The Effects of Uniaxial Strain on the Percolation Threshold of Fibers in Polymer Composites through

Monte Carlo Simulation

by

Eunse Chang

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Graduate Department of Mechanical & Industrial Engineering University of Toronto

© Copyright by Eunse Chang 2014

ii

The Effects of Uniaxial Strain on the Percolation Threshold of

Fibers in Polymer Composites through Monte Carlo Simulation

Eunse Chang

Master of Applied Science

Department of Mechanical and Industrial Engineering

University of Toronto

2014

Abstract

Non-structural applications of composite materials have been rapidly growing in

modern industry, and as a consequence functional polymer composites have emerged as ideal

candidates. Despite their superior properties, high cost of new generation of fillers such as

carbon nanotube is a huge drawback. Efforts are being devoted to investigate the feasibility

of foaming as a potential strategy to induce percolation networks, thereby achieving ideal

properties at low filler content.

In this research, Monte Carlo model is built to examine the effects of compression

and tension on the percolation threshold of fibers in polymer composites to partially simulate

cell growth. Fiber orientation and displacement effects are studied under numerous

simulation conditions with various aspect ratios in 2-D and 3-D systems. The results in both

systems confirm that increasing aspect ratio dramatically reduces critical concentration, and

potential improvement of fiber interconnectivity through biaxial stretching is observed.

iii

Acknowledgments

I would like to first thank my supervisor Prof. Chul B. Park for his great support and

guidance throughout my research, as well as granting me this wonderful opportunity.

My gratitude is extended to Dr. Amir Ameli, who has guided and inspired me with

many thought-provoking discussions and advice.

I would also like to express my gratitude to the members of my Thesis Committee,

Prof. Hani Naguib and Prof. Craig Steeves for their invaluable comments and kindness to

serve on my examination.

I would like to express my special thanks to Lun Howe Mark for his remarkable

contributions in the development and modification of the simulation models.

I wish to acknowledge the support and friendship from my colleagues in the

Microcellular Plastics Manufacturing Laboratory, especially the conductivity group for the

discussions and encouragement

Finally, I would like to express my greatest gratitude to my family and my friends for

their endless love, patience and encouragement which made it all possible for me.

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Table of Contents

Abstract .................................................................................................................................... ii

Acknowledgments .................................................................................................................. iii

List of Tables ......................................................................................................................... vii

List of Figures ....................................................................................................................... viii

CHAPTER 1. INTRODUCTION .......................................................................................... 1

1.1 Preamble ............................................................................................................................. 1

1.2 Polymer Matrix Composites ............................................................................................... 1

1.2.1 Functional Polymer Matrix Composites ...................................................................... 2

1.3 Microcellular Foaming........................................................................................................ 2

1.3.1 Foaming Processes ....................................................................................................... 3

1.3.1.1 Batch Foaming Process ............................................................................................. 3

1.3.1.2 Continuous Processes ................................................................................................ 3

1.3.2 Blowing Agent ............................................................................................................. 5

1.3.3 Characterization of Foam ............................................................................................. 6

1.4 Thesis Objective.................................................................................................................. 8

1.5 Thesis Overview ................................................................................................................. 9

CHAPTER 2. LITERATURE REVIEW AND THEORETICAL BACKGROUND ..... 10

2.1 Percolation Theory ............................................................................................................ 10

2.2 Percolative Properties ....................................................................................................... 11

2.2.1 Electrical Conductivity .................................................................................................. 12

2.2.2 Thermal Conductivity ................................................................................................... 16

2.2.3 Mechanical Properties .................................................................................................... 19

2.2.4 Rheological Properties ................................................................................................... 20

2.3 Simulation Model.............................................................................................................. 22

CHAPTER 3. DESIGN OF PERCOLATION MODEL ................................................... 24

3.1 Overview ........................................................................................................................... 24

3.2 Foaming Effects ................................................................................................................ 25

3.3 2-D Model ......................................................................................................................... 27

3.3.1 Fiber Generation ......................................................................................................... 29

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3.3.2 Fiber Interconnection ................................................................................................. 29

3.3.3 Percolation Network Formation ................................................................................. 30

3.3.4 Strain Effect Simulation ............................................................................................. 30

3.3.5 Reliability ................................................................................................................... 32

3.4 3-D Model ......................................................................................................................... 33

3.4.1 Fiber Generation ......................................................................................................... 33

3.4.2 Fiber Interconnection ................................................................................................. 35

3.4.3 Percolation Network Formation ................................................................................. 35

3.4.4 Strain Effect Simulation ............................................................................................. 36

3.4.5 Reliability ................................................................................................................... 37

3.5 General Assumptions and Discussion ............................................................................... 37

CHAPTER 4. RESULTS OF 2-D PERCOLATION MODEL AND DISCUSSION ...... 42

4.1 Overview ........................................................................................................................... 42

4.2 Sensitivity Analysis .......................................................................................................... 42

4.2.1 Ideal Number of Iterations ......................................................................................... 43

4.2.2 Scale Effect ................................................................................................................ 46

4.3 Poisson’s Ratio.................................................................................................................. 48

4.4 Effect of Aspect Ratio ....................................................................................................... 52

4.5 Fiber Alignment Effect ..................................................................................................... 55

4.6 Strain Effect ...................................................................................................................... 58

CHAPTER 5. RESULTS OF 3-D PERCOLATION MODEL AND DISCUSSION ...... 60

5.1 Overview ........................................................................................................................... 60

5.2 Sensitivity Analysis .......................................................................................................... 60

5.2.1 Scaling Effects without Deformation ......................................................................... 61

5.2.2 Scaling Effects with Deformation .............................................................................. 63

5.3 Effect of Aspect Ratio ....................................................................................................... 66

5.4 Fiber Alignment Effect ..................................................................................................... 69

5.5 Percolation Direction ........................................................................................................ 70

5.6 Strain Effect ...................................................................................................................... 72

5.6.1 Compression Effect .................................................................................................... 73

vi

5.6.2 Tensile Effect ............................................................................................................. 74

5.6.3 Comparison Analysis ................................................................................................. 77

CHAPTER 6. CONCLUSION AND FUTURE WORK .................................................... 78

6.1 Conclusion ........................................................................................................................ 78

6.2 Future Work ...................................................................................................................... 78

References .............................................................................................................................. 80

Appendices ............................................................................................................................. 87

vii

List of Tables

CHAPTER 4

Table 4-1 Statistics of Samples with Aspect Ratio = 100 ........................................................47

Table 4-2 Statistics of Samples with Aspect Ratio = 200 ........................................................48

Table 4-3 System dimensions at various strains and Poisson’s ratios ....................................50

Table 4-4 Degree of alignment at various strains and Poisson’s ratios ..................................50

CHAPTER 5

Table 5-1 Scaling Effect in 3-D Simulations ..........................................................................62

Table 5-2 Scaling Effect in 3-D Simulations with Compression Strain of 0.6 .......................65

Table 5-3 Scaling Effect in 3-D Simulations with Tensile Strain of 0.6 ................................65

Table 5-4. Degree of alignment values and their corresponding * and strain ......................70

viii

List of Figures

Fig. 1-1. Schematic of extrusion foaming system ....................................................................5

Fig. 2-1. Filler network below and above the percolation threshold .......................................11

Fig. 2-2. Microstructural changes in a polymer foam containing carbon nanoparticles with

foaming and the effects in electrical conduction ....................................................................15

Fig. 2-3. Electrical conductivity as a function of relative density at different CNT content ...16

Fig. 2-4. Thermal conductivity of PEI/graphene nanocomposite foams at 50 and 200 °C ....18

Fig. 2-5. A summary of literature data showing the change of relative modulus of polymer

foams with the change of relative density................................................................................20

Fig. 2-6. Complex viscosity versus nanotube content at different frequencies .......................21

Fig. 3-1. Effect of cell growth on the interconnectivity: a) before b) after cell growth ..........25

Fig. 3-2. Effect of cell growth on fiber orientation and position: a) before b) after cell growth

..................................................................................................................................................26

Fig. 3-3. Effect of cell growth in different regions of polymer matrix ....................................27

Fig. 3-4. Schematic of 2-D percolation model structure in 3 steps..........................................28

Fig. 3-5. Geometries of fiber a) before and b) after deformation ...........................................31

Fig. 3-6. Carbon nanotube and its representation as a capped cylinder ..................................34

Fig. 3-7. Cartesian representation of a long fiber in 3-D .........................................................34

Fig. 3-8. Geometries of fibers distributed in a matrix. a) before and b) after deformation .....39

Fig. 3-9. Intersection of interpenetrating fiber with another fiber and system boundary ........40

Fig. 4-1. Simulation statistics when sample pool size = 50 .....................................................44

Fig. 4-2. Simulation statistics when sample pool size = 100 ...................................................44

Fig. 4-3. Simulation statistics when sample pool size = 200 ...................................................45

Fig. 4-4. Simulation statistics when sample pool size = 500 ...................................................45

Fig. 4-5. Effect of Poisson’s ratio on percolation threshold under various strains ..................51

ix

Fig. 4-6. Effect of Poisson’s ratio on percolation threshold under various strains ..................51

Fig. 4-7. Simulation output graphs at different aspect ratio ....................................................53

Fig. 4-8. Simulation results with various aspect ratios ............................................................54

Fig. 4-9. Fiber networks of different type of fiber ...................................................................55

Fig. 4-10. Alignment effect on the percolation threshold ........................................................57

Fig. 4-11. Probability function of percolation at various degree of alignment ........................57

Fig. 4-12. Combined effect of alignment and displacement of fibers on the percolation

threshold ..................................................................................................................................59

Fig. 5-1. Representation of 3-D fiber model ...........................................................................61

Fig. 5-2. Probability density plot of critical concentration with low and high fiber size ........63

Fig. 5-3. 3-D Simulation results with various aspect ratios .....................................................67

Fig. 5-4. Critical concentration for the 3D systems of randomly oriented soft-core sticks .....67

Fig. 5-5. 2-D and 3-D simulation results with various aspect ratios........................................68

Fig. 5-6. Alignment effect on the percolation threshold ..........................................................69

Fig. 5-7. Alignment effect on the percolation threshold in the planar direction ......................71

Fig. 5-8. Alignment effect on the percolation threshold in the vertical direction ....................72

Fig. 5-9. Compression effect on the percolation threshold at low aspect ratio (a = 10) .........73

Fig. 5-10. Compression effect on the percolation threshold at medium aspect ratio (a = 30) 74

Fig. 5-11. Compression effect on the percolation threshold at high aspect ratio (a = 100) .....74

Fig. 5-12. Tensile effect on the percolation threshold at low aspect ratio (a = 10) ................75

Fig. 5-13. Tensile effect on the percolation threshold at medium aspect ratio (a = 30) ..........76

Fig. 5-14. Tensile effect on the percolation threshold at high aspect ratio (a =100) ..............76

Fig. 5-15. Strain effect on the percolation threshold at various aspect ratios ..........................77

1

CHAPTER 1. INTRODUCTION

1.1 Preamble

In modern life, plastic products can be found everywhere in various applications,

owing to their light weight, low cost, ease of manufacture, as well as versatilities to name a

few. Plastic material refers to any organic solids, both synthetic and semi-synthetic, that are

typically capable of being molded or formed. However, as the plastics industry continues to

grow to replace conventionally used metals in many applications, these polymers often lack

in areas where certain material characteristics, such as electrical, thermal, and mechanical

properties are required. One potential strategy to overcome these obstacles is to fill the

polymer matrix with reinforcement particles or fibers to create a composite material that

exhibits combined properties of the components. The main benefit of using such polymer

composite materials is that superior material properties can be achieved without

compromising desirable properties of the polymer matrix.

1.2 Polymer Matrix Composites

Composite materials are defined as multi-phase materials obtained by combination of

two or more materials. When the components are combined to form a composite material, it

may exhibit properties that the individual components alone cannot attain. Polymer matrix

and cement matrix composites are most commonly used due to their low processing cost [1].

One of the main advantages of polymer matrix composites is that they are generally

much easier to fabricate compared to metal, carbon, or ceramic-based composites due to their

2

relatively low processing temperatures. Fibrous polymer matrix composites can be classified

as either discontinuous or continuous, where continuous or long fibers provide superior

properties at the cost of anisotropy. Major applications of polymer matrix composites include

lightweight structures for aircraft and sporting goods, asphalt, as well as emerging functional

applications.

1.2.1 Functional Polymer Matrix Composites

The main benefit of traditional composite materials was limited to improvement in

mechanical properties due to the dominance of structural applications, but as non-structural

applications, such as electronic, thermal, biomedical, and other industries have been rapidly

growing, other functionalities of composite materials also came into the picture [1]. For

instance, electrically insulative nature of polymer can be altered by addition of electrically

conducting particles such as carbon nanotubes. Another example would be to embed alumina

fillers in a polymer matrix to create a composite material that is electrically insulating but

thermally conducting [2].

1.3 Microcellular Foaming

Microcellular plastic can be classified as polymeric foams with a cell density in

excess of 108

cells /cm3 and a cell size under 10 µm [3]. Microcellular plastics can help

significantly reducing the material cost as well as the weight while maintaining the desirable

properties with fine cell size [4]. Mechanical properties are one of such design criteria, where

microcellular foaming could result in potential improvements. For instance, toughness of

microcellular foam compared to its solid counterpart may be up to 5 times higher [5].

3

Moreover, surface quality of microcellular foam is superior through having sharper contours,

and corners [6]. In addition, microcellular structure of the foams makes them good thermal

insulators [7].

1.3.1 Foaming Processes

For both batch and continuous foaming processes, the three general steps are the

formation of solution, cell nucleation, and cell growth [8].

1.3.1.1 Batch Foaming Process

Batch foaming process is very suitable for research and development purposes. It

however has low commercial value due to high saturation time as well as repeatability issues.

Formation of polymer-gas solution: Pre-shaped thermoplastic parts are placed in a

pressurized foaming chamber filled with blowing agent at elevated temperature and pressure.

Depending on the type of polymer and blowing agent, the gas impregnation time varies and it

typically is several hours [9].

Cell Nucleation/Growth: the thermodynamic instability of the polymer/gas solution

is induced by rapidly decreasing the chamber pressure via a release valve. At above the glass

transition temperature, the reduction in gas solubility will generate small bubbles inside the

polymer matrix. These bubbles will eventually form a cellular structure as they grow in size

[10].

1.3.1.2 Continuous Processes

4

Continuous processes of microcellular foaming include extrusion foaming and

injection mold foaming. Figure 1-1 displays a schematic of an extrusion foaming process that

is cost effective and highly productive.

