![1
Overview
Eui-Hyeok Yang1
, Dibakar Datta2
, Grzegorz (Greg) Hader3
, Junjun Ding4
1
MECHANICAL ENGINEERING DEPARTMENT, STEVENS INSTITUTE OF TECHNOLOGY,
HOBOKEN, NJ, UNITED STATES 2
DEPARTMENT OF MECHANICAL AND INDUSTRIAL
ENGINEERING, NEWARK COLLEGE OF ENGINEERING, NEW JERSEY INSTITUTE OF
TECHNOLOGY, NEWARK, NJ, UNITED STATES 3
US ARMY CCDC, ARMAMENTS CENTER,
PICATINNY ARSENAL, NJ, UNITED STATES 4
KAZUO INAMORI SCHOOL OF ENGINEERING, NEW
YORK STATE COLLEGE OF CERAMICS, ALFRED UNIVERSITY, ALFRED, NJ, UNITED STATES
1.1 Overview of two-dimensional materials and the scope of
the book
In December of 1959, the physicist, Richard P. Feynman, gave a lecture titled, “There’s
Plenty of Room at the Bottom: An Invitation to Enter a New Field of Physics.” This lecture
would become the advent to the scientific field of nanotechnology. Feynman’s lecture on the
manipulation of atoms would eventually become reality when researchers demonstrated the
precise placement of individual atoms by synthesizing graphene nanoribbons into specific
patterns [1]. His radical idea to make machines at a small scale would eventually culminate
in the development of microelectromechanical systems (MEMS) starting in the 1980s.
Fabrication processes of microscaled electromechanical devices were based on techniques
adapted from the integrated circuit (IC) industry. It was this synergy between the IC industry
and the need for MEMS that would bring consumers a wealth of technology advancements
in cell phones, automobiles, gaming, robotics, fitness/health trackers, airplanes, many mili-
tary applications, and last but not least, drones, which would not have been possible without
MEMS technology. This synergistic relationship is now being explored between two-
dimensional (2D) materials and the silicon-based semiconductor industry [2]. Due to the
extraordinary properties of atomically thin 2D materials, they have now made their way to
the forefront of several research areas, including electronics, photonics, electrophotonics,
catalysis, and energy. There have been extensive research efforts on the mechanical, thermal,
optical, and electrical properties, including modeling, synthesis, and their applications. As
the need for new high-performance materials continues to push toward the mantra of ligh-
ter, stronger, and faster, bringing credence to the lecture by Feynman, that there is still
“Plenty of Room at the Bottom,” leaves one to image what new technology lies beyond the
horizon.
3
Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures.
DOI: https://doi.org/10.1016/B978-0-12-818475-2.00001-5
© 2020 Elsevier Inc. All rights reserved.](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-22-320.jpg)
![Over a decade has passed since the seminal work in isolating graphene by Sir Andre
Geim and Sir Konstantin Novoselov, which started a revolution in the research of a new fam-
ily of materials with atomic thickness and planar dimensionality. Graphene is a monolayer of
carbon atoms arranged in a hexagonal lattice. Its high degree of crystallinity and outstanding
electronic, mechanical, thermal, and optical properties leads to the term the new wonder
material [3] and makes graphene an ideal candidate for novel high-speed (GHz THz)
optoelectronic devices [4,5]. Graphene is a gapless semimetal with a linear dispersion rela-
tion in the low bias transport regime. The research on graphene has opened the floodgates
to a vast library of other 2D-layered materials [6], including the fabrication of heterostruc-
tures, all at atomic thicknesses. Although the micromechanical exfoliation technique has
been adopted for rapid material characterization and demonstration of innovative device
ideas based on these 2D systems, significant advances have recently been made in large-
scale homogeneous and heterogeneous growth of these materials. The emergence of these
new 2D materials dramatically broadens the spectrum of properties. Unlike the zero-
bandgap graphene, hexagonal-boron nitride (h-BN) is an insulator with a similar atomic
structure to graphene, while monolayer transition metal dichalcogenides such as molybde-
num disulfide (MoS2), molybdenum diselenide (MoSe2), tungsten disulfide (WS2), and tung-
sten diselenide (WSe2) are direct bandgap semiconductors. The diverse properties of these
2D material systems make it flexible for the use of various applications. Mechanical, thermal,
optical, and electrical properties of 2D materials will be further discussed in Chapters 2, 3, 4,
and 5, respectively.
With the constant discovery of new 2D materials and 2D heterostructures, the develop-
ment of 2D materials opens up a completely new territory for both experimental studies and
computational studies. Recent advances in the modeling of phenomena during the nanofab-
rication and mechanics of controllable synthesis of 2D materials have paved the way for vari-
ous applications. With the continuous increase in computing power and significant
advancements of theoretical methods and algorithms, the modeling for physical properties
of 2D materials and 2D heterostructures has shown comparable accuracy to experiments,
while keeping the cost down. The advantages of computational materials databases are not
limited to speed and cost as compared to experimental efforts. The computational work
makes it possible for sharing and comparison of research data with reduced duplication of
research efforts. The increasing volume of databases enables the application of machine-
learning techniques for the discovery of new 2D materials and designing materials with tai-
lored properties. Modeling topics on atomistic modeling, molecular dynamics simulation,
Monte Carlo methods, and continuum modeling are covered in Chapters 6, 7, 8, and 9,
respectively.
To characterize the layer-dependent properties of 2D properties, it is essential to synthe-
size 2D materials in a controllable manner. Other than the micromechanical exfoliation tech-
nique, many strategies have been reported to synthesize monolayer or few-layer 2D
materials, such as chemical vapor deposition (CVD) method, chemical exfoliation, and
hydrothermal method. These methods show their advantages and disadvantages in terms of
quality, production volume, and layer control, which determines the applications of these
4 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-23-320.jpg)
![synthesized 2D materials. The synthesis of 2D heterostructures often requires a more com-
plicated process, which integrates two or more synthesis methods for each layer of 2D mate-
rials. The synthesis methods of graphene, h-BN, TMD, and 2D heterostructures are
introduced in depth in Chapters 10, 11, 12, and 13, respectively. Chapter 14 discusses the
characterization techniques utilized for the confirmation and analysis of natural and syn-
thesized 2D materials. This chapter outlines transmission electron microscopy (TEM),
Raman Spectroscopy, atomic force microscopy (AFM), including other surface and atomic
characterization tools to gain insight into the 2D materials physics, chemistry, and material
science.
The understanding of the physical properties of 2D materials and the synthesis of 2D
materials and their heterostructures make it possible to design electrical and optoelectronic
devices with superb performance. The photodetector, which converts photons into electrical
signals, can be redesigned with 2D materials other than the conventional semiconductors,
such as silicon and indium gallium arsenide. The 2D materials and 2D heterostructures
enable new photoresponse effects at much greater sensitivities and provide photodetection
covering UV, visible, IR, and THz ranges. The unique mechanical properties also ensure the
fascinating processing of photodetection in flexible electronics as well as bioelectronics.
Detailed discussion on 2D material-based photodetectors is shown in Chapter 15, 2D
Materials and Hybrid Systems for Photodetection. In addition to optoelectronics, 2D materi-
als have found a wide range of applications in electronic devices such as conductors, thin-
film transistors, sensors, and energy storage devices. Solution-processed 2D materials bear
high potential due to the advantages of low cost and high-volume production, which is criti-
cal for the fast-growing demands of printed electronics and other electronics applications.
The exfoliated 2D materials are solution-processable so that the 2D materials can be easily
assembled into designed layered structures on arbitrary substrates, which is important for
flexible electronics. Most 2D materials can be chemically exfoliated, while more researchers
are trending toward synthesis by other methods such as CVD. Therefore potential applica-
tions with 2D heterostructures can be realized by solution-processed 2D materials using
methods such as layer-by-layer assembly, Langmuir Blodgett assembly, spin coating, elec-
trophoretic deposition, inkjet printing, and vacuum filtration. The details on solution-
processed 2D materials for electronic applications are discussed in Chapter 16, Electronic
Devices Based on Solution-Processed 2D Materials.
Due to the extraordinary electrical properties, 2D materials have been extensively
explored as additives in composites for electrodes in energy storage devices in order to
increase electronic conductivity and mechanical stability and provide additional Li storage
sites for lithium-based batteries. The 2D materials and 2D heterostructures are excellent can-
didates as anodes and help provide high porosity, good electron mobility, lightweight, high
charge capacity, high rate capability, and increased operational voltage. Many researchers
have reported improvements in the performance of anodes in lithium-ion batteries and offer
2D materials as an alternative option to anodes fabricated with Li metal, which is prone to
deadly dendrite formation [7]. While monolayers of most 2D materials are not ideal candi-
dates for battery electrodes, van der Waals layered heterostructures offer possibilities to
Chapter 1 • Overview 5](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-24-320.jpg)
![design battery electrodes for fast diffusion kinetics, high structural integrity, and excellent
electron conductivity. However, there are many questions to be answered to fully understand
the mechanics of 2D electrodes and how to optimize the performance for energy storage
devices. Chapter 17, 2D Materials and Its Heterostructures for Energy Storage, provides a sys-
tematic review on the state-of-the-art of 2D materials and their heterostructures for energy
storage applications. The World Health Organization (WHO) announced a global pandemic
on March 11, 2020, due to the uncontrolled outbreak of the novel coronavirus (COVID-19).
The study of low dimensional materials in virology and living organisms is discussed in
Chapter 18 and provides insight into how these materials are prime candidates for the cap-
ture, detection, and analysis of biological systems.
Despite extensive research efforts of 2D materials in the last two decades, the 2D materi-
als and their heterostructures have greatly expanded their territory for more opportunities to
explore. With the development of computational power and algorithms, computational
modeling has grown into an important tool for the discovery of new 2D materials and predic-
tion of their physical properties [8]. The increasingly large database of 2D materials and their
2D heterostructures offers endless possibilities in designing electronic, photonic, optoelec-
tronic, and energy storage devices [6,8]. Chapter 19, Machine Learning in Materials
Modeling—Fundamentals and the Opportunities in 2D Materials, discusses the emerging
field of machine learning for 2D materials research. This book provides an overview of the
synthesis, modeling, and characterization of 2D materials and their heterostructures.
Applications are provided to the reader throughout the text as well as current technological
breakthroughs, outlining recent scientific progress in the fast-paced research environment of
2D materials.
References
[1] J. Cai, et al., Atomically precise bottom-up fabrication of graphene nanoribbons, Nature 466 (2010)
470 473. Available from: https://doi.org/10.1038/nature09211
[2] D. Akinwande, et al., Graphene and two-dimensional materials for silicon technology, Nature 573 (2019)
507 518. Available from: https://doi.org/10.1038/s41586-019-1573-9
[3] A.K. Geim, Graphene: status and prospects, Science 324 (2009) 1530 1534. Available from: https://doi.
org/10.1126/science.1158877
[4] F. Wang, et al., Gate-variable optical transitions in graphene, Science 320 (2008) 206 209. Available
from: https://doi.org/10.1126/science.1152793
[5] F. Bonaccorso, Z. Sun, T. Hasan, A.C. Ferrari, Graphene photonics and optoelectronics, Nat. Photonics 4
(2010) 611 622. Available from: https://doi.org/10.1038/nphoton.2010.186
[6] G. Cheon, et al., Data mining for new two- and one-dimensional weakly bonded solids and lattice-
commensurate heterostructures, Nano Lett. 17 (2017) 1915 1923. Available from: https://doi.org/
10.1021/acs.nanolett.6b05229
[7] L. Li, et al., Self-heating induced healing of lithium dendrites, Science 359 (2018) 1513 1516.
[8] N. Mounet, et al., Two-dimensional materials from high-throughput computational exfoliation of experi-
mentally known compounds, Nat. Nanotechnol. 13 (2018) 246 252. Available from: https://doi.org/
10.1038/s41565-017-0035-5
6 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-25-320.jpg)
![2
Mechanical properties of two-
dimensional materials: atomistic
modeling and future directions
M.A.N. Dewapriya1,2
, R.K.N.D. Rajapakse1
, S.A. Meguid2
1
SCHOOL OF ENGINEERING SCIENCE, SIMON FRASER UNIVERSITY, BURNABY, BC, CANADA
2
MADL, MECHANICAL AND INDUSTRIAL ENGINEERING, UNIVERSITY OF TORONTO, TORONTO,
ON, CANADA
2.1 Introduction
Research in two-dimensional (2D) materials is receiving worldwide attention from the scientific
and engineering communities due to extraordinary properties of these materials and their poten-
tial to serve as the building blocks of next-generation materials [13]. Since the isolation of gra-
phene from bulk graphite in 2004, several other 2D materials, such as hexagonal boron nitride
(h-BN) and molybdenum disulfide (MoS2), have been developed (see Fig. 21).
This newly emerging family of 2D materials offers a wide range of multiphysical properties. For
example, electrical conductivity of the 2D materials varies from conducting graphene to insulating
h-BN, while MoS2 is a semiconductor. The unique 2D crystal structures of these materials render
distinct combinations of mechanical properties, such as high in-plane stiffness combined with
FIGURE 2–1 Atomic structure of graphene, hexagonal boron nitride (h-BN) and molybdenum disulfide (MoS2).
9
Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures.
DOI: https://doi.org/10.1016/B978-0-12-818475-2.00002-7
© 2020 Elsevier Inc. All rights reserved.](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-26-320.jpg)
![extremely low flexural rigidity, which are promising for a wide range of novel applications [4].
For example, graphene has already demonstrated great potentials in a rich variety of engineering
applications, such as flexible electronics [5], nanoelectromechanical systems (NEMS) [6], and
multifunctional nanocomposites [7,8]. The mechanical properties of this class of materials, which
depend on their nanostructure, play a critical role in their utility and the service life of their pro-
ducts. Recent advances in nano-manufacturing offer the possibilities for manipulating the micro-
structure of these atomically thin membranes to achieve desirable characteristics, and thus
opening a new design paradigm of nanoscale engineering.
This chapter focuses on the mechanical integrity and the fracture behavior of plane
nanostructured materials such as graphene and h-BN. Three aspects of the work were
accordingly examined—the first with the use of molecular dynamics (MD) to predict the
mechanical and fracture behavior of graphene; the second with the atomistic interaction of a
crack in close proximity to an inhomogeneity or a vacancy, and the resulting stress shielding
and amplification effects at the crack tip; and the third with the fracture characteristics of
atomistic grapheneh-BN heterostructures. In addition, we offer some insights into future
research directions in this area pertaining to the modeling, characterization, and application
of 2D materials, accounting for their topological design and potential application in machine
learning in nanomaterial design.
The chapter is divided into five sections. Following this introduction, Section 2.2 provides
a brief overview of the current state of research. Section 2.3 highlights important aspects of
MD simulations of 2D materials. Several atomistic modeling studies, related to fracture of
graphene and grapheneh-BN heterostructures, are discussed in Section 2.4, which is fol-
lowed by an overview of future research directions.
2.2 Current state of research
Plane or 2D materials resemble a thin membrane and therefore predominantly demonstrate
two fundamental deformation modes: (1) in-plane stretching and (2) out-of-plane bending.
As a result, both in-plane and bending moduli are required to properly characterize the
deformation of these nanostructured materials. Moreover, a set of coupling moduli can also
be theoretically defined to further characterize the deformation patterns of 2D materials [9].
A recent experiment by Blees et al. [10] revealed that the bending modulus of graphene is
orders of magnitude higher than the theoretically predicted value. This result poses a ques-
tion on the applicability of well-established mechanics of ultrathin membranes for 2D mate-
rials [11]. Several experimental studies have revealed that suspended graphene membranes
demonstrate intrinsic ripples at finite temperatures (see for, e.g., Ref. [12]). These ripples
could be responsible for the observed high bending modulus. In addition, such ripples could
have profound effects on the in-plane modulus as well as the thermal expansion [13]. These
findings suggest the possibility of designing 2D microstructures membrane that resists the
inherent out-of-plane deformation leading to an improved bending rigidity.
In general, 2D materials demonstrate significantly high fracture strength. For example, a
pristine single crystalline graphene has a fracture strength of 130 GPa [14]. However, atomic
10 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-27-320.jpg)
![imperfections such as vacancies and defects are difficult to avoid in the fabrication of gra-
phene and other 2D nanostructured materials. In addition, graphene contains grains and
their boundaries contain numerous defects, such as pentagons, heptagons, and dislocations.
In fact, defects may be intentionally introduced into 2D materials in order to tailor their elec-
tromechanical properties [15]. These defect sites could generate stress concentrations result-
ing in fracture under significantly small levels of applied stresses.
Indeed, low fracture toughness of graphene (B4 MPaOm) poses serious limitations in its
use for structural applications [16]. Therefore identifying potential toughening mechanisms
for 2D materials is critical for their widespread use in engineering application. Commonly
prevailing defects such as vacancies, dislocations, and grain boundaries may be manipulated
to achieve novel topologies with improved resistance to fracture. For example, proper
manipulation of grain boundaries (e.g., size/orientation of nanocrystals) or the effective use
of interacting microdefects could significantly improve the fracture resistance of 2D materials
[1719]. Even though graphene demonstrates a purely brittle fracture, its derivative gra-
phene oxide has demonstrated some plasticity [20,21]. The atomistic mechanisms associated
with the plasticity of graphene oxide, however, have not been understood or characterized
yet. Moreover, the applicability of the classical continuum concepts of plasticity at the atomic
scale should be carefully examined due to the inherent discreteness of the matter and the
quantum manifestations at this scale [22,23]. On the other hand, brittle-to-ductile transfor-
mation of 2D materials such as graphene and h-BN could be achieved through topological
design.
Developing hybrid/hierarchical materials is one of the innovative routes of designing
novel 2D materials with unique properties. Most natural materials possess hierarchy and are
hybrid (e.g., bone and nacre). Similarly, the lattice structure of graphene and h-BN allows
the fabrication of graphene/h-BN heterostructures with unique electronic and magnetic
properties [24,25]. More importantly, the physical properties of the graphene/h-BN hetero-
structures can be effectively tailored through the selection of the relative domain size of each
material, which is quite beneficial for advanced applications in engineering [26,27]. On the
other hand, these 2D materials can be arranged one on top of the other using the layer-by-
layer assembly techniques to design novel multilayered hierarchical materials with unique
and improved physical properties [28]. Multiphysical properties of these recently emerging
hybrid and multilayered materials have to be thoroughly understood before integrating them
into device applications.
Extremely high strength to weight ratio and stiffness of graphene make it a superior ballis-
tic armor candidate for aerospace and defense-related applications, as confirmed by ballistic
impact tests in Refs. [14,29]. The test revealed that the specific penetration energy of multi-
layered graphene is about 10 times the corresponding value of a microscopic steel sheet.
Moreover, a recent experiment demonstrated that graphene-based polyvinyl alcohol contain-
ing a relatively low volume fraction of graphene has the potential to reach three times the
ballistic impact resistance of existing high-performance composites [30]. However, experi-
ments are unable to reveal the complex atomic mechanisms of energy dissipation at the
nanoscale, which can only be realized through atomistic simulations. A recant atomistic
Chapter 2 • Mechanical properties of two-dimensional materials 11](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-28-320.jpg)
![simulation revealed that graphene could transform polyethylene into a high-performance
ballistic material, where a single coat of graphene improves the ballistic performance of poly-
ethylene over eightfolds [31].
2.3 Molecular dynamics simulations of two-dimensional
materials
MD simulations are widely used to study mechanical properties of 2D materials. A compre-
hensive overview of MD simulations can be found in Ref. [32]. When MD simulations are
employed to study fracture characteristics of graphene, the proper selection of cutoff para-
meters in interatomic potentials is critically important. However, the influence of the cutoff
function on the fracture characteristics has been overlooked by several studies. This section
briefly describes the influence of cutoff parameters on the computed fracture stress of gra-
phene samples using Tersoff-type potential, such as reactive bond order (REBO) interatomic
potential.
In order to shed light on the effect of the cutoff function, we present a set of MD simula-
tions of uniaxial tensile tests of several graphene samples. The simulations were conducted
using large-scale atomic/molecular massively parallel simulator [33]. The adaptive intermo-
lecular REBO (AIREBO) potential [34] was used for the simulations. The AIREBO potential
consists of three subpotentials—the REBO [35], the Lennard-Jones, and the torsional. The
REBO potential evaluates energy stored in atomic bonds. The Lennard-Jones and the tor-
sional potentials include energies due to nonbonded and torsional interactions between
atoms, respectively. The REBO potential expresses energy stored in a bond between atoms i
and j as:
EREBO
ij 5 f rij
VR
ij 1 bijVA
ij
h i
; (2.1)
where VR
ij and VA
ij are the respective repulsive and attractive potentials, bij is the bond order
term, which modifies the attractive potential depending on local bonding environment, rij is
the distance between atoms i and j, and f(rij) is the cutoff function, which limits the inter-
atomic interactions to the nearest neighbors, and it is expressed as follows:
f ðrijÞ 5
1; rij , Rð1Þ
1
2
1
1
2
cos
π rij 2 Rð1Þ
Rð2Þ 2 Rð1Þ
ð Þ
2
4
3
5; Rð1Þ
, rij , Rð2Þ
0; Rð2Þ
, rij
;
8
:
(2.2)
where R(1)
and R(2)
are the two cutoff radii—1.7 and 2 Å, respectively. The values of the cutoff
radii were originally selected by considering the first and the second nearest neighboring dis-
tances of relevant hydrocarbons [35]. However, these cutoff radii introduce erroneous non-
physical strain hardening effect in the stressstrain curve of carbon-based structures, such
12 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-29-320.jpg)
![as diamond [36,37] and carbon nanotubes [36,37]. Therefore modified cutoff radii, ranging
from 1.9 to 2.2 Å, have been used in the literature to eliminate this nonphysical strain hard-
ening [3841].
In order to obtain an insight into the effect of the cutoff function on the fracture of an
individual bond, the forcestrain curve of a bond between two carbon atoms was obtained
by increasing the bond length r [37], as depicted in the inset of Fig. 22. In the case of the
default cutoff radii, the values of R(1)
and R(2)
are 1.7 and 2 Å, respectively. In the case of the
modified cutoff, the values of the two radii were set to be 2 Å. As shown in Fig. 22, a signifi-
cant strain hardening of the forcestrain curve can be observed when the default values of
the cutoff radii were used, whereas the strain hardening disappears when the modified cutoff
radii were used. A detailed investigation on the topic is presented in Ref. [42].
