Auxetic Behaviour of Re-entrant Cellular Structured Kirigami … · Auxetic Behaviour of Re-entrant Cellular Structured Kirigami at The Nanoscale . Jing Luo . B.E. (Beihang University,

Auxetic Behaviour of Re-entrant Cellular

Structured Kirigami at The Nanoscale

Jing Luo

B.E. (Beihang University, P.R. China)

Supervised by Prof. Qinghua Qin

February 2017

A thesis submitted for the degree of Master of Philosophy

of The Australian National University

Research School of Engineering

College of Engineering & Computer Science

The Australian National University

To my family

For their endless love and support

Declaration

This thesis contains no material which has been previously accepted for the

award of any other degree or diploma in any university, institute or college. To

the best of the author’s knowledge, it contains no material previously published

or written by another person, except where due reference is made in the text.

Jing Luo

Research School of Engineering

College of Engineering & Computer Science

The Australian National University

28 February 2017

Signature_____________________

© Jing Luo 2017

All Rights Reserved

iv

Acknowledgments

First and foremost, I would like to show endless and sincere gratitude to my

primary supervisor and also the chair of my supervisory panel, Prof. Qinghua

Qin, without whom my research journey would not have been possible. He has

always been supportive and taught me so much about how to do high-quality

research. Besides, I also would like to express my thanks and appreciation to my

associate supervisor, Dr. Yi Xiao, for being so enthusiastic, supportive and

helpful over my past years at the Australian National University (ANU). I will

surely benefit from this unique training and experience throughout my whole

life.

I am also in debt to Prof. Kun Cai, a visiting research fellow of my

research group, for his advice, guidance and kindness throughout my research

work and thesis manuscript writing. In addition, I am grateful to Jing Wan, an

outstanding student of Prof. Cai, for his encouragements and advices on MD

simulation scripts programming, too.

My sincere thanks also go to our group members for their helpful

suggestions and sincere friendships throughout my research and daily life at

Research Group of Engineering Mechanics, ANU. They are Prof. Hui Wang,

Prof. Hongzhi Cui, Mr. Cheuk-Yu Lee, Mr. Haiyang Zhou, Mr. Song Chen, Ms.

Shuang Zhang, Mr. Bobin Xing, Mr. Shaohua Yan, Ms. Yiru Ling and Ms. Ting

Wang. For all of their help, interesting and valuable hints and comments.

I am also indebted to the people who accompany me at Canberra, Mr.

Tong Zhang, Dr. Xin Yu, Mr. Zhixun Li, Mr. Yan Zhang and Xiaosong Li, who

are my lifetime friends. We have been through both difficulties and

achievements all the way through. There are many great memories that we will

cherish for a long time, such as a journey to the Batemans Bay and ANU Kioloa

Coastal Campus. I sincerely wish all of you a successful future and wonderful

life.

v

Last but not least, especially, I would like to give my special thanks to my

family, my parents and my elder sister, for encouraging me and convincing me

to believe myself and for always trusting me even sometimes I particularly do

not. I promise I will spend my lifetime to reciprocate your endless love and

support.

vi

Abstract

Some typical two-dimensional (2D) materials are active elements used in nano-

electro-mechanical systems (NEMS) design, owing to their excellent in-plane

physical properties on mechanical, electrical and thermal aspects. Considering a

component with a negative Poisson’s ratio used in NEMS, the adoption of

kirigamis made of periodic re-entrant honeycomb structures at the nanoscale

would be a feasible method. The focus of this thesis work is to investigate the

specific auxetic behaviour of this kind of structures from typical tailored 2D

materials. By employing the numerical simulation method: molecular dynamics

simulation, the auxetic behaviour of re-entrant cellular structured kirigami is

discussed thoroughly and concretely.

Three main effects of a re-entrant cellular structured kirigami are

systematically simulated, and then analysed and discussed here. They are size

effect, surface effect and matrix effect of 2D materials. The study begins with a

demonstration that a kirigami with specific auxetic property obtained by

adjusting the sizes of its honeycombs. Making use of molecular dynamics

experiments, the size effect on auxetic behaviour of the kirigami is discussed.

The results show that, in some cases, the auxetic difference between the

microscopic structured kirigami and macroscopic structure kirigami is

negligible, which means the results from macro-kirigami could be used to

predict the auxetic behaviour of nano-kirigami. Surface effect of kirigami is also

illustrated from two aspects. The one is to identify the difference of mechanical

responses between pure kirigami and hydrogenated kirigami at some geometry

and loading condition. And another is from the difference of mechanical

responses between microstructure kirigami and continuum kirigami under the

same loading condition and geometric configuration. Graphene is selected as the

major 2D material in the study. As kirigami tailored from various 2D materials

would exhibit different mechanical behaviour, graphene, single-layer hexagonal

boron nitride (h-BN) and single-layer molybdenum disulphide (MoS2) are

vii

selected as representative 2D materials to investigate the influence of this effect,

without loss of generality.

viii

ContentsDeclaration ............................................................................................................................................................ iii

Acknowledgments ................................................................................................................................................. iv

Abstract ................................................................................................................................................................. vi

Contents ............................................................................................................................................................... viii

List of Figures ........................................................................................................................................................ x

List of Tables ....................................................................................................................................................... xiii

List of Abbreviations .......................................................................................................................................... xiv

Chapter 1 Introduction ................................................................................................................................... 1

1.1 BACKGROUND AND MOTIVATION ............................................................................................................... 1 1.2 OBJECTIVE ................................................................................................................................................. 5 1.3 OUTLINE .................................................................................................................................................... 6

Chapter 2 Literature Review ......................................................................................................................... 7

2.1 PREVIOUS WORK ........................................................................................................................................ 7 2.2 2D MATERIALS ........................................................................................................................................... 8

2.2.1 Graphene ......................................................................................................................................... 8 2.2.2 Hexagonal boron nitride ................................................................................................................. 9 2.2.3 Molybdenum disulphide ................................................................................................................. 10

2.3 AUXETIC MATERIALS ............................................................................................................................... 11 2.3.1 Mechanism of auxetic materials .................................................................................................... 11 2.3.2 Natural auxetic materials .............................................................................................................. 13 2.3.3 Artificial auxetic materials ............................................................................................................ 17 2.3.4 Properties of auxetic materials ...................................................................................................... 20 2.3.5 Applications of auxetic materials .................................................................................................. 21

2.4 SUMMARY ................................................................................................................................................ 23

Chapter 3 Research Methodology ................................................................................................................ 24

3.1 MD SIMULATION...................................................................................................................................... 24 3.1.1 Modelling the N-atom physical system .......................................................................................... 25 3.1.2 Potentials ....................................................................................................................................... 26 3.1.3 Time integration algorithm ............................................................................................................ 28 3.1.4 Ensembles ...................................................................................................................................... 29 3.1.5 Energy minimization ...................................................................................................................... 30 3.1.6 Heating baths ................................................................................................................................. 30 3.1.7 Periodic boundary condition ......................................................................................................... 31 3.1.8 MD software package .................................................................................................................... 32

3.2 STRAIN CALCULATION IN 2D MATERIAL KIRIGAMI .................................................................................. 33

ix

3.2.1 Deformation control method.......................................................................................................... 33 3.2.2 Stress method ................................................................................................................................. 34

3.3 DENSITY FUNCTIONAL THEORY ................................................................................................................ 35 3.4 THEORETICAL ANALYSIS OF RE-ENTRANT CELLULAR STRUCTURE KIRIGAMI ........................................... 36 3.5 SUMMARY ................................................................................................................................................ 38

Chapter 4 Size and surface effects of graphene kirigami .......................................................................... 39

4.1 MODEL DESCRIPTIONS ............................................................................................................................. 39 4.1.1 Geometric model of kirigami ......................................................................................................... 39 4.1.2 Methods for numerical experiments .............................................................................................. 41 4.1.3 Schemes for size effect analysis ..................................................................................................... 43

4.2 NUMERICAL RESULTS AND DISCUSSIONS .................................................................................................. 45 4.2.1 Results on included angle between vertical and oblique bar varying ........................................... 45 4.2.2 Results on the width of oblique bar ............................................................................................... 46 4.2.3 Results on different width of vertical bar ....................................................................................... 47 4.2.4 Results on different length of vertical bar ..................................................................................... 48 4.2.5 Results on different length of oblique bar ...................................................................................... 49

4.3 SUMMARY ................................................................................................................................................ 53

Chapter 5 Effect of different matrix 2D materials ...................................................................................... 55

5.1 MODEL DESCRIPTIONS ............................................................................................................................. 55 5.1.1 Geometric model of kirigami ......................................................................................................... 56 5.1.2 Methods for numerical experiments .............................................................................................. 56

5.2 NUMERICAL RESULTS AND DISCUSSIONS .................................................................................................. 57 5.2.1 Results on different angle between vertical and oblique bar ......................................................... 57 5.2.2 Results on different width of oblique bar ....................................................................................... 59 5.2.3 Results on different width of vertical bar ....................................................................................... 61

5.3 SUMMARY ................................................................................................................................................ 63

Chapter 6 Conclusions and Future Works .................................................................................................. 65

6.1 CONCLUSIONS .......................................................................................................................................... 65 6.2 LIMITATIONS............................................................................................................................................ 66 6.3 FUTURE WORKS ....................................................................................................................................... 67

Bibliography ........................................................................................................................................................ 68

Publications .......................................................................................................................................................... 79

Appendix A .......................................................................................................................................................... 81

Appendix B ........................................................................................................................................................... 84

Appendix C .......................................................................................................................................................... 92

Appendix D .......................................................................................................................................................... 98

x

List of Figures

Figure 1-1: Schematic diagram of different Poisson’s ratio material (a) Positive

ν (b) Negative ν. ............................................................................................. 2

Figure 1-2: Auxetic materials (a) Natural (b) Artificial. ....................................... 3

Figure 1-3: Schematic diagram of tailoring 2D material to re-entrant cellular

structures at the nanoscale. ............................................................................. 4

Figure 2-1: Schematic diagram of graphene53. ..................................................... 9

Figure 2-2: Schematic diagram of h-BN53. ......................................................... 10

Figure 2-3: Schematic diagram of MoS253. ......................................................... 10

Figure 2-4: Auxetic materials exist from molecular level to macroscopic level64.

...................................................................................................................... 12

Figure 2-5: Deformation mechanism of typical auxetic material. The blue dash

lines represent the initial structure of this material. ..................................... 13

Figure 2-6: Cristobalite crystal66 ......................................................................... 13

Figure 2-7: Schematic diagram of cristobalite crystal. (a) Crystal structure of α-

cristobalite (b) The variation in Poisson’s ratio, maximum auxetic behaviour

in the (1 0 0) and (0 1 0) planes (YZ, XZ) at 45° to the major axes67. .......... 14

Figure 2-8: Schematic diagram of zeolite crystal72. ............................................ 15

Figure 2-9: Schematic diagram of zeolite crystal. (a) Crystal structure of zeolite

(b) The value of Poisson’s ratio obtained from different force-field-based

molecular simulations72. ............................................................................... 15

Figure 2-10: The structure of B.C.C. solid metal74. ............................................ 16

Figure 2-11: Three typical cellular structure models81. A: Model with re-entrant

units, B: Model with chiral units, C: Model with rotating units. (a)

Undeformed structure and (b) Deformed structure. ..................................... 18

Figure 2-12: (a) Microstructure details of PTFE obtained by SEM82, (b)

undeformed and deformed structure of PTFE81. .......................................... 19

Figure 2-13: A periodic fibre reinforced composite with star-shaped

encapsulated inclusions86. ............................................................................ 20

xi

Figure 2-14: Indentation test for (a) non-auxetic and (b) auxetic materials64. ... 21

Figure 2-15: Schematic diagram of artificial blood vessels89. (a) Non-auxetic

materials (b) Auxetic materials. ................................................................... 22

Figure 3-1: A schematic diagram of 3D PBC simulation box113. ....................... 32

Figure 3-2: A flow chart of MD simulation script programming. ...................... 33

Figure 3-3: A part of re-entrant cellular structured kirigami. The right half part

of (a) is a quarter of the unit cell. The grey area represents the domain

whose mechanical property is determined by the oblique bars and the

vertical bars. The vertical bars determine the mechanical property of the

yellow area. (b) The homogenised model of the right half part of (a). E1 and

E2 (<E1) are the in-plane equivalent moduli of the yellow and grey areas,

respectively. (c) The final deformation. ....................................................... 36

Figure 3-4: Simplified homogenized zone. ......................................................... 37

Figure 4-1: Schematic diagram of graphene kirigami with detailed geometric

parameters121. (a) The atomic system with periodic microstructure in the

solid yellow frame. (b) The geometry of local microstructure (in the solid

yellow frame in (a)) with detailed geometric parameters of the kirigami. The

microstructure has one more vertical rod than the unit cell. ........................ 39

Figure 4-2: The finite element model of the continuum kirigami and the local

finite element (FE) mesh in the solid black frame shown right121. .............. 40

Figure 4-3: Schematic diagram of different surface energy model. ................... 42

Figure 4-4: Configurations of the local microstructure of GK with different

geometric parameters in five schemes121. Here N(w) is denoted as the

number of the basic honeycomb atoms along the direction of variable w,

e.g., N(w)=3 in Figure 4-1(b). (a) Scheme 1, θ changes; (b) N(wθ) changes;

(c) N(w) changes; (d) N(lG) changes; (e) N(lθ) changes. .............................. 44

Figure 4-5: Comparison of configurations before and after deformation with

respective to different lengths of oblique bar. In each inserted figure, the

mid microstructure is labelled with a solid lime frame. In each case, the

configuration of the structure is formed with three layers of microstructures.

xii

The upper figure in each case is the initial configuration after relaxation, the

lower figure in each case is the final stable configuration after loading and

relaxation. ..................................................................................................... 51

Figure 4-6: Different contour plot of electron density nearby the neighbour ends

of vertical bars in the initial configuration of the pure carbon GK with (a)

N(lθ)=24 (see the upper layer of Figure 4-5(b)) and (b) N(lθ)>36 obtained by

the calculation using first principles 126. ....................................................... 52

Figure 5-1: Cell model of re-entrant cellular structured kirigami made of

different 2D materials, Graphene Kirigami, h-BN Kirigami and MoS2

Kirigami. For the convenience of observation, same length scale is not

adopted here.................................................................................................. 55

Figure 5-2: Configurations of local microstructure of the kirigami with different

geometric parameters in three schemes. Here N(w) is denoted as the number

of the honeycomb atoms along the direction of variable w, e.g., N(w) = 3 in

Figure 4-1(b). (a) Scheme 1, θ changes; (b) N(wθ) changes; (c) N(w)

changes. ........................................................................................................ 56

Figure 5-3: Results on included angle between vertical and oblique bar varying.

(a) Rp (b) Rm .................................................................................................. 58

Figure 5-4: Results on width of oblique bar varying. (a) Rp (b) Rm .................... 60

Figure 5-5: Results on width of vertical bar varying. (a) Rp (b) Rm .................... 62

xiii

List of Tables

Table 4-1: Comparisons of Poisson’s ratio and the modulus ratio for the three

models in Scheme 1. ..................................................................................... 45

Table 4-2: Comparisons of Poisson’s ratio and the modulus ratio for the three

models in Scheme 2. ..................................................................................... 46

Table 4-3: Comparisons of Poisson’s ratio and the modulus ratio for the three

models in Scheme 3. ..................................................................................... 47

Table 4-4: Comparisons of Poisson’s ratio and the modulus ratio for the three

models in Scheme 4. ..................................................................................... 49

Table 4-5: Comparisons of Poisson’s ratio and the modulus ratio for the three

models in Scheme 5. ..................................................................................... 50

Table 5-1: Comparisons of Poisson’s ratio and the modulus ratio for the three

models in Scheme 1. ..................................................................................... 58

Table 5-2: Comparisons of Poisson’s ratio and the modulus ratio for the three

models in Scheme 2. ..................................................................................... 60

Table 5-3: Comparisons of Poisson’s ratio and the modulus ratio for the three

models in Scheme 3. ..................................................................................... 62

xiv

List of Abbreviations 2D …… Two-Dimensional

AIREBO …… Adaptive Intermolecular Reactive Empirical Bond Order

AMBER …… Assisted Model Building with Energy Refinement

B.C.C. …… Body Centred Cubic

BK …… Hexagonal boron nitride Kirigami

BP …… Black Phosphorus

CG …… Conjugate Gradient

CHARMM …… Chemistry at HArvard Macromolecular Mechanics

CK …… Continuum Kirigami

DFT …… Density Functional Theory

F.C.C …… Face Centred Cubic

FEM …… Finite Element Method

FET …… Field-Effect Transistor

fs …… Femtosecond

GK …… Graphene Kirigami

GROMACS …… GROningen MAchine for Chemical Simulations

h-BN …… hexagonal Boron Nitride

LAMMPS …… Large-scale Atomic/Molecular Massively Parallel Simulator

MD …… Molecular Dynamics

MEMS …… Micro-Electro-Mechanical Systems

MK …… Molybdenum disulphide Kirigami

MoS2 …… Molybdenum Disulphide

NAMD …… NAnoscale Molecular Dynamics

NEMS …… Nano-Electro-Mechanical Systems

nm …… Nanometre

NPT …… Isothermal–Isobaric Ensemble

NVE …… Microcanonical Ensemble

NVT …… Canonical Ensemble

PBC …… Periodic Boundary Conditions

ps …… Picosecond

xv

PTFE …… Polytetrafluoroethylene

SADS …… Sil AD Spezial

SEM …… Scanning Electron Microscope

SW …… Stillinger-Weber

UHMWP …… Ultra-High-Molecular-Weight Polyethylene

ν …… Poisson’s ratio

1

Chapter 1 Introduction

1.1 BACKGROUND AND MOTIVATION

Nowadays, auxetic materials and two-dimensional (2D) materials have been two

of the most active research fields for many years in material science1. Recently,

the two formerly independent fields have started to intersect in quite new and

interesting ways. Consequently, it becomes possible to some extent to take full

advantages of the auxetic property in 2D materials by optimizing their

microstructures, and particularly to apply the structure in the rapidly developed

micro-electromechanical systems (MEMS) and nano-electromechanical systems

(NEMS)2.