Formation of polymer/gas solution: This step will significantly affect the cell density

of the foam. For uniformly sized and distributed bubble formation, one must make sure that

the blowing agent is well mixed within the polymer melt [11]. Solubility limit of the blowing

agent in polymer melt at the operating conditions must be examined ahead of time and taken

into account, so that excessive amount of blowing agent is not injected. When the gas content

is above the saturation point, it will not completely dissolve into the polymer solvent, and as

a result, processing instabilities occur and large voids form [12].

Cell Nucleation: Cell nucleation takes place when small clusters of gas molecules are

transformed into energetically stable pockets [13]. To create bubbles in the polymer melt, a

minimum amount of energy that can break the free energy barrier is required. Sudden

increase in temperature or decrease in pressure triggers this mechanism in the continuous

foaming process.

Cell Growth: Once cells are nucleated in the polymer matrix, they start to expand

because pressure inside the cells is larger than outside. The cell growth is heavily affected by

parameters such as viscosity, diffusion coefficient, blowing agent concentration, and the

number of cells. Controlling the operating temperatures closely is therefore crucial in order to

achieve good cell growth and desirable expansion ratio [14]. In microcellular foams, cell

walls are relatively thinner than conventional foams, so cell coalescence may take place as a

5

result. This phenomenon is undesirable since it will deteriorate cell density, as well as other

good mechanical properties.

Fig. 1-1. Schematic of extrusion foaming system

1.3.2 Blowing Agent

Blowing agent must be carefully selected before an experiment can be performed.

The blowing agent type as well as the amount plays a crucial role in any foam process since

excessive un-dissolved gas may cause cell deterioration. The processing conditions must also

be taken into account when choosing the right type of blowing agent.

There are two main types of blowing agents, namely chemical blowing agents and

physical blowing agents. Chemical blowing agents are substances that decompose at

processing temperatures and they are usually used to produce high to medium density plastic

and rubber foams [15]. Chemical blowing agents were conventionally used for general

foaming purposes because of their high solubility in polymer melts. Physical blowing agents

such as butane, pentane, and carbon dioxide are another class of blowing agents and

6

generally used due to their low cost, and with some modification, they can be directly

injected to the extruder barrel.

1.3.3 Characterization of Foam

There are a number of parameters that determine the nature of a foam product. These

include cell morphology, cell density, and expansion ratio. Such parameters are heavily

dependent on the resin and also governed by the processing conditions during the foaming

process.

Cell morphology of foam can be described in terms of its cell size, cell size

distribution, and cell density. Scanning electron microscope (SEM) is often used to

characterize a cross-section of foam. SEM is preferred over the conventional optical

microscopes for characterization purposes, especially for microcellular and nanocellular

foams due to its superior resolution and magnification control. The cell density, N0, is the

number of cells per cubic centimeter relative to the unfoamed material and can be calculated

using:

𝑁0 = (𝑁𝑀2

𝑎)

3 2⁄

(1

1 − 𝑉𝑓) (1 − 1)

where N, M, and a represent the number of cells, the magnification, and the area under

investigation, respectively.

The parameter Vf refers to the void fraction of foam, which is related to the expansion

ratio and foam density. It indicates the amount of void in the foam and can be calculated

from the following equation:

7

𝑉𝑓 = 1 −𝜌𝑓

𝜌 (1 − 2)

where ρ and ρf are the density of unfoamed precursor and foamed sample, respectively.

Volume expansion ratio is another parameter similar to void fraction that indicates the

change in final volume with respect to the initial volume prior to foaming. It can be

calculated from the following equation:

𝜌𝑓 =𝜌

𝛷 (1 − 3)

where Φ is the expansion ratio.

Foam density, ρf, is a structural parameter that describes the reduction in density of

the sample upon foaming. It is usually determined by water volume displacement technique

[7]. Theoretically, it can be determined from the following formula:

𝜌𝑓 =𝑀

𝑉 (1 − 4)

where M is the mass, and V is the volume of the foam.

Although these structural parameters characterize polymeric foams reasonably well,

when the cells are non-uniform and non-spherical with possibility of having open-cell

content, these parameters are not sufficient to describe the cell morphology. This is

especially true for composite materials with high fiber loading, in which case the cell size

distribution curve is commonly used to better interpret the cell morphology [16].

8

1.4 Thesis Objective

Although functional polymer composites that contain new generation of materials

such as carbon nanotubes are ideal for maintaining excellent material properties as well as

functionalities [17], the apparent drawback has been the high material cost of such fillers.

Consequently, efforts are being made to develop an economical strategy to maintain good

properties and functionalities at low filler loadings to reduce the cost.

Incorporation of foaming can be one strategy as foaming has recently shown promises

in promoting the conductive nanocomposites for various applications [18-22]. The matrix

weight can be reduced significantly through foaming, but also the electrical properties may

be positively affected [23]. It was also reported that when physical blowing agent such as

supercritical carbon dioxide (scCO2) is used, the dissolved gas in the matrix can improve the

dispersion and distribution of the fillers [24-27]. In addition, physical foaming is known to

reduce the chance of fibre breakage in foam injection molding process due to plasticizing and

lubricating effects of blowing agent [27, 28].

To investigate the percolation phenomenon in electrically conductive polymer matrix

composite foams, a computer-aided percolation model that can simulate matrix/fiber

interaction is to be established. The percolation threshold of a system at given conditions

could be calculated with this model to understand the percolation behaviour of the system.

Furthermore, a system under compression and tension is to be examined, in order to partially

explain the foaming effects on percolation threshold.

9

1.5 Thesis Overview

Chapter 2 focuses on the literature review on the percolation theory as well as

percolation simulation models. The various properties that exhibit percolative behavior are

studied. Different computer simulation models that visualize the system of polymer matrix

composite are thoroughly examined.

Chapter 3 introduces the modelling of percolation simulation code by MATLAB. A

number of reasonable assumptions and generalizations were to be made for simplification of

the model. These assumptions were then assessed and the accuracy of the 2-D and 3-D

models was verified by comparing with established data from previous works by others.

Chapter 4 presents the results of 2-D model to investigate the effects of various

conditions such as no strain, fiber alignment only, fiber alignment and displacement, and

constant area at different aspect ratio values on the percolation threshold. The results were

analyzed to find out the physical meaning and to predict the trends in 3-D model.

Chapter 5 shows the results of 3-D model under various conditions. The results were

analyzed in a similar fashion as in Chapter 4. A comparison analysis between the two models

was also carried out.

Chapter 6 provides a summary of the research with conclusion. Moreover,

recommendations for future work are presented.

10

CHAPTER 2. LITERATURE REVIEW AND THEORETICAL BACKGROUND

2.1 Percolation Theory

Percolation theory is often employed to describe the movement of classical particles

through a medium. Percolation is said to be achieved when a continuous fiber network is

formed so that opposite faces of the polymer matrix are connected [29]. Figure 2-1 represents

filler network in a matrix below and above the percolation threshold. A transportation-related

physical property, K, is said to exhibit percolation behavior when the following universal

power law can describe their relationship with the material composition:

𝐾

𝐾𝑓= (

𝜑 − 𝜑𝑐

1 − 𝜑𝑐)𝑡, (2 − 1)

Where 𝐾𝑓 refers to the physical property of the filler, and 𝜑 and 𝜑𝑐 are the filler volume

fraction and critical volume fraction (percolation threshold), respectively [30]. 𝑡 is a constant

that varies from different eccentricity for spheroid fillers, and ranges from 1.6 to 2 [31,32].

The interesting characteristic of the equation 2-1 is that a slight increase in the filler content

may lead to a huge increase in the physical property K.

To model such continuum percolation problem where randomly oriented fillers are

dispersed in an isotropic system, a computer simulation method is normally used to aid the

research [29].

11

Fig. 2-1. Filler network below and above the percolation threshold [33]

2.2 Percolative Properties

Percolative properties are properties that dramatically change in value upon

percolation. A variety of factors such as fiber type, aspect ratio, fiber loading, fiber alignment,

purity, and interfacial adhesion may affect these properties [34]. This section discusses in

detail material properties that exhibit percolation behavior or properties that are reported to

have potential percolation behavior in a number of literature.

12

2.2.1 Electrical Conductivity

In general, electrical conductivity of a polymer matrix composite material with

conductive fillers depends on size, shape, concentration, distribution, as well as the surface

treatment of fillers. Especially, aspect ratio and volume fraction of the fillers are very crucial

factors that decide the electrical conductivity of these composites as they form percolation

network. Conductive fillers such as MWCNTs can help reach the percolation pathway at a

relatively low volume fraction owing to their very high aspect ratio in the range of 100–1000

[35], compared with other types of fillers.

Another possible strategy that has been suggested is to accommodate foaming

technology in conductive composites to improve the low electrical conductivity of polymer

composites, and by doing so extending their applications to a wider range of sectors such as

electronics. Carbon nanotubes in the form of multi-walled carbon nanotubes (MWCNT) have

been extensively used in developing this type of polymer foams. More economical

alternative filler in the form of carbon fibers and carbon nanofibers have also been

considered. Other fillers that have been mostly used for research purposes include graphene

nanoplatelets, and expandable graphite. As for the polymer matrix in such applications, the

most common material that has been used is Polyurethane (PU), which facilitates the

incorporation of solid fillers.

Research conducted by You et al. [18] suggested that the addition of as low as 0.1

php of MWCNT led to dramatic increase in electrical conductivity of PU foams. Furthermore,

additional loading of MWCNT did not result in significant increase in electrical conductivity,

which supports the percolation theory of electrically conductive fillers.

13

Another research group that studied PU foams with conductive fillers led by Yan

came up with electrically conductive PU foams with a relatively low percolation threshold by

using around 1.2 wt% of carbon nanotubes. They were able to carefully control the filler

dispersion in the cell wall and strut regions during processing to effectively create conductive

path at relatively low filler loading [19]. This study indicates a sign that foamed cells may be

beneficial by reducing the loading of filler, which can in turn save material cost and also help

preserve good processability.

In addition, conductive PU foams with densities as low as 0.05 g/cm3 prepared by Xu

et al. appeared to form a conductive percolation path with only 2 wt% of MWCNT content,

owing to a highly effective pathway in the cell strut regions [20]. As the expansion ratio of

the foam continued to increase, the electrical conductivity suddenly dropped. This conductor

to insulator transition was explained by reduction in cell wall thickness, which limits the

amount of filler in the cell wall and also the filler contact. This phenomenon was described as

3D to 2D percolation transition.

However, this hypothesis that the filler content in cell wall upon cell growth is

reduced is proven wrong, as it was found that the CNT loading in cell walls does not change

because the particles do not leave cell walls as the foam density changes [21]. Therefore, the

decrease in density and cell wall thickness indicated the 3D to 2D transition, which increased

the percolation threshold while also reducing the cell wall conductivity. Another point to note

was that lower density foams exhibited more porous structures, which limited the number of

CNT bridging among the cells, leading to difficulty in formation of overall conductive paths.

Contrary to what has been argued, F. Du et al. claimed that a possible reason for such

reduction in conductivity when the cell expansions was increased is that the cell wall

14

stretching significantly affects the alignment of CNT in the cell wall, thereby interfering the

fiber contacts and resulting in lower conductivity [36].

For the case of PP-based foams with carbon nanofiber (CNF) reinforcement, the

foaming process could also be adopted to decrease the CNF loading without compensating its

electrical conductivity. Antunes et al. were able to lower the critical concentration of CNF,

which was 6 vol % for solid, down to 5 vol % for its foamed counterpart, which showed

around 3 times larger expansion ratio [37]. They claimed that the foaming process may lead

to two different effects on the overall electrical conductivity percolation threshold. At the cell

nucleation stage, the nanoparticles are pushed together by the excluded volume of cells. As

the cells grow, strong extensional flow is generated, causing the particles to reorient [38].

The other effect is also caused by the volume expansion of the cells, and the adjacent

nanoparticles move further apart.

Therefore, the degree of volume expansion is a key factor in deciding the electrical

conductivity percolation of polymer composite foams. According to a study by H.B. Zhang et

al., the foaming process can reduce the electrical conductivity percolation threshold of

PS/graphene nanocomposite slightly [39]. The same trend was found for the case of

microcellular PEI/graphene nanocomposite, where the foaming process resulted in a

reduction of electrical conductivity percolation threshold from 0.21 to 0.18 vol % [40]. J.

Ling et al. claimed that this phenomenon was possibly caused by the orientation and

enrichment of graphene during the cell growth stage. Figure 2-2 demonstrates the overall

process of cell growth and how it can possibly affect the electrical conductivity [41].

15

Fig. 2-2. Microstructural changes in a polymer foam containing carbon nanoparticles with

foaming and the effects in electrical conduction [41]

Figure 2-3 shows how the electrical conductivity evolves with foam density of PP-

CNT composite at different CNT content in a batching foaming process [42]. A. Ameli et al.

have also conducted foam injection molding experiments using PP-carbon fiber composites

containing various contents of carbon fiber [27]. They explained the increased inter-

connectivity of fibers induced by cell growth as a combined effect of biaxial stretching of the

polymer matrix and the fiber re-orientation. Foaming enhanced the through-plane electrical

conductivity up to 6 orders of magnitude, while reducing the fiber content as well.

16

Fig. 2-3. Electrical conductivity as a function of relative density at different CNT content [42]

2.2.2 Thermal Conductivity

Thermal Conductivity is a controversial area where its behaviour as a function of

filler loading is still unclear. S. Choi et al. showed through nanotube-in-oil suspensions that

the increase in thermal conductivity with addition of filler is percolative and abnormally

greater than theoretical estimation, contrary to the expected monotonic increase [43]. J. Ling

et al. however claimed that the polyetherimide/graphene composite foams dropped its

thermal conductivity as more conductive fillers were added [40].

It has been studied that the thermal conductivity of polymer matrix composite

increases linearly with the filler loading, which is quite different from the electrical

conductivity percolation behavior. According to Lazarenko et al. [44], as the content of

nanocarbon filler was increased, the value of thermal conductivity also was increased. Such

conductivity growth was monotonic throughout the entire interval of concentrations, and

many different factors contributed to it, including the particle distribution, orientation with

17

respect to the heat flux direction, as well as the ability to form chains. The effect of fiber

orientation was especially significant when the fiber aspect ratio was high (100 – 1000) [44].

Thermal conductivity is of greater interest when foaming technology is involved,

because one of the major functions of polymeric foams is to provide thermal insulation [45].

In general, the effective thermal conductivity of polymeric foams break down to a number of

factors: a heat flow in the polymer and the cell, radiation, and convection. Depending on the

foam properties such as cell size and the foam density, the contribution of these factors vary

greatly. For instance, polymer foams with small cells would experience minimal convective

heat transfer [46], whereas for low density foams, the gas properties play an important role

due to its high volume fraction. Therefore, for thermal insulation foams, which usually have

small cell size with low foam density, the thermal conductivity is heavily reliant on the heat

flow in the cell and radiation.