Numerous studies have used a value of R(1)
between 1.9 and 1.94 Å to study the fracture of
graphene samples. However, our simulation results demonstrate that R(1)
has a great influence
on the computed fracture stress of graphene samples when the value of R(1)
is below 1.96 Å.
Fig. 23A compares stressstrain curves of a pristine graphene sample and a sample contain-
ing a crack; several simulations were performed with different values of R(1)
, where the value
FIGURE 2–2 Forcestrain curves of the carboncarbon bond between atoms 1 and 2 marked on inset. The two
horizontal arrows indicate the loading direction. Reprinted from K.G.S. Dilrukshi, M.A.N. Dewapriya, U.G.A.
Puswewala, Size dependency and potential field influence on deriving mechanical properties of carbon nanotubes
using molecular dynamics, Theor. Appl. Mech. Lett. 5 (2015) 167172. https://doi.org/10.1016/j with permission
from Elsevier.
Chapter 2 • Mechanical properties of two-dimensional materials 13](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-30-320.jpg)
![of R(2)
was kept constant at 2 Å. The curves pertinent to pristine sample demonstrate that the
influence of the cutoff is significant even when R(1)
is 1.94 Å. Fig. 23B shows that the com-
puted critical stress intensity factor (SIF) from MD simulations is significantly higher than the
experimental value when R(1)
is below 1.96 Å. This suggests that the accuracy of the fracture
simulations highly depends on the proper selection of R(1)
. These results confirm that, accord-
ing to the AIREBO potential, fracture of carboncarbon bond occurs when the bond length is
approximately 1.95 Å. When the value R(1)
is less than 1.95 Å, the cutoff function impedes the
bond breaking process and introduces the erroneous nonphysical strain hardening effect (see
Fig. 22).
2.4 Fracture characteristics of two-dimensional materials
2.4.1 Effect of functionalization and temperature on graphene
In many engineering applications the surface of graphene has to be modified by introducing
various defects (e.g., vacancies and adatoms) in order to achieve certain desired functionalities.
As an example, a 2D amorphous graphene membrane can be obtained by means of electron
irradiationinduced vacancies, which opens new possibilities to engineer graphene-based
NEMS [43]. Chemical functionalization, which involves the addition of foreign atoms or func-
tional groups, could induce better interaction between graphene and a host composite matrix,
leading to improved electromechanical properties [44,45]. Moreover, hydrogen functionaliza-
tion creates new bandgap openings in graphene [45], and carbon adatoms significantly modi-
fies their electronic and magnetic properties [4648].
Numerous studies concerning functionalized graphene have focused on its electronic and
magnetic properties [4648]. On the other hand, understanding the influence of adatoms
and functional groups on the mechanical properties of graphene-based systems is vitally
FIGURE 2–3 Effects of the cutoff radius R(1)
: (A) stressstrain curves of an armchair sheet at various cutoff radii.
Length of the considered crack is 20.8 nm, and (B) variation of the computed critical value of the stress intensity
factor at different values of the cutoff radius as compared with an experimental result [16].
14 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-31-320.jpg)
![important in many applications such as in graphene-based structural composites, where the
adsorption of adatoms and functional groups is unavoidable [44,4952]. In graphene-based
composites, interaction between graphene and the composite matrix is governed by non-
bonded interactions, which is mainly van der Waals force [4951,53]. The adsorption of ada-
toms could have a significant impact on the nonbonded interactions between graphene and
composite matrix. MD simulation studies have revealed that hydrogen adsorption can have a
significant impact on the strength of graphene and its allotropes [54,55].
On the other hand, graphene could be subjected to high temperatures (B1000K) during
the synthesis and fabrication of graphene-based composite materials [56]. Several MD simula-
tions have been conducted on the temperature-dependent mechanical properties of graphene
[5759]. However, the behavior of functionalized graphene sheets could be significantly different
from that of pristine sheets, since functional groups (or adatoms) transform the hybridization of
carbon in graphene from sp2
to sp3
. This section briefly describes the development of an analyti-
cal model, based on Bailey durability criterion and the Arrhenius equation, to study
temperature-dependent fracture strength of functionalized graphene along various chiral direc-
tions [60]. The predicted fracture strength depends on temperature, strain rate, and hydrogen
functionalization.
Baileys durability criterion [61], given in Eq. (2.3), provides a framework for calculating
the lifetime of materials at various temperatures [62]. Let t be time and T be temperature:
ðtf
0
dt
τ T; t
ð Þ
5 1; (2.3)
where tf is the time taken to fracture and τ(T,t) is the time and temperature-dependent dura-
bility function, which is generally obtained from experiments [62]. However, in the absence
of experimental data on the durability of carboncarbon bonds in graphene, the Arrhenius
equation is a good approximation for the durability function [58].
The Arrhenius equation [63] expresses the temperature-dependent rate of a chemical reac-
tion (k) as k5 A 3 exp[ΔE/(kBT)], where A is a constant that depends on the type of chemical
bonding, ΔE is the activation energy barrier, and kB is the Boltzmann constant. When a mechan-
ical force F is applied to a molecule, the activation energy barrier is reduced by an amount of
FΔx, where Δx is the change in the atomic coordinates due to F [64]. A durability function for
carboncarbon bonds in graphene can then be defined in the form of Arrhenius equation as:
τ T; t
ð Þ 5
τ0
n
exp
U0 2 vγσ t
ð Þ
βkBT
; (2.4)
where τ0 is the vibration period of the atoms, n is the number of bonds in the sheet, U0 is the
interatomic bond dissociation energy (4.93 eV for a carboncarbon bond [65]), v is the repre-
sentative volume of a carbon atom in graphene, which is approximately 8.6 Å3
, and γ is a
directional constant that takes into account the different bond orientation along different chiral
directions. According to Ref. [38], the strength (S) along a chiral direction, at an angle θ mea-
sured from the armchair direction, can be approximated as Sθ 5 Sac/cos θ, where Sac is the
Chapter 2 • Mechanical properties of two-dimensional materials 15](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-32-320.jpg)
![strength along the armchair direction. The chiral angle between armchair and zigzag directions
is π/6. According to the proposed strength relation in Ref. [38], the strength ratio of pristine
graphene along the zigzag and armchair directions can be obtained as Sac/Szz 5 cos(π/6) 5
0.87. An independent MD simulations have revealed that the ratio Sac/Szz is 0.85 [60].
Therefore the chirality-dependent strength can be introduced into the proposed atomistic
model by setting γ 5 cosθ.
The stress at time t, σ(t), is expressed in terms of the strain rate _
ε
ð Þ as follows:
σ t
ð Þ 5 a _
εt
ð Þ 1 b _
εt
ð Þ2
; (2.5)
where a and b are the second- and the third-order elastic moduli, respectively. The values of
a and b were obtained from regression analysis of the stressstrain curves given by MD
simulations at 300K. The regression analysis determined a and b to be 1.11 and 23.20 TPa
for armchair graphene.
The constant β describes the reduction of activation energy barrier due to the presence of
hydrogen adatoms, which is defined in terms of adatom concentration (α) as being:
β 5
1; α 5 0
0:023α 1 1:11; α . 0
:
(2.6)
The governing equation of the system can be obtained by substituting Eq. (2.4) into
Eq. (2.3), which yields:
ðtF
0
exp
γσ t
ð Þ 2 U0
βkBT
dt 5
τ0
n
: (2.7)
Then Eq. (2.7) is solved for the time taken to fracture (tF), and the solution can be
expressed as:
tF 5
erf21
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 b=π
q
2λτ0 _
ε
ð Þ=n
exp λ2
U0=γ
1 a2
=4b
2 erf χ
ð Þ
n o
1 χ
ffiffiffiffiffiffiffiffiffi
2 b
p
λ_
ε
; (2.8)
where erf is the error function [66], with λ 5
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γ=βkBT
p
and χ 5 λa=
ffiffiffiffiffiffiffiffiffiffiffi
2 4b
p
. Once tF is
obtained from Eq. (2.8), the fracture strength, σ(tF), can be obtained from Eq. (2.5).
Fig. 24A and B compares the fracture strength given by the analytical model with MD
simulations of armchair and zigzag graphene at various adatom concentrations and tempera-
tures. The figure shows that the results obtained from the analytical model agree quite well
with the MD simulation predictions.
After the analytical model has been verified by MD simulations, the model can be used to
predict the strength of hydrogen functionalized graphene under various temperatures, strain
rates, and chirality. Fig. 25A shows that highly functionalized graphene completely loses the
strength when it is subjected to higher temperatures. This strength loss could be an indication of
sublimation of graphene. A Monte Carlo simulation study [67] has revealed that the melting
16 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-33-320.jpg)
![temperature of graphene is B4900K. However, according to Fig. 25A, melting of graphene
highly depends on the hydrogen functionalization suggesting that highly functionalized graphene
is not suitable for high temperature applications. Fig. 25B shows that the reduction of strength
due to the functionalization is less independent on the loading direction.
2.4.2 Out-of-plane deformation of crack surfaces
2D materials such as graphene and h-BN are only a single-atom thick. Therefore even a
small out-of-plane perturbation could disturb its planar geometry by creating ripples and
wrinkles, thus influencing its fracture behavior [68,69]. It has also been demonstrated that
FIGURE 2–4 Temperature and hydrogen adatom concentration-dependent fracture strength of (A) armchair and
(B) zigzag graphene. The dashed lines indicate the values given by the proposed analytical model. Reprinted from
M.A.N. Dewapriya, R.K.N.D. Rajapakse, N. Nigam, Influence of hydrogen functionalization on the fracture strength
of graphene and the interfacial properties of graphenepolymer nanocomposite, Carbon 93 (2015) 830842.
doi:10.1016/j.carbon.2015.05.101 with permission from Elsevier.
FIGURE 2–5 The variation of the strength of hydrogen functionalized graphene with (A) temperature and (B) the
loading direction (i.e., chirality). Results for armchair graphene are shown in (A) and the simulation temperature was
maintained at 300K in (B). Reprinted from M.A.N. Dewapriya, R.K.N.D. Rajapakse, N. Nigam, Influence of hydrogen
functionalization on the fracture strength of graphene and the interfacial properties of graphenepolymer
nanocomposite, Carbon 93 (2015) 830842. doi:10.1016/j.carbon.2015.05.101 with permission from Elsevier.
Chapter 2 • Mechanical properties of two-dimensional materials 17](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-34-320.jpg)
![deliberately induced geometrical distortions can be effectively used to tailor the mechanical
properties of graphene [10,17,41,7072].
Three-dimensional (3D) MD simulations of the nanoscale uniaxial tensile test of a gra-
phene sheet with a central crack demonstrate that the crack surfaces experience significant
out-of-plane deformation during the uniaxial tensile test. This phenomenon has been observed
in recent MD simulations of the nanoscale uniaxial tensile test [73,74]. The out-of-plane defor-
mation of the crack surfaces increases with the applied uniaxial strain (see Fig. 26). The max-
imum out-of-plane deformation occurs at the onset of crack propagation, and its magnitude
increases with the crack length. Both armchair and zigzag crack surfaces show approximately
similar out-of-plane deformation during the uniaxial tensile test, and the out-of-plane defor-
mation could be attributes to the poor transverse stiffness of the graphene sheet as well as the
biaxiality of the stress field at the crack tips [75].
Griffith’s thermodynamic failure criterion has been widely used to characterize the frac-
ture properties of nanoscale materials, including graphene [17,40,70,76,77]. When applying
Griffith’s criterion to 2D materials, a planar configuration of the cracked sample is typically
assumed. A recent molecular MD study [76] revealed that Griffith’s criterion is applicable to
graphene when the crack length is greater than 100 Å. Below this limit, the criterion over
predicts the fracture stress. This observation was attributed to the presence of local effects at
the crack tip. In addition, the out-of-plane deformation of the crack surfaces (see Fig. 26)
also has a significant influence on the fracture characteristics of graphene.
According to Griffith’s thermodynamic criterion of crack propagation [78], the fracture
stress σf can be expressed in terms of Young’s modulus (E), the surface energy (γ), and the
initial crack length (2a) such that:
FIGURE 2–6 Out-of-plane deformation of zigzag crack surfaces at two strain εyy levels. The crack length is 50.8 Å.
Reprinted from M.A.N. Dewapriya, S.A. Meguid, Atomistic modeling of out-of-plane deformation of a propagating
Griffith crack in graphene, Acta Mech. 228 (2017) 30633075. doi:10.1007/s00707-017-1883-7 with permission from
Springer Nature.
18 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-35-320.jpg)
![σf 5
ffiffiffiffiffiffiffiffi
2γE
πa
r
: (2.9)
By rearranging the terms of Eq. (2.9), an expression for the critical SIF (KIC), which is gen-
erally expressed as KIC 5 σf
ffiffiffiffiffiffi
πa
p
, can be obtained in terms of γ and E as:
KIC 5
ffiffiffiffiffiffiffiffi
2γE
p
(2.10)
for plane stress condition.
If breaking of an individual carboncarbon bond results in a crack advance of Δa, the
surface energy γ can be expressed as:
2γ 5
PEbond
Δat
; (2.11)
where PEbond is the equilibrated potential energy of a carboncarbon bond of graphene,
which is 4.916 eV according to AIREBO potential, and t is the thickness of graphene. The cal-
culated values of 2γ for the zigzag and the armchair crack configurations are 9.58 and
11.05 J/m2
, respectively. These values are in agreement with the ones calculated using REBO
potential in Refs. [16,65].
Substituting the computed values of E and γ into Eq. (2.10), KIC of the zigzag and the
armchair crack configurations were calculated to be 3.1 and 3.32 MPaOm, respectively.
Fig. 27 shows that KIC obtained using Eq. (2.10) is considerably higher than the critical
KI given by the expression KI 5 σf
ffiffiffiffiffiffi
πa
p
, when the out-of-plane deformations are allowed
to take place by relaxing the boundary conditions (i.e., 3D). Reference [76] suggests that
Griffith’s criterion of fracture is applicable to graphene with B15% accuracy when the
crack length is around 100 Å. Fig. 27 clearly demonstrates that when the out-of-plane
deformations of the crack surfaces are restrained, the critical value of KI approaches the
Griffith’s value [Eq. (2.10)] at a significantly smaller crack length (B60 Å). These results
FIGURE 2–7 Change in the critical value of KI with crack length as compared with Griffith’s crack for (A) armchair
and (B) zigzag crack configurations. Reprinted from M.A.N. Dewapriya, S.A. Meguid, Atomistic modeling of out-of-
plane deformation of a propagating Griffith crack in graphene, Acta Mech. 228 (2017) 30633075. doi:10.1007/
s00707-017-1883-7 with permission from Springer Nature.
Chapter 2 • Mechanical properties of two-dimensional materials 19](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-36-320.jpg)
![reveal that the out-of-plane deformation of graphene significantly reduces the fracture
stress below the stress predicted using Griffith’s energy balance approach.
2.4.3 Crackdefect interactions
In continuum fracture mechanics, it has been well established that the interaction between
crack and a microdefect in close proximity plays an important role in the overall failure
mechanism of quasibrittle materials [7985]. It has also been demonstrated that the crack
tip stress field in a linear elastic continuum can be controlled by strategically placing a
microdefect near the crack tip [86,87]. Recent nanoindentation tests of graphene containing
vacancies have revealed that the catastrophic failure of graphene can be transformed into a
local failure by controlling its defect concentration [88].
Most of the recent efforts have been focused on enhancing the fracture strength of gra-
phene by introducing topological defects such as pentagonheptagon [41,72,73,89,90] and
grain boundaries [70,91]. Studies on the complex stress state surrounding crackdefect inter-
actions could provide new insights into improving fracture resistance of 2D materials. In
addition, advanced continuum-based design tools for characterizing crackdefect interaction
[7987] have not been thoroughly tested at the atomic scale, which is critically important con-
sidering the limited applicability of continuum concepts at the nanoscale [22,23,58,60,76,92,93].
If applicable, the continuum tools such as design envelope to ascertain crack tip stress shielding,
and amplification zones due to the crackdefect interactions [86,87] can be very useful for nano-
scale design of 2D materials [2].
Fig. 28 shows a typical MD simulation sample of graphene containing an atomic vacancy
interacting with an edge crack. The origins of the two rectangular coordinate systems xy and
FIGURE 2–8 Typical MD simulation of graphene containing an edge crack interacting with an atomic vacancy. MD;
Molecular dynamics. Reprinted from M.A.N. Dewapriya, S.A. Meguid, Tailoring fracture strength of graphene,
Comput. Mater. Sci. 141 (2018) 114121. doi:10.1016/j.commatsci.2017.09.005 with permission from Elsevier.
20 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-37-320.jpg)
![x0y0 are taken at the tip of the edge crack and at the center of the vacancy, respectively. The
orientation angle of the vacancy is φ, and the distance between the tip of the crack and the
center of the vacancy is taken to be r. The inclination angle between the x-axis and the line
joining the tip of the crack and the center of the vacancy is θ. The value of 2c is selected to be
3.6 nm (see Fig. 28).
The stress distribution of individual carbon atoms at the crack tip can be very informative
in characterizing the fracture behavior of graphene. In order to obtain the time-averaged stress
of atoms at the incipient crack propagation, two sequential MD simulations can be conducted
as outlined in Ref. [75]. Fig. 29A and B shows the stress distributions at the armchair and zig-
zag crack tips, respectively. These stress distributions resemble the ones predicted by the con-
tinuum linear elastic fracture mechanics [76]. The peak stress at the tip of the armchair crack
at the incipient crack propagation is 17% higher than the corresponding stress of the zigzag
crack. This is due to the high far-field (or applied) stress σ0 level of the armchair crack configu-
ration and, more importantly, the different bond arrangements at the crack tips. In contrast to
the isolated bond perpendicular to the crack at the zigzag crack tip (see inset of Fig. 29A),
the two inclined bonds at the armchair crack tip (inset of Fig. 29B) accommodate part of the
applied tensile strain by adjusting the bond angles, which allows the atom at the crack tip to
carry a higher strain prior crack growth leading to a higher atomic stress.
According to linear elastic fracture mechanics, the critical SIF of a single-edge cracked
sample under mode I loading KIC can be defined as follows [94]:
KIC 5 1:12σf
ffiffiffiffiffiffi
πa
p
(2.12)
where a is the initial crack length and σf is the fracture stress. The computed KIC for arm-
chair and zigzag cracks in Ref. [18] are 4.04 and 3.97 MPaOm, respectively, which are in
excellent agreement with the experimentally measured value 4 MPaOm [16].
Earlier, Gong and Meguid studied the interaction between a semiinfinite crack and an
elliptical vacancy located near its tip (see Fig. 28) under mode I loading [86]. In the
absence of the vacancy, the singular stress field near the crack tip can be described by using
the corresponding SIF KI 5 1:12σ0
ffiffiffiffiffiffi
πa
p
. However, the presence of the elliptical vacancy in
close proximity to the crack tip influences the crack tip stress field and leads to a modified
SIF, we will call Kðc2vÞ
I . When a collinear elliptical vacancy is located ahead of the crack, that
is, θ 5 0 and φ 5 0, the solution for the normalized SIF under mode I loading Kðc2vÞ
I =KI can
be explicitly expressed up to the order (c/r)4
as follows [86]:
KðcvÞ
I
KI
5 1 1
1
4
2
1 1 β2
c
r
2
1
1
128
2
23 1 46β2
1 12β3
2 49β4
c
r
4
1 ? (2.13)
where β is b/c. For the case of a circular vacancy (i.e., β 5 1), Eq. (2.13) reduces to:
Kðc2vÞ
I
KI
5 1 1
1
2
c
r
2
1
1
4
c
r
4
1 ? (2.14)
Chapter 2 • Mechanical properties of two-dimensional materials 21](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-38-320.jpg)
![where C and G are explicitly expressed as follows:
C 5 cos
3θ
2
cos
θ
2
(2.16a)
G 5 2cos 2ϕ 1 θ
ð Þ 1 4cos 2ϕ 2 θ
ð Þ 1 8cos 2ϕ 2 2θ
ð Þ 2 6cos 2ϕ 2 3θ
ð Þ
2 8cos 2ϕ 2 4θ
ð Þ 2 3cos 3θ
ð Þ 1 3cos θ
ð Þ
(2.16b)
In order to characterize the crackvacancy interaction, normalized crack tip stress
σðc2vÞ
tip =σc
tip can be employed, where σðc2vÞ
tip is the crack tip stress along the y direction in the
presence of an interacting vacancy, and σc
tip is the crack tip stress in the absence of any
interacting vacancy. The normalized crack tip stress σðc2vÞ
tip =σc
tip was computed at an applied
tensile strain εyy level of 1% for various arrangements of the interacting vacancies. The
values of σc
tip for the armchair and zigzag cracks are 63.9 and 56 GPa, respectively.
Fig. 29C to G demonstrates that the presence of vacancies greatly influences the stress
field of zigzag crack compared to that of the armchair crack, as a result of their underlying
crystal structures at the crack tips. It can be seen in Fig. 29C and F that the collinear
(i.e., θ 5 φ 5 0) nanocracks and circular vacancies result in an increase in the crack tip
stress field (known as stress amplification effect, i.e., σðc2vÞ
tip =σc
tip . 1). The oriented vacancies
(see Fig. 29D and G), with the orientation angle θ . 60 degrees, result in a decrease in
the crack tip stress field (known as stress shielding effect, i.e., σðc2vÞ
tip =σc
tip , 1). More impor-
tantly, Fig. 29C to G shows that the continuum-based analytical solutions given in
Eqs. (2.14) and (2.15) are able to accurately capture the trends of the crack tip stress fields
obtained from the atomistic simulations. Here, Kðc2vÞ
I =KI given by the analytical solutions
was compared with σðc2vÞ
tip =σc
tip obtained from the atomistic simulations. The normalized
crack tip stress σðc2vÞ
tip =σc
tip is a comparable quantity to the corresponding normalized SIFs
[94]. However, the continuum expressions are unable to predict the influence of the under-
lying crystal structures (i.e., armchair and zigzag) on the crack tip stress field, which sets a
limit on developing a unified continuum fracture mechanics framework at the atomic
scale.
2.4.4 Hybrid two-dimensional materials
Besides having a similar lattice structure of graphene, h-BN possesses electromechanical
properties that are comparable to those of graphene [95,96]. In contrast to the zero-
bandgap semimetal nature of graphene, h-BN is a finite-bandgap semiconductor [97,98].