First of all, the 2D materials are a class of nanomaterials defined by their

property of being merely one or two atoms thick. These categories of

nanomaterials are thinned to their physical limits, and thus exhibit novel

properties different from their bulk counterpart3. These 2D materials mainly

include graphene, hexagonal boron nitride (h-BN) and single-layer Molybdenum

Disulphide (MoS2), etc4. Since the (re)discovery of 2D material, the single-layer

graphene, in 2004 by Novoselov and Geim5, they have gained extensive

attention in a variety of fields of nano-engineering6–8, due to their remarkable

mechanical, electrical, and thermal properties9. All of these properties make 2D

materials a good candidate in the application to micro-electromechanical

systems (MEMS) and nano-electromechanical systems (NEMS)2. Some research

efforts have been devoted to discovering the potential applications of the 2D

materials in the fabrication of nano-devices. As a result, they could, as raw

materials, be made into gigahertz oscillators10, nano-pumps11, nano-bearings12,

nano strain-sensors13–15, and many other nano-devices16–20.

Moreover, the auxetic materials are those engineering structures that have a

negative Poisson’s ratio21. Here, the Poisson’s ratio, ν, could be expressed as,

2

.trans

axial

εν

ε= − (1.1)

where εtrans represents transverse strain and εaxial represents axial strain. It

characterizes the negative ratio of the transverse normal strain to the axial

normal strain for a sample of material subjected to an axial loading22,23. When

stretched, these specific materials become thicker in the direction perpendicular

to the applied force. For illustration, Figure 1-1 intuitively presents the behavior

of different Poisson’s ratio materials under tensile test.

Figure 1-1: Schematic diagram of different Poisson’s ratio material (a) Positive ν (b) Negative ν.

As aforementioned, since the auxetic materials are able to exhibit novel

behaviour under deformation, they are of great interest in the field of material

science. Recently, the design and synthesis of such auxetic materials at the

nanoscale level is a research hotspot. For most traditional materials, the

Poisson’s ratio is typically a positive number and has a value around 0.3 in a

large number of engineering materials, such as steels. While, a lot of natural

materials have been reported to be auxetic24,25, they exist extensively in actual

engineering. In Figure 1-2, the two major types of auxetic materials in the actual

engineering is shown, with the first one the natural and another fabricated from

industry, 69% of the cubic metal crystals and some face-centred cubic(F.C.C)

rare gas solids for instance behaved auxetic when they were stretched along a

non-axial direction26,27. As for artificial materials, there are also many auxetic

materials developed and invented, which is very popular in practical engineering

field. And they could be found in many structures, such as honeycomb28,

3

foams29, microporous polymers30, composites31, ceramic32 and origami33. At the

nanoscale, although some auxetic material had been reported as early as twenty

years ago34. But, for the applications in NEMS, most of existed natural auxetic

materials are unusable, due to their special phase form18 and length scale19.

Besides, according to the elastic strain theory35 and thermodynamics

considerations, the value of Poisson’s ratio varies mostly within -1.0 to 0.529, for

the natural material. However, the wider range could be easily achieved for

artificial materials by designing the specific microstructure in the matrix

material.

Figure 1-2: Auxetic materials (a) Natural (b) Artificial.

Considering the significant demand of auxetic nano-devices in NEMS, some

artificial manufacturing technologies could be employed to fabricate this kind of

auxetic nano-devices36. For example, by implementing some technique methods,

like tailoring, some conventional non-auxetic materials could be directly

converted into an auxetic material with specific Poisson’s ratio value37–39.

4

The re-entrant cellular structure is a special type of tailoring materials

with a negative Poisson’s ratio, which has been well studied by researchers for

decades25. An innovative tailoring method at the nanoscale is presented in the

thesis work, which introduces the macroscopic re-entrant cellular structure into

nanoscale 2D materials. In Figure 1-3, it shows how to tailor a piece of 2D

material into re-entrant cellular structured kirigami at the nanoscale.

Figure 1-3: Schematic diagram of tailoring 2D material to re-entrant cellular structures at the nanoscale.

Due to its specific characteristics, the auxetic behaviour in a re-entrant

cellular structured kirigami at the nanoscale is very important to the application

in nano-devices such as NEMS. It is well known that Poisson’s ratio and

modulus are the most representative mechanical properties showing auxetic

behaviour. In case that an auxetic material is under tensile loading, the value of

its Poisson’s ratio could govern the extent of its expansion. And the modulus

ratio, which is the ratio of equivalent modulus to matrix modulus, could

determine the tensile strength of auxetic materials. Therefore, studies on effects

of Poisson’s ratio or the modulus ratio on materials’ mechanical behaviour may

provide a guidance to the design of auxetic nano-device in NEMS. Regarding

the specific microstructure of kirigami, following three major factors could

affect aforementioned two mechanical properties:

(1) Different arrangements of microstructure (size effect);

(2) Surface energy at different length scale (surface effect); and

5

(3) Matrix effect of 2D materials.

However, at nano length scale, the classical continuum mechanics theory is

no longer suitable for investigating those effects on the mechanical properties of

auxetic kirigami. In addition, due to the large consumptions and limitations of

real experiments, there are still many essential issues remain puzzled. Here,

molecular dynamics (MD) simulation, which is the most effective method for

the study of mechanical behaviour at the nanoscale, is employed to study the

influence of the factors discussed above.

The analysis of mechanical properties in re-entrant cellular structured

kirigami in this study is expected to provide some constructive suggestions to

the potential applications to nano-devices.

1.2 OBJECTIVE

As discussed in Section 1.1, the innovative idea of this research work is to

introduce the macroscopic re-entrant cellular structure into nanoscale structures

by tailoring the single-layer 2D materials. As a result, the major objective of this

work is to explore how the auxetic behaviour of tailored 2D material kirigami is

affected by its geometric arrangements, surface energy and matrix 2D materials

using MD simulation. Some theoretical analyses are also described to verify and

illustrate the influences mentioned above.

There are three major effects covered in this thesis. According to the

geometric arrangement of a single unit re-entrant cellular structure in the

tailored kirigami, size effects of the five main variables showing in the unit cell

(Figure 4-1(b)) are investigated. Next, in order to show the effect of surface

energy on the mechanical property of the Kirigami, the hydrogenation schemes

and continuum schemes are employed. At the end, to figure out the effect of

manufacturing 2D raw materials, further discussions on this issue are given.

Besides, some shortcomings and future work in the field of this thesis project are

presented.

6

1.3 OUTLINE

This thesis consists of six chapters and the rest of dissertation is organized as

follows: Firstly, Chapter 2 begins with a brief review on the existing work

related to this research. Then, a comprehensive summary of 2D materials,

auxetic materials and their unique properties, as well as their applications are

described. Chapter 3 describes the research methodology used in this work,

which includes MD simulations, density functional theory and theoretical

mechanics analysis of composite materials. In particular, details of these

methods are described in this chapter for the reference and notation of late

chapters. In Chapter 4, the results of MD simulations on the graphene kirigami

are reported. Detailed discussions regarding the size effect and surface effect are

presented in this chapter. A comprehensive study on the effect of kirigami

tailored from different matrix 2D materials is presented in Chapter 5. In the end,

chapter 6 presents a summary of the outcomes of the thesis work is provided.

Furthermore, some future work in this field and limitations of this work are

demonstrated.

7

Chapter 2 Literature Review

2.1 PREVIOUS WORK

Materials at the nanoscale with a negative Poisson’s ratio, i.e., auxetics, are

becoming increasingly popular as a result of their remarkable properties. In

recent years, significant advances have been made in the design of nano-devices

utilising the specific characteristics of this category of materials. These advances

are either numerical or experimental.

In the respect of real experiment, the findings of artificial materials at the

nanoscale and with a negative Poisson’s ratio could be dated back to 1999, Xu et

al. used soft lithography to fabricate the auxetic materials with re-entrant cellular

microstructures40. Later, Hall et al. found that the in-plane Poisson’s ratio of

carbon nanotube sheets could be tuned from positive to negative by mixing

single-walled and multi-walled nanotubes41. In 2010, Bertoldi et al. discovered

that the pattern transformations in additional curing silicone rubber Sil AD

spezial (SADS) could lead to unidirectional negative Poisson’s ratio behaviour

only under compression test42. By employing the contemporary advanced

fabrication techniques, some of the materials could be made auxetic. However,

all of the aforementioned real experiments are at micrometre scale, which is

1000 times longer than nanoscale. To bypass this hurdle, a kind of kirigami at

the nanoscale is introduced here. Here, the kirigami is a variation of origami that

includes cutting of a paper43. It originated from Japan, where “kiru” means “cut”

and “kami” means paper. Blees et al. used patterning methods to manipulate

graphene kirigami in building robust nanoscale structures with tunable

mechanical behaviour43.

In numerical respect, Grima et al. presented some MD simulations that

showed how the conformation of graphene could be modified through the

introduction of defects so as to make it amenable to exhibit auxetic38. Jiang et al.

discovered that edge induced warping in monolayer graphene and single-layer

black phosphorus would make them show auxetic behaviour24,44. In contrast to

8

the work of Blees’ experiments, researchers including Qi45, Hanakata46, Gao47

and Wei48, utilised MD simulations to investigate the properties of graphene

kirigami. It can be concluded that MD simulation is a very powerful tool in the

study on mechanical behaviour of nanoscale kirigami.

In this work, the macroscopic re-entrant cellular structures will be

introduced at the nanoscale by tailoring the single-layer 2D material. By

employing MD approach, the auxetic behaviour of this specific micro-structured

materials at the nanoscale is investigated.

2.2 2D MATERIALS

Many layered materials contain strong in-plane covalent bonds and weak

coupling interactions between layers. Such layered structures could be exfoliated

into individual atomic layers. Those layers with one dimension strictly limited to

one single-layer are named 2D material49. The background of 2D materials can

be traced back to 2004 attributed to graphene50. Since then, extensive research

efforts are dedicated to making significant progress in basic science and

applications of this category of material51. The Kirigami made of 2D materials

including graphene, single-layer hexagonal boron nitride and single-layer

molybdenum disulphide is our focus in this thesis.

2.2.1 Graphene

Among the family of 2D materials, graphene is most popular around the world.

Due to generating exfoliation manufacturing method of graphene, Geim and

Novoselov were awarded jointly the Nobel Prize in Physics in 201052. As a

popular 2D material, the physical properties of graphene have been extensively

investigated.

Graphene is a single atomic plane of graphite, which consists of a single-

layer of carbon atoms arranged in a 2D honeycomb lattice. Despite being a 2D

material that is only one plane of atoms thick, monolayer graphene exhibits

9

some desirable mechanical properties, such as large surface area, high Young’s

modulus and excellent thermal conductivity.

Figure 2-1: Schematic diagram of graphene53.

Graphene has attracted intensive attentions not only for its unusual

physical properties aforementioned, but also for its potential as a basic building

block for a wealth of device applications. Some examples of the structural and

geometric diversity that can be achieved by using kirigami for graphene have

already been demonstrated experimentally43.

2.2.2 Hexagonal boron nitride

Similar to graphene, the h-BN has received considerable attentions too. Boron

nitride has a similar structural lattice as that found in the carbon of graphene in

that it consists of equal numbers of boron and nitrogen atoms. The h-BN is the

most widely used polymorph and presents the same honeycomb morphology as

graphite, with very close bond length values, being known also as the inorganic

graphite (or white graphite). It is composed of alternative boron and nitrogen

atoms in a honeycomb arrangement, as shown in Figure 2-2.

This unique structure could make h-BN sheet as an excellent lubricant.

Nowadays, 2D h-BN sheets are attracting a lot of attentions, not only because of

the popularity of graphene technologies but also their superb chemical stability

and intrinsic insulation54.

10

Figure 2-2: Schematic diagram of h-BN53.

2.2.3 Molybdenum disulphide

Monolayer molybdenum disulphide (MoS2) is a naturally occurring molybdenite,

which has been studied in recent years as an alternative 2D material to graphene.

It is a kind of single-layered transition metal dichalcogenides, consisting of two

atomic layers of close-packed S atoms separated by one close-packed Mo

atomic layer55, which is shown in Figure 2-3.

Figure 2-3: Schematic diagram of MoS253.

MoS2 exhibits many intriguing physical and chemical properties with a

wide range of potential applications. Because of its direct bandgap and its well-

known properties as a lubricant, this kind of 2D material has attracted interest

for diverse applications.

As mentioned above, compared with 2D materials’ potential applications, even

though they have so many applications for different areas of industry presently,

those are just simply as some tiny drops in the bucket.

11

2.3 AUXETIC MATERIALS

Auxetic materials are a kind of materials that have a unique characteristics

where they would expand laterally when stretched, or shrink laterally when they

are compressed. Following subsections would discuss the mechanism of auxetic

materials and some typical auxetic materials.

2.3.1 Mechanism of auxetic materials

When a material is uniaxially loaded in tension, it extends in the direction of the

applied load, which extension is accompanied by a lateral deformation. The

lateral deformations are quantified by a property known as the Poisson’s ratio

which is defined in mathematical terms as the negative ratio of the transverse

over longitudinal strains, as mentioned in equation(1.1). In particular, for

stretching in the x-direction, the Poisson’s ratio in the xy plane of the material is

defined by equation(1.1):

.yxy

x

εν

ε= (2.1)

Even though, the value of Poisson’s ratio was always assumed to be

positive since most everyday materials get thinner when stretched. Today, it has

been shown through numerous studies that negative Poisson’s Ratio materials

exist, meaning that referring to Figure 1-2(b), for the xy plane, a uniaxial load in

the x-direction results in an extension in the y-direction. The existence of these

materials confirms what had been predicted by theory for a long time, the

classical theory of elasticity23 states that the Poisson’s ratio for three-

dimensional isotropic materials may range between -1≤ν≤0.529, while for two-

dimensional isotropic materials, it may range between -1≤ν≤156, with no upper

or lower bounds for anisotropic materials. Some terms have been produced to

describe these counterintuitive materials, such as auxetic21, dilational57, ‘anti-

rubber’58 and ‘self-expanding’59, however, today, the word auxetic is commonly

used in the material science area.

12

Up to now, a lot of auxetic materials have been discovered, the negative

Poisson’s behaviour could be revealed by the intrinsic of the micro or

nanostructure of the materials and the way that this geometric arrangement

deforms on the tensile test. In fact, there are various geometric arrangements

based deformation mechanisms presented to illustrate the auxetic behaviour in

different natural occurring auxetic materials, including cubic metal crystals26, α-

cristobalite60 and liquid crystalline polymers61 and in different artificial auxetic

materials, such as micro/nanostructured polymers62 and foams63. Auxetic

behaviour is scale independent and the similar combination of geometry and

deformation mechanism can operate at any length scale level. Figure 2-4 shows

various discovered auxetic materials at different length scale level.

Figure 2-4: Auxetic materials exist from molecular level to macroscopic level64.