When fillers are introduced, however, a very interesting phenomenon is detected. As

opposed to electrical conductivity, which has been shown to increase with higher filler

loading with percolation behavior, this was not the case for thermal conductivity with

addition of thermally conductive fillers. Thermal conductivity of PEI/graphene

nanocomposite foam was measured at 2 different temperatures (50°C and 200°C) with filler

loading ranging from 0 to 7 wt % [40], and Figure 2-4 displays the corresponding thermal

conductivity values. With the addition of graphene, which is a thermally conductive filler, the

nanocomposite foam was expected to experience an increase in the thermal conductivity as

was the case in unfoamed samples, but instead the overall conductivity was reduced with

increased filler loading. One possible explanation is that the addition of graphene resulted in

decrease in cell size, which in turn decreased the thermal conductivity of the foam with the

18

same density [47, 48].

Another hypothesis is that the graphene may absorb and reflect the infrared radiation,

depressing the thermal radiation of the composite foam. It was previously found that strong

absorption and reflection of IR radiation caused the thermal conductivity of

polystyrene/carbon foams to be lower than the pure polystyrene foam [49]. Since graphene

has a much higher specific surface area, the statement may be valid.

In another research [41], PP-CNF foams presented linear increases in thermal

conductivity with high concentration of CNF, while the unfoamed counterparts displayed

constant values at different filler loading, which indicates that the introduction of foaming

somewhat led to the formation of thermally conductive network. However, the increase in

thermal conductivity was still relatively insignificant compared to the expected theoretical

value, and this is because the heat transfer mechanism is different from that of electrical

conduction. It has been shown that an intimate contact between the conductive fillers needs

to be made to form a thermally conductive network [50].

Fig. 2-4. Thermal conductivity of PEI/graphene nanocomposite foams at 50 and 200 °C [40].

19

2.2.3 Mechanical Properties

It is found that some mechanical properties change dramatically at points called

rigidity percolation and particle percolation point in sand-filled polyethylene composites [51].

This is a lesser studied phenomenon where the order of magnitude of property change is not

as dramatic as in electrical conductivity percolation behavior. Rigidity percolation point is

the volume fraction of filler where there is just enough resin present to yield a rigid structure.

Particles may form percolative pathways below this point but these networks have no rigidity.

Polymeric foams are also good potential candidate for structural applications due to

their low density. However, the benefits of foaming usually come at a cost of inferior

mechanical properties resulting from reduction in density [52]. For this reason, developing a

light weight and high strength polymeric foam remains a major research objective and

studies have been conducted to investigate effects of nanofiller addition.

The Gibson–Ashby model is one of the most commonly used model, and it is known

to be applicable on low relative density foams (ρr< 0.2). The mechanical properties of

polymer foams depend on the mechanical properties of the bulk solid polymer and the

relative density of the polymer foam. In the case of nanocomposite foams, the nanofillers can

reinforce the solid matrix polymer, resulting in foams with enhanced mechanical properties

[53-55]. Although other models have been developed for high density foams as well, it is

difficult to define an explicit relationship between the cell size and the mechanical properties

of polymeric foams [45].

Based on numerous studies [56-61], it can be concluded that addition of nanofillers

can significantly enhance the mechanical properties of polymeric foams since they enhance

the polymer matrix as well as the foam structure. However, better understanding is required

20

to fully reveal the foam properties of polymer nanocomposite foams and verify their

mechanical percolation behavior. Figure 2-5 summarizes the literature data on change of

relative modulus with respect to relative density [45].

Fig. 2-5. A summary of literature data showing the change of relative modulus of polymer

foams with the change of relative density [45, 62-66]

2.2.4 Rheological Properties

Given that the filler dispersion is good, the storage modulus (G¢) for a typical

response of polymer nanotube composites at a fixed frequency exhibits a percolation

behavior [67]. The rheological percolation is known to depend on parameters such as fiber

dispersion, aspect ratio, and alignment, which is quite similar to electrical percolation.

Potschke et al. confirmed that the complex viscosity (|η*|) increases with CNT

content in their experiment with PC/MWCNT composites [68]. The effect of increasing filler

loading was more pronounced at low frequencies due to shear thinning behavior at higher

21

frequencies as shown in Figure 2-6 [69-71].

Fig. 2-6. Complex viscosity versus nanotube content at different frequencies [68]

In the same literature, G’ and G’’ plots show that the increase in G’ is much greater

than that of G” with the nanofiller content. As the nanofiller content is increased, the filler-

filler interactions begin to dominate the composite system and eventually formulate a

percolative network system [72].

In addition, rheological percolation is also strongly dependent on temperature, as

discovered by Potschke et al. [73]. They used PC/SWCNT composites to claim that the

percolation threshold can decrease by up to 10 times upon increasing the temperature from

170°C to 280°, and that the nanotube network does not solely determine the rheological

properties, but also the entangled polymer network must be taken into account. For this

reason, the rheological percolation threshold must be lower than the corresponding electrical

percolation threshold. Also, Du et al. confirmed that this was actually the case by using

PMMA/SWCNT composites. They found that the rheological percolation threshold of the

composite was as low as 0.12 wt %, as opposed to 0.39 wt % for the electrical percolation

22

threshold [67].

2.3 Simulation Model

In order to numerically study the many continuum percolation problems, researchers

worldwide have adopted the computer simulation method over the years. The main

advantage of employing a computer-aided simulation program is that sensitivity or

comparison analysis can be performed by altering many different critical parameters with

relative ease.

This type of simulation model is classed as a Monte Carlo Method. Monte Carlo

Methods, also known as Monte Carlo Experiments, are statistical algorithms that rely on

random sampling iterations. Since the continuum percolation model involves randomly

generated fibers that are randomly dispersed in a system, repeated simulations are required to

accurately obtain the distribution of entities, such as percolation threshold.

Continuum percolation models generally consist of three steps: filler generation, filler

interconnection, and percolation network formation. First thing to consider in fiber

generation process is the physical attributes of the filler. Given the size and shape, these

fillers can be randomly generated inside a cube which represents the polymer matrix [29].

Then the program detects fillers intersection by filtering out fillers that are overlapping and

they are grouped together. Once the filler intersection process is complete, the program

determines whether a percolation path between opposite faces of the system cube has been

formed. The program continues its iterations until percolation is reached. Various parameters

such as filler or system size may be altered to better simulate the proper dimensions filler and

matrix. When sufficient data have been collected to predict the percolation threshold value

23

with high confidence level, the program stops its loops. Due to the statistical nature of this

method, repetition of simulations is required to generate a solid prediction [29].

24

CHAPTER 3. DESIGN OF PERCOLATION MODEL

3.1 Overview

As introduced in Chapter 2, one technique to study various continuum percolation

problems is computer simulation. Although the form and algorithm varies from one study to

another, Monte Carlo method is widely adopted to understand the fundamentals of polymer

matrix composite. The main advantage of using Monte Carlo method to model polymer

matrix composite system is that the effect of each critical parameter can be closely examined.

Because percolation conduction mechanism is chance-based, a Monte Carlo simulation

model can offer huge benefits in computing statistical average with high confidence level,

whereas real experiments can prove to be inflexible, time-consuming, and also expensive.

Such simulation model, under the assumption that it is error-free and therefore

functioning properly, can provide a good prediction of what to expect prior to actual

experiments. It may help the user set the initial point of experiments. Alternatively, when

experimental results are to be analysed, the argument can be theoretically supported with the

simulation results.

For this reason, Monte Carlo models that are able to predict percolation threshold and

also potentially analyse the effects of foaming action, which have shown positive signs to

lower the critical concentration of polymer matrix composites, have been generated in both

2-D and 3-D spaces. 2-D model is a good tool to quickly measure percolation threshold in

various conditions owing to its simplicity. It can also provide a 2-D image to help visualize

the filler network upon reaching the percolation limit. 3-D model is a more realistic

25

representation of physical experiments, although more time-consuming, and it can help the

user analyze fiber connectivity with higher accuracy.

3.2 Foaming Effects

In order to accurately simulate foaming action, it is required to closely examine the

change of microstructure of composite upon foaming. A. Ameli et al. have schematically

illustrated that when foaming is introduced, parameters such as fiber orientation, length,

interconnectivity, and skin layer thickness were changed in injection molding process [27].

Especially, fiber interconnectivity and fiber orientation appeared to be closely related as is

demonstrated in Figure 3-1.

Fig. 3-1. Effect of cell growth on the interconnectivity: a) before b) after cell growth [27]

The figure suggests that introduction of foaming leads to biaxial stretching of matrix,

where the degree of displacement and orientation vary for different locations. Expansion of

cell reorients the fibers and thus the connectivity can be improved as the increased number of

26

intersections may be created. This phenomenon can be locally described as compression

around the cell wall region as the polymer matrix experiences squeezing effect in the

direction normal to cell radius.

In other words, when a single fiber is locally inspected before and after foaming, the

orientation and position of fiber is altered as shown in Figure 3-2. As the cell expand, they

essentially exert compressive force on the cell walls along the cell growth direction (Z

direction in Figure 3-2), and as a result, the fiber is displaced and rotated as shown in Figure

3-2b.

Fig. 3-2. Effect of cell growth on fiber orientation and position: a) before b) after cell growth

[74]

When more than a single cell is taken into account, however, different regions of the

polymer matrix undergo different types of stress. Figure 3-3 is a simple schematic that

demonstrates how expansion of cell may affect the system as a whole. As discussed earlier,

cell walls are exposed to biaxial stretching, whereas in the cell strut region surrounded by

numerous cells uniaxial stretching takes place.

27

Fig. 3-3. Effect of cell growth in different regions of polymer matrix [74]

It is therefore crucial to fully understand the effects of uniaxial as well as biaxial stretching

on the percolation behaviour of polymer matrix composite prior to analysing foaming actions.

The 2-D and 3-D Monte Carlo models were created in a compatible way such that they can

be equipped with modules to study the uniaxial and biaxial stretching effects on the

interconnection and percolation of fibers through compression and tension simulations.

3.3 2-D Model

2-D Monte Carlo model was generated to assist in calculating the percolation

threshold (critical area fraction) of a 2-D surface using the MATLAB software. The generally

adopted 3-step process was implemented as shown in Figure 3-4.

28

Fig. 3-4. Schematic of 2-D percolation model structure in 3 steps

A point with random coordinates is first created. With this random point as a mid-

point, a fiber of pre-defined length is created at a random angle relative to the horizontal axis.

Another fiber is generated following the same procedure, and the two fibers are scanned for

interconnection through bsxfun(@hypot) toolbox from MATLAB, which measures the

displacement between two points. If two or more fibers are in contact, they are grouped

together in a cluster. Finally, every cluster is scanned for contact with the system boundaries,

and if such cluster exists it is declared as a percolation network. This 3-step process is

repeated until a percolation network is formed, and the number of fibers generated is

recorded.

29

3.3.1 Fiber Generation

A number of different algorithms have been previously used by researchers to

randomly create fillers in a 2-D system. Shape of fillers is a major factor that decides the

most efficient way of generating fillers. To model continuum percolation problem involving

particle fillers, M. Kortschot et al. randomly assigned sites in a given system that represent

spheres of given size [29]. For our purpose, fibers with high aspect ratio were of greater

interest, and therefore slender stick figures were generated as done in a number of literatures.

Firstly, the simulation conditions are to be assigned by the user. These include the length of

fibers, L, and the size of system boundary. The initial system size is set 100 x 100 unit2. Then,

the midpoint of a fiber is randomly assigned its coordinates through random number

generator within the system boundary. In other words, the x and z co-ordinate of the midpoint

can take any value from 0 to 100 or the system dimensions if altered. Next, another random

number between 0 and 2pi is generated the same way, and is labeled as . Then, a line

equation with tan as the slope that passes the midpoint can be generated from these random

numbers. A fiber (line segment) of length L can be generated on this line, and the coordinates

of end points can be calculated and stored along with other fiber information.

3.3.2 Fiber Interconnection

The fiber generation process continues, and each time a new fiber is generated, then,

any possible intersections with previously generated fibers are examined. The main algorithm

that enables this process is quite simple. Since the line information from filler generation

process is stored in a linear equation format, the intersection point of extension lines of two

fibers can be easily calculated using a set of two linear equations, corresponding to the two

30

fibres. If there is an intersection and the distances from the midpoint of each fiber segment to

the intersection point are both less than half of the fiber length, 𝑙

2, the intersection point lies

within both of the fiber segments and therefore the two fibers are in contact. Once such

connection is identified, the two fibers are grouped into the same cluster. If the new fiber has

no intersection with the previously created fibers, a new cluster is generated.

Upon contact, it is very important that the intersection scanning process continues

before moving on to generating a new fiber. This is because some of the newly generated

fiber may be in contact with more than one cluster. In such scenario, this fiber in inquiry will

form a bridge between two or more clusters and they can be grouped together as one new

cluster.

3.3.3 Percolation Network Formation

At the end of filler intersection process, each fiber cluster is tested to see if they cross

the boundaries of the system. This process is achieved by examining the minimum and

maximum value of both x and z coordinates in each cluster. If both extremes are outside the

boundary condition for either x or z coordinates, the cluster is declared as a percolation

network and the number of fibers generated at the time is recorded as the percolation

threshold.

3.3.4 Strain Effect Simulation

When a 3-D system is under compressive stress along one dimesion, it is

experiencing biaxial stretching effect in the other two dimensions. Similarly, in 2-D system,

application of compressive force results in stetching along the other dimension. As

31

mentioned in section 2.2.1, matrix under compression promotes two different deformation

modes: displacement and rotation. To account for the displacement deformation of fibers

upon compression, C. Lin et al. used the following equations to describe the relationship

between M and m from Figure 3-5:

𝑥 = 𝑋(1 + 𝑣𝛾), 𝑧 = 𝑍(1 − 𝛾) , (3 − 1)

where 𝛾 and 𝑣 are compression strain and the 2D Poisson’s ratio, respectively [76]. Poisson’s

ratio is a constant that varies from a material to another depending on the type and structure.

For our purpose, 2-D model was built to serve as a stepping stone for 3-D model and a

slightly modified equation was implemented in strain effect simulations.

