Estimation of Effective Thermal Conductivity ofGraphite flakes/Al Composites by Using TwoDimensional Microstructure Images（二次元微細組織画像を用いたグラファイトフレーク/ Al
複合材料の有効熱伝導率の推定）

Department of Mechanical Science and Engineering
Graduate School of Engineering
Hiroshima University, Japan

Advisor: Prof. Kenjiro Sugio

Zhao Yan

CONTENTSCONTENTSi

List of Figuresv

List of Tables

viii

Chapter 1
1.1

Background and Objective

Introduction

2

1.2 Experimental research on ETC of GFs/Al composites
1.2.1

Fabrication methods of GFs/Al composites

88

1.2.1.1

Powder metallurgy processing

8

1.2.1.2

Liquid metallurgy processing

10

1.2.2

Problems of previous studies

11

1.3 Theoretical research on interfacial thermal resistance

13

1.3.1

Macro theory

13

1.3.2

Micro theory

14

1.3.3

Theoretical calculation for the interfacial thermal resistance at GFs-Al
interface17

1.4 Theoretical and simulation calculations for the ETC of composites

19

1.4.1

Theoretical calculation for the ETC of composites

19

1.4.2

Theoretical calculation for the ETC of GFs/Al composites

21

1.4.3

Simulation calculation for the ETC of composites

23

1.4.3.1

Microscale model

23

1.4.3.2

Mesoscale model

24

1.4.3.3

Macroscale and multiscale model26

1.5 Objective of this thesis28

1.6 Outline of this thesis

30

1.7

32

ReferencesiChapter 2

Fabrication and Thermal Properties of GFs/Al composites

2.1

Introduction

46

2.2

Experimental procedure

48

2.2.1

Raw materials

48

2.2.2

Preparation of graphite flakes and Al powders mixtures49

2.2.3 Fabrication of graphite flakes/Al composites by sparking plasma sintering51
2.3

Evaluation methods

53

2.3.1

Microstructures and relative density

53

2.3.2

X-ray powder diffraction method

53

2.3.3

Effective thermal conductivity

53

2.3.4

Coefficient of thermal expansion

55

2.4 Results and discussion

56

2.4.1

Microstructure characterization and orientation of GFs in composites 56

2.4.2

Interface reaction

60

2.4.3

Effective thermal conductivity and coefficient of thermal expansion

62

2.5

Summary69

2.6

References

71

Chapter 3

Relationship between the thermal conductivity of GFs in 3D and2D microstructure images

3.1

Introduction

76

3.2

Calculation Procedure

78

3.2.1

Creating 3D and 2D models

78

3.2.2

TC calculation of GFs

80

3.2.3

Image-based simulation method

81

3.3 Results and Discussions

84ii

3.3.1Comparison of GFs’s orientations in 2D and 3D models

84

3.3.2

Relationship between aspect ratio of GFs and the orientation of GFs

86

3.3.3

ETC calculation of GFs/Al composites

88

3.4

Summary91

3.5

References

92

Chapter 4

Effect of the orientation of GFs and interfacial thermal resistanceon the effective thermal conductivity of GFs/Al composites

4.1

Introduction

95

4.2 Experimental and calculation procedures

97

4.2.1

Experiments

97

4.2.2

Numerical methods

98

4.2.2.1

Temperature distribution

98

4.2.2.2

Calculation of effective thermal conductivity

100

4.2.2.3

Thermal conductivity of GFs

101

4.2.2.4

Interfacial thermal resistance102

4.3 Results and discussion

104

4.3.1

Microstructures

104

4.3.2

Effect of porosity on TC of Al

107

4.3.3

Effect of the orientation of GFs on the effective thermal conductivity 108

4.3.4

Effect of interfacial thermal resistance on the effective thermal
conductivity

111

4.4

Summary114

4.5

References

115

Chapter 5Conclusions

119

Acknowledgements

123

iii

Published Papers in Regards to This Thesis

124

Presentations

125

iv

List of Figures

Figure 1-1 The development of microelectronics and modern smart products.
Figure 1-2 TC as a function of CTE for composites.
Figure 1-3 Schematic diagram of (a) spark plasma sintering, (b) vacuum hot pressing.
Figure 1-4 Schematic diagram of the fabrication process of GFs/Al composites, gaspressure infiltration method (a), squeeze casting method (b).
Figure 2-1 Microstructures of raw materials: (a) Al powder and (b) GFs.
Figure 2-2 Flowchart of GFs and Al powder mixing.
Figure 2-3 Sedimentation velocity of Al powders with 30µm diameter in ethanol,
butanol and acetone.
Figure 2-4 Schematic of the SPS equipment.
Figure 2-5 Flow chart of the spark sintering process.
Figure 2-5 Schematic diagram of a steady-state thermal conductivity measurement
device.
Figure 2-6 SEM images of the mixtures of Al and 10-40 vol. % GFs. 10 vol.% GFs (a),20 vol.% GFs (b), 30 vol.% GFs (c), and 40 vol.% GFs (d).
Figure 2-7 Schematic of the GFs/Al composites sintered by SPS. The grey arrow
indicates that the heat flow was from B to D when measuring the ETC of the sample.
Figure 2-8 OM images of 10-40 vol.% GFs/Al composites.
Figure 2-9 Distribution of GFs orientation in the 10-40 vol.% GFs/Al composites, theorientation was described by the angle between GFs basal-plane direction and heat flow
direction.
Figure 2-10 Statistical counting of the aspect ratio of GFs.
Figure 2-11 XRD analysis of GFs/Al composites.
Figure 2-12 HRTEM image of the Al-GFs interface [9]: (a) HRTEM image; (b) FFT ofamorphous layer; (c–d) FFT of crystalline graphite; (e) FFT aluminum.

v

Figure 2-13 Temperature variation in the ETC measurement process of 1040vol.%GFs/Al composites, (a-d) relationship between temperature and time, and (a1d1) relationship between temperature and distance of the measuring points.
Figure 2-14 Comparison of measured and theoretical ETC of 10-40 vol.% GFs/Al
composites.
Figure 2-15 Thermal expansion variation versus temperature (a), coefficient of thermal
expansion (CTE) versus GFs content (b).
Figure 3-1. Flowchart of the calculation procedure for comparing the orientations of
GFs in two- and three-dimensional models.
Figure 3-2. Three-dimensional model and orientation of GFs. a) 3D model of GFs/Al
composites; b) Orientation of GFs in 3D model; c) Orientation of GFs in the extracted
cross-section; The heat flow is parallel to the X-axis; The short dash line (z) is thenormal vector of GFs basal-plane; The short dash line (z’ ) is the translation of the shortdash line (z); θ’ is the angle between z’ and the X-axis; Rectangle O’ABC is thesmallest circumscribed rectangle of GFs, O’P∥O’A, θ2D

is the angle between X-

axis and O’P.
Figure 3-3. Simulation model for the ETC calculation.
Figure3-4. Comparison of the ª2D obtained from mathematical calculation and 2D
cross-sectional image.
Figure 3-5. Orientation difference of GFs in the 2D cross-sections and 3D models (a);TC difference of GFs in the 2D cross-sections and 3D models (b).
Figure 3-6. Relationship between the θ2D, θ2D-3D and 𝑅 . Scatter image (a); Nonlinear
polynomial fitting surface (b).
Figure 3-7. TC of GFs calculated by equation (1), θ was equal to the θ3D for the solid
black line and θfitted-3D for the grey circle.
Figure 3-8. Microstructure images of the 10 vol % GFs /Al composite; (a) The OM
images; (b) Distribution of the GFs TC.
Figure 3-9. ETC of the 10 vol.% GFs/Al composite considering the interfacial thermal
conductance (h) from 1.1 × 103 to 1.1 × 109 W m-2 K-1.
vi

Figure 3-10. A SEM image of the microstructure at the Al-GFs interface.
Figure 4-1. The shape of the sample sintered by SPS; P1, P2, and P3 at the A plane
denotes the regions corresponding to the optical micrographs. The grey arrow illustrates
that the heat flow was from B to D during ETC measurement, C is the plane bearing
the pressure during sintering.
Figure 4-2. Schematic diagram of heat flow at the interface. The dashed line denotes
the Al–GF interface, qn denotes the heat flow in the direction of the dashed arrows.
Figure 4-3 Simulation model for the ETC calculation.
Figure 4-4 TC of GFs as a function of the angle (θ). The nominal TC of the Al matrix,
λAl, was 236 W m–1 K–1 (grey horizontal dotted line).
Figure 4-5 Optical micrographs of Samples 1–4. Regions P1, P2, and P3 correspond tothe black circles at the A-plane in Figure 1. The grey arrow represents the heat flow
direction during the ETC measurements. The brighter phases correspond to the Al
matrix, and the elongated dark phases to the GFs.
Figure 4-6 Distribution of the GF orientations in the Al matrix. Gray and white
rectangles in (a) and (b) represented the smallest circumscribed rectangles and wereused to measure the angle (θ) between the basal plane of GFs and the heat flow direction;in case –90 < θ < 0, θ was taken as the absolute value.
Figure 4-7 TC values of GFs in Samples1-4. Regions denoted as P1, P2, and P3
correspond to the microstructures in Figure 5; the color bar shows the TCs of GFs in
different orientations.
Figure 4-8 ETCs of the composite samples. λeff denotes the measured ETC, λROM is theETC calculated using the rule of mixture, and λs-eff is the ETC calculated using the 2D
image-based simulation considering GF orientations.
Figure 4-9 ETC as a function of the heat transfer coefficient, h. The dashed lines
represent the experimental ETCs, and the arrows denote the h value at the Al-GF
interface.

vii

List of Tables
Table 1-1 Thermal properties of semiconductor materials.
Table 1-2 Thermal properties of traditional electronic packaging materials.
Table 1-3 Thermal properties of reinforcements.
Table 1-4 Theoretical models using to calculate the ETC of composites.
Table 2 -1 Chemical components of Al and GFs.
Table 2-2 Shape and physical properties of Al and GFs.
Table 2-3 Viscosity and density of ethanol, butanol, and acetone.
Table 2-4 Measurement conditions of XRD.
Table 2-5 Relative density and effective thermal conductivity (ETC) of GFs/Al
composites.
Table 2-6 Calculation parameters in equation (2-7).
Table 2-7 Theoretical ETC of GFs/Al composites.
Table 2-8 Coefficient of thermal expansion.
Table 3-1. Calculation parameters in equation (3-5)
Table 4-1 Volume fractions of GFs (vGFs), measured ETCs (λeff), and relative densities
of Samples 1–4 and pure Al sample.
Table 4-2 Calculation parameters in equation (10) and calculated effective thermal
conductivities of composites.
Table 4-3 Material parameters of the DMM calculation, interfacial thermal conductance(h), and interfacial thermal resistance (R); Sim. is the simulated h using 2D-image, Cal.
is calculated h using DMM method, Ref. is the reference value of h.

viiiChapter 1

Background and Objective

1.1

Introduction

2

1.2 Experimental research on ETC of GFs/Al composites
1.2.1

Fabrication methods of GFs/Al composites

88

1.2.1.1

Powder metallurgy processing

8

1.2.1.2

Liquid metallurgy processing

10

1.2.2

Problems of previous studies

11

1.3 Theoretical research on interfacial thermal resistance

13

1.3.1

Macro theory

13

1.3.2

Micro theory

14

1.3.3

Theoretical calculation for the interfacial thermal resistance at GFs-Al
interface17

1.4 Theoretical and simulation calculations for the ETC of composites

19

1.4.1

Theoretical calculation for the ETC of composites

19

1.4.2

Theoretical calculation for the ETC of GFs/Al composites

21

1.4.3

Simulation calculation for the ETC of composites

23

1.4.3.1

Microscale model

23

1.4.3.2

Mesoscale model

24

1.4.3.3

Macroscale and multiscale model26

1.5 Objective of this thesis28

1.6 Outline of this thesis

30

1.7

32

References

1Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

1.1 Introduction
Today is an era of the rapid development of electronic technology. As shown in
Figure 1-1, various electronic products are gradually integrated into people's daily lives,
making people's lives convenient, comfortable, and safe. Therefore, ensuring theelectronic products' efficiency, stability, and safety is essential to people's lives.

Figure 1-1 The development of microelectronics and modern smart products.

However, with the rapid development of electronic technology in the direction ofhigh power, high frequency, miniaturization, and integration, the heat generated by theincrease in the power density of electronic components is increased dramatically. As aresult, the generated heat would heat electronic components, affecting their working
effectiveness and life span, and even resulting in a malfunction of the device if notdissipation in time[1]. In other words, the mismatch in the heat dissipation performanceand the thermal expansion coefficient between the materials used to make thecomponents cause the electronic components' damage and failure. The key to solving
this problem is to develop new electronic materials with high thermal conductivity (TC)and lower coefficient of thermal expansion (CTE) [2]. Semiconductors are the corecomponent of electronic products; hence, electronic packaging materials must match
2

Chapter 1 Background and Objective

thermal properties to semiconductors. Table 1-1 lists the thermal properties ofsemiconductors materials.
Table 1-1 thermal properties of semiconductor materials [1, 3-5].
Materials

Density / g·m-3

TC / W·m-1·K-1

CTE /10-6 /K-1

Silicon (Si)

2.3-2.34

139-150

2.8-4.2

Silicon carbide (SiC)

3.1-3.21

80-4902.8-5.0

Gallium arsenide (GaAs)

5.3

46-58

5.8-5.9

Gallium nitride (GaN)

6.1

150

3.2

Traditional electronic packaging materials contain metals, ceramics, and composite
materials. Table1-2 shows the TC and CTE of traditional electronic packaging materials.
However, the traditional electronic packaging materials, such as Copper (Cu),
Argentum (Ag), Aurum (Au), Alumina (Al2O3), W-Cu, etc., all have high density or
high TC or high CTE. As a result, they can no longer meet electronic products' current
comprehensive development requirements, such as high integration, efficient heatdissipation, and green environmental protection[6, 7].

Table 1-2 thermal properties of traditional electronic packaging materials.
Materials

Density / g·m-3

TC / W·m-1·K-1

CTE /10-6 /K-1

Al

2.7

236

23.0

Cu

8.9

400

17.0

Ag

10.49

429

18.9

Au

19.3

318

14.2

Al2O3

3.9

20

6.5-6.7

BeO

2.9

250-275

6.7-8.0

SiC

3.2

270

4.5

AlN

3.3-4.5

220

3.3-4.5

W-Cu

8.4

167

6.5

220-270

7.9-9.3

Mo-Cu
3

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

To cater for the future development characteristics of electronic products: e.g., highintegration of integrated circuits and the miniaturization, portability, efficiency, highpower, high stability, and low cost, the new generation of electronic packaging
materials is required to have the following mechanical and thermal properties [8]:1, High TC: Timely homogenize and dissipate the heat generated by thesemiconductor.
2, Low CTE: Reduce the thermal stress between the semiconductor and electronic
packaging materials.
3, Sufficient strength, stiffness, and toughness: Play a good role in supporting andprotecting semiconductors and devices.
4, Lightweight: The density is as low as possible to facilitate the lightweight designof the device structure.
5, Cost control and competitive requirements: high yield, suitable for mass
production, low price.
Metal-matrix composites have a high TC, low expansion of non-metallic
reinforcement, easy processing of metal-matrix, and good plastic toughness. Therefore,
metal matrix composites have attracted significant attention from researchers. Al is of
low density, low cost, and high TC, and it is the commonly used metal matrix of thecomposites. The reinforcements include mainly carbon materials (carbon fiber, graphite,diamond), silicon carbide particles (SiC), silicon particles (Si), etc. Table 1-3 shows the
TC, CTE, and density of those reinforcements.
SiC/Al, Si/Al, Diamond/Al are the most mature metal matrix composites. These
composites have been widely used in the world and have gradually replaced the
traditional electronic packaging materials. However, the metal matrix composites
currently used in electronic packaging, e.g., SiC/Al, Si/Al, Diamond/Al, show highprices and TC lower than expected in large-scale industrial production. Figure 1-2
shows the density, TC, and CTE of the composites used in electronic packaging.

4Chapter 1 Background and Objective

Table 1-3 Thermal properties of reinforcements[1, 7, 9-11].
Materials

Density/ g·cm-3

TC / W·m-1·K-1

CTE / 10-6/ K

Carbon fiber

1.75 ~ 2.19

530-1200-1.45 ~ 7

Graphite

2.23~3000 (xy)
6 ~ 38 (z)-1.5 ~ -1.0 (xy)29 (z)

Diamond

3.51

~2500

1.3

SiC

3.2

270

4.5

Si

2.3

150

4.1

BeO

2.9

250

6.7

Where a and b: the TC and ETC could be less than those of crystalline graphite due tothe defects and impurities.

Figure 1-2 TC as a function of CTE for composites.

It is fortunate that carbon fibers and graphite flakes also have excellent thermal
properties and are low-cost. Particularly for graphite flake, their TCs are values (~2200,~38) W m–1 K–1 [9, 12, 13] in the basal-plane and the out-plane, respectively, and their CTE
are (- 1.0, 8) × 10－6 K－1 [10, 14, 15] in the basal-plane and the out-plane, respectively. In

5Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

view of GFs' excellent thermal properties, GFs/Al composites are considered to be oneof the most promising candidates for electronic packaging. Many experimental studiesof the effective thermal conductivity (ETC) of GFs/Al composites have been conducted
extensively, e.g., Chen et al. [14] reported that the ETC of 80 vol.% GFs/Al composites(783 W m–1 K–1) was higher than that of 80 vol.% diamond/copper composites [16] (724W m–1 K–1). Li et al. [17] reported that the ETC of 70 vol.% GFs (graphite flakes)/Al
composite was 714 W m–1 K–1 in the plane parallel to the GFs layers. Comparison with
SiC/Al, Si/Al, Diamond/Al composites, GFs/Al composites show higher TC, lower
CTE, lower cost, and better machining performance. However, the ETC of GFs/Al
composites is not high as the expected theoretical value due to the following
deficiencies[10, 18-21]:(1) GFs show strong anisotropic TC, resulting in the orientation of GFs has a great
effect on the effective thermal conductivity (ETC) of GFs/Al composites.
(2) Interface reaction can occur readily during the fabricating process of GFs/Al
composites, and the harmful formation of Al4C3 is often observed at the GFs-Al
interface. The Al4C3 is a brittle compound, and it can reduce the plasticity of thecomposites and promote accelerated fatigue crack growth rates.
(3) The interfacial thermal resistance at the interface between GFs and Al
considerably affects composites' ETC.
Additionally, to predict the ETC of composites, many theoretical models areproposed in previous studies [10, 11, 13, 14, 17, 18, 22-32]. However, those theoretical models
are not suitable for calculating the ETC of GFs/Al composites due to the diversity
orientation of the GFs in composites. Moreover, to evaluate the interfacial thermal
resistance at the GFs-Al interface, the commonly used methods include the phonon
diffuse mismatch model (DMM) [33], acoustic mismatch model (AMM)[34], etc.
However, those models are based on an idea interface to calculate interfacial thermal
resistance. In reality, the contact interface microstructure of two solids is extremelycomplex. Contact interfaces are affected by many factors, such as the thermophysicalproperties, the hardness, shape, roughness of the material, the gap at the interface, the6

Chapter 1 Background and Objective

roughness of the contact interface, as well as the pressure and temperature duringcomposite fabrication. Furthermore, it is almost impossible to obtain the direct
relationship between those influence factors and the contact interface microstructure.
Thus, those models have specific limitations for calculating interface thermal resistance,
and they are not entirely reliable for calculating interfacial thermal resistance.
This study reviews many previous studies on the ETC of GFs/Al composites anddevelops a new method to calculate the ETC of GFs/Al composites. Research method:a newly developed two-dimensional image-based model. Research contents: 1,
Comparing the thermal conductivity of GFs in the 3D model and the 2D model, anddetermining the relationship between the thermal conductivity of GFs in the 3D modeland the 2D model; 2, Study the effect degrees of orientation of GFs and interfacial
thermal resistance on the effective thermal conductivity of GFs/Al composites using
two-dimensional image-based simulations.