The similarity of the lattice structures of graphene and h-BN allows the construction of
grapheneh-BN heterostructures with unique electronic and magnetic properties [24,25].
For these advanced applications, a clear understanding of the mechanical behavior of these
2D heterostructures is vital. Especially, the fracture characteristics of such a hybrid
Chapter 2 • Mechanical properties of two-dimensional materials 23](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-40-320.jpg)
![structure are critically important, because both graphene and h-BN have relatively low frac-
ture toughness [99].
Several MD studies have focused on the stability, fracture, and thermal properties of
grapheneh-BN heterostructures. For example, MD study [100] revealed that interfacial
defects can have a significant influence on the structural configuration and thermal
conductance of these nanostructures. In addition, the presence of atomic defects in
hybrid grapheneh-BN sheets results in significantly reduced failure strength and
Young’s modulus [101]. However, cracks in a grapheneh-BN sheet have a much lower
effect on Young’s modulus, when compared to the effect on the failure strength [102].
The edge configuration of graphene and h-BN, that is, armchair or zigzag, has a signifi-
cant influence on their mechanics and interfacial properties [103,104]. For example, the
tensile strength of grapheneh-BN sheets, with perfect armchair or zigzag interfaces, is
approximately similar to that of pristine graphene [105]. However, the tensile strength
of misorientated interfaces highly depends on the mismatch angle between graphene
and h-BN domains [106].
Fig. 210A shows the simulated graphene sample containing a circular h-BN inclu-
sion with a diameter of 10 nm. The h-BN inclusion does not generate a significant
eigenstrain in the sample due to the fact that the lengths of both CC and BN bonds
are 1.44 Å according to the Tersoff potential [99]. Moreover, the stress distribution
within the inclusion is constant (see Fig. 210C). This observation agrees with Eshelby
theory of the ellipsoidal inclusion problem [108,109], which states that a uniformly
applied far-field stress induces a constant stress state within the inclusion. A complex
stress state is observed at the grapheneh-BN interface, where the atomic stress
ranges from 0 to 35 GPa due to an externally applied far-field stress of 20 GPa. This
complex stress distribution is attributed to (1) heterogeneous atomic bonds at the
interface and (2) the change of chirality along the grapheneh-BN interface. The inter-
atomic bonds within the h-BN inclusion and the surrounding graphene sheet are BN
and CC, respectively. However, atoms at the h-BNgraphene interface form four
types of atomic bonds: BC, NC, BN, and CC. This highly heterogeneous bond
arrangement at the interface contributes to the observed complex stress state. In addi-
tion, chirality of the interface gradually changes from armchair to zigzag when the
angle β (see Fig. 210C) increases from 0 to π/6 [60]. The chirality further changes
gradually back to zigzag when β further increases from π/6 to π/3. This change in the
underlying crystal structure along the interface also results in a complex stress state at
the interface. Fig. 210D shows that the uniform stress field within the inclusion is
approximately 17 GPa. In addition, a stress concentration of approximately 1.2 can
be observed in graphene at the interface due to the relatively low elastic modulus of
h-BN [107].
24 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-41-320.jpg)
![2.5 Future directions
2.5.1 Topological design of two-dimensional materials
The idea of creating bulk materials by manipulating nanometer-sized microstructures and
defects has been illustrated for materials such as metals and ceramics [110]. Taking this idea
further, the assembly of bulk materials using nanometer-sized crystallites has been demon-
strated in Ref. [111]. A microstructurally heterogeneous nanostructured material can be cre-
ated by using nanometer-sized building blocks (crystallites) that have different atomic
FIGURE 2–10 Stress field around a circular hexagonal boron nitride (h-BN) inclusion in graphene: (A) the simulated
sample, where the inset demonstrates the selected origin of the Cartesian coordinate system. (B) and (C) show the
stress σyy fields of the graphene sheet and the h-BN inclusion due to an applied tensile strain εyy of 2%. (D)
Variation of the atomic stress εyy along the x-axis. Reprinted from M.A.N. Dewapriya, S.A. Meguid, R.K.N.D.
Rajapakse, Atomistic modelling of crack-inclusion interaction in graphene, Eng. Fract. Mech. 195 (2018) 92103.
doi:10.1016/j.engfracmech.2018.04.003 with permission from Elsevier.
Chapter 2 • Mechanical properties of two-dimensional materials 25](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-42-320.jpg)
![structure, different crystallographic orientation, and/or different chemical composition as
illustrated in Fig. 211.
The concept of nanostructured materials directly fits with 2D materials such as graphene
because their basic building block consists of hexagonally packed atomic layers of carbon
and/or other atoms. Although pristine graphene has the hexagonal atomic structure, defects
are difficult to avoid during fabrication [15]. These defects are typically pentagons, hepta-
gons, dislocations, and grain boundaries (series of pentagons and heptagons) that produce
out-of-plane displacements and alter the 2D material properties [112]. Recent experimental
and modeling studies demonstrate that these topological defects could either enhance or
weaken the properties of 2D materials [113]. Although attempts are taking place to under-
stand the role of defects in 2D materials and their influence on properties, our overall under-
standing of their effects is still in its infancy. Like in the case of bulk materials with
nanometer-sized microstructure, 2D materials present an exciting opportunity for tailoring
their multiphysical properties of their single- and/or multilayered sheets through the topo-
logical design of their nanometer-sized microstructures.
2.5.2 Piezoelectricity of two-dimensional materials
Superior multiphysical properties of 2D materials could serve as the foundation for creating
next-generation smart composites. Some 2D materials such as h-BN and MoS2, which are
nonpiezoelectric in their bulk form, display piezoelectric behavior when their thickness is
reduced to a one atom thick [114,115]. In addition to hierarchical optimization, the piezo-
electric properties of individual 2D membranes could be manipulated through topological
design. For instance, piezoelectricity can be engineered in nonpiezoelectric materials (e.g.,
graphene) through the selective adsorption of atoms or by introducing atomic vacancies
FIGURE 2–11 Material containing nanometer-sized microstructure.
26 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-43-320.jpg)
![[116]. Even though the piezoelectric coefficients of 2D materials are lower than the high-
performance bulk piezoelectric materials such as lead zirconate titanate the high strength,
stiffness, and flexibility of the 2D materials have unique advantages over the conventional
bulk materials, when it comes to device integration.
For example, superior mechanical properties along with the intrinsic piezoelectricity of h-
BN have attracted significant attention of the research community. The highly stretchable
nature of h-BN allows us to modulate its polarization through elastic strain, which opens a
new class of strain-engineered piezoelectric materials. Moreover, h-BN has the simplest crys-
tal structure among the piezoelectric materials. It has also been demonstrated that the appli-
cation of a strain gradient through curvature result in polarization in bilayer h-BN (i.e.,
flexoelectricity) [117]. In the case of multilayered h-BN assemblies, the piezoelectric proper-
ties significantly depend on the number of layers in the stack [118] and the type of load
applied. A similar phenomenon has been observed for the case of multilayered MoS2 [119].
Moreover, piezoelectric properties of 2D materials significantly depend on the crystal struc-
ture and its orientation. For example, Ref. [120] demonstrated that, under the same electric
field, armchair and zigzag h-BN sheets experience significantly different deformation pat-
terns. This observation confirms that piezoelectricity of 2D materials can be engineered
through topological design.
2.5.3 Application of machine learning methods
The highly complex process of the bottom-up design of 2D materialbased systems at the
nanoscale is often limited by the available computational power. For example, the first prin-
ciples computational methods (e.g., density functional theory) are impractical to employ in
studying crack propagation of 2D materials due to the extremely high computational cost
associated with them. In recent years, the field of deep learning has demonstrated applica-
tions in various disciplines of engineering [121]. Deep learning is a subset of machine learn-
ing, which structures algorithms in layers to create deep neural networks that can learn from
a set of examples and make intelligent predictions [122]. Combination of atomistic modeling
techniques with deep learning could significantly reduce the computational burden associ-
ated with atomistic simulations. For example, computationally expensive calculations of
interatomic potential energy could be replaced with properly trained neural networks at a
fraction of the initial computational cost [123]. It has been recently demonstrated that deep
learning techniques can be used to efficiently solve numerical problems in continuum
mechanics (e.g., finite elements) [124]. In addition, machine learning has been used to pre-
dict dynamic fracture growth in brittle material [125,126] and to characterize fracture stress
of defective graphene samples [127]. These recent studies suggest that the application of
machine learning techniques could significantly revolutionize computer-aided design of 2D
materials.
Chapter 2 • Mechanical properties of two-dimensional materials 27](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-44-320.jpg)
![Acknowledgments
The authors thank NSERC for supporting the research. Computing resources were provided by Compute/
Calcul Canada.
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acs.jpclett.5b01815
32 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-49-320.jpg)
![Coach is a Stage Coach, that is drawn by 6 Horses, and will
sometimes run 90 or 100 English Miles on one day.
'It may also be noted that Carriage by Waggon or Pack Horses, is
about 5 Shillings for carrying 112 Pound Weight 100 Miles; and so in
proportion; though 'tis something cheaper in the Summer than
Winter.'[597]
The Hackney coach was a very useful institution, in spite of all
said against it. We have read Ward's description of the bumping he
had in one; in another part of the London Spy he says: 'Would you
have me, said I, undergo the Punishment of a Coach again, when
you Know I was so great a Sufferer by the last, that it made my
Bones rattle in my Skin, and has brought as many Pains about me,
as if troubled with the Rheumatism. That was a Country Coach, says
he, and only fit for the Road; but London Coaches are hung more
loose, to prevent your being Jolted by the roughness of the
pavement.'
The ordinary hackney coaches do not seem to have been provided
with glasses. 'For want of Glasses to our Coach, having drawn up our
Tin Sashes, pink'd like the bottom of a Cullender, that the Air might
pass thro' the holes, and defend us from Stifling.'
By the 5 6 Will. and Mary, cap. 22, the number of hackney
coaches was fixed at 700, and a tax was imposed of 4l. per annum
each, 1l. to be paid every quarter day, besides a fine of 50l. for their
first licence for 21 years; and 8l. per annum on stage coaches.
To look after these hackney carriages there were five
commissioners, at a salary each of 200l., and their office was in
Surrey Street, Strand. The fares were not very heavy, even taking
the difference of the value of money into consideration, and the fact
that they had two horses.
s. d.
For one day of 12 Hours 10 —
For one Hour 1 6
For every hour after the first 1 —](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-57-320.jpg)
![From any of the Inns of Court to any part of St. James's or City of
Westminster, except beyond Tuttle Street 1 —
From the Inns of Court, or thereabouts, to the Royal Exchange 1 —
From any of the Inns of Court, to the Tower, Aldgate, Bishopsgate Street
or thereabouts 1 6
And the same Rates back again, or to any Place of the like Distance.
And, if any Coachman shall refuse to go at, or exact more, for Hire, than the
Rates hereby limited, he shall for every such Offence forfeit 40 Shillings
In 1710 the number of coaches was increased to 800 by the 9
Anne, cap. 23, which also provided that they were to pay five
shillings weekly, and were to go a mile and a half for one shilling,
two miles for one shilling and sixpence, above two miles two
shillings, and greater distances in the same proportion.
The hackney coachmen petitioned against the tax, and said they
were willing to pay the old one. One petition was entitled 'Some
Reasons most humbly Offered to the Consideration of the Right
Honourable the House of LORDS and the Honourable the House of
Commons; by all the 700 Hackney Coachmen and their Widows to
Enable them to pay the Great Tax laid upon them;' and another was
'The Hackney Coachmen's case. Humbly presented to the Right
Honourable House of Commons, with a proposal to raise for her
Majesty 200,000l. per annum.' This was proposed, very coolly, to be
done by laying a tax on all coaches and carriages not licensed, on
passengers going by stage coaches, and on goods carried by
waggons and packhorses.
The coaches were numbered, although I can only find one notice
of it: 'So that, rather than to stand a Vapulation, one of them took
Notice of his Number;'[598] and the coachmen were noted for their
incivility. Of course they did not come from a very high class, and
the habits and language of the lower class of that time were
extremely coarse. 'We discharged our Grumbling Coachman, who
Mutter'd heavily, according to their old Custome, for t'other
Sixpence; till at last moving us a little beyond our Patience, we gave
an Angry Positive Denial to his Unreasonable Importunities; for we](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-58-320.jpg)
![Somerset House in great state and splendour, their Coach of State
embroidered with gold, and the richest that ever was seen in
England: they had two with 8 horses, and eight with 6 horses,
trimm'd very fine with ribbons, 48 footmen in blew velvet cover'd
with gold lace, 24 gentlemen and pages on horseback, with feathers
in their hats.' And the novelty does not seem to have worn off, for,
four years afterwards, Swift writes to Stella: 'This evening I saw the
Venetian Ambassador coming from his first public audience. His
coach is the most monstrous, huge, fine, rich, gilt thing that ever I
saw.' He also writes her, Feb. 6, 1712: 'Nothing has made so great a
noise as one Kelson's Chariot, that cost nine hundred and thirty
Pounds, the finest was ever seen. The rabble huzzaed him as much
as they did Prince Eugene.'
Anybody with any pretension to wealth and fashion drove six
horses, as says Mrs. Plotwell[599]: 'I must have Six Horses in my
Coach, four are fit for those that have a Charge of Children, you and
I shall never have any;' and Lucinda tells Sir Toby Doubtful[600]:
'You'll at least keep Six Horses Sir Toby, for I wou'd not make a Tour
in High Park with less for the World; for me thinks a pair looks like a
Hackney.' The coachman, however, did not drive all six, one of the
leaders being always ridden by a postilion. These carriage horses
were heavy, long-tailed Flemish ones, and naturally went at a sedate
and sober trot.
STATE COACH.](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-60-320.jpg)
![with a Round Cypher in the Pannel, Lin'd with White Cloath embos'd
with Red, having a Glass in one Frame, and White Canvas in another,
with Red Strings to both Frames. Whosoever hath taken it up are
desir'd to bring it to Mr. Jacob's a Coachmaker at the corner of St.
Mary Ax near London Wall, where they shall receive 30s. Reward if
all be brought with it; or if offer'd to be Pawn'd or Sold, desire it may
be stop'd and notice given, or if already Pawn'd or Sold, their money
again.'
In very many advertisements of the sale of second-hand carriages,
it is mentioned that the glasses are complete. One would imagine
from this that glass was dear, but it was not particularly so. 'These
are to give notice to all Persons that have occasions for Coach
Glasses, or Glasses for Sash Windows, that they may be furnished
with all sorts, at half the prises they were formerly sold for.' And it
goes on to say that 12 inches square was 2s. 6d., and increased up
to 22 inches, nearly at the rate of 6d. per inch, or 8s. 6d.; 23 inches
was 10s. 6d., and so on at about 2s. 6d. per inch to 28 inches,
which was 20s., until it culminated at 36 inches square for £2 10s. If
this, really, was half the previous cost, and if we reckon the
difference in the value of money then and now perhaps some
economical people would think twice before having a broken glass
repaired.
There were also a 'Chasse marée Coach,' and a 'Curtin Coach for
Six People.'
They used to take nice little drives, too, in these clumsy old
carriages—but they took their time over the journey. Thoresby's
'kind friend Mr. Boulter, brought his chariot from Chelsea, purposely
to carry him to see Hampton Court.' They started about eleven, and,
'having passed through the City, we passed the Gravel Pits,[601] and
had a clear air (whither the Consumptive are sent by the physicians)
and delicate pleasant Country to Acton and Brentford; the Duke of
Somerset's Seat at Sion House looked most charmingly, and was the
first time I had observed the lime trees in the avenues cut in a
pyramidal form, even to a great distance from the palace, which
looked very Noble; thence through Thistleworth and Twitnam, a very](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-62-320.jpg)
![1s. per mile. Next year, another Act (10 Anne, c. 19) was passed,
licensing 100 more, but keeping the fares unaltered.
Like coaches, their adornment was indicative of the wealth and
position of their owners—although, perhaps, none ever came up to
Anne's royal present.[602] 'The Queen has made a present of a
chair value £8000 to the King of Prussia, which is ordered for Berlin.'
Still they were highly ornamental, as the following, which was the
property of Sir Joseph Williamson, deceased, will show. 'A Cedan (or
Chair) lin'd with Crimson Velvet, trim'd with Gold and Silver, and a
new Mourning Chair c.'
The prefix 'Sedan' was seldom used, and these conveyances were
generally termed 'Chairs.' That they were considered somewhat of a
novelty in Anne's reign is evidenced by that line of Gay's 'Nor late
invented Chairs perplex'd the way,' and also by the fact that then the
public chairs were first licensed, and the number, a very small one,
regulated.
They were not particularly comfortable, as the Marquis of Hazard
describes[603]: 'Hey, let my three Footmen wait with my Chair there
—the Rascals have come such a high trot—they've jolted me worse
than a Hackney Coach—and I am in as much disorder as if I had not
been drest to day.' And they were sometimes dangerous too.
Or, box'd within the Chair, contemn the Street,
And trust their Safety to another's Feet.
. . . . . . .
The drunken Chairman in the Kennel Spurns,
The Glasses shatters, and his Charge o'erturns.
Gay evidently did not like either chairs or chairmen, for he warns
his reader thus:—](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-65-320.jpg)
![Brown, in his 'Letters from the Dead to the Living,' says in that
'From Bully Dawson[604] to Bully W....n': 'Therefore if ever you
intend to be my Rival in Glory, you must fight a Bailiff once a Day,
stand Kick and Cuff once a Week, Challenge some Coward or other
once a Month, Bilk your Lodgings once a quarter, and Cheat a Taylor
once a year, crow over every Coxcomb you meet with, and be sure
you kick every jilt you bully into submission and a compliance of
treating you; never till then will the fame of W....n ring like Dawson's
in every Coffee House, or be the merry subject of every Tavern Tittle
Tattle.'
There seem to have been two special outbreaks of Mohocks—one
in 1709, and the other in 1712. Of the first Steele says:[605] 'When
I was a middle-aged Man, there were many societies of Ambitious
young men in England, who, in their pursuits after same, were every
night employed in roasting Porters, smoaking Coblers, knocking
down Watchmen, overturning Constables, breaking Windows,
blackening Sign Posts, and the like immortal enterprizes, that
dispersed their Reputation throughout the whole Kingdom. One
could hardly find a knocker at a door in a whole street after a
midnight expedition of these Beaux Esprits. I was lately very much
surprised by an account of my Maid, who entered my bed chamber
this morning in a very great fright, and told me, she was afraid my
parlour was haunted; for that she had found several panes of my
Windows broken, and the floor strewed with half-pence. I have not
yet a full light into this new way, but am apt to think, that it is a
generous piece of wit, that some of my Contemporaries make use of,
to break windows, and leave money to pay for them.'
Gay notices the Mohocks, and their window-breaking thus:—](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-68-320.jpg)
![Now is the Time that Rakes their Revells keep;
Kindlers of Riot, Enemies of Sleep.
His scatter'd Pence the flying Nicker flings,
And with the Copper Show'r the Casement rings.
Who has not heard the Scowrer's Midnight Fame?
Who has not trembled at the Mohock's Name?
Was there a Watchman took his hourly Rounds,
Safe from their Blows, or new invented Wounds?
I pass their desp'rate Deeds, and Mischiefs done,
Where from Snow Hill black Steepy Torrents run;
How Matrons, hoop'd within the Hogshead's Womb,
Were tumbled furious thence, the rolling Tomb
O'er the Stones thunders, bounds from Side to Side.
So Regulus to save his Country dy'd.
The greatest scare, however, was in March 1712, and that
exercised the popular mind as much as the garotters of modern
times. People, of course, were more frightened than hurt, and there
is very little doubt but that this outbreak was much exaggerated.
Still, we can only take the contemporary accounts, and this is one of
them.
[606]'The Town Rakes, or the Frolicks of the Mohocks or
Hawkubites. With an Account of their Frolicks last night, and at
several other Times: shewing how they slit the Noses of several Men
and Women, and wounded others; Several of which were taken up
last Night by the Guards, and committed to several Prisons, the
Guards being drawn out to disperse them.
'There are a certain set of Persons, amongst whom there are
some of too great a Character, to be nam'd in these barbarous and
ridiculous Encounters, did they not expose themselves by such mean
and vulgar Exploits.
'These Barbarities have been carry'd on by a Gang of 'em for a
considerable time, and many innocent Persons have receiv'd great
Injury from them, who call themselves Hawkubites; and their
mischievous Invention of the Word is, that they take people betwixt
Hawk and Buzzard, that is, betwixt two of them, and making them
turn from one to the other, abuse them with Blows and other](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-69-320.jpg)
![Scoffings; and, if they pretend to speak for themselves, they then
Slit their Noses, or cut them down the Back.
'The Watch in most of the Out-parts of the town stand in awe of
them, because they always come in a body, and are too strong for
them, and when any Watchman presumes to demand where they
are going, they generally misuse them.
'Last Night they had a general Rendezvouz, and were bent upon
Mischief; their way is to meet People in the Streets and Stop them,
and begin to Banter them, and if they make any Answer, they lay on
them with Sticks, and toss them from one to another in a very rude
manner.
'They attacked the Watch in Devereux Court and Essex Street,
made them scower; they also slit two Persons' Noses, and cut a
Woman in the Arm with a penknife that she is lam'd. They likewise
rowled a Woman in a Tub down Snow Hill, that was going to Market,
set other Women on their Heads, misusing them in a barbarous
manner.
'They have short Clubs or Batts that have Lead at the End, which
will overset a Coach, or turn over a Chair, and Tucks[607] in their
Canes ready for Mischief.
'One of these Persons suppos'd to be of the Gang, did formerly slit
a Drawer's Nose at Greenwich, and has committed many such
Frolicks since. They were so outrageous last Night, that the Guards
at White Hall was alarm'd, and a Detachment order'd to Patrole; and
'tis said, the Train Bands will be order'd to do Duty for the future, to
prevent these Disorders; several of them were taken up last Night,
and put into the Round Houses till order is taken what to do with
them.'