𝑥 = 𝑋

√(1−𝛾), 𝑧 = 𝑍(1 − 𝛾) , (3 − 2)

Fig. 3-5. Geometries of fiber a) before and b) after deformation

The angle of rotation can be derived from equation 3-1 through simple trigonometry,

and the following equation describes the relationship between the initial and final degree of

orientation:

tan 𝜃∗ tan 𝜃⁄ = (1 − 𝛾) (1 + 𝑣𝛾)⁄ , (3 − 3)

For strain effect simulations, this equation was also modified to accommodate the

32

new displacement equation 3-2.

tan 𝜃∗ tan 𝜃⁄ = (1 − 𝛾)32, (3 − 4)

With this set of equations, percolation threshold under compression at various strains

can be explored.

3.3.5 Reliability

The 2-D Monte Carlo model serves its purpose well as a quick measure to simulate

analogous physical systems. However, over-simplification of the model may be responsible

for some sources of error. Other general assumptions made in both 2-D and 3-D models are

discussed in section 3.5. In addition, because the simulation is conducted in 2-D space, the

output is not a very accurate representation of physical systems.

First, the displacement equation 3-1 is a good approximation at low strains, but it

appears to lose the accuracy greatly towards higher strain. 2-D Poisson’s ratio of 1 by

definition is for incompressible material, which means the total volume before and after

deformation does not change [76]. This assumption is logical because the aim of this work is

to eventually investigate foaming actions, in which case polymer matrix deforms at melt-like

state. The system area after deformation is not far off from the original size when the applied

strain is relatively low (ɛ ~<0.3), but as the strain increases, the constant area assumption is

not valid. Modified equations 3-2 and 3-4 were used for this reason, but there is still margin

for error.

Another widely disputed point in 2-D line percolation models is that the generated

fibers are essentially line segments with zero thickness value. This is a problem when the

aspect ratio of fibers is concerned, so a thickness value of 1/30 unit was assigned when

33

calculating aspect ratios. To prevent any interference that the assignment of thickness may

cause in the filler intersection process, a tolerance of 1/30 in the fiber interconnection stage,

but this feature tends to overestimate fiber contacts in some situations.

In addition, a fiber network is declared as percolating when it sticks out of the system

boundary, which is not the case in reality. However, this issue can be accounted for through

careful sensitivity analysis.

3.4 3-D Model

3-D percolation model was similarly created based on the Monte Carlo method. The

main difference from the 2-D counterpart is that the 3-D model offers a more realistic

prediction at the cost of computing time. For better time efficiency, the visualization feature

was completely removed. The same 3-step procedure was taken as in the formulation of the

2-D model.

3.4.1 Fiber Generation

Analogous with the 2-D model, many different approaches were practiced to model

different types of filler. A. Behnam et al. used stacks of 2-D stick fibers to model single-

walled carbon nanotube films [78], while particle fillers can simply be modelled as spheres.

The most notable model for carbon nanotube was conducted by M. Foygel et al., who used a

cylindrical shape with a hemisphere capped on each end of the cylinder to describe carbon

nanotube as shown in Fig. 3-6 [77]. This cylindrical 3-D representation was adopted in 3-D

model because it represents a long fiber or carbon nanotube very well and also greatly

simplifies the computation, which is addressed in the next section in more detail.

34

Fig. 3-6. Carbon nanotube and its representation as a capped cylinder [77]

To create a cylinder, a line of length L is first generated from a midpoint that has

random x, y, and z coordinates as in Figure 3-7, and let d represents the diameter of the

cylinder. This line segment is equivalent to the dotted centre line from Figure 3-6. One thing

to note is that another randomized angle β is added along with the z coordinate from the 2-D

model. These parameters are sufficient to describe any line segment in the 3-D space.

Fig. 3-7. Cartesian representation of a long fiber in 3-D

(X1, Y

1, Z

1)

35

3.4.2 Fiber Interconnection

Detection of filler intersection is more complex in the 3-D system and therefore this

process is quite challenging. Fortunately, the capped cylinder model simplifies this process

significantly because the shortest distance from any point on the generated line segment to

the boundary of the cylinder is always constant at d/2. In other words, when two cylinders are

in contact the shortest distance between their corresponding centre lines can never exceed d.

Therefore, if the shortest distance is less than or equal to d, the two cylinders in question are

intersecting. This algorithm is applied each time a new line segment (centre line of new

cylinder) is created.

The shortest distance between two line segments can be calculated using D. Sunday’s

algorithm [79]. A code initially developed by N. Gravish was modified for compatibility with

the Monte Carlo model [80].

3.4.3 Percolation Network Formation

The formation of percolation network is detected in a similar fashion as is done in the

2-D model. Any cluster that crosses the boundary (the opposing faces) of the system is

considered as percolating. The difference is there are three opposing faces in the 3-D model,

as opposed to two in the 2-D model. Each time a new fiber is generated and grouped into its

corresponding cluster through step 1 and 2, the model checks the minimum and maximum x,

y, z values from every cluster. If a cluster exists where the minimum and maximum values of

x, y, or z are out of the system boundaries simultaneously, the loop discontinues. Unless

otherwise stated (directionality analysis in section 5.5), the number of fibers required to

36

formulate the first percolation network is recorded regardless of the direction.

The number of fibers required to form the first percolating network can then be used

to estimate the percolation threshold.

3.4.4 Strain Effect Simulation

A new equation was developed to account for the change of matrix dimensions upon

deformation. In the same way that the 2-D equations were built, the constant volume

deformation mode is assumed since it well describes the matrix deformation for the purpose.

Based on this assumption, the displacement equation in the 3-D system is derived as the

following:

𝑥 = 𝑋

√(1 − 𝛾), y =

𝑌

√(1 − 𝛾), 𝑧 = 𝑍(1 − 𝛾) , (3 − 5)

where (X, Y, Z ) and (x, y, z ) are the initial and final matrix dimensions, respectively. The

strain 𝛾 is defined as the ratio of change in height to the original height, and applied along the

z direction.

The randomly generated angles β and are to undergo a change upon deformation as

well. However, the base angle β is essentially a projection of the fiber segment onto the x-y

plane and is not expected to change when the stress is uniformly applied in the perpendicular

z direction. Rotation of the height angle in the 3-D system can also be developed by

trigonometry:

37

tan 𝜃∗ tan 𝜃⁄ = (1 − 𝛾)32, (3 − 6)

where and * are the initial and final height angle, respectively. For tension simulations,

the same equations can be used with negative values of 𝛾, which is essentially a tensile strain.

This would result in a degree of alignment value greater than 1.

3.4.5 Reliability

3-D Monte Carlo model was established based on a number of fundamental

assumptions that may differ from the physical systems. For instance, in the compression and

tension simulations it was assumed that the load is applied evenly across the entire matrix,

but in fact different stress is applied at different location. As a result, the deformed matrix

cannot actually be the ideal box-shape.

Moreover, the equations 3-5 and 3-6 are only valid when the assumption of constant

volume deformation holds true. Poisson’s ratio of polymer widely varies from one to another

and it is also dependent on the operating temperature. For polymer melts with Poisson’s ratio

close to 0.5 (incompressible), it is reasonable to assume that the initial and final volume of

polymer matrix stays constant.

3.5 General Assumptions and Discussion

In the modelling of 2-D and 3-D Monte Carlo models, some assumptions had to be

made to have a simple and feasible Monte Carlo model. Although most of these assumptions

were reasonable based on the literature, they led to inevitable computational errors.

38

Fillers are assumed to be rigid body fibers of uniform size and shape throughout the

simulations, but this is far from the case in reality. Starting from the manufacturing process

of fillers, differences in size and shape within a tolerated range are present. This type of

simplification can be overcome to an extent in the simulation model by assigning the length

and thickness ranges of fillers through random distribution rather than using constant values.

However, the problem becomes much more complex when the fiber breakage is also

deliberated. When handling polymer matrix composites, fillers may experience excessive

stresses leading to breakage in both compounding and processing stages. This results in a

reduction of aspect ratio, as well as deterioration of size uniformity and may affect the

overall material properties. In fact, A. Ameli et al. claimed that by reducing the shear stress

applied to the fibers through addition of gas in the foam processing of PP-CF composites,

they were able to reduce the fiber breakage phenomenon along with fiber alignment in the

flow direction [27]. They also were able to significantly reduce the fiber breakage as well as

the percolation threshold in the foam processing of polypropylene/stainless-steel fiber

composites through foam injection molding with 3 wt % CO2 as a lubricant. In this

experiment, the main reason for dramatic decrease in the percolation threshold was reduction

of fiber breakage because the steel fibers were much longer than the cell size. Therefore, the

effect of fiber breakage must not be neglected, especially for longer fibers. [28].

Another factor that may influence the fiber shape is the buckling effect. When

compressive strain is introduced in the model, mechanical buckling of fibers is inevitable.

This is more evident in the case of using high aspect ratio fillers, since they are more prone to

mechanical buckling due to their slender structure. Efforts have been made to study its

correlation with the electrical conductivity and H. Hu et al. found out that piezoresistance

39

behaviour of polymer-CNT composites was partially due to the mechanical buckling effect.

To accommodate this, C. Lin et al. designed a Monte Carlo simulation model that generates a

combination of straight and curved fillers as displayed in Figure 3-8 [76]. However, for our

purpose, the focus is on the simulation of compression and tension at elevated temperature of

foaming conditions, where the buckling effect will be a lot more insignificant, and therefore

the assumption of rigid fiber is reasonable.

Fig. 3-8. Geometries of fibers distributed in a matrix. a) before and b) after deformation [76]

In addition, fibers are treated as soft core material, meaning that upon contact they

interpenetrate each other. This assumption was made to simplify the computation but it

causes the system to overestimate the critical concentration, and the exclusion of overlapping

volume of fibers is required. However, M. Foygel et al. argued in their simulation model that

this assumption is in the acceptable range as the margin of error is negligible, especially for

high aspect ratio fibers (less than 1% for a > 100) [77]. Figure 3-9 describes how fibers are

positioned upon contact with each other and the boundary of the polymer matrix.

40

Fig. 3-9. Intersection of interpenetrating fiber with another fiber and system boundary [77]

There are also some limitations with the simulation model that distinguish the

simulation results from those of real life experiments. For instance, this model takes into

account only the geometrical aspect of filler connection, and therefore thermodynamic effects

are neglected.

According to Miyasaka et al. [75], surface energy of the polymer matrix has

significant effect on the critical concentration. They have tested a variety of resins with the

same carbon particles and were able to conclude that the percolation concentration varied

widely for polymers with different level of surface energy. Polymers that have high surface

energy caused isolation in carbon particles due to wetting effect, and therefore the resulting

critical concentration was high. Low surface energy polymers, on the other hand, showed

lower percolation thresholds because carbon particles were segregated. However, the

comparison between fiber interconnectivity of un-pressed and deformed samples is still valid,

since the type of polymer stays the same in each case.

41

Also, the chance of a fiber to be found near the boundaries of the system is less than

in the center regions of the systems, because the midpoints of fibers cannot lie outside the

system boundary. In other words, the chance of detecting a fiber on the edges and the corners

of a 2-D system is only a half, and a quarter, respectively. Although such regions are

relatively small, fibers found in these locations play very crucial roles in the detection of

percolation networks, and therefore may result in overestimation of percolation threshold.

Lastly, the sensitivity of the models (e.g. scale effects, sample size) is potentially the

largest source of error throughout the simulations. Therefore, the first section in each of

Chapter 4 and 5 is dedicated to extensive sensitivity analysis.

42

CHAPTER 4. RESULTS OF 2-D PERCOLATION MODEL AND DISCUSSION

4.1 Overview

2-D Monte Carlo model was used to estimate the critical concentration and generate

simulation graphs under various conditions. Each run yields an output that is equal to the

number fibers required to create the first percolation network. This number is different from

one run to another because the fibers are randomly generated every time. The average

number of fibers required to form a percolation network under a particular condition can thus

be calculated from a set of data obtained by iteration. Then, the average value can be

converted to critical area fraction from dividing the total area occupied by the fibers by the

system area. The program also generates graphs that display the fiber network structure in a

2-D system very clearly.

Before any simulations are conducted, it is very important to find out the optimum

system size through sensitivity analysis. Once these system parameters are defined,

simulations under different types of conditions to assess aspect ratio effect, alignment effect,

and strain effect can be analyzed.

4.2 Sensitivity Analysis

Sensitivity analysis is carried out in this section to find out the effects of changing

system parameters such as system dimensions and the fiber size (aspect ratio remains

constant). The purpose of this section is to figure out the operating system conditions that

will allow error margin in the acceptable range and time-efficiency without compensating the

43

accuracy of the model. One of the main challenges while conducting the simulations is to

accurately predict the percolation threshold in a timely manner. Therefore sensitivity analysis

must be carefully carried out in order to maximize the productivity as well as the credibility

of the model.

4.2.1 Ideal Number of Iterations

Due to the statistical nature of these simulations, one strategy to estimate the critical

concentration with high accuracy is to increase the number of iterations. This strategy assures

that the average value of the iterations lies closer to the true mean value. The main drawback

of this approach is that the time efficiency will be compensated. Therefore, in order to

efficiently estimate the critical concentration of systems with various conditions, it is very

crucial to determine the appropriate sample pool size. Other parameters such as fiber size and

system size were fixed at constant values, while simulations with different pool size were

conducted and repeated 10 times, creating 10 different sample pools. Below are the

simulation conditions that were used in this particular sensitivity analysis.

Simulation Conditions:

Aspect Ratio = 100 (L = 10/3 unit, W = 1/30 unit)

System Dimensions = 50 x 50 unit2

Orientation: Isotropic

Sample Pool Size = 50, 100, 200, 500

44

1 2 3 4 5 6 7 8 9 10

1200

1220

1240

1260

1280

1300

# o

f fib

er

Sample Pool #

Sample Pool Size = 50

Mean = 1232.534

Std. Dev. = 7.705

Fig. 4-1. Simulation statistics when sample pool size = 50

Starting off with a pool size of 50, a mean value of 1232.534 fibers and standard

deviation of 7.705 were calculated as shown in Figure 4-1. The difference between the

minimum and the maximum value was almost as high as 40, which is more than 3% margin

of error. To minimize the error, it was concluded that 50 iterations are not sufficient.

Fig. 4-2. Simulation statistics when sample pool size = 100

1 2 3 4 5 6 7 8 9 10

1200

1220

1240

1260

1280

1300

# o

f fib

er

Sample Pool #

Sample Pool Size = 100

Mean = 1225.185

Std. Dev. = 7.034

45

1 2 3 4 5 6 7 8 9 10

1200

1220

1240

1260

1280

1300

Sample Pool #

# o

f fib

er

Sample Pool Size = 200

Mean = 1229.816

Std. Dev. = 5.884

Fig. 4-3. Simulation statistics when sample pool size = 200

1 2 3 4 5 6 7 8 9 10

1200

1220

1240

1260

1280

1300

Sample Pool Size = 500

Mean = 1229.278

Std. Dev. = 4.274

# o

f fib

er

Sample Pool #

Fig. 4-4. Simulation statistics when sample pool size = 500

Sample pool of 100 and 200 iterations were analysed in a similar fashion, and the

corresponding standard deviation decreased slightly as shown in Figures 4-2 and 4-3. Lastly,

46

500 iterations were carried out and the mean value appeared to be stable and the standard

deviation also dropped to 4.3. Figure 4-4 displays that the difference between the minimum

and the maximum value was less than 15, which is around 1% of the sample mean. Thus, it

was concluded that 500 iterations are sufficient and it was adopted throughout the entire

chapters 4 and 5.