7

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

1.2

Experimental research on ETC of GFs/Al composites

1.2.1 Fabrication methods of GFs/Al composites
Metal matrix composites can be fabricated via a variety of methods like stir casting,squeeze casting, ultrasonic infiltration, pressure infiltration, spark plasma sintering, andhot pressing, etc. However, not all fabrication techniques are suitable for fabricating
GFs/Al composites since there are some challenges to overcome, such as the oriented
GFs, avoiding the formation of Al4C3, and reducing the interfacial thermal resistance.
In this part, some common methods for fabricating GFs/Al composites are reviewed,
along with their advantages and disadvantages. The fabrication methods can beclassified into two routes: powder metallurgy process and liquid metallurgy process.

1.2.1.1 Powder Metallurgy Processing
The powder metallurgy process (PMP) is a low-temperature sintering technologythat can effectively inhibit the interfacial reaction and the formation of Al4C3. The PMP
methods can be divided into spark plasma sintering (SPS) [35-37] and vacuum hotpressing[14, 23, 26, 27]. Figure 1-3 shows the schematic diagram for SPS and vacuum hotpressing.
The process of using the PMP methods to prepare GFs/Al composites is to fill the
mixtures of raw materials Al and GFs into a graphite mold for sintering. Sometechniques in past studies were proposed to ensure GFs have a consistent orientation incomposites. Such as that, to stack the GFs by layers, Chen et al. [14] tapped the graphite
mold while filling naturally; Xue et al. [26] used a shaking device to vibrate the graphite
mold filled with mixtures of Al and GFs, hoping to obtain the oriented GFs; Huang etal. [23] fill the graphite mold with layers of artificial graphite films, and Al foils, theorientation of graphite film and Al foil can be well controlled.
The formation of Al4C3 at the GFs-Al interface is thermodynamically favorable,
and it can damage the mechanical and thermal properties. To avoid the formation of
Al4C3, Chen et al. [14] used some approaches, including a lower temperature than8

Chapter 1 Background and Objective

melting point of Al during the hot-pressing process, and GFs with nearly 100%graphitization degree and a larger flat surface, to prevent the formation of Al4C3. Salvo[38]

and Candan et al.[39] filled a certain minimum critical Si content to prevent theformation of Al4C3. Similarly, Xue et al. [26] coated GFs surface with Si to prevent theformation of Al4C3, and Kurita et al.[27] added Al-Si11.3at% alloy into the raw materials
Al powders to avoid the formation of Al4C3 during fabricated GFs/Al composites. The
introduction of silicon not only prevents the formation of Al4C3, but silicon caneffectively fill the pores at the interface between aluminum and graphite. Meanwhile,the disadvantages of the introduction of silicon were also revealed, which caused themicrostructure of GFs to disarray in proximity to the GFs-Al interface, which canweaken the interlaminar strength between GF in proximity to the Al/GF interface and
Al/GF interface [27].
(a)

(b)

Figure 1-3 Schematic diagram for (a) spark plasma sintering [40], (b) vacuum hotpressing[41].
9Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

1.2.1.2 Liquid metallurgy processing
The liquid metallurgy process is also widely used to fabricate GFs/Al composites.
The commonly used methods include gas pressure infiltration [11, 13, 31] and squeeze
casting [17, 28, 29]. Figure 1-4 shows the schematic diagram of gas pressure infiltrationand squeeze casting.
(a)

(b)

Figure 1-4 Schematic diagram of the fabrication process of GFs/Al composites, (a) gas
pressure infiltration method [11], (b) squeeze casting method [17].

The common features of gas pressure infiltration and squeeze casting in GF/Al
composite fabrication are: First, it is to put the raw material GFs into a mold and press
GFs into preforms; Then, the molten Al is poured into a preheated mold containing theGFs preforms, and apply pressure to infiltrate the molten Al into the preforms. As aresult, GFs are able to achieve uniform orientations through the fabrication of GFs
preforms, and as reported in the papers [11, 13, 17, 28, 29], GFs were stacked by layers in
GFs/Al composites.
10Chapter 1 Background and Objective

As known that high temperature is an essential condition for liquid metallurgy to
improve the fluidity of molten Al, but conversely, it promotes the interface reaction as
well. According to Okura’s studies [42] on the rate of formation of intermetallic
compounds in Al matrix-carbon fiber composites, the interface reaction rate can beevaluated by: X = [2kt·exp(-Q/RT)]1/2. X is the thickness of Al4C3 reaction layer, k is aconstant, t is heating time, Q is the activation energy, and T is temperature. The formula
shows that the formation of Al4C3 is related to the temperature T and the heating timet. Consequently, the rapid cooling method[17, 29, 31] in the pressure infiltration process
and squeeze casting process is one of the most used methods to suppress the formationof Al4C3. In addition, coating Si on the surface of GFs, or mixing a small amount of Siwith raw GFs, or replacing pure aluminum with Al-Si alloys are also often used toprevent the formation of Al4C3 [11, 13, 20, 28, 29].
1.2.2 Problems of previous studies
Lots of experimental studies, as mentioned above 1.2.1 and 1.2.2, have tried toconquer the problems of the orientation of GFs and the formation of Al4C3. But themeasured ETCs of GFs/Al composites were not as high as the expected theoretical ETC.
It is evident that the high TC of GFs cannot be fully utilized in GFs/Al composites. The
reasons can be analyzed from the following aspects.
Firstly, it is the method to control the orientation of GFs. The vibration method isoften used to obtain oriented GFs in powder metallurgy processing. However, it isalmost impossible to ensure that all the graphite flakes are entirely parallel to each otherby vibrating the mold containing the mixtures of Al and GFs. Moreover, the pressure
applied during the thermoforming process is usually given along one direction, theorientations of GFs cannot be well controlled, and angles between GFs basal-plane andthe surface of GFs/Al composites are easy to form. Therefore, for the GFs/Al
composites with a high-volume fraction of Gfs, it is more difficult to control GFs
orientation by the vibration method. In the liquid metallurgy processing, obtaining theoriented GFs is to pre-press the GFs into a preform. However, the infiltration process
11Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Imageof molten Al is challenging because the distance between GFs is minimal. Especiallythe prefabricated GFs preforms with a preferred orientation, and there is almost no
space between GFs[10, 11, 13, 14, 17, 29]. In order to enhance the infiltration process, the third
phases (SiC particles, Si particles, and short carbon fiber) were added into GFs
preforms[10, 11, 13, 29, 43-46]. But the third phase would reduce the ETC of GFs/Al
composites due to the low TC of the third phase itself, and it may destroy the parallel
stacking of GFs in the preform.
Secondly, it is the approach to avoid the formation of Al4C3. There are two
approaches used usually in experiments: 1 using highly graphitized GF and low
sintering temperature.; 2 introducing the third phase (Si or SiC). However, according
to Huang [23] and Kurita et al. [27], low sintering temperature can cause low density andmore pores embedded in GFs/Al composites. The pores can reduce the ETC ofcomposites [47]. In addition, the low TC of the third phase (Si or SiC) also lowers the
ETC of GFs/Al composites.
Thirdly, it is the effect of interfacial thermal resistance on the ETC of composites.
Many studies have proposed that the interfacial thermal resistance between carbon andAl dramatically affects the ETC of composites [13, 23, 29, 31, 48-54]. Besides, previous
studies have investigated the effects of the shape [55-57] and size [14, 17, 18, 52, 53, 58, 59] ofenhanced phase on the ETC of composites. For example, according to the studies of theeffect of particle size on the ETC, the smaller filler particles can result in a lower ETC
of composites because they caused a more extensive interface area [14, 17, 18].

12

Chapter 1 Background and Objective

1.3 Theoretical research on interfacial thermal resistanceInterfacial thermal resistance is derived from the Kapitsa[60] thermal resistancediscovered in 1941. Interfacial thermal resistance is defined as the ratio of thetemperature discontinuity at the interface to the power per unit area flowing across thatinterface [33]. Up to now, lots of scholars have studied the interface thermal resistance[33, 34, 61-71].

To determine interfacial thermal resistance, the commonly used experimental
methods include steady-state measurement[72-74], transient measurement (such as laser
photothermal measurement[75, 76] and photoacoustic method[77-79], thermal imaging
method[72], “flash” flash method[80, 81], etc.), and micro/nano-scale measurement
methods (such as transient thermoreflectance (TTR) technology and 3ω method [33, 82,
83]. However, those measurements are very strict, and the measurement results are notaccurate enough because those measurements are susceptible to many external factors.
Therefore, many theoretical models are developed to predict interfacial thermal
resistance, and interface thermal resistance is evaluated mainly from macro and micro
perspectives.

1.3.1 Macro theory
At the macroscopic level, the interfacial thermal resistance comes from theincomplete contact of the material interface [84, 85]. The interfacial thermal resistance,
Rc, can be expressed by Rc = Rd + RL. Rd is the thermal resistance caused by heatdiffusion and heat shrinkage resistance at the contact interface, RL is the thermal
resistance of the contact area at the contact interface [86]. Microstructure at the contact
interface dominates the interfacial thermal resistance. However, the microstructure at
the contact interface is affected by many factors, such as surface roughness,microscopic morphology, elastoplastic deformation of surface peaks, and stages in thesurfaces of contact materials. Based on those influencing factors, some theoretical
13

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

methods have been proposed to calculate interface thermal resistance, including CMY
plastic contact model[61, 86], Multi-scale contact model[87, 88], Monte-Carlosimulation[89], Micro-scale contact model [90], etc. But those macroscopic models andsimulations all have specific usage restrictions due to the complexity of the interface
microstructure.

1.3.2 Micro theory
At the micro-level, heat is transferred by phonons, electrons, and photons.
Therefore, the calculation mechanism of interfacial thermal resistance is to calculate
phonon and electron couplings. In the past decades, different microscale models were
proposed to predict interfacial thermal resistance. The common models include
interfacial layer model (ILM) [91, 92], AMM model[34], DMM model and Boltzmann
mesoscopic simulation[68].
ILM model considers that the contact interface consists of subsurface layers on thetwo sides of the contact interface and the middle contact interface layer. The thicknessof the subsurface layer is 1 micron. At the contact interface, hot carriers transfer energy
in the form of radiation. The subsurface layer is mainly composed of local materials,
but it also contains the atoms diffused from the other side of the interface and someother microstructures such as microprotrusions, dislocations, and oxidation from theother side of the interface. These microstructures cause the hot carriers scattering, thelattice defect scattering, dislocation scattering, and others scattering. The hot carriers
scattering at the interface caused the interfacial thermal resistance. The interfacial
thermal resistance, Rc, can be calculated by the following formulas:
𝑅 =

𝑅 +𝑅
…… (1.2.1.2-1)
𝑅 =

∆𝑇
𝑞
…… (1.2.1.2-2)

14Chapter 1 Background and Objective

𝑞

=

∑ 𝑉,

𝛼→

𝑇 ∫

ℏ ⁄
( )

−𝑇 ∫

ℏ ⁄
( )

…… (1.2.1.2-3)
𝑞

=

𝐶𝑉𝑙∆𝑇
…… (1.2.1.2-4)

where subscripts CI and Sub are the contact interface layer and subsurface layer,
respectively; R is thermal resistance; △T and q are the temperature difference of theinterface layer and the heat flux density through the interface, respectively; KB is
Boltzmann constant; V is mean sounding velocity; α is the transmittance of phonon; T
is temperature; ħ is Planck’s constant; ω is angular frequency; z is the acoustic
resistance of solid materials; d is the thickness of the subsurface layer; C is volumespecific heat; l is the mean free path of phonon scattering process; j is the phonon modefor 1 and 2. However, the interface thickness d is closely related to the interface heat
flux density, contact area, load pressure on the contact surface, and other factors.
Therefore, the correlation between them cannot be obtained directly. Thus, it is not easy
to obtain the interface thickness d.
AMM model assumes that the contact interface of two solid materials is a plane,
and phonons are treated as continuous plane waves. The propagation of phonons on theinterface is complete specular reflection and refraction, and the propagation is no
possibility of diffuse scattering. The interfacial thermal resistance can be calculated asfollows:𝑅 =𝜋 𝐾 Γ→
15ℏ

𝑉, Γ,

𝑇
…… (1.2.1.2-5)

/

Γ, =

𝛼 → (𝜃, 𝑗) sin 𝜃 cos 𝜃𝑑…… (1.2.1.2-6)

where θ is phonon incident angle. Under low-temperature conditions, phonon
15

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

wavelength is much larger than interface roughness size and interface defect size, andthe contact interface can be approximately regarded as a plane. The AMM model isreliable to calculate the interfacial thermal resistance under low-temperature conditions.
However, under high-temperature conditions, the wavelength of phonons is reduced toclose to the size of interface roughness, so the contact interface of two solids cannot beapproximated as a plane, and the possibility of phonons scattering at the interface isvery high. The interfacial thermal resistance calculated by the AMM model may havea more significant error.
DMM model assumes that phonons are completely elastically scattered at thecontact interface of two solid materials, and the transmittance of phonons (α) is related
to the phonon density of states of the material. Therefore, the calculation expressionsof the interface thermal resistance are as follows:𝑅 =𝜋 𝐾 𝑉 𝛼→30ℏ

𝑇…… (1.2.1.2 -7)

𝛼→ =

∑ 𝑉,
∑ 𝑉,

+∑ 𝑉 ,…… (1.2.1.2-8)

Comparing with the AMM model, the DMM model can better predict theinterfacial thermal resistance under high-temperature conditions. For example, Stevens[93]

and Stoner[94] reported that the value of interfacial thermal resistances calculatedwith the DMM model under room temperature was one order of magnitude less thanthe experimental value of interfacial thermal resistances. AMM and DMM models bothhave a specific temperature range to calculate interfacial thermal resistance. The
reasons are that AMM ignores the phonons scattering at the interface, while DMM
assumes that phonons are completely scattered at the interface.
The heat transfer mechanism of hot carriers (phonons and electrons) at the contact
interface is very complicated. Therefore, it is difficult to obtain ideal results if only the16Chapter 1 Background and Objective

macro or micro model calculates the interface heat conduction. Fortunately, Lattice
Boltzmann proposed a mesoscopic simulation at multi-scale levels. The model
combines the discrete macroscopic simulation and the continuous microscopic method,
and the calculation of interface thermal resistance considers the scattering and radiationof phonons and electrons. Han et al. [68, 70] calculated the interfacial thermal resistanceusing the thermal lattice Boltzmann method framework.

1.3.3 Theoretical calculation for the interfacial thermal resistance at GFs-Al
interface
For GFs/Al composites, the interfacial thermal resistance is considered to have aconsiderable effect on the ETC, and the effect cannot be eliminated. The interfacial
thermal resistance at the GFs-Al interface is commonly calculated using AMM andDMM models. Interfacial thermal resistance is equal to the reciprocal of interfacial
thermal conductance (h). Chen et al. [32] obtained the experimental value of h to be 5.0◊107 W m-2 K-1, and the calculated value by the DMM model showed the same orderof magnitude as 107. In addition, The values of h calculated by AMM were as 4.5◊107[11, 29]

W m-2 K-1, 4.8◊107 [26] W m-2 K-1, and 7.4◊107 [13] W m-2 K-1.

However, as mentioned in 1.3.1 and 1.3.2 parts, the AMM and DMM models are
suitable for calculating interfacial thermal resistance in different temperature ranges.
Moreover, the calculation results are not completely reliable because the calculations
are based on an ideal contact interface. Therefore, it is crucial to understand how heat
conducts at the interfacial surface to obtain the accurate value of interfacial thermal
resistance. Researchers considered that the hot carriers of metal heat transfer are mainly
electrons at the micro-level, and the hot carriers of nonmetal heat transfer are mainly
phonons. Hence, the heat transfer across the metal-nonmetal interface has two possible
pathways. One is the coupling between electrons of metal and phonons of nonmetal
through a harmonic interaction at the contact interfaces. Another is the coupling17

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

between electrons and phonons in the metal matrix and then the couplings of the metalelectrons and the nonmetal phonons [95]. As Battabyal et al.[96] reported that the heat
conduction mechanisms at the diamond-Al interface were the coupling between Al
electrons and Al phonons and the coupling between diamond and Al phonons that
occurred at the interface. Similarly, the heat conduction mechanism at the diamond-Al
should be suitable for the heat conduction at the GFs-Al interface.

18

Chapter 1 Background and Objective

1.4 Theoretical and simulation calculations for the ETC ofcomposites
1.4.1 Theoretical calculation for the ETC of composites
The influential factors of the ETC of composite include the size, shape, volume
fraction, dispersion and orientation of the filling phase, porosity, and interfacial thermal
resistance. In past studies, many models were proposed to calculate the effects of thosefactors on the ETC of composites. The models[18, 48, 49, 57, 97-105] can be divided into
theoretical and empirical models. Table 1-4 shows some summarized theoreticalmodels.
Maxwell’s model is the original theoretical model used to predict the ETC ofcomposites [106, 107]. Maxwell’s model assumes that the shape of the filled phase isspherical, the volume fraction of the filled phase is small, and the filled phases are
uniform dispersion and independent of each other. However, Maxwell’s model is a very
simple theoretical model, and its application has significant limitations. For example,
the ETC value calculated by Maxwell’s model would significantly differ from theexperimental ETC value if the filled phase has a high-volume fraction, or the shape ofthe filled phase was ellipsoid, fiber, and sheet, or considering the effects of interfacial
thermal resistance and porosity. Therefore, many new theoretical models extended
based on Maxwell’s model were proposed, such as Bruggeman’s model [58, 108], Fricke’s
model [109], Hamilton-Crosser’s model [110], Every’s model [58], Nan’s model[111], etc.
For example, Bruggeman’s model was developed by integrating the differential
equation of Maxwell’s model, and it is suitable for the ETC calculations of composites
with a high-volume fraction of the filled phase. Fricke’s model is suitable for the ETC
calculations of composites with filled elliptical phases and a high volume fraction of
filled phase. Hamilton-Crosser’s model is suitable for calculating the effect of the filled
phase shape on the ETC. Furthermore, Every’s model was extended based on
Bruggeman’s model, and it further considers the effect of the interfacial thermal
resistance on ETC. Nan’s model can consider the effects of size, shape, orientation,
19

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

dispersion, the volume fraction of filled phase, and interfacial thermal resistance on
ETC.
Besides, many scholars also developed other theoretical models, such as theCheng-Vachon model [112, 113], Nielsen model [114, 115], etc. Meanwhile, the empiricalmodels have the Springer-Tsai model [116], Y. Agari’s models[107, 117, 118], etc.
Table 1-4 Theoretical models using to calculate the ETC of composites.
ModelsEquation

Maxwell

𝜆 = 𝜆Bruggeman

1−𝑉Fricke∑HamiltonCrosser
Every

)

(

)

;

;

=
⁄

𝜆 =𝜆

𝑆= ∑

(

(

;

)

;

1 + 𝜆 ⁄𝜆 − 1 𝑓

𝑓 = 1; fi is the half-axis length of elliptical particles.
(

𝜆 =𝜆

)
(

(

)

;

)

n = 3/ψ; ψ is Particle sphericity.
((1 − 𝑉 ) =

)⁄(

)

(

)

(

)

⁄(

)

;

a is the radius of the reinforcement particles.
Cheng-=+

Vachon

⁄

ln
⁄

𝐵=

Nielsen

𝜆 =𝜆

(

⁄

⁄ [

] ⁄

⁄ [

] ⁄

;

, 𝐶=𝐵 ;)();

KE is the Einstein coefficient.
where λ and V represent the TC and volume fraction; Subscript c represents composite,and m and p represent the matrix and filled phase of composites, respectively.