As aforementioned, auxetic behaviour is mainly caused by the internal

structure of this kind of material itself. Thus, to show the mechanism of auxetic

material, one of the most typical models, re-entrant cellular structure, is selected.

In Figure 2-5, when the structure is being stretched horizontally, the re-entrant

angle (marked in orange in Figure 2-5) would begin to move outwards, i.e., the

structure expands vertically. Conversely, once the structure is being compressed

horizontally, the re-entrant angle would start to move in the inward direction,

which makes structure shrink vertically.

13

Figure 2-5: Deformation mechanism of typical auxetic material. The blue dash lines represent the initial

structure of this material.

It should be mentioned that many natural materials have been found to be

auxetic in the practical engineering65. In the category of artificial or man-made

materials, auxetic materials are also popular.

2.3.2 Natural auxetic materials

2.3.2.1 Cristobalite

A popular naturally-occurring auxetic is the α-cristobalite, which has attracted a

lot of attentions in recent years. The cristobalite is a sort of mineral that has a

high-temperature of silica, which means that it shares same chemical formula

with quartz, SiO2. This kind of crystalline consists of two phases, one is with

high-temperature phase (α-cristobalite) and the other is with low-temperature

phase (β-cristobalite)58,66. Figure 2-6 shows the crystal structure of cristobalite

crystal existing in nature.

Figure 2-6: Cristobalite crystal66

14

Looking at the example of cristobalite in Figure 2-7, the auxetic behaviour

has been measured in single crystalline form. It's anisotropic in different extent

at different place and the signs of Poisson’s ratio in different directions are

presented.

Figure 2-7: Schematic diagram of cristobalite crystal. (a) Crystal structure of α-cristobalite (b) The

variation in Poisson’s ratio, maximum auxetic behaviour in the (1 0 0) and (0 1 0) planes (YZ, XZ) at 45° to

the major axes67.

Interestingly, for the α-cristobalite, when doing the averaging to come up

with a polycrystalline aggregate Poisson’s ratio, an overall isotropic auxetic

material is ended up. In Figure 2-7(a), the unit crystal structure of α-cristobalite

is shown. It has been reported that the extent of auxeticity in the unit single

crystal is such that isotropic polycrystalline aggregates of α-cristobalite are also

generated to exhibit a negative Poisson’s ratio. Figure 2-7(b) shows the range of

Poisson’s ratio obtained from the experiments of α-cristobalite60.

2.3.2.2 Zeolite

Zeolite is another kind of important naturally-occurring auxetic material. It is a

type of microporous and aluminosilicate mineral, which is commonly used as

commercial adsorbents and catalysts. Zeolites are aluminosilicate framework

structures containing molecular-sized cages and channels formed from an array

of corner-sharing SiO44- and AlO4

5- tetrahedral. Figure 2-8 shows the structure of

zeolite crystal existing in nature.

15

Figure 2-8: Schematic diagram of zeolite crystal72.

Due to its highly geometric nanostructures in the crystal structure, auxetic

behaviour of zeolite would be easy to generate. In Figure 2-9(a), the unit crystal

structure of zeolite is shown. The negative Poisson’s ratio could be to some

extent explained by a simple geometry-deformation mechanism relationship. It

should be mentioned there are very few experimental data available on the

single crystalline mechanical properties of zeolite. The value of its Poisson’s

ratio derived from different force-field-based molecular simulations is listed in

Figure 2-9(b), which ranges from -0.55 to -0.33.

Figure 2-9: Schematic diagram of zeolite crystal. (a) Crystal structure of zeolite (b) The value of Poisson’s

ratio obtained from different force-field-based molecular simulations72.

Force-field νxy νyx

Burchart68 -0.55 -0.55

BKS69 -0.33 -0.53

Universal70 -0.33 -0.40

CVFF71 -0.46 -0.46

16

2.3.2.3 Cubic metals

Some metals may exhibit auxetic properties as they are stretched or compressed

in certain direction. For instance, Baughman reported that up to 69% of all cubic

materials and some face-centred cubic rare gas solids behaved auxetic when

they were stretched along the [110] off-axis direction73,74. Figure 2-10 shows a

typical cubic metal possessing a negative Poisson’s ratio.

Figure 2-10: The structure of B.C.C. solid metal74.

Two body centred cubic (B.C.C.) unit cells, which are defined by the

dotted lines in Figure 2-10, are chosen as a crystallographic reference for

describing the relative directions of atomic displacements (marked as black

arrows) of atoms 1–6 in response to an applied force in the [110] direction (F,

marked as two white arrows). To avoid increasing the nearest-neighbour

separations (marked as solid black lines), the only way is to elongate the crystal

in the [110] stress direction. Therefore, the separation distance between atoms 1

and 3 begins to decrease, providing a positive ν(110,001). This angle decrement

17

closes part of the rhombus, by making atoms 5 and 6 apart and approach to a

negative ν(110,1 1 0).

Many other natural auxetic materials may exist in the real world. Due to the

limitation of space, however, we won’t go further on the discussion in this area.

2.3.3 Artificial auxetic materials

Although some natural auxetic materials have been reported recently, artificial

auxetic materials are also very popular in the practical engineering. They can be

classified into following types, cellular structure, microporous structure and

composites.

2.3.3.1 Auxetic cellular structure

In the cellular structure, there are various honeycomb models which could show

auxetic characteristics. Some models have re-entrant units or chiral units, and

some have rotating units. Figure 2-11 shows three typical cellular structure

models. For re-entrant units structure in honeycomb, the first traditional one was

first reported by Gibson in 198275. When a stress load is applied in either

direction, hereby the diagonal ribs move in such a way, which leads to the

auxetic effect in the other direction. Next, in terms of the chiral type, the word

“chiral” means non-superimposable on the mirror image76,77. The auxetic

behaviour is owing to the rotation of rigid rings, which makes ligaments on the

rings wind or unwind, hereby obtaining this unique characteristic78. Finally, in

the model with rotating units, the auxetic behaviour is acquired from the rotation

of rigid polygons joined with each other hinges79. Also, the geometry of auxetic

re-entrant honeycombs was also developed with the numerical analysis of

eigenvalues and natural frequencies of various non-auxetic ones80.

18

Figure 2-11: Three typical cellular structure models81. A: Model with re-entrant units, B: Model with

chiral units, C: Model with rotating units. (a) Undeformed structure and (b) Deformed structure.

Due to its remarkable property and special microstructure, re-entrant

cellular units are adopted to achieve negative Poisson’s ratios of 2D materials in

this work.

2.3.3.2 Auxetic microporous structure

Auxeticity in the microporous structure was first reported by Evans et al. in

1989, which is the expanded form of polytetrafluoroethylene (PTFE)82. Besides,

19

ultra-high molecular weight polyethylene (UHMWP) is another well-known

microporous polymer83. Figure 2-12 shows the scanning electron micrograph

(SEM) of PTFE and mechanism of its auxetic behaviour.

Figure 2-12: (a) Microstructure details of PTFE obtained by SEM82, (b) undeformed and deformed

structure of PTFE81.

Under the uniaxial tension, the fibrils of the microporous structure rotate

with respect to the connection of the nodules and the overall structure then

undergoing expansion. Hence, microporous structure possesses auxetic

properties, because their porous structures allow sufficient space for the nodules

to spread apart.

2.3.3.3 Auxetic Composites

It is well known that continuum materials with specific microstructural

characteristics and composite structures have been confirmed to be auxetic

analytically84. To an extent, some composites could exhibit auxetic behaviour by

adding fibre reinforced laminates. In addition, another kind of auxetic composite

material is that composites with inclusions85. The term “inclusions” here means

that materials in different shapes which are embedded in a matrix. Materials

used to be non-auxetic are able to achieve auxeticity. Figure 2-13 shows a

schematic diagram of a periodic fibre reinforced composite with star-shaped

encapsulated inclusions.

20

Figure 2-13: A periodic fibre reinforced composite with star-shaped encapsulated inclusions86.

In summary, there are still a lot of other artificial auxetic materials developed by

scientists and technicians. However, due to limitation of space, no more

discussions are presented here.

2.3.4 Properties of auxetic materials

As aforementioned, auxetic materials possess enhanced properties, owing to the

counterintuitive characteristics subjected to the applied stress load. A lot of

theoretical analyses and experiments have shown that mechanical properties can

be reinforced by auxeticity87. Materials with a negative Poisson’s ratio have the

following special properties:

(1) High in-plane indentation resistance;

(2) Good fracture toughness;

(3) High transverse shear modulus; and,

(4) High dynamic properties, etc.

2.3.4.1 Indentation Resistance

Unlike non-auxetic materials, auxetic materials do not dent very easily, so that

they have more resistance to indentations. Figure 2-14 shows the schematic of

deformation behaviour when both non-auxetic and auxetic materials are

subjected to impact compressive loading. It reveals that different materials

possess different indentation resistance.

21

Figure 2-14: Indentation test for (a) non-auxetic and (b) auxetic materials64.

For that reason, auxetic materials are more resistant to indentations than

traditional materials.

2.3.4.2 Facture toughness

Compared with traditional materials, auxetic materials have other special and

desirable mechanical properties87. For example, once the material has a crack, it

expands and closes up the crack, after being stretched apart. That is to say, this

kind of material should have more resistance to fracture. Besides, it also has

high material resistance to shear strain.

2.3.4.3 Dynamic properties

Some foam structures with a negative Poisson’s ratio exhibit an overall

superiority regarding damping and acoustic properties, compared with the

conventional ones63. Their dynamic crushing performance is superior to normal

foams, indicating that a potential application in structural integrity compliant

elements.

2.3.5 Applications of auxetic materials

Generally speaking, applications of auxetic materials are mostly based on

following properties:

22

(1) Unique negative Poisson’s ratios;

(2) Superior mechanical properties; and

(3) Acoustic absorption properties.

The Poisson’s ratio affects deformation kinematics in various ways, some of

them are useful and could influence the distribution of stress.

2.3.5.1 Sensors

Since the low bulk modulus of auxetic materials makes them more sensitive to

hydrostatic pressure, they can be adopted in the design of hydrophones or other

precision sensors. Due to the high in-plane indentation resistance, the sensitivity

of sensor made by auxetic materials is increased by almost one order of

magnitude, compared with traditional ones88.

2.3.5.2 Biomedicine

In the area of clinical application, the man-made blood vessel is a representative

use for the biomedicine. As shown in Figure 2-15, if the blood vessel in a human

body is made from auxetic materials, the wall would become thicker when a

pulse of blood flows through it. Otherwise, the wall thickness would decrease,

when the blood vessel is made from conventional materials.

Figure 2-15: Schematic diagram of artificial blood vessels89. (a) Non-auxetic materials (b) Auxetic

materials.

23

Besides, some more potential applications, such as surgical implants90, and

suture anchors or muscle/ligament anchors, where the porous structure could

contribute to promoting bone growth89.

2.3.5.3 Aerospace and defence

Some structures with the unique properties of auxetic materials (i.e. negative

Poisson’s ratio) have already been adopted in real military defense applications.

For example, a kind of pyrolytic graphite (ν=-0.21) has been employed for

thermal protection in aerospace area91. And some large single crystals of Ni3Al

(νmin=-0.18) has been used in Vanessa for aircraft gas turbine engines74.

2.4 SUMMARY

To sum up, in this chapter, an in-depth summary of previous studies related to

this work is presented. Inspiring from their essence and discarding the dross, an

innovative idea of this thesis is proposed. For better studying the auxetic

kirigami made of 2D materials, an intensive study of 2D materials and auxetic

materials is also discussed here to help comprehend their characteristics and

applications.

24

Chapter 3 Research Methodology

In nano length scale, since the walls becoming more dominant, the classical

continuum mechanics theory breaks down gradually. Besides, due to the huge

consumption in experiments, there are still a lot of important questions remain

unsolved. Thus, the numerical simulation may provide an alternative approach

that could help to circumvent these problems. MD simulation is the most

common method for the study of mechanical behaviour at the nanoscale because

of relatively high accuracy, low computational cost92–94. Simulations at different

length scale considering different physics require different simulation methods.

To study the mechanical behaviour, MD simulation and molecular mechanics

are used to calculate the mechanical deformation of different 2D materials.

3.1 MD SIMULATION

In recent years, computational engineering have been extensively developed for

different length scales and material properties. In this work, the length scale

spans from nano to macro, and MD simulation is one of the most powerful tools

for computational mechanics study. MD simulation was first introduced by

Berni Alder for transition problem in 195792, and afterwards it was developed

for various studies in physics, biology, chemistry, material science and nano-

mechanics. MD computes time evolution of a cluster of interacting atoms by

integrating their equations of motions. The classical molecular dynamics is

employed here, and the word “classical” represents particles in simulation

system respects the classical mechanics, in other words, Newton’s Laws95, are

followed:

,i im=iF a (3.1)

For the atom i in a N-atom-system, where mi is the atom’s mass,

2

2 ,ii

ddt

=ra (3.2)

25

The term ai is the acceleration, and Fi is the force acting on the atom. Thus and

so, the force is derived as the gradients of potential with respect to atomic

displacements:

( )1, .ii NV= −∇rF r r (3.3)

Therefore, MD simulation is a deterministic method: given initial

positions and velocities, the Fi following time evolution is completely

determined and the computer calculates a trajectory in a 6N-dimensional phase

space (3N positions and 3N momenta). Besides, the MD simulation is a type of

statistical mechanics method which contains a set of distributions according to

determined statistical ensembles. Physical quantities could be easily evaluated

by taking the arithmetic average among different instantaneous values during the

MD simulation. In the limit of long simulation time, the phase space could be

regarded as fully sampled, and all the mechanical properties would be generated

during the averaging process. In summary, the MD simulation can be applied in

to calculate mechanical properties of microstructure made of different 2D

materials.

3.1.1 Modelling the N-atom physical system

The major part of a MD simulation is the physical model, which means selecting

the potential in simulation: a function, V(r1,…,rN). Forces are then derived from

the equation(3.3), which implies the conservation of total energy, and term V has

been simply written as a sum of pairwise interactions.

When it comes to the practical use, various types of many-body potentials

are now very popular in condensed matter simulation, because the

approximation has been recognized to be inadequate. The improvement of

discovering accurate potentials is remarkable, which includes the development

of Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO)

potential96,97, Tersoff potential98–100 and Stillinger-Weber (SW) potential101,102,

etc. In this thesis, AIREBO potential is applied to the MD simulation about

carbon nanostructures; SW potential is employed to conduct MD simulation

26

related to single-layer MoS2; and Tersoff potential is involved in the MD

simulation on the single-layer h-BN.

3.1.2 Potentials

In general, materials acquire their properties from the interactions among their

component atoms103. These basic interactions maintain the atoms congregate in a

specific crystalline structure. The same interactions also define how the atoms

prefer to arranging themselves on the surface or around a vacancy.

In order to build an appropriate model for MD simulation, the potential

function is required to describe the potential energy and force fields for the

whole system. The potential function is to model the interactions between

particles (atoms, molecules and proteins, etc.), and all electron interactions are

lumped into the potential form. It usually takes an empirical description where

the parameters are fitted from more accurate Density Functional Theory (DFT)

calculations or real experiments. The general analytic form of an intramolecular

potential is originally derived by Abell from chemical pseudopotential theory104.

Beginning with a local basis of unperturbed atomic orbitals, Abell shows that

chemical binding energy can be simply written as a sum of nearest neighbours:

( ) ( ) .R Abonding ij ij ij

i j iE V b V

≠

= − ∑∑ r r (3.4)

The functions VR(r) and VR(r) are pairwise interactions that denote all

interatomic repulsion and attraction from valence electrons. The quantity bij

denotes a bond order between atoms i and j that is derived from electronic

structure theory. As mentioned in Subsection 3.1.1, three major potentials are

our focus in this thesis, which are AIREBO, Tersoff and SW potential,

independently.

3.1.2.1 AIREBO potential

The AIREBO potential computes the Adaptive Intermolecular Reactive

Empirical Bond Order (AIREBO) potential for a system of carbon and/or

27

hydrogen atoms96. The AIREBO potential consists of three terms and could be

written as：

, , ,

1 .2

REBO LJ TORSIONij ij ijkl

i j i k i j l i j kE E E E

≠ ≠ ≠

= + +

∑∑ ∑ ∑ (3.5)

All of three terms are included in the simulation by default. The EREBO

term gives the model its reactive capabilities and only describes short-ranged C-

C, C-H and H-H interactions. The EREBO term adds longer-ranged interactions by

using a form similar to the standard Leonardo Jones potential. And the cutoff of

C-C is 0.34 nm, with a scale factor of 3.0, the resulting ELJ cutoff would be

1.02nm. The ETORSION term denotes an explicit 4-body potential that describes

various dihedral angle preferences in hydrocarbon configurations.