4.2.2 Scale Effect

Another factor that may influence the outcome of simulation significantly is the scale

effect. For instance, given the system dimensions and aspect ratio of filler, let us assume that

the length and the width of the filler are doubled. Theoretically, for a system with infinite

size, the critical fraction must then be the same since the aspect ratio also remains the same.

However, this may not be true if the size of the filler is too big relative to the system.

Changing the system dimensions would bring the same scale effects as tweaking with the

filler size. For a smaller system, number of fillers required to form a percolation network will

be relatively low while the standard deviation of the critical concentration will be high. As

with the sample pool size, increasing the system size or decreasing the fiber size would

apparently result in a more accurate estimation at the cost of computation time. Therefore, it

is essential that the acceptable range of scale is identified, so that with given filler size, the

system dimensions can be determined, or vice versa.

In the 2-D Monte Carlo model, fiber width is a fixed constant, so it is difficult to

change the fiber size without affecting the aspect ratio. Alternatively, system size can be

altered in a way that the ratio of the fiber length to the system size varies, while the fiber size

47

remains the same. To investigate the scale effect, 4 different system dimensions were tested

at 2 different aspect ratios.

Simulation Conditions:

Aspect Ratio = 100 (L = 10/3 unit, W = 1/30 unit), 200 (L= 20/3 unit, W = 1/30 unit)

System Dimensions = 25x 25, 50 x 50, 100 x 100 unit2

Orientation: Isotropic

Table 4-1 shows the statistics of samples of different system dimensions when the

aspect ratio is 100. The criteria were to have less than 1% margin of error at 99% confidence

level, assuming normal distribution of samples. After calculating the confidence interval for

each system, it was concluded that the system size of 50 x 50 unit2 is sufficient when the

fiber length is 10/3. From this result, a rule can be made so that the ratio of system dimension

to the fiber length must be at least 15 in the 2-D system.

Table 4-1. Statistics of Samples with Aspect Ratio = 100

Mean

Threshold (%)

Std. Dev. Minimum

(%)

Median

(%)

Maximum

(%)

Margin of Error

25 x 25 5.34918 0.6553 2.98547 5.29566 7.21489 ± 0.08

50 x 50 5.47574 0.43149 4.07837 5.49336 6.62846 ± 0.05

100 x 100 5.51987 0.2461 4.6859 5.52557 6.21973 ± 0.03

200 x 200 5.56707 0.15106 5.07491 5.56222 5.9329 ± 0.02

48

To confirm this rule, another sensitivity analysis was carried out at aspect ratio of 200,

and all the other parameters were fixed. Table 4-2 displays the statistics of the results, and the

confidence interval was calculated under the same criterion.

Table 4-2. Statistics of Samples with Aspect Ratio = 200

Mean

Threshold (%)

Std. Dev. Minimum

(%)

Median

(%)

Maximum

(%)

Margin of Error

25 x 25 2.61229 0.49282 1.10211 2.5953 4.23069 ±0.06

50 x 50 2.6998 0.33473 1.58206 2.71084 3.71518 ±0.04

100 x 100 2.71888 0.20055 2.04646 2.72862 3.42855 ±0.02

200 x 200 2.76566 0.11805 2.44864 2.7675 3.12135 ±0.01

A margin of error less than 1% was achieved at the system size of 100 x 100 unit2,

which also indicates that the rule is valid.

This rule was applied throughout all the 2-D simulations to minimize the effect of

scaling. One important factor to consider in compression and tension simulations is the

change in system dimensions with applied strain. As a safety factor, this rule was applied

with respect to the smallest system dimension in this type of simulations.

4.3 Poisson’s Ratio

Revisiting the equations 3-1: 𝑥𝑖 = 𝑋𝑖(1 + 𝑣𝛾), 𝑧𝑖 = 𝑍𝑖(1 − 𝛾) , and 3-2:

tan 𝜃∗ tan 𝜃⁄ = (1 − 𝛾) (1 + 𝑣𝛾)⁄ from Chapter 3, it is notable that the change of system

49

dimension and fiber alignment due to strain are heavily dependent on the Poisson’s ratio of

the composite. 2-D Poisson’s ratio of 1 indicates incompressible material [75], whereas 0

theoretically means the system does not change its dimension along the planar direction

when strain is applied perpendicular to it. To thoroughly investigate how this number would

affect the calculated percolation threshold, simulations were conducted at 4 strain levels

using 4 different Poisson’s ratio.

Simulation Conditions:

Aspect Ratio = 300 (L = 10 unit, W = 1/30 unit)

System Dimensions = 100 x 100 unit2 to 0.3 strain

Orientation: Isotropic to 0.3 strain

2-D Poisson’s Ratio = 1, 0.5, 0.3, 0

For clarification, strain values, 𝛾, used throughout Chapter 4 and 5 are defined as the

following:

𝛾 =∆L

L=

𝐿 − 𝑙

𝐿, (4 − 1)

where L and l are the initial and final length of the height, respectively. Degree of alignment

is another parameter defined as tan 𝜃∗ tan 𝜃⁄ , which can be calculated using the equation 3-4.

Degree of alignment of 1 means the system is completely isotropic and alignment in the

direction perpendicular to strain application is introduced as this value goes down, until it

reaches 0 where the fiber will be perfectly aligned in that direction. Table 4-3 and 4-4 list

50

different strain levels for simulation and their corresponding system dimensions as well as

degree of alignment.

Table 4-3. System dimensions at various strains and Poisson’s ratios

ε = 0 ε = 0.1 ε = 0.2 ε = 0.3

v = 1 100 x 100 110 x 90 120 x 80 130 x 70

v = 0.5 100 x 100 105 x 90 110 x 80 115 x 70

v = 0.2 100 x 100 102 x 90 104 x 80 106 x 70

v = 0 100 x 100 100 x 90 100 x 80 100 x 70

Table 4-4. Degree of alignment at various strains and Poisson’s ratios

ε = 0 ε = 0.1 ε = 0.2 ε = 0.3

v = 1 1 0.8181 0.6666 0.5385

v = 0.5 1 0.8571 0.7272 0.6087

v = 0.2 1 0.8824 0.7692 0.6604

v = 0 1 0.9 0.8 0.7

With given operating conditions, 500 iterations were performed to generate data

points for Poisson’s ratio effect on the percolation threshold, and the points were plotted as

shown in Figure 4-5. The figure indicates that as the strain level was increased, the critical

area fraction either increased (v = 1) or decreased (v = 0.5, 0.3, 0).

51

0.00 0.05 0.10 0.15 0.20 0.25 0.30

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

Cri

tica

l F

ractio

n

(%)

Strain

v = 1

v = 0.5

v = 0.3

v = 0

Fig. 4-5. Effect of Poisson’s ratio on percolation threshold under various strains

This result is in good agreement with that of the Monte Carlo simulation conducted

by C. Lin et al. using high aspect ratio fibers displayed in figure 4-6 [76]. Their finding from

this study was that for 2-D Poisson’s ratio of 0.5 or less, the percolation threshold decreased

as strain was applied.

Fig. 4-6. Effect of Poisson’s ratio on percolation threshold under various strains [76]

52

The reason behind this relationship can be explained by looking at the individual

components of Poisson’s ratio effect. Data from Tables 4-3 and 4-4 suggest that v governs

the rate at which system dimensions and degree of alignment change with respect to

increment in strain. At the same strain level, total system area decreases with v, while degree

of alignment increases. Reduction in system size indicates that it will require less number of

fibers for percolation network to be formed. Moreover, increase in degree of alignment also

possibly is responsible for the drop in percolation threshold, and therefore it may play a

significant role in deciding the critical concentration. Further inspection is carried out in

section 4.5 to study the alignment effect.

Another noteworthy point from Table 4-3 is that the equation 3-1 is a good

approximation for low strain values, but significantly drops in its accuracy from ε > 0.3

onwards. For instance, the idea of constant area deformation (v = 1) is no longer valid at

relatively high strain as the system dimensions suggest. Therefore, equations 3-1 and 3-3

have been modified to accommodate low strain deformation modes as well as the high strains

in a way that the constant volume deformation assumption is valid throughout (equations 3-2,

3-4). These modified equations are extensively discussed in Chapter 5.

4.4 Effect of Aspect Ratio

Aspect ratio of filler is another very important parameter that governs the critical

concentration of polymer matrix composites [77]. Figure 4-7 displays sample output graphs

of the 2-D Monte Carlo model at aspect ratio of 100 and 300, and system dimensions of 100

x 100 unit2. Fibers of the same color belong to the same cluster, meaning that they form a

53

fiber network of their own, while the black colored cluster is the first percolation network

formulated in these simulations. These graphs indicate that the lower aspect ratio sample is

much denser than the other one, leading to the prediction that the critical concentration and

the aspect ratio are in an inversely proportional relationship.

Fig. 4-7. Simulation output graphs at different aspect ratio

To find out in more detail the role that aspect ratio plays in determining the critical

concentration, 2-D simulations with 7 different aspect ratios were conducted to generate the

trend.

Sdf

ASPECT RATIO = 300 ASPECT RATIO = 100

54

Simulation Conditions:

Aspect Ratio = 200, 100, 50, 30, 20, 10, 5

System Dimensions = 25x 25 or 50 x 50 or 100 x 100 unit2

Orientation: Isotropic

Figure 4-8 shows the simulation result. The calculated critical area fraction was

2.77%, 5.52%, 10.92%, 18.38%, 27.65%, 55.40%, and 77.41%, at aspect ratio of 200, 100,

50, 30, 20, 10, and 5, respectively. As was expected, the percolation threshold dropped as the

aspect ratio increased. Another interesting finding is that the percolation threshold and the

aspect ratio showed a linear relationship in logarithmic scales, up to a very low aspect ratio.

The result is in general agreement with the well-established trend claimed by many other

researchers [29, 36, 76, 77].

1 10 1001

10

100

Cri

tica

l C

on

. (%

)

Aspect Ratio

Fig. 4-8. Simulation results with various aspect ratios

55

Figure 4-9 explains this trend through a simple schematic. Black and red solid lines

represent a long fiber of length AB and a short fiber of length << AB, respectively.

Assuming that they share the same fiber width and a fiber network path between point A and

B within a polymer matrix exists for both types of fibers, the total fiber length required for

the short fiber system is always greater than that required by the long fiber case. Moreover,

the number of contacts required for this path to form is much larger for the short fiber case,

meaning the chance of network formation is much smaller.

Fig. 4-9. Fiber networks of different type of fiber

As can be observed from Figure 4-8, the critical concentration of 2-D percolation

model is a very sensitive function of the aspect ratio of filler. For this reason, in the more

advanced simulations that are to be discussed in the later sections, effect of aspect ratio is

further investigated.

4.5 Fiber Alignment Effect

As discussed briefly in section 4.3, the fiber orientation of fiber composite affects its

percolation threshold. It is generally accepted that alignment of fibers will increase the

56

percolation threshold as there will be less number of contacts between fibers. This

relationship was shown in a number of experiments involving foaming, where growth of cells

resulted in thinning of the cell walls, exerting extensional stress and causing fibers to re-align

[37-40]. Consequently, electrical conductivity has dropped due to loss of fiber contacts.

Other studies however claimed that the highest electrical conductivity is achieved

when the fibers are slightly aligned rather than fully isotropic [36, 78]. They argued that there

exists an optimum alignment angle at which maximum connectivity occurs because as

initially isotropic nanotubes become more aligned, shorter path and fewer junctions are

required for a conductive percolation network to be formed [78]. Numerous studies have

been completed to explore this area for the sake of maximizing the functional properties at

low filler loading.

Although 3-D simulations are necessary to thoroughly investigate the alignment

effect, 2-D Monte Carlo model can serve its purpose to estimate the trend and provide insight

on the subject.

Simulation Conditions:

Aspect Ratio = 100 (L = 10/3 unit, W = 1/30 unit)

System Dimensions = 100 x 100 unit2

Orientation: Isotropic to 0.7 degree of alignment

Figure 4-10 displays simulation results at 4 different degree of alignment. The decrease in

the number of fibers at 0.9 indicates that maximum connectivity occurs at slightly aligned

configuration, albeit rather small. This finding is in agreement with the statements made by

57

other researchers [36, 78], and it shows promising signs that fiber re-orientation through

foaming can potentially be a valid strategy indeed.

Fig. 4-10. Alignment effect on the percolation threshold

To further investigate the alignment effect, percolation probability plot was generated

in full range of degree of alignment from 1 to 0 in figure 4-11. The figure displays the

probability of detecting a percolation cluster when 4700, 4800, 5000, 5500, and 7000 fibers

were randomly generated at different degree of fiber alignment.

Fig. 4-11. Probability function of percolation at various degree of alignment

1.0 0.9 0.8 0.7

4960

4980

5000

5020

5040

5060

# o

f fib

er

Degree of alignment

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

0.2

0.4

0.6

0.8

1.0

4700

4800

5000

5500

7000

Pe

rco

latio

n p

rob

ab

ility

Degree of alignment

58

The percolation probability plot also suggests that slight alignment of fibers tend to

increase the chance of percolation at various fiber loadings. After the optimum point around

0.9 degree of alignment, further alignment of fibers sharply reduces the number of fiber

contacts. The effect is more profound at low degree of alignment. The simulation data

showed that with 0.1 degree of alignment, even 7000 fibers had 0% chance of forming a

percolation network.

4.6 Strain Effect

It was briefly discussed in section 3.4.4 that application of compressive force results

in two different deformation modes: displacement of fibers and rotation of fibers. The effect

of rotation of fibers was covered in section 4.5, and this section is dedicated to finding out

the combined effect of both fiber displacement and rotation. Revisiting section 3.4.4, the new

position of mid-point of a fiber and fiber angle with respect to the x-axis under strain of 𝛾 can

be calculated by the equations 𝑥 = 𝑋

√(1−𝛾), 𝑦 = 𝑌(1 − 𝛾) , and tan 𝜃∗ tan 𝜃⁄ = (1 − 𝛾)

3

2,

respectively. Simulation conditions were maintained from the alignment effect simulation for

consistency, although direct comparison is difficult because the degree of alignment in this

simulation is dependent on the strain. Also, the system size was doubled in order to minimize

the scaling effect as per the sensitivity analysis.