20

Chapter 1 Background and Objective

1.4.2 Theoretical calculation for the ETC of GFs/Al composites
As above summarized, many theoretical models have been developed to predict
the ETC of composites. However, according to the structural characteristics of GFs/Al
composites, the mixing rule and Nan’s model are the favorite of researchers [14, 18, 26, 28,29].
The mixing rules can be expressed:
𝒙 𝒚

λ𝒄

= 𝑉

+ (1 − 𝑉×𝜆)×𝜆…… (1.3.2-1)

1
𝑉
=
λ
𝜆

+

1−𝑉
𝜆
…… (1.3.2-2)

where λc is the ETC of GFs/Al composites, 𝜆plane, 𝜆is the TC of GFs along the basal-is the TC of GFs along the out-plane, λAl is the TC of Al matrix, and the

VGFs is the volume fraction of GFs. Equation 1.3.2-1 uses to calculate the ETC of
GFs/Al composites when the basal-pane of GFs is parallel to the heat flow direction;
Equation 1.3.2-2 uses to calculate the ETC of GFs/Al composites when the out-pane of
GFs is parallel to the heat flow direction. However, it is worth noting that the mixing
rule is a simple theoretical model to calculate the ETC of composites, and the mixing
rules only consider the effect of volume fraction of the filled phase on the ETC.
Therefore, the ETC of GFs/Al composites calculated using the mixing rules would behigher than the ETC obtained by experiment methods, mainly attributed to the effectsof the orientation and the dispersion of GFs and interfacial thermal resistance.
To consider the effect of interfacial thermal resistance, TC of GFs, λGFs-eff, can becalculated by the following equation1.3.2-3, as follows:

𝜆

𝜆

=

1+

2𝜆
ℎ𝐷
…… (1.3.2-3)

where i is the x-y plane or z-axis direction, h is the interfacial thermal conductance, and21

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

D is the thickness and diameter of GFs. h is the reciprocal of interfacial thermal
resistance (R), h = 1/ R. To calculate ETC of composites considering the interfacial
thermal resistance, the equations can be obtained using λGFs-eff to replace the 𝜆
the 𝜆

and

in the mixing rules.

Nan’s model[111] is an effective medium approximate model (EMA) to predict theETC of GFs/Al composites. It includes the effects of the size, shape, and orientation [28,29]

. The equations can be expressed as:

𝜆

=𝜆

2+𝑉

(1 − 𝑆 )(1 + 〈𝑐𝑜𝑠 𝜃〉) + 𝜆

𝜆2−𝑉𝑆 (1 + 〈𝑐𝑜𝑠 𝜃〉) + 𝜆

𝜆(1 − 𝑆 )(1 − 〈𝑐𝑜𝑠 𝜃〉)
𝑆 (1 − 〈𝑐𝑜𝑠 𝜃〉)…… (1.3.2-4)

𝜆 =𝜆

1+𝑉

(1 − 𝑆 )(1 − 〈𝑐𝑜𝑠 𝜃〉) + 𝜆

𝜆
1−𝑉

𝜆

𝑆 (1 − 〈𝑐𝑜𝑠 𝜃〉) + 𝜆(1 − 𝑆 )〈𝑐𝑜𝑠 𝜃〉𝑆 〈𝑐𝑜𝑠 𝜃〉…… (1.3.2-5)〈𝑐𝑜𝑠 𝜃〉 =∫ 𝜌(𝜃)𝑐𝑜𝑠 𝜃 sin 𝜃 𝑑𝜃
∫ 𝜌(𝜃) sin 𝜃 𝑑𝜃…… (1.3.2-6)

where θ is the angle between heat flow direction and the basal plane of GFs; Sij is thegeometrical factors (such as size and shape) of GFs. To consider the effects of thegeometrical factors Sij of GFs on the ETC of GFs/Al composites, the thermal
conductivities of GFs along the basal-plane ( 𝜆replaced by 𝜆

and 𝜆

.

𝜆

=

𝜆
𝜆 +𝑆

) and the out-plane (𝜆

) are

−𝜆
𝜆

−𝜆
…… (1.3.2-7)

22Chapter 1 Background and Objective

𝜆

=

𝜆
−𝜆
−𝜆 )
𝜆 + 𝑆 (𝜆……. (1.3.2-8)

Assuming that GFs is an ellipsoidal shape, the semi-axes a and b of GFs on the twodimensional plane are the same (a = b), and the thickness of GFs is c. According to theratio (p) between c and semi-axes a or b, Sii can be obtained using the following
equations[119]:

𝑆

=𝑆

=

𝑝
𝑝
⎧
−
⎪2(𝑝 − 1) 2(𝑝 − 1)𝑐𝑜𝑠ℎ 𝑝, 𝑝 > 1𝑝𝑝⎨⎪ 2(𝑝 − 1) +2(𝑝 − 1)⎩𝑐𝑜𝑠

𝑝, 𝑝 < 1…… (1.3.2-9)

If GFs are highly oriented in the Al matrix, the <cos2θ> is infinitely close to 1. Moreover,to regard the shape of GFs as a disk, Sij were calculated[119] as, S11 = S22 = πp/4 and S33= 1 - πp/2.
1.4.3 Simulation calculation for the ETC of composites
With the rapid development of computer technology, simulation methods havebecome a new and effective research tool to calculate the ETC of composites. Many
simulation methods have been developed to predict the ETC of composites, such as
molecular dynamics (MD), lattice Boltzmann method (LBM), finite volume method(FVM), finite element method (FEM), etc. Those methods analyze the ETC ofcomposites at different simulation scales from micro-scale, mesoscale, to macro-scale.
1.4.3.1 Microscale model
The microscopic simulation is an atomic level calculation, and it is used to analyzes
the interaction between atoms through first principles. The calculation method's essence
is to use the Monte Carlo method (MC) or MD to solve the approximate solution of the
Schrödinger equation. MD is the most used model to solve the Schrödinger equation.
MD includes equilibrium and non-equilibrium molecular dynamics (EMD and NEMD).
23

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

EMD calculates the ETC of composites under the steady temperature field, and its
theoretical function to calculate the ETC(λc) is based on the Green-Kubo equation, as
follow:
𝜆 =

1
3𝑉𝐾 𝑇

〈𝐽(𝑡)𝐽(0)〉𝑑𝑡…… (1.3.3.1)

where V is the volume of the system, KB is Boltzmann constant, T is temperature, J is
microscopic heat flow, and t is time. However, using EMD to calculate ETC would
generate amounts of calculation and spend lots of time because of the very slow
convergence calculation of J and its related functions. As a comparison with EMD,
NEMD shows a better convergence calculation. NEMD establishes a non-equilibrium
heat conduction process by applying disturbances to the system, and then to calculate
ETC of composites using the Fourier function of heat conduction, J = - λ△T. NEMD
method is generally used to calculate the ETC of composites[120-124]. Tian et al[120].
studied the effects of the filled phase's orientation, dispersion, and arrangement, theinterface mismatch, the interface density, and the filled phase polydispersity on the ETC
of nanocomposites. Yang et al. [124] employed the NEMD method to calculate the TC of
defective graphene oxide. However, NEMD also has the disadvantages, such as poor
model homogeneity, momentum, energy conservation, etc. Müller-plathe et al[125].
proposed a reverse perturbation non-equilibrium molecular dynamic method (RNEMD)
for calculating the thermal conductivity. RNEMD can correct the disadvantages in the
momentum and energy conservation of NEMD mothed, making it more suitable forcalculating ETCs.

1.4.3.2 Mesoscale model
Compared with the microscopic simulation, mesoscopic simulation models
contain huge numbers of atoms (about 1023 /cm3), and they are much larger than the
microscopic simulation models. The theoretical equation is different from the24

Chapter 1 Background and Objective

Schrödinger equation, and it belongs to the average basic characteristic structure
relationship. Due to the complex diversity of mesoscopic scale mechanisms andconstitutive laws, different mesoscopic simulation models were developed[126]: 1, Spaceand time discretization dislocation dynamics model; 2, Phase-field dynamics orgeneralized Ginzburg-Landau model; 3, Deterministic or probabilistic meta-packet
automata model; 4, Polymorphic Dynamics Potts Model; 5, Geometry topology andgrouping model; 6, Topological mesh and vertex model. Although the mesoscopic
simulation models are diverse, they have some common characteristics. E.g., themesoscopic model does not contain obvious atomic-scale dynamics but an idealized
assumption that the material is a continuum. So, the lattice defects have a common
uniform matrix, and the matrix naturally couples the interactions between lattice defects.
Thus, intrinsic space and time scales are not explicitly included in the corresponding
governing equations. Moreover, a continuum mesoscopic scale model containing asingle lattice defect usually has a series of phenomenological partial differential rates
and intrinsic structural equations. Those differential equations are usually solved by thefinite difference method, finite element, or Monte Carlo method.
In the mesoscopic scale model, the determinations of space-time scale parametersand dispersion are determined by the differential equation and its coefficients and thecharacteristics and properties of the variables. The control of differential equations isused for local or global situations according to the interaction between lattice defects.
State variables (atomic concentration, dislocation density, structural parameters,
displacement, lattice orientation) are usually combined with spatial grid coordinates.
Because the phenomenological equation and the law of structural evolution have been
well studied on the mesoscopic scale, it is of great significance to use the continuum
approximation method to predict the structural evolution on the mesoscopic scale.
Additionally, the experimental data obtained from the mesoscale model is easier thanthe data obtained from the micro-scale, and it is more detailed than the information
obtained from the macro-scale model. The methods to solve the mesoscopic scalemodel mainly include DM, MC, and finite element method (FEM).
Mesoscopic scale models used to calculate the ETC of composites have been
25

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

developed, and the most commonly used calculation methods include dissipative
particle dynamics (DPD), LBM, off-lattice MC, etc. Such as DPD was proposed by
Hoogerbrugge and Koelman in the 1992 year, and it was used to solve the difference
between the lattice automata method and the reality and the fluid problem on themesoscopic time and space scale that MD cannot solve. DPD is a coarse-grained
technology based on MD, which means that each particle represents one or more
molecules or a fluid region of a fragment of a polymer rather than a single atom.
Moreover, the particle interaction is expressed by a pair of conservative, dissipation,and random forces. R. Qiao et al. [127] calculated the TC of nanocomposites by energyconserving DPD, and the obtained results were well agreed with the results calculatedby the Maxwell-Garnett model. Bing Zhou et al.[128] studied the effect of volume
fraction, alignment, dispersion and size of CNT, and polymer size on the ETC of
CNT/polymer composites with random and aligned CNT by using coupling smoothed
particle hydrodynamics and DPD methods.

1.4.3.3 Macroscale and multiscale model
The macroscopic simulation model treats the composite material as a single
orthotropic material or completely anisotropic material. The macroscopic models ofcomposites are commonly used to simulate the overall performances of the composites.
Thus, the Non-linearity of materials and local failures are not considered. Furthermore,compared with microscopic and mesoscopic scale models, macroscopic models do notcontain the calculations at the atomic and molecular level, which can save lots ofcalculations and calculation time. Using macroscopic models to calculate the ETC ofcomposites, FEM[105, 129-133], FVM[49], and FDM[134] are the most used methods.
Multi-scale simulation is a combination of microsimulation and macro simulation.
Macroscopic simulation has the advantage of being a small amount of calculation andsaving time, but it is difficult to reflect the effect of the microscopic behavior of eachmaterial on the overall properties of the composite. On the other hand, although the26

Chapter 1 Background and Objective

microscopic simulation can well understand the influence of the micro-behavior of eachmaterial on the overall behavior of the composite material, it has a considerable amountof calculation. As a result, because of the limitation of computer computing power andthe structural complexity of composites, it is only possible to establish a micro-model
based on the local characteristics of the composite. In contrast, the multi-scale model
can well coordinate the collaborative calculation of the micro model and the macromodel to study better the influence of the micro behavior of the material on the overallbehavior and the overall properties of the material. Based on those advantages, multiscale simulation has been widely used in the ETC calculation of composites[135-140].

27

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

1.5 Objective of this thesis
GFs are low-cost, lightweight, and possess excellent thermal properties, including
high thermal conductivity and low thermal expansion coefficient. GFs/Al composites
are thus chosen as the candidates of thermal management materials. Many experimental
and theoretical studies have been conducted to investigate the ETC of GFs/Al
composites. However, previous experimental studies found that the orientation of GFs,
the interface reaction, and the interfacial thermal resistance greatly affect thecomposites' thermal properties.
Theoretical calculations and three-dimensional(3D) simulations are used toevaluate the ETC of GFs/Al composites considering the orientation of GFs and theinterfacial thermal resistance. However, both theoretical calculations and 3D
simulations exhibit the disadvantages of calculating the ETC of GFs/Al composites.
For theoretical calculations, it is difficult for those theoretical models to calculate the
ETC of GFs/Al composites due to the diversity of GFs’ orientations in composites.
Additionally, the 3D simulation calculations would take lots of time and be very
expensive because these 3D models are constructed using images from ultra-highresolution X-ray computed tomography. On the other hand, although two-dimensional(2D) images simulations are low-cost and high timeliness, few studies have used 2D
images simulations to calculate the ETC of GFs/Al composites. The reason can be
attributed to the orientation of GFs in 2D images may not be equivalent to theorientation of GFs in the corresponding experimental samples, which can result in the
ETC of GFs/Al composites calculated by 2D images simulations are not reliable.
This study aims to study the effects of the orientation of GFs and the interfacial
thermal resistance on the ETC of GFs/Al composites using 2D images simulations. The
following steps did the works: In the first step, GFs/Al composites were fabricatedusing SPS. We measured the relative density of GFs/Al composites, analyzed the
distribution and orientation of GF, investigated the formation of Al4C3, and evaluatedthe ETC and thermal expansion using the experimental and theoretical methods.
28

Chapter 1 Background and Objective

Secondly, we studied the relationship between the TCs of the GFs in 3D and 2D images.
As a result, we obtained a fitted function used to convert TC of GFs from 2D to 3D.
ETC of 10%vol GFs/Al composite was calculated using 2D image simulations,employing the fitted function. Thirdly, the effect degrees of the orientation of GFs andthe interfacial thermal resistance on the ETC of GFs/Al composites were calculated by
2D image simulations.

29

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

1.6 Outline of this thesis
Chapter 1 Background and Objective
This chapter discusses the development of electronic packaging materials, thefabrication methods of the GFs/Al composites, some experimental studies of thethermal properties of GFs/Al composites, some theoretical and simulation methods forcalculating the ETC of GFs/Al composites, and the objectives of this thesis.

Chapter 2 Experiment process and Fundamental properties of GFs/Al
composites
In this chapter, the process of preparing GFs/Al composites using SPS was showed.
The orientation, distribution, and aspect ratio of GFs in the Al matrix were analyzed by
OM images. The relative density of composite was determined by the Archimedes
method. The interface reaction was checked by XRD method. The thermal expansion
coefficient and the ETC of GFs/Al composite were obtained by theoretical andexperimental methods.

Chapter 3 Investigation the relationships between the effective thermal
conductivity of GFs in the three-dimensional and two-dimensional model
In this chapter, we constructed 3D and 2D models of GFs/Al composites. The
orientations and TCs of GFs in the 3D and 2D models were compared. Furthermore,
the relationships between the orientations of GFs in 3D and 2D models were analyzed.
As a result, a fitted equation was derived and used to convert the TC of GFs from 2D
to 3D. Moreover, the effective thermal conductivity of the graphite flake/aluminumcomposite with 10 vol% graphite flakes was calculated using 2D image simulation andthe fitted equation.

Chapter 4 Effect of the orientation of GFs and interfacial thermal
resistance on the effective thermal conductivity of GFs/Al composites
In this chapter, the ETC of GFs/Al composites were calculated by using 2D image
30

Chapter 1 Background and Objective

simulations, including the effect of GFs' orientation and interfacial thermal resistance.
The effect degree of the orientation of GFs and the interfacial thermal resistance on the
ETC of GFs/Al composites were analyzed. The interfacial thermal resistances at the
GFs-Al interface were calculated.

Chapter 5 Conclusions
The results from the studies mentioned above were summarized in this chapter.

