes radio and antenna as a hobby, plays chess regularly and strongly, speaks

northern Europa language occasionally and does sports every single day! This is not only a

figure of my professor but an idol of my life, whom I would like to be after 20 years.

I really enjoy the talks with you. Topics about life, about love, about parents, about politicians,

and about almost everything! Unfortunately, I cannot recall every conversation we had together,

but truly speaking, I was, am and have been, strongly influenced by your values. It is just like,

people cannot remember each piece of protein they have eaten in their life, but it was just those

protein, which makes people finally grow to be strong.

A hidden head poem from me in Chinese, is written for Prof. Schubert:

The first row of the eight characters: I am going to leave my supervisor, and I am bowing

down and kowtowing, showing my grateful appreciation.

* Duanmu Ci (520–456 BC), one of the most famous student of Confucius.

Acknowledgement 125

• To Prof. Fritjof Nilsson at KTH

I was always grateful that I can have Prof. Nilsson as my 2nd supervisor. But when I was first

introduced to him in Stockholm, I did not expect too much- why should a Swedish professor

supervise or even discuss with a Chinese student in Germany? As the time went by, I knew I

was wrong. Prof. Nilsson is noble and generous, he always listens to the others carefully and

respectfully. We have met each other only four times, but the emails between me and Prof.

Nilsson, ranks No.1 in my email box, counting both frequency and content quantity. He has

really spent a lot of time and energy on me, especially the Chapter 5 in this thesis.

After my 1st paper is published, I believed I was somebody. When Prof. Nilsson told me that

my 2nd paper needed some “polish”, I was, of course, waiting for some praises. However, the

refinement of the paper was 80% different than the last version and most crucial is, I knew the

later version was definitely better! Since then, I was deeply impressed by his profound

knowledge and humility. Prof. Nilsson has helped me to generate a novel equation and taught

me how to use MATLAB software step by step. After the submission of our 2nd paper to CSTE,

we received 45 comments from one single reviewer. I have generated a response letter with 16

pages, and Prof. Nilsson carefully read them all, extended it to 22 pages- which was even longer

than our manuscript!

The 3rd paper of mine, I was not even satisfied or convinced by the results, and Prof. Nilsson

has pointed out the value of it with the first glance! That was a novel approach to explain the

phenomena, where words must be carefully chosen to explain. Prof. Nilsson and I have worked

for weeks for that together!

Last time when I met Prof. Nilsson was November 2017. I was starting to write my Ph.D.

thesis already, and Prof. Nilsson encouraged me to write the rest results on a paper. I did not

126 Acknowledgement

see any value of that. But when we finished the manuscript one month later, I knew it really

worth!

I would like to thank Prof. Nilsson sincerely, for your kind supervision and your selfless help

during my last two years of Ph.D. study. In the context of my ability in the future, I will be

always there for you, because you were once there for me, too.

A hidden head poem from me in Chinese, is written for Prof. Nilsson:

First row of the eight characters: Peaches and plums do not speak, but they are so attractive

that a path is formed below the trees. **

• To all the other important people

I would like to thank Prof. Werner A. Goedel at TU Chemnitz. It was him who took me to

the Ph.D. position at FAU. And he always used “scientific nobody” to describe himself, which

encouraged me always working hard.

I want to thank my colleges, especially the three former guys sitting in Room 193: Dr. Peter

Kunzelmann, Dr. Mathias Bechert, Dr. Michael Härth. Without them, I could not get the

progress so quickly. Thanks are also given to Dr. Xianhu Liu, Dr. Yamin Pan, Dr. Zdeněk Starý

and Dr. Joachim Kaschta for our scientific discussions. My further gratitude goes to Ms.

Jennifer Reiser for TGA test, Ms. Susanne Michler for arranging all in the laboratory. Ms.

** This sentence was used for Li Guang, who was a famous general in China 2100 years ago.

Acknowledgement 127

Magdalena Papp is specially thanked for taking SEM pictures, she was even more excited than

me when we found some valuable pictures. Ms. Brigitte Saigge is thanked for all kinds of help

in my daily life at LSP, Mr. Harald Rost and Mr. Marko Heyder are thanked for the technical

support, they made a model for me to explain “fibers in fiber” structure.

I would like to thank guys from Panda cage, especially to Jonas for those days in Stockholm,

and to Andi for the night at Christmas Market. I am grateful to Florian for those days in Kuala

Lumpur and to Piotrej for the great support of the DCFII association.

Prof. Chunhua Zhang at HIT is thanked for her advice and idea on the ternary composites

with CF and CB. M.Sc. Huagen Xu is thanked for those talks about the meaning of life and

purpose of living. M.Sc. Yu Wang is thanked for her support in the SketchUp Software. The

three roommates from my college HIT are thanked for the encouragement during my Ph.D.

study. Even though they all achieved their Ph.D. in Singapore two years ago, Dr. Yupang Tang,

Dr. Chenzhong Mu, Dr. Shiji Hao, they are still offering me their advice and helps. Especially

thanks to Dr. Jing Wang, I lost his contact after my bachelor study. However, 8 years later, this

kind and helpful guy is still doing well in US.

My special and ultimate thanks are for my family: My parents, my girl (M.Sc. Yijing Qin),

and my bosom friend (M. Sc. Guanda Yang). Thanks for your sacrifice and the support behind

me. I am still scientific nobody, but it is your love, making me more and more complete.

characterization and modeling of anisotropic electrically

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