Simulation Conditions:

Aspect Ratio = 100 (L = 10/3 unit, W = 1/30 unit)

System Dimensions = 200 x 200 unit2

Strain = 0 to 0.9

59

As can be observed from Figure 4-12, the number of fibers required decreases

gradually as higher strain is applied until an optimum point around 𝛾 = 0.6 where

connectivity is maximized. Then it starts to grow sharply, as shown in the figure.

However, this result should only be considered as a reference curve for 3-D model

simulation results. Even though the modified equations may be able to predict the strain

effect in 3-D system, they still violate the constant area deformation assumption. Therefore, it

is premature to assign any physical meaning to these numbers.

0.0 0.2 0.4 0.6 0.8 1.0

15000

16000

17000

18000

19000

20000

# o

f fib

er

Strain

Fig. 4-12. Combined effect of alignment and displacement of fibers on the percolation

threshold

60

CHAPTER 5. RESULTS OF 3-D PERCOLATION MODEL AND DISCUSSION

5.1 Overview

In the previous chapter, effects of different parameters such as aspect ratio, fiber

alignment, and strain were analyzed using 2-D simulations. Aspect ratio showed linear

inversely proportional relationship with the percolation threshold, while fiber alignment and

compressive strain showed positive signs in terms of improving connectivity.

However, these 2-D simulation results are far from what really happens in physical

systems, so a 3-D Monte Carlo model that can better represent the fundamentals of

continuum percolation problems was built upon the 2-D model.

Sensitivity analysis was again carried out first to eliminate any potential error that

may rise from scaling effect. Percolation threshold was then measured for composite systems

with fibers of different aspect ratio and a comparison with the 2-D model simulation results

was made. Effect of fiber alignment was studied, followed by effect of strain through

compressional and tensional modes.

5.2 Sensitivity Analysis

Due to the code structure, 3-D simulations are much more time-consuming so it is

more challenging to find the right balance of time efficiency and accuracy than that for 2-D

model. The importance of sensitivity analysis is thereby highlighted even more in the 3-D

model.

61

In order to ensure that simulations under deformation are also safe from any scaling

effect, a subsection is dedicated to investigate scaling effects when the system undergoes

different types of deformation.

The fiber dimensions of L and W represent the length and width of the cylindrical

shape, respectively, as shown in Figure 5-1.

Fig. 5-1. Representation of 3-D fiber model

The total length of the fiber model is therefore L + W, rather than just L, and this was

considered when calculating the aspect ratio in the following sections.

5.2.1 Scaling Effects without Deformation

Contrary to the 2-D model, length and width of fibers can be easily altered by the user

in the 3-D model. Change of system size is also another way to investigate scale effects.

Simulations using 5 different fiber sizes were conducted first to determine the allowable

range of fiber dimensions.

Simulation Conditions:

Aspect Ratio = 10

62

L x W = (4.5 x 0.5), (9 x 1), (13.5 x 1.5), (22.5 x 2.5), (27 x 3)

System Dimensions = 100 x 100 x 100 unit3

Orientation: Isotropic

Table 5-1 summarizes all the statistics acquired from the simulations with 5 different

fiber sizes. 500 simulation samples were recorded for each fiber size. It is noticeable that the

sample mean stayed quite consistent throughout, but the standard deviation grew rapidly as

the fiber size was increased.

Table 5-1. Scaling Effect in 3-D Simulations

Mean

Threshold (%)

Std. Dev. Margin of Error

L = 4.5 W = 0.5 7.55946 0.15291 ± 0.02

L =9 W = 1 7.56975 0.33407 ± 0.04

L = 13.5 W = 1.5 7.53336 0.50137 ± 0.06

L = 22.5 W = 2.5 7.59 0.89876 ± 0.1

L = 27 W = 3 7.56737 1.05186 ± 0.12

Based on the criterion of maintaining less than 1% margin of error at 99% confidence

level, which is consistent with the Chapter 4 scaling effect analysis, fiber length of 13.5 unit

and fiber width of 1.5 unit were within the tolerable range with margin for error of ± 0.06.

To highlight the importance of reducing standard deviation, a probability density plot

of simulation results with L = 4.5 unit and L = 27 unit was composed in Figure 5-2. As can

63

be observed, calculated critical concentration in each repetition can widely vary when the

fiber size is high, while the values are very consistent in small fiber simulations. Critical

volume fraction when L = 27 unit ranges from 3% to as high as 11%, indicating reliability

can be a huge issue when simulations are conducted with large-sized fibers.

2 3 4 5 6 7 8 9 10 11 120.0

0.1

0.2

0.3

0.4

0.5

L=27 , W=3

L=4.5, W=0.5

Pro

ba

bili

ty d

en

sity

Critical concentration (%)

Fig. 5-2. Probability density plot of critical concentration with low and high fiber size

5.2.2 Scaling Effects with Deformation

One of the main objectives of this research is to find out the effect of compression

and tension on the percolation threshold. Therefore, for simulations involving deformation of

the matrix, an additional analysis must be done to assess the proper fiber size range prior to

64

the simulations, because matrix dimensions vary as a function of degree of deformation and

so does the ratio of system dimensions to the fiber size.

A high strain value (0.6) was chosen to test the scaling effects upon both compression

and tension simulations, with the reason being high strain simulations are more prone to

scaling effect.

Simulation Conditions:

Aspect Ratio = 10

L x W = (4.5 x 0.5), (6 x 0.667), (9 x 1), (13.5 x 1.5), (22.5 x 2.5)

System Dimensions

= 158.11 x 158.11 x 40 unit3 (compression), 79.06 x 79.06 x 160 unit

3 (tension)

Strain: 0.6 (compression), -0.6 (tension)

Table 5-2 lists the compression simulation results when the applied strain is 0.6.

Contrary to the simulations without deformation, the compression simulation results were

highly sensitive to the scaling effect as the significant drop in mean values suggests. Fiber

length of 6 and fiber width of 0.667 yielded reasonable mean value as well as standard

deviation. In other words, any simulation involving strain greater than or equal to 0.6 may

not be reliable if the fiber size exceeds L =6, and W = 0.667.

65

Table 5-2. Scaling Effect in 3-D Simulations with Compression Strain of 0.6

Mean

Threshold (%)

Std. Dev. Margin of Error

L = 4.5 W = 0.5 7.70236 0.25554 ± 0.03

L =6 W = 0.667 7.65026 0.29694 ± 0.03

L =9 W = 1 7.28839 0.49556 ± 0.06

L = 13.5 W = 1.5 6.8758 0.75151 ± 0.09

L = 22.5 W = 2.5 6.25389 1.28908 ± 0.15

In addition to the sensitivity analysis under compression, tensional mode was also

investigated and the results are summarized in Table 5-3. Under the same amount of strain

(0.6), it was revealed that the scaling effect in tension simulations is not as significant as in

the compression condition due to relatively mild decrease in system dimensions. Simulations

with L =9 unit and W = 1 unit were within safe range of mean, and also with reasonable

standard deviation.

Table 5-3. Scaling Effect in 3-D Simulations with Tensile Strain of 0.6

Mean

Threshold (%)

Std. Dev. Margin of Error

L = 4.5 W = 0.5 7.81066 0.17841 ± 0.02

L =6 W = 0.667 7.79243 0.26442 ± 0.03

L =9 W = 1 7.77718 0.38114 ± 0.04

66

L = 13.5 W = 1.5 7.71991 0.59213 ± 0.07

L = 22.5 W = 2.5 7.55256 1.05114 ± 0.12

5.3 Effect of Aspect Ratio

In the previous chapter, it was confirmed that aspect ratio plays a critical role in

determining the percolation threshold in different types of simulations. In section 4.4, the

effect of altering aspect ratio in 2-D system was extensively studied and it is expected that

the trend shown in Figure 4-8 is preserved when transformed into 3-D system. To unveil the

aspect ratio effect in 3-D system, similar approach as in section 4.4 was made using the 3-D

Monte Carlo model with 7 different aspect ratios.

Simulation Conditions:

Aspect Ratio = 5, 10, 20, 30, 50, 100, 200

System Dimensions = 100 x 100 x 100 unit3

Orientation: Isotropic

3-D simulations with no strain and aspect ratio of 5, 10, 20, 30, 50, 100, and 200 were

conducted and the results are plotted in Figure 5-3.

67

1 10 100

0.1

1

10

100

Cri

tica

l co

n.

(%)

Aspect ratio

Fig. 5-3. 3-D Simulation results with various aspect ratios

Critical volume fractions of 0.276%, 0.599%, 1.27%, 2.23%, 3.60%, 7.56%, and

14.73% were estimated at aspect ratios of 200, 100, 50, 30, 20, 10, and 5, respectively. The

same trend with the 2-D results was observed in the 3-D simulation results, where the critical

concentration dropped linearly with growing aspect ratio in logarithmic scales. The plot is

also in excellent agreement with the results from the Monte Carlo model developed by M.

Foygel et al. [77], where they claimed the critical volume fraction at aspect ratio of 10 and

100 are around 7.5% and 0.6%, respectively. Figure 5-4 illustrates their critical fractional

volume results with respect to a wide range of aspect ratios.

68

Fig. 5-4. Critical concentration for the 3D systems of randomly oriented soft-core sticks [77]

Furthermore, critical concentrations at varying aspect ratio of 2-D and 3-D system

were plotted together for comparison in Figure 5-5. It was observed that critical

concentrations of the 2-D model are always around an order of magnitude larger than those

of the 3-D model. They also both exhibit a linear logarithmic relationship from aspect ratio of

200 down to 5. Although 3-D model results are better representation of the physical models,

it can be concluded that the 2-D model can also estimate the trend very well.

Fig. 5-5. 2-D and 3-D simulation results with various aspect ratios

1 10 100

0.1

1

10

100

Cri

tica

l co

n.

(%)

Aspect ratio

3 - D

2 - D

69

5.4 Fiber Alignment Effect

In section 4.5, how orientation of fibers can affect fiber networks and the percolation

threshold was discussed in 2-D systems. Similar pattern is expected in 3-D systems as the

same ideas are applied. Degree of orientation of fibers is calculated by the same formula that

was used in 2-D systems. 6 strains were tested with constant aspect ratio and system size.

Simulation Conditions:

Aspect Ratio = 10 (L = 9 unit, W = 1 unit)

System Dimensions = 100 x 100 x 100 unit3

Orientation: Isotropic to 0.4 degree of alignment

Figure 5-6 illustrates the fiber alignment effect on the percolation threshold when

aspect ratio is 10. The decrease in the critical concentration can be observed from 1 to 0.6

degree of alignment, albeit rather small. Compared to the 2-D system, the optimum strain at

which connectivity is maximized occurs at a different location, but direct comparison is

unsuitable because the simulation conditions differ.

Fig. 5-6. Alignment effect on the percolation threshold

1.0 0.9 0.8 0.7 0.6 0.5 0.4

9850

9900

9950

10000

10050

10100

10150

10200

10250

# o

f fib

er

Degree of alignment

70

5.5 Percolation Direction

The effect of fiber alignment on the critical concentration is further investigated in

this section. The range of degree of alignment is extended to values greater than 1 to account

for the fiber re-orientation under both compressive and tensile conditions.

Another factor that is considered in this section is the direction of measurement. The

percolation threshold values at various degree of fiber alignment from 0.4 to 1.6 were

estimated in the planar (y) and the vertical (z) directions separately (refer to Figure 3-7 for the

definition of 3-D coordinate system). Degree of alignment values less than 1 indicates that

fibers are aligned perpendicular to the vertical direction, whereas those greater than 1 point to

alignment in the vertical direction.

Prior to the simulations, Table 5-4 was produced to clarify the physical meaning of

degree of alignment. Positive values of strain refer to the compressive strain in the vertical

direction, and negative values are for the tensile simulations.

Table 5-4. Degree of alignment values and their corresponding * and strain

Degree of alignment 0.4 0.6 0.8 1 1.2 1.4 1.6

* when = 45°

Strain 0.4572 0.2886 0.1382 0 -0.1292 -0.2515 -0.3680

Simulation Conditions:

Aspect Ratio = 10 (L = 6 unit, W = 0.667 unit)

71

System Dimensions = 100 x 100 x 100 unit3

Orientation: 0.4 to 1.6 degree of alignment

Direction of measurement: y, z

The percolation model was first modified to detect percolation network in the y

direction only. The quadrant 2 of Figure 5-7 suggests that as the fibers are aligned along the

measurement direction, there exists an optimum point that yields the lowest percolation

threshold. This trend is similar to that shown in Figure 5-6. As the fibers are forced to align

perpendicular to the measurement direction however (quadrant 1), percolation threshold

exhibited monatomic increase.

Fig. 5-7. Alignment effect on the percolation threshold in the planar direction

Following the measurement in y direction, the percolation model was then modified

to identify only z direction percolation networks. Figure 5-8 proposes that the same

relationship can be claimed as the previous case, where the percolation threshold is minimal

72

when fibers are slightly aligned in the measurement direction. This trend is more prominent

in the y direction measurements, and the optimum point occurs at higher alignment of fibers.

Fig. 5-8. Alignment effect on the percolation threshold in the vertical direction

5.6 Strain Effect

The effect of fiber alignment in 3-D systems was extensively studied in the previous

sections. Under both compressive and tensile condition, combined effect of fiber alignment

and fiber displacement needs to be addressed. In both tensile and compression simulations,

the coordinates of midpoint of fibers after deformation can be calculated using the equation

3-3: 𝑥 = 𝑋

√(1−𝛾), y =

𝑌

√(1−𝛾), 𝑧 = 𝑍(1 − 𝛾). From the initial height angle θ, a new height

angle can also be calculated using the equation 3-4: tan 𝜃∗ tan 𝜃⁄ = (1 − 𝛾)3

2 . For

comparison analysis between the two modes of deformation, consistent simulation conditions

were used in the next two sections with low, medium, and high aspect ratios. The strain 𝛾

73

was again defined as 𝛾 =∆L

L=

𝐿−𝑙

𝐿, where L and l represent the initial and the final length of

system dimension.

5.6.1 Compression Effect

The simulation methodology in this section is equivalent to that in section 4.6.

Simulation Conditions:

Aspect Ratio = 10, 30, 100

System Dimensions = 100 x 100 x 100 unit3

Strain = 0 to 0.6

Figure 5-9, 5-10, and 5-11 display compression simulations in 3-D system at aspect

ratio of 10, 30, and 100, respectively. Fiber size was determined as per the sensitivity

analysis criterion to eliminate potential errors. In all three plots, the critical concentration

slightly decreases until it reaches an optimum strain. After the optimum point, the critical

concentration starts to increase as higher strain is applied. This behavior is to be discussed in

section 5.5.3.