31

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

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44Chapter 2
Fabrication and Thermal Properties of GFs/Al composites
2.1

Introduction

46

2.2

Experimental procedure

48

2.2.1

Raw materials

48

2.2.2

Preparation of graphite flakes and Al powders mixtures

49

2.2.3

Fabrication of graphite flakes/Al composites by sparking plasma sintering51

2.3

Evaluation methods

53

2.3.1

Microstructures and relative density

53

2.3.2

X-ray powder diffraction method

53

2.3.3

Effective thermal conductivity

53

2.3.4

Coefficient of thermal expansion

55

2.4 Results and discussion

56

2.4.1

Microstructure characterization and orientation of GFs in composites 56

2.4.2

Interface reaction

60

2.4.3

Effective thermal conductivity and coefficient of thermal expansion

62

2.5

Summary69

2.6

References71

45

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

2.1 Introduction
Carbon materials reinforced Al-matrix composites attract great interest fromresearchers of electron packaging materials due to their lightweight, superiormechanical, and thermal properties [1-7]. Especially for graphite flakes (GFs) reinforced
Al-matrix composites, their excellent thermal properties, such as high thermal
conductivity (TC) and low coefficient of thermal expansion [5, 8-10], are able to meet thethermal performance requirements of electron packaging materials.
Lots of experiments have been performed to study the effective thermal
conductivity (ETC) of GFs/Al composites and found that the high thermal conductivityof GFs cannot be fully displayed due to the following characteristics: 1, the thermal
conductivity of GFs is significant anisotropic, its orientation in the composites hinder
the effective thermal conductivity of GFs/Al composites; 2, GFs tends to react with Al
during the fabricating the GFs/Al composites, the formation of Al4C3 is bad for themechanical properties and reduce the effective thermal conductivity of GFs/Al
composites; 3, the interfacial thermal resistance is considered to have a great effect on
the effective thermal conductivity of GFs/Al composites. Therefore, a suitablefabrication process of GFs/Al composites is essential to take advantage of the highthermal conductivity properties of GFs.
In previous studies on the effective thermal conductivity of GFs/Al composite, thecommonly used fabrication methods of GFs/Al composite include the vacuum hotpressing[9-12], squeeze casting[13-15], and gas pressure infiltration[8, 16]. The interface
reaction can be effectively avoided using those methods. For example, Chen et al. [12]fabricated the GFs/Al composites by hot pressure method, and a low sintering
temperature, lower than the melting point of Al, was used to avoid the interface reaction.
The ETC of GFs/Al composites with 80 vol. % GFs achieved up to 783 W m-1 K-1. Xueet.al[10]. fabricated GFs/Al composites by vacuum hot pressing method, the surface of
GFs were pre-processed with SiC particles coating to inhibit the interface reaction. The
ETC of GFs/Al composites with 70 vol. % GFs was up to 735 W m-1 K-1. Li et al.[13]46Chapter 2 Fabrication and Thermal Properties of GFs/Al composites

prepared GFs/Al composites by squeeze casting method, and rapid cooling method wasused to hinder the interface reaction. The effective thermal conductivity of the sample
with 70 vol. % GFs was as high as 714 W m-1 K-1. Zhou et al.[8] used pressure infiltration
to prepare GFs/Al composites. Si particles were introduced to suppress the interface
reaction. The effective thermal conductivity of the sample with 71.1 vol. % GFs was
526 W m-1 K-1. The interface reaction can be hinted at by low sintering temperature,
rapid cooling, or adding Si or SiC particles. However, the effective thermal
conductivities of GFs/Al composites were lower than the expected theoretical effective
thermal conductivities. The main reasons can be attributed to the effects of theorientation of GFs and the interfacial thermal resistance between Al and GFs.
This study employed the spark plasma sintering (SPS) method to prepare GFs/Al
composites. SPS is a low-temperature sintering technology. The low sintering
temperature can avoid the interface reaction, and SPS can generate a plasma
atmosphere to effectively destroy the oxide film on the surface of Al powders, increase
new metallic contacts and necks, and reduce the porosity [17-20]. The GFs/Al composites
with 10-40 vol. % of GFs were prepared. Orientations of GFs in GFs/Al composites
were analyzed using the OM images; The interface reaction was detected by the XRD
method; The effective thermal conductivity and the coefficient of thermal expansion of
GFs/Al composites were evaluated by experimental and theoretical methods.

47

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

2.2 Experimental procedure
2.2.1 Raw materials
The raw materials consisted of Al powders and GFs. The institute of high purity
chemistry offered Al powders in Japan, and Ito Graphite Industry Co., Ltd. offered thehighly crystalline GFs in Japan. Figure 2-1 shows the microstructure of Al powder and
GFs. Al powders show a rugby shape, and their size is uneven, while GFs show a flat
surface and uniform thickness. Table 2-1 shows the chemical components of the raw
materials. The purity is as high as 99.9% for Al powder and 98.8% for GFs. GFs arehighly crystalline, which can hinder the chemical reaction between Al and GFs. Table
2-2 shows some physical properties and thermal conductivity of Al powder and GFs.

(a)

(b)

Figure 2-1 Microstructures of raw materials: (a) Al powder and (b) GFs.

Table 2 -1 Chemical components of Al and GFs.
CarbonAshVolatile

Material

H2O

Cu

Fe

Si

%

Al
GFs (XD150)

Al

99.9 0.001 0.06 0.03
98.78

0.61

0.61

48

0.27

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites

Table 2-2 Shape and physical properties of Al and GFs.
d

ρ

Cp

TC

µm

g·cm-3

J· kg-1· K-1

W·m-1·K-1

powder

30

2.7

880

236

flake

137.02

2.23

710

Materials

Shape

Al
GFs

880 (in-plane)
38 (out-plane)

where d is the average diameter, ρ is density, and Cp is the specific heat.
2.2.2 Preparation of graphite flakes and Al powders mixtures
The wet mixing process was used to mix Al powders and GFs. First, the Al powders,
GFs, and alumina balls were filled into an aluminum jar. Subsequently, the mixtures
were mixed for 2 h using a V-type mixer at 50rpm. Finally, the mixtures were dried ina constant temperature oven at 70 ℃. The mixing procedure of raw material Al powder
and GFs was shown in Figure 2-2.

Figure 2-2 Flowchart of GFs and Al powder mixing.

According to Xu’s studies [21], the morphology of the Al particle would not be
destroyed when the diameter of the Alumina ball is 10 mm. Thus, the Alumina ball with
10 mm diameter was used in the mixing process of Al powder and GFs. Besides, thechoice of dispersion solution is crucial for the uniform mixing of GFs and Al powder.
Because the density of GFs is lower than that of Al, the low-viscosity dispersion
solution may cause Al particles and GFs to separate during the mixing process; Al
particles settle at the bottom of the mixed solution, while GFs are in an aggregate state.
Commonly used hybrid solutions are ethanol, butanol, and acetone, and their viscosity49

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Imageand density are shown in Table 2-3. The sedimentation velocity (V) of Al powders with
30µm diameter in ethanol(Ve), butanol (Vb), and acetone (Va) can be evaluated by thefollowing equation [22]:𝑉 = 𝑑 ∙ 𝑔 ∙ (𝜌 − 𝜌 )/18𝜂
…… (2-1)
where the d is the diameter of Al particle, g is gravity, ρAl and ρs are the density of the
Al powders and dispersion solution, and η is the viscosity of the dispersion solution.

Table 2-3 Viscosity and density of ethanol, butanol and acetone.
Dispersion solution

Butanolethanol

AcetoneViscosity (η, 10-3 Pa·s)

2.95

1.07

0.32

Density (ρ, 106 g/m3)

0.81

0.80

0.79

Figure 2-3 shows the calculated result. The sedimentation velocity is Vb < Ve < Va.
Therefore, butanol should be the most suitable dispersion solution in the wet mixing
process. However, the higher viscosity of butanol can cause a lower drying speed of themixed solution of GFs and Al powders, and there may be residual butanol in the
mixture[21]. Moreover, the difference between Vb and Ve is not large (as shown in Figure2-3). Thus, ethanol is used as the dispersion solution in the mixing process of GFs andAl powders.

Figure 2-3 Sedimentation velocity of Al powders with 30µm diameter in ethanol,
butanol, and acetone [21].
50Chapter 2 Fabrication and Thermal Properties of GFs/Al composites

2.2.3 Fabrication of graphite flakes/Al composites by sparking plasma sintering
Figure 2-4 shows the equipment of SPS. The mixed powders were filled into agraphite mold (graphite die) to fabricate GFs/Al composites. The mold was tapped
during the filling of mixed powders in order to stack GFs in layers. Then, spark plasma
sintering (SPS) was conducted at 873 K, and 60 MPa pressure for 0.5 h under a vacuumof 1.3 × 10–2 Pa. Sintering temperature was kept below the melting point of Al toprevent the reaction between Al and GF. The temperature was increased at a rate of 200
K/min. The details of the sintering procedure were shown in Figure 2-5. A pure Al
sample was fabricated by SPS under the same condition as those composites. The
dimensions of the sintered GFs/Al composites were 65 ◊ 10 ◊ 15 mm3.

Figure 2-4 Schematic of the SPS equipment.

51

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

Removing the oxide
layer on the surface of
Al powders

Heating
process

Isothermal process

Cooling

Temperature curve

Pressure

15 MPa

Time

900 s

60 MPa≤ 300 s1800 s

Temp.

873 K
Figure 2-5 Flow chart of spark sintering process.

52< 323 K

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites

2.3 Evaluation methods
2.3.1 Microstructure and relative density
The electron probe micro-analyzer (EPMA) and an optical microscope (OM) wereused to observe the microstructure of the composites and the mixtures of Al and GFs
powders; the relative densities of composites were measured using the Archimedes
method.

2.3.2 X-ray powder diffraction method
The formation of Al4C3 was investigated by the X-ray powder diffraction method(XRD) (D/MAX-2500/PC). The measurement conditions are listed in Table 2-4.
Table 2-4 Measurement conditions of XRD.
TargetCu

Wavelength (10-10 m)

1.54056 (Kα1)

Power (kW)≤ 18

Voltage (kV)40

Current (mA)200

Sampling step (degree)

0.02

Scan speed (degree/min)

1

2.3.3 Effective thermal conductivity
The effective thermal conductivities of GFs/Al composites were measured at 25 ℃
using a steady-state thermal-conductivity measuring device[23], as shown in Figure 2-5.
The device consists of a heating end and a cooling end. The heating end includes aheating plate and a hot bar. The heating plate is connected to the cartridge heaters, anda heater controller controls the temperature (TH). The cooling end also includes twocomponents: a cooling plate and a cold bar. The cooling plate is connected to the water53

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

cooled chiller through the cooling pipes. The water-cooled chiller controls thetemperature of the cooling plate (TC). There are three thermocouples fixed on the hot
bar and the cold bar, respectively. The hot bar and cool bar are cylindrical copper rods
with a diameter of 10 mm. When measuring the effective thermal conductivity, thesample is sandwiched between the hot bar and cold bar and contacts threethermocouples. After the temperature of the nine thermocouples reaches a steady state,
the temperatures for no less than 300 seconds at a 1-second interval are collected by thedata acquisition system. Subsequently, the temperature gradients of the hot bar, coldbar, and sample ( △ Th, △ Ts, △ TC) are calculated by a least-square method as
follows[23]:

𝛥𝑇 =

∑

(𝑍 − 𝑍̅ )(𝑇 − 𝑇 )1, 𝑍̅ =
̅∑ (𝑍 − 𝑍 )3

𝑍 ,𝑇 =1
3

𝑇…… (2-2)

𝛥𝑇 =∑(𝑍 − 𝑍)(𝑇 − 𝑇 ),∑ (𝑍 − 𝑍 )

𝑍̅ =

1
3

𝑍 ,

𝑇 =

1
3

𝑇
…… (2-3)

𝛥𝑇 =∑(𝑍 − 𝑍̅ )(𝑇 − 𝑇 ),∑ (𝑍 − 𝑍̅ )

𝑍̅ =

1
3

𝑍 ,

𝑇 =

1
3

𝑇
…… (2-4)

where Zi is the distance between the i-th thermocouple and the heating plate, and Ti isthe temperature of the i-th thermocouple. Finally, the effective thermal conductivity ofthe sample (λeff) can be calculated by fowling equation:

𝜆= 𝜆 𝐶(𝛥𝑇 + 𝛥𝑇 )/2𝛥𝑇…… (2-5)

where λCu is the thermal conductivity of copper bar (λCu = 385 Wm-1K-1); C is acorrection factor, and it is a constant value (i.e., C = 236/257).

54

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites

Heating Plate
TH

HeaterController
z1
S
z2
U
P
z3
P
z4
O
z
z5
R
T
z
6

Water-cooled
Chiller

B
A
R

z7
z8
z9

T1

Hot
Bar
λR

T2
T3
T4

Sample
λx

T5

Cold
Bar
λR

T7

Data Acquisition
SystemT6
T8
T9

Cooling PlateTC

Figure 2-5 Schematic diagram of the steady-state thermal conductivity measurement
device [23].

2.3.4 Coefficient of thermal expansion
The heat expansion of GFs/Al composites was measured from 25 to 200 ℃ by athermal expansion meter (NETZSCH DIL 402 C).

55

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

2.4 Results and discussion
2.4.1 Microstructure characterization and orientation of GFs in composites
Figure 2-6 shows the microstructure images of the mixtures of Al and 10-40 vol.%GFs. GFs were evenly dispersed in the Al powders, and GFs in Figure 2-6 (a-d) havethe same particle size. Moreover, compared with Figure 2-1, GFs did not show the
obvious fracture, its average particle size was still about 130μm, and the morphologyof Al powder was no obvious change.
(a)

(b)

(c)

(d)

Figure 2-6 SEM images of the mixtures of Al and 10-40 vol. % GFs. 10 vol.%GFs (a), 20 vol.% GFs (b), 30 vol.% GFs (c), and 40 vol.% GFs (d).

10-40 vol. % GFs/Al composites and one pure Al sample were prepared by SPS.
Their relative densities were over 98%, except for the pure Al sample being at 97%.
The high relative density indicated that the fabricated GFs/Al composites were
densified. The detailed values of the relative densities were listed in Table 2-5.

56

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites

Table 2-5 Relative density and effective thermal conductivity (ETC) of GFs/Al
composites.
SampleAl

GFs/Al composites

Volume fraction of GFs

0

10 vol.%20 vol.%30 vol.%40 vol.%Relative density

97.0%

99.4%

99.3%

98.6%

97.1%

ETC (λeff, Wm-1K-1)

217

248

280

313

346

Figure 2-7 depicts the shape of the samples sintered via SPS. The black arrows
represent that C-plane was bearing the pressure during sintering. Therefore, GFs in the
Al-matrix were stacked by layers, and the basal plane of GFs is parallel to the C surface.
In order to better observe the orientation of GFs in the Al-matrix, the OM images of the
GFs/Al composites were obtained from the A surface.

Figure 2-7 Schematic of the GFs/Al composites sintered by SPS. The grey arrow
indicates that the heat flow was from B to D when measuring the ETC of the sample.

Figure 2-8 shows the OM images of 10-40 vol.% GFs/Al composites. The grey phases
are of the Al matrix, and the elongated dark phases are GFs. The GFs were
homogeneously distributed in the Al matrix and stacked by layers as expected. Most
GFs were parallel to the heat flow direction, except for a few GFs whose basal plane
direction formed an angle with the direction of the heat flow. In addition, as shown in
Figure 2-8, with the increase of GFs content from 10 vol.% to 40 vol.%, GFs exhibits
aggregation along the direction perpendicular to the heat flow direction, and GFs tend
to form a network system by connecting GFs. As Chen et al. [12] reported, networks
57

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

40vol.%GFs/Al

30vol.%GFs/Al

20vol.%GFs/Al

10vol.%GFs/Al

formed by connecting GFs can effectively improve the ETC of composites.
(a

(b)

(c)

(d)

200μm
Figure 2-8 OM images of 10-40 vol.% GFs/Al composites.

An angle between GFs basal plane and heat flow direction was used to characterize
the orientation of GFs in composites. The angles were counted statistically, and theresults were shown in Figure 2-9. The x-axis showed that almost 70% of the angles
were less than 10 degrees, which meant that nearly 70% of the GFs were oriented
roughly parallel to the direction of heat flow. GFs parallel to each other is expected toenhance the overall thermal conduction.

58

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites(a) 10 vol.% GFs/Al(b) 20 vol.% GFs/Al(c) 30 vol.% GFs/Al(d) 40 vol.% GFs/Al

Figure 2-9 Distribution of GFs’ orientation in the 10-40 vol.% GFs/Al composites, theorientation was described by the angle between GFs basal-plane direction and heatflow direction.

Figure 2-10 shows the statistical averages of the aspect ratio from the GFs in Figure
2-8. The aspect ratios values were mainly located at (1/10, 3/10]. Besides, the average
diameters of GFs in composites (d) were calculated by equivalent circle area (i.e., 𝑑 =2 𝑆⁄𝜋, S is the area), the values were 1.62, 2.54, 3.89, and 5.41 mm in 10~40 vol.%GFs/Al composites, respectively.

59

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

Figure 2-10 Statistical counting of the aspect ratio of GFs.

2.4.2 Interface reaction
As Etter et al. [24] reported if Al and GFs undergo a chemical reaction during the
manufacturing process of GFs/Al composites, the formation of Al4C3 exists at the AlGFs interface. Therefore, XRD was used to determine the interface reaction at the AlGFs interface, as shown in Figure 2-11. The result showed that the Al4C3 phase wasabsent in 10-40 vol.% GFs/Al composites. The reason can be attributed to the following
two points: first, the sintering temperature was less than the melting point of Al[25];second, the raw material GFs was high graphitization[26]. On the other hand, hightemperature sintering methods were also commonly used to fabricate GFs/Al
composites in the liquid metallurgy process. For example, Huang[9] and Chen [12] usedvacuum hot pressing process to fabricate GFs/Al composites, Zhou[8] and Chang[16]used the pressure infiltration method to fabricate GFs/Al composites, and Li[13] used
60

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites

squeeze casting technique to fabricate GFs/Al composites. The vacuum hot pressing
process, pressure infiltration, and squeeze casting belong to the liquid metallurgy
process, and a higher temperature than the melting point of Al is required to fabricate
GFs/Al composites. Therefore, in order to hinder the formation of Al4C3, a high coolingrate was used in the liquid metallurgy process to reduce the reaction time between Al
and GFs. Because the formation of Al4C3 is associated with the reaction temperatureand reaction time, according to the equation 𝑋 = √2𝐾𝑡, where X is the thickness ofreaction layer, t is the reaction time, and K is the reaction rate related to the reaction
temperature[27, 28].

Figure 2-11 XRD analysis of GFs/Al composites.

Moreover, Huang et al. [9], Chen et al. [12], Zhou et al. [8], Chang et al. [16], and Li etal. [13] examined the microstructure and composition of the Al-GFs interface by
HRTEM. Their studies revealed that an amorphous layer interface was formed at the
Al-GFs interface, and the amorphous layer can be frequently observed, as shown in
Figure 2-12 [9]. Therefore, we assumed the amorphous layers were formed at the Al-Gfs
61

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

interfaces in the 10-40 vol.% GFs/Al composites.

Figure 2-12 HRTEM image of the Al-GFs interface [9]: (a) HRTEM image; (b) FFT ofamorphous layer; (c–d) FFT of crystalline graphite; (e) FFT aluminum.

2.4.3 Effective thermal conductivity and coefficient of thermal expansion
The effective thermal conductivities of 10-40 vol. % GFs/Al composites and thepure Al sample were measured by the steady-state thermal conductivity measuring
device (see Figure 2-5). Figure 2-13 (a-d) shows the temperature variation in the ETC
measurement process. The temperatures were a minor fluctuation and approximately
maintained constant value, indicating that the measuring temperature was steady.
Figure 2-13 (a1-d1) shows the average temperature with respect to the distance between
thermocouples and the heating plate. The solid black dot is the average temperature of
the thermocouples, the solid black line is the temperature gradient and is calculatedusing the equations (2-2) to (2-4). The dashed line is the location of the interface
between the sample and hot bar or cold bar, and the corresponding temperature gap
indicates the temperature difference at the interface.

