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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Serie: Microsystem and Nanotechnology Series
- Sprache: Englisch
Contains the latest research advances in computational nanomechanics in one comprehensive volume
- Covers computational tools used to simulate and analyse nanostructures
- Includes contributions from leading researchers
- Covers of new methodologies/tools applied to computational nanomechanics whilst also giving readers the new findings on carbon-based aggregates (graphene, carbon-nanotubes, nanocomposites)
- Evaluates the impact of nanoscale phenomena in materials
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Seitenzahl: 615
Veröffentlichungsjahr: 2015
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Table of Contents
Cover
Microsystem and Nanotechnology Series
Title Page
Copyright
List of Contributors
Series Preface
Preface
Chapter 1: Thermal Conductivity of Graphene and Its Polymer Nanocomposites: A Review
1.1 Introduction
1.2 Graphene
1.3 Thermal Conductivity of Graphene–Polymer Nanocomposites
1.4 Concluding Remarks
References
Chapter 2: Mechanics of CNT Network Materials
2.1 Introduction
2.2 Experimental Studies on Mechanical Characterization of CNT Network Materials
2.3 Theoretical Approaches Toward CNT Network Modeling
2.4 Molecular Dynamics Study of Heat-Welded CNT Network Materials
References
Chapter 3: Mechanics of Helical Carbon Nanomaterials
3.1 Introduction
3.2 Theory of HN-Tubes
3.3 Experiment of HN-Fibers
3.4 Perspective and Possible Applications
References
Chapter 4: Computational Nanomechanics Investigation Techniques
4.1 Introduction
4.2 Fundamentals of the Nanomechanics
4.3 Molecular Dynamics Method
4.4 Tight Binding Method
4.5 Hartree–Fock and Related Methods
4.6 Density Functional Theory
4.7 Multiscale Simulation Methods
4.8 Conclusion
References
Chapter 5: Probabilistic Strength Theory of Carbon Nanotubes and Fibers
5.1 Introduction
5.2 A Probabilistic Strength Theory of CNTs
5.3 Strength Upscaling from CNTs to CNT Fibers
5.4 Conclusion
References
Chapter 6: Numerical Nanomechanics of Perfect and Defective Hetero-junction CNTs
6.1 Introduction
6.2 Theory and Simulation
6.3 Results and Discussion
6.4 Conclusion
References
Chapter 7: A Methodology for the Prediction of Fracture Properties in Polymer Nanocomposites
7.1 Introduction
7.2 Literature Review
7.3 Atomistic
J
-Integral Evaluation Methodology
7.4 Atomistic
J
-Integral at Finite Temperature
7.5 Cohesive Contour-based Approach for
J
-Integral
7.6 Numerical Evaluation of Atomistic
J
-Integral
7.7 Atomistic
J
-Integral Calculation for a Center-Cracked Nanographene Platelet
7.8 Atomistic
J
-Integral Calculation for a Center-Cracked Nanographene Platelet at Finite Temperature (
T
= 300 K)
7.9 Atomistic
J
-Integral Calculation for a Center-Cracked Nanographene Platelet Using ReaxFF
7.10 Atomistic
J
-Integral Calculation for a Center-Cracked EPON 862 Model
7.11 Conclusions and Future Work
Acknowledgments
References
Chapter 8: Mechanical Characterization of 2D Nanomaterials and Composites
8.1 Discovering 2D in a 3D World
8.2 2D Nanostructures
8.3 Mechanical Assays
8.4 Mechanical Properties and Characterization
8.5 Failure
8.6 Multilayers and Composites
8.7 Conclusion
Acknowledgment
References
Chapter 9: The Effect of Chirality on the Mechanical Properties of Defective Carbon Nanotubes
9.1 Introduction
9.2 Carbon Nanotubes, Their Molecular Structure and Bonding
9.3 Methods and Modelling
9.4 Results and Discussions
9.5 Conclusions
References
Chapter 10: Mechanics of Thermal Transport in Mass-Disordered Nanostructures
10.1 Introduction
10.2 Equilibrium Molecular Dynamics to Understand Vibrational Spectra
10.3 Nonequilibrium Molecular Dynamics for Property Prediction
10.4 Quantum Mechanical Calculations for Phonon Dispersion Features
10.5 Mean-Field Approximation Model for Binary Mixtures
10.6 Materials Informatics for Design of Mass-Disordered Structures
10.7 Future Directions in Mass-Disordered Nanomaterials
References
Chapter 11: Thermal Boundary Resistance Effects in Carbon Nanotube Composites
11.1 Introduction
11.2 Background
11.3 Techniques to Enhance the Thermal Conductivity of CNT Nanocomposites
11.4 Dual-Walled CNTs and Composites with CNTs Encapsulated in Silica
11.5 Discussion and Conclusions
11.6 Acknowledgments
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Thermal Conductivity of Graphene and Its Polymer Nanocomposites: A Review
Figure 1.1 Lattice structure of graphene
Figure 1.2 Bonding mechanism between carbon atoms in graphene
Figure 1.3 Unit cell and directionality of graphene
Figure 1.4 Impact of interfacial thermal resistance between filler and matrix on effective thermal conductivity of graphene–polymer nanocomposites
Figure 1.5 A graphene–polymer composite system and steady-state temperature gradient using molecular dynamics simulation [148]
Figure 1.6 Vibrational density of states of polymer atoms and graphene atoms [148]
Chapter 5: Probabilistic Strength Theory of Carbon Nanotubes and Fibers
Figure 5.1 Schematic of hexagonal lattice structure and the set-up of weakest link elements in a single-walled CNT
Figure 5.2 Schematic of a log–log plot of the flaw size distribution of weakest link elements.
Figure 5.3 Schematic of the Weibull plot for the strength distribution of CNTs.
Figure 5.4 Number of weakest link elements
n
versus the relative error of the generalized Weibull distribution (1.18) when approximating the weakest link distribution (5.17) with
m
= 3 and
σ
0
= 150 GPa [2]
Figure 5.5 (a) Weibull plot of Group 2; (b) normal plot of Group 2; (c) normal plot of Group 1; (d) normal plot of Group 3 (dashed lines delimit 95% confidence band) (Reproduced with permission from [13]. Copyright [2007], AIP Publishing LLC)
Figure 5.6 Weibull plot of the strength test data showing a rising upper tail, for multiwall WS
2
nanotubes that are 15–30 nm in diameter and 2–5 µm long [25] (Copyright (2006) National Academy of Sciences, U.S.A.)