Fig. 5-9. Compression effect on the percolation threshold at low aspect ratio (a = 10)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

7.50

7.52

7.54

7.56

7.58

7.60

7.62

7.64

7.66

7.68

7.70

7.72

Cri

tica

l co

nc.

(%)

Strain

74

Fig. 5-10. Compression effect on the percolation threshold at medium aspect ratio (a = 30)

Fig. 5-11. Compression effect on the percolation threshold at high aspect ratio (a = 100)

5.6.2 Tensile Effect

Because foaming actions are not only limited to compression effect, tensile

simulations were also conducted to better comprehend their effect on critical concentration.

Same simulation conditions as in the previous section were applied for comparison.

Simulation Conditions:

0.0 0.1 0.2 0.3 0.4 0.5 0.6

2.37

2.38

2.39

2.40

2.41

2.42

2.43

2.44

Cri

tica

l co

nc.

(%)

Strain

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.590

0.592

0.594

0.596

0.598

0.600

Cri

tica

l co

nc.

(%)

Strain

75

Aspect Ratio = 10, 30, 100

System Dimensions = 100 x 100 x 100 unit3

Strain = 0 to 0.6

Figure 5-12, 5-13, and 5-14 display the tension simulations in 3-D system at aspect

ratio of 10, 30, and 100, respectively. Sensitivity analysis data on tensile mode from section

5.2.2 was used to determine the fiber sizes.

0.0 0.1 0.2 0.3 0.4 0.5 0.6

7.55

7.60

7.65

7.70

7.75

7.80

7.85

Cri

tica

l co

nc.

(%)

Strain

Fig. 5-12. Tensile effect on the percolation threshold at low aspect ratio (a = 10)

76

0.0 0.1 0.2 0.3 0.4 0.5 0.6

2.38

2.40

2.42

2.44

2.46

2.48

Cri

tica

l co

nc (

%)

Strain

Fig. 5-13. Tensile effect on the percolation threshold at medium aspect ratio (a = 30)

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.600

0.605

0.610

0.615

0.620

Cri

tica

l co

nc.

(%)

Strain

Fig. 5-14. Tensile effect on the percolation threshold at high aspect ratio (a =100)

As can be observed from the tension simulation plots, the critical concentration

gradually increases without achieving a minimum critical concentration point, indicating that

initial point is the optimum point in terms of connectivity. Although compression and tension

77

effects both appear to increase the percolation threshold at high strain, their behaviour at

lower strain is quite different.

5.6.3 Comparison Analysis

Results from section 5.6.1 and 5.6.2 were combined to generate a plot (Fig. 5-15) that

enables comparison analysis. Data on the right side of dotted line represent compression

effects and data on the left are obtained from tensile simulations. One intriguing phenomenon

observed from the plot is that the optimum connectivity through compression occurs at

different strain for systems with different aspect ratio. As the aspect ratio increases, the

optimum point tends to shift to the right, i.e., higher strains. However, simulations must also

be repeated for smaller fiber size to generalize this statement. Also, more data points at other

strains are also required to find out the exact optimum point.

Fig. 5-15. Strain effect on the percolation threshold at various aspect ratios

78

CHAPTER 6. CONCLUSION AND FUTURE WORK

6.1 Conclusion

Monte Carlo models in 2-D system were initially designed to estimate the percolation

threshold of polymer matrix composites. The validity of the model was substantiated through

comparison with similar studies conducted by other researchers. The model was then

expanded to simulate system under compression. The effects of numerous parameters such as

strain, degree of alignment, aspect ratio of fiber, and Poisson’s ratio on the percolation

threshold were analysed with simulation results from this 2-D model. 3-D Monte Carlo

model was also built based on the 2-D model to conduct similar analysis and the effect of

tension was also addressed. The results in both 2-D and 3-D system indicated that aspect

ratio and percolation threshold show inverse proportional linear logarithmic relationship,

with an order of magnitude difference between the results from two systems. Reducing the

degree of orientation of fibers resulted in slight decrease of percolation threshold, but further

alignment of fibers caused the percolation threshold to increase sharply. Tensile simulations

led to monatomic increase in the percolation threshold, whereas compression simulations

exhibited optimum strains where the percolation threshold was the minimum.

6.2 Future Work

Although the 2-D simulation results and part of 3-D simulation results are in excellent

agreement with those from other studies, strain analysis data from the 3-D simulation model

needs further verification because it is very sensitive to the scale effect. Other

recommendations for future work include:

79

i. Debug or modify the current 3-D model so that the simulation time is further reduced.

This will allow 3-D strain analysis of higher aspect ratio fibers (a >>100) without

major scale effect.

ii. Conduct simulations at strain values of smaller interval in order to find out the precise

optimum strain for various aspect ratios.

iii. Develop a new module to estimate percolative properties (e.g., electrical conductivity)

based on the calculated percolation threshold.

iv. Expand the model to accommodate mixture of fibers in a system, in a way that

randomly generated fibers may have different size or shape.

80

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87

Appendices

2-D Model MATLAB Code

% set(0,'DefaultFigureWindowStyle','docked') % rng(1); % set reset random seed to a fixed

clc clear all; close all;

%%%%%%%%%%%%% % set values fibreNum = 50000; % number of fibres fibreLen = 1; % 1/2 fibre length boundsX = [0, 100]; % X axis limits boundsY = [0, 100]; % Y axis limits %%%%%%%%%%%%% %% DO NOT MODIFY %% %pts = [x y theta groupNum x1 y1 x2 y2, A, B] X = 1; Y = 2; THETA = 3; GROUPNUM = 4; X1 = 5; Y1 = 6; X2 = 7; Y2 = 8; A = 9; B = 10; GRPNUM= 1; GRPX1 = 2; % x max GRPX2 = 3; GRPY1 = 4; % y max GRPY2 = 5; filename = ['test_' num2str(fibreLen) '.xlsx']; fibreTol = fibreLen * 3; % 2 * sqrt(2) %DO NOT MODIFY %%%%%%%%%%%%%%%%%%%%%%% %% for iteration=1:500 tic solution = ''; pts = rand([3 fibreNum ])' .* repmat([boundsX(2) boundsY(2)

pi],[fibreNum, 1]); %pts(:,THETA) = atan(tan(pts(:,THETA))*0.5857); %for strain

experiments only %disp(pts(:,THETA)); pts(:,GROUPNUM:B) = 0; % initialize matrix pts(:,X1) = pts(:,X) + cos(pts(:,THETA))*fibreLen; pts(:,X2) = pts(:,X) - cos(pts(:,THETA))*fibreLen; pts(:,Y1) = pts(:,Y) + sin(pts(:,THETA))*fibreLen; pts(:,Y2) = pts(:,Y) - sin(pts(:,THETA))*fibreLen;

88

%calculate y = Ax + B pts(:,A) = (pts(:,Y2)-pts(:,Y1))./(pts(:,X2)-pts(:,X1)); pts(:,B) = pts(:,Y2) - pts(:,A).*pts(:,X2); %%%%%%%%% Track limits of groups groupCount = 0; groupLimits = zeros(fibreNum, 5); %% % tic for i=1:fibreNum inRange = find(and(and(pts(1:i,X) - fibreTol < pts(i,X) ,

pts(1:i,X) + fibreTol > pts(i,X)) , and(pts(1:i,Y) - fibreTol < pts(i,Y) ,

pts(1:i,Y) + fibreTol> pts(i,Y))));

if inRange == i % tested point lies outside all groups,

add another group groupCount = groupCount + 1; pts(i, GROUPNUM) = groupCount; groupLimits(groupCount,GRPNUM) = groupCount; groupLimits(groupCount,GRPX1) = max(pts(i, [X1 X2])); groupLimits(groupCount,GRPX2) = min(pts(i, [X1 X2])); groupLimits(groupCount,GRPY1) = max(pts(i, [Y1 Y2])); groupLimits(groupCount,GRPY2) = min(pts(i, [Y1 Y2])); else % possible points found within range inRange(inRange == i) = []; %remove the current point (i) from

inRange %y = a1x+b1=a2x+b2 xInter = (pts(inRange,B)- pts(i,B))./(pts(i,A)-

pts(inRange,A)); yInter = pts(i,A)*xInter + pts(i,B); %distances for i to all Intercepts iPos = [xInter - pts(i,X), yInter - pts(i,Y)]; iDist = bsxfun(@hypot, iPos(:,1), iPos(:,2)); %distances for other fibres to Intercepts relPos = [xInter - pts(inRange,X), yInter - pts(inRange,Y)]; relDist = bsxfun(@hypot, relPos(:,1), relPos(:,2)); %fibres connecting with i fibresConnect = inRange(and(iDist < fibreLen, relDist <

fibreLen)); [m n] = size(fibresConnect); if m == 0 groupCount = groupCount + 1; pts(i, GROUPNUM) = groupCount;

groupLimits(groupCount,GRPNUM) = groupCount; groupLimits(groupCount,GRPX1) = max(pts(i, [X1 X2])); groupLimits(groupCount,GRPX2) = min(pts(i, [X1 X2])); groupLimits(groupCount,GRPY1) = max(pts(i, [Y1 Y2])); groupLimits(groupCount,GRPY2) = min(pts(i, [Y1 Y2])); else for j = 1:m if j == 1 %add point to first group groupNumber = pts(fibresConnect(1), GROUPNUM); pts(i, GROUPNUM) = groupNumber;

89

groupLimits(groupNumber,GRPX1) = max([pts(i, [X1

X2]), groupLimits(groupNumber,GRPX1)]); groupLimits(groupNumber,GRPX2) = min([pts(i, [X1

X2]), groupLimits(groupNumber,GRPX2)]); groupLimits(groupNumber,GRPY1) = max([pts(i, [Y1

Y2]), groupLimits(groupNumber,GRPY1)]); groupLimits(groupNumber,GRPY2) = min([pts(i, [Y1

Y2]), groupLimits(groupNumber,GRPY2)]); else %bridge groups bridgeGroup = pts(fibresConnect(j), GROUPNUM); groupLimits(groupNumber,GRPX1) =

max([groupLimits(groupNumber,GRPX1), groupLimits(bridgeGroup,GRPX1)]); groupLimits(groupNumber,GRPX2) =

min([groupLimits(groupNumber,GRPX2), groupLimits(bridgeGroup,GRPX2)]); groupLimits(groupNumber,GRPY1) =

max([groupLimits(groupNumber,GRPY1), groupLimits(bridgeGroup,GRPY1)]); groupLimits(groupNumber,GRPY2) =

min([groupLimits(groupNumber,GRPY2), groupLimits(bridgeGroup,GRPY2)]);

pts(pts(:,GROUPNUM) == bridgeGroup,GROUPNUM) =

pts(fibresConnect(1), GROUPNUM);

if (bridgeGroup ~= pts(fibresConnect(1), GROUPNUM)) groupLimits(bridgeGroup,:) = 0; % erase old

group end end end end end %condition for formation of percolation network exceedX = find((groupLimits(1:groupCount,GRPX1) > boundsX(2) &

groupLimits(1:groupCount,GRPX2) < boundsX(1) )); exceedY = find((groupLimits(1:groupCount,GRPY1) > boundsY(2) &

groupLimits(1:groupCount,GRPY2) < boundsY(1) )); if ~isempty(exceedX) solution = 'X'; elseif ~isempty(exceedY) solution = 'Y'; end

if solution == 'X' | solution == 'Y' disp('*************'); display([solution ' ' num2str(i)])

numRow = ['A',num2str(iteration)]; xyRow = ['B',num2str(iteration)]; xlswrite(filename,i,'Sheet1',numRow); xlswrite(filename,solution, 'Sheet1',xyRow);

break; end

end

90

% toc %%

2-D Visualization Module % tic % figure % hold on % xlim([boundsX(1) - 2*fibreLen, boundsX(2) + 2*fibreLen ]) % ylim([boundsY(1) - 2*fibreLen, boundsY(2) + 2*fibreLen ]) % axis equal % [m n] = size(pts); % groups = unique(pts(:,GROUPNUM)); % % cMap = colormap(hsv); % % if groups(1) == 0 % groups(1) = []; % end % % for i = 1:size(groups) % display groups with same color % groupMembers = pts(:,GROUPNUM) == groups(i); % line([pts(groupMembers,X1) pts(groupMembers,X2)]',

[pts(groupMembers,Y1), pts(groupMembers,Y2)]', 'Marker', 'none', 'Color',

cMap(mod(i-1,64)+1,:)); % % pause(1) % end % % if solution > 0 % groupMembers = pts(:,GROUPNUM) == solution; % line([pts(groupMembers,X1) pts(groupMembers,X2)]',

[pts(groupMembers,Y1), pts(groupMembers,Y2)]', 'Marker', 'none', 'Color',

'k', 'LineWidth', 1.5); % end % % rectangle('Position',[boundsX(1),boundsY(1),boundsX(2) -

boundsX(1) ,boundsY(2) - boundsY(1)]); % toc %% draw only large groups % % figure % hold on

% xlim([boundsX(1) - 2*fibreLen, boundsX(2) + 2*fibreLen ]) % ylim([boundsY(1) - 2*fibreLen, boundsY(2) + 2*fibreLen ]) % axis equal % [m n] = size(pts); % groups = unique(pts(:,GROUPNUM)); % % cMap = colormap(lines); % % scatter([pts(:,X) ], [pts(:,Y)], 'Marker', 'o', 'MarkerEdgeColor',

cMap(1,:)); % % if groups(1) == 0 % groups(1) = []; % end

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% % colourCount = 0; % for i = 1:size(groups) % display groups with same color % groupMembers = pts(:,GROUPNUM) == groups(i); % if sum(groupMembers) > 100 % colourCount = colourCount +1; % line([pts(groupMembers,X1) pts(groupMembers,X2)]',

[pts(groupMembers,Y1), pts(groupMembers,Y2)]', 'Marker', 'none', 'Color',

cMap(mod(colourCount-1,64)+1,:)); % end % % pause(1) % end % % rectangle('Position',[boundsX(1),boundsY(1),boundsX(2) -

boundsX(1) ,boundsY(2) - boundsY(1)]);