62

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites(a)(a1) 10 vol.% GFs/Al(b)(b1) 20 vol.% GFs/Al(c)(c1) 30 vol.% GFs/Al(d)(d1) 40 vol.% GFs/Al

Figure 2-13 Temperature variation in the ETC measurement process of 1040vol.%GFs/Al composites, (a-d) relationship between temperature and time, and (a1d1) relationship between temperature and distance of the measuring points.
63

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure ImageThe ETCs of 10-40 vol.% GFs/Al composites and the pure Al sample were
calculated by equations (2-2) to (2-5) and listed in Table 2-5. As a result, the TC of Al
was 217 Wm-1K-1, and the ETC of GFs/Al composites λeff was increased from 248 to
346 Wm-1K-1 as GFs volume fraction increasing from 10 to 40 vol.%. Moreover, thetheoretical ETC (λTheo.) of GFs/Al composites were calculated using the layer-inparallel model[29] and the effective medium approximation (EMA) model.
The layer-in-parallel model considers the basal-plane of GFs parallel to heat flow
direction, ETC (λTheo.) of GFs/Al composites is calculated as follow:

𝜆

. =

𝑣 _

𝜆 _

+𝑣

𝜆
…… (2-6)

where v is volume fraction, λ is thermal conductivity, the subscript Al_matrix and GFs
are Al-matrix and GFs, respectively. If to consider the effect of interfacial thermal
resistance (h) on ETC, the TC of GFs (λGFs) can be replaced by 𝜆

𝜆

= 𝜆

/(

, given by[15]:

2𝜆
+ 1)
ℎ𝑑
…… (2-7)

where a value of h, 5 × 107 W m-2 K-1 [30] was used. Thus, equation (2-6) can be
expressed as:
𝜆

. =

𝑣 _

𝜆 _

+𝑣

𝜆

/(

2𝜆
+ 1)
ℎ𝑑
…… (2-8)

The EMA model considers the geometrical factor and h to evaluate the ETC ofcomposites, and the equation can be expressed as [15]:𝜆= 𝜆 (1 +𝑣
𝜋(14𝑝 − 𝑣

)+

1
𝜆
𝜆

−1
(2-9)

where p is the aspect ratio of GFs, the value was 5 (see Figure 2-10).
64

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites

Moreover, the measured relative densities of GFs/Al composites and pure Al sample
were less than 1, indicating that some pores were embedded in the samples. However,equations (2-6), (2-8), and (2-9) do not include the effect of pores on the ETC. Therefore,
the pores in this study were presumed to be embedded within the Al matrix. The TC of
the pure Al sample and the Al-matrix, λAl_eff, can be calculated by the equation derived
by Landauer [31]. The equation was as follow:

𝜆 _1= [𝜆 3𝑣 − 1 + 𝜆 (3𝑣 − 1)4+

𝜆 3𝑣 − 1 + 𝜆 (3𝑣 − 1)

+ 8𝜆 𝜆]…… (2-10)

where the λAl is the thermal conductivity of Al powder, λp is the thermal conductivity ofpores, vp and vAl are the volume fraction of Al and pores. In addition, the values of λAl,λp, vp, and vAl were listed in Table 2-6.

Table 2-6 Calculation parameters in equation (2-7).
λp

λAl

λAl_eff

λAl_matrix

0.03

0.214

228

217

-

0.994

0.06

0.214

228

225.9

225.9

20 Vol.% GFs/Al

0.993

0.07

-

-

225.6

225.6

30 Vol.% GFs/Al

0.986

0.014

-

-

223.2

223.2

40 Vol.% GFs/Al

0.971

0.029

-

-

218.1

218.1

Samples

vAl

vp

Al

0.97

10 Vol.% GFs/Al

First, by substituting the measured TC (217 Wm-1K-1) and the relative density (0.97)
of the pure Al sample into the equation (2-10), the TC of Al powder (λAl) was calculated
for 228 Wm-1K-1. Second, the TCs of the Al-matrix for 10-40 vol.% GFs/Al composites
were calculated by substituting the λAl, λp, vp, and vAl from Table 2-6 into equation (210). The calculated values of λAl_eff were 225.9, 225.6, 223.2, and 218.1 Wm-1K-1 for
10-40 vol.% GFs/Al composites, respectively. λAl_eff was equal to λAl_matrix. Therefore,65

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

the theoretical ETC of the 10-40 vol.% GFs/Al composites can be calculated byequations (2-6), (2-8) and (2-9), and the values were listed in Table 2-7.

Table 2-7 Theoretical ETC of GFs/Al composites.
GFs/Al composites
ETC

10 Vol.% GFs 20 Vol.% GFs 30 Vol.% GFs 40 Vol.% GFs

λeff. (Wm-1K-1)

248

280

313

346

λTheo. (Wm-1K-1)

291

357

420

483

K-1)

289

353

417

480

λEMA (Wm-1K-1)

271

320

370

422

𝜆

-1

. (Wm

Figure 2-14 shows a comparison between the measured and theoretical ETCs. The
theoretical value was considerably higher than the measured value λeff, and the deviation
between the measured and the theoretical ETCs increased with GFs content. The largest
deviation between the measured and theoretical ETCs was up to 28.4%.

Figure 2-14 Comparison of measured and theoretical ETC of 10-40 vol.% GFs/Al
composites.

66

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites

The value of λTheo was approximately equal to that of 𝜆

, indicating that theeffect of interfacial thermal resistance on the ETC was small. The deviation between
λTheo and λeff can be attributed to the effects of the orientation of GFs and the interface
thermal resistance. In addition, Zhou et al.[15] calculated the ETC of GFs/Al composites
using the EMA approach, and the calculated values were agreed with the experimental
values. However, the λEMA were greater than the λeff. The deviation between them canbe attributed to the effect of the orientation of GFs.
The coefficient of thermal expansion of 10-40 vol.% GFs/Al composites were
measured along the direction parallel to the basal plane of GFs, and the results weresummarized in Table 2-7. Figure 2-15 (a) shows the thermal expansion at temperaturesfrom 25 to 200 ℃. The thermal expansion of GFs/Al composites decreased with the
GFs content under the temperature from 25 to 200 ℃, indicating that the coefficient ofthermal expansion of GFs/Al composites decreases with the increase of GFs content.

Table 2-7 Coefficient of thermal expansion.
SamplesAl

αExp. (ppm/K)
αCom. (ppm/K)

23.8[12]

GFs content (vol.%)

GFs-1.5[34-35]
25[37]

10

20

30

40

26.2

24.4

23.4

22.1

21.27

18.74

16.21

13.68

23.92

24.04

24.16

24.8

(a)

(b)

Figure 2-15 Thermal expansion variation versus temperature (a), coefficient ofthermal expansion (CTE) versus GFs content (b).

67

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

To calculate the coefficient of thermal expansion of composite, the commonly usedtheoretical models include the rule of mixture [12], Turner model [32], Kerner model [33],and Schapery model [34]. But Chen et al.[12] and Kurita et al.[11] revealed that the Turnerand Kerner models were failed to predict the coefficient of thermal expansion of 10-40
vol.% GFs/Al composites. Thus, the mixture rule here was used to evaluate thecoefficient of thermal expansion of 10-40 vol.% GFs/Al composites. The rule of the
mixture is described as follows:𝛼. = 𝑣 _𝛼 _+𝑣

𝛼…… (2-8)

where α is the coefficient of thermal expansion, and the subscript Com. is composite.
The coefficient of thermal expansion is 23.8 ppm/K[12] for Al and -1.5ppm/K[35, 36] for
GFs along the basal plane direction and 25ppm/K [37] for GFs along the direction
perpendicular to the basal plane. The calculated values were list in Table 2-7. Figure 215 (b) shows the measured and calculated coefficient of thermal expansion. The blackline represents the experimental coefficient of thermal expansion, the experimental
values of thermal expansion decreased with the increase of GFs content. The blue line
presents the coefficient of thermal expansion parallel to GFs basal plane, the red line
presents the coefficient of thermal expansion perpendicular to GFs basal plane. Because
the orientations of some GFs were disorder in composites, the coefficient of thermal
expansion of composite should be located between the red and bule lines. However, theblack line was intersected with the blue line. The experimental values for thecomposites with 10-20 vol.% GFs were higher than the calculated values, and theexperimental values exceeded even the coefficient of thermal expansion of Al and GFs(25 ppm/K[37] along the direction perpendicular to the basal plane). Besides, the blackline was around the blue line, the reasons can be attributed to pores embedded in thecomposites and the disorder orientations of GFs.

68

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites

2.5 Summary
GFs/Al composites with 10-40 vol.% GFs were fabricated by using spark plasma
sintering in this chapter. Subsequently, the following studies on the GFs/Al composites
were completed: 1) The relative density of composites was measured by theArchimedes method; 2) The microstructure images analyzed the distribution andorientation of GFs in the composites; 3) The formation of Al4C3 was investigated by
XRD method; 4), the effective thermal conductivities of the composites were measured
using a steady-state thermal conductivity measurement device and calculated bytheoretical methods; 5) The coefficient of thermal expansions of the composites were
measured by a thermal dilatometer (NETZSCH DIL 402 C) and calculated by the ruleof mixture. Finally, the following conclusions were obtained.
1) The relative density was up to 99% for the GFs/Al composites; with the GFs
content up to 40%, the relative density of composites decreased to 97.1%.
2) GFs were homogeneously distributed and stacked by layers in the Al matrix.
The orientations of nearly 70% GFs were parallel to each other. Moreover, with
the GFs content up to 40 vol.%, GFs exhibits aggregation and tend to form the
network system by connecting GFs.
3) The formation of Al4C3 was absent in the 10-40 vol.% GFs/Al composites, but
amorphous layers at the Al-GFs interface may be formed at the interface.
4) The measured effective thermal conductivities were 248, 280, 313, and 346 W
m-1 K-1 for 10-40 vol.% GFs/Al composites, respectively. The effects oforientation of GFs and interface thermal resistance caused the ETC of GFs/Al
composites to be lowered by 28.4%.
5) The experimental values of the coefficient of thermal expansion were 26.2,
24.4, 23.4, and 22.1 ppm/K for 10-40 vol.% GFs/Al composites, respectively.
Moreover, because of the effects of the pores embedded in composites and thedisorder orientations of GFs, the experimental values for the composites with
10-20 vol.% GFs were higher than the calculated values perpendicular to GFs
basal plane, and the experimental values exceeded even the coefficient of69

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

thermal expansion of Al and GFs (25 ppm/K along the direction perpendicular
to the basal plane.

70

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites

2.6 References[1] Chung D. Properties of Carbon Fibers, Carbon Fiber Composites. ButterworthHeinemann, Newton 1994.
[2] Miracle D. Metal matrix composites–from science to technological significance.
Composites science and technology. 2005;65(15-16):2526-40.
[3] Shirvanimoghaddam K, Hamim SU, Karbalaei Akbari M, Fakhrhoseini SM,
Khayyam H, Pakseresht AH, et al. Carbon fiber reinforced metal matrix composites:
Fabrication processes and properties. Composites Part A: Applied Science andManufacturing. 2017;92:70-96.
[4] Qu X-h, Zhang L, Wu M, Ren S-b. Review of metal matrix composites with highthermal conductivity for thermal management applications. Progress in Natural Science:
Materials International. 2011;21(3):189-97.
[5] Inagaki M, Kaburagi Y, Hishiyama Y. Thermal management material: graphite.
Advanced Engineering Materials. 2014;16(5):494-506.
[6] Biercuk M, Llaguno MC, Radosavljevic M, Hyun J, Johnson AT, Fischer JE.
Carbon nanotube composites for thermal management. Applied physics letters.
2002;80(15):2767-9.
[7] Kidalov S, Shakhov F. Thermal Conductivity of Diamond Composites. Materials.
2009;2(4):2467-95.
[8] Zhou C, Ji G, Chen Z, Wang M, Addad A, Schryvers D, et al. Fabrication, interface
characterization and modeling of oriented graphite flakes/Si/Al composites for thermal
management applications. Materials & Design. 2014;63:719-28.
[9] Huang Y, Ouyang Q, Guo Q, Guo X, Zhang G, Zhang D. Graphite film/aluminum
laminate composites with ultrahigh thermal conductivity for thermal management
applications. Materials & Design. 2016;90:508-15.
[10] Xue C, Bai H, Tao PF, Wang JW, Jiang N, Wang SL. Thermal conductivity andmechanical properties of flake graphite/Al composite with a SiC nano-layer on graphite
surface. Materials & Design. 2016;108:250-8.
[11] Kurita H, Miyazaki T, Kawasaki A, Lu Y, Silvain J-F. Interfacial microstructure of71

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Microstructure Image

graphite flake reinforced aluminum matrix composites fabricated via hot pressing.
Composites Part A: Applied Science and Manufacturing. 2015;73:125-31.
[12] Chen JK, Huang IS. Thermal properties of aluminum–graphite composites bypowder metallurgy. Composites Part B: Engineering. 2013;44(1):698-703.
[13] Li W, Liu Y, Wu G. Preparation of graphite flakes/Al with preferred orientation andhigh thermal conductivity by squeeze casting. Carbon. 2015;95:545-51.
[14] Zhou C, Chen D, Zhang XB, Chen Z, Zhong SY, Wu Y, et al. The roles of geometry
and topology structures of graphite fillers on thermal conductivity of thegraphite/aluminum composites. Physics Letters A. 2015;379(5):452-7.
[15] Zhou C, Huang W, Chen Z, Ji G, Wang ML, Chen D, et al. In-plane thermalenhancement behaviors of Al matrix composites with oriented graphite flake alignment.
Composites Part B: Engineering. 2015;70:256-62.
[16] Chang J, Zhang Q, Lin Y, Wu G. Layer by layer graphite film reinforced aluminumcomposites with an enhanced performance of thermal conduction in the thermal
management applications. Journal of Alloys and Compounds. 2018;742:601-9.
[17] KURAMOTO H, MATSUGI K, KAWAHARA K, YANAGISAWA O.
Densification rate of Cu-Al2O3 composite in the spark sintering process. Nippon
Kinzoku Gakkaishi (1952). 2003;67(10):528-37.
[18] Xu Z-F, Choi Y-B, Matsugi K, Li D-C, Sasaki G. Mechanical and thermal
properties of vapor-grown carbon fiber reinforced aluminum matrix composites byplasma sintering. Materials transactions. 2010:1002010992-.
[19] Kubota M. Properties of nano-structured pure Al produced by mechanical grinding
and spark plasma sintering. Journal of Alloys and Compounds. 2007;434:294-7.
[20] Xie G, Ohashi O, Yoshioka T, Song M, Mitsuishi K, Yasuda H, et al. Effect ofinterface behavior between particles on properties of pure Al powder compacts by spark
plasma sintering. Materials transactions. 2001;42(9):1846-9.
[21] Xu Z. Fabrication and Properities Evaluations of VGCNFs/Al Composites[D].
Department of Mechanical Science and Engineering Graduate School of Engineering
Hiroshima University, Japan 2010.
72

Chapter 2 Fabrication and Thermal Properties of GFs/Al composites[22] Bochniak W. The Cottrell-Stokes law for FCC single crystals. Acta metallurgica etmaterialia. 1993;41(11):3133-40.
[23] Sugio K, Yamada R, Choi YB, Sasaki G. Effect of the Interfacial Thermal
Resistance on Effective Thermal Conductivity of Al/SiC Particle-Dispersed
Composites. Materials Science Forum. 2016;879:1889-94.
[24] Etter T, Schulz P, Weber M, Metz J, Wimmler M, Löffler JF, et al. Aluminium
carbide formation in interpenetrating graphite/aluminium composites. Materials
Science and Engineering: A. 2007;448(1-2):1-6.
[25] Mizuuchi K, Inoue K, Agari Y, Morisada Y, Sugioka M, Tanaka M, et al. Processingof diamond particle dispersed aluminum matrix composites in continuous solid–liquid
co-existent state by SPS and their thermal properties. Composites Part B: Engineering.
2011;42(4):825-31.
[26] Maire J, Mering J. Graphitization of soft carbons. Chemistry and physics of carbon.
1970;6:125-90.
[27] Genlgl L, Chiai S, Peng H, Gao L, Sun J, Sun Z. FABRICATION OF
NANOCRYSTALLINE ZrO, PARTICLE REINFORCED ALUMINUM ALLOYCOMPOSITE BY SQUEEZE CASTING ROUTE. 1997.
[28] Bakshi SR, Agarwal A. An analysis of the factors affecting strengthening in carbon
nanotube reinforced aluminum composites. Carbon. 2011;49(2):533-44.
[29] Truong H, Zinsmeister GE. Experimental study of heat transfer in layered
composites. International Journal of Heat and Mass Transfer. 1978;21(7):905-9.
[30] Schmidt AJ, Collins KC, Minnich AJ, Chen G. Thermal conductance and phonontransmissivityofmetal–graphite

interfaces.

JournalofApplied

Physics.

2010;107(10):104907.
[31] Landauer R. The electrical resistance of binary metallic mixtures. Journal ofApplied Physics. 1952;23(7):779-84.
[32] Turner PS. Thermal-Expansion Stresses in Reinforced Plastics. Research of the
National Bureau of Standards. 1946;37:239-50.
[33] KERNER EH. The Elastic and Thermo-elastic Properties of Composite Media.
Proc Phys Soc Sec B. 1956;69(8):808-13.
73

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image[34] Schapery RA. Thermal Expansion Coefficients of Composite Materials Based on
Energy Principles. Journal of Composite Materials. 1968;2(3):380-404.
[35] Nelson J, Riley D. The thermal expansion of graphite from 15 c. to 800 c.: Part I.
Experimental. Proceedings of the Physical Society. 1945;57(6):477.
[36] Riley D. The thermal expansion of graphite: part II. Theoretical. Proceedings of
the Physical Society. 1945;57(6):486.
[37] Tsang DKL, Marsden BJ, Fok SL, Hall G. Graphite thermal expansion relationshipfor different temperature ranges. Carbon. 2005;43(14):2902-6.

74

Chapter 3

Relationship between the thermal conductivity of GFs in 3D
and 2D microstructure images
3.1

Introduction

76

3.2

Calculation Procedure

78

3.2.1

Creating 3D and 2D models

78

3.2.2

TC calculation of GFs

80

3.2.3

Image-based simulation method

81

3.3 Results and Discussions

84

3.3.1

Comparison of GFs’s orientations in 2D and 3D models

84

3.3.2

Relationship between aspect ratio of GFs and the orientation of GFs

86

3.3.3

ETC calculation of GFs/Al composites

88

3.4

Summary91

3.5

References

9275

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

3.1 Introduction
Graphite flakes (GFs) reinforced aluminum (Al) matrix composites (GFs/Al) arelight-weight, low cost, and exhibit superior thermal properties, such as high effective
thermal conductivity (ETC) and low coefficient of heat expansion (CTE). Therefore,
GFs/Al composites are considered as one of the most promising thermal management
materials. However, the ETCs of GFs/Al composites are affected by the orientation of
GFs, the interfacial thermal resistance, and the interface reaction between Al and GFs[1-5]. In particular, the interfacial thermal resistance and the orientation of GFs have agreat effect on the ETCs of GFs/Al composites, which results in lower experimental
ETCs than expected.
The commonly used theoretical methods using to calculate the ETC of GFs/Al
composites include mixing rules [1, 2, 6], Fricke equation[7], effective medium approach(EMA)[8], and three-dimensional simulations [9, 10]. However, the orientations of GFs incomposites are uncontrollable diverse in the composites. Therefore, obtaining thereliable ETC of GFs/Al composites is challenging using those theoretical models.
Three-dimensional (3D) image simulations can be used to calculate the ETC of GFs/Al
composites well. But the 3D-images simulation is expensive and has low timelinessbecause the 3D models are constructed using the ultra-high-resolution X-ray computed
tomography images [11]. In addition, two-dimensional (2D) image simulations are also
often used to evaluate the ETC of composites [12, 13], and it is simple, saves time, and is
low cost. However, the orientation of GFs in the 2D microstructure images may not be
equivalent to that in the 3D microstructure images due to a 2D microstructure image
cannot provide the information in the depth direction with respect to the viewing
surface. Therefore, the ETC calculated by the 2D image-based simulation is not reliable.
In order to develop a 2D image simulation method for calculating the ETC of
GFs/Al composites considering the orientation of GFs and the interfacial thermal
resistance, the relationship between the orientation of GFs in 2D and 3D models wasinvestigated. Moreover, the ETC of the 10 vol % GFs/Al composite was calculated76Chapter 3 Relationship between the thermal conductivity of GFs in 3D and 2D microstructure images

using 2D image-based simulation, and the interfacial thermal conductance at the GFsAl interface was calculated using the reversed method.