3.1.2.2 Tersoff potential

The Tersoff potential computes 3-body potential and its analytical form for the

pair potential:

( ) ( ) ( )

12

,

iji j i

ij C ij R ij ij A ij

E V

V f f b f

≠

=

= +

∑∑

r r r (3.6)

where the function fC(r) denotes a cutoff term, which ensures only nearest-

neighbour interactions, functions fR(r) and fA(r) are competing attractive and

repulsive pairwise terms, describes the 2-body interaction and the 3-body

interaction respectively. In addition, quantity bij is the bond angle term. The

hybrid B-N-C Tersoff potential105 is adopted in this thesis. More details of this

potential refer to Appendix A.

3.1.2.3 SW potential

Recently, researchers have selected the SW potential developed by Jiang in

2015106, for the simulation on single-layer MoS2. The SW style potential

computes a 3-body potential for the total energy. The total energy interactions in

SW consists of LJ, 2-body and 3-body contributions, which could be expressed

as following:

28

,total LJ two threeE E E E= + + (3.7)

where the Etwo term denotes all 2-body interactions with all possible pairs, i.e.

Mo-S, Mo-Mo and S-S:

( ) ( )max/ 4/ 1 ,r rtwoE Ae B rρ − = − (3.8)

And Ethree term covers the atoms involved in the ϕ, θ and ψ angles:

( ) ( ) ( )1 12 max12 2 13 max13 2/ /0cos cos .r r r r

threeE Ke ρ ρ θ θ − + − = − (3.9)

It is noted that the ELJ only takes effect when there are two or more layers

of MoS2 in simulation models and it denotes the van der Waals interaction

between S atoms in a different layer of multilayers MoS2. More details of this

potential refer to Appendix A.

3.1.3 Time integration algorithm

It is well known that the time integration algorithm is the most powerful

numerical method to integrate the equations of motion of the various interacting

particles and following their trajectory. The basic theory of time integration

algorithm is finite difference methods, where time is discretised on finite small

time step, ∆t. Once the positions, velocities and accelerations of particles are

given, the integration algorithm could derive all of these quantities at the later

time.

In nanoscale MD simulation, one of the most commonly used time

integration algorithms is the Verlet algorithm107,108:

( ) ( ) ( ) ( ) ( )2 42 .t t t t t t t O t+ ∆ = − −∆ + ∆ + ∆r r r a (3.10)

This is the basic form of Verlet algorithm, where a(t) is just the acceleration,

which is the force divided by the mass:

( ) ( )( ).

V tt

m∇

= −r

a (3.11)

As shown in equation(3.10), the truncation error of the Verlet algorithm is of the

order of ∆t4.

29

Despite the fact that algorithm is simple in use, as well as it is very

accurate and stable in the simulation, the problem with this kind of version is

that velocities are not generated directly. To eliminate this weakness, a better

version of the same algorithm is developed, named the velocity Verlet

algorithm. In this version of the algorithm, the positions, velocities and

accelerations at time t+∆t are derived from the same quantities at the time t as

following:

( ) ( ) ( ) ( )

( ) ( )

( ) ( )( )

( ) ( )

21 ,2

1 ,2 2

1 ,

1 .2 2

t t t t t a t t

tt t t t t

t t V t tm

tt t t t t t

+ ∆ = + ∆ + ∆

∆ + = ∆ + ∆

+ ∆ = − ∇ + ∆

∆ + ∆ = + + + ∆ ∆

r r v

r v a

a r

v v a

(3.12)

It is noted that if one wants to save 3N positions, velocities and

accelerations, 9N memory is occupied, but there is no need to simultaneously

store the values at two different times.

3.1.4 Ensembles

In MD simulation, a statistical ensemble is required to control some quantities

related to the thermodynamics, like pressure, temperature and energy.

Traditional MD simulation differs from most other experimental studies is that

the energy and volume are fixed, instead of temperature and pressure94.Common

ensembles consist of canonical ensemble (NVT), micro-canonical ensemble

(NVE) and isothermal-isobaric ensemble (NPT). MD simulation produces NVE

ensemble averages, while constant-temperature experiment is comparable to the

NVT ensemble; suppose constant pressure is maintained as well, it is the NPT

ensemble.

The NPT ensemble is applied in this work to reach a stable thermal

equilibrium with a heating bath at a fixed temperature109. Thereupon, the Nosé-

Hoover thermostat will perform in the NPT ensemble.

30

3.1.5 Energy minimization

In the field of MD simulation, energy minimization is the process of discovering

a stable arrangement in space of a congregation of atoms, according to the

second law thermodynamics. Performing an energy minimization of the

simulation system is to adjust atoms positions iteratively. The objective function

being minimized is the total potential energy of the system as a function of the

N-atom coordinates:

( ) ( ) ( ) ( )

( ) ( ) ( )

1, , , ,

, , , , , ,

, , , , ,

, , , , , , ,

N pair i j bond i j angle i j ki j i j i j k

angle i j k l improper i j k l fix ii j k l i j k l i

E E E E

E E E

= + + +

+ +

∑ ∑ ∑

∑ ∑ ∑

r r r r r r r r r

r r r r r r r r r

(3.13)

where the first term is the sum of non-bonded pairwise interactions, the second

to fifth terms are bond, angle, dihedral and improper interactions respectively,

and the last term is a supplementary term to equation.

Due to its high computational efficiency, Conjugate Gradient (CG)

algorithm often used in an atomistic simulation. Hence, CG algorithm is utilized

in this thesis for minimizing the whole simulation system110. The CG algorithm

mainly depends on the atomic forces and atomic energy. And it goes through a

series of search directions. The local minimum energy point along each search

direction is reached before algorithm proceeding to the next search.

3.1.6 Heating baths

The reliability of simulations significantly depends on an appropriate modelling

of the interaction with thermal reservoirs. In a study of the non-equilibrium

process, stationary non-equilibrium states are required to reach, by that to obtain

the relevant determined thermodynamic properties. In order to gain the stable

simulation, the reservoirs on both sides are linear, however, the central system is

nonlinear.

A conventional way to employ the interaction with reservoirs is to

introduce random forces and dissipation as stated in the fluctuation dissipation

31

theorem. Taking the case of one-dimension chain for example, this equals to the

following form of Langevin equations:

( ) ( ) ( ) ( )1 1 1 ,i i i i i i i i i iNm F F ξ λ δ ξ λ δ− + + + − −= − − − + − + −r r r r r r r (3.14)

where ξ+ and ξ- denote independent variables respecting Wiener processes with

zero mean and variance 2 BK Tλ± ± , KB is the Boltzmann constant and T± means the

temperature of two different heating reservoirs.

For the sake of providing a self-consistent description of out-of-

equilibrium processes, definitive heating baths have been introduced, among

which the Nosé-Hoover thermostat has been employed by most of researchers in

the MD simulation111,112.

In the heating bath, the evolution of the particles is controlled by the

following equation:

( ) ( )1 1

, i S,

, i Si

i i i i ii

m F Fςς+ +

− +− −

∈= − − − − ∈

rr r r r r

r

(3.15)

where ς± are supplementary variables which model the microscopic behaviours

of the thermostat, and S± stand for two sets of N± particles onto reservoirs.

3.1.7 Periodic boundary condition

The appropriate boundary condition is essential to the success of MD

simulation103. Fundamentally, the total number of atoms in an MD simulation is

usually up to many orders of magnitude smaller than required in most situations

of interest for materials simulation. By choosing the suitable periodic boundary

condition (PBC), one can mimic the effects of atoms outside the simulation box

and assist to remove the unwanted artefacts associated with the unavoidably

small size of simulation box. In a typical MD simulation, PBC could remove its

surface effects and maintain translational invariance. Usually, PBC could be

applied to one, two or three directions of simulation box. Figure 3-1 shows a

schematic diagram of PBC simulation box.

32

Figure 3-1: A schematic diagram of 3D PBC simulation box113.

3.1.8 MD software package

Nowadays, there are a lot of popular MD simulator software are implemented in

the research fields, such as LAMMPS, GROMACS, CHARMM, AMBER, NAMD

and MD++, etc.

In this work, MD simulations were conducted with the open source code

package LAMMPS. LAMMPS is a classical molecular dynamics code, and an

acronym for Large-scale Atomic/Molecular Massively Parallel Simulator114. It is

designed to be used for running efficiently on parallel computers. This kind of

simulator software package is distributed and developed by Sandia National

Laboratories, a US Department of Energy laboratory. It has potentials for solid-

state materials (metals, semiconductors) and soft matter (biomolecules,

polymers) and coarse-grained or mesoscopic systems, and could be used to

model atoms or, more generically, as a parallel particle simulator at the atomic

scale. To illustrate the procedure of MD simulation and structure of

programming scripts simply and clearly, a flow chart is shown as in Figure 3-2.

33

Figure 3-2: A flow chart of MD simulation script programming.

3.2 STRAIN CALCULATION IN 2D MATERIAL KIRIGAMI

In this thesis, one of the most important mechanical quantities derived from MD

simulations is the strain calculation in 2D material kirigami. Based on the

continuum mechanics theory, there are two different methods were used to

calculate strain field of single-layer 2D material kirigami from atomistic

simulation results. The deformation control method and stress method are

discussed as followings.

3.2.1 Deformation control method

According to the continuum mechanics theory, uniaxial tensile tests are chosen

to obtain some essential mechanical properties. In the deformation control

34

method, the strain increment with a constant strain rate is applied to simulation

model every constant time step115.

In the uniaxial tensile test, the engineering (nominal) strain and stress are

related by the following equations:

0

0

,xt xx

x

L LL

ε −= (3.16)

0

0

,yt yy

y

L LL

ε−

= (3.17)

0

0 0 0

1

,

xx

x y

UV

V l l T

σε∂

=∂

= ⋅ ⋅ (3.18)

where lx0 and ly0 are the initial lengths of nanoribbon in x- and y-directions, lxt

and lyt are the lengths of the nanoribbon after deformation, term U denotes the

strain energy, term V0 is the initial volume of the simulation model and term T in

the equation means the assumed thickness of given 2D materials. Hence, the

equivalent Young’s modulus Eequiv and Poisson’s ratio ν are defined as follows:

2

20 0

1 ,x

equivx

UEV

εε

=

∂=

∂ (3.19)

.y

x

εν

ε= − (3.20)

In LAMMPS, the strain-control method is utilized by changing the volume

or shape of the simulation box during a dynamics run. Besides, engineering

strain rate could be maintained by using this method.

3.2.2 Stress method

In the MD simulations conducted on the LAMMPS, the virial stress could be

acquired on a per-atom basis. These stresses could then be derived to the strain

through a linear constitutive relationship116. In a stress plane model, the in-plane

strains could be written as:

35

( )

( )

1 ,

1 ,

.

xx xx yy

yy yy xx

xyxy

E

E

G

ε σ µσ

ε σ µσ

σε

= −

= −

=

(3.21)

However, it is noteworthy that, since the relationship between stress and

strain is assumed to be linear, then the resulting strain could be mostly

underestimated, especially at large deformation117.

The virial stress calculation consists of both potential and kinetic parts.

Moreover, the plane stress constitutive model was utilized to calculate the strain

by the equations of

1 0

1 0 ,

10 02

xx xx

yy yy

zz zz

E E

E E

G

µ

ε σµε σ

ε σ

− = − ⋅

(3.22)

It is noted that the values output from LAMMPS are in units of

“Pressure∙Volume”. Accordingly, the real value of stress should take the

“Volume” factor into calculation.

In general, the resultant strain is required in this research work, rather than

stress distribution. In addition, in order to better control and measure the rate of

applied stretched strain. The first method, which is deformation-control method,

is utilized here.

3.3 DENSITY FUNCTIONAL THEORY

Sometimes, the continuum mechanics theories become invalid to illustrate some

phenomenon at the nanoscale. To solve this problem, the density functional

theory (DFT) is introduced here, to investigate the electronic structure of atoms,

molecules and the condensed phases. By employing DFT, the properties of

nanostructure system could be determined by using the spatial electron density

contour. The electron density is the measurement of the probability of an

36

electron being present at a specific location. In quantum chemical calculations,

the electron density, ρ(r) is a function of the coordinates r, thus r∙ρ(r) is the

number of electrons in a small value. The function ρ(r) could be written as:

( ) ( ) ( ) ,Pµν µ νµ ν

ρ φ φ=∑∑r r r (3.23)

where, P denotes the density matrix and ϕ represents basic function.

3.4 THEORETICAL ANALYSIS OF RE-ENTRANT CELLULAR STRUCTURE

KIRIGAMI

To estimate the mechanical behaviour of re-entrant cellular structure kirigami, a

theoretical analysis is developed as following. As shown in Figure 3-3, the right

part with yellow and grey areas (including adjacent bars) is a quarter of the unit

cell of kirigami. We noticed that the equivalent mechanical properties,

equivalent modulus and Poisson’s ratio, are determined by the mechanical and

geometric parameters of the oblique and vertical bars.

Suppose that the total strain along y-direction, εy is specified and denote

the equivalent moduli of yellow and grey areas as E1 and E2, respectively.

Besides, the area fraction of the two areas are w1 and w2= (1-w1), respectively.

Figure 3-3: A part of re-entrant cellular structured kirigami. The right half part of (a) is a quarter of the

unit cell. The grey area represents the domain whose mechanical property is determined by the oblique

bars and the vertical bars. The vertical bars determine the mechanical property of the yellow area. (b)

The homogenised model of the right half part of (a). E1 and E2 (<E1) are the in-plane equivalent moduli of

the yellow and grey areas, respectively. (c) The final deformation.

37

The structure shown in Figure 3-3(b) could be further simplified as the structure

shown in Figure 3-4.

Figure 3-4: Simplified homogenized zone.

According to the composite materials theory118, the equivalent modulus of

the kirigami could be easily obtained as

1 2

1 2

1 ,equiv

w wE E E

= + (3.24)

then the equivalent modulus could be represented by Eequiv=E1∙E2/(E1∙w2+E2∙w1).

Let εy1 and εy2 represent the line strains of the yellow and grey parts along y-

direction, respectively. εy could be then written as:

1 1 2 2 ,y y yw wε ε ε+ = (3.25)

and the total resultant force along y-direction applied to yellow and grey parts is

the same:，

1 1 2 2.y y yF E Eε ε= ⋅ = ⋅ (3.26)

Making use of equations (3.25) and (3.26), the εy1 and εy2 could be expressed in

terms of εy as:

1

1 2 1 2

22 1 2 1

1/

.1

/

y y

y y

w w E E

w w E E

ε ε

ε ε

= + ⋅ = + ⋅

(3.27)

Besides, line strain of the half part along x-direction, εx could also be calculated

via the equation(3.25),

38

1 2 ,x x xlw

l w l wθ

θ θ

ε ε ε= ⋅ + ⋅+ +

(3.28)

where εx<0 is the x-strain of the vertical bar adjacent to the yellow part, εx2

denotes the x-strain of the grey part. For the structure with small deformation, it

can be obtained using the following formulation,

22 2

1

cot .Gx y

lww lθ

ε ε θ≈ ⋅ ⋅ (3.29)

Therefore, the equivalent Poisson’s ratio could be written as

2 1

2 1 2 1

1 1~ cot .1 / /

x xp

y y

wRw l w w E E lθ θ

ε εθε ε

= − ≈ − ⋅ + + + ⋅

(3.30)

By employing the theoretical analysis developed in this section, the size

effect of re-entrant cellular structure kirigami could be illustrated clearly.

3.5 SUMMARY

In summary, MD simulations are conducted within the Sandia-developed open

source code LAMMPS114. The Verlet algorithm is used for simulation time

integration with a time step of 0.5fs, and different ensembles are applied to the

given physical situations. For the potential aspects, interactions between carbon

atoms are described by the AIREBO potential96 with a cutoff at 1.02nm, which

has been adopted to measure the mechanical properties of carbon-based

nanostructures119,120. And the force field to describe interactions between Mo

and S atoms is a modified version of SW potential developed by Jiang106, in

2015. Besides, the interactions between B and N atoms in h-BN are set to be B-

N-C Tersoff potential105. All the simulations are carried out at 8K with Nosé-

Hoover thermostat. The deformation-control method with the applied strain ratio

of 5%/fs is used to perform uniaxial tensile tests in MD simulations.

39

Chapter 4 Size and surface effects of graphene kirigami

4.1 MODEL DESCRIPTIONS

Strictly speaking, some typical 2D materials, such as graphene and black

phosphorus, are not auxetic materials, or not in-plane auxetic more precisely,.