%% double check the answer % tic % groups = unique(pts(:,GROUPNUM)); % if groups(1) == 0 % groups(1) = []; % end % lims = zeros(size(groups,1),4); % for i = 1:size(groups) % display groups with same color % groupMembers = pts(:,GROUPNUM) == groups(i); % lims(i,1) = max(max(pts(groupMembers, [X1 X2]))); % lims(i,2) = min(min(pts(groupMembers, [X1 X2]))); % lims(i,3) = max(max(pts(groupMembers, [Y1 Y2]))); % lims(i,4) = min(min(pts(groupMembers, [Y1 Y2]))); % % if lims(i,1) > 100 && lims(i,2) < 0 % display(['X ' num2str(i) ' ' num2str(groups(i)) ' '

num2str(lims(i,1)) ' ' num2str(lims(i,2))]) % % numRow = ['A',num2str(iteration)]; % xyRow = ['B',num2str(iteration)]; % xlswrite(filename,fibre,'Sheet1'numRow); % xlswrite(filename,'X', 'Sheet1', xyRow); % elseif lims(i,3) > 100 && lims(i,4) < 0 % display(['Y ' num2str(i) ' ' num2str(groups(i)) ' '

num2str(lims(i,3)) ' ' num2str(lims(i,4))]) % % numRow = ['A',num2str(iteration)]; % xyRow = ['B',num2str(iteration)]; % xlswrite(filename,fibre,'Sheet1',numRow); % xlswrite(filename,'Y','Sheet1',xyRow); % end % end

toc end

92

3-D Model MATLAB Code

% set(0,'DefaultFigureWindowStyle','docked') % rng(1); % set reset random seed to a fixed

clc clear all; close all; %%%%%%%%%%%%% % set values fibreNum = 50000; % number of fibres fibreLen = 2; % 1/2 fibre length fibreWid = 0.1; boundsX = [0, 100]; % X axis limits boundsY = [0, 100]; boundsZ = [0, 100]; % Y axis limits %%%%%%%%%%%%% %% DO NOT MODIFY %% %pts = [x y z alpha theta groupNum x1 y1 z1 x2 y2 z2] X = 1; Y = 2; Z = 3; ALPHA = 4; THETA = 5; GROUPNUM = 6; X1 = 7; Y1 = 8; Z1 = 9; X2 = 10; Y2 = 11; Z2 = 12;

GRPNUM= 1; GRPX1 = 2; % x max GRPX2 = 3; GRPY1 = 4; % y max GRPY2 = 5; GRPZ1 = 6; GRPZ2 = 7;

filename = ['test.xlsx']; fibreTol = fibreLen * 3; % 2 * sqrt(2)

for iteration=1:500 tic solution = '';

pts = rand([5 fibreNum ])' .* repmat([boundsX(2) boundsY(2) boundsZ(2)

pi pi],[fibreNum, 1]); %disp(pts);

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pts(:,THETA) = atan(tan(pts(:,THETA))*1.3145); %for strain

experiments only

pts(:,GROUPNUM:Z2) = 0; % initialize matrix pts(:,X1) = pts(:,X) + fibreLen./sqrt(1+tan(pts(:,ALPHA)).^2 +

tan(pts(:,THETA)).^2); pts(:,X2) = pts(:,X) - fibreLen./sqrt(1+tan(pts(:,ALPHA)).^2 +

tan(pts(:,THETA)).^2); pts(:,Y1) = pts(:,Y) + (fibreLen./sqrt(1+tan(pts(:,ALPHA)).^2 +

tan(pts(:,THETA)).^2)).*tan(pts(:,ALPHA)); pts(:,Y2) = pts(:,Y) - (fibreLen./sqrt(1+tan(pts(:,ALPHA)).^2 +

tan(pts(:,THETA)).^2)).*tan(pts(:,ALPHA)); pts(:,Z1) = pts(:,Z) + (fibreLen./sqrt(1+tan(pts(:,ALPHA)).^2 +

tan(pts(:,THETA)).^2)).*tan(pts(:,THETA)); pts(:,Z2) = pts(:,Z) - (fibreLen./sqrt(1+tan(pts(:,ALPHA)).^2 +

tan(pts(:,THETA)).^2)).*tan(pts(:,THETA));

%%%%%%%%% Track limits of groups groupCount = 0; groupLimits = zeros(fibreNum, 7);

%% % tic

for i=1:fibreNum fibresConnect = []; inRange = find(and(pts(1:i,X) - fibreTol < pts(i,X) , pts(1:i,X) +

fibreTol > pts(i,X)) & and(pts(1:i,Y) - fibreTol < pts(i,Y) , pts(1:i,Y) +

fibreTol> pts(i,Y)) & and(pts(1:i,Z) - fibreTol < pts(i,Z) , pts(1:i,Z) +

fibreTol > pts(i,Z)));

if inRange == i % tested point lies outside all groups,

add another group groupCount = groupCount + 1; pts(i, GROUPNUM) = groupCount;

groupLimits(groupCount,GRPNUM) = groupCount; groupLimits(groupCount,GRPX1) = max(pts(i, [X1 X2])); groupLimits(groupCount,GRPX2) = min(pts(i, [X1 X2])); groupLimits(groupCount,GRPY1) = max(pts(i, [Y1 Y2])); groupLimits(groupCount,GRPY2) = min(pts(i, [Y1 Y2])); groupLimits(groupCount,GRPZ1) = max(pts(i, [Z1 Z2])); groupLimits(groupCount,GRPZ2) = min(pts(i, [Z1 Z2]));

else % possible points found within range inRange(inRange == i) = []; %remove the current point (i) from

inRange %y = a1x+b1=a2x+b2

p1 = repmat([pts(i,X1) pts(i,Y1) pts(i,Z1)],size(inRange,1),1); p2 = repmat([pts(i,X2) pts(i,Y2) pts(i,Z2)],size(inRange,1),1);

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%distances for i to all Intercepts p3 = [pts(inRange,X1) pts(inRange,Y1) pts(inRange,Z1)]; p4 = [pts(inRange,X2) pts(inRange,Y2) pts(inRange,Z2)];

%disp(p1); %disp(p2); %disp(p3); %disp(p4); distance = DistBetween2Segment(p1,p2,p3,p4); fibresConnect = inRange(distance < fibreWid); %fibresConnect = fibresConnect + inRange(distance < fibreWid);

[m n] = size(fibresConnect);

if m == 0 groupCount = groupCount + 1; pts(i, GROUPNUM) = groupCount;

groupLimits(groupCount,GRPNUM) = groupCount; groupLimits(groupCount,GRPX1) = max(pts(i, [X1 X2])); groupLimits(groupCount,GRPX2) = min(pts(i, [X1 X2])); groupLimits(groupCount,GRPY1) = max(pts(i, [Y1 Y2])); groupLimits(groupCount,GRPY2) = min(pts(i, [Y1 Y2])); groupLimits(groupCount,GRPZ1) = max(pts(i, [Z1 Z2])); groupLimits(groupCount,GRPZ2) = min(pts(i, [Z1 Z2]));

else for j = 1:m if j == 1 %add point to first group groupNumber = pts(fibresConnect(1), GROUPNUM); pts(i, GROUPNUM) = groupNumber;

groupLimits(groupNumber,GRPX1) = max([pts(i, [X1

X2]), groupLimits(groupNumber,GRPX1)]); groupLimits(groupNumber,GRPX2) = min([pts(i, [X1

X2]), groupLimits(groupNumber,GRPX2)]); groupLimits(groupNumber,GRPY1) = max([pts(i, [Y1

Y2]), groupLimits(groupNumber,GRPY1)]); groupLimits(groupNumber,GRPY2) = min([pts(i, [Y1

Y2]), groupLimits(groupNumber,GRPY2)]); groupLimits(groupNumber,GRPZ1) = max([pts(i, [Z1

Z2]), groupLimits(groupNumber,GRPZ1)]); groupLimits(groupNumber,GRPZ2) = min([pts(i, [Z1

Z2]), groupLimits(groupNumber,GRPZ2)]);

else %bridge groups bridgeGroup = pts(fibresConnect(j), GROUPNUM);

groupLimits(groupNumber,GRPX1) =

max([groupLimits(groupNumber,GRPX1), groupLimits(bridgeGroup,GRPX1)]);

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groupLimits(groupNumber,GRPX2) =

min([groupLimits(groupNumber,GRPX2), groupLimits(bridgeGroup,GRPX2)]); groupLimits(groupNumber,GRPY1) =

max([groupLimits(groupNumber,GRPY1), groupLimits(bridgeGroup,GRPY1)]); groupLimits(groupNumber,GRPY2) =

min([groupLimits(groupNumber,GRPY2), groupLimits(bridgeGroup,GRPY2)]); groupLimits(groupNumber,GRPZ1) =

max([groupLimits(groupNumber,GRPZ1), groupLimits(bridgeGroup,GRPZ1)]); groupLimits(groupNumber,GRPZ2) =

min([groupLimits(groupNumber,GRPZ2), groupLimits(bridgeGroup,GRPZ2)]);

pts(pts(:,GROUPNUM) == bridgeGroup,GROUPNUM) =

pts(fibresConnect(1), GROUPNUM);

if (bridgeGroup ~= pts(fibresConnect(1), GROUPNUM)) groupLimits(bridgeGroup,:) = 0; % erase old

group end end end end end

%condition for formation of percolation network exceedX = find((groupLimits(1:groupCount,GRPX1) > boundsX(2) &

groupLimits(1:groupCount,GRPX2) < boundsX(1) )); exceedY = find((groupLimits(1:groupCount,GRPY1) > boundsY(2) &

groupLimits(1:groupCount,GRPY2) < boundsY(1) )); exceedZ = find((groupLimits(1:groupCount,GRPZ1) > boundsZ(2) &

groupLimits(1:groupCount,GRPZ2) < boundsZ(1) )); if ~isempty(exceedX) solution = 'X'; elseif ~isempty(exceedY) solution = 'Y'; elseif ~isempty(exceedZ) solution = 'Z'; end

if solution == 'X' | solution == 'Y' | solution == 'Z' disp('*************'); display([solution ' ' num2str(i)]);

numRow = ['A',num2str(iteration)]; xyzRow = ['B',num2str(iteration)]; xlswrite(filename,i,'Sheet1',numRow); xlswrite(filename,solution, 'Sheet1',xyzRow);

%disp(i); %disp(solution); break; end

end

96

Distance Between 2 Line Segments Module

%Copyright (c) 2011, Gravish %All rights reserved. % Modified by Eunse Chang

%Redistribution and use in source and binary forms, with or without %modification, are permitted provided that the following conditions are %met:

% * Redistributions of source code must retain the above copyright % notice, this list of conditions and the following disclaimer. % * Redistributions in binary form must reproduce the above copyright % notice, this list of conditions and the following disclaimer in % the documentation and/or other materials provided with the

distribution

%THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS

IS" %AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE %IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR

PURPOSE %ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE %LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR %CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF %SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS %INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN %CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) %ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF

THE %POSSIBILITY OF SUCH DAMAGE.

% Computes the minimum distance between two line segments. Code % is adapted for Matlab from Dan Sunday's Geometry Algorithms originally % written in C++ %

http://softsurfer.com/Archive/algorithm_0106/algorithm_0106.htm#dist3D_Seg

ment_to_Segment

% Usage: Input the start and end x,y,z coordinates for two line segments. % p1, p2 are [x,y,z] coordinates of first line segment and p3,p4 are for % second line segment.

% Output: scalar minimum distance between the two segments.

% Example: % P1 = [0 0 0]; P2 = [1 0 0]; % P3 = [0 1 0]; P4 = [1 1 0]; % dist = DistBetween2Segment(P1, P2, P3, P4) % dist =

97

% % 1 % function [distance varargout] = DistBetween2Segment(p1, p2, p3, p4) distance = zeros(size(p1,1),1); sN = zeros(size(p1,1),1); tN = zeros(size(p1,1),1); sc = zeros(size(p1,1),1); tc = zeros(size(p1,1),1); dP = zeros(size(p1,1),3); u = p1 - p2; v = p3 - p4; w = p2 - p4;

a = dot(u,u,2); b = dot(u,v,2); c = dot(v,v,2); d = dot(u,w,2); e = dot(v,w,2);

D = a.*c - b.*b; % ac -b^2 sD = D; tD = D;

SMALL_NUM = 0.00000001;

% compute the line parameters of the two closest points for n=1:size(p1,1); if (D(n,1) < SMALL_NUM) % the lines are almost parallel sN(n,1) = 0.0; % force using point P0 on segment S1 sD(n,1) = 1.0; % to prevent possible division by 0.0 later tN(n,1) = e(n,1); tD(n,1) = c(n,1); else % get the closest points on the infinite lines sN(n,1) = (b(n,1)*e(n,1) - c(n,1)*d(n,1)); tN(n,1) = (a(n,1)*e(n,1) - b(n,1)*d(n,1)); if (sN(n,1) < 0.0) % sc < 0 => the s=0 edge is visible sN(n,1) = 0.0; tN(n,1) = e(n,1); tD(n,1) = c(n,1); elseif (sN(n,1) > sD(n,1))% sc > 1 => the s=1 edge is visible sN(n,1) = sD(n,1); tN(n,1) = e(n,1) + b(n,1); tD(n,1) = c(n,1); end end

if (tN(n,1) < 0.0) % tc < 0 => the t=0 edge is visible tN(n,1) = 0.0; % recompute sc for this edge if (-d(n,1) < 0.0) sN(n,1) = 0.0; elseif (-d(n,1) > a(n,1)) sN(n,1) = sD(n,1);

98

else sN(n,1) = -d(n,1); sD(n,1) = a(n,1); end elseif (tN(n,1) > tD(n,1)) % tc > 1 => the t=1 edge is visible tN(n,1) = tD(n,1); % recompute sc for this edge if ((-d(n,1) + b(n,1)) < 0.0) sN(n,1) = 0; elseif ((-d(n,1) + b(n,1)) > a(n,1)) sN(n,1) = sD(n,1); else sN(n,1) = (-d(n,1) + b(n,1)); sD(n,1) = a(n,1); end end % finally do the division to get sc and tc if(abs(sN(n,1)) < SMALL_NUM) sc(n,1) = 0.0; else sc(n,1) = sN(n,1) / sD(n,1); end

if(abs(tN(n,1)) < SMALL_NUM) tc(n,1) = 0.0; else tc(n,1) = tN(n,1) / tD(n,1); end % get the difference of the two closest points dP(n,:) = w(n,:) + (sc(n,1) * u(n,:)) - (tc(n,1) * v(n,:)); % = S1(sc)

- S2(tc) distance(n,1) = norm(dP(n,:)); end

% for m = 1:size(p1,1) % distance(m,1) = norm(dP(m,:)); % end

outV = dP;

varargout(1) = {outV}; % vector connecting the closest points %varargout(2) = {p2+sc*u}; % Closest point on object 1 %varargout(3) = {p4+tc*v}; % Closest point on object 2

end