77

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

3.2 Calculation Procedure
3.2.1

Creating 3D and 2D models

Figure 3-1 shows the flowchart of the calculation procedure for comparing theorientations of GFs in the 2D model and the corresponding 3D model. First, some 3Dmodels were created, as shown in Figure 3-2 (a).

Figure 3-1. Flowchart of the calculation procedure for comparing the orientations of
GFs in two- and three-dimensional models.

The dimensions of the 3D model were 110 × 100 × 100 elements in the X-, Y-, andZ-axis. One GFs was embedded in the center of one 3D model. The shape of GFs wasan elliptical cylinder, the semi-major axis(D) was 40 elements, and the semi-minor axis(d) was 20 elements. The thickness of the ellipse (T) was 10 elements. As shown in
Figure 3-2 (b), the GFs in the 3D model can rotate θx, θy, and θz around the X-, Y-, andZ-axis, respectively, according to the right-hand rotation rule. The rotated coordinates(x, y, z) were calculated using the following rotation matrix:𝑥𝑐𝑜𝑠𝜃
𝑦 = 𝑠𝑖𝑛𝜃
𝑧
0

−𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃
0

0
0
1

𝑐𝑜𝑠𝜃
0
−𝑠𝑖𝑛𝜃

0 𝑠𝑖𝑛𝜃
100 𝑐𝑜𝑠𝜃
78

1
0
0 𝑐𝑜𝑠𝜃
0 𝑠𝑖𝑛𝜃

𝑥0−𝑠𝑖𝑛𝜃 𝑦𝑧𝑐𝑜𝑠𝜃…… (3-1)

Chapter 3 Relationship between the thermal conductivity of GFs in 3D and 2D microstructure images

where (x0, y0, z0) is the initial coordinates. The x and y are set to coincide with the semimajor axis(D) and semi-minor axis(d) of GFs, respectively. In this study, the orientationof GF in 3D models was represented by the angles α, β, and γ, which is the difference
between the global coordinate X-Y-Z and the local coordinate x-y-z. In other words, α,β, and γ were the angles between the axes (X-x, Y-y, and Z-z), and their values were
calculated as follows:

𝛼 = arccos

⃗∙
⃗|

⃗
⃗|

, 𝛽 = arccos

⃗∙ ⃗
⃗|

⃗|

, 𝛾 = arccos

⃗∙ ⃗
⃗|

⃗|

…… (3-2)
where 𝑂𝑋⃗ , 𝑂𝑌⃗ and 𝑂𝑍⃗ are the unit coordinate vectors, and their values are (1,0,0),(0,1,0), and (0,0,1), respectively. 𝑂𝑥⃗ , 𝑂𝑦⃗, and 𝑂𝑧⃗ were calculated using equation (31).

Figure 3-2. Three-dimensional model and orientation of GFs. a) 3D model of GFs/Al
composites; b) Orientation of GFs in 3D model; c) Orientation of GFs in the 2D crosssectional image; (d) Describing the angle between of GFs basal plane and heat flow; α,β, γ and θ’ are the angles between the axes, x-X, y-Y, z-Z, and z-X, respectively; α, β and79

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

γ are within [0,90°]; OM is the line perpendicular to OY and Oz, and θ2D is the angle
between OX and OM; Rectangle O’ABC is the minimum bounding rectangle of GFs,
θ2D is the angle between X-axis and AB; XP is perpendicular to the basal plane of GFs.
Second, as many 2D cross-sections were extracted from each 3D model as possible,
the cross-sections were extracted parallel to the X-O-Z plane, as shown in Figure 3-2(b). The shape of GFs may be irregular in the cross-sectional images; therefore, aminimum bounding rectangle (O’ABC, see Figure 3-2 (c)) of the GFs was introduced
to measure the orientation of the GFs. The orientation of GFs in cross-sectional images
was defined by the angle (θ2D) between GFs and the X-axis, as shown in Figure 3-2 (c).
θ2D was calculated from the 2D cross-sectional images, excluding GFs smaller than 5× 5 elements. Moreover, the aspect ratio (R) of GFs in each extracted 2D cross-sectional
image was also calculated.
3.2.2

TC calculation of GFs

Here, the heat flow direction in the 3D model and the 2D cross-sectional image
was set to be along the X-axis. The TC of GFs along the X-axis thus can be calculated
as follows [14]:𝜆)𝑠𝑖𝑛 𝜃]𝜆𝜆
𝜆 = 𝜆 [1 − (1 − )𝑐𝑜𝑠 𝜃]
𝜆𝜆// = 𝜆 [1 − (1 −…… (3-3)
where λ// is the TC of GFs parallel to the X-axis direction, λa is the TC of GFs in thebasal-plane direction (880 W m-1 K-1), and λc is the TC of GFs in the out-of-plane
direction (38 W m-1 K-1). θ in the 3D models refers to the orientations of GFs relative
to the heat flow direction, which are equal to the angles between the X-axis and basal
plane of the GFs (Figure 3-2 (d)), and it is denoted as θ3D. Thus, θ3D can be obtained asfollows:
θ3D = |90-θ’|, 𝜃 = arccos

⃗∙

⃗

⃗|

⃗|

…… (3-4)80

Chapter 3 Relationship between the thermal conductivity of GFs in 3D and 2D microstructure images

where θ’ is the angle between X-axis and z-axis (Figure 3-2 (b)). In other words, θ’ isthe angle between the heat flow direction and the normal vector of the basal plane, andθ3D is the angle between the heat flow direction and basal plane.
In the 2D cross-section images, θ refers to the angle between the cross-section of
the GFs and the X-axis, i.e., θ = θ2D. Therefore, θ2D can be calculated mathematically,
as follows:𝜃= arccos (𝑂𝑀⃗ ∙ 𝑂𝑋⃗)|𝑂𝑀⃗||𝑂𝑋⃗|…… (3-5)

where OM is the line perpendicular to OY and Oz (Figure 3-2 (b)), and 𝜃

is in therange [0, 90).
3.2.3

Image-based simulation method

The finite-volume method was employed to calculate the 2D temperature
distributions. The temperature of the elements can be calculated using the following
equation [13]:
𝑇,

=𝑇, +

∆𝑡 𝑞
𝜌𝑐

−𝑞
∆𝑥

,

,

+

𝑞 ,

−𝑞 ,
∆𝑦
…… (3-6)

where 𝑇 ,is the temperature of the element at (x, y) coordinates, 𝑇 ,is thetemperature of the element at (x, y) coordinates after a time ∆t, ρ is the density, c is the
specific heat, and △x and △y are the sizes of the elements. The heat flow qn can becalculated as follows:
𝑞
𝑞 ,

,

,

= 𝜆(
= 𝜆

,

△
,

,

△

),

𝑞

,

𝑞 ,

,

= 𝜆(

,

= 𝜆

,

△
,

)

,

△

When the heat flow qn move at the GFs-Al interface:
𝑞

,

= ℎ(𝑇

,

− 𝑇 , ),𝑞81,

= ℎ(𝑇 , − 𝑇,)Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image𝑞 ,= ℎ(𝑇 ,

− 𝑇 , ),𝑞 ,= ℎ(𝑇 , − 𝑇 ,)…… (3-7)

where h is the interfacial thermal conductance between different materials.

Figure 3-3 Simulation model for the ETC calculation.

Figure 3-3 shows the schematics of the simulation model. The model is a sandwich
structure consisting of two heat sources and a composite part. The composite part was
obtained from the microstructure image of the GFs/Al composites. The size of thecomposite part (Nx◊Ny) was 450◊600 elements, and that of the heat source was 5 ◊600 elements (NL = NR = 5 elements). The size of each element was 1.18◊ 10–6 m.
Both the upper and lower sides were the periodic boundary, while the left and rightsides were the adiabatic boundary. The temperatures of the left and right edge elements
were fixed at 301 K and 300 K, respectively. The initial temperature of the otherelements was set at 300 K, and the temperature was iteratively updated until thetemperature variation was lower than 10–13 K. The temperature distribution at thismoment was in a steady state, and the ETC of the GFs/Al composite, λeff, was calculated
as follows:
𝜆

=

△𝑇

𝜆 △𝑇 𝑁−𝑁 △𝑇 −𝑁 △𝑇…… (3-8)

where λAl is the TC of the Al matrix, △T12 is the average value of the temperature
82

Chapter 3 Relationship between the thermal conductivity of GFs in 3D and 2D microstructure images

difference between the first and second columns, and NL, NR, Nx, and Ny are the numberof elements along the corresponding direction.

83

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

3.3 Results and Discussions
3.3.1

Orientations of GFs in 2D and 3D models

The 2D orientations of GFs, i.e., θ2D, were obtained from Equation (5) and the 2Dcross-sectional images, respectively. Moreover, the θ2D from the 2D-cross-sectional
images extracted from the same 3D model showed the same values. Figure 3-4 shows
a comparison of θ2D obtained from the mathematical calculation and the 2D-crosssectional images. The solid line is the θ2D from Equation (3-5), and the open circles arethe θ2D from the 2D-cross-sectional images. Thus, the open circles are on the solid line
(x = y), confirming that the θ2D values obtained from the 2D cross-sectional images
were reliable.

Figure 3-4 Comparison of the θ2D obtained from the mathematical calculation and 2D
cross-sectional image.

The angle difference between θ2D and θ3D (θ2D-3D) was calculated. Figure 3-5 (a)shows θ2D-3D in a standard regular triangle. The black dots represent the calculated
points. The black color indicates that the value of θ2D-3D is zero. The other colors
indicate that the value of θ2D-3D is between 0° and 90°. A few black dots were in the84Chapter 3 Relationship between the thermal conductivity of GFs in 3D and 2D microstructure images

black region, whereas the rest were distributed in other colored regions. In the triangle,
most of the θ2D-3D in the triangle was under 20°, but there were some θ2D-3D thatexceeded 40°. Additionally, the difference between the TCs of GFs in the 2D and 3Dmodels (i.e., TC3D-2D) was shown in Figures 3-5 (b). The maximum value of TC3D-2D
was up to 840 W m-1 K-1. Moreover, comparing Figure 3-5 (a) and Figure 3-5 (b), thevalue of TC3D-2D can be large, even if the value of θ2D-3D was small in the triangle.

(a)

(b)

Figure 3-5. Orientation difference of GFs in the 2D cross-sections and 3D models (a);TC difference of GFs in the 2D cross-sections and 3D models (b).

85

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

3.3.2 Relationship between the aspect ratio of GFs and the orientation of GFs
The aspect ratio of the GFs (R) in each 2D cross-section extracted from the same
3D model showed different values, while the corresponding θ2D was the same as eachother. In order to investigate the relationship between the aspect ratios and theorientations, the average arithmetic value of R was calculated and marked as 𝑅, i.e.,∑

𝑅= ∑(i was the number of R). Figure 3-6 shows the relationships between θ2D-

3D, 𝑅,and θ2D.

(a)

(b)

Figure 3-6. Relationship between the θ2D, θ2D-3D and 𝑅 . Scatter image (a); Nonlinear
polynomial fitting surface (b).
86

Chapter 3 Relationship between the thermal conductivity of GFs in 3D and 2D microstructure images

Figure 3-6 (a) shows that all data was distributed on a curved surface, which meansthat anyone of the θ2D-3D, 𝑅 and θ2D can be expressed by using the curved surfacefunction. Subsequently, a similar curved surface was obtained by the surface fitting
method, as shown in Figures 3-6 (b). The function of the fitted curved surface can be
expressed as:𝜃=𝜃− [A + A × 𝜃

𝑓=(0° < 𝜃×𝑓−𝐴 ×𝜃

× 𝑒(

× )

]

1
1+𝑒 (

×

)

< 90° , 1 < 𝑅 < 6.7)…… (3-9)

The coefficients, A0, A1, A2, A3, A4, A5, were listed in Table 3-1, θfitted-3D represents
the angle between GFs basal-plane and heat flow direction in the 3D model. Thisfunction can effectively convert θ2D into θ3D and improve the reliability of the ETC
calculated by 2D image simulations. The deviation between the θfitted-3D and θ3D waswithin 6°.

Table 3-1. Calculation parameters in equation (3-9)
Equation

A0

A1

A2

A3

A4

A5

(3-6)

0.64

0.86

0.62

0.059

1.80

3.15

To determine the reliability of function Equation (3-9), a comparison between theTCs of GFs calculated with θ3D and θfitted-3D was conducted. Figure 3-7 shows the GFs’
TCs calculated with θ3D and θfitted-3D by Equation (3-3). The solid black line represents
TCs calculated with θ3D, and grey circles were for θfitted-3D. The circles were tightly
bound around the solid black line, and the errors were so minor that they can be ignored.
Thus, Figure 3-7 indicated that equation (3-6) is successful in converting θ2D to θ3D.

87

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

Figure 3-7. TC of GFs calculated by equation (3-3), θ was equal to the θ3D for theblack solid line and θfitted-3D for the gray circle.

3.3.3

ETC calculation of GFs/Al composites

10 vol.% GFs/Al composites were fabricated via spark plasma sintering. The
diameter of the GFs was 137.02 µm (98% purity), and the average particle of Al powder
was 30 µm (99.9% purity). The relative density of the samples was 99.4%. The ETC of
the sample, measured using a steady-state thermal-conductivity-measuring device, was
238 W m-1 K-1. The microstructures of the samples were observed using an optical
microscope. Figure 3-8 (a) shows the microstructures of the 10 vol.% GFs/Al
composites. To evaluate the TC distributions of GFs in the microstructures, different
colors were used to mark the GFs according to the orientations of the GFs, as shown in
Figure 3-8(b). For example, the red color indicates that the orientation of the GFs wasparallel to the heat flow direction, and the GFs had high TC; the other colors indicatethat the GFs tilt from the heat flow direction, decreasing TC.
88

Chapter 3 Relationship between the thermal conductivity of GFs in 3D and 2D microstructure images

Figure 8 Microstructure images of the 10 vol.% GFs/Al composite; (a) The OM images;(b) Distribution of the GFs TC.

Figure 9. ETC of the 10 vol.% GFs/Al composite considering the interfacial thermal
conductance (h) from 1.1 × 103 to 1.1 × 109 W m-2 K-1.
To calculate the ETC of 10 vol.% GFs/Al composites using 2D image-based
simulation, the orientations of GFs in the microstructures were corrected using
Equation (9), and the TC of GFs was calculated using Equation (3-3). The TC of the Al89

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Imagematrix (λAl-matrix) was calculated using the following equation [15]:𝜆=1[𝜆 3𝑣 − 1 + 𝜆 (3𝑣 − 1)4+

𝜆 3𝑣 − 1 + 𝜆 (3𝑣 − 1)

+ 8𝜆 𝜆 )…… (3-10)

where the TCs of the raw material Al powder (λAl) was taken as 228 W m-1 K-1 as in ourprevious works, TC of the pores (λp) was 0.214 W m-1 K-1, volume fraction of Al (vAl)
was 0.994, and volume fraction of pores (vp) was 0.006. The pores in this study wereassumed to be embedded in the Al matrix. The λAl-matrix was 226 W m-1 K-1. Figure 3-9
shows the calculated ETC considering the interfacial heat conduction (h) from 1.1 ×
103 to 1.1 × 109 W m–2 K–1. When the calculated ETC was equal to the experimental
ETC, h was 1.8 × 107 W m–2 K–1. This value (1.8 × 107 W m–2 K–1) was the same orderof magnitude as the h given by Xue, Zhou, and Schmidt (4.5 ~ 5 × 107 W m–2 K–1) [2, 8,
16]. Moreover, the calculated ETC was 250 W m–2 K–1 when h was in 4.5 ~ 5 × 107 W

m–2 K–1.
To view the Al-GFs interface (see Figure 3-10), pores can be observed.
Additionally, Li [4], Zhou [5], and Huang [17] reported that an amorphous layer interface
could be readily formed at the Al-GFs interface. Therefore, the small difference
between h calculated in this work and h reported by Xue, Zhou, and Schmidt can be
attributed to the pores and amorphous layer at the interface.

Figure 10. A SEM image of the microstructure at the Al-GFs interface.
90

Chapter 3 Relationship between the thermal conductivity of GFs in 3D and 2D microstructure images

3.4 Summary
This study aimed to calculate the ETC of GFs/Al composites using 2D image-based
simulations. However, the ETC calculated from the microstructure images might not
be equivalent to that measured using experimental methods. Because the 2D
microstructure image cannot reveal the depth information related to the observation
surface, the orientation of GFs in the 2D microstructure image differs from that of theexperimental sample. Thus, we studied the relationships between the orientation of GFs
in 2D cross-sectional images, the orientation of GFs in 3D models, and the aspect ratioof GFs in the 2D cross-sectional images.
The GFs in the modeled 3D models are elliptical cylinders. In light of themorphological characteristics of GFs, the conclusions of this study are limited to thestudy of the thermal conductivity of flake graphite reinforced aluminum matrix
composites. GFs orientation was defined by the angle between GFs basal plane andheat flow direction in this study. Angles were marked as θ2D in the 2D cross-section andθ3D in the 3D model. Based on the research results, the following conclusions weredrawn:
1) θ2D was generally larger than θ3D, and the difference between them could cause theTC error to be as high as 840 Wm-1K-1. No regularity was observed between theθ2D and θ3D differences.
2) The data for θ2D, θ3D, and aspect ratio were on a curved surface, and the curved
surface function can be used to convert θ2D into θ3D. The difference between theactual θ3D and θ3D values calculated using the curved surface function was less than
6°.
3) The ETC of the 10 vol.% GFs/Al composite was evaluated using 2D image-based
simulation, where the suggested converting θ2D into θ3D was employed.
4) The calculated interfacial thermal conductance was 1.8◊107 W m-2 K-1, and it is
the same order of magnitude with the reference value (4.5~5◊107 W m-2 K-1).