For example, they are either out-of-plane auxetic24,65, or wrinkle-open-up

auxetic38. Once a kirigami of 2D material is tailored into a re-entrant cellular

structure, the auxetic behaviour will be generated. The size and surface effects

on the performance are of importance and are our focus in this chapter.

4.1.1 Geometric model of kirigami

Figure 4-1 shows the schematic diagram of kirigami tailored from the 2D

material. The size effect is mainly due to the five main variables in the cell

honeycomb structure, which are,

Figure 4-1: Schematic diagram of graphene kirigami with detailed geometric parameters121. (a) The

atomic system with periodic microstructure in the solid yellow frame. (b) The geometry of local

microstructure (in the solid yellow frame in (a)) with detailed geometric parameters of the kirigami. The

microstructure has one more vertical rod than the unit cell.

(1) θ, the internal angle between vertical bar and oblique bar;

(2) wθ, the width of oblique bar;

(3) w, the width of vertical bar;

(4) lG, the length of vertical bar; and

40

(5) lθ, the length of oblique bar.

The continuum model of graphene kirigami (GK) is shown in

Figure 4-2: The finite element model of the continuum kirigami and the local finite element (FE) mesh in

the solid black frame shown right121.

The deformation of the atomic system in Figure 4-1(a) will be calculated

by using the result from MD simulations. In Figure 4-2, the deformation of the

continuum kirigami (CK) will be solved via finite element method (FEM)122–125

in working with the other members in our research group. The engineering strain

along y-direction is controlled within 5%, which belongs to the category of

small deformation. The engineering strain along x-direction, i.e., εx, is calculated

by the relative expansion of the blue points in Figure 4-1(a) as,

0

0

,xt xx

x

L LL

ε−

= (4.1)

where, Lx0 and Lxt are the initial and current horizontal distance of the red and

blue points in Figure 4-1(a). And the engineering strain along y-direction, i.e., εy

, is calculated via the time integration of the velocity of the upper side of

specimen,

,y rt eε = ⋅ (4.2)

where, term t denotes the time of strain loading and er denotes the applied strain

ratio. Hence, the equivalent modulus of kirigami can be calculated using the

principle of minimum potential energy,

2

equiv 20 y

1 ,ePEV ε

∂= ⋅

∂ (4.3)

41

where, term V0 denotes volume occupied by the atoms in statistic area for

finding the potential energy of the atoms. The thickness of graphene is set to be

0.335nm in this study. According to the equation(1.1), the Poisson’s ratio in x-y

plane could be represented by the ratio between -εx and εy. To reveal the surface

effect on the equivalent modulus in equation(4.3), modulus ratio, i.e., the ratio

between the equivalent modulus and the in-plane modulus of the two-

dimensional matrix, is defined as

equivm

matrix

.E

RE

= (4.4)

For the computation accuracy, the modulus of idea graphene nanoribbon

is obtained from MD simulation. The graphene nanoribbon (10nm×10nm) is

selected as simulation object with the 5% deformation. And the ratio of applied

strain is set to be 5%/fs. Therefore, based on the theories aforementioned, the

modulus of ideal graphene is obtained, which is 995.8GPa.

4.1.2 Methods for numerical experiments

For the nanostructure of GK, the open sources for large scale MD simulator,

LAMMPS114, is employed. In simulation, the AIREBO potential function96 is

used to describe the interaction between the hydrogen and carbon atoms. The

time step for integration is 0.5fs. The procedure of MD experiment contains

following steps:

(1) Establish the kirigami model with specified geometry shown in the green

frame in Figure 4-1(a). Once the hydrogenation model is adopted to the

simulation, the red bond of edge in Figure 4-1(b) will be bonded with

hydrogen atoms;

(2) Maintain the both two vertical sides of the nanostructure, which are left

and right sides, as free boundaries, and the two horizontal sides as

periodic boundaries, in other words, make la=lb+lc);

42

(3) Adjust the structure by minimizing the potential energy of whole

nanostructure. Here, conjugated-gradient algorithm is utilized for

updating the coordinates of the atoms;

(4) Relax the nanostructure in a Nosé-Hoover thermostat for 200ps at the

temperature of 8K;

(5) Apply specified displacement along y-direction: the lower side is fixed

along the y-direction and the upper side moves with a strain ratio of

5%/fs. The total motion lasts 1000ps; and

(6) Obtain the engineering strains along the x- and y-directions according to

the positions of the blue points in Figure 4-1(a). Compute the Poisson’s

ratio according to equation(4.5). Acquire the equivalent elastic modulus

of the kirigami by using equation(4.6). And the ratio between the

equivalent modulus and that of the ideal graphene could be easily gained.

Moreover, in order to show the effect of surface energy on the mechanical

properties of GK, hydrogenation schemes are added (see Figure 4-3).

Figure 4-3: Schematic diagram of different surface energy model.

43

If there is no carbon atom on the graphene bonded with the hydrogen atom, the

model is labelled as “+0H”; if each carbon atom on the edge (i.e., unsaturated

carbon atom) is bonded to a hydrogen atom, the model is labelled as “+1H”. The

potential energy and volume of the system with respect to the “+1H” model are

the summation of those of carbon atoms only. All the results will be compared

with those of a macrostructure with geometric similarity to the GK. The

Poisson’s ratio and the modulus ratio of the structure of CK are analysed by

finite element method. The analysis procedure of a deformed continuum using

FEM includes the following steps:

(1) Build the CK structure (shown in Figure 4-2) which has the same

geometric configuration with that of the GK;

(2) Mesh the CK structure with finite elements (plane stress elements);

(3) Add displacement constraints: the lower side is fixed along the y-

direction. The node located at the centre of the lower side is fully fixed.

Provide the nodes on the upper side of the structure with specified

displacement, which reflects the same engineering strain along the y-

direction, i.e., no more than 5%;

(4) Find the deformation of the CK structure by solving its equilibrium

equation; and

(5) Compute the Poisson’s ratio by using the displacement of the labelled

points in Figure 4-2. Obtain the modulus ratio by using equations (4.7)

and (4.8).

4.1.3 Schemes for size effect analysis

As shown in Figure 4-4, to study the size effect of the GK, the following

schemes are considered:

44

Figure 4-4: Configurations of the local microstructure of GK with different geometric parameters in five

schemes121. Here N(w) is denoted as the number of the basic honeycomb atoms along the direction of

variable w, e.g., N(w)=3 in Figure 4-1(b). (a) Scheme 1, θ changes; (b) N(wθ) changes; (c) N(w) changes; (d)

N(lG) changes; (e) N(lθ) changes.

(1) Variation of θ: Set N(wθ)=3, N(w)=3, N(lG)=6, N(lθ)=12, N(la)=55. θ

changes from 30° to 60°, 90°, 120° and 150°. The local microstructures

are shown in Figure 4-4(a);

(2) Variation of N(wθ): Set θ =60°, N(w)=3, N(lθ)=12, N(la)=33. N(wθ) is

chosen from {2, 3, 4, 5, 6, 7}, and the related value of N(lG) is set to be

16, 14, 12, 10, 8 and 6, respectively. The local microstructures are shown

in Figure 4-4(b);

(3) Variation of w: Set θ=60°, N(wθ)=3, N(lG)=6, N(lθ)=12, N(la)=25. N(w) is

chosen from {1, 3, 5, 7}. The local microstructures are shown in Figure

4-4(c);

(4) Variation of lG: Set θ=60°, N(wθ)=3, N(w) =3, N(lθ)=12. N(lG) is chosen

from {6, 12, 18, 24, 30, 36}, which leads N(la) equal to 25, 31, 37, 43, 49

and 55, respectively. The local microstructures are shown in Figure

4-4(d); and

(5) Variation of lθ: Set θ=60°, N(wθ)=3, N(w) =3, N(lG)=12. N(lθ) is chosen

from {12, 15, 18, 21, 24, 27}, which makes N(la) equal 25, 28, 31, 34, 37

45

and 40, correspondingly. The local microstructures are shown in Figure

4-4(e).

4.2 NUMERICAL RESULTS AND DISCUSSIONS

4.2.1 Results on included angle between vertical and oblique bar varying

In the first scheme, θ is chosen from the angles, i.e., 30, 60, 90, 120 and 150

degrees, respectively. The other parameters are given in Subsection 4.1.3. The

Poisson’s ratio and the modulus ratio are calculated using MD and FEM

simulations, and the results are shown in Table 4-1. Table 4-1: Comparisons of Poisson’s ratio and the modulus ratio for the three models in Scheme 1.

θ =30° θ =60° θ =90° θ =120° θ =150° Rp (GK+0H) -2.074 -0.980 0.318 1.868 6.302 Rp (GK+1H) -1.442 -0.669 0.149 1.667 5.077 Rp (CK) -1.139 -0.722 0.035 1.034 2.907

Rm (GK+0H) 5.16% 14.40% 17.91% 23.14% 31.27% Rm (GK+1H) 7.08% 16.86% 22.31% 25.63% 37.91% Rm (CK) 1.88% 2.05% 2.19% 2.59% 4.47%

Table 4-1 lists the Poisson’s ratios and modulus ratios of the GK with

different values of θ. For the Poisson’s ratio of GK in the three models, the

value increases monotonously along with an increase in θ. When θ is less than

90°, the Poisson’s ratio is negative, i.e., GK shows auxetic behaviour. At θ=90°,

the Poisson’s ratios are positive, rather than zero. The reason is that the oblique

bar, which currently is horizontal, has a rotary angle which leads to θ>90°. The

table also demonstrates the value of Rp is sensitive to the surface modification.

For example, for “+0H” model, Rp is 0.318, which is over twice of 0.149 of

“+1H” model, and is far greater than 0.035 of CK model. Similar differences

among the values of Rp of the three models can also be found at any other values

of θ. It is concluded that the GK without hydrogenation (+0H model) has the

highest absolute value of Rp among the three models. The hydrogenated GK

(+1H model) has a lower absolute value of Rp than the pure carbon kirigami

does, but the value is higher than that of the CK (calculated using FEM) if θ is

not 60°.

46

For the modulus ratios, Rm, of kirigami in the three models, their increase

is observed with the increase of θ. The reason is that the deformation of the

structure is caused by the rotation of the oblique bars when θ is small, while, at

the large value of θ, the deformation of GK is mainly caused by the axial linear

strain of bars. So, an increment of potential energy is larger at a larger value of

θ. According to equation(4.9), a larger modulus is associated with a larger value

of θ. Comparing the three models’ modulus ratios with the same value of θ, the

GK with “+1H” model has the largest modulus. It is about 10%~20% larger than

that of GK with “+0H” model. The reason can be found from the comparison of

potential energy of the two models, i.e., in the “+1H” model, the carbon atoms

on the edge have more steep increases in potential energy as they are bonded

with hydrogenation atoms. The modulus ratios of the GK with or without

hydrogenation are far greater than that of the CK model under the same

geometric parameter setting. It demonstrates that the surface effect should also

be considered at the nanoscale when evaluating the modulus of a 2D material.

4.2.2 Results on the width of oblique bar

In this subsection, only N(wθ) changes from 2 to 7, indicating the width of

oblique bars increases monotonously (Figure 4-4(b)) while the other parameters

are kept unchanged. Making use of MD and FEM simulations, the Poisson’s

ratio and the modulus ratio are obtained and shown in Table 4-2. Table 4-2: Comparisons of Poisson’s ratio and the modulus ratio for the three models in Scheme 2.

N(wθ) =2 N(wθ)=3 N(wθ)=4 N(wθ)=5 N(wθ)=6 N(wθ)=7 Rp (GK+0H) -0.835 -0.593 -0.470 -0.333 -0.235 -0.171 Rp (GK+1H) -0.558 -0.457 -0.336 -0.239 -0.171 -0.119 Rp (CK) -0.517 -0.413 -0.311 -0.226 -0.163 -0.118

Rm (GK+0H) 3.96% 6.54% 8.21% 9.43% 9.91% 9.92% Rm (GK+1H) 6.41% 7.96% 9.08% 10.74% 11.10% 11.34% Rm (CK) 0.47% 1.16% 2.04% 2.94% 3.75% 4.46% When comparing Rp and Rm of the three models with the same geometric

configurations, the surface effect can also be found. Similar conclusions as those

in Scheme 1 can be found. Interestingly, the difference between Rp of GK with

47

“+1H” model and that of the CK model is very small whatever the width of

oblique bar is. It evidences that the auxetic behaviour of a 2D material shown in

Figure 4-4(b) can be estimated using experiment on a macro-continuum model.

Now, we examine the size effect on the mechanical properties of kirigami

through adjusting the width of the oblique bars. It is noted that, when θ=60°

(less than 90°), the kirigami of all the three models shows auxetic behaviour no

matter how the width of oblique bar is. It is also found that the absolute value of

Rp decreases with an increase in the width of oblique bar. The reason can be

explained from equation(4.10). In the model, the geometric parameters, e.g., w,

lθ, θ, lG, and E1 and εy are assumed to be constant. Both w2/w1 and E2 increase

with an increase in the width of oblique bar. Negative scalar εx1 decreases with

an increase in the width of oblique bar. Consequently, the absolute value of the

Rp is smaller when the width of oblique bar is wider. The modulus ratios of the

three models are also different. The difference between the moduli of the GKs

with and without hydrogenation is smaller than that between the GK models and

CK model.

4.2.3 Results on different width of vertical bar

In this part, N(w) is chosen from 1, 3, 5, 7, i.e., the width of vertical bar

increases monotonously. The rest parameters are kept unchanged. Employing

MD and FEM simulations, the Poisson’s ratio and the modulus ratio are

obtained and listed in Table 4-3. Table 4-3: Comparisons of Poisson’s ratio and the modulus ratio for the three models in Scheme 3.

N(w)=1 N(w)=3 N(w)=5 N(w)=7 Rp (GK+0H) -0.406 -0.455 -0.416 -0.372 Rp (GK+1H) -0.311 -0.345 -0.337 -0.305 Rp (CK) -0.256 -0.280 -0.254 -0.225

Rm (GK+0H) 4.67% 3.58% 2.82% 2.36% Rm (GK+1H) 5.56% 4.35% 3.46% 3.35% Rm (CK) 0.83% 0.79% 0.68% 0.60%

In the scheme, the absolute values of Rp do not vary monotonously, and

the value reaches its peak at N(w)=3. The reason can be revealed via the analysis

48

using equation(4.11), for example, when N(w)=1, the width of vertical bar is less

than that of oblique bar, the vertical bar plays a major role in the deformation of

the GK. The rotary angle of the oblique bar is small because of small tension

component on the vertical bar (σy). As N(w)=3, the vertical bar has a stronger

tensile force which leads to a higher rotary angle of oblique bar. Therefore, the

Rp at N(w)=3 is larger than that at N(w)=1. When N(w)≥5, the rotary angle of

oblique bar is slightly larger than that at N(w)=3. Simultaneously, w/lθ is over

1.6 times of that at N(w)=3. From the second item in the left part of

equation(4.12), we can find that the wider width of the vertical bar (N(w)≥5)

leads to smaller Poisson’s ratio of the GK. On the other hand, the difference

between the Poisson’s ratio of GK with “+1H” model and CK model is obvious

no matter if θ=60° or not in the models. It implies that the surface effect cannot

be neglected when using the continuum model to estimate the auxetic behaviour

of GK with different widths of vertical bar.

For the modulus ratio of the three models, it decreases along with an

increase in the width of vertical bar. The reason is that the deformation of the

GK mainly depends on the rotation of the oblique bar during loading in y-

direction. Hence, the variation of potential energy of the vertical bar decreases

along with an increase in its width. Therefore, the variation of potential energy

of the system decreases, too, which results in decrease in the modulus according

to equation(4.13).

4.2.4 Results on different length of vertical bar

In this scheme, the length of vertical bar N(lG) is chosen from 6, 12, 18, 24, 30

and 36. The other parameters are maintained invariant. The results of Poisson’s

ratio and the modulus ratio obtained from MD and FEM simulations are listed in

Table 4-4.

49

Table 4-4: Comparisons of Poisson’s ratio and the modulus ratio for the three models in Scheme 4.