The Spectator, whose living was by making the most of any
popular subject of the hour, was specially exercised over the
Mohocks. 'An outrageous Ambition of doing all possible hurt to their
Fellow-Creatures, is the great Cement of their Assembly and the only
Qualification required in the Members. In order to exert this Principle
in its full Strength and Perfection, they take Care to drink themselves](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-70-320.jpg)
![to a pitch that is beyond the Possibility of attending to any Motives
of Reason and Humanity; then make a general Sally, and attack all
that are so unfortunate as to walk the Streets through which they
patrole. Some are knock'd down, others stabb'd, others cut and
carbonado'd. To put the Watch to a total Rout, and mortify some of
those inoffensive Militia, is reckon'd a Coup d'éclat. The particular
Talents by which these Misanthropes are distinguished from one
another, consist in the various kinds of Barbarities which they
execute upon their Prisoners. Some are celebrated for a happy
Dexterity in tipping the Lion upon them; which is perform'd by
squeezing the Nose flat to the Face, and boring out the Eyes with
their Fingers; Others are called the Dancing Masters, and teach their
Scholars to cut Capers by running Swords thro' their Legs; a new
Invention, whether originally French I cannot tell; A third sort are
the Tumblers, whose office it is to set Women on their Heads, and
commit certain Indecencies or rather Barbarities on the Limbs which
they expose.'[608]
Sir Roger de Coverley was even somewhat nervous about them
when he went to the play—and 'asked me, in the next place whether
there would not be some danger in coming home late, in case the
Mohocks should be Abroad'; and we learn how, finally, the party
went to the theatre. 'The Captain, who did not fail to meet me there
at the appointed Hour, bid Sir Roger fear nothing, for that he had
put on the same Sword which he made use of at the Battle of
Steenkirk. Sir Roger's Servants, and among the rest my old Friend,
the Butler, had, I found, provided themselves with good Oaken
Plants, to attend their Master upon this occasion.'
Swift was in mortal fear of them, and, in his 'History of the Four
Last Years of Queen Anne,' declares it was part of a deliberate plan
to raise riot, during which Harley might have been assassinated—
and accuses Prince Eugene of setting it afloat. He writes Stella—in
Letter 43—fragmentary jottings of his feelings during this period of
terror. 'Did I tell you of a race of Rakes, called the Mohocks, that
play the devil about this town every night, slit peoples noses, and
bid them, c.... Young Davenant was telling us at Court how he was](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-71-320.jpg)
![THE WATCH.
Can the following advertisement have any possible relation to the
midnight orgies of the Mohocks? Post Boy, Dec. 18/20, 1712: 'Lately
found, several Pair of Stockings, some Night Caps, and several Pair
of Shooes, with two Brazill Rolling Pins, and some Brass Knockers of
Doors.'
Brass knockers evidently were attractive, for in 1714 we find a
genius advertising, 'There is to be Sold at the Sign of the Plow on
Fleet Ditch, New Fashion Brass Knockers of all Sizes that cannot be
broke off so easily as any that have yet been made. However, this is
to Satisfy all Gentlemen and others that do buy any of them, that if
any should be broke off, upon their bringing me a Piece of that
which I sold, I will give them gratis one as good and as large as they
bought.'
The fright soon passed off, for we find Budgell[609] writing on
April 8, 1712, that some began to doubt 'whether indeed there were
ever any such Society of Men. The Terror which spread itself over
the whole Nation some Years since, on account of the Irish, is still](https://image.slidesharecdn.com/33775-250403123237-5c4d2095/85/Synthesis-Modelling-and-Characterization-of-2D-Materials-and-their-Heterostructures-Micro-and-Nano-Technologies-1st-Edition-Eui-Hyeok-Yang-Editor-75-320.jpg)
Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures (Micro and Nano Technologies) 1st Edition Eui-Hyeok Yang (Editor)
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- 6. Synthesis, Modeling, and Characterization of 2D Materials, and Their Heterostructures
- 7. Synthesis, Modeling, and Characterization of 2D Materials, and Their Heterostructures Editor-in-Chief Eui-Hyeok Yang Edited by Dibakar Datta Junjun Ding Grzegorz Hader
- 8. Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States Copyright © 2020 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-818475-2 For Information on all Elsevier publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Matthew Deans Acquisitions Editor: Simon Holt Editorial Project Manager: Charlotte Rowley Production Project Manager: Kamesh Ramajog Cover Designer: Greg Harris Typeset by MPS Limited, Chennai, India
- 9. Contents List of contributors xiii About the editors xvii Part I Introduction to 2D materials and their heterostructures 1 1. Overview 3 EUI-HYEOK YANG, DIBAKAR DATTA, GRZEGORZ (GREG) HADER AND JUNJUN DING 1.1 Overview of two-dimensional materials and the scope of the book 3 References 6 Part II Properties of 2D materials and their heterostructures 7 2. Mechanical properties of two-dimensional materials: atomistic modeling and future directions 9 M.A.N. DEWAPRIYA, R.K.N.D. RAJAPAKSE AND S.A. MEGUID 2.1 Introduction 9 2.2 Current state of research 10 2.3 Molecular dynamics simulations of two-dimensional materials 12 2.4 Fracture characteristics of two-dimensional materials 14 2.5 Future directions 25 Acknowledgments 28 References 28 v
- 10. 3. Thermal transport properties of two-dimensional materials 37 FAN YANG 3.1 Introduction to thermal transport 37 3.2 Thermal transport in two-dimensional materials 39 3.3 Simulation methods for thermal transport properties in two-dimensional materials 45 3.4 Experimental methods for thermal transport property in two-dimensional materials 49 3.5 Conclusion 52 References 52 4. Optical properties of semiconducting transition metal dichalcogenide materials 57 IBRAHIM SARPKAYA 4.1 Introduction 57 4.2 Photophysics of excitons and other excitonic complexes 58 4.3 Quantum emitters in semiconducting transition metal dichalcogenides 67 References 70 5. Electronic properties of two-dimensional materials 77 GERARDO G. NAUMIS 5.1 Introduction and outline 77 5.2 Structure and diffraction of two-dimensional materials 79 5.3 Electronic properties of Dirac and Weyl materials 82 5.4 Two-dimensional materials made from group IV, V, and VI elements 93 5.5 Multilayered two-dimensional materials 97 Acknowledgments 103 References 103 vi Contents
- 11. Part III Computational modeling of two-dimensional materials 111 6. Atomistic modeling by density functional theory of two-dimensional materials 113 DEQUAN ER AND KAMALIKA GHATAK 6.1 Introduction 113 6.2 Theoretical background 114 6.3 Implementation of density functional theory in two-dimensional systems 117 References 121 7. Molecular dynamics simulations of two-dimensional materials 125 SOUVICK CHAKRABORTY AND HEMANT KUMAR 7.1 Introduction 125 7.2 Historical background 126 7.3 Molecular dynamics algorithm 126 7.4 Scope and limitations of molecular dynamics simulations in the context of two-dimensional materials 135 7.5 Summary 146 References 146 8. Monte Carlo method in two-dimensional materials 149 SWASTIK BASU AND VIDUSHI SHARMA 8.1 Introduction 149 8.2 Metropolis Monte Carlo method 150 8.3 Grand canonical Monte Carlo simulations to study the effect of substrates on lithiation-induced fracture of silicon electrode 151 8.4 Kinetic Monte Carlo method 155 References 161 Contents vii
- 12. 9. Lattice and continuum based modeling of 2D materials 165 T. MUKHOPADHYAY, A. MAHATA AND S. ADHIKARI 9.1 Introduction 165 9.2 Mechanical equivalence of atomic bonds 166 9.3 Equivalent elastic moduli of two-dimensional materials 168 9.4 Results and discussion 170 9.5 Summary 173 References 173 Part IV Synthesis and characterization of 2D materials and their heterostructures 179 10. Synthesis of graphene 181 GRZEGORZ (GREG) HADER 10.1 Early history 181 10.2 Existence of two-dimensional crystals 181 10.3 Properties of carbon, graphite, and graphene 183 10.4 Graphene suppliers 190 10.5 Raman spectroscopy—graphene fingerprints 190 10.6 Visibility of graphene 191 10.7 Automated visualization and identification of two-dimensional layers 194 10.8 Graphene synthesis 194 10.9 Graphene on SiC 204 10.10 Liquid-phase exfoliation 205 10.11 Molecular assembly 206 10.12 Cold-wall reactor 207 10.13 Atmospheric pressure chemical vapor deposition 209 10.14 Summary of graphene synthesis 212 viii Contents
- 13. 10.15 Autonomous robotic assembly of van der Waals heterostructure superlattices 212 10.16 Synthesis methods and reviews 214 10.17 Applications of graphene and beyond 214 References 217 11. Synthesis of two-dimensional hexagonal boron nitride 223 MINJIE WANG, JUNJUN LYV, FEI GUO AND YI LI 11.1 Introduction 223 11.2 Synthesis of two-dimensional hexagonal boron nitride 225 11.3 Summary and outlook 239 Acknowledgment 239 References 239 12. Synthesis of transition metal dichalcogenides 247 KYUNGNAM KANG, SIWEI CHEN AND EUI-HYEOK YANG 12.1 Introduction 247 12.2 Mechanical exfoliation 247 12.3 Liquid-phase exfoliation 250 12.4 Chemical vapor deposition 252 12.5 Molecular-beam epitaxy 256 12.6 Doping/alloy of transition metal dichalcogenides 257 12.7 Summary 260 References 260 13. Synthesis of heterostructures based on two-dimensional materials 265 YUQI GAO AND JUNJUN DING 13.1 Introduction 265 13.2 Synthesis of heterostructures 266 13.3 Summary 279 References 280 Contents ix
- 14. 14. Characterization of two-dimensional materials 289 DANIEL KAPLAN, RAYMOND FULLON AND NICHOLAS A. SIMONSON 14.1 Introduction 289 14.2 Visualization—microscopy 289 14.3 X-ray photoelectron spectroscopy 295 14.4 Raman spectroscopy 302 14.5 Why scanning probe microscopy? 309 References 318 Part V Mechanical, Optical, and Electrical Devices 323 15. Two-dimensional materials and hybrid systems for photodetection 325 ZE XIONG AND JINYAO TANG 15.1 Introduction 325 15.2 Fundamentals of photodetectors 326 15.3 Materials in photodetectors 335 15.4 Classification of photodetectors 343 15.5 Prospect of two-dimensional photodetectors in flexible electronics and bioelectronics 344 15.6 Conclusion 345 References 345 16. Electronic devices based on solution-processed two-dimensional materials 351 PEI HE, JIANYUN CAO, HUI DING, XIN ZHAO AND ZHELING LI 16.1 Introduction 351 16.2 Preparation of two-dimensional materials via solution process 353 x Contents
- 15. 16.3 Device fabrication techniques for two-dimensional material based inks 359 16.4 Electronic applications based on two-dimensional nanosheets 370 16.5 Conclusion 378 References 378 17. Two-dimensional materials and its heterostructures for energy storage 385 VIDUSHI SHARMA, KAMALIKA GHATAK AND DIBAKAR DATTA 17.1 Current non two-dimensional material based batteries and their shortcomings 385 17.2 Two-dimensional material based anodes for Li/Na-based batteries 386 17.3 Two-dimensional heterostructures for energy storage 391 17.4 Progress made in two-dimensional materials as cathode 393 17.5 Potential of two-dimensional heterostructures for promising performance 395 References 396 18. The application of low-dimensional materials in virology and in the study of living organisms 403 YIN-TING YEH, VENKATARAMAN SWAMINATHAN AND MAURICIO TERRONES 18.1 Viral infectious disease 403 18.2 Nitrogen-doped carbon nanotubes 405 18.3 Device performance in virology 412 18.4 A portable virus capture and detection microplatform 423 18.5 Cellular digestion of transition metal dichalcogenide monolayers 431 18.6 Future prospects 435 References 437 Contents xi
- 16. Part VI Future Perspectives 443 19. Machine learning in materials modeling—fundamentals and the opportunities in 2D materials 445 SHREEJA DAS, HANSRAJ PEGU, KISOR KUMAR SAHU, AMEEYA KUMAR NAYAK, SEERAM RAMAKRISHNA, DIBAKAR DATTA AND S. SWAYAMJYOTI 19.1 The launch platform for machine learning 445 19.2 Nature-inspired engineering: the birth of artificial intelligence and machine learning 447 19.3 Data collection and representation 450 19.4 Model selection and validation 455 19.5 Model optimization and quality assessment 459 19.6 Opportunities of machine learning for two-dimensional materials 461 References 465 Index 469 xii Contents
- 17. List of contributors S. Adhikari College of Engineering, Swansea University, Swansea, United Kingdom Swastik Basu Rensselaer Polytechnic Institute (RPI), Troy, NY, United States Jianyun Cao Department of Materials, University of Manchester, Manchester, United Kingdom Souvick Chakraborty School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Odisha, India Siwei Chen Mechanical Engineering Department, Stevens Institute of Technology, Hoboken, NJ, United States Shreeja Das School of Minerals, Metallurgical and Materials Engineering, Indian Institute of Technology (IIT) Bhubaneswar, Bhubaneswar, India Dibakar Datta Department of Mechanical and Industrial Engineering, Newark College of Engineering, New Jersey Institute of Technology, Newark, NJ, United States M.A.N. Dewapriya School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada; MADL, Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, Canada Hui Ding Department of Materials, University of Manchester, Manchester, United Kingdom Junjun Ding Kazuo Inamori School of Engineering, New York State College of Ceramics, Alfred University, Alfred, NY, United States Dequan Er Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA, United States Raymond Fullon Department of Materials Science and Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ, United States xiii
- 18. Yuqi Gao Kazuo Inamori School of Engineering, New York State College of Ceramics, Alfred University, Alfred, NY, United States Kamalika Ghatak Department of Mechanical and Industrial Engineering, New Jersey Institute of Technology (NJIT), Newark, NJ, United States Fei Guo Department of Initiators and Pyrotechnics, Institute of Chemical Materials in China Academy of Engineering Physics, Mianyang, Sichuan, China Grzegorz (Greg) Hader US Army CCDC, Armaments Center, Picatinny Arsenal, NJ, United States Pei He School of Physics and Electronics, Central South University, Changsha, P.R. China Kyungnam Kang Mechanical Engineering Department, Stevens Institute of Technology, Hoboken, NJ, United States Daniel Kaplan Advanced Materials Technology Branch, U.S. Army CCDC-AC, Picatinny Arsenal, NJ, United States Hemant Kumar School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Odisha, India Yi Li Department of Initiators and Pyrotechnics, Institute of Chemical Materials in China Academy of Engineering Physics, Mianyang, Sichuan, China Zheling Li Department of Materials, University of Manchester, Manchester, United Kingdom Junjun Lyv Department of Initiators and Pyrotechnics, Institute of Chemical Materials in China Academy of Engineering Physics, Mianyang, Sichuan, China A. Mahata Department of Materials Science and Engineering, Missouri University of Science and Technology, Rolla, MO, United States S.A. Meguid MADL, Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, Canada T. Mukhopadhyay Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India Gerardo G. Naumis Complex Systems Department, Physics Institute, Universidad Nacional Autonoma de Mexico (UNAM), CDMX, Mexico xiv LIST OF CONTRIBUTORS
- 19. Ameeya Kumar Nayak Department of Mathematics, Indian Institute of Technology (IIT) Roorkee, Roorkee, India Hansraj Pegu School of Minerals, Metallurgical and Materials Engineering, Indian Institute of Technology (IIT) Bhubaneswar, Bhubaneswar, India R.K.N.D. Rajapakse School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada Seeram Ramakrishna Department of Mechanical Engineering, National University of Singapore (NUS), Singapore, Singapore Kisor Kumar Sahu School of Minerals, Metallurgical and Materials Engineering, Indian Institute of Technology (IIT) Bhubaneswar, Bhubaneswar, India Ibrahim Sarpkaya Bilkent University, UNAM, Ankara, Turkey Vidushi Sharma Department of Mechanical and Industrial Engineering, New Jersey Institute of Technology (NJIT), Newark, NJ, United States Nicholas A. Simonson Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA, United States Venkataraman Swaminathan Department of Physics, The Pennsylvania State University, University Park, PA, United States S. Swayamjyoti School of Minerals, Metallurgical and Materials Engineering, Indian Institute of Technology (IIT) Bhubaneswar, Bhubaneswar, India; NetTantra Technologies, Bhubaneswar, India Jinyao Tang Department of Chemistry, The University of Hong Kong, Hong Kong, P.R. China Mauricio Terrones Department of Physics, The Pennsylvania State University, University Park, PA, United States Minjie Wang Department of Initiators and Pyrotechnics, Institute of Chemical Materials in China Academy of Engineering Physics, Mianyang, Sichuan, China Ze Xiong Department of Chemistry, The University of Hong Kong, Hong Kong, P.R. China Eui-Hyeok Yang Mechanical Engineering Department, Stevens Institute of Technology, Hoboken, NJ, United States List of contributors xv
- 20. Fan Yang Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, NJ, United States Yin-Ting Yeh Department of Physics, The Pennsylvania State University, University Park, PA, United States Xin Zhao BTR New Material Group Co., Ltd., Shenzhen, P.R. China xvi LIST OF CONTRIBUTORS
- 21. About the editors Dibakar Datta is an assistant professor of Mechanical Engineering at the New Jersey Institute of Technology (NJIT), United States. He received his PhD from Brown University in 2015 with a major in Solid Mechanics and minors in Physics and Chemistry. His current research includes the areas of mechanics of nanomaterials, imperfections in crystalline solids, and modeling of energy storage systems. He received funding from federal agencies such as NSF. Junjun Ding is an assistant professor of Materials Science and Engineering, Inamori School of Engineering, New York State College of Ceramics at Alfred University, United States. His current research focuses on flexible electronics and advanced manufacturing, including large-scale nanomanufacturing and additive manufacturing of ceramics, polymer, and their composites. Grzegorz (Greg) Hader is a mechanical engineer at the U.S. Army Combat Capabilities Development Command Armaments Center, located at Picatinny Arsenal, NJ, United States. He graduated with his BS in Mechanical Engineering from Virginia Polytechnic Institute and State University in 2002. Areas of research include numerical modeling, nanofabrication, and characterization of NEMS and MEMS sensors, and flexible devices utilizing 1D and 2D materials. Eui-Hyeok Yang is a professor of the Mechanical Engineering Department at Stevens Institute of Technology, United States. He joined Stevens in 2006 following tenure as a senior member of the engineering staff at NASA Jet Propulsion Laboratory. He has secured more than 35 federal grants and contracts, including funding from the National Science Foundation, Air Force Office of Scientific Research, National Reconnaissance Office, US Army, and NASA. xvii
- 22. 1 Overview Eui-Hyeok Yang1 , Dibakar Datta2 , Grzegorz (Greg) Hader3 , Junjun Ding4 1 MECHANICAL ENGINEERING DEPARTMENT, STEVENS INSTITUTE OF TECHNOLOGY, HOBOKEN, NJ, UNITED STATES 2 DEPARTMENT OF MECHANICAL AND INDUSTRIAL ENGINEERING, NEWARK COLLEGE OF ENGINEERING, NEW JERSEY INSTITUTE OF TECHNOLOGY, NEWARK, NJ, UNITED STATES 3 US ARMY CCDC, ARMAMENTS CENTER, PICATINNY ARSENAL, NJ, UNITED STATES 4 KAZUO INAMORI SCHOOL OF ENGINEERING, NEW YORK STATE COLLEGE OF CERAMICS, ALFRED UNIVERSITY, ALFRED, NJ, UNITED STATES 1.1 Overview of two-dimensional materials and the scope of the book In December of 1959, the physicist, Richard P. Feynman, gave a lecture titled, “There’s Plenty of Room at the Bottom: An Invitation to Enter a New Field of Physics.” This lecture would become the advent to the scientific field of nanotechnology. Feynman’s lecture on the manipulation of atoms would eventually become reality when researchers demonstrated the precise placement of individual atoms by synthesizing graphene nanoribbons into specific patterns [1]. His radical idea to make machines at a small scale would eventually culminate in the development of microelectromechanical systems (MEMS) starting in the 1980s. Fabrication processes of microscaled electromechanical devices were based on techniques adapted from the integrated circuit (IC) industry. It was this synergy between the IC industry and the need for MEMS that would bring consumers a wealth of technology advancements in cell phones, automobiles, gaming, robotics, fitness/health trackers, airplanes, many mili- tary applications, and last but not least, drones, which would not have been possible without MEMS technology. This synergistic relationship is now being explored between two- dimensional (2D) materials and the silicon-based semiconductor industry [2]. Due to the extraordinary properties of atomically thin 2D materials, they have now made their way to the forefront of several research areas, including electronics, photonics, electrophotonics, catalysis, and energy. There have been extensive research efforts on the mechanical, thermal, optical, and electrical properties, including modeling, synthesis, and their applications. As the need for new high-performance materials continues to push toward the mantra of ligh- ter, stronger, and faster, bringing credence to the lecture by Feynman, that there is still “Plenty of Room at the Bottom,” leaves one to image what new technology lies beyond the horizon. 3 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures. DOI: https://doi.org/10.1016/B978-0-12-818475-2.00001-5 © 2020 Elsevier Inc. All rights reserved.
- 23. Over a decade has passed since the seminal work in isolating graphene by Sir Andre Geim and Sir Konstantin Novoselov, which started a revolution in the research of a new fam- ily of materials with atomic thickness and planar dimensionality. Graphene is a monolayer of carbon atoms arranged in a hexagonal lattice. Its high degree of crystallinity and outstanding electronic, mechanical, thermal, and optical properties leads to the term the new wonder material [3] and makes graphene an ideal candidate for novel high-speed (GHz THz) optoelectronic devices [4,5]. Graphene is a gapless semimetal with a linear dispersion rela- tion in the low bias transport regime. The research on graphene has opened the floodgates to a vast library of other 2D-layered materials [6], including the fabrication of heterostruc- tures, all at atomic thicknesses. Although the micromechanical exfoliation technique has been adopted for rapid material characterization and demonstration of innovative device ideas based on these 2D systems, significant advances have recently been made in large- scale homogeneous and heterogeneous growth of these materials. The emergence of these new 2D materials dramatically broadens the spectrum of properties. Unlike the zero- bandgap graphene, hexagonal-boron nitride (h-BN) is an insulator with a similar atomic structure to graphene, while monolayer transition metal dichalcogenides such as molybde- num disulfide (MoS2), molybdenum diselenide (MoSe2), tungsten disulfide (WS2), and tung- sten diselenide (WSe2) are direct bandgap semiconductors. The diverse properties of these 2D material systems make it flexible for the use of various applications. Mechanical, thermal, optical, and electrical properties of 2D materials will be further discussed in Chapters 2, 3, 4, and 5, respectively. With the constant discovery of new 2D materials and 2D heterostructures, the develop- ment of 2D materials opens up a completely new territory for both experimental studies and computational studies. Recent advances in the modeling of phenomena during the nanofab- rication and mechanics of controllable synthesis of 2D materials have paved the way for vari- ous applications. With the continuous increase in computing power and significant advancements of theoretical methods and algorithms, the modeling for physical properties of 2D materials and 2D heterostructures has shown comparable accuracy to experiments, while keeping the cost down. The advantages of computational materials databases are not limited to speed and cost as compared to experimental efforts. The computational work makes it possible for sharing and comparison of research data with reduced duplication of research efforts. The increasing volume of databases enables the application of machine- learning techniques for the discovery of new 2D materials and designing materials with tai- lored properties. Modeling topics on atomistic modeling, molecular dynamics simulation, Monte Carlo methods, and continuum modeling are covered in Chapters 6, 7, 8, and 9, respectively. To characterize the layer-dependent properties of 2D properties, it is essential to synthe- size 2D materials in a controllable manner. Other than the micromechanical exfoliation tech- nique, many strategies have been reported to synthesize monolayer or few-layer 2D materials, such as chemical vapor deposition (CVD) method, chemical exfoliation, and hydrothermal method. These methods show their advantages and disadvantages in terms of quality, production volume, and layer control, which determines the applications of these 4 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures
- 24. synthesized 2D materials. The synthesis of 2D heterostructures often requires a more com- plicated process, which integrates two or more synthesis methods for each layer of 2D mate- rials. The synthesis methods of graphene, h-BN, TMD, and 2D heterostructures are introduced in depth in Chapters 10, 11, 12, and 13, respectively. Chapter 14 discusses the characterization techniques utilized for the confirmation and analysis of natural and syn- thesized 2D materials. This chapter outlines transmission electron microscopy (TEM), Raman Spectroscopy, atomic force microscopy (AFM), including other surface and atomic characterization tools to gain insight into the 2D materials physics, chemistry, and material science. The understanding of the physical properties of 2D materials and the synthesis of 2D materials and their heterostructures make it possible to design electrical and optoelectronic devices with superb performance. The photodetector, which converts photons into electrical signals, can be redesigned with 2D materials other than the conventional semiconductors, such as silicon and indium gallium arsenide. The 2D materials and 2D heterostructures enable new photoresponse effects at much greater sensitivities and provide photodetection covering UV, visible, IR, and THz ranges. The unique mechanical properties also ensure the fascinating processing of photodetection in flexible electronics as well as bioelectronics. Detailed discussion on 2D material-based photodetectors is shown in Chapter 15, 2D Materials and Hybrid Systems for Photodetection. In addition to optoelectronics, 2D materi- als have found a wide range of applications in electronic devices such as conductors, thin- film transistors, sensors, and energy storage devices. Solution-processed 2D materials bear high potential due to the advantages of low cost and high-volume production, which is criti- cal for the fast-growing demands of printed electronics and other electronics applications. The exfoliated 2D materials are solution-processable so that the 2D materials can be easily assembled into designed layered structures on arbitrary substrates, which is important for flexible electronics. Most 2D materials can be chemically exfoliated, while more researchers are trending toward synthesis by other methods such as CVD. Therefore potential applica- tions with 2D heterostructures can be realized by solution-processed 2D materials using methods such as layer-by-layer assembly, Langmuir Blodgett assembly, spin coating, elec- trophoretic deposition, inkjet printing, and vacuum filtration. The details on solution- processed 2D materials for electronic applications are discussed in Chapter 16, Electronic Devices Based on Solution-Processed 2D Materials. Due to the extraordinary electrical properties, 2D materials have been extensively explored as additives in composites for electrodes in energy storage devices in order to increase electronic conductivity and mechanical stability and provide additional Li storage sites for lithium-based batteries. The 2D materials and 2D heterostructures are excellent can- didates as anodes and help provide high porosity, good electron mobility, lightweight, high charge capacity, high rate capability, and increased operational voltage. Many researchers have reported improvements in the performance of anodes in lithium-ion batteries and offer 2D materials as an alternative option to anodes fabricated with Li metal, which is prone to deadly dendrite formation [7]. While monolayers of most 2D materials are not ideal candi- dates for battery electrodes, van der Waals layered heterostructures offer possibilities to Chapter 1 • Overview 5
- 25. design battery electrodes for fast diffusion kinetics, high structural integrity, and excellent electron conductivity. However, there are many questions to be answered to fully understand the mechanics of 2D electrodes and how to optimize the performance for energy storage devices. Chapter 17, 2D Materials and Its Heterostructures for Energy Storage, provides a sys- tematic review on the state-of-the-art of 2D materials and their heterostructures for energy storage applications. The World Health Organization (WHO) announced a global pandemic on March 11, 2020, due to the uncontrolled outbreak of the novel coronavirus (COVID-19). The study of low dimensional materials in virology and living organisms is discussed in Chapter 18 and provides insight into how these materials are prime candidates for the cap- ture, detection, and analysis of biological systems. Despite extensive research efforts of 2D materials in the last two decades, the 2D materi- als and their heterostructures have greatly expanded their territory for more opportunities to explore. With the development of computational power and algorithms, computational modeling has grown into an important tool for the discovery of new 2D materials and predic- tion of their physical properties [8]. The increasingly large database of 2D materials and their 2D heterostructures offers endless possibilities in designing electronic, photonic, optoelec- tronic, and energy storage devices [6,8]. Chapter 19, Machine Learning in Materials Modeling—Fundamentals and the Opportunities in 2D Materials, discusses the emerging field of machine learning for 2D materials research. This book provides an overview of the synthesis, modeling, and characterization of 2D materials and their heterostructures. Applications are provided to the reader throughout the text as well as current technological breakthroughs, outlining recent scientific progress in the fast-paced research environment of 2D materials. References [1] J. Cai, et al., Atomically precise bottom-up fabrication of graphene nanoribbons, Nature 466 (2010) 470 473. Available from: https://doi.org/10.1038/nature09211 [2] D. Akinwande, et al., Graphene and two-dimensional materials for silicon technology, Nature 573 (2019) 507 518. Available from: https://doi.org/10.1038/s41586-019-1573-9 [3] A.K. Geim, Graphene: status and prospects, Science 324 (2009) 1530 1534. Available from: https://doi. org/10.1126/science.1158877 [4] F. Wang, et al., Gate-variable optical transitions in graphene, Science 320 (2008) 206 209. Available from: https://doi.org/10.1126/science.1152793 [5] F. Bonaccorso, Z. Sun, T. Hasan, A.C. Ferrari, Graphene photonics and optoelectronics, Nat. Photonics 4 (2010) 611 622. Available from: https://doi.org/10.1038/nphoton.2010.186 [6] G. Cheon, et al., Data mining for new two- and one-dimensional weakly bonded solids and lattice- commensurate heterostructures, Nano Lett. 17 (2017) 1915 1923. Available from: https://doi.org/ 10.1021/acs.nanolett.6b05229 [7] L. Li, et al., Self-heating induced healing of lithium dendrites, Science 359 (2018) 1513 1516. [8] N. Mounet, et al., Two-dimensional materials from high-throughput computational exfoliation of experi- mentally known compounds, Nat. Nanotechnol. 13 (2018) 246 252. Available from: https://doi.org/ 10.1038/s41565-017-0035-5 6 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures
- 26. 2 Mechanical properties of two- dimensional materials: atomistic modeling and future directions M.A.N. Dewapriya1,2 , R.K.N.D. Rajapakse1 , S.A. Meguid2 1 SCHOOL OF ENGINEERING SCIENCE, SIMON FRASER UNIVERSITY, BURNABY, BC, CANADA 2 MADL, MECHANICAL AND INDUSTRIAL ENGINEERING, UNIVERSITY OF TORONTO, TORONTO, ON, CANADA 2.1 Introduction Research in two-dimensional (2D) materials is receiving worldwide attention from the scientific and engineering communities due to extraordinary properties of these materials and their poten- tial to serve as the building blocks of next-generation materials [13]. Since the isolation of gra- phene from bulk graphite in 2004, several other 2D materials, such as hexagonal boron nitride (h-BN) and molybdenum disulfide (MoS2), have been developed (see Fig. 21). This newly emerging family of 2D materials offers a wide range of multiphysical properties. For example, electrical conductivity of the 2D materials varies from conducting graphene to insulating h-BN, while MoS2 is a semiconductor. The unique 2D crystal structures of these materials render distinct combinations of mechanical properties, such as high in-plane stiffness combined with FIGURE 2–1 Atomic structure of graphene, hexagonal boron nitride (h-BN) and molybdenum disulfide (MoS2). 9 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures. DOI: https://doi.org/10.1016/B978-0-12-818475-2.00002-7 © 2020 Elsevier Inc. All rights reserved.
- 27. extremely low flexural rigidity, which are promising for a wide range of novel applications [4]. For example, graphene has already demonstrated great potentials in a rich variety of engineering applications, such as flexible electronics [5], nanoelectromechanical systems (NEMS) [6], and multifunctional nanocomposites [7,8]. The mechanical properties of this class of materials, which depend on their nanostructure, play a critical role in their utility and the service life of their pro- ducts. Recent advances in nano-manufacturing offer the possibilities for manipulating the micro- structure of these atomically thin membranes to achieve desirable characteristics, and thus opening a new design paradigm of nanoscale engineering. This chapter focuses on the mechanical integrity and the fracture behavior of plane nanostructured materials such as graphene and h-BN. Three aspects of the work were accordingly examined—the first with the use of molecular dynamics (MD) to predict the mechanical and fracture behavior of graphene; the second with the atomistic interaction of a crack in close proximity to an inhomogeneity or a vacancy, and the resulting stress shielding and amplification effects at the crack tip; and the third with the fracture characteristics of atomistic grapheneh-BN heterostructures. In addition, we offer some insights into future research directions in this area pertaining to the modeling, characterization, and application of 2D materials, accounting for their topological design and potential application in machine learning in nanomaterial design. The chapter is divided into five sections. Following this introduction, Section 2.2 provides a brief overview of the current state of research. Section 2.3 highlights important aspects of MD simulations of 2D materials. Several atomistic modeling studies, related to fracture of graphene and grapheneh-BN heterostructures, are discussed in Section 2.4, which is fol- lowed by an overview of future research directions. 2.2 Current state of research Plane or 2D materials resemble a thin membrane and therefore predominantly demonstrate two fundamental deformation modes: (1) in-plane stretching and (2) out-of-plane bending. As a result, both in-plane and bending moduli are required to properly characterize the deformation of these nanostructured materials. Moreover, a set of coupling moduli can also be theoretically defined to further characterize the deformation patterns of 2D materials [9]. A recent experiment by Blees et al. [10] revealed that the bending modulus of graphene is orders of magnitude higher than the theoretically predicted value. This result poses a ques- tion on the applicability of well-established mechanics of ultrathin membranes for 2D mate- rials [11]. Several experimental studies have revealed that suspended graphene membranes demonstrate intrinsic ripples at finite temperatures (see for, e.g., Ref. [12]). These ripples could be responsible for the observed high bending modulus. In addition, such ripples could have profound effects on the in-plane modulus as well as the thermal expansion [13]. These findings suggest the possibility of designing 2D microstructures membrane that resists the inherent out-of-plane deformation leading to an improved bending rigidity. In general, 2D materials demonstrate significantly high fracture strength. For example, a pristine single crystalline graphene has a fracture strength of 130 GPa [14]. However, atomic 10 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures
- 28. imperfections such as vacancies and defects are difficult to avoid in the fabrication of gra- phene and other 2D nanostructured materials. In addition, graphene contains grains and their boundaries contain numerous defects, such as pentagons, heptagons, and dislocations. In fact, defects may be intentionally introduced into 2D materials in order to tailor their elec- tromechanical properties [15]. These defect sites could generate stress concentrations result- ing in fracture under significantly small levels of applied stresses. Indeed, low fracture toughness of graphene (B4 MPaOm) poses serious limitations in its use for structural applications [16]. Therefore identifying potential toughening mechanisms for 2D materials is critical for their widespread use in engineering application. Commonly prevailing defects such as vacancies, dislocations, and grain boundaries may be manipulated to achieve novel topologies with improved resistance to fracture. For example, proper manipulation of grain boundaries (e.g., size/orientation of nanocrystals) or the effective use of interacting microdefects could significantly improve the fracture resistance of 2D materials [1719]. Even though graphene demonstrates a purely brittle fracture, its derivative gra- phene oxide has demonstrated some plasticity [20,21]. The atomistic mechanisms associated with the plasticity of graphene oxide, however, have not been understood or characterized yet. Moreover, the applicability of the classical continuum concepts of plasticity at the atomic scale should be carefully examined due to the inherent discreteness of the matter and the quantum manifestations at this scale [22,23]. On the other hand, brittle-to-ductile transfor- mation of 2D materials such as graphene and h-BN could be achieved through topological design. Developing hybrid/hierarchical materials is one of the innovative routes of designing novel 2D materials with unique properties. Most natural materials possess hierarchy and are hybrid (e.g., bone and nacre). Similarly, the lattice structure of graphene and h-BN allows the fabrication of graphene/h-BN heterostructures with unique electronic and magnetic properties [24,25]. More importantly, the physical properties of the graphene/h-BN hetero- structures can be effectively tailored through the selection of the relative domain size of each material, which is quite beneficial for advanced applications in engineering [26,27]. On the other hand, these 2D materials can be arranged one on top of the other using the layer-by- layer assembly techniques to design novel multilayered hierarchical materials with unique and improved physical properties [28]. Multiphysical properties of these recently emerging hybrid and multilayered materials have to be thoroughly understood before integrating them into device applications. Extremely high strength to weight ratio and stiffness of graphene make it a superior ballis- tic armor candidate for aerospace and defense-related applications, as confirmed by ballistic impact tests in Refs. [14,29]. The test revealed that the specific penetration energy of multi- layered graphene is about 10 times the corresponding value of a microscopic steel sheet. Moreover, a recent experiment demonstrated that graphene-based polyvinyl alcohol contain- ing a relatively low volume fraction of graphene has the potential to reach three times the ballistic impact resistance of existing high-performance composites [30]. However, experi- ments are unable to reveal the complex atomic mechanisms of energy dissipation at the nanoscale, which can only be realized through atomistic simulations. A recant atomistic Chapter 2 • Mechanical properties of two-dimensional materials 11
- 29. simulation revealed that graphene could transform polyethylene into a high-performance ballistic material, where a single coat of graphene improves the ballistic performance of poly- ethylene over eightfolds [31]. 2.3 Molecular dynamics simulations of two-dimensional materials MD simulations are widely used to study mechanical properties of 2D materials. A compre- hensive overview of MD simulations can be found in Ref. [32]. When MD simulations are employed to study fracture characteristics of graphene, the proper selection of cutoff para- meters in interatomic potentials is critically important. However, the influence of the cutoff function on the fracture characteristics has been overlooked by several studies. This section briefly describes the influence of cutoff parameters on the computed fracture stress of gra- phene samples using Tersoff-type potential, such as reactive bond order (REBO) interatomic potential. In order to shed light on the effect of the cutoff function, we present a set of MD simula- tions of uniaxial tensile tests of several graphene samples. The simulations were conducted using large-scale atomic/molecular massively parallel simulator [33]. The adaptive intermo- lecular REBO (AIREBO) potential [34] was used for the simulations. The AIREBO potential consists of three subpotentials—the REBO [35], the Lennard-Jones, and the torsional. The REBO potential evaluates energy stored in atomic bonds. The Lennard-Jones and the tor- sional potentials include energies due to nonbonded and torsional interactions between atoms, respectively. The REBO potential expresses energy stored in a bond between atoms i and j as: EREBO ij 5 f rij VR ij 1 bijVA ij h i ; (2.1) where VR ij and VA ij are the respective repulsive and attractive potentials, bij is the bond order term, which modifies the attractive potential depending on local bonding environment, rij is the distance between atoms i and j, and f(rij) is the cutoff function, which limits the inter- atomic interactions to the nearest neighbors, and it is expressed as follows: f ðrijÞ 5 1; rij , Rð1Þ 1 2 1 1 2 cos π rij 2 Rð1Þ Rð2Þ 2 Rð1Þ ð Þ 2 4 3 5; Rð1Þ , rij , Rð2Þ 0; Rð2Þ , rij ; 8 : (2.2) where R(1) and R(2) are the two cutoff radii—1.7 and 2 Å, respectively. The values of the cutoff radii were originally selected by considering the first and the second nearest neighboring dis- tances of relevant hydrocarbons [35]. However, these cutoff radii introduce erroneous non- physical strain hardening effect in the stressstrain curve of carbon-based structures, such 12 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures
- 30. as diamond [36,37] and carbon nanotubes [36,37]. Therefore modified cutoff radii, ranging from 1.9 to 2.2 Å, have been used in the literature to eliminate this nonphysical strain hard- ening [3841]. In order to obtain an insight into the effect of the cutoff function on the fracture of an individual bond, the forcestrain curve of a bond between two carbon atoms was obtained by increasing the bond length r [37], as depicted in the inset of Fig. 22. In the case of the default cutoff radii, the values of R(1) and R(2) are 1.7 and 2 Å, respectively. In the case of the modified cutoff, the values of the two radii were set to be 2 Å. As shown in Fig. 22, a signifi- cant strain hardening of the forcestrain curve can be observed when the default values of the cutoff radii were used, whereas the strain hardening disappears when the modified cutoff radii were used. A detailed investigation on the topic is presented in Ref. [42]. Numerous studies have used a value of R(1) between 1.9 and 1.94 Å to study the fracture of graphene samples. However, our simulation results demonstrate that R(1) has a great influence on the computed fracture stress of graphene samples when the value of R(1) is below 1.96 Å. Fig. 23A compares stressstrain curves of a pristine graphene sample and a sample contain- ing a crack; several simulations were performed with different values of R(1) , where the value FIGURE 2–2 Forcestrain curves of the carboncarbon bond between atoms 1 and 2 marked on inset. The two horizontal arrows indicate the loading direction. Reprinted from K.G.S. Dilrukshi, M.A.N. Dewapriya, U.G.A. Puswewala, Size dependency and potential field influence on deriving mechanical properties of carbon nanotubes using molecular dynamics, Theor. Appl. Mech. Lett. 5 (2015) 167172. https://doi.org/10.1016/j with permission from Elsevier. Chapter 2 • Mechanical properties of two-dimensional materials 13
- 31. of R(2) was kept constant at 2 Å. The curves pertinent to pristine sample demonstrate that the influence of the cutoff is significant even when R(1) is 1.94 Å. Fig. 23B shows that the com- puted critical stress intensity factor (SIF) from MD simulations is significantly higher than the experimental value when R(1) is below 1.96 Å. This suggests that the accuracy of the fracture simulations highly depends on the proper selection of R(1) . These results confirm that, accord- ing to the AIREBO potential, fracture of carboncarbon bond occurs when the bond length is approximately 1.95 Å. When the value R(1) is less than 1.95 Å, the cutoff function impedes the bond breaking process and introduces the erroneous nonphysical strain hardening effect (see Fig. 22). 2.4 Fracture characteristics of two-dimensional materials 2.4.1 Effect of functionalization and temperature on graphene In many engineering applications the surface of graphene has to be modified by introducing various defects (e.g., vacancies and adatoms) in order to achieve certain desired functionalities. As an example, a 2D amorphous graphene membrane can be obtained by means of electron irradiationinduced vacancies, which opens new possibilities to engineer graphene-based NEMS [43]. Chemical functionalization, which involves the addition of foreign atoms or func- tional groups, could induce better interaction between graphene and a host composite matrix, leading to improved electromechanical properties [44,45]. Moreover, hydrogen functionaliza- tion creates new bandgap openings in graphene [45], and carbon adatoms significantly modi- fies their electronic and magnetic properties [4648]. Numerous studies concerning functionalized graphene have focused on its electronic and magnetic properties [4648]. On the other hand, understanding the influence of adatoms and functional groups on the mechanical properties of graphene-based systems is vitally FIGURE 2–3 Effects of the cutoff radius R(1) : (A) stressstrain curves of an armchair sheet at various cutoff radii. Length of the considered crack is 20.8 nm, and (B) variation of the computed critical value of the stress intensity factor at different values of the cutoff radius as compared with an experimental result [16]. 14 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures
- 32. important in many applications such as in graphene-based structural composites, where the adsorption of adatoms and functional groups is unavoidable [44,4952]. In graphene-based composites, interaction between graphene and the composite matrix is governed by non- bonded interactions, which is mainly van der Waals force [4951,53]. The adsorption of ada- toms could have a significant impact on the nonbonded interactions between graphene and composite matrix. MD simulation studies have revealed that hydrogen adsorption can have a significant impact on the strength of graphene and its allotropes [54,55]. On the other hand, graphene could be subjected to high temperatures (B1000K) during the synthesis and fabrication of graphene-based composite materials [56]. Several MD simula- tions have been conducted on the temperature-dependent mechanical properties of graphene [5759]. However, the behavior of functionalized graphene sheets could be significantly different from that of pristine sheets, since functional groups (or adatoms) transform the hybridization of carbon in graphene from sp2 to sp3 . This section briefly describes the development of an analyti- cal model, based on Bailey durability criterion and the Arrhenius equation, to study temperature-dependent fracture strength of functionalized graphene along various chiral direc- tions [60]. The predicted fracture strength depends on temperature, strain rate, and hydrogen functionalization. Baileys durability criterion [61], given in Eq. (2.3), provides a framework for calculating the lifetime of materials at various temperatures [62]. Let t be time and T be temperature: ðtf 0 dt τ T; t ð Þ 5 1; (2.3) where tf is the time taken to fracture and τ(T,t) is the time and temperature-dependent dura- bility function, which is generally obtained from experiments [62]. However, in the absence of experimental data on the durability of carboncarbon bonds in graphene, the Arrhenius equation is a good approximation for the durability function [58]. The Arrhenius equation [63] expresses the temperature-dependent rate of a chemical reac- tion (k) as k5 A 3 exp[ΔE/(kBT)], where A is a constant that depends on the type of chemical bonding, ΔE is the activation energy barrier, and kB is the Boltzmann constant. When a mechan- ical force F is applied to a molecule, the activation energy barrier is reduced by an amount of FΔx, where Δx is the change in the atomic coordinates due to F [64]. A durability function for carboncarbon bonds in graphene can then be defined in the form of Arrhenius equation as: τ T; t ð Þ 5 τ0 n exp U0 2 vγσ t ð Þ βkBT ; (2.4) where τ0 is the vibration period of the atoms, n is the number of bonds in the sheet, U0 is the interatomic bond dissociation energy (4.93 eV for a carboncarbon bond [65]), v is the repre- sentative volume of a carbon atom in graphene, which is approximately 8.6 Å3 , and γ is a directional constant that takes into account the different bond orientation along different chiral directions. According to Ref. [38], the strength (S) along a chiral direction, at an angle θ mea- sured from the armchair direction, can be approximated as Sθ 5 Sac/cos θ, where Sac is the Chapter 2 • Mechanical properties of two-dimensional materials 15