Figure 5.7 Extrapolation of test data in Region II to Regions I and III [3]
Figure 5.8 Carbon nanotubes appear aligned at the focused ion beam cut cross section of an aerogel-spun fiber (Reproduced with permission from [28]. Copyright [2012], AIP Publishing LLC)
Figure 5.9 SEM micrographs of CNT yarns with a low-twist angle (a) and a high-twist angle (b) [29] (Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission)
Figure 5.10 Moore's neighborhood (gray area on (a)) and Von Neumann's neighborhood (gray area on (b)) of the broken bond (black square), with the first- and second-order zones indicated with dash and bold lines, respectively [1]
Figure 5.11 Calculated load distribution factors for examples: (a) single broken bond; (b) two simultaneously broken bonds using first-order LLS; (c) two simultaneously broken bonds using second-order LLS; (d) four simultaneously broken bonds using first-order LLS; (e) four simultaneously broken bonds using second-order LLS [1]
Figure 5.12 A sample set of stress–strain curves and snapshots (white-broken bonds, dark-intact bonds, left to right from zeroth- to fourth-order LLS) at the peak load for a CNT rope consisting of CNTs [1]
Figure 5.13 Schematic of hierarchical structure of CNT-fibers (the parameter values shown correspond to a benchmark model for simulation) [1]
Figure 5.14 Normal probability paper plot for the strengths of CNT bundles using the 2D zeroth to fourth-order LLS rules on 5-µm-long CNTs (10,000 Monte Carlo samples or each order of the LLS) [1]
Chapter 8: Mechanical Characterization of 2D Nanomaterials and Composites
Figure 8.1 Dimensional classification of nanomaterials. (a) 0D with all dimensions in the three directions are in the nanoscale ( 10
−9
m). Examples of this kind are nanoparticles, quantum dots, and clusters. (b) 1D where one dimension of the nanostructure will be outside the nanometer range. Nanorods, nanotubes, and nanofibers are included in this classification. (c) 2D with two dimensions outside the nanometer range. These include different types of thin films and plates. (d) 3D with all dimensions outside the nanoscale. Bulk amorphous materials and materials with a nanocrystalline structure are included in this category
Figure 8.2 Atomic structures of (a) graphite and (b) graphene
Figure 8.3 (a) Classification of graphyne family. (b) Schematic of graphene to graphyne
Figure 8.4 Atomic structures of graphene allotropes. (a) Graphyne, (b) graphdiyne, (c) graphene allotrope with Stone–Wales defects, (d) supergraphene
Figure 8.5 Schematic of silicene
Figure 8.6 Atomic structure of boron nitride (BN)
Figure 8.7 Atomic structure of molybdenum disulfide (MoS
2
)
Figure 8.8 Atomic structure of phosphorene
Figure 8.9 Schematic of an indentation experimental setup showing a nanostructure on substrate, suspended over open holes. The mechanical properties are probed by deforming and breaking the resulting suspended free-standing sheet with an atomic force microscope (AFM)
Figure 8.10 Simple spring–network model representation for scaling law. (a) Serial spring representation for the acetylene links. (b) The linear spring stiffness can be associated to an effective modulus via an effective cross-sectional area
Figure 8.11 (a) Stable graphyne structure after minimization and equilibration of 0.5 ns at a temperature of 300 K. (b) Schematic illustration of a 2D silicene sheet bended with an imposed radius of curvature
κ
= 1/
r
. (c) Energy versus curvature for a silicene sheet with 95% confidence bounds
Figure 8.12 Nanoparticles trapped in a circular blister at graphene–silicon interface. (a) SEM image, where the trapped particles appear dark. (b) 3D representation. (c) Blister model sketch. [73]
Figure 8.13 Folded configurations of graphene sheet. (a) Schematic of a folded single-layer graphene. (b) Flat state and (c) folded configuration of monolayer graphene from MD model. (d) Enlarged view of the folded graphene edge. [79, 85]
Figure 8.14 Nanocomposite configurations: (a) single graphdiyne with copper substrate, (b) bilayer graphdiyne with copper substrate, (c) single graphdiyne with copper sandwich, (d) bilayer graphdiyne with copper sandwich. [126]
Chapter 10: Mechanics of Thermal Transport in Mass-Disordered Nanostructures
Figure 10.1 (a) Density of vibrational states (DOS) of CNTs containing 0%, 40%, and 100% of the
14
C isotopes in the material structure. (b) The vibrational spectra are shifted in the vertical direction to facilitate comparison. Reproduced by permission from Royal Society of Chemistry [7]
Figure 10.2 Density of vibrational states (DOS) of CNTs containing 0%, 10%, 20%, 40%, 80%, and 100% of the
14
C isotopes in the material structure. The wave numbers are rescaled according to Eq. 10.2. The intensity in the plot is reported in arbitrary units. Reproduced by permission from Royal Society of Chemistry [7]
Figure 10.3 The steady-state temperature distribution along the direction of heat transfer obtained from the MD simulations is presented for (a) CNT and (b) GNR, each of length
L
= 50 nm. Reproduced by permission from Elsevier [12]
Figure 10.4 Thermal conductivity (in W/m K) of the pure CNTs with respect to the inverse of the square root of the mass of the atom (atomic mass unit) constituting the structure. The dotted line shows a fit to the data that result in a linear profile. Reproduced by permission from American Institute of Physics [13]
Figure 10.5 (a) The one-dimensional phonon dispersion behavior for a
12
C nanotube as determined by LDA calculations. (b) The classical DOS of pure
X
C nanotube, where
X
= 12, 14, 20, and 40, with wave numbers rescaled by (12/
X
)
0.5
. Reproduced by permission from American Institute of Physics [13]
Figure 10.6 Comparison between RNEMD calculations and other simulations of carbon nanostructures (shown as data points), and the mean-field approximation model represented as a dashed curve shows good agreement. Reproduced by permission from Royal Society of Chemistry [7]
Figure 10.7 (a) Comparison between the model and the MD simulations of silicon crystals with
28
Si and
29
Si substituted with
29
Si and
30
Si atoms, respectively, shows good agreement.
x
denotes the fraction of the lightest isotope, that is,
28
Si or
29
Si. The ratio
k
r
(
x
)/
k
r
(0.5) is presented instead of
k
r
(
x
) because data for the pure components are unavailable. The dashed line corresponds to mean-field approximation model described in Eq. 10.6. (b) Comparison between MD simulations in silicon–germanium nanowires and the prediction of Eq. 10.6 shows differences that are eliminated by correcting effects of multielemental composition. Reproduced by permission from Royal Society of Chemistry [7]
Figure 10.8 There is a good agreement between informatics predictions with MD simulation, and both capture the same minima in conductivity with 50% isotope concentration. The predictions also capture the same trends as previous computational and experimental results, with differences existing due to different atomic masses of the carbon isotopes. Using informatics, we are able to predict the thermal conductivity of isotope-substituted graphene of all possible binary compositions without requiring additional MD calculations or measurements. Reproduced by permission from American Institute of Physics [27]
Figure 10.9 The modes of the vibrational spectra most affected by changing isotope percentages and resulting in thermal conductivity variation. In isotope-substituted materials, high-frequency modes are strongly influenced to induce a reduction in thermal conductivity. (a) The localized modes are defined as those with wavenumber greater than 800 cm
−1
, while those below 800 cm
−1
are delocalized. In the loadings spectra, the circled peaks (highest magnitude) are those that are (shown in (b)) most impacted by changing composition and (shown in (c)) most significant in determining thermal conductivity. Without employing informatics, we could not have otherwise picked up on the relationship between localized modes and thermal conductivity. Reproduced by permission from American Institute of Physics [27]
Chapter 11: Thermal Boundary Resistance Effects in Carbon Nanotube Composites
Figure 11.1 Snapshot of the DWNT coated by amorphous SiO
2
. (a) View from the side and along the z-axis; (b) Close up view showing the inserted nanotube to
β
-cristobalite SiO
2
Figure 11.2 Kapitza resistance between outer wall of DWNT and amorphous SiO
2
. “ACAC” and “ACZZ” indicate (5,5)–(10,10) and (6,6)–(19,0) DWNTs, respectively
Figure 11.3 Kapitza resistance between the inner wall and outer wall of the DWNT
Figure 11.4 The ratio of Kapitza resistance between the full-length outer wall of DWNTs and amorphous silica (
R
fl
), and the Kapitza resistance between the edges of the outer wall and amorphous silica (
R
e
)
Figure 11.5 The ratio of Kapitza resistance between the full-length inner wall and outer wall of the DWNTs, and the Kapitza resistance between their edges
List of Tables
Chapter 5: Probabilistic Strength Theory of Carbon Nanotubes and Fibers
Table 5.1 Test data of 19 MWCNTs
Table 5.2 Weibull strength fits for 19 MWCNTs
Table 5.3 Three strength regions of CNTs
Table 5.4 Mean values and c.o.v obtained from ELS and LLS rules on 5-µm-long CNT bundles [1]
Table 5.5 Mean values and c.o.v obtained from the weakest link model on 50-µm-long CNT bundles [1]
Table 5.6 Mean values and c.o.v (%) obtained from LL rules on 50-µm-long CNT fibers
Chapter 8: Mechanical Characterization of 2D Nanomaterials and Composites
Table 8.1 Mechanical properties for extended graphynes [11]
Microsystem and Nanotechnology Series
Series Editors–Ron Pethig and Horacio Dante Espinosa
Advanced Computational Nanomechanics
Silvestre, February 2016
Micro-Cutting: Fundamentals and Applications
Cheng, Huo, August 2013
Nanoimprint Technology: Nanotransfer for Thermoplastic and Photocurable Polymer
Taniguchi, Ito, Mizuno and Saito, August 2013
Nano and Cell Mechanics: Fundamentals and Frontiers
Espinosa and Bao, January 2013
Digital Holography for MEMS and Microsystem Metrology
Asundi, July 2011
Multiscale Analysis of Deformation and Failure of Materials
Fan, December 2010
Fluid Properties at Nano/Meso Scale
Dyson et al., September 2008
Introduction to Microsystem Technology
Gerlach, March 2008
AC Electrokinetics: Colloids and Nanoparticles
Morgan and Green, January 2003
Microfluidic Technology and Applications
Koch et al., November 2000
Advanced Computational Nanomechanics
Edited by
Nuno Silvestre
University of LisbonPortugal
This edition first published 2016
© 2016 John Wiley & Sons Ltd
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Library of Congress Cataloging-in-Publication Data
Advanced computational nanomechanics / edited by Nuno Silvestre.
pages cm
Includes bibliographical references and index.
ISBN 978-1-119-06893-8 (cloth)
1. Nanotechnology–Mathematics. 2. Nanoelectromechanical systems–Mathematical models. 3. Nanostructures–Mathematical models. 4. Micromechanics–Mathematics. I. Silvestre, Nuno, editor.
T174.7.A385 2016
620′.5–dc23
2015035009
A catalogue record for this book is available from the British Library.
List of Contributors
Avinash Akepati
, Graduate Research Assistant, Department of Aerospace Engineering and Mechanics, University of Alabama, Tuscaloosa, AL, USA
Ganesh Balasubramanian
, Assistant Professor, Department of Mechanical Engineering, Iowa State University, Ames, IA, USA
Irene J. Beyerlein
, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
Khoa Bui
, School of Chemical, Biological and Materials Engineering, The University of Oklahoma, Norman, OK, USA
Steven W. Cranford
, Assistant Professor, Department of Civil and Environmental Engineering, Northeastern University, Boston, MA, USA
Ghasem Ghadyani
, Faculty of Mechanical Engineering, University Malaya, Kuala Lumpur, Malaysia
Ali Ghavamian
, Research Assistant, School of Engineering, Griffith University, Gold Coast Campus, Southport, Queensland, Australia
Yuantong Gu
, Professor, School of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology, Brisbane, QLD, Australia
Mesut Kirca
, Assistant Professor, Department of Mechanical Engineering, Istanbul Technical University, Istanbul, Turkey
Huong Nguyen
, School of Chemical, Biological and Materials Engineering, The University of Oklahoma, Norman, OK, USA
Andreas Öchsner
, Prof. Dr.-Ing., School of Engineering, Griffith University, Gold Coast Campus, Southport, Queensland, Australia
Dimitrios V. Papavassiliou
, Professor, School of Chemical, Biological and Materials Engineering, The University of Oklahoma, Norman, OK, USA; Division for Chemical, Bioengineering and Environmental Transport Systems, National Science Foundation, Arlington, VA, USA
Nicola M. Pugno
, Professor, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy; Centre of Materials and Microsystems, Bruno Kessler Foundation, Trento, Italy; School of Engineering and Materials Science, Queen Mary University, London, UK
Moones Rahmandoust
, Dr., Protein Research Center, Shahid Beheshti University, G.C., Velenjak, Tehran, Iran; School of Engineering, Griffith University, Gold Coast Campus, Southport, Queensland, Australia; Deputy Vice Chancellor Office of Research and Innovation, Universiti Teknologi Malaysia, Johor Bahru, Johor, Malaysia
Ruth E. Roman
, Graduate Research Assistant, Department of Civil and Environmental Engineering, Northeastern University, Boston, MA, USA
Samit Roy
, William D Jordan Professor, Department of Aerospace Engineering and Mechanics, University of Alabama, Tuscaloosa, AL, USA
Hiroyuki Shima
, Associate Professor, Department of Environmental Sciences,University of Yamanashi, Kofu, Yamanashi, Japan
Yoshiyuki Suda
, Associate Professor, Department of Electrical and Electronic Information Engineering, Toyohashi University of Technology, Toyohashi, Aichi, Japan
Keka Talukdar
, HOD, Department of Physics, Nadiha High School, Durgapur, West Bengal, India
Albert C. To
, Associate Professor, Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, USA
Chien Ming Wang
, Professor, Engineering Science Programme, Faculty of Engineering, National University of Singapore, Singapore, Singapore
Yu Wang
, PhD Student, School of Computing, Engineering and Mathematics, University of Western Sydney, Sydney, NSW, Australia
Xi F. Xu
, Professor, School of Civil Engineering, Beijing Jiaotong University, Beijing, China
Yingyan Zhang
, Senior Lecturer, School of Computing, Engineering and Mathematics, University of Western Sydney, Sydney, NSW, Australia
Series Preface
Books in the Wiley's Microsystem and Nanotechnology Series are intended, through scholarly works of the highest quality, to serve researchers and scientists wishing to keep abreast of advances in this expanding and increasingly important field of technology. Each book in the series is also intended to be a rich interdisciplinary resource for not only researchers but also teachers and students of specialized undergraduate and postgraduate courses.
Past books in the series include the university textbook Introduction to Microsystem Technology by Gerlach and Dötzel, covering the design, production and application of miniaturized technical systems from the viewpoint that for engineers to be able to solve problems in this field, they need to have interdisciplinary knowledge over several areas as well as the capability of thinking at the system level. In their book Fluid Properties at Nano/Meso Scale, Dyson et al. take us step by step through the fluidic world bridging the nanoscale, where molecular physics is required as our guide, and the microscale where macro continuum laws operate. Jinghong Fan in Multiscale Analysis of Deformation and Failure of Materials provides a comprehensive coverage of a wide range of multiscale modelling methods and simulations of the solid state at the atomistic/nano/submicron scales and up through those covering the micro/meso/macroscopic scale. Most recently, Nano and Cell Mechanics: Fundamentals and Frontiers, edited by Espinosa and Bao, assembled through their own inputs and those of 47 other experts of their chosen fields of endeavour, 17 timely and exciting chapters that represent the most comprehensive coverage yet presented of all aspects of the mechanics of cells and biomolecules.
In this book edited by Professor Nuno Silvestre, who is well known for his own work in the modelling and simulation of carbon nanotube structures, he has assembled 11 chapters written by international experts that comprehensively cover the experimental, modelling and theoretical studies of the mechanical and thermal properties of carbon nanotubes, polymer nanocomposites and other nanostructures. The research literature on Computational Nanomechanics is spread among many journals specialising in mechanics, computer science, materials science and nanotechnology. The latest advances in this rapidly moving field of nanotechnologies have been collected together in a single book!
Although the chapters are primarily intended for established scientists, research engineers and PhD students who have knowledge in materials science and numerical simulation methods, the clarity of writing and pedagogical style of the various chapters make much of this book's content suitable for inclusion in undergraduate and postgraduate courses.
Ronald PethigSchool of EngineeringThe University of Edinburgh
Preface
During the last decade, nanomechanics emerged on the crossroads of classical mechanics, solid-state physics, statistical mechanics, materials science, and quantum chemistry. As an area of nanoscience, nanomechanics provides a scientific foundation of nanotechnology, that is, an applied area with a focus on the mechanical properties of engineered nanostructures and nanosystems. Owing to smallness of the studied objects (atomic and molecular systems), nanomechanics also accounts for discreteness of the object, whose size is comparable with the interatomic distances, plurality, but finiteness, of degrees of freedom in the object, thermal fluctuations, entropic effects, and quantum effects. These quantum effects determine forces of interaction between individual atoms in physical objects, which are introduced in nanomechanics by means of some averaged mathematical models called interatomic potentials. Subsequent utilization of the interatomic potentials within the classical multibody dynamics provides deterministic mechanical models of nanostructures and systems at the atomic scale/resolution. This book focuses on a variety of numerical and computational methods to analyze the mechanical behavior of materials and devices, including molecular dynamics, molecular mechanics, and continuum approaches (finite element simulations). In resume, several computational methods exist to model and simulate the behavior of nanostructures. All of them present advantages and drawbacks. This book presents a survey on the computational modeling of the mechanical behavior of nanostructures, with particular emphasis on CNTs, graphene, and composites. It includes 11 chapters and each chapter is an independent contribution by scientists with worldwide expertise and international reputation in the technological area.
In Chapter 1, Y. Y. Zhang, Y. Wang, C.M. Wang, and Y.T. Gu present a state-of-the art review on the thermal conductivity of graphene and its polymer nanocomposites. It is known that some progress has been achieved in producing graphene–polymer nanocomposites with good thermal conductivity, but interfacial thermal resistance at the graphene/polymer interfaces still hinders this improvement. This chapter reports recent research studies, which have shown that covalent and noncovalent functionalization techniques are promising in reducing the interfacial thermal resistance. The authors argue that further in-depth research studies are needed to explore the mechanisms of thermal transport across the graphene/polymer interfaces and achieve graphene–polymer nanocomposites with superior thermal conductivity.
In Chapter 2, M. Kirca and A.C. To describe the mechanics of CNT network materials. Recently, the application of CNTs has been extended to CNT networks in which the CNTs are joined together in two- or three-dimensional space and CNT networks. In this chapter, a thorough literature review including the most recent theoretical and experimental studies are presented, with a focus on the mechanical characteristics of CNT network materials. As a supporting material, some recent studies of authors are also introduced to provide deeper understanding in mechanical behavior of CNT network materials.
In Chapter 3, H. Shima and Y. Suda present the “Helical carbon nanomaterial,” which refers to exotic nanocarbons having a long, thin, and helical morphology. Their spiral shape can be exploited for the development of novel mechanical devices such as highly sensitive tactile nanosensors, nanomechanical resonators, and reinforced nanofibers in high-strain composites. Because the quantitative determination of their mechanical properties and performance in actual applications remains largely unexamined, advanced computational techniques will play an important role, especially for the nanomaterials that hold the promise for use in next-generation nanodevices that have been unfeasible by the current fabrication techniques. This chapter gives a bird's eye view on the mechanical properties of helical carbon nanomaterials, paying particular attention to the latest findings obtained by both theoretical and experimental efforts.
In Chapter 4, G. Ghadyani and M. Rahmandoust present a review of the fundamental concepts of the Newtonian mechanics including Lagrangian and Hamiltonian functions, and then the developed equations of motion of a system with interacting material points are introduced. After that, based on the physics of nanosystems, which can be applicable in any material phases, basic concepts of molecular dynamic simulations are introduced. The link between molecular dynamics and quantum mechanics is explained using a simple classical example of two interacting hydrogen atoms and the major limitations of the simulation method are discussed. Length and timescale limitation of molecular dynamic simulation techniques are the major reasons behind opting multiscale simulations rather than molecular dynamics, which are explained briefly at the final sections of this chapter.
In Chapter 5, a probabilistic strength theory is presented by X. Frank Xu to formulate and model probability distribution for strength of CNTs and CNT fibers. A generalized Weibull distribution is formulated to explain statistical features of CNT strength, and a multiscale method is described to show how to upscale strength distribution from nanoscale CNTs to microscale CNT fibers. The probabilistic theory is considered applicable to fracture strength of all brittle materials, and with certain modifications, to failure of non-brittle materials as well. The benchmark model on strength upscaling from CNTs to fibers indicates that the full potential of CNT fibers for exploitation is expected to be in the range between 10 and 20 GPa with respect to mean strength, due to universal thermodynamic effects and inherent geometric constraints.
In Chapter 6, A. Ghavamian, M. Rahmandoust, and A. Öchsner review and present the latest developments on nanomechanics of perfect and defective heterojunction CNTs. Homogeneous and heterojunction CNTs are of a great importance because of their exceptional properties. In this chapter, the studies on considerable number of different types of perfect and atomically defective heterojunction CNTs with all possible connection types as well as their constructive homogeneous CNTs of different chiralities and configurations are presented and their elastic, torsional, and buckling properties are numerically investigated based on the finite element method with the assumption of linear elastic behavior. They conclude that the atomic defects in the structure of heterojunction CNTs lead to an almost linear decrease in the mechanical stability and strength of heterojunction CNTs, which appeared to be considerably more in the models with carbon vacancy rather than Si-doped models.
In Chapter 7, S. Roy and A. Akepati present a methodology for the prediction of fracture properties in polymer nanocomposites. These authors propose a new methodology to compute J-integral using atomistic data obtained from LAMMPS (Large-Scale Atomic/Molecular Massively Parallel Simulator). As a case study, the feasibility of computing the dynamic atomistic J-integral over the MD domain is evaluated for a graphene nano-platelet with a central crack using OPLS (Optimized Potentials for Liquid Simulations) potential. For model verification, the values of atomistic J-integral are compared with results from linear elastic fracture mechanics (LEFM) for isothermal crack initiation at 300 K. Computational results related to the path-independence of the atomistic J-Integral are also presented. Further, a novel approach that circumvents the complexities of direct computation of entropic contributions in polymers is also discussed.
In Chapter 8, R.E. Roman, N.M. Pugno, and S.W. Cranford characterize the mechanical behavior of 2D nanomaterials and composites. They relate the recent isolation of graphene from graphite with the discovery and synthesis of a multitude of similar two-dimensional crystalline. These single-atom thick materials (2D materials) have shown great promise for emerging nanotechnology applications, and a full understanding of their mechanistic behavior, properties, and failure modes is essential for the successful implementation of 2D materials in design. R.E. Roman, N.M. Pugno, and S.W. Cranford focus on the fundamental nanomechanics of 2D materials, describing how to characterize basic properties such as strength, stiffness, bending rigidity, and adhesion using computational methods such as molecular dynamics (MD). Failure is discussed in terms of the local, atomistic interpretation of quantized fracture mechanics (QFM) and extended to bulk systems via Nanoscale Weibull Statistics (NWS).
In Chapter 9, K. Talukdar studied the effect of chirality on the mechanical properties of defective CNTs. In spite of many existing studies, the experimental values of the mechanical properties of the CNTs vary in magnitude up to one order as the actual internal structure of the CNTs is still not completely known and the properties vary largely with their chirality and size. For better understanding of the internal molecular details, computer modeling and simulation may serve as an important tool, and MD simulations provide the internal dynamical change that the system is running through in course of time under various external forces. In this chapter, a comprehensive investigation is presented to find out the influence of chirality on the defective CNTs.
In Chapter 10, G. Balasubramanian describes the mechanics of thermal transport in mass disordered nanostructures. Its scope is to provide a flavor of a set of computational approaches that are employed to investigate heat transfer mechanisms in mass disordered nanostructures, and in particular those containing isotope impurities. Carbon nanomaterials have been attractive case studies for their remarkable properties. Although the examples below demonstrate effects of isotopes mostly in carbon nanotubes and graphene, the approaches are generic and applicable to a wide array of nanomaterials. This chapter begins from the fundamentals of thermal transport in nanomaterials, moves to engineering material properties and their computation, and finally concludes with data-driven methods for designing defect-engineered nanostructures.
In Chapter 11, D.V. Papavassiliou, K. Bui, and H. Nguyen present a study on the thermal boundary resistance (TBR) effects in CNT composites. In this work, the authors review briefly predictive models for the thermal behavior of CNTs in composites, and simulation efforts to investigate improvements in Keff with an increase in CNT volume fraction. Emphasis is placed in the investigation of the TBR in CNTs coated with silica in order to reduce the overall TBR of the composites. The authors propose the use of multiwalled CNTs (instead of single-walled CNTs), because heat can be transferred mainly through the outer wall offering the advantage of a larger area of heat transfer. A better approach proposed by the authors might be to design composites where the CNT orientation, when it leads to anisotropic thermal properties, or thermal rectification could be explored.
Finally, the editor would like to thank all authors for having accepted the challenge of writing these chapters and also for they support to achieve this high-quality book. The editor also acknowledges Wiley professionals (Clive Lawson and Anne Hunt) and SPi Global (Durgadevi Shanmughasundaram) for their enthusiastic and professional support.
Nuno SilvestreLisbon, May 2015
Chapter 1Thermal Conductivity of Graphene and Its Polymer Nanocomposites: A Review
Yingyan Zhang1, Yu Wang1, Chien Ming Wang2 and Yuantong Gu3
1School of Computing, Engineering and Mathematics, University of Western Sydney, Sydney, NSW, Australia
2Engineering Science Programme, Faculty of Engineering, National University of Singapore, Singapore, Singapore
3School of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology, Brisbane, QLD, Australia
1.1 Introduction
In recent years, there have been many research studies conducted on the properties and applications of graphene. Experimental and theoretical studies have revealed that graphene is hitherto the best thermal conductor ever known in nature. This extraordinary thermal transport property of graphene makes it promising in various applications. One of the popularly pursued applications is its use as thermally conductive fillers in polymer nanocomposites. It is expected that the novel graphene–polymer nanocomposites possess superior thermal conductivity and thus hold an enormous potential in thermal management applications. In this chapter, we review the recent research progress of graphene, with an emphasis on its thermal conductivity and its polymer nanocomposites.
1.2 Graphene
1.2.1 Introduction of Graphene
Graphene is the latest discovered carbon allotrope in 2004, following the discoveries of fullerenes and carbon nanotubes in the past three decades. Before its discovery, graphene actually has, for many years, been deemed as an academic material that could not stand-alone in the real world because of the thermal instability of two-dimensional structures at the nanoscale level [1, 2]. However, this perception was quashed in 2004 by the ground-breaking research of Geim and Novoselov [3–5]. For the first time, they succeeded in producing real samples of graphene. Owing to their pioneering work, they were awarded the 2010 Nobel Prize in Physics. Ever since the discovery of graphene, its extraordinary properties and huge potential applications have attracted immense attentions from the scientific world. This section gives a brief overview on graphene. In Sections 1.2.1.1 and 1.2.1.2, the structure and synthesis of graphene are explained. Section 1.2.2 gives an overview on its electronic, mechanical and optical properties. The emphasis is given on its thermal conductivity in Section 1.2.3.
1.2.1.1 Structure
Graphene is a single two-dimensional flat layer of carbon atoms, where the carbon atoms are arranged in a honeycomb-like hexagonal crystalline lattice [3–10]. Figure 1.1 shows an illustration of the lattice structure of graphene.
Figure 1.1 Lattice structure of graphene
The properties of graphene are determined by its bonding mechanism. Carbon is the sixth element in the periodic table, and one carbon atom possesses six electrons. In the ground state, its configuration of electrons is 1s22s22p2, where s and p indicate different atomic orbitals, the numbers in front of the orbital represent different electron shells and the superscript numbers denote the number of electrons in the shells. This electronic configuration of carbon means that there are two electrons filling, respectively, in 1s, 2s and 2p orbitals. The two electrons in 1s orbital are close to the nucleus and are irrelevant for chemical reactions. The 2s orbital is approximately 4 eV lower than the 2p orbitals; thus, it is energetically favourable for a carbon atom to keep two electrons in the 2s orbital and the remaining two electrons in the 2p orbital [11]. The 2p orbital can be divided into 2px, 2py and 2pz orbitals for designation purposes. In the ground state, each of the two electrons in the 2p orbital occupies one single 2p orbital, that is 2px and 2py. While in a presence of other carbon atoms, it is favourable for a carbon atom to excite one electron from the 2s orbital to the third 2p orbital (i.e. 2pz orbital) to form covalent bonds. It costs 4 eV for this excitation, but the energy gain from the covalent bond is actually higher than the cost. Therefore, in an excited state, a carbon atom has four equivalent orbitals 2s, 2px, 2py and 2pz available to form covalent bonds with other atoms. The superposition of the 2s orbital with n (=1, 2 or 3) 2p orbitals is called spn hybridisation [12]. Hybridisation plays an essential role in covalent carbon bonds, and it is the reason why carbon can form a variety of materials such as diamond, graphite, graphene, carbon nanotubes and fullerenes. When carbon atoms combine to form graphene, sp2 hybridisation or the superposition of the 2s orbital with two 2p orbitals (i.e. 2px and 2py) occurs. The 2s orbital and two 2p orbitals form three hybrid sp2 orbitals within one plane and have a mutual angle of 120°. These three sp2 orbitals interact with the nearest neighbours of the carbon atoms to form in-plane σ bonds. The σ bonds are strong covalent bonds that bind the carbon atoms in the plane. The remaining 2pz orbital forms a π bond, which is perpendicular to the atomic plane. The out-of-plane π bond is much weaker than the in-plane σ bond. The π bond plays an important role in the out-of-plane interactions, such as van der Waals interaction. Figure 1.2 illustrates the bonding mechanism between carbon atoms in graphene. Two adjacent hexagonal lattices are displayed, and the solid circles denote the nuclei of carbon atoms.
Figure 1.2 Bonding mechanism between carbon atoms in graphene
As shown in Figure 1.3, the unit cell of the graphene hexagonal crystalline lattice is made of two nonequivalent carbon atoms. Within the atomic plane, the distance between two nearest carbon atoms is 0.142 nm [13]. Graphene has two main types of edge terminations, zigzag and armchair, which are commonly used for referring to the directions of the graphene lattice. With a thickness of single atomic layer, graphene is the thinnest material known to mankind. Graphene can be regarded as the basic building unit of other carbon allotropes. For example, it can be wrapped up to form zero-dimensional fullerenes, rolled into one-dimensional carbon nanotubes or stacked up to form three-dimensional graphite.
Figure 1.3 Unit cell and directionality of graphene
1.2.1.2 Synthesis
The initial approach used by Geim and Novoselov to produce graphene was simply mechanical exfoliation (i.e. repeated peeling) of highly oriented pyrolytic graphite [3]. In the recent years, both academia and industry have invested great effort in developing the techniques for mass production of graphene. In general, the methods for graphene synthesis can be classified into two categories, namely bottom-up method and top-down method.
Bottom-Up Method
The bottom-up methods for synthesising graphene mainly include chemical vapour deposition (CVD) and epitaxial growth. Abundant reports are available on the growth of graphene using CVD [14–30]. In this process, a substrate is exposed to thermally decomposed precursors, and the desired graphene layers are deposited on to the substrate surface at high temperatures. Various carbon-rich materials such as hydrocarbons can be used as the precursors. The substrate acts as a transition surface and it is usually made of metals such as nickel, copper, cobalt, ruthenium and platinum. On cooling the substrate, the solubility of carbon in the transition metal decreases, and the graphene layer is expected to precipitate from the surface. In 2009, Reina et al. [14] demonstrated the growth of continuous single- to few-layered graphene by CVD on polycrystalline nickel and transfer of graphene to a large variety of substrates using a wet etching method. They used diluted hydrocarbon gas flow as the precursor, and the growth was done at temperatures of 900 °C to 1000 °C and at ambient pressure. The resulting film exhibited a large fraction of single- and bilayered graphene regions with up to approximately 20 µm in lateral dimension. In 2010, Srivastava et al. [15] reported the growth of centimetre-sized, uniform and continuous single- and few-layered graphene by CVD on polycrystalline copper foils with liquid hexane precursor. Structural characterisations using the Raman spectroscopy, transmission electron microscopy and atomic force microscopy suggested that the as-grown graphene has single to few layers over large areas and is highly continuous. This study demonstrated that these graphene layers can be easily transferred to any desired substrate without damage by dissolving the copper foil in diluted nitric acid.
Epitaxial growth is another substrate-based method for synthesising graphene, and it has been widely explored in the last few years [31–40]. In this method, the graphene layer is obtained from the high temperature reduction of silicon carbide substrates in an ultrahigh vacuum. As silicon is removed around 1000 °C, a carbon-rich layer is left behind, which later undergoes graphitisation to form graphene. Berger et al. [31] produced one- to three-layered graphene by epitaxial growth on the silicon-terminated (0001) surface of single-crystal 6H–SiC. In their work, the process started with surface preparation by H2 etching. Next, the 6H–SiC samples were heated by electron bombardment in ultrahigh vacuum to approximately 1000 °C. Finally, the graphene layers were formed when the samples were heated to temperatures of 1250 °C to 1450 °C for up to 20 min. In 2006, Rollings et al. [32] used a similar epitaxial process and synthesised graphene down to one atom layer on the (0001) surface of 6H–SiC.
The synthesis of graphene by CVD and by epitaxial growth has several major advantages. Firstly, these methods produce graphene of high quality and high purity. Secondly, both methods are able to produce graphene in large sizes, in theory, up to an entire wafer. Thirdly, they are compatible with current semiconductor processes. By controlling the process parameters, graphene in desired morphology, shape and size is achievable. However, both methods suffer the disadvantages of high costs and complex processes.
Top-Down Method
The top-down methods for synthesising graphene mainly include mechanical, chemical and thermal exfoliation/reduction of graphite or graphite oxide. Mechanical exfoliation of graphite is the first recognised method for the synthesis of graphene. Graphite is actually graphene layers stacked together by weak van der Waals forces. The interlayer distance is 3.34 Å and the van der Waals binding energy is approximately 2 eV/nm2. Mechanical cleaving of graphite into graphene of single atomic layer is achievable by applying an external normal force of about 300 nN/µm2 [41]. Novoselov et al. [3] used a scotch tape to gradually peel off the graphite flakes and release them in acetone. The flakes were then transferred from the acetone solution to a silicon substrate. After cleaning with water and propanol, thin flakes of graphene including single-layered graphene were obtained. The mechanical exfoliation method can produce graphene of large size and high quality but in very limited quantities, which makes it only suitable for experimental purposes. In order to be applied in industry, this process needs to be further improved.
Chemical exfoliation of graphite is a process in which chemicals are intercalated with the graphite structure to isolate the graphene layers. Recent research studies reported that graphite can be chemically exfoliated into single- and few-layered graphene in the presence of perfluorinated aromatic solvents [42], organosilane [43], N-methyl-pyrrolidone [44] and chlorosulphonic acid [45]. The chemical exfoliation processes are usually assisted with sonication that facilitates the solubilisation and exfoliation of graphite. Chemical exfoliation has several advantages. Firstly, it produces graphene at a low cost. Secondly, the process is relatively simple and does not rely on complex infrastructure. Thirdly, the process is capable of depositing graphene on a wide variety of substrates and it can be extended to produce graphene-based composites. Most importantly, the process is scalable for high-volume manufacturing.
The reduction of graphite oxide is another promising method for graphene synthesis. This process starts from oxidising the raw material of graphite to graphite oxide. Through oxidisation, the interlayer distance between graphene layers increases from 3.34 Å to 7 Å because of the presence of oxygen moieties [46]. The interlayer van der Waals interactions in graphite oxide become much weaker in comparison to the original graphite structure. It is, therefore, much easier to exfoliate graphite oxide than it is to exfoliate graphite directly into graphene. Abundant reports have demonstrated that exfoliation and reduction of graphite oxide to graphene can be achieved through chemical and/or thermal processes [46–57]. The synthesis of graphene through reduction of graphite oxide has basically the same advantages as the chemical exfoliation method. Thus, the reduction of graphite oxide is another promising method for achieving mass production of graphene at a low cost.
To sum up, a number of methods have already been well developed for the synthesis of graphene, some of which are transferable to mass production at low cost. Nowadays, graphene has become a readily available material for use in various applications.
1.2.2 Properties of Graphene
Owing to its unique two-dimensional structure and sp2 carbon bonding mechanism, graphene has been discovered to possess several extraordinary properties that endow it for a variety of potential applications.
1.2.2.1 Electrical Property
The charge carriers experience a very weak scattering while transporting in graphene. Recent experiments reported that the charge carrier mobility reaches more than 20,000 cm2/V s at room temperature. This value is an order of magnitude higher than the most commonly used electronic material – silicon [58]. Graphene is one of the most promising replacements for silicon in electronic applications. It is believed that graphene-based field-effect transistors will exceed the frequency limits of conventional planar transistors, thereby paving the way for high-speed electronic devices. Intensive research studies have been carried out on this application of graphene and significant progress has been made. In 2010, Lin et al. [59] presented field-effect transistors fabricated on a 2-in. graphene wafer with a cut-off frequency as high as 100 GHz. Liao et al. [60] fabricated graphene-based field-effect transistors with a record of 300-GHz performance. In spite of the promising application of graphene in electronics, one of the major obstacles is that pristine graphene acts as a semi-metal with a zero electronic band gap at room temperature [61]. This feature of pristine graphene makes it unsuitable for the transistor digital circuits because of the relatively strong inter-band tunnelling in the field-effect transistor off state [62, 63]. Research efforts have been devoted to develop approaches to open the band gap of graphene and fabricate graphene-based field-effect transistor with a sufficiently large on/off ratio [64–66].
1.2.2.2 Mechanical Property
Graphene has been confirmed to be the strongest material ever discovered. In 2008, Lee et al. [67] measured the elastic properties and intrinsic breaking strength of a free-standing monolayer graphene by nanoindentation in an atomic force microscopy. They demonstrated that defect-free graphene possesses a Young's modulus of 1 TPa and an ultimate strength of 130 GPa. The experimental findings have been validated by theoretical and simulation works [68, 69]. The outstanding mechanical properties of graphene have attracted interest in a wide range of applications. For instance, the light but stiff graphene serves as a great reinforcement material for building nano-electro-mechanical systems [70, 71]. Many studies have been reported that graphene–polymer composites possess significantly enhanced Young's modulus and ultimate strength over their pristine polymer counterparts [72–76].
1.2.2.3 Optical Property
Another interesting property of graphene is its optical transparency, which is particularly important for optoelectronic applications. Experimental reports confirmed that the optical absorption of a monolayer graphene over the visible spectrum is 2.3%, and it linearly increases with an increase in the layer number [77–79]. This optical transparency of graphene in conjunction with its exceptional electrical and mechanical properties could lead to numerous novel photonic devices. The most direct applications of graphene are flexible electronic products, such as flexible screen displays, electronic papers and organic light-emitting diodes [80–82]. In these applications, the properties of graphene make it superior to the traditional material – indium tin oxide. Other potential applications of graphene include its use in photodetectors, solar cells, mode-locked lasers and optical modulators [83, 84].
1.2.3 Thermal Conductivity of Graphene
Conduction is one of the three ways of heat transfer, with the other two ways being convection and radiation. The basic principle of heat transfer by conduction is governed by Fourier's law. According to Fourier's law, the time rate of heat transfer through a material is directly proportional to the cross-sectional area and the negative temperature gradient in the heat transfer direction. Fourier's law may be expressed as
where is the heat, the time, is the cross-sectional area, is the temperature gradient along the heat transfer direction . In Eq. (1.1), is the thermal conductivity, which is a material-dependent constant. The thermal conductivity of a material is given by
where is the specific heat, is the average velocity of the thermal energy carriers and is the mean free path of thermal energy carriers. The thermal energy carriers in non-metals are mainly phonons. In SI units, the thermal conductivity is measured in W/m K.
One of the superior properties of graphene is its extremely high thermal conductivity. In 2008, Balandin and his co-workers [85, 86] reported the first experimental investigation on the thermal conductivity of suspended single-layered graphene. The measurement was conducted in a non-contact manner using the confocal micro-Raman spectroscopy. Graphene has clear signatures in Raman spectra, and the G peak in graphene spectra manifest strong temperature dependence [87–91]. The temperature sensitivity of G peak allows one to monitor the local temperature change produced by the variation of the laser excitation power focused on a graphene layer. In a properly designed experiment, the local temperature rises as a function of the laser power and it can be utilised to extract the thermal conductivity. The single-layered graphene samples were suspended on a 3-µm- wide trench fabricated on a Si/SiO2 substrate. Laser light of 488 nm in wavelength was chosen for this experiment, because it gives both efficient heating on the sample surface and clear Raman signatures for graphene. Using the aforementioned methodology, they discovered that the thermal conductivity value of a suspended single-layered graphene is up to 5300 W/m K at room temperature. The phonon mean free path in graphene was calculated to be approximately 775 nm. This experimental data was later confirmed by theoretical work. In 2009, Nika et al. [92, 93] conducted a theoretical study on the phonon thermal conductivity of a single-layered graphene by using the phonon dispersion obtained from the valence force field method, and treating the three-phonon Umklapp processes accounting for all phonon relaxation channels in the two-dimensional Brillouin zone of graphene. The uniqueness of graphene was reflected in the two-dimensional phonon density of states and restrictions on the phonon Umklapp scattering phase-space. They found that the thermal conductivity of a single-layered graphene at room temperature is in the range of 2000–5000 W/m K depending on the flake width, defect concentration and roughness of the edges, which is in good agreement with the experimental data.
Significant research studies have been pursued to further understand the thermal properties of graphene. In 2010, by using the same experimental technique based on confocal micro-Raman spectroscopy [85, 86], Ghosh et al. [94] showed that the thermal conductivity of graphene is dependent on the number of graphene layers. As the number of graphene layers increases from 2 to 4, the thermal conductivity changes from 2800 to 1300 W/m K. They explained the evolution of thermal transport from two-dimensional to bulk by the cross-plane coupling of phonons and the change in phonon Umklapp scattering. In a single-layered graphene, phonon Umklapp scattering is quenched and the thermal transport is limited mostly by the in-plane edge boundary scattering. While in a bilayered graphene, thermal conductivity decreases because the phonon Umklapp scattering increases as a result of cross-plane coupling. When the layer number increases to 4, the thermal conductivity is even lower because of stronger extrinsic effects resulted from the non-uniform thickness of samples. Also due to increased Umklapp scattering and extrinsic effects, several studies reported that the thermal conductivity of graphene supported by substrate is lower when compared to the suspended one [95–97]. In 2012, Chen et al. [98] reported their experimental study of the isotope effects on the thermal conductivity of graphene. They found that the thermal conductivity of graphene sheets composed of a 50:50 mixture of 12C and 13C is about half that of pure 12C graphene. The reason is that isotopes, defects, impurities or vacancies reduce the point-defect-limited phonon lifetime, and thus reduce the phonon mean free path and the thermal conductivity.
In general, graphene outperforms all the known materials in heat conduction. The high thermal conductivity, together with its exceptional high specific surface area (2630 m2/g), makes graphene a very promising material to be applied in thermal management applications [85, 86]. The reported value of greater than 5300 W/m K is more than a magnitude higher than any of the conventional filler materials used in nanocomposites. Therefore, graphene has been recognised as one of the most promising thermally conductive fillers for making high-performance nanocomposites with superior thermal conductivity.
1.3 Thermal Conductivity of Graphene–Polymer Nanocomposites
1.3.1 Measurement of Thermal Conductivity of Nanocomposites
Thermal conductivity of nanocomposites can be measured by either the steady-state methods or the transient methods.
1.3.1.1 Steady-State Method
The principle of the steady-state methods is to establish a steady temperature gradient over a known thickness of the sample and control the heat flow through the sample. The thermal conductivity may be determined from Fourier's law of heat conduction given in Eq. (1.1). The guarded hot plate method is the most representative steady-state method. In this method, the sample is first prepared as a disc with known thickness and area. Then it is placed between two parallel plates with constant but different temperatures, as a hot plate and a cold plate. Both plates have temperature sensors that measure the temperature gradient across the sample. When the equilibrium state is established, the thermal conductivity can be determined on the basis of the temperature gradient across the sample, the electrical power input, the thickness and the area of the sample. In order to minimise the effect of heat transfer through convection, the whole set-up is placed in a high vacuum environment. The effect of heat transfer through radiation is negligible. The details of the guarded hot plate method can be found in ASTM C177 [99] and ISO 8302 [100].
1.3.1.2 Transient Method
Among the transient methods, the most commonly used one is the laser flash method. This method does not measure thermal conductivity directly. Instead, it measures thermal diffusivity from which the thermal conductivity can be calculated through
where is the thermal diffusivity, is the specific heat capacity and is the density. In the laser flash method, the sample is a disc with a few millimetres in thickness. During the measurement, the front surface of the sample is irradiated with a pulse of energy from a laser and the subsequent temperature rise on the rear surface is recorded. The shape of the temperature–time curve on the rear surface of the sample can be used for extracting the thermal diffusivity of the sample. The details of laser flash method can be found in ASTM E1461 [101].
1.3.2 Modelling of Thermal Conductivity of Nanocomposites
Modelling is not only a powerful tool to predict the thermal conductivity of nanocomposites before synthesis, but it also interprets the experimental results during the development of new nanocomposites. Appropriately