91

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

3.5 References[1] Chen JK, Huang IS. Thermal properties of aluminum–graphite composites bypowder metallurgy. Composites Part B: Engineering. 2013;44(1):698-703.
[2] Xue C, Bai H, Tao PF, Wang JW, Jiang N, Wang SL. Thermal conductivity andmechanical properties of flake graphite/Al composite with a SiC nano-layer on graphite
surface. Materials & Design. 2016;108:250-8.
[3] Etter T, Schulz P, Weber M, Metz J, Wimmler M, Löffler JF, et al. Aluminium
carbide formation in interpenetrating graphite/aluminium composites. Materials
Science and Engineering: A. 2007;448(1-2):1-6.
[4] Li W, Liu Y, Wu G. Preparation of graphite flakes/Al with preferred orientation andhigh thermal conductivity by squeeze casting. Carbon. 2015;95:545-51.
[5] Zhou C, Ji G, Chen Z, Wang M, Addad A, Schryvers D, et al. Fabrication, interface
characterization and modeling of oriented graphite flakes/Si/Al composites for thermal
management applications. Materials & Design. 2014;63:719-28.
[6] Xue C, Bai H, Tao PF, Jiang N, Wang SL. Analysis on Thermal Conductivity ofGraphite/Al Composite by Experimental and Modeling Study. Journal of Materials
Engineering and Performance. 2016;26(1):327-34.
[7] Kurita H, Miyazaki T, Kawasaki A, Lu Y, Silvain J-F. Interfacial microstructure ofgraphite flake reinforced aluminum matrix composites fabricated via hot pressing.
Composites Part A: Applied Science and Manufacturing. 2015;73:125-31.
[8] Zhou C, Huang W, Chen Z, Ji G, Wang ML, Chen D, et al. In-plane thermalenhancement behaviors of Al matrix composites with oriented graphite flake alignment.
Composites Part B: Engineering. 2015;70:256-62.
[9] Gori F, Corasaniti S. Effective thermal conductivity of composites. International
Journal of Heat and Mass Transfer. 2014;77:653-61.
[10] Wejrzanowski T, Grybczuk M, Chmielewski M, Pietrzak K, Kurzydlowski KJ,
Strojny-Nedza A. Thermal conductivity of metal-graphene composites. Materials &Design. 2016;99:163-73.
92Chapter 3 Relationship between the thermal conductivity of GFs in 3D and 2D microstructure images[11] Czabaj MW, Riccio ML, Whitacre WW. Numerical reconstruction ofgraphite/epoxy composite microstructure based on sub-micron resolution X-ray
computed tomography. Composites Science and Technology. 2014;105:174-82.
[12] Sugio K, Kono K, Choi YB, Sasaki G. Evaluation of Effective Thermal
Conductivity of Metal Matrix Composites by Using Image-Based Calculation.
Materials Science Forum: Trans Tech Publ; p. 1939-43.
[13] Sugio K, Choi Y-B, Sasaki G. Effect of the Interfacial Thermal Resistance on the
Effective Thermal Conductivity of Aluminum Matrix Composites. Materials
Transactions. 2016;57(5):582-9.
[14] Kelly B. Physics of Graphite (Applied Science. London/New Jersey. 1981:267.
[15] Landauer R. The electrical resistance of binary metallic mixtures. Journal ofApplied Physics. 1952;23(7):779-84.
[16] Schmidt AJ, Collins KC, Minnich AJ, Chen G. Thermal conductance and phonontransmissivityofmetal–graphite

interfaces.

JournalofApplied

Physics.

2010;107(10):104907.
[17] Huang Y, Ouyang Q, Guo Q, Guo X, Zhang G, Zhang D. Graphite film/aluminum
laminate composites with ultrahigh thermal conductivity for thermal management
applications. Materials & Design. 2016;90:508-15.

93Chapter 4

Effect of the orientation of GFs and interfacial thermal
resistance on the effective thermal conductivity of GFs/Al
composites

4.1 Introduction

95

4.2 Experimental and calculation procedures

97

4.2.1 Experiments

97

4.2.2 Numerical methods

98

4.2.2.1 Temperature distribution

98

4.2.2.2 Calculation of effective thermal conductivity

100

4.2.2.3 Thermal conductivity of GFs

101

4.2.2.4 Interfacial thermal resistance102

4.3 Results and discussion

104

4.3.1 Microstructures

104

4.3.2 Effect of porosity on TC of Al

107

4.3.3 Effect of the orientation of GFs on the effective thermal conductivity

108

4.3.4 Effect of interfacial thermal resistance on the effective thermal conductivity
111
4.4 Summary

114

4.5 References

115

94

Chapter 4 Effect of the orientation of GFs and interfacial thermal resistance on the effective thermal conductivityof GFs/Al composites

4.1 Introduction
Graphite flakes (GFs) have excellent thermal properties, including high thermal
conductivity (TC) and low coefficient of thermal expansion (CTE). In the basal plane,
GFs TC and CTE are 2800 W m-1 K-1[1-4] and -1.5 ppm K-1 [5, 6], respectively. Therefore,
GFs/Al composites have attracted much attention from researchers of thermal
management materials. Many experimental and theoretical studies have been
conducted to study the effective thermal conductivities of GFs/Al composites. GFs/Al
composites exhibit high effective thermal conductivities [7-20], e.g., Chen et al.[7]reported that the ETC of 80 vol.% GFs/Al composites (783 W m-1 K-1) was higher thanthat of 80 vol.% diamond/copper composites (724 W m-1 K-1). Li et al. [14] reported that
the ETC of 70 vol.% GFs /Al composite was 714 W m-1 K-1 in the plane parallel to the
GFs layers.
However, the past studies [7, 11, 19, 21-23] also revealed that the ETCs of GFs/Al
composites were affected by the orientation of GFs, interfacial thermal resistance, andinterface reaction. The ETCs of GFs/Al composites are not as high as the expected
theoretical values. The TCs of GFs are significant anisotropy, and its TC in the outplane is 38 W m-1 K-1 [3, 4, 12]. The anisotropic TC of GFs causes its orientation to
influence the ETC of GFs/Al composites greatly. Moreover, it is almost impossible tocontrol perfectly the orientation of GFs; thus, the orientations of GFs may be diverse incomposites. On the other hand, the interfacial thermal resistance at the Al-GFs interface
may be anisotropic due to the anisotropic crystal structure of GFs. Therefore, if theorientations of GFs in composites cannot be obtained, it would be difficult to calculate
the effects of the orientation of GFs and the interfacial thermal resistance on the ETC
using theoretical models, such as the rule of mixture [24, 25], Fricke model [26], Nan modeland effective medium approximation (EMA) approach [27].
The interface reaction is a chemical reaction. Al4C3 [10, 21, 28, 29] and amorphouslayers [9, 14, 17] are easily formed at the Al-GFs interface. Al4C3 phase is brittle andharmful for the bonding between Al and GFs. Meanwhile, Al4C3 has low TC and can95

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

hinder the heat transfer in the Al-GFs interface [7, 11, 21]. Fortunately, some approaches
can suppress the interface reaction while fabricating the GFs/Al composites. The
approaches include lowing the sintering temperature [7, 13, 17, 19] or reducing the reaction
time [9, 21, 30, 31], or adding Si or SiC particles to the starting materials [8, 9, 12, 13, 15, 19], etc.
Besides, the amorphous layer at the Al-GFs interface may act as the nucleation site for
Al4C3 [32]. The amorphous layer thus can also hinder the heat transfer itself [9, 33].
Therefore, in this chapter, 10~20 vol.% GFs/Al composites were prepared. twodimensional(2D) image-based simulation method [34] is used to calculate the ETC of
GFs/Al composites considering the effect of the orientations of GFs and the interfacial
thermal resistance. In addition, the interfacial thermal resistances in the in-plane andthe out-of-plane of GFs were calculated.

96Chapter 4 Effect of the orientation of GFs and interfacial thermal resistance on the effective thermal conductivityof GFs/Al composites

4.2 Experimental and calculation procedures
4.2.1 Experiments
Al powder (99.9%, 30 µm in diameter) and GFs (98%, average particle size:
137.02 µm) were used. The nominal TC of the Al powder was 236 W m–1 K–1, whilethat of GFs was 880 and 38 W m–1 K–1 at the basal-plane and out-of-plane directions,
respectively. Al powders were mixed with 10 vol.% and 20 vol.% of GFs, respectively.
Subsequently, appropriate amounts of alumina balls were added through a wet process
for 2 h using a V-type mixer at 50 rpm. Then, spark plasma sintering (SPS) was
conducted at 873 K and 60 MPa pressure for 0.5 h under a vacuum of 1.3 × 10–2 Pa. In
order to avoid the interface reaction between Al and GF, the sintering temperature wasmaintained lower than the melting point of Al. The temperature was raised by 200
K/min: one pure Al sample, 10 vol.% GF/Al (Samples 1-2), and 20 vol.% GF/Al(Samples 3-4) composites were fabricated. Figure 4-1 depicts the shape of the samples
sintered using SPS.

Figure 4-1. Shape of the sample sintered by SPS; P1, P2, and P3 at the A plane denote
the regions that correspond to the optical micrographs. The grey arrow illustrates thatthe heat flow was from B to D during ETC measurement, C is the plane bearing thepressure during sintering.

The microstructure images of the sintered samples were obtained via optical

97

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

microscopy from the P1, P2, and P3 regions on A plane in Figure 4-1. The relative
densities of the sintered samples were measured using the Archimedes method. ETCsof the sintered samples were measured at 25 ℃ using a steady-state thermalconductivity device.

4.2.2 Numerical methods
2D image-based simulations were performed in two steps. First, the ETCs ofcomposites were calculated considering the effect of the orientations of GFs. Then,
ETCs of the composites were calculated considering the orientations of the GFs and theinterfacial thermal resistance at the Al-GFs interface. The obtained microstructure
images of the experimental samples were used to perform 2D image-based simulations.

4.2.2.1 Temperature distribution
We calculated 2D-temperature distributions using the finite volume method as
follows:

𝑇,

=𝑇, +

∆𝑡 𝑞
𝜌𝑐

−𝑞 , 𝑞
∆𝑥

,

,

+

𝑞 ,

−𝑞 ,
∆𝑦
…… (4-1)

where x and y denote the coordinate positions of the element, 𝑇 ,of element at (x, y) coordinates, and 𝑇 ,is the temperatureis the element's temperature at (x, y)

coordinates after a time △t, ρ is density, c is specific heat, △x and △y denote thesizes of elements along the x-axis and the y-axis, respectively, qn denotes heat flow.
When qn is the heat conduction between adjacent elements (e.g., as depicted in Figure
4-2, qn moves from the E to adjacent elements M or N), qn can be calculated as follows:

𝑞

,

= 𝜆

,

(

,

△

,

),

𝑞
98

,

= 𝜆

,

(

,

,

△

)

Chapter 4 Effect of the orientation of GFs and interfacial thermal resistance on the effective thermal conductivityof GFs/Al composites

𝑞 ,

𝜆

𝜆 ,

,

=

=

,

,

= 𝜆 ,

△

,

,

,

,

,
,

,
,

,

𝑞 ,

,

= 𝜆 ,

,

𝜆

,

𝜆 ,

,

=

=

,

△

,

,

,

,

,
,

,
,

…… (4-2)
where λ(x, y) denotes the TC of element at (x, y) coordinates, λ(x+1,y), λ(x-1,y), λ(x,y+1), andλ(x,y-1) denote the TC of elements adjacent to (x, y), and λ(x+1/2,y), λ(x-1/2,y), λ(x,y+1/2), andλ(x,y-1/2) denote the harmonic mean of TC.

Figure 4-2. Schematic diagram of heat flow at the interface. The dashed line denotes
the Al–GF interface, qn denotes the heat flow in the direction of the dashed arrows.

Considering the effect of the interfacial thermal resistance in the composite, if qn
moves by heat transfer between different types of elements, the interfacial heat transfer
coefficient (h) must be considered. For example, as depicted in Figure 4-2, consideringthat qn moves from E to M and through the interface between Al and GFs (Al-GF), qn
can be calculated as follows:99Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

𝑞
𝑞 ,

,

= ℎ(𝑇
= ℎ(𝑇 ,,

− 𝑇 , ),𝑞

− 𝑇 , ),𝑞 ,,= ℎ(𝑇 , − 𝑇= ℎ(𝑇 , − 𝑇 ,,))…… (4-3)

4.2.2.2 Calculation of effective thermal conductivity
As depicted in Figure 4-3, the simulation model is comprised of two heat sources
and composite components. The top and bottom surfaces correspond to the periodic
boundary, while the left and right sides correspond to the adiabatic boundary. The
composite part is based on the microstructure images of Samples 1–4. Therefore, thesize of the composite part (Nx × Ny) was 450 × 600 elements for Samples 1, 2, and 4,
and it was 570 × 450 elements for Sample 3. The heat source size was 5 × 600
elements for Samples 1, 2, and 4, and it was 5 × 450 elements for sample 3 (NL = NR =5 elements). The size of each element was 1.18 × 10–6 m. The initial TC of the GFs was
set to be (880, 38) W m–1 K–1 in the (x, y) directions.

Figure 4-3 Simulation model for the ETC calculation.

The temperature of the left edge elements was set to 301 K, and the initial
100

Chapter 4 Effect of the orientation of GFs and interfacial thermal resistance on the effective thermal conductivityof GFs/Al composites

temperature of the other elements was set to 300 K. The temperatures at the left andright sides were fixed. The temperature of the remaining elements was iterated until thetemperature variation was lower than 10–13 K, and the temperature distribution in thesteady-state was obtained. The ETC values of the GF/Al composites, λs-eff, were
calculated when the temperature distribution was in the steady-state, as follows:

𝜆

=

△𝑇

𝜆
△𝑇 𝑁
−𝑁 △𝑇 −𝑁 △𝑇…… (4-4)

where λAl-eff denotes the TC of the Al matrix considering the effect of pores, and △T12
is the average temperature difference between the first and the second columns; NL and
NR denote the numbers of elements in the heat sources, Nx and Ny are the numbers ofelements in the composite region.
4.2.2.3 Thermal conductivity of GFs
This study used an angle (θ) between the basal plane of GFs and the direction ofheat flow to characterize the GFs’ orientation in composites. Thus, by using thefollowing equation (4-5) [18], the TC of GFs can be calculated along the direction of thebasal plane of GFs parallel (λ//) and perpendicular (λ┴) to the heat flow direction.
𝜆)𝑠𝑖𝑛 𝜃]𝜆𝜆
𝜆 = 𝜆 [1 − (1 − )𝑐𝑜𝑠 𝜃]
𝜆𝜆// = 𝜆 [1 − (1 −…… (4-5)
where λa and λc are the TC in the basal-plane and the out-of-plane directions of GFs,
respectively.
Figure 4-4 illustrates the relationship between the TC of GFs and the angle θ. It
can be observed that TC decreased significantly from 880 to 38 W m–1 K–1 when theangle increased from 0° to 90°, revealing that the orientation of the GFs significantly

101

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Imageaffects the ETC of composites. In addition, Figure 4-4 shows that the TC of GFs was
lower than that of the Al matrix (236 W m-1 K-1) if the angles were greater than 61°,
indicating that the ETC of composites could be enhanced at critical angles smaller thanor equal to 61°.

Figure 4-4 TC of GFs as a function of angle (θ). The nominal TC of Al matrix, λAl, was
236 W m–1 K–1 (gray horizontal dotted line).

4.2.2.4 Interfacial thermal resistance
The interfacial thermal conductance in the in-plane and the out-of-plane may bedifferent due to the anisotropic crystal structure of GFs. Thus, the interfacial thermal
conductance, h, at the Al-GFs interface was calculated by the DMM method. h can becalculated as follows:
ℏ
ℎ=
8𝜋 𝐾𝑇

⎡
1
⎢
𝑣,
⎢𝑣 , 𝑣 , (𝑣 , + 𝑣 , )
⎣

𝜔 𝑒

,

𝑒

ℏ

ℏ

−1

⎤
𝑑𝜔⎥
⎥
⎦

…… (4-6)
where ℏ is Planck’s constant, K is Boltzmann constant, T is temperature, ω is phonon
angular frequency, and 𝜔 , is the maximum phone angular frequency. The phonosound velocity, v, can be written as:
102

Chapter 4 Effect of the orientation of GFs and interfacial thermal resistance on the effective thermal conductivityof GFs/Al composites𝑣 = 𝑣 + 2𝑣

𝑣 =(1 − µ)𝐸∙𝜌 (1 + 𝜇)(1 − 2µ)

𝑣 =

𝐸
1
∙
𝜌 2(1 + 𝜇)…… (4-7)

where vL is longitudinal sound velocity, vT is transverse sound velocity, E is Young’s
modulus, ρ is density, µ Poisson's ratio. The maximum phone angular frequency is
calculated:𝜔 , =𝑣 ,6𝜋

𝜌𝑁𝑀…… (4-8)

where NA is Avogadro constant, M1 is relative molecular mass. The material parameters
are given in Table 4-3.

103

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

4.3 Results and discussion
4.3.1 Microstructures
Figure 4-5 illustrates the microstructures of the samples. The GFs were
homogeneously distributed in the Al matrix. However, GFs’ orientations were notuniform in the samples. As shown in Figures 4-5, the orientations of GFs were almost
parallel to each other in Samples 1 and 4. However, GFs’ orientations were partiallydisordered in Samples 2 and 3, forming different angles between GFs’ basal-plane andheat flow direction. The orientations of GFs in Figures 4-5 were determined and shownin Figures 4-6. As depicted in Figures 4-6 (a), GFs were bounded by the smallest
circumscribed rectangle, and the angle θ represents the orientation of GFs. The θ values
in the microstructures were measured as depicted in Figure 4-6 (b), excluding GFs
smaller than 5 × 5 pixels. The measured |θ| values were shown in Figure 4-6 (c)–(f).
Most of the angles ranged from 0° to 30° in all the samples. Nevertheless, for the angles
ranging between 50° and 90°, the relative frequency in Samples 1 and 4 was < 10%,
while it was > 10% in Samples 2 and 3. Additionally, as depicted in Figures 4-4, the TC
of GFs along the heat flow direction was smaller than that of the Al matrix when theangle was > 61.0°. Thus, it can be derived from Figure 4-6 that the average TCs of GFs
in Samples 2 and 3 along the heat flow direction were smaller than the average TCs of
GFs in Samples 1 and 4. Moreover, the average angle was calculated for each sample,
as shown in Figure 4-6 (c)–(f).

104Chapter 4 Effect of the orientation of GFs and interfacial thermal resistance on the effective thermal conductivityof GFs/Al composites

Figure 4-5 Optical micrographs of Samples 1–4. Regions P1, P2, and P3 correspond tothe black circles at the A-plane in Figure 1. The gray arrow represents the heat flow
direction during the ETC measurements. The brighter phases correspond to the Al

matrix and the elongated dark phases to the GFs.
105

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

(b)

(a)
90 º

0º

GFs
θ
θ Heat flow direction
GFs

-90 º

(c)

(d)

(e)

(f)

Figure 4-6 Distribution of the GF orientations in the Al matrix. Gray and white
rectangles in (a) and (b) represent the smallest circumscribed rectangles and were used

to measure the angle (θ) between the basal-plane of GFs and the heat flow direction; incase –90° < θ < 0°, θ was taken as the absolute value.