N(lG)=6 N(lG)=12 N(lG)=28 N(lG)=24 N(lG)=30 N(lG)=36 Rp (GK+0H) -0.455 -0.559 -0.662 -0.763 -0.845 -0.980 Rp (GK+1H) -0.345 -0.430 -0.505 -0.560 -0.623 -0.669 Rp (CK) -0.280 -0.381 -0.475 -0.563 -0.645 -0.722

Rm (GK+0H) 3.58% 5.71% 7.94% 9.97% 12.29% 14.40% Rm (GK+1H) 4.35% 6.93% 9.58% 11.93% 14.36% 16.86% Rm (CK) 0.79% 1.07% 1.34% 1.59% 1.82% 2.05%

When the length of vertical bar increases or N(lG) changes from 6 to 36,

the absolute values of Poisson’s ratio of the three models increase

monotonously. The reason can be found by checking either equation(4.14), or

the deformation of Figure 3-3(b). For example, for the second item of the left

part of equation(4.15), when w, lθ, E1 and E2(<E1), are kept constant, an increase

in w1 leads to the increase of Rp. It suggests that the 2D material can achieve

higher auxetic effect through use of longer vertical bars. In this scheme, the

difference between the values of Rp of the hydrogenated GK and CK model is

very small. One can estimate the auxetic behaviour of the GK using the

experiments on the continuum model with the same geometric configurations

according to this principle.

The modulus ratio of the three models increase with an increase in the

length of vertical bar. But the maximal value is reached when w1 tends to be 1.0.

The maximal value will be E1/Ematrix=w/(w+lθ)∙(Evert_bar/Ematrix)=~20%, according

to equations (4.16) and (4.17) and Figure 3-3(b). The hydrogenated GK still has

the highest modulus among the three models. The CK model has the lowest

modulus, which is no more than 20% of that of the hydrogenated GK.

4.2.5 Results on different length of oblique bar

At last, the change in the length of oblique bar are achieved by selecting

different N(lθ). In this work, N(lθ) is chosen from 12, 15, 18, 24 and 27. The

remaining parameters are kept constant. The corresponding results of Poisson’s

ratio and the modulus ratio are listed in Table 4-5.

50

Table 4-5: Comparisons of Poisson’s ratio and the modulus ratio for the three models in Scheme 5.

N(lθ)=12 N(lθ)=15 N(lθ)=18 N(lθ)=21 N(lθ)=24 N(lθ)=27 Rp (GK+0H) -0.455 -0.484 -0.437 -0.431 -0.551 -0.544 Rp (GK+1H) -0.345 -0.362 -0.377 -0.383 -0.350 -0.305 Rp (CK) -0.280 -0.278 -0.271 -0.264 -0.256 -0.249

Rm (GK+0H) 3.58% 2.16% 1.41% 1.17% 5.03% 9.99% Rm (GK+1H) 4.35% 2.82% 1.92% 1.61% 1.26% 0.90% Rm (CK) 0.79% 0.42% 0.25% 0.16% 0.11% 0.08% The results of Rp of CK with different geometric parameters show that the

absolute value of Rp decreases slightly with an increase in the length of oblique

bar. Although similar surface effects still exist, demonstrated by the absolute

value of Rp of hydrogenated GK is smaller than that of the pure carbon GK, but

larger than that of CK model, the value of Rp does not change monotonously

with respect to the variation of the length of oblique bar. For example, the

absolute value of pure carbon GK peaks at N(lθ)=24. While, the maximal

absolute value of hydrogenated GK appears at N(lθ)=21. The reason can be

revealed from the configurations of the models shown in Figure 4-5. For

instance, at N(lθ)=21, the two vertical bars on the same column are not attracted

to be bonded together. When N(lθ)=24 or 27, the two vertical bars get close to

each other, with the distance of only 0.298 nm for N(lθ)=24 (Figure 4-6(a)) or

0.265nm for N(lθ)=27. The two adjacent ends of vertical bars are not bonded.

However, the van der Waals interaction between the two adjacent ends becomes

very strong, which leads to the decrease of the value of θ.

51

Figure 4-5: Comparison of configurations before and after deformation with respective to different

lengths of oblique bar. In each inserted figure, the mid microstructure is labelled with a solid lime frame.

In each case, the configuration of the structure is formed with three layers of microstructures. The upper

figure in each case is the initial configuration after relaxation, the lower figure in each case is the final

stable configuration after loading and relaxation.

Thus, the Poisson’s ratio at N(lθ)=24 is greater than that at N(lθ)=21. When lG

keeps unchanged, larger lθ leads to smaller variation of θ after relaxation. That is

why the Poisson’s ratio at N(lθ)=24 is higher than that at N(lθ)=27.

The electron density distribution nearby the adjacent ends of vertical bars

shown in Figure 4-6(a) reveals that no new carbon-carbon covalent bond is

generated. Thence, after deformation, the two ends are separated. However, the

variation of potential energy is greater than that of the model with N(lθ)=21. It

results in higher modulus according to equation(4.18). This is verified by the

values of the Rm listed in Table 4-4. On the other hand, if the value of N(lθ) is far

greater than 24, the oblique bars are softer and the two ends of the vertical bars

can get closer, which provide a chance to generate new covalent bonds among

the unsaturated carbon atoms at the ends (Figure 4-6(b)). In such condition, the

52

topology of the kirigami changes and the new GK may alter mechanical

properties, such as becoming stiffer or even non-auxetic.

Figure 4-6: Different contour plot of electron density nearby the neighbour ends of vertical bars in the

initial configuration of the pure carbon GK with (a) N(lθ)=24 (see the upper layer of Figure 4-5(b)) and (b)

N(lθ)>36 obtained by the calculation using first principles 126.

When the GK is hydrogenated (“+1H” model), the repulsion exists between

adjacent vertical bar and oblique bar at the joint whilst attraction exists between

the two adjacent vertical bars. A model with a longer oblique bar will have

higher deformation of the oblique bar due to repulsion between the oblique and

vertical bars and stronger attraction at adjacent ends of vertical bars. Hence, the

peak value of Poisson’s ratio appears at N(lθ)=21. On the other hand, the two

adjacent ends of vertical bars are attracted due to van der Waals interaction,

rather than covalent bonds, the potential energy of the system changes slightly.

The modulus ratio decreases with an increase in N(lθ). The modulus ratio of CK

model has similar variation trend. But the modulus ratio of hydrogenated GK is

still much higher than that of CK.

53

4.3 SUMMARY

In conclusion, after analysing the Poisson’s ratio and modulus ratio of GK with

re-entrant honeycomb microstructure, the dependence of the two mechanical

properties on the sizes of the microstructure is revealed. And the mechanical

response of the CK with geometric similarity to the GK is also calculated for

exploring the surface effect of GK through comparisons of the related responses.

The results show that the specified Poisson’s ratio and modulus of GK can be

obtained by adjusting the sizes of the microstructure. The results presented in

this chapter has been published in Nature Scientific Report127. Some conclusions

can be made as follows:

(1) Of all the schemes, pure carbon GK, being with the highest surface energy,

has the highest absolute value of Poisson’s ratio among the three models. The

absolute value of Poisson’s ratio of the hydrogenated GK is higher than that

of the CK due to the higher surface energy of the GK than that of the CK.

Regarding the effect of θ illustrated in scheme 1, when θ is less than 90°, GK

shows auxetic.

(2) In general, the modulus ratio of hydrogenated GK is about 10%~20% higher

than that of the pure carbon GK. The modulus ratio of CK is far less than that

of the GK when they have the same geometric configurations. It

demonstrates the significant surface effect on the modulus of 2D

nanomaterials.

(3) In Scheme 2, the difference between the Poisson’s ratios of hydrogenated GK

and CK models is very small, meaning that the Poisson’s ratios depend on

the width of oblique bar, slightly. It indicates that the auxetic behaviour of a

2D nanomaterial can be estimated using the experiment on a macro-

continuum model with respect to the width of oblique bar. For the modulus

ratios of GK and CK models, when the width of oblique bar increases, the

absolute value of Rp decreases, but the modulus ratio increases,

monotonically.

54

(4) When increasing the width of vertical bar as was done in scheme 3, the peak

value of Poisson’s ration appears at N(w)=3. The difference between the

Poisson’s ratios of GK models and CK model is obvious. As for the modulus

ratios of the three models, they decrease with an increase in the width of

vertical bar.

(5) When the length of vertical bar N(lG) increases from 6 to 36, the absolute

values of Poisson’s ratio of the three models increases monotonously. It

suggests that the 2D material can have higher auxetic effect with longer

vertical bars as illustrated in Scheme 4. In this condition, the difference

between the values of Poisson’s ratio of the hydrogenated GK and CK model

is very small. The auxetic behaviour of GK can be estimated using the

experiments on the CK model with the same geometric configurations. The

modulus ratio of the three models increases with an increase in the length of

vertical bar.

(6) In Scheme 5, the absolute value of Poisson’s ratio varies slightly with the

increase in the length of oblique bar (or N(lθ)). If the value of lG is small (i.e.,

vertical bars are short), the two adjacent vertical bars in pure carbon GK

model may be bonded together. It results in the sharp increase of both

Poisson’s ratio and modulus ratio. But it does not happen in either the

hydrogenated GK model or the CK model.

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Chapter 5 Effect of different matrix 2D materials

5.1 MODEL DESCRIPTIONS

Graphene, single-layer h-BN and single-layer MoS2 are three of the most typical

2D materials. They are typical isotropy material and have attracted extensive

attentions due to their excellent mechanical properties, since they were found in

the laboratory. Figure 5-1 shows the different cell models of re-entrant cellular

structured kirigami made of different 2D materials: graphene kirigami (GK),

single-layer hexagonal boron nitride kirigami (BK), and single-layer

molybdenum disulphide kirigami (MK).

Literally, all of them have the similar intrinsic microstructure, which is

the assemblies of the hexagonal atoms in an in-plane view. Therefore, these 2D

materials could be tailored to the re-entrant cellular structure. Kirigami made of

different 2D material may possess different mechanical properties. Further

discussions of this effect is discussed in this chapter.

Figure 5-1: Cell model of re-entrant cellular structured kirigami made of different 2D materials,

Graphene Kirigami, h-BN Kirigami and MoS2 Kirigami. For the convenience of observation, same length

scale is not adopted here.

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5.1.1 Geometric model of kirigami

In order to show the effect of different parameters, huge MD computations are

required. To save computational time, geometric size effect is considered by

varying the three independent geometric variables only, which are,

(1) θ, the included angle between vertical bar and oblique bar;

(2) wθ, the width of oblique bar; and

(3) w, the width of vertical bar.

Further details of the model microstructure are shown in Figure 5-2.

Figure 5-2: Configurations of local microstructure of the kirigami with different geometric parameters in

three schemes. Here N(w) is denoted as the number of the honeycomb atoms along the direction of

variable w, e.g., N(w) = 3 in Figure 4-1(b). (a) Scheme 1, θ changes; (b) N(wθ) changes; (c) N(w) changes.

The atomic system of this chapter is the same as that shown in Figure

4-1(a). And the deformation of the whole system will be calculated using the

result from MD simulation. All of the relevant physical quantities, such as

engineering strain along the x-direction, engineering strain along the y-direction,

Poisson’s ratio and modulus ratio, are calculated by the methods described in

Section 4.1. The configuration employed in this chapter is the same as that in the

previous chapter and the nanoribbons (10nm×10nm) are, thus, selected as

simulation objects with 5% deformation. And the ratio of applied strain is set to

be 5%/fs. Accordingly, based on the theories aforementioned, the modulus of

idea graphene, single-layer h-BN and single-layer MoS2 are obtained, which are

995.8, 784.0 and 150.3GPa, respectively.

5.1.2 Methods for numerical experiments

To investigate the microstructure of different kirigami, the open sources for

large scale MD simulator, LAMMPS114, is employed. In the simulation, the

57

AIREBO potential function96 is used to describe the interaction between the

hydrogen and carbon atoms; the B-N-C Tersoff potential function98 is used to

describe the interaction between the boron and nitrogen atoms; and the SW

potential function102 is used to describe the interaction between molybdenum

and disulphide atoms. The time step used in the simulation is 0.5fs. The

procedure of MD experiment is listed as follows:

(1) Build the kirigami model (within the green frame in Figure 4-1(a)) with

specified geometry;

(2) Maintain the two vertical sides (left and right sides) of the nanostructure

as free boundaries, and the two horizontal sides as periodic boundaries

(Maintain la=lb+lc);

(3) Reshape the structure by minimizing the potential energy of the

nanostructure. The conjugated-gradient algorithm is adopted for updating

the positions of the atoms;

(4) Relax the nanostructure in a Nosé-Hoover thermostat for 200ps with

temperature of 8K;

(5) Provide displacement load along y-direction: the lower side is fixed along

the y-direction and the upper side moves with a strain ratio of 5%/fs. The

total motion costs 1000ps; and

(6) Compute the engineering strains along x- and y-directions according to the

positions of the blue points in Figure 4-1(a). Calculate the Poisson’s ratio

according to equation(5.1). Acquire the equivalent elastic modulus of the

kirigami by using equation(5.2). And the ratio between the equivalent

modulus and that of the ideal graphene, single-layer h-BN and single-

layer MoS2 could emerge easily.

5.2 NUMERICAL RESULTS AND DISCUSSIONS

5.2.1 Results on different angle between vertical and oblique bar

For the first scheme, the internal angle between vertical and oblique bars, θ is

chosen from following angles, i.e., 30, 60, 90, 120 and 150 degrees,

58

respectively. Other parameters in Figure 4-1(b) are kept invariant. The results of

Poisson’s ratio and modulus ratio are listed in Figure 5-3 and Table 5-1. Table 5-1: Comparisons of Poisson’s ratio and the modulus ratio for the three models in Scheme 1.

θ=30° θ=60° θ=90° θ=120° θ=150° Rp(GK) -2.074 -0.980 0.318 1.868 6.302 Rp(BK) -2.153 -1.006 0.275 2.013 7.304 Rp(MK) -2.086 -0.914 0.277 1.923 6.635

Rm(GK) 5.16% 14.40% 16.82% 23.13% 31.27% Rm(BK) 5.50% 15.52% 17.91% 25.03% 32.85% Rm(MK) 5.77% 16.32% 18.27% 25.66% 34.63%

Figure 5-3: Results on included angle between vertical and oblique bar varying. (a) Rp (b) Rm

59

From the results of the Poisson’s ratio for all three kinds of kirigami, it is

fund that the auxetic behaviour could only be observed at the condition of

θ<90°. Besides, in the most of angle conditions of this scheme, all the three

types of kirigami possess the same value of Poisson’s ratio. When oblique bars

perpendicular to vertical bars or in other words, θ=90°, the value of Poisson’s

ratio in three models show positive, instead of zero. Therefore, we can conclude

that main reason that the internal angle affects the auxetic behaviour of this kind

of re-entrant cellular structured kirigami is that rotation of oblique bars.

However, when the angle increases up to 150°, the BK model has the highest

absolute value of Rp, and then, the MK model is in the middle. GK model has

the smallest absolute value of Rp, among the three models.

As for the Modulus ratio in all three types of kirigami, an increase in Rm

could be observed when θ increases. In other words, we could get a higher

modulus when θ becomes higher. The reason could be revealed by the

microstructure of re-entrant cellular structured kirigami. Its deformation results

mainly from the axial linear strain of all the bars when at a higher value of θ.

While, as the θ becomes smaller, rotation of oblique bars would take effects on

the modulus of re-entrant cellular structured kirigami. Accordingly, the structure

with greater θ could obtain larger increment of potential energy. According to

the equation(5.3), it could be concluded that the modulus is larger when θ is

larger. Comparing the value of modulus ratio of different 2D materials, we can

observe that the MK model has the highest value of Rm, the modulus ratio of BK

model is the smallest among three models. With the same geometric

configurations, the re-entrant cellular structured kirigami in MoS2 could get the

largest modulus ratio among three different 2D materials.

5.2.2 Results on different width of oblique bar

In this part, the effect of oblique bars is our major focus. To this end, different

value of N(wθ) is employed (from 2 to 7 is selected). The other parameters in

60

Figure 4-1(b) are kept unchanged. Figure 5-4 and Table 5-2 show the results of

Poisson’s ratio and modulus ratio. Table 5-2: Comparisons of Poisson’s ratio and the modulus ratio for the three models in Scheme 2.

N(wθ)=2 N(wθ)=3 N(wθ)=4 N(wθ)=5 N(wθ)=6 N(wθ)=7 Rp(GK) -0.810 -0.593 -0.470 -0.333 -0.235 -0.178 Rp(BK) -0.835 -0.655 -0.499 -0.371 -0.275 -0.200 Rp(MK) -0.782 -0.555 -0.453 -0.331 -0.237 -0.171

Rm(GK) 3.96% 6.54% 8.21% 9.43% 9.91% 9.92% Rm(BK) 4.12% 6.79% 8.97% 10.38% 11.04% 11.42% Rm(MK) 4.38% 7.39% 9.61% 11.00% 11.68% 11.76%

Figure 5-4: Results on width of oblique bar varying. (a) Rp (b) Rm

61

Firstly, for the Poisson’s ratio, as seen in Figure 5-4 and Table 5-2, the

absolute values of Rp in three models decrease along with an increase in the

width of the oblique bar. The value of Poisson’s ratios ranges from ~-0.8 to ~-

0.2. Besides, due to the internal angle equals to 60°, which is less than 90°, all

the Poisson’s ratios in this scheme are negative. Hence, the auxetic behaviour

could be observed in all the simulations. As for the difference in the three re-

entrant kirigami models, the influence of bulk matrix materials is very obvious.

