The Effects of Uniaxial Strain on the Percolation ...€¦ · applications. 1.2.1 Functional...
Transcript of The Effects of Uniaxial Strain on the Percolation ...€¦ · applications. 1.2.1 Functional...
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
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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.
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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
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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
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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
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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
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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
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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
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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].
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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
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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
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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
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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:
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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].
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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.
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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.
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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].
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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.
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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.
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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
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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].
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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.
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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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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
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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);
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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