106Chapter 4 Effect of the orientation of GFs and interfacial thermal resistance on the effective thermal conductivityof GFs/Al composites

4.3.2 Effect of porosity on TC of Al
The measured ETCs and the relative densities of the experimental samples werelisted in Table 4-1. The ETC of Sample 1 was higher than that of Sample 2, and the
ETC of Sample 3 was lower than that of Sample 4. These measurement results agreewith the results shown in Figures 4-6.

Table 4-1 Volume fraction of GFs (vGFs), measured ETCs (λeff), and relative densities ofSamples 1–4 and pure Al sample.
vGFs

λeff

%

Wm–1K–1

1

10

248

99.2%

2

10

238

99.4%

3

20

273

98.0%

4

20

280

99.3%

Al

0

217

97.0%

SampleRelative density

The measured relative density (Table 4-1) indicated that some pores existed in thecomposites. Therefore, in this study, the pores for all the samples were assumed to bein the Al matrix. The following equation derived by Landauer [35] was used to eliminate
the pores’ effect on the TC of Al-matrix:

𝜆=1[𝜆 3𝑣 − 1 + 𝜆 (3𝑣 − 1)4+

𝜆 3𝑣 − 1 + 𝜆 (3𝑣 − 1)

+ 8𝜆 𝜆 )…… (4-9)

where λAl-eff denotes the TC of Al matrix, λAl and λp are the TCs of the raw material Al
powder and pores, respectively, vAl is the volume fraction of Al, and vp is the volume
fraction of the pores. In addition, vAl, vp, and λp are listed in Table 4-2. First, the107Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

measured TC of the pure Al sample (λAl-eff, 217 W m-1 K–1) and its relative density (97%)
were substituted in equation (4-9), and λAl was determined to be 228 W m–1 K–1, asshown in Table 4-2. Second, the TCs of the Al-matrix for Samples 1–4, λAl-eff, were
calculated by substituting vAl, vp, λp, and λAl from Table 4-2 in equation (4-9). The
calculated values of λAl-eff are 225.3, 225.9, 221.2, and 225.6 W m–1 K–1 for Samples1–
4, respectively. The ETCs (λROM) of Samples 1–4 were calculated using the rule of
mixture (ROM), i.e., λROM = (vGFs)( λGFs)+ (1-vGFs)(λAl-eff), where the orientations of all
the GFs were assumed to be aligned with the heat flow direction in the composites. The
TC of the GFs (λGFs) was set to 880 W m–1 K–1 along the heat flow direction. The valuesof λROM were 291, 291, 353, and 357 Wm–1K–1 for Samples 1–4, respectively. The
calculated λROM was considerably higher than the measured ETC. Furthermore, We
performed 2D image-based simulations to confirm the effect of the orientation of GFs
on the ETC of composites.

Table 4-2 Calculation parameters in equation (10) and calculated effective thermal
conductivities of composites.
λROM

λs-eff

225.3

291

270

225.9

291

263

0.02

221.2

353

318

0.007

225.6

357

323

λp
Sample

vAl

vp

Al

0.97

0.03

1

0.992

0.008

2

0.994

0.006

3

0.98

4

0.993

λAl

λAl-eff
Wm–1K–1

0.214

0.214

228

217

4.3.3 Effect of the orientation of GFs on the effective thermal conductivity
The TC values of GFs in the experimental samples were calculated using equation
108

Chapter 4 Effect of the orientation of GFs and interfacial thermal resistance on the effective thermal conductivityof GFs/Al composites(4-5), and Figure 4-7 illustrated the results. GFs were denoted in different colors
according to the orientations of GFs in the Al matrix. When the orientation of GFs wasparallel to the heat flow direction, GFs exhibited high TC, and they were marked in red.

Figure 4-7 TC values of GFs in Samples1-4. Regions denoted as P1, P2, and P3
correspond to the microstructures in Figure. 5; The color bar shows the TCs of GFs in
different orientations; The red arrow indicates the heat flow direction.
109

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

In contrast, if the GFs tilted from the heat flow direction, color changes indicateda decrease in TC. Figure 4-7 showed that most GFs were colored in red for Samples 1
and 4, which implied that GFs had a high TC, while the yellow, blue, and other colors
appeared in the P2 and P3 regions for Samples 2 and 3. Table 4-2 listed the simulated
ETCs (λs-eff) based on 2D image-based simulations. Figure 4-8 further compared themeasured ETC, λROM, and λs-eff of Samples 1–4. The values of λs-eff were smaller thanthose of λROM, and the deviations between λROM and λs-eff for Samples1–4 were 7.2%,
9.6%, 9.9%, and 9.5%, respectively. Comparing the ETC loss in Samples1–4, thedifference between Samples 1 and 2 in ETC loss was 2.4%, while it was only 0.4%
between Samples 3 and 4. Those deviations may be explained by the angle deviations
depicted in Figure 4-6 (c)-(f), where the angle difference between samples 1 and 2 was
17°, and between samples 3 and 4 was 3°. Furthermore, compared the λs-eff values andthe measured ETC values, the deviations between λs-eff and the measured ETCs were8.1%, 9.5%, 14.2%, and 13.3% for Samples1–4, respectively. These deviations can be

λattributed to the effect of the interfacial thermal resistance between Al and GFs.

Figure 4-8 ETCs of the composite samples. λeff denotes the measured ETC, λROM is theETC calculated using the rule of mixture, and λs-eff is the ETC calculated using the 2D
image-based simulation considering GF orientations.
110Chapter 4 Effect of the orientation of GFs and interfacial thermal resistance on the effective thermal conductivityof GFs/Al composites

4.3.4 Effect of interfacial thermal resistance on the effective thermal conductivity
A series of 2D image-based simulations were performed to study further the effectof the interfacial thermal resistance on the ETC of the composites. The heat transfer
coefficient, h, ranged from 103 to 109 W m–2 K–1 at the Al–GF interface. Figure 4-9
illustrated ETC values as a function of h for composites.

Figure 4-9 ETC as a function of the heat transfer coefficient, h. The dashed lines
represent the experimental ETCs and the arrows denote the h value at the Al-GF
interface.

111

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure ImageThe ETC curve was similar to KJMA [36] equation, and ETC values rapidly
increased as h increased from 105 to 109. Moreover, the ETC curves for Sample 1 andSample 2 did not overlap in the h value range between 103 and 109, further proving thatthe orientations of GFs affected the composite ETC significantly. In Figure 4-9, thedashed lines represented the experimental ETC, and the arrows denoted the heattransfer coefficients (h) at the Al-GF interface. Table 4-3 showed the h values evaluated
via inverse analysis, and the values showed the same order of magnitude for all samples,i.e., 107. Moreover, a reference value of h calculated using the AMM model was listed
in Table 4-3. The order of magnitude of the reference value is also 107, and the minor
differences between the h reference value and the simulated h for Samples 1–4 can be
attributed to the misfits at the interface. However, the values of h calculated by DMM
were 1.1 × 108 Wm-2K-1 in the in-plane and 1.32 × 109 Wm-2K-1 in the out-of-plane.
Thus, the values were greater than the simulated h.

Table 4-3 Material parameters of the DMM calculation[37], interfacial thermal
conductance (h), and interfacial thermal resistance (R); Sim. is the simulated h using
2D-image, Cal. is calculated h using DMM method, Ref. is the reference value of h.
Sample
Al

T

E

Ρ

(K)

(GPa)

(g/cm3)

70.0

/
In-

GFs

plane 300
Out-ofplaneh / 107 Wm–2K–1

R / 10-8

Sim. Cal. Ref.

m2KW-1

μ

M1

2.7

0.3

27

1153.0

2.2

0.195

12

11 4.8[19]39.5112.2

0.0002

12

130

1

1.62

6.17

2

1.56

6.41

3

1.64

6.10

4

1.78

5.62

112

Chapter 4 Effect of the orientation of GFs and interfacial thermal resistance on the effective thermal conductivityof GFs/Al composites

The interfacial thermal resistance, R, is the reciprocal of heat transfer coefficient,i.e., R = 1 / h, R values for Samples 1–4 were listed in Table 4-3. It should be noted that
R values were sufficiently small and even negligible. However, as shown in Figure 48, R affected the ETC of the composites significantly, leading to a significant decreasein ETC. The reason can be attributed to the considerable number of interfaces formedin the GF/Al composites.

113

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

4.4 Summary
10 vol.% GF/Al and 20 vol.% GF/Al composites were fabricated via SPS. The
microstructures images of the composites were observed, and their relative density and
ETCs were measured. The measured ETC values were 248, 238, 273, and 280 W m–1
K–1 for Samples 1–4, which were smaller than the ETCs calculated through the ROM(291, 291, 353, and 357 W m–1 K–1, for Samples 1–4, respectively).
The average angles of the GFs with respect to the heat flow direction were
calculated to be 16°, 33°, 14°, and 11° for Samples 1–4, respectively. The ETCs ofcomposites considering the orientations of GFs were evaluated using 2D image-based
simulations. Compared with the ETC calculated using ROM, the results showed that
ETCs decreased due to the anisotropic TC of GFs by 7.2%, 9.6%, 9.9%, and 9.5% for
Samples 1–4, respectively. Due to the interfacial thermal resistance effect, the ETCs
decreased by 8.1, 9.5, 14.2, and 13.3% for Samples 1–4, respectively. The R values at
the Al–GF interface were evaluated to be 6.17 × 10–8, 6.41 × 10–8, 6.10 × 10–8, and 5.62
× 10–8 W m–2 K–1 for Samples 1–4, respectively.

114

Chapter 4 Effect of the orientation of GFs and interfacial thermal resistance on the effective thermal conductivityof GFs/Al composites

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Microstructure Image[32] Khalid FA, Beffort O, Klotz UE, Keller BA, Gasser P. Microstructure andinterfacial characteristics of aluminium–diamond composite materials. Diamond andRelated Materials. 2004;13(3):393-400.
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118

Chapter 5

Conclusions
Graphite flake reinforced aluminum matrix composites (GFs/Al composite) arelightweight, low cost, and exhibit outstanding thermal conductivity and low coefficientof thermal expansion. Especially for the excellent thermal conductivity, GFs/Al
composites have been proposed as one of the most promising electronic packaging
materials.
Lots of studies have been carried out to explore the thermal properties of GFs/Al
composites. However, past studies reported that the high thermal conductivity of GFs
could not be fully displayed in the GFs/Al composites. The effective thermal
conductivity of GFs/Al composites was influenced by the volume fraction, dispersion,
size, orientation of GFs, interface reaction, and interface thermal resistance, inparticular the orientation of GFs and the interfacial thermal resistance. Unfortunately,
until today, no method has been proposed to control the orientation of GFs well, andthe theoretical models such as mixture rules, Maxwell, Fricke, Every, Nielsen, and theeffective medium approximation cannot be considered the effect of the orientation of
GFs. Moreover, interface thermal resistance seems to be an inherent property of carbon
materials reinforced metal-matrix composites, and the value of interface thermal
resistance is affected by the structure, composition, and state of the Al-GFs interface.
AMM and DMM models are commonly used to calculate interface thermal resistance.
However, the AMM and DMM models calculate the interface thermal resistance byassuming an ideal interface, but in reality, there may be gaps and lattice defects at theinterface. Therefore, the interface thermal resistance evaluated by the AMM and DMM
methods is not completely reliable.
This study aims to study the effective thermal conductivity of GFs/Al composites
119

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Imageby two-dimensional image simulations, considering the orientation of GFs and theinterface thermal resistance. The works were performed in three steps: first, 10-40 vol.%GFs/Al composites were prepared by spark plasma sintering. Moreover, weinvestigated the relative density of composites, orientation and distribution of GFs,
interface reaction between Al and GFs, effective thermal conductivity, and thermal
expansion coefficient of GFs/Al composites. Second, since the orientation of GFs in
two-dimensional images may differ from that of GFs in three-dimensional images,some three-and two-dimensional models of GFs/Al composites were created toinvestigate the relationships between the orientations of GFs in three- and twodimensional models. As a result, a fitted function was obtained to convert the twodimensional orientation to three-dimensional orientation. Third, the effects of GFs’
orientation and the interfacial thermal resistance on the effective thermal conductivities
of GFs/Al composites were calculated by two-dimensional image simulations, and theinterface thermal resistances at the Al-GFs interface were calculated by inverse analysis,
AMM, and DMM methods. Through the above research, this thesis draws the following
conclusions:
1) 10-40 vol.% GFs/Al composites were fabricated by spark plasma sintering; Mostof the GFs were stacked by layers and parallel to each other in the composites;
With GFs content up to 40%, GFs exhibit a state of aggregation and tends to form
the network by connecting GFs; The formation of Al4C3 was absent in the 10-40
vol.% GFs/Al composites; The relative density was up to 99% for the GFs/Al
composites; With GFs content up to 40%, the relative density of composites
decreased to 97.1%. The measured effective thermal conductivities were 248, 280,
313, and 346 W m-1 K-1 for 10-40 vol.% GFs/Al composites, respectively; Due tothe orientation of GFs and the interface thermal resistance, the experimental valuesof effective thermal conductivity of the GFs/Al composites were lower than thetheoretical values. The experimental values of the coefficient of thermal expansion
were 26.2, 24.4, 23.4, and 22.1 ppm/K for 10-40 vol.% GFs/Al composites,
respectively. Moreover, because of the effects of the pores embedded in composites
120

Chapter 5 Conclusions

and the disorder orientations of GFs, the experimental values for the composites
with 10-20 vol.% GFs were higher than the calculated values perpendicular to GFs
basal plane, and the experimental values exceeded even the coefficient of thermal
expansion of Al and GFs (25 ppm/K along the direction perpendicular to the basal
plane.
2) A small difference between the orientations can cause the difference in the thermal
conductivity of GFs to be as high as 840 Wm-1K-1; The orientations of GFs in twodimensional and three-dimensional images and two-dimensional aspect ratio of
GFs have a strong correlation, and the correlation function can be obtained and use
to convert the orientation of GFs from two-dimensional to three-dimensional.
Moreover, the ETC of 10 vol% GFs/Al composite was evaluated using 2D imagebased simulation and the correlation function. The results showed that thecalculated ETC was equal to the experimental ETC 238 W m-1 K-1 when theinterfacial thermal conductance at the GFs-Al interface was 1.8 × 107 W m-2 K-1,
and the calculated ETC was 250 Wm-1K-1 when the interfacial thermal conductancevalue was the same as that in a reference (4.5 ~5 × 107 W m-2 K-1).
3) A two-dimensional image simulation code was developed to calculate the effective
thermal conductivity of GFs/Al composites considering the effects of theorientation of GFs and interfacial thermal resistance; Orientations of GFs withrespect to the heat flow direction were from 11° to 33° in 10-20 vol.% GFs/Al
composites, and the orientation of GFs resulted in a loss of the effective thermal
conductivity by7.2% to 9.9%. According to the DMM method, the interfacial
thermal conductance at the Al-GFs interface was 1.1 × 108 W m-2 K-1 in the inplane and 1.3 × 109 W m-2 K-1 in the out-plane. In contrast, the result (1.1 × 108 W
m-2 K-1 in the in-plane) was much larger than the result of two-dimensional image
simulation (1.56 ~ 1.78 × 107 Wm-2K-1) and reference value (4.8 × 107 Wm-2K-1).

121

Estimation of Effective Thermal Conductivity of Graphite flakes/Al Composites by Using Two-Dimensional
Microstructure Image

122

Acknowledgments

I would like to express my sincere gratitude to Assistant Professor Kenjiro Sugio
for his guidance and insight throughout this study. He gave me full support in myresearch and life, which has played an important role in my professional and personal
development. I am also greatly indebted to Professor Gen Sasaki and AssistantProfessor Yongbum Chio. They gave me much positive help in the research, theaccumulation of the knowledge, and my life, and I have learned much from them. In
addition, I want to express my heartfelt thanks to Professor Kazuhiro Matsugi at
Hiroshima University and Professor Jinku Yu at Yanshan University for their technicaladvice and experimental assistance. Finally, I would like to acknowledge the helpfulcomments and suggestions from Assistant Professor Zhefeng Xu.
Then, I would like to thank Mr. Lu Yang, Mr. Yongqiang Yao, Mr. Shaoming Kang,
Mr. Fei Gao, Mr. Xilong Ma, and Mr. Yujiao Ke for their considerable assistance andhelp in my experiments. Moreover, I would like to thank all the past and present
members of the Materials physics Laboratory in the Department of MechanicalPhysical Engineering, Hiroshima University, for their enthusiastic help to my life andstudy.
Finally, my thanks would go to my beloved family for their loving considerations
and great confidence in me all through these years. I also owe my sincere gratitude to
my friends who gave me their help and time in helping me work out my problems
during the challenging course of the thesis.

123

Published Papers in Regards to This Thesis
1. Yan Zhao, Kenjiro Sugio, Sasaki Gen, Zhefeng Xu, Jinku Yu. Evaluation of Effective
Thermal Conductivity of Graphite Flake/Aluminum Composites by Two-Dimensional
Image Simulation under a Correction Function. Material transactions, 62 (2021) 16251631. (Chapter 3)

2. Yan Zhao, Kenjiro Sugio, Yongbum Choi, Sasaki Gen, Zhefeng Xu, Jinku Yu. Effectof Anisotropic Thermal Conductivity of Graphite Flakes and Interfacial Thermal
Resistance on the Effective Thermal Conductivity of Graphite Flakes/Aluminum
Composites. Material transactions, 62 (2021) 98-104. (Chapter 4)

3. Yan Zhao, Kenjiro Sugio, Yong Bum Choi, Gen Sasaki and Jinku Yu. The relationshipbetween the effective thermal conductivity and the orientation of graphite flakes in Al
matrix composites. Processing of the 10th Pacific Rim International Conference on
Advanced Materials and Processing (PRICM10). (2019) 815-820. (Chapter 4)

124Presentations
1. Yan Zhao, Kenjiro Sugio, Yong Bum Choi, Gen Sasaki and Jinku Yu. Effect ofAnisotropic Thermal Conductivity of Carbon Fiber in Aluminum Matrix Composite.
日本金属学会 2018 年春期講演大会. March 19-21, 2018, Tokyo, Japan; Chiba
Institute of Technology University.
2. Yan Zhao, Kenjiro Sugio, Yong Bum Choi, Gen Sasaki and Jinku Yu. The
relationship between the effective thermal conductivity and the orientation ofgraphite flakes in Al matrix composites. The Tenth Pacific Rim International
Conference on Advanced Materials and Processing (PRICM10). August 18-22,
2019, Xian, China, Qujiang International Conference Center.
3. Yan Zhao, Kenjiro Sugio, Sasaki Gen. Effect of anisotropic thermal conductivity
and interfacial thermal resistance on the effective thermal conductivity of graphite
flake/Al composites. 日本金属学会 2019 年秋期講演大会. September 11-13,
2019, Okayama, Japan; Okayama University Tsushima Campus.
4. Yan Zhao, Kenjiro Sugio, Sasaki Gen. Kenjiro Sugio, Sasaki Gen. Effect ofanisotropic thermal conductivity and interfacial thermal resistance on the effective
thermal conductivity of graphite flake/Al composites. 2018 年 12 月 日本金属学会 日本鉄鋼協会 中国四国支部 第 43 回若手フォーラム 広島, 日本.

125