The kirigami tailored from single-layer h-BN possesses the highest absolute

value of Poisson’s ratio, while the one originated from single-layer MoS2 has the

lowest value.

With regard of the modulus ratio, an obvious increase could be found in

all three models when N(wθ) becomes larger. And the value of Modulus Ratio

ranges from ~4% to ~11%. It could be concluded that the re-entrant

microstructure with wider oblique bars always has a larger value of Rm. As for

the comparison of different modulus ratio for three kinds of 2D materials, the

MK model has the highest value of Rm, the GK model has the lowest value, and

the value of BK model is between the MK model and GK model, which is the

same as the result in scheme 1.

5.2.3 Results on different width of vertical bar

In this scheme, N(w) is chosen to be 1,3,5,7, i.e., the width of vertical bar

increases monotonously, otherwise the other parameters in schematic diagram

are maintained constant. The tendency of Poisson’s ratio and modulus ratio

could be seen in Figure 5-5, and their accurate value are shown in Table 5-3.

62

Table 5-3: Comparisons of Poisson’s ratio and the modulus ratio for the three models in Scheme 3.

N(w)=1 N(w)=3 N(w)=5 N(w)=7

Rp(GK) -0.430 -0.455 -0.416 -0.372

Rp(BK) -0.455 -0.479 -0.460 -0.381

Rp(MK) -0.405 -0.439 -0.390 -0.363

Rm(GK) 4.67% 3.58% 2.82% 2.36%

Rm(BK) 5.08% 3.71% 2.92% 2.39%

Rm(MK) 5.26% 3.97% 3.72% 3.33%

Figure 5-5: Results on width of vertical bar varying. (a) Rp (b) Rm

63

Unlike the former two schemes, the absolute values of Rp do not vary

monotonously with the variations of width vertical bars. And from the

information in Figure 5-5 and Table 5-3, the absolute value comes to a peak at

N(w)=3. The reason could be revealed by the ratio between width of vertical bar

and length of oblique bar, which is discussed in Subsection 4.2.3. In regards to

the difference among three different 2D materials, the BK model owns the

highest absolute value of Poisson’s ratio, and the absolute value of Poisson’s

ratio in MK is less than that in GK. This kind of variation tendency and

regularity could also be found in scheme 2 and 150° in scheme 1.

As for the modulus ratio among all three models, there is an obvious

decrease in the value of Rm when the vertical bars becomes wider and wider in

the microstructure. The reason could be revealed by the variation of potential

energy of the whole model. As N(w) becoming larger, the decrease of variation

emerges, which results in decreasing of the modulus. In respect to the

comparison of different modulus ratio for three kinds of 2D materials, the MK

model has the highest value of Rm, the GK model has the lowest value. This kind

of tendency and regularity could also be found in the previous two schemes.

5.3 SUMMARY

To conclude, after the analysis of Poisson’s ratios and modulus ratios in three

different kirigami with re-entrant structure, the dependence of the two

mechanical properties on the sizes of microstructure is demonstrated again. And

the mechanical response of different kirigami with geometric similarity is also

calculated for investigating the effect of different matrix 2D materials through

comparisons of the responses.

The results show that the specified Poisson’s ratio and modulus can be

obtained by adjusting the arrangements of microstructure. Some conclusions

could be drawn as following:

(1) From the results of all schemes, when the internal angle between vertical

and oblique bar, θ is less than 90°, all kirigamis show auxetic. On the

contrary, once the internal angle is obtuse, the specific auxetic behaviour

could not happen.

(2) Among all the schemes, in most conditions, the kirigami tailored from

single-layer h-BN owns the highest absolute value of Poisson’s ratio,

while the one originated from single-layer MoS2 has the lowest. As a

result, it can be concluded that, with the similar geometric arrangements

in microstructure, the single-layer h-BN could help the re-entrant cellular

structured kirigami obtain higher absolute Poisson’s ratio value.

(3) From the results shown in all the schemes, the kirigami tailored from

single-layer MoS2 possesses the highest value of modulus ratio, while the

one derived from graphene has the lowest. As a result, it could be

concluded that, with the similar geometric arrangements in

microstructure, the single player MoS2 could make the re-entrant cellular

structured kirigami show stiffer mechanical behaviour.

65

Chapter 6 Conclusions and Future Works

6.1 CONCLUSIONS

In this thesis, an innovative method for achieving nanomaterials with a negative

Poisson’s ratio is demonstrated. After some preliminary investigations on 2D

materials and auxetic materials, we decide to pay more attention on auxetic

behaviour of the re-entrant cellular structured kirigami tailored from 2D

materials.

In general, in order to investigate the size and surface effect on the specific

auxetic behaviour of re-entrant cellular structured kirigami, this work conducts a

systematic study on the different conformational structures. Graphene is selected

as the research subject here. There are five numerical analysis schemes of

different geometric atomic system being adopted to discover the size effect.

Meanwhile, three types of kirigami models including pure GK model,

hydrogenated GK model and macroscopic continuum model, are employed to

explore the surface effect. Some conclusions is obtained as:

(1) Once the internal angle between vertical and oblique bar, θ is less than

90°, the kirigami always show auxetic behaviour.

(2) Of all the conditions, pure carbon GK, being with the highest surface

energy, has the highest absolute value of Poisson’s ratio among the

three models. The absolute value of Poisson’s ratio of the hydrogenated

GK is higher than that of the CK due to the higher surface energy of the

GK than that of the CK.

(3) The modulus ratio of hydrogenated GK is about 10%~20% higher than

that of the pure carbon GK. The modulus ratio of CK is far less than that

of the GK when they have the same geometric configurations. It

demonstrates the surface influence on the modulus of 2D nanomaterials.

In addition, the effect of matrix 2D materials on the performance of re-

entrant cellular structured kirigami is also illustrated in this thesis. There are

66

three kinds of kirigami models being used in this work, including GK model,

BK model and MK model. Some conclusions could be drawn:

(1) Once the internal angel between vertical and oblique bar, θ is less than

90°, the kirigami always show auxetic behaviour.

(2) Among all the models, in most conditions, the kirigami tailored from

single-layer h-BN owns the highest absolute value of Poisson’s ratio,

while the one originated from single-layer MoS2 has the lowest. Thus, it

can be concluded that, with the similar geometric arrangements in

microstructure, the single-layer h-BN could help the re-entrant cellular

structured kirigami obtain higher absolute Poisson’s ratio value.

(3) Of all the models, the kirigami tailored from single-layer MoS2

possesses the highest value of modulus ratio, while the one derived from

graphene has the lowest. As a result, it could be concluded that, with the

similar geometric arrangements in microstructure, the single player

MoS2 could make the re-entrant cellular structured kirigami show

stiffer mechanical behaviour.

6.2 LIMITATIONS

As demonstrated in Section 2.2, auxetic materials have many applications,

however, they also have their own limitations like other materials. Due to the

substantial porosity in these materials, they are normally not stiff enough or not

dense for load-bearing applications. For example, all of the results from Chapter

4 and Chapter 5 show that the maximum modulus ratio is just ~38%, which

proves that the tailoring method makes the re-entrant cellular structured kirigami

is not as stiff as its matrix 2D material.

Besides, MD simulation is not only capable of simulating many

sophisticated systems within modern high-performance research computing

clusters128,129; but also suitable for solving many well-known issues in research

fields. Firstly, the biggest problem of MD simulations is that the length of time

step is small92, which is usually on the order of femtoseconds. Therefore, MD

67

simulations are not capable of simulating experimental time scales of seconds or

even longer95,130. Besides, since the deformation in an MD simulation is

achieved in a very short time frame, the resulting strain ratio is usually nearly 10

orders of magnitude larger than it would be seen in real experiments131,132. In

such a way, an appropriate choice of applied strain ratio is able to describe the

mechanical deformation of materials133–135.

At last, this research work mainly concentrates on the discussions and

analysis on results from numerical simulations. Due to the limitations of

experimental technique nowadays, experimental verification to the most

numerical results obtained in this work is still of problem.

6.3 FUTURE WORKS

As aforementioned, there are some shortcomings in this thesis. Hence, some

improvements are expected to be done in the future to make this research work

more perfect. Next, aiming at the weakness of re-entrant structure solid, which is

not stiff enough, the tension test curve for the kirigami should be further

demonstrated. It usually contains following physical quantities, i.e., the linear

range of plastic strange, the nonlinear range of elastic strain, yield stress, facture

stress and ultimate stress. By studying this clearly and distinctly, some

improvements to modify this weakness could be raised. At last, the new

advanced real experiments to verify the results of this thesis are and always will

be our new research focus in the future.

68

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Publications

1. Cai, K., Luo, J., Ling, Y., Wan, J. & Qin, Q.H. Effects of size and surface

on the auxetic behaviour of monolayer graphene kirigami. Sci. Rep. 6,

(2016).

2. Luo, J., Cai, K., Qin, Q.H. Auxetic behaviour of re-entrant cellular

structured kirigami tailored from different 2D materials. To be submitted.

80

81

Appendix A Table A- 1: The parameters of B-N-C Tersoff potential105.

Element 1

Element2

Element3 m γ λ3

(Å-1) c D (Å) cosθ n β λ2

(Å-1) B

（eV） R

(Å) D λ 1 (Å-1)

A（eV）

N B B 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.199 340 1.95

0.05 3.568 1380

N B N 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.199 340 1.95

0.05 3.568 1380

N B C 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.199 340 1.95

0.05 3.568 1380

B N B 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.199 340 1.95

0.05 3.568 1380

B N N 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.199 340 1.95

0.05 3.568 1380

B N C 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.199 340 1.95

0.05 3.568 1380

N N B 3 1 0 17.7959 5.9484 0 0.618443

0.019251

2.627272

138.7787 2 0.1 2.82930

9 128.868

7

N N N 3 1 0 17.7959 5.9484 0 0.618443

0.019251

2.627272

138.7787 2 0.1 2.82930

9 128.868

7

N N C 3 1 0 17.7959 5.9484 0 0.618443

0.019251

2.627272

138.7787 2 0.1 2.82930

9 128.868

7

B B B 3 1 0 0.52629 0.001587 0.5 3.99290

6 1.60E-

06 2.07749

8 43.1320

2 2 0.1 2.237258

40.05202

B B N 3 1 0 0.52629 0.001587 0.5 3.99290

6 1.60E-

06 2.07749

8 43.1320

2 2 0.1 2.237258

40.05202

B B C 3 1 0 0.52629 0.001587 0.5 3.99290

6 1.60E-

06 2.07749

8 43.1320

2 2 0.1 2.237258

40.05202

C C C 3 1 0 3.80E+04 4.3484 -0.93 0.72751 1.57E-

07 2.2119 430 1.95

0.15 3.4879 1393.6

Continued

82

Continued

C C B 3 1 0 3.80E+04 4.3484 -0.93 0.72751 1.57E-

07 2.2119 430 1.95

0.15 3.4879 1393.6

C C N 3 1 0 3.80E+04 4.3484 -0.93 0.72751 1.57E-

07 2.2119 430 1.95

0.15 3.4879 1393.6

C B B 3 1 0 3.80E+04 4.3484 -0.93 0.72751 1.57E-

07 2.2054 339.0689

1.95 0.1 3.5279 1386.78

C B N 3 1 0 3.80E+04 4.3484 -0.93 0.72751 1.57E-

07 2.2054 339.0689

1.95 0.1 3.5279 1386.78

C B C 3 1 0 3.80E+04 4.3484 -0.93 0.72751 1.57E-

07 2.2054 339.0689

1.95 0.1 3.5279 1386.78

C N B 3 1 0 3.80E+04 4.3484 -0.93 0.72751 1.57E-

07 2.2054 387.5752

1.95 0.1 3.5279 1386.78

C N N 3 1 0 3.80E+04 4.3484 -0.93 0.72751 1.57E-

07 2.2054 387.5752

1.95 0.1 3.5279 1386.78

C N C 3 1 0 3.80E+04 4.3484 -0.93 0.72751 1.57E-

07 2.2054 387.5752

1.95 0.1 3.5279 1386.78

B C C 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.2054 339.0689

1.95 0.1 3.5279 1386.78

B C B 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.2054 339.0689

1.95 0.1 3.5279 1386.78

B C N 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.2054 339.0689

1.95 0.1 3.5279 1386.78

N C C 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.2054 387.5752

1.95 0.1 3.5279 1386.78

N C B 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.2054 387.5752

1.95 0.1 3.5279 1386.78

N C N 3 1 0 25000 4.3484 -0.89 0.72751 1.26E-

07 2.2054 387.5752

1.95 0.1 3.5279 1386.78

83

Table A- 2: The parameters of Mo-S SW potential106.

Two Body SW Potential max[ /( )] 4( / 1)r rtwoE Ae B rρ −= −

A(eV) ρ(Å) B(Å4) rmin(Å) rmax(Å) Mo-S 6.918 1.252 17.771 0 3.16

Three Body SW potential 1 12 max12 2 13 max13[ / ) /( )] 20(cos cos )r r r r

threeE Ke ρ ρ θ θ− + −= −

K(eV) θ(°) ρ1(Å) ρ2(Å) rmin12(Å) rmax12(Å) rmin13(Å) rmax13(Å) rmin23(Å) rmax23(Å) Mo-S-S 67.883 81.788 1.252 1.252 0 3.16 0 3.16 0 3.78

S-Mo-Mo 62.449 81.788 1.252 1.252 0 3.16 0 3.16 0 4.27

84

Appendix B

MD simulation script for ideal graphene nanoribbon

# ---------- Initialize Simulation ---------------------

clear

dimension 3

units metal

processors 4 4 1

boundary p p p

atom_style full

neighbor 2.0 nsq

neigh_modify every 1 delay 0 check yes

# ---------- Define Interatomic Potential --------------

read_data cellwe.data

pair_style airebo 3.0 1 1

pair_coeff * * CH.airebo C C C C C

# ---------- Create Atom Definition---------------------

group Gbase type 1

group GL1 type 2

group GL2 type 3

group GR1 type 4

group GR2 type 5

variable L1X equal xcm(GL1,x)

variable L1Y equal xcm(GL1,y)

variable L2X equal xcm(GL2,x)

variable L2Y equal xcm(GL2,y)

variable R1X equal xcm(GR1,x)

variable R1Y equal xcm(GR1,y)

85

variable R2X equal xcm(GR2,x)

variable R2Y equal xcm(GR2,y)

variable LX equal "((v_R1X+v_R2X)-(v_L1X+v_L2X))/2"

variable LY equal "((v_L1Y+v_R1Y)-(v_L2Y+v_R2Y))/2"

# ---------- Define Computation Settings ---------------

compute sta all stress/atom NULL

compute PE all pe

variable PE equal c_PE

fix data all ave/time 2000 1 2000 v_L1X v_L1Y v_L2X v_L2Y v_R1X v_R1Y

v_R2X v_R2Y v_LX v_LY v_PE file cellwe.data.txt

# ---------- Energy Minimization ----------------------

timestep 0.0005

dump 1 all custom 1000 relax.cellwe.lammpstrj id mol type x y z fx fy fz

thermo 500

thermo_style custom step temp v_LX v_LY ly

thermo_modify lost ignore

# ---------- Energy Equilibration -------------------

min_style cg

minimize 1.0e-12 1.0e-12 1500 1500

fix fxnpt all npt temp 8.0 8.0 0.1 iso 0.0 0.0 1.0

run 400000

undump 1

unfix fxnpt

write_restart relax.cellwe.restart

# ---------- Set Simulation --------------------

dump 2 all custom 1000 load.cellwe.lammpstrj id mol type x y z fx fy fz

variable Ly equal "ly"

print "Initial Length, Ly=${Ly}"

86

fix fxnvt all nvt temp 8.0 8.0 0.1

variable srate equal 0.05

variable erate equal "v_srate/1.0e3"

fix fxdeform all deform 1 y erate ${erate} units box

run 2000000

write_restart load.cellwe.restart

# ---------- Simulation Done ---------------------------

print "All done"