- 33. strength along the armchair direction. The chiral angle between armchair and zigzag directions is π/6. According to the proposed strength relation in Ref. [38], the strength ratio of pristine graphene along the zigzag and armchair directions can be obtained as Sac/Szz 5 cos(π/6) 5 0.87. An independent MD simulations have revealed that the ratio Sac/Szz is 0.85 [60]. Therefore the chirality-dependent strength can be introduced into the proposed atomistic model by setting γ 5 cosθ. The stress at time t, σ(t), is expressed in terms of the strain rate _ ε ð Þ as follows: σ t ð Þ 5 a _ εt ð Þ 1 b _ εt ð Þ2 ; (2.5) where a and b are the second- and the third-order elastic moduli, respectively. The values of a and b were obtained from regression analysis of the stressstrain curves given by MD simulations at 300K. The regression analysis determined a and b to be 1.11 and 23.20 TPa for armchair graphene. The constant β describes the reduction of activation energy barrier due to the presence of hydrogen adatoms, which is defined in terms of adatom concentration (α) as being: β 5 1; α 5 0 0:023α 1 1:11; α . 0 : (2.6) The governing equation of the system can be obtained by substituting Eq. (2.4) into Eq. (2.3), which yields: ðtF 0 exp γσ t ð Þ 2 U0 βkBT dt 5 τ0 n : (2.7) Then Eq. (2.7) is solved for the time taken to fracture (tF), and the solution can be expressed as: tF 5 erf21 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b=π q 2λτ0 _ ε ð Þ=n exp λ2 U0=γ 1 a2 =4b 2 erf χ ð Þ n o 1 χ ffiffiffiffiffiffiffiffiffi 2 b p λ_ ε ; (2.8) where erf is the error function [66], with λ 5 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ=βkBT p and χ 5 λa= ffiffiffiffiffiffiffiffiffiffiffi 2 4b p . Once tF is obtained from Eq. (2.8), the fracture strength, σ(tF), can be obtained from Eq. (2.5). Fig. 24A and B compares the fracture strength given by the analytical model with MD simulations of armchair and zigzag graphene at various adatom concentrations and tempera- tures. The figure shows that the results obtained from the analytical model agree quite well with the MD simulation predictions. After the analytical model has been verified by MD simulations, the model can be used to predict the strength of hydrogen functionalized graphene under various temperatures, strain rates, and chirality. Fig. 25A shows that highly functionalized graphene completely loses the strength when it is subjected to higher temperatures. This strength loss could be an indication of sublimation of graphene. A Monte Carlo simulation study [67] has revealed that the melting 16 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures
- 34. temperature of graphene is B4900K. However, according to Fig. 25A, melting of graphene highly depends on the hydrogen functionalization suggesting that highly functionalized graphene is not suitable for high temperature applications. Fig. 25B shows that the reduction of strength due to the functionalization is less independent on the loading direction. 2.4.2 Out-of-plane deformation of crack surfaces 2D materials such as graphene and h-BN are only a single-atom thick. Therefore even a small out-of-plane perturbation could disturb its planar geometry by creating ripples and wrinkles, thus influencing its fracture behavior [68,69]. It has also been demonstrated that FIGURE 2–4 Temperature and hydrogen adatom concentration-dependent fracture strength of (A) armchair and (B) zigzag graphene. The dashed lines indicate the values given by the proposed analytical model. Reprinted from M.A.N. Dewapriya, R.K.N.D. Rajapakse, N. Nigam, Influence of hydrogen functionalization on the fracture strength of graphene and the interfacial properties of graphenepolymer nanocomposite, Carbon 93 (2015) 830842. doi:10.1016/j.carbon.2015.05.101 with permission from Elsevier. FIGURE 2–5 The variation of the strength of hydrogen functionalized graphene with (A) temperature and (B) the loading direction (i.e., chirality). Results for armchair graphene are shown in (A) and the simulation temperature was maintained at 300K in (B). Reprinted from M.A.N. Dewapriya, R.K.N.D. Rajapakse, N. Nigam, Influence of hydrogen functionalization on the fracture strength of graphene and the interfacial properties of graphenepolymer nanocomposite, Carbon 93 (2015) 830842. doi:10.1016/j.carbon.2015.05.101 with permission from Elsevier. Chapter 2 • Mechanical properties of two-dimensional materials 17
- 35. deliberately induced geometrical distortions can be effectively used to tailor the mechanical properties of graphene [10,17,41,7072]. Three-dimensional (3D) MD simulations of the nanoscale uniaxial tensile test of a gra- phene sheet with a central crack demonstrate that the crack surfaces experience significant out-of-plane deformation during the uniaxial tensile test. This phenomenon has been observed in recent MD simulations of the nanoscale uniaxial tensile test [73,74]. The out-of-plane defor- mation of the crack surfaces increases with the applied uniaxial strain (see Fig. 26). The max- imum out-of-plane deformation occurs at the onset of crack propagation, and its magnitude increases with the crack length. Both armchair and zigzag crack surfaces show approximately similar out-of-plane deformation during the uniaxial tensile test, and the out-of-plane defor- mation could be attributes to the poor transverse stiffness of the graphene sheet as well as the biaxiality of the stress field at the crack tips [75]. Griffith’s thermodynamic failure criterion has been widely used to characterize the frac- ture properties of nanoscale materials, including graphene [17,40,70,76,77]. When applying Griffith’s criterion to 2D materials, a planar configuration of the cracked sample is typically assumed. A recent molecular MD study [76] revealed that Griffith’s criterion is applicable to graphene when the crack length is greater than 100 Å. Below this limit, the criterion over predicts the fracture stress. This observation was attributed to the presence of local effects at the crack tip. In addition, the out-of-plane deformation of the crack surfaces (see Fig. 26) also has a significant influence on the fracture characteristics of graphene. According to Griffith’s thermodynamic criterion of crack propagation [78], the fracture stress σf can be expressed in terms of Young’s modulus (E), the surface energy (γ), and the initial crack length (2a) such that: FIGURE 2–6 Out-of-plane deformation of zigzag crack surfaces at two strain εyy levels. The crack length is 50.8 Å. Reprinted from M.A.N. Dewapriya, S.A. Meguid, Atomistic modeling of out-of-plane deformation of a propagating Griffith crack in graphene, Acta Mech. 228 (2017) 30633075. doi:10.1007/s00707-017-1883-7 with permission from Springer Nature. 18 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures
- 36. σf 5 ffiffiffiffiffiffiffiffi 2γE πa r : (2.9) By rearranging the terms of Eq. (2.9), an expression for the critical SIF (KIC), which is gen- erally expressed as KIC 5 σf ffiffiffiffiffiffi πa p , can be obtained in terms of γ and E as: KIC 5 ffiffiffiffiffiffiffiffi 2γE p (2.10) for plane stress condition. If breaking of an individual carboncarbon bond results in a crack advance of Δa, the surface energy γ can be expressed as: 2γ 5 PEbond Δat ; (2.11) where PEbond is the equilibrated potential energy of a carboncarbon bond of graphene, which is 4.916 eV according to AIREBO potential, and t is the thickness of graphene. The cal- culated values of 2γ for the zigzag and the armchair crack configurations are 9.58 and 11.05 J/m2 , respectively. These values are in agreement with the ones calculated using REBO potential in Refs. [16,65]. Substituting the computed values of E and γ into Eq. (2.10), KIC of the zigzag and the armchair crack configurations were calculated to be 3.1 and 3.32 MPaOm, respectively. Fig. 27 shows that KIC obtained using Eq. (2.10) is considerably higher than the critical KI given by the expression KI 5 σf ffiffiffiffiffiffi πa p , when the out-of-plane deformations are allowed to take place by relaxing the boundary conditions (i.e., 3D). Reference [76] suggests that Griffith’s criterion of fracture is applicable to graphene with B15% accuracy when the crack length is around 100 Å. Fig. 27 clearly demonstrates that when the out-of-plane deformations of the crack surfaces are restrained, the critical value of KI approaches the Griffith’s value [Eq. (2.10)] at a significantly smaller crack length (B60 Å). These results FIGURE 2–7 Change in the critical value of KI with crack length as compared with Griffith’s crack for (A) armchair and (B) zigzag crack configurations. Reprinted from M.A.N. Dewapriya, S.A. Meguid, Atomistic modeling of out-of- plane deformation of a propagating Griffith crack in graphene, Acta Mech. 228 (2017) 30633075. doi:10.1007/ s00707-017-1883-7 with permission from Springer Nature. Chapter 2 • Mechanical properties of two-dimensional materials 19
- 37. reveal that the out-of-plane deformation of graphene significantly reduces the fracture stress below the stress predicted using Griffith’s energy balance approach. 2.4.3 Crackdefect interactions In continuum fracture mechanics, it has been well established that the interaction between crack and a microdefect in close proximity plays an important role in the overall failure mechanism of quasibrittle materials [7985]. It has also been demonstrated that the crack tip stress field in a linear elastic continuum can be controlled by strategically placing a microdefect near the crack tip [86,87]. Recent nanoindentation tests of graphene containing vacancies have revealed that the catastrophic failure of graphene can be transformed into a local failure by controlling its defect concentration [88]. Most of the recent efforts have been focused on enhancing the fracture strength of gra- phene by introducing topological defects such as pentagonheptagon [41,72,73,89,90] and grain boundaries [70,91]. Studies on the complex stress state surrounding crackdefect inter- actions could provide new insights into improving fracture resistance of 2D materials. In addition, advanced continuum-based design tools for characterizing crackdefect interaction [7987] have not been thoroughly tested at the atomic scale, which is critically important con- sidering the limited applicability of continuum concepts at the nanoscale [22,23,58,60,76,92,93]. If applicable, the continuum tools such as design envelope to ascertain crack tip stress shielding, and amplification zones due to the crackdefect interactions [86,87] can be very useful for nano- scale design of 2D materials [2]. Fig. 28 shows a typical MD simulation sample of graphene containing an atomic vacancy interacting with an edge crack. The origins of the two rectangular coordinate systems xy and FIGURE 2–8 Typical MD simulation of graphene containing an edge crack interacting with an atomic vacancy. MD; Molecular dynamics. Reprinted from M.A.N. Dewapriya, S.A. Meguid, Tailoring fracture strength of graphene, Comput. Mater. Sci. 141 (2018) 114121. doi:10.1016/j.commatsci.2017.09.005 with permission from Elsevier. 20 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures
- 38. x0y0 are taken at the tip of the edge crack and at the center of the vacancy, respectively. The orientation angle of the vacancy is φ, and the distance between the tip of the crack and the center of the vacancy is taken to be r. The inclination angle between the x-axis and the line joining the tip of the crack and the center of the vacancy is θ. The value of 2c is selected to be 3.6 nm (see Fig. 28). The stress distribution of individual carbon atoms at the crack tip can be very informative in characterizing the fracture behavior of graphene. In order to obtain the time-averaged stress of atoms at the incipient crack propagation, two sequential MD simulations can be conducted as outlined in Ref. [75]. Fig. 29A and B shows the stress distributions at the armchair and zig- zag crack tips, respectively. These stress distributions resemble the ones predicted by the con- tinuum linear elastic fracture mechanics [76]. The peak stress at the tip of the armchair crack at the incipient crack propagation is 17% higher than the corresponding stress of the zigzag crack. This is due to the high far-field (or applied) stress σ0 level of the armchair crack configu- ration and, more importantly, the different bond arrangements at the crack tips. In contrast to the isolated bond perpendicular to the crack at the zigzag crack tip (see inset of Fig. 29A), the two inclined bonds at the armchair crack tip (inset of Fig. 29B) accommodate part of the applied tensile strain by adjusting the bond angles, which allows the atom at the crack tip to carry a higher strain prior crack growth leading to a higher atomic stress. According to linear elastic fracture mechanics, the critical SIF of a single-edge cracked sample under mode I loading KIC can be defined as follows [94]: KIC 5 1:12σf ffiffiffiffiffiffi πa p (2.12) where a is the initial crack length and σf is the fracture stress. The computed KIC for arm- chair and zigzag cracks in Ref. [18] are 4.04 and 3.97 MPaOm, respectively, which are in excellent agreement with the experimentally measured value 4 MPaOm [16]. Earlier, Gong and Meguid studied the interaction between a semiinfinite crack and an elliptical vacancy located near its tip (see Fig. 28) under mode I loading [86]. In the absence of the vacancy, the singular stress field near the crack tip can be described by using the corresponding SIF KI 5 1:12σ0 ffiffiffiffiffiffi πa p . However, the presence of the elliptical vacancy in close proximity to the crack tip influences the crack tip stress field and leads to a modified SIF, we will call Kðc2vÞ I . When a collinear elliptical vacancy is located ahead of the crack, that is, θ 5 0 and φ 5 0, the solution for the normalized SIF under mode I loading Kðc2vÞ I =KI can be explicitly expressed up to the order (c/r)4 as follows [86]: KðcvÞ I KI 5 1 1 1 4 2 1 1 β2 c r 2 1 1 128 2 23 1 46β2 1 12β3 2 49β4 c r 4 1 ? (2.13) where β is b/c. For the case of a circular vacancy (i.e., β 5 1), Eq. (2.13) reduces to: Kðc2vÞ I KI 5 1 1 1 2 c r 2 1 1 4 c r 4 1 ? (2.14) Chapter 2 • Mechanical properties of two-dimensional materials 21
- 39. Considering the leading order solution up to order (c/r)2 , a general solution for any com- bination of r, θ, and φ can be given as: Kðc2vÞ I KI 5 1 1 c 8r 2 1 2 β2 G 1 c 2r 2 1 1 β2 C (2.15) FIGURE 2–9 The effect of atomic vacancies on the crack tip stress field. (A) and (B) show averaged stress σyy distributions at the tips of zigzag and armchair cracks at the incipient crack propagation, respectively. (CG) show variation of the normalized crack tip stress σðc2vÞ tip =σc tip with r, θ, and φ. (C), (D), and (E) are for the collinear (θ 5 φ 5 0), oriented (r is fixed and φ 5 0), and oblique (r and θ are fixed) nanocracks, respectively. (F) and (G) are for the collinear and oriented circular vacancies, respectively. Insets in (CG) show the stress σyy distribution at an armchair crack tip due to an applied tensile strain εyy of 1% in the presence of an atomic vacancy at the specified location. Reprinted from M.A.N. Dewapriya, S.A. Meguid, Tailoring fracture strength of graphene, Comput. Mater. Sci. 141 (2018) 114121. doi:10.1016/j.commatsci.2017.09.005 with permission from Elsevier. 22 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures
- 40. where C and G are explicitly expressed as follows: C 5 cos 3θ 2 cos θ 2 (2.16a) G 5 2cos 2ϕ 1 θ ð Þ 1 4cos 2ϕ 2 θ ð Þ 1 8cos 2ϕ 2 2θ ð Þ 2 6cos 2ϕ 2 3θ ð Þ 2 8cos 2ϕ 2 4θ ð Þ 2 3cos 3θ ð Þ 1 3cos θ ð Þ (2.16b) In order to characterize the crackvacancy interaction, normalized crack tip stress σðc2vÞ tip =σc tip can be employed, where σðc2vÞ tip is the crack tip stress along the y direction in the presence of an interacting vacancy, and σc tip is the crack tip stress in the absence of any interacting vacancy. The normalized crack tip stress σðc2vÞ tip =σc tip was computed at an applied tensile strain εyy level of 1% for various arrangements of the interacting vacancies. The values of σc tip for the armchair and zigzag cracks are 63.9 and 56 GPa, respectively. Fig. 29C to G demonstrates that the presence of vacancies greatly influences the stress field of zigzag crack compared to that of the armchair crack, as a result of their underlying crystal structures at the crack tips. It can be seen in Fig. 29C and F that the collinear (i.e., θ 5 φ 5 0) nanocracks and circular vacancies result in an increase in the crack tip stress field (known as stress amplification effect, i.e., σðc2vÞ tip =σc tip . 1). The oriented vacancies (see Fig. 29D and G), with the orientation angle θ . 60 degrees, result in a decrease in the crack tip stress field (known as stress shielding effect, i.e., σðc2vÞ tip =σc tip , 1). More impor- tantly, Fig. 29C to G shows that the continuum-based analytical solutions given in Eqs. (2.14) and (2.15) are able to accurately capture the trends of the crack tip stress fields obtained from the atomistic simulations. Here, Kðc2vÞ I =KI given by the analytical solutions was compared with σðc2vÞ tip =σc tip obtained from the atomistic simulations. The normalized crack tip stress σðc2vÞ tip =σc tip is a comparable quantity to the corresponding normalized SIFs [94]. However, the continuum expressions are unable to predict the influence of the under- lying crystal structures (i.e., armchair and zigzag) on the crack tip stress field, which sets a limit on developing a unified continuum fracture mechanics framework at the atomic scale. 2.4.4 Hybrid two-dimensional materials Besides having a similar lattice structure of graphene, h-BN possesses electromechanical properties that are comparable to those of graphene [95,96]. In contrast to the zero- bandgap semimetal nature of graphene, h-BN is a finite-bandgap semiconductor [97,98]. The similarity of the lattice structures of graphene and h-BN allows the construction of grapheneh-BN heterostructures with unique electronic and magnetic properties [24,25]. For these advanced applications, a clear understanding of the mechanical behavior of these 2D heterostructures is vital. Especially, the fracture characteristics of such a hybrid Chapter 2 • Mechanical properties of two-dimensional materials 23
- 41. structure are critically important, because both graphene and h-BN have relatively low frac- ture toughness [99]. Several MD studies have focused on the stability, fracture, and thermal properties of grapheneh-BN heterostructures. For example, MD study [100] revealed that interfacial defects can have a significant influence on the structural configuration and thermal conductance of these nanostructures. In addition, the presence of atomic defects in hybrid grapheneh-BN sheets results in significantly reduced failure strength and Young’s modulus [101]. However, cracks in a grapheneh-BN sheet have a much lower effect on Young’s modulus, when compared to the effect on the failure strength [102]. The edge configuration of graphene and h-BN, that is, armchair or zigzag, has a signifi- cant influence on their mechanics and interfacial properties [103,104]. For example, the tensile strength of grapheneh-BN sheets, with perfect armchair or zigzag interfaces, is approximately similar to that of pristine graphene [105]. However, the tensile strength of misorientated interfaces highly depends on the mismatch angle between graphene and h-BN domains [106]. Fig. 210A shows the simulated graphene sample containing a circular h-BN inclu- sion with a diameter of 10 nm. The h-BN inclusion does not generate a significant eigenstrain in the sample due to the fact that the lengths of both CC and BN bonds are 1.44 Å according to the Tersoff potential [99]. Moreover, the stress distribution within the inclusion is constant (see Fig. 210C). This observation agrees with Eshelby theory of the ellipsoidal inclusion problem [108,109], which states that a uniformly applied far-field stress induces a constant stress state within the inclusion. A complex stress state is observed at the grapheneh-BN interface, where the atomic stress ranges from 0 to 35 GPa due to an externally applied far-field stress of 20 GPa. This complex stress distribution is attributed to (1) heterogeneous atomic bonds at the interface and (2) the change of chirality along the grapheneh-BN interface. The inter- atomic bonds within the h-BN inclusion and the surrounding graphene sheet are BN and CC, respectively. However, atoms at the h-BNgraphene interface form four types of atomic bonds: BC, NC, BN, and CC. This highly heterogeneous bond arrangement at the interface contributes to the observed complex stress state. In addi- tion, chirality of the interface gradually changes from armchair to zigzag when the angle β (see Fig. 210C) increases from 0 to π/6 [60]. The chirality further changes gradually back to zigzag when β further increases from π/6 to π/3. This change in the underlying crystal structure along the interface also results in a complex stress state at the interface. Fig. 210D shows that the uniform stress field within the inclusion is approximately 17 GPa. In addition, a stress concentration of approximately 1.2 can be observed in graphene at the interface due to the relatively low elastic modulus of h-BN [107]. 24 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures
- 42. 2.5 Future directions 2.5.1 Topological design of two-dimensional materials The idea of creating bulk materials by manipulating nanometer-sized microstructures and defects has been illustrated for materials such as metals and ceramics [110]. Taking this idea further, the assembly of bulk materials using nanometer-sized crystallites has been demon- strated in Ref. [111]. A microstructurally heterogeneous nanostructured material can be cre- ated by using nanometer-sized building blocks (crystallites) that have different atomic FIGURE 2–10 Stress field around a circular hexagonal boron nitride (h-BN) inclusion in graphene: (A) the simulated sample, where the inset demonstrates the selected origin of the Cartesian coordinate system. (B) and (C) show the stress σyy fields of the graphene sheet and the h-BN inclusion due to an applied tensile strain εyy of 2%. (D) Variation of the atomic stress εyy along the x-axis. Reprinted from M.A.N. Dewapriya, S.A. Meguid, R.K.N.D. Rajapakse, Atomistic modelling of crack-inclusion interaction in graphene, Eng. Fract. Mech. 195 (2018) 92103. doi:10.1016/j.engfracmech.2018.04.003 with permission from Elsevier. Chapter 2 • Mechanical properties of two-dimensional materials 25
- 43. structure, different crystallographic orientation, and/or different chemical composition as illustrated in Fig. 211. The concept of nanostructured materials directly fits with 2D materials such as graphene because their basic building block consists of hexagonally packed atomic layers of carbon and/or other atoms. Although pristine graphene has the hexagonal atomic structure, defects are difficult to avoid during fabrication [15]. These defects are typically pentagons, hepta- gons, dislocations, and grain boundaries (series of pentagons and heptagons) that produce out-of-plane displacements and alter the 2D material properties [112]. Recent experimental and modeling studies demonstrate that these topological defects could either enhance or weaken the properties of 2D materials [113]. Although attempts are taking place to under- stand the role of defects in 2D materials and their influence on properties, our overall under- standing of their effects is still in its infancy. Like in the case of bulk materials with nanometer-sized microstructure, 2D materials present an exciting opportunity for tailoring their multiphysical properties of their single- and/or multilayered sheets through the topo- logical design of their nanometer-sized microstructures. 2.5.2 Piezoelectricity of two-dimensional materials Superior multiphysical properties of 2D materials could serve as the foundation for creating next-generation smart composites. Some 2D materials such as h-BN and MoS2, which are nonpiezoelectric in their bulk form, display piezoelectric behavior when their thickness is reduced to a one atom thick [114,115]. In addition to hierarchical optimization, the piezo- electric properties of individual 2D membranes could be manipulated through topological design. For instance, piezoelectricity can be engineered in nonpiezoelectric materials (e.g., graphene) through the selective adsorption of atoms or by introducing atomic vacancies FIGURE 2–11 Material containing nanometer-sized microstructure. 26 Synthesis, Modelling and Characterization of 2D Materials and their Heterostructures
- 44. [116]. Even though the piezoelectric coefficients of 2D materials are lower than the high- performance bulk piezoelectric materials such as lead zirconate titanate the high strength, stiffness, and flexibility of the 2D materials have unique advantages over the conventional bulk materials, when it comes to device integration. For example, superior mechanical properties along with the intrinsic piezoelectricity of h- BN have attracted significant attention of the research community. The highly stretchable nature of h-BN allows us to modulate its polarization through elastic strain, which opens a new class of strain-engineered piezoelectric materials. Moreover, h-BN has the simplest crys- tal structure among the piezoelectric materials. It has also been demonstrated that the appli- cation of a strain gradient through curvature result in polarization in bilayer h-BN (i.e., flexoelectricity) [117]. In the case of multilayered h-BN assemblies, the piezoelectric proper- ties significantly depend on the number of layers in the stack [118] and the type of load applied. A similar phenomenon has been observed for the case of multilayered MoS2 [119]. Moreover, piezoelectric properties of 2D materials significantly depend on the crystal struc- ture and its orientation. For example, Ref. [120] demonstrated that, under the same electric field, armchair and zigzag h-BN sheets experience significantly different deformation pat- terns. This observation confirms that piezoelectricity of 2D materials can be engineered through topological design. 2.5.3 Application of machine learning methods The highly complex process of the bottom-up design of 2D materialbased systems at the nanoscale is often limited by the available computational power. For example, the first prin- ciples computational methods (e.g., density functional theory) are impractical to employ in studying crack propagation of 2D materials due to the extremely high computational cost associated with them. In recent years, the field of deep learning has demonstrated applica- tions in various disciplines of engineering [121]. Deep learning is a subset of machine learn- ing, which structures algorithms in layers to create deep neural networks that can learn from a set of examples and make intelligent predictions [122]. Combination of atomistic modeling techniques with deep learning could significantly reduce the computational burden associ- ated with atomistic simulations. For example, computationally expensive calculations of interatomic potential energy could be replaced with properly trained neural networks at a fraction of the initial computational cost [123]. It has been recently demonstrated that deep learning techniques can be used to efficiently solve numerical problems in continuum mechanics (e.g., finite elements) [124]. In addition, machine learning has been used to pre- dict dynamic fracture growth in brittle material [125,126] and to characterize fracture stress of defective graphene samples [127]. These recent studies suggest that the application of machine learning techniques could significantly revolutionize computer-aided design of 2D materials. Chapter 2 • Mechanical properties of two-dimensional materials 27
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- 51. That Walker, who regardless of his Pace, Turns oft' to pose upon the Damsel's Face From Side to Side by thrusting Elbows tost, Shall strike his aking Breast against the Post; Or Water, dash'd from fishy Stalls, shall stain His hapless Coat with Spirts of Scaly Rain. But if unwarily he chance to stray, Where twirling Turnstiles intercept the Way, The thwarting Passenger shall force them round, And beat the Wretch half breathless to the Ground. Let constant Vigilance thy Footsteps guide, And wary circumspection guard thy Side; Then shalt thou walk unharm'd the dang'rous Night, Nor need th' officious Link-Boy's smoaky Light. Thou never wilt attempt to cross the Road, Where Alehouse Benches rest the Porter's load, Grievous to heedless Shins; No Barrow's Wheel, That bruises oft the Truant School Boy's Heel, Behind thee rolling, with insidious Pace, Shall mark thy Stocking with a miry Trace. Let not thy vent'rous Steps approach too nigh, Where gaping wide, low steepy Cellars lie; Should thy Shoe wrench aside, down, down you fall And overturn the scolding Huckster's Stall. The scolding Huckster shall not o'er thee moan, But Pence exact for Nuts and Pears o'er thrown. . . . . . . . . . Where the nail'd Hoop defends the painted Stall Brush not thy sweeping Skirt too near the Wall; Thy heedless Sleeve will drink the colour'd Oil, And Spot indelible thy Pocket soil.
- 52. CHAPTER XXXVI. CARRIAGES, ETC. Smithfield — Horse courses — Waggons — Stage coaches: travelling in them described — Bad roads — Posting — Hackney coaches: their Fares — Hackney coachmen — State coaches — Other carriages — Suburban drives — A Mechanical coach — Mourning coaches — Harness — Sedan chairs — Chairmen. Among the many places swept away, and yet which many of us well remember, is Smithfield, where both cattle and horses were sold; and Ward gives a very amusing account of the horse sales there. 'From thence we proceeded to the Rails, where Country Carters stood Arm'd with their Long Whips, to keep their Teams (upon Sale in a due Decorum,) who were drawn up into the most sightly order with their fore feet Mounted on a Dunghill, and their Heads dress'd up to as much advantage as an Inns of Court Sempstress, or the Mistress of a Boarding School: Some with their Manes Frizzled up, to make 'em appear high Wither'd, that they look'd as Fierce as one of Hungess's Wild Boars. Others with their Manes Plaited, as if they had been ridden by the Nightmare: And the fellows that attended 'em made as uncouth Figures as the Monsters in the Tempest; amongst these Cattel, here and there, was the Conductor of a Dung Cart, in his Dirty Surplice, wrangling about the Price of a Beast, as a wary Purchaser; and that he ought not to be deceived in the Goodness of the Creature, he must see him stand three fair Pulls at a Post, to which the Poor Jade is ty'd, that he may exert his Strength, and shew the Clown his excellencies; for which he strokes him on the Head, or claps him on the Buttocks, to recompence his Labour.
- 53. 'We went a little further, and there we saw a parcel of Ragged Rapscallions, mounted upon Scrubbed Tits, scowring about the Rounds, some Trotting, some Galloping, some Pacing, and others Stumbling. 'Pray friend, said I, what are those Eagle Look'd Fellows in their Narrow Brimm'd White Beavers, Jockeys Coats, a Spur in one Heel, and Bended Sticks in their Hands, that are so busily peeping into every Horses Mouth?... Those Blades, says my friend, are a Subtle Sort of Smithfield Foxes, called Horse Coursers, who Swear every Morning by the Bridle, that they will not, from any Man, suffer a Knavish trick, or ever do an Honest one. They are a sort of English Jews, that never deal with any Man but they Cheat him; and have a rare Faculty of Swearing a man out of his Senses, Lying him out of his Reason, and Cozening him out of his Money; If they have a Horse to sell that is Stone Blind, they'll call a Hundred Gods to Witness he can see as well as you can. If he be downright Lame, they will use all the Asseverations that the Devil can assist 'em with, that it is nothing but a Spring Halt; and if he be Twenty Years old, they'll Swear he comes but Seven next Grass, if they find the Buyer has not Judgment enough to discover the Contrary.' This horse market was of importance to the metropolis, which was supplied from the country fairs, from which the horses came up in droves. 'A Set of Geldings and Mares, just from a Journey to be sold Cheap.' So many were wanted for riding, carriages, and draught purposes. Horse-stealing was a crime so extremely prevalent, that is difficult to take up a paper that does not contain an advertisement respecting a lost or stolen horse. Some of the inland traffic was still done by means of packhorses. 'These are to give Notice to all Gentlemen or others that have occasion to send Goods, or travel from London to Exeter or Plymouth, or from Exeter and Plymouth, or any parts of Cornwall or Devonshire to London; that they may be accommodated for Expedition by Pack Horse Carriage, who set out from the Cross Keys Inn in Wood Street London every Saturday, and from the Mermaid Inn in Exon every Monday. Perform'd, if God permit, by Ebenezer
- 54. Brookes.' But there were also waggons, which, by the divine permission, started for every town of note in England. Stage coaches ran to most of the towns; and we may judge of the time they took over their journeys, Gloucester, 82 miles, in one day, and Hereford, 134 miles, in one day and a half. Their fares may be somewhat approximately guessed at: Bath, 16s.; Bristol, 15s. to 18s.; and Gosport, 9s. Steele gives an amusing description in the Spectator (No. 132) of stage-coach travelling: how the captain was subdued by the good plain sense of Ephraim the Quaker. 'We can not help it, Friend, I say; if thou wilt, we must hear thee.... To speak indiscreetly what we are obliged to hear, by being hasped up with thee in this publick Vehicle is in some degree assaulting on the high road.' The captain took the rebuke in good part, and thorough good- fellowship prevailed. 'Faith, Friend, I thank thee: I should have been a little impertinent if thou hadst not reprimanded me.' In 'A Step to the Bath' we get an insight into stage-coach travelling. 'Enquiring of the Tapster what Company I was like to have, he said more he believ'd than I desir'd; for there was four Places taken just after I went, and three of the Passengers were in the House, and to Lye there that Night; the other was for a Merchant of Bristol. Then asking what those in the House were, he told me two Gentlewomen and their Maid Servant, who were just going to Supper. Whereupon I bid him go and give my Service to 'em, and tell 'em I was to Travel with 'em to morrow, and should take it as a great favour, if they would please to Honour me so far, as to admit me of their Company, for I was alone. The Fellow brought word they desir'd me to walk in, and they should be very glad of mine.... Supper being ended, they call'd for a Bill, which was presently brought; out I lugg'd and was going to Discharge, but they begg'd my Pardon, and would by no means suffer me; telling me that I must submit to the Rule that is generally observ'd in Travelling, for the Major of either Sex to Treat the Minor.' They breakfasted at Colebrook, dined at Reading, and then drained the merchant's bottle of 'Right Nants'; after which one of the ladies told a story. They stopped at Theale to taste Old Mother
- 55. Cleanly's bottled ale and plum cake; then the merchant told a story; and the day's journey terminated at Newbury. There they supped, and grumbled loudly at the bill. 'For a brace of Midling Trouts they charged us but a Leash of Crowns, Six Shillings for a Shoulder of Mutton and a Plate of Gerkins, Three and Sixpence for Six Rowles, and three Nipperkins of Belch; and two Shillings more for Whip in drinking our Healths. Their Wine indeed was good, so was their price; and in a Bill of two pound four Shillings, they made a mistake but of Nine; I ask'd what Countrey-man my Landlord was? answer was made, Full North; and Faith 'twas very Evident, for he had put the Yorkshire most damnably upon us.' Next morning one of the ladies presented them with a pot of chocolate of her own preparing; they refilled the merchant's bottle, and started, beguiling the way with stories. Came to Marlborough, where the road was so bad that the brandy bottle was broken; and there they breakfasted. They seem to have dined at Calne or Chippenham, complaining bitterly of the roads, the last portion of which was got over at the rate of two miles in three hours. Here they stopped at a famous house, where 'there was more Coaches and Waggons drawn up before her Gate, than Hacks in Palace Yard, during the Session of Parliament, or Term Time. All her Entertainment is Loins of Mutton or Rabbits; and she makes more Broth in a day than all the Chop Houses in Castle Alley in a Week.' 'Having Din'd, we proceeded on our Journey, but with a great deal of difficulty; for the Road was so Rocky, Unlevel, and Narrow in some places, that I am persuaded the Alps are to be passed with less Danger,' and they finally reached Bath that evening. The roads were bad almost everywhere, and no one travelled more than they could help. The coaches were heavy and strong, to stand the fearful wear and tear; but, to the passengers, a journey was simply the time spent in torture. Even in London the stones jolted terribly. Says Ward, 'When our Stratford Tub by the assistance of its Carrionly Tits of different Colours, had outrun the Smoothness of the Road, and enter'd upon London Stones, with as frightful a rumbling as an empty Hay Cart, our Leathern Conveniency being
- 56. bound in the Braces to its good Behaviour, had no more Sway than a Funeral Herse, or a Country Waggon; That we were jumbled about like so many Pease in a Child's Rattle, running at every Kennel Jolt a great hazard of a Dislocation: This we endured till we were brought within White Chappel Bars, where we lighted from our stubborn Caravan with our Elbows and Shoulders as Black and Blew as a Rural Man that has been under the pinches of an angry Fairy.' Posthouses were at convenient stages all over the kingdom, and the postmaster was bound to provide horses for all comers, either to ride or drive. His duties and tariffs were as follows:— 'The Post Master is obliged to receive of every Person, Riding Post with Horses and Guide, thus 3d. per Mile for each Horse Hire and 4d. per Stage for a Guide. 'And no Person carrying a Bundle that doth not exceed 80 lbs. Averdupoise, shall be charged for it. 'If through the default of the Post Master, any Person Riding Post shall fail of being furnished, he shall forfeit 5l. Or if the Post Master cannot, or do not furnish any Person with Horses for Riding Post, then they are at Liberty to provide Horses for themselves; but no Horses to be seized without the Owner's Consent. HACKNEY COACH. 'The other way that Gentlemen commonly Travel is in Stage Coaches, which is from about 2d. Farthing to 3d. per Mile. The Flying
- 57. Coach is a Stage Coach, that is drawn by 6 Horses, and will sometimes run 90 or 100 English Miles on one day. 'It may also be noted that Carriage by Waggon or Pack Horses, is about 5 Shillings for carrying 112 Pound Weight 100 Miles; and so in proportion; though 'tis something cheaper in the Summer than Winter.'[597] The Hackney coach was a very useful institution, in spite of all said against it. We have read Ward's description of the bumping he had in one; in another part of the London Spy he says: 'Would you have me, said I, undergo the Punishment of a Coach again, when you Know I was so great a Sufferer by the last, that it made my Bones rattle in my Skin, and has brought as many Pains about me, as if troubled with the Rheumatism. That was a Country Coach, says he, and only fit for the Road; but London Coaches are hung more loose, to prevent your being Jolted by the roughness of the pavement.' The ordinary hackney coaches do not seem to have been provided with glasses. 'For want of Glasses to our Coach, having drawn up our Tin Sashes, pink'd like the bottom of a Cullender, that the Air might pass thro' the holes, and defend us from Stifling.' By the 5 6 Will. and Mary, cap. 22, the number of hackney coaches was fixed at 700, and a tax was imposed of 4l. per annum each, 1l. to be paid every quarter day, besides a fine of 50l. for their first licence for 21 years; and 8l. per annum on stage coaches. To look after these hackney carriages there were five commissioners, at a salary each of 200l., and their office was in Surrey Street, Strand. The fares were not very heavy, even taking the difference of the value of money into consideration, and the fact that they had two horses. s. d. For one day of 12 Hours 10 — For one Hour 1 6 For every hour after the first 1 —
- 58. From any of the Inns of Court to any part of St. James's or City of Westminster, except beyond Tuttle Street 1 — From the Inns of Court, or thereabouts, to the Royal Exchange 1 — From any of the Inns of Court, to the Tower, Aldgate, Bishopsgate Street or thereabouts 1 6 And the same Rates back again, or to any Place of the like Distance. And, if any Coachman shall refuse to go at, or exact more, for Hire, than the Rates hereby limited, he shall for every such Offence forfeit 40 Shillings In 1710 the number of coaches was increased to 800 by the 9 Anne, cap. 23, which also provided that they were to pay five shillings weekly, and were to go a mile and a half for one shilling, two miles for one shilling and sixpence, above two miles two shillings, and greater distances in the same proportion. The hackney coachmen petitioned against the tax, and said they were willing to pay the old one. One petition was entitled 'Some Reasons most humbly Offered to the Consideration of the Right Honourable the House of LORDS and the Honourable the House of Commons; by all the 700 Hackney Coachmen and their Widows to Enable them to pay the Great Tax laid upon them;' and another was 'The Hackney Coachmen's case. Humbly presented to the Right Honourable House of Commons, with a proposal to raise for her Majesty 200,000l. per annum.' This was proposed, very coolly, to be done by laying a tax on all coaches and carriages not licensed, on passengers going by stage coaches, and on goods carried by waggons and packhorses. The coaches were numbered, although I can only find one notice of it: 'So that, rather than to stand a Vapulation, one of them took Notice of his Number;'[598] and the coachmen were noted for their incivility. Of course they did not come from a very high class, and the habits and language of the lower class of that time were extremely coarse. 'We discharged our Grumbling Coachman, who Mutter'd heavily, according to their old Custome, for t'other Sixpence; till at last moving us a little beyond our Patience, we gave an Angry Positive Denial to his Unreasonable Importunities; for we
- 59. found, like the rest of his Fraternity, he had taken up the Miserly Immoral rule, viz. Never to be Satisfied.' Gay gently hints at their incivility:— If Wheels bar up the Road, where Streets are Crost, With gentle Words the Coachman's Ear accost: He ne'er the Threat, or harsh Command obeys, But with Contempt the spatter'd Shoe surveys. And, according to him, they were not only surly but pugnacious. Now Oaths grow loud, with Coaches, Coaches jar, And the smart Blow provokes the sturdy War; From the high Box they whirl the Thong around, And with the twining Lash their Shins resound: Their Rage ferments, more dang'rous Wounds they try, And the Blood gushes down their painful Eye. And now on Foot the frowning Warriors light, And with their pond'rous Fists renew the Fight; Blow answers Blow, their Cheeks are 'smeared with Blood 'Till down they fall, and grappling, roll in Mud. STATE COACH. State coaches were very handsome, being elaborately painted, carved, and gilt, a fine coach and many servants being indispensable to a person of rank. But even in that age of luxuriously appointed equipages everyone was astonished at the magnificence of that of the Venetian ambassador. Luttrell notes it on May 20, 1707: 'Yesterday the Venetian ambassadors made their publick entry thro' this citty to
- 60. Somerset House in great state and splendour, their Coach of State embroidered with gold, and the richest that ever was seen in England: they had two with 8 horses, and eight with 6 horses, trimm'd very fine with ribbons, 48 footmen in blew velvet cover'd with gold lace, 24 gentlemen and pages on horseback, with feathers in their hats.' And the novelty does not seem to have worn off, for, four years afterwards, Swift writes to Stella: 'This evening I saw the Venetian Ambassador coming from his first public audience. His coach is the most monstrous, huge, fine, rich, gilt thing that ever I saw.' He also writes her, Feb. 6, 1712: 'Nothing has made so great a noise as one Kelson's Chariot, that cost nine hundred and thirty Pounds, the finest was ever seen. The rabble huzzaed him as much as they did Prince Eugene.' Anybody with any pretension to wealth and fashion drove six horses, as says Mrs. Plotwell[599]: 'I must have Six Horses in my Coach, four are fit for those that have a Charge of Children, you and I shall never have any;' and Lucinda tells Sir Toby Doubtful[600]: 'You'll at least keep Six Horses Sir Toby, for I wou'd not make a Tour in High Park with less for the World; for me thinks a pair looks like a Hackney.' The coachman, however, did not drive all six, one of the leaders being always ridden by a postilion. These carriage horses were heavy, long-tailed Flemish ones, and naturally went at a sedate and sober trot. STATE COACH.
- 61. It was not everyone that could afford six, or even four, horses, so there were lighter vehicles, as the chariot, the calash, and the chaize or chaise. The latter was adapted for one or two horses, and sometimes was highly ornamented. 'A very fine Chaize, very well Carved, gilded and painted, and lined with blue Velvet, and a very good Horse for it, are to be sold together, or apart c.—The Horse is also a very good Horse for the Saddle.' 'A very fine pair of young Stone Horses, and a very neat Chaize, well Carved, gilt and painted, and lined with Scarlet, and but little the worse for using to be sold.' 'A Curious 4 Wheel Shaze, Crane Neck'd, little the worse for wearing, it is to be used with one or 2 Horses, and there is a fine Harness for one Horse, and a Reputable Sumpture Laopard Covering.' The ordinary chaises, however, were much plainer, and they were built strongly, to stand the strain of bad pavements and roads; but it is probable that very few were put to such a severe test as the following: 'At the Greyhound Inn in West Smithfield is to be sold a Two Wheel Chaise, with a Pair of Horses well match'd: It has run over a Bank and a Ditch 5 Foot High; and likewise through a deep Pit within the Ring at Hide Park, in the presence of several Persons of Quality; which are very satisfied it cannot be overturn'd with fair Driving. It is to be Lett for 7s. 6d. a Day, with some Abatement for a longer Time.' There should be a history attaching to the following advertisement: 'Whereas, upon the 10th of Octob. last, a Gentleman brought a Calash and one Horse, to the Duke of Grafton's Head at Hide Park Corner, and on the 20th of the same Month fetched away the Horse, but left the Calash as a pawn for what was due for the same. If the Owner will come and pay what is due, he may have his Calash again, else it will be appraised and sold in 10 days time.' The innkeeper had waited six months before he advertised. Here is another curious advertisement connected with coaches. 'Lost the 26th of February, about 9 a Clock at Night, between the Angel and Crown Tavern in Threadneedle Street, and the end of Bucklers Berry, the side Door of a Chariot, Painted Coffee Colour,
- 62. with a Round Cypher in the Pannel, Lin'd with White Cloath embos'd with Red, having a Glass in one Frame, and White Canvas in another, with Red Strings to both Frames. Whosoever hath taken it up are desir'd to bring it to Mr. Jacob's a Coachmaker at the corner of St. Mary Ax near London Wall, where they shall receive 30s. Reward if all be brought with it; or if offer'd to be Pawn'd or Sold, desire it may be stop'd and notice given, or if already Pawn'd or Sold, their money again.' In very many advertisements of the sale of second-hand carriages, it is mentioned that the glasses are complete. One would imagine from this that glass was dear, but it was not particularly so. 'These are to give notice to all Persons that have occasions for Coach Glasses, or Glasses for Sash Windows, that they may be furnished with all sorts, at half the prises they were formerly sold for.' And it goes on to say that 12 inches square was 2s. 6d., and increased up to 22 inches, nearly at the rate of 6d. per inch, or 8s. 6d.; 23 inches was 10s. 6d., and so on at about 2s. 6d. per inch to 28 inches, which was 20s., until it culminated at 36 inches square for £2 10s. If this, really, was half the previous cost, and if we reckon the difference in the value of money then and now perhaps some economical people would think twice before having a broken glass repaired. There were also a 'Chasse marée Coach,' and a 'Curtin Coach for Six People.' They used to take nice little drives, too, in these clumsy old carriages—but they took their time over the journey. Thoresby's 'kind friend Mr. Boulter, brought his chariot from Chelsea, purposely to carry him to see Hampton Court.' They started about eleven, and, 'having passed through the City, we passed the Gravel Pits,[601] and had a clear air (whither the Consumptive are sent by the physicians) and delicate pleasant Country to Acton and Brentford; the Duke of Somerset's Seat at Sion House looked most charmingly, and was the first time I had observed the lime trees in the avenues cut in a pyramidal form, even to a great distance from the palace, which looked very Noble; thence through Thistleworth and Twitnam, a very
- 63. pleasant road.' After their visit to Hampton, they stopped for the night at Richmond Wells, returning next day viâ Kew, Mortlake, Putney, and Wandsworth. His friend Boulter, on another occasion, 'took me in his Coach to Hampstead, where we dined with his mother; and after viewing that pleasant town, and taking a view of the Country from the Hill beyond it, we took a tour to Highgate, Mussel Hill and other Country villages, and a pleasant Country, and returned by Islington and Newington home again.' There was a mechanical curiosity which appeared in 1711, and of which the following is an advertisement. 'An Invention of a wonderful Chariot, in which Persons may travel several Miles an Hour, without the assistance of Horses, and Measure the Miles as they go; it turns or goes back; having the Praise of all Persons of Quality, and ingenious Men that have seen it. Note that it is convenient for any Gentleman that is incapable of walking thro' lameness, to ride about his Park or Garden, without damaging his Tarris-Walks or Grass-Plats. The Invention is so highly approv'd that there is one already bespoke by a Person of Quality, which is to go on four Wheels, and swing in the Nature of a large Coach; which according to a modest Computation, will travel at the Rate of 7 or 8 Miles an Hour. If any Person of Quality is desirous to use them with Horses, they may either travel as far again in a Day as they can with another Coach, or can go as far with a Pair of Horses, as the Coaches hitherto in Use can with 6. Note that such as are bespoke for Parks or Gardens only, will come very reasonable, others at proportionable Prices.' It was the fashion to use a mourning coach all the time mourning was worn, and this rendered it incumbent upon people to possess such a vehicle; consequently they were frequently advertised for sale —'At Mr. Harrison's, Coach Maker, in the Broadway, Westminster, is a Mourning Coach and Harness, never used, with a whole Fore Glass, and two Door Glasses and all other Materials (the Person being deceased); also a Mourning Chariot, being little used, with all Materials likewise, and a Leather Body Coach, being very fashionable
- 64. with a Coafoay Lining and 4 Glasses, and several sorts of Shazesses, at very reasonable Rates.' The reins were not of leather, but of worsted, and sometimes of gay colours. Pepys, on that Memorable May Day in 1669 when he started his pretty gilt coach, and had the horses' manes and tails tied with red ribbons, had 'green reins, that people did mightily look upon us.' French harness seems to have been most fashionable, although there is 'a pair of fine new Rumpee Town Harness' advertised; and hammer-cloths were used on the coach-boxes. A singular industry sprang up—that of stealing these hammer-cloths. 'Lost off a Gentleman's Coach Box a Crimson Coffoy Hammer Cloth, with 2 yellow Laces about it.' 'Lost off a Gentleman's Coach Box, a Blue Hammer Cloth, trimm'd with a Gold colour'd Lace that is almost turn'd yellow.' 'Lost a Red Shag Hammock Cloth, with white Silk Lace round it, embroider'd with white and blue, and 3 Bulls Heads and a Squirrel for the Coat of Arms. THE SEDAN CHAIR. The sedan chair was a conveyance that was getting into vogue in Anne's reign. Taking its name from the town of Sedan in France, it was first used in England in 1581, and in London in 1623. In 1711 an Act (9 Anne, c. 23) was passed licensing 200 public sedan chairs at ten shillings each yearly, and their fare was settled at
- 65. 1s. per mile. Next year, another Act (10 Anne, c. 19) was passed, licensing 100 more, but keeping the fares unaltered. Like coaches, their adornment was indicative of the wealth and position of their owners—although, perhaps, none ever came up to Anne's royal present.[602] 'The Queen has made a present of a chair value £8000 to the King of Prussia, which is ordered for Berlin.' Still they were highly ornamental, as the following, which was the property of Sir Joseph Williamson, deceased, will show. 'A Cedan (or Chair) lin'd with Crimson Velvet, trim'd with Gold and Silver, and a new Mourning Chair c.' The prefix 'Sedan' was seldom used, and these conveyances were generally termed 'Chairs.' That they were considered somewhat of a novelty in Anne's reign is evidenced by that line of Gay's 'Nor late invented Chairs perplex'd the way,' and also by the fact that then the public chairs were first licensed, and the number, a very small one, regulated. They were not particularly comfortable, as the Marquis of Hazard describes[603]: 'Hey, let my three Footmen wait with my Chair there —the Rascals have come such a high trot—they've jolted me worse than a Hackney Coach—and I am in as much disorder as if I had not been drest to day.' And they were sometimes dangerous too. Or, box'd within the Chair, contemn the Street, And trust their Safety to another's Feet. . . . . . . . The drunken Chairman in the Kennel Spurns, The Glasses shatters, and his Charge o'erturns. Gay evidently did not like either chairs or chairmen, for he warns his reader thus:—
- 66. Let not the Chairman with assuming Stride, Press near the Wall, and rudely thrust thy Side: The Laws have set him Bounds; his servile Feet Should ne'er encroach where Posts defend the Street. Yet who the Footman's Arrogance can quell, Whose Flambeau gilds the Sashes of Pell Mell? When in long Rank a Train of Torches flame, To light the Midnight Visits of the Dame? Others, perhaps, by happier Guidance led, May where the Chairman rests, with Safety tread; When e'er I pass, their Poles, unseen below, Make my Knee tremble with the jarring Blow.
- 67. CHAPTER XXXVII. THE MOHOCKS. Scourers, etc. — Bully Dawson — Two outbreaks — That in 1712 — Hawkubites — Exploits of the Mohocks — Sir Roger de Coverley — Swift's fear of them — Emperor of the Mohocks — Gog and Magog — The Queen's proclamation — Decline of the scare — Constables and watchmen. In every age and country young blood is hot blood, and in this reign it was particularly so. The wild blood of the Cavaliers still danced in the veins of the beaus in Anne's time, and nightly frolics and broils were of frequent occurrence. They had their predecessors in this work—as Sir Tope says in Shadwell's play of 'The Scowrers': 'Puh, this is nothing, why I knew the Hectors, and before them the Muns and the Titire Tus, they were brave fellows indeed; in those days a man could not go from the Rose Tavern to the Piazza once, but he must venture his life twice.' And Whackum, in the same play, describes the doings of the fraternity of Scourers. 'Then how we Scour'd the Market People, overthrew the Butter Women, defeated the Pippin Merchants, wip'd out the Milk Scores, pull'd off the Door Knockers, dawb'd the Gilt Signs.' In Anne's reign these roysterers were called Mohocks—why, I know not, except that it was then a sort of generic term for North American Indians. In a later age this furore was termed Tom and Jerryism; but then it had an intelligible origin, from Pierce Egan's 'Life in London, or the Day and Night Scenes of Jerry Hawthorn Esq. and his elegant Friend Corinthian Tom c.' It still exists, although it has no special name.
- 68. Brown, in his 'Letters from the Dead to the Living,' says in that 'From Bully Dawson[604] to Bully W....n': 'Therefore if ever you intend to be my Rival in Glory, you must fight a Bailiff once a Day, stand Kick and Cuff once a Week, Challenge some Coward or other once a Month, Bilk your Lodgings once a quarter, and Cheat a Taylor once a year, crow over every Coxcomb you meet with, and be sure you kick every jilt you bully into submission and a compliance of treating you; never till then will the fame of W....n ring like Dawson's in every Coffee House, or be the merry subject of every Tavern Tittle Tattle.' There seem to have been two special outbreaks of Mohocks—one in 1709, and the other in 1712. Of the first Steele says:[605] 'When I was a middle-aged Man, there were many societies of Ambitious young men in England, who, in their pursuits after same, were every night employed in roasting Porters, smoaking Coblers, knocking down Watchmen, overturning Constables, breaking Windows, blackening Sign Posts, and the like immortal enterprizes, that dispersed their Reputation throughout the whole Kingdom. One could hardly find a knocker at a door in a whole street after a midnight expedition of these Beaux Esprits. I was lately very much surprised by an account of my Maid, who entered my bed chamber this morning in a very great fright, and told me, she was afraid my parlour was haunted; for that she had found several panes of my Windows broken, and the floor strewed with half-pence. I have not yet a full light into this new way, but am apt to think, that it is a generous piece of wit, that some of my Contemporaries make use of, to break windows, and leave money to pay for them.' Gay notices the Mohocks, and their window-breaking thus:—
- 69. Now is the Time that Rakes their Revells keep; Kindlers of Riot, Enemies of Sleep. His scatter'd Pence the flying Nicker flings, And with the Copper Show'r the Casement rings. Who has not heard the Scowrer's Midnight Fame? Who has not trembled at the Mohock's Name? Was there a Watchman took his hourly Rounds, Safe from their Blows, or new invented Wounds? I pass their desp'rate Deeds, and Mischiefs done, Where from Snow Hill black Steepy Torrents run; How Matrons, hoop'd within the Hogshead's Womb, Were tumbled furious thence, the rolling Tomb O'er the Stones thunders, bounds from Side to Side. So Regulus to save his Country dy'd. The greatest scare, however, was in March 1712, and that exercised the popular mind as much as the garotters of modern times. People, of course, were more frightened than hurt, and there is very little doubt but that this outbreak was much exaggerated. Still, we can only take the contemporary accounts, and this is one of them. [606]'The Town Rakes, or the Frolicks of the Mohocks or Hawkubites. With an Account of their Frolicks last night, and at several other Times: shewing how they slit the Noses of several Men and Women, and wounded others; Several of which were taken up last Night by the Guards, and committed to several Prisons, the Guards being drawn out to disperse them. 'There are a certain set of Persons, amongst whom there are some of too great a Character, to be nam'd in these barbarous and ridiculous Encounters, did they not expose themselves by such mean and vulgar Exploits. 'These Barbarities have been carry'd on by a Gang of 'em for a considerable time, and many innocent Persons have receiv'd great Injury from them, who call themselves Hawkubites; and their mischievous Invention of the Word is, that they take people betwixt Hawk and Buzzard, that is, betwixt two of them, and making them turn from one to the other, abuse them with Blows and other
- 70. Scoffings; and, if they pretend to speak for themselves, they then Slit their Noses, or cut them down the Back. 'The Watch in most of the Out-parts of the town stand in awe of them, because they always come in a body, and are too strong for them, and when any Watchman presumes to demand where they are going, they generally misuse them. 'Last Night they had a general Rendezvouz, and were bent upon Mischief; their way is to meet People in the Streets and Stop them, and begin to Banter them, and if they make any Answer, they lay on them with Sticks, and toss them from one to another in a very rude manner. 'They attacked the Watch in Devereux Court and Essex Street, made them scower; they also slit two Persons' Noses, and cut a Woman in the Arm with a penknife that she is lam'd. They likewise rowled a Woman in a Tub down Snow Hill, that was going to Market, set other Women on their Heads, misusing them in a barbarous manner. 'They have short Clubs or Batts that have Lead at the End, which will overset a Coach, or turn over a Chair, and Tucks[607] in their Canes ready for Mischief. 'One of these Persons suppos'd to be of the Gang, did formerly slit a Drawer's Nose at Greenwich, and has committed many such Frolicks since. They were so outrageous last Night, that the Guards at White Hall was alarm'd, and a Detachment order'd to Patrole; and 'tis said, the Train Bands will be order'd to do Duty for the future, to prevent these Disorders; several of them were taken up last Night, and put into the Round Houses till order is taken what to do with them.' The Spectator, whose living was by making the most of any popular subject of the hour, was specially exercised over the Mohocks. 'An outrageous Ambition of doing all possible hurt to their Fellow-Creatures, is the great Cement of their Assembly and the only Qualification required in the Members. In order to exert this Principle in its full Strength and Perfection, they take Care to drink themselves
- 71. to a pitch that is beyond the Possibility of attending to any Motives of Reason and Humanity; then make a general Sally, and attack all that are so unfortunate as to walk the Streets through which they patrole. Some are knock'd down, others stabb'd, others cut and carbonado'd. To put the Watch to a total Rout, and mortify some of those inoffensive Militia, is reckon'd a Coup d'éclat. The particular Talents by which these Misanthropes are distinguished from one another, consist in the various kinds of Barbarities which they execute upon their Prisoners. Some are celebrated for a happy Dexterity in tipping the Lion upon them; which is perform'd by squeezing the Nose flat to the Face, and boring out the Eyes with their Fingers; Others are called the Dancing Masters, and teach their Scholars to cut Capers by running Swords thro' their Legs; a new Invention, whether originally French I cannot tell; A third sort are the Tumblers, whose office it is to set Women on their Heads, and commit certain Indecencies or rather Barbarities on the Limbs which they expose.'[608] Sir Roger de Coverley was even somewhat nervous about them when he went to the play—and 'asked me, in the next place whether there would not be some danger in coming home late, in case the Mohocks should be Abroad'; and we learn how, finally, the party went to the theatre. 'The Captain, who did not fail to meet me there at the appointed Hour, bid Sir Roger fear nothing, for that he had put on the same Sword which he made use of at the Battle of Steenkirk. Sir Roger's Servants, and among the rest my old Friend, the Butler, had, I found, provided themselves with good Oaken Plants, to attend their Master upon this occasion.' Swift was in mortal fear of them, and, in his 'History of the Four Last Years of Queen Anne,' declares it was part of a deliberate plan to raise riot, during which Harley might have been assassinated— and accuses Prince Eugene of setting it afloat. He writes Stella—in Letter 43—fragmentary jottings of his feelings during this period of terror. 'Did I tell you of a race of Rakes, called the Mohocks, that play the devil about this town every night, slit peoples noses, and bid them, c.... Young Davenant was telling us at Court how he was
- 72. set upon by the Mohocks, and how they ran his chair through with a Sword. It is not safe being in the streets at Night for them. The Bishop of Salisbury's son is said to be of the gang. They are all Whigs; and a great lady sent to me, to speak to her father and to lord treasurer, to have a care of them, and to be careful likewise of myself; for she heard they had malicious intentions against the Ministers and their friends.... I walked in the Park this evening, and came home early to avoid the Mohocks.... Here is the devil and all to do with these Mohocks. Grub Street papers about them fly like lightning, and a list printed of near eighty put into several prisons, and all a lie; and I almost begin to think there is no truth, or very little, in the whole Story. He that abused Davenant was a Drunken gentleman; none of that gang. My man tells me, that one of the lodgers heard in a Coffee House, publicly, that one design of the Mohocks was upon me, if they Could Catch me; and though I believe nothing of it, I forbear walking late, and they have put me to the Charge of some shillings already.... I came home in a Chair for fear of the Mohocks.... I came afoot but had my Man with me. Lord treasurer advised me not to go in a Chair, because the Mohocks insult Chairs more than they do those on foot. They think there is some mischievous design in those villains. Several of them, lord- treasurer told me, are actually taken up. I heard at dinner, that one of them was killed last night.... Lord Winchelsea told me to day at Court, that two of the Mohocks caught a maid of old Lady Winchelsea's at the door of their house in the Park, with a candle, and had just lighted out somebody. They Cut all her face, and beat her without any provocation.... I staid till past twelve, and could not get a Coach, and was alone, and was afraid enough of the Mohocks.' This dreaded association was supposed to be under the orders of a chief or 'Emperor,' who wore a crescent on his forehead, and is so described both in the Spectator and in Gay's very amusing play of 'The Mohocks,' which is a delicious burlesque on the scare. Here is a sample of it. Some of the watch are talking about this dreaded band, and their doings. Says one: 'I met about five or six and thirty of these Mohocks—by the same token 'twas a very windy Morning—
- 73. they all had Swords as broad as Butcher's Cleavers, and hack'd and hew'd down all before them—I saw—as I am a Man of credit, in the Neighbourhood—all the Ground covered with Noses—as thick as 'tis with Hail Stones after a Storm.' Says another: 'That is nothing to what I have seen—I saw them hook a Man as cleverly as a Fisher Man would a great Fish—and play him up and down from Charing Cross to Temple Bar—they cut off his Ears, and eat them up, and then gave him a swinging slash in the Arm—told him bleeding was good for a fright, and so turned him loose.' A third relates his experience: 'Poh! that's nothing at all—I saw them cut off a Fellow's Legs, and, if the poor Man had not run hard for it, they had Cut off his Head into the bargain.' And the fourth tells how, 'Poor John Mopstaff's Wife was like to Come to damage by them—for they took her up by the Heels, and turn'd her quite inside out—the poor Woman, they say, will ne'er be good for anything More.' Gay also wrote another skit on these awful beings. 'An Argument proving from History, Reason, and Scripture, that the present Mohocks and Hawkubites are the Gog and Magog mentioned in the Revelations, and therefore, That this vain and Transitory World will shortly be brought to its final Dissolution.' It is not particularly amusing, being a parody on scriptural prophecy, and it winds up with the following:— From Mohocks and from Hawkubites Good Lord deliver me, Who wander through the Streets by Night Committing Cruelty. They slash our Sons with Bloody knives, And on our Daughters fall; And if they ravish not our Wives, We have good Luck withal. Coaches and Chairs they overturn, Nay Carts most easily; Therefore from Gog and eke Magog Good Lord, deliver me. Public feeling on the matter, however, was so strong, that on March 17, 1712, the Queen issued a Royal Proclamation.
- 74. 'Anne, R. The Queen's Most Excellent Majesty being watchful for the Publick Good of Her Loving Subjects, and taking notice of the great and unusual Riots and Barbarities which have lately been committed in the Night time, in the open Streets, in several parts of the Cities of London and Westminster, and Parts adjacent, by numbers of Evil dispos'd Persons, who have combined together to disturb the Publick Peace, and in an inhuman manner, without any Provocation, have Assaulted and Wounded many of her Majesty's good Subjects, and have had the Boldness to insult the Constables and Watchmen, in the Execution of their Office, to the great Terror of Her Majesty's said Subjects, and in Contempt and Defiance of the Laws of this Realm, to the Dishonour of Her Majesty's Government, and the Displeasure of Almighty God c. c. c.... Her Majesty doth hereby promise and declare, That whosoever shall before the First Day of May now next ensuing discover to any of Her Majesty's Justices of Peace, any Person who, since the First Day of February last past, hath, without any Provocation, Wounded, Stabb'd or Maim'd, or who shall before the said First Day of May, without any Provocation, Wound, Stab, or Maim, any of Her Majesty's Subjects within the said Cities of London and Westminster, and Parts adjacent, so as such Offender be brought to Justice, shall have and receive the Reward of One Hundred Pounds, c.'
- 75. THE WATCH. Can the following advertisement have any possible relation to the midnight orgies of the Mohocks? Post Boy, Dec. 18/20, 1712: 'Lately found, several Pair of Stockings, some Night Caps, and several Pair of Shooes, with two Brazill Rolling Pins, and some Brass Knockers of Doors.' Brass knockers evidently were attractive, for in 1714 we find a genius advertising, 'There is to be Sold at the Sign of the Plow on Fleet Ditch, New Fashion Brass Knockers of all Sizes that cannot be broke off so easily as any that have yet been made. However, this is to Satisfy all Gentlemen and others that do buy any of them, that if any should be broke off, upon their bringing me a Piece of that which I sold, I will give them gratis one as good and as large as they bought.' The fright soon passed off, for we find Budgell[609] writing on April 8, 1712, that some began to doubt 'whether indeed there were ever any such Society of Men. The Terror which spread itself over the whole Nation some Years since, on account of the Irish, is still
- 76. A CONSTABLE. fresh in most Peoples Memories, tho' it afterwards appeared there was not the least Ground for that general Consternation. The late Panick Fear was, in the opinion of many deep and penetrating Persons, of the same Nature.' But there is no doubt there was a substratum of reality, mixed with a great deal of exaggeration. The civil power was utterly unable to cope with riots of this description. What were the watchmen like? From the time of Dogberry to the institution of the present police they have ever been a laughing-stock. Old, infirm men, badly paid, incumbered with a long staff and a lantern, perambulated the streets under the authority of a constable. Who cared for them? Certainly not a Mohock. Nay, their very honesty was called in question. 'Two of them like honest fellows, handed me home to my Chambers, without so much as stealing my Hat or picking my pockets which was a Wonder.' Ward gives an amusing little sketch of their venality. 'Civil and Sober Persons, said he, how do I know that, Mr. Prattle Box? You may be Drunk for ought I know, and only feign yourselves Sober before my presence to escape the penalty of the Act. 'My Friend puts his Hand in his Pocket, plucks out a Shilling, Indeed, Mr. Constable, says he, we tell you nothing but the Naked Truth. There is something for your Watch to Drink; We know it is a late Hour, but hope you will detain us no longer. 'With that Mr. Surly Cuff directs himself to his right hand Janizary, Hem, hah, Aminidab, I believe they are Civil Gentlemen; Ay, ay, said he, Master, you need not question it; they don't look as if they had Fire balls about 'em. Well, Gentlemen, you may pass; but Pray go civilly home. Here, Colly, light the Gentlemen down the Hill, they may chance to Stumble in the Dark, and break their Shins against the Monument.'
- 77. What sings Gay of watchmen? Yet there are Watchmen, who with friendly Light, Will teach thy reeling Steps to tread aright; For Sixpence will support thy helpless Arm, And Home conduct thee, safe from nightly Harm; But if they shake their Lanthorns, from afar, To call their Breth'ren to confed'rate War, When Rakes resist their Pow'r; if hapless you Should chance to wander with the Scow'ring Crew; Though Fortune yield thee Captive, ne'er despair, But seek the Constable's consid'rate Ear; He will reverse the Watchman's harsh Decree, Mov'd by the Rhetrick of a Silver Fee. Thus, would you gain some fav'rite Courtier's Word; Fee not the petty Clarks, but bribe my Lord.
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