MD simulation script for de-hydrogenated graphene kirigami

# ---------- Initialize Simulation ---------------------

clear

dimension 3

units metal

processors 4 4 1

boundary p p p

atom_style full

neighbor 2.0 nsq

neigh_modify every 1 delay 0 check yes

# ---------- Define Interatomic Potential --------------

read_data cellw3.data

pair_style airebo 3.0 1 1

pair_coeff * * CH.airebo C C C C C

# ---------- Create Atom Definition---------------------

group Gbase type 1

group GL1 type 2

group GL2 type 3

group GR1 type 4

group GR2 type 5

87

group GC type 1 2 3 4 5

variable L1X equal xcm(GL1,x)

variable L1Y equal xcm(GL1,y)

variable L2X equal xcm(GL2,x)

variable L2Y equal xcm(GL2,y)

variable R1X equal xcm(GR1,x)

variable R1Y equal xcm(GR1,y)

variable R2X equal xcm(GR2,x)

variable R2Y equal xcm(GR2,y)

variable LX equal "((v_R1X+v_R2X)-(v_L1X+v_L2X))/2"

variable LY equal "((v_L1Y+v_R1Y)-(v_L2Y+v_R2Y))/2"

# ---------- Define Computation Settings ---------------

compute sta all stress/atom NULL

compute PE all pe

compute PECa GC pe/atom

compute PEC GC reduce sum c_PECa

fix data all ave/time 2000 1 2000 v_L1X v_L1Y v_L2X v_L2Y v_R1X v_R1Y

v_R2X v_R2Y v_LX v_LY c_PE c_PEC file relax.cellw3.data.txt

# ---------- Energy Minimization ----------------------

timestep 0.0005

dump 1 all custom 1000 relax.cellw3.lammpstrj id mol type x y z fx fy fz

thermo 500

thermo_style custom step temp pe etotal v_LX v_LY ly

thermo_modify lost ignore

# ---------- Energy Equilibration --------------------

min_style cg

minimize 1.0e-12 1.0e-12 1500 1500

88

fix fxnpt all npt temp 8.0 8.0 0.1 iso 0.0 0.0 1.0

run 5000

undump 1

unfix fxnpt

write_restart relax.cellw3.restart

# ---------- Set Simulation --------------------

reset_timestep 0

unfix data

fix data all ave/time 2000 1 2000 v_L1X v_L1Y v_L2X v_L2Y v_R1X v_R1Y

v_R2X v_R2Y v_LX v_LY c_PE c_PEC file load.cellw3.data.txt

dump 2 all custom 1000 load.cellw3.lammpstrj id mol type x y z fx fy fz

variable Ly equal "ly"

print "Initial Length, Ly=${Ly}"

fix fxnvt all nvt temp 8.0 8.0 0.1

variable srate equal 0.05 #5%

variable erate equal "v_srate/1.0e3" #2000000*0.0005=1000ps deform

fix fxdeform all deform 1 y erate ${erate} units box

run 2000000

write_restart load.cellw3.restart

# ---------- Simulation Done ---------------------------

print "All done"

MD simulation script for hydrogenated graphene kirigami

# ---------- Initialize Simulation ---------------------

clear

dimension 3

units metal

processors 4 4 1

boundary p p p

89

atom_style full

neighbor 2.0 nsq

neigh_modify every 1 delay 0 check yes

# ---------- Define Interatomic Potential --------------

read_data cellw3.data

pair_style airebo 3.0 1 1

pair_coeff * * CH.airebo C C C C C H

# ---------- Create Atom Definition---------------------

group Gbase type 1

group GL1 type 2

group GL2 type 3

group GR1 type 4

group GR2 type 5

group GC type 1 2 3 4 5

group GH type 6

variable L1X equal xcm(GL1,x)

variable L1Y equal xcm(GL1,y)

variable L2X equal xcm(GL2,x)

variable L2Y equal xcm(GL2,y)

variable R1X equal xcm(GR1,x)

variable R1Y equal xcm(GR1,y)

variable R2X equal xcm(GR2,x)

variable R2Y equal xcm(GR2,y)

variable LX equal "((v_R1X+v_R2X)-(v_L1X+v_L2X))/2"

variable LY equal "((v_L1Y+v_R1Y)-(v_L2Y+v_R2Y))/2"

# ---------- Define Computation Settings ---------------

compute sta all stress/atom NULL

compute PE all pe

90

compute PECa GC pe/atom

compute PEC GC reduce sum c_PECa

compute PEHa GH pe/atom

compute PEH GH reduce sum c_PEHa

fix data all ave/time 2000 1 2000 v_L1X v_L1Y v_L2X v_L2Y v_R1X v_R1Y

v_R2X v_R2Y v_LX v_LY c_PE c_PEC c_PEH file relax.cellw3.data.txt

# ---------- Energy Minimization ----------------------

timestep 0.0005

dump 1 all custom 1000 relax.cellw3.lammpstrj id mol type x y z fx fy fz

thermo 500

thermo_style custom step temp pe etotal v_LX v_LY ly

thermo_modify lost ignore

# ---------- Energy Equilibration --------------------

min_style cg

minimize 1.0e-12 1.0e-12 1500 1500

fix fxnpt all npt temp 8.0 8.0 0.1 iso 0.0 0.0 1.0

run 400000

undump 1

unfix fxnpt

write_restart relax.cellw3.restart

# ---------- Set Simulation --------------------

reset_timestep 0

unfix data

fix data all ave/time 2000 1 2000 v_L1X v_L1Y v_L2X v_L2Y v_R1X v_R1Y

v_R2X v_R2Y v_LX v_LY c_PE c_PEC c_PEH file load.cellw3.data.txt

dump 2 all custom 1000 load.cellw3.lammpstrj id mol type x y z fx fy fz

variable Ly equal "ly"

print "Initial Length, Ly=${Ly}"

91

fix fxnvt all nvt temp 8.0 8.0 0.1

variable srate equal 0.05

variable erate equal "v_srate/1.0e3"

fix fxdeform all deform 1 y erate ${erate} units box

run 2000000

write_restart load.cellw3.restart

# ---------- Simulation Done ---------------------------

print "All done"

92

Appendix C

MD simulation script for ideal h-BN nanoribbon

# ---------- Initialize Simulation ---------------------

clear

dimension 3

units metal

processors 4 4 1

boundary p p p

atom_style full

neighbor 2.0 nsq

neigh_modify every 1 delay 0 check yes

# ---------- Define Interatomic Potential --------------

read_data SLBN.data

pair_style tersoff

pair_coeff * * BNC.tersoff B N

# ---------- Create Atom Definition---------------------

region GS block 32.5 96.2 18.7 56.4 INF INF

group GS region GS

region GT block 32.5 96.2 56.3 56.4 INF INF

group GT region GT

region GB block 32.5 96.2 18.7 18.8 INF INF

group GB region GB

region GL block 32.5 32.6 18.7 56.4 INF INF

group GL region GL

region GR block 96.1 96.2 18.7 56.4 INF INF

group GR region GR

variable SL equal xcm(GL,x)

93

variable SR equal xcm(GR,x)

variable ST equal xcm(GT,y)

variable SB equal xcm(GB,y)

variable WS equal "v_SR-v_SL"

variable LS equal "v_ST-v_SB"

variable TS equal 3.3306

variable VS equal "v_WS*v_LS*v_TS*1.0e-30"

variable Vall equal "lx*ly*v_TS*1.0e-30"

# ---------- Define Computation Settings ---------------

#all PE

compute PE all pe

#all PES

compute PESa GS pe/atom

compute PES GS reduce sum c_PESa

fix data all ave/time 2000 1 2000 v_LS v_WS v_VS v_Vall c_PE c_PES file

relax.SLBN.data.txt

# ---------- Energy Minimization ----------------------

timestep 0.0005

dump 1 all custom 1000 relax.SLBN.lammpstrj id mol type x y z fx fy fz

thermo 500

thermo_style custom step temp pe ke etotal v_WS v_LS v_VS v_Vall

thermo_modify lost ignore

# ---------- Energy Equilibration --------------------

min_style cg

minimize 1.0e-12 1.0e-12 1500 1500

fix fxnpt all npt temp 8.0 8.0 0.1 iso 0.0 0.0 1.0

run 400000

undump 1

unfix fxnpt

94

write_restart relax.SLBN.restart

# ---------- Set Simulation --------------------

reset_timestep 0

unfix data

fix data all ave/time 2000 1 2000 v_LS v_WS v_VS v_Vall c_PE c_PES file

load.SLBN.data.txt

dump 2 all custom 1000 load.SLBN.lammpstrj id mol type x y z fx fy fz

variable Ly equal "ly"

print "Initial Length, Ly=${Ly}"

fix fxnvt all nvt temp 8.0 8.0 0.1

variable srate equal 0.05

variable erate equal "v_srate/1.0e3"

fix fxdeform all deform 1 y erate ${erate} units box

run 2000000

write_restart load.SLBN.restart

unfix fxdeform

run 600000

# ---------- Simulation Done ---------------------------

print "All done"

MD simulation script for h-BN kirigami

# ---------- Initialize Simulation ---------------------

clear

dimension 3

units metal

processors 4 4 1

boundary p p p

atom_style full

neighbor 2.0 nsq

95

neigh_modify every 1 delay 0 check yes

# ---------- Define Interatomic Potential --------------

read_data BN_w3.data

pair_style tersoff

pair_coeff * * BNC.tersoff B N

# ---------- Create Atom Definition---------------------

group GB type 1

group GN type 2

variable CX1 equal 68.6698

variable CX2 equal 263.8365

variable CY1 equal 68.8598

variable CY2 equal 11.268

variable CR equal 1.45

region GL1 cylinder z ${CX1} ${CY1} ${CR} INF INF units box

region GL2 cylinder z ${CX1} ${CY2} ${CR} INF INF units box

region GR1 cylinder z ${CX2} ${CY1} ${CR} INF INF units box

region GR2 cylinder z ${CX2} ${CY2} ${CR} INF INF units box

group GL1 region GL1

group GL2 region GL2

group GR1 region GR1

group GR2 region GR2

variable L1X equal xcm(GL1,x)

variable L1Y equal xcm(GL1,y)

variable L2X equal xcm(GL2,x)

variable L2Y equal xcm(GL2,y)

variable R1X equal xcm(GR1,x)

variable R1Y equal xcm(GR1,y)

variable R2X equal xcm(GR2,x)

96

variable R2Y equal xcm(GR2,y)

variable LX equal "((v_R1X+v_R2X)-(v_L1X+v_L2X))/2"

variable LY equal "((v_L1Y+v_R1Y)-(v_L2Y+v_R2Y))/2"

# ---------- Define Computation Settings ---------------

#all PE

compute PE all pe

#all PEMo

compute PENa GN pe/atom

compute PEN GN reduce sum c_PENa

#all PES

compute PEBa GB pe/atom

compute PEB GB reduce sum c_PEBa

fix data all ave/time 2000 1 2000 v_L1X v_L1Y v_L2X v_L2Y v_R1X v_R1Y

v_R2X v_R2Y v_LX v_LY c_PE c_PEB c_PEN file relax.BN_w3.data.txt

# ---------- Energy Minimization ----------------------

timestep 0.0005

dump 1 all custom 1000 relax.BN_w3.lammpstrj id mol type x y z fx fy fz

thermo 500

thermo_style custom step temp pe ke etotal v_LX v_LY ly

thermo_modify lost ignore

# ---------- Energy Equilibration --------------------

min_style cg

minimize 1.0e-12 1.0e-12 1500 1500

fix fxnpt all npt temp 8.0 8.0 0.1 iso 0.0 0.0 1.0

run 400000

undump 1

unfix fxnpt

97

write_restart relax.BN_w3.restart

# ---------- Set Simulation --------------------

reset_timestep 0

unfix data

fix data all ave/time 2000 1 2000 v_L1X v_L1Y v_L2X v_L2Y v_R1X v_R1Y

v_R2X v_R2Y v_LX v_LY c_PE c_PEB c_PEN file load.BN_w3.data.txt

dump 2 all custom 1000 load.BN_w3.lammpstrj id mol type x y z fx fy fz

variable Ly equal "ly"

print "Initial Length, Ly=${Ly}"

fix fxnvt all nvt temp 8.0 8.0 0.1

variable srate equal 0.05

variable erate equal "v_srate/1.0e3"

fix fxdeform all deform 1 y erate ${erate} units box

run 2000000

write_restart load.BN_w3.restart

# ---------- Simulation Done ---------------------------

print "All done"

98

Appendix D

MD simulation script for ideal MoS2 nanoribbon

# ---------- Initialize Simulation ---------------------

clear

dimension 3

units metal

processors 4 4 1

boundary p p p

atom_style full

neighbor 2.0 nsq

neigh_modify every 1 delay 0 check yes

# ---------- Define Interatomic Potential --------------

read_data SLMD.data

pair_style sw

pair_coeff * * mos2.sw Mo S

# ---------- Create Atom Definition---------------------

region GS block 41.0 121.4 23.7 71.2 INF INF

group GS region GS

region GT block 41.0 121.4 71.0 71.2 INF INF

group GT region GT

region GB block 41.0 121.4 23.7 23.8 INF INF

group GB region GB

region GL block 41.0 41.1 23.7 71.2 INF INF

group GL region GL

region GR block 121.3 121.4 23.7 71.2 INF INF

group GR region GR

99

variable SL equal xcm(GL,x)

variable SR equal xcm(GR,x)

variable ST equal xcm(GT,y)

variable SB equal xcm(GB,y)

variable WS equal "v_SR-v_SL"

variable LS equal "v_ST-v_SB"

variable VS equal "v_WS*v_LS*6.092*1.0e-30"

variable Vall equal "lx*ly*6.092*1.0e-30"

# ---------- Define Computation Settings ---------------

#all PE

compute PE all pe

#all PES

compute PESa GS pe/atom

compute PES GS reduce sum c_PESa

fix data all ave/time 2000 1 2000 v_LS v_WS v_VS v_Vall c_PE c_PES file

relax.SLMD.data.txt

# ---------- Energy Minimization ----------------------

timestep 0.0005

dump 1 all custom 1000 relax.SLMD.lammpstrj id mol type x y z fx fy fz

thermo 500

thermo_style custom step temp pe ke etotal v_WS v_LS v_VS v_Vall

thermo_modify lost ignore

# ---------- Energy Equilibration --------------------

min_style cg

minimize 1.0e-12 1.0e-12 1500 1500

fix fxnpt all npt temp 8.0 8.0 0.1 iso 0.0 0.0 1.0

run 400000

100

undump 1

unfix fxnpt

write_restart relax.SLMD.restart

# ---------- Set Simulation --------------------

reset_timestep 0

unfix data

fix data all ave/time 2000 1 2000 v_LS v_WS v_VS v_Vall c_PE c_PES file

load.SLMD.data.txt

dump 2 all custom 1000 load.SLMD.lammpstrj id mol type x y z fx fy fz

variable Ly equal "ly"

print "Initial Length, Ly=${Ly}"

fix fxnvt all nvt temp 8.0 8.0 0.1

variable srate equal 0.05

variable erate equal "v_srate/1.0e3"

fix fxdeform all deform 1 y erate ${erate} units box

run 2000000

write_restart load.SLMD.restart

unfix fxdeform

run 600000

# ---------- Simulation Done ---------------------------

print "All done"

MD simulation script for MoS2 kirigami

# ---------- Initialize Simulation ---------------------

clear

dimension 3

units metal

processors 4 4 1

boundary p p p

101

atom_style full

neighbor 2.0 nsq

neigh_modify every 1 delay 0 check yes

# ---------- Define Interatomic Potential --------------

read_data mos2_w3.data

pair_style sw

pair_coeff * * mos2.sw Mo S

# ---------- Create Atom Definition---------------------

group GMo type 1

group GS type 2

fix fixallz GMo setforce NULL NULL 0.0

group Gbase molecule 1

group GL1 molecule 2

group GL2 molecule 3

group GR1 molecule 4

group GR2 molecule 5

variable L1X equal xcm(GL1,x)

variable L1Y equal xcm(GL1,y)

variable L2X equal xcm(GL2,x)

variable L2Y equal xcm(GL2,y)

variable R1X equal xcm(GR1,x)

variable R1Y equal xcm(GR1,y)

variable R2X equal xcm(GR2,x)

variable R2Y equal xcm(GR2,y)

variable LX equal "((v_R1X+v_R2X)-(v_L1X+v_L2X))/2"

variable LY equal "((v_L1Y+v_R1Y)-(v_L2Y+v_R2Y))/2"

# ---------- Define Computation Settings ---------------

#all PE

102

compute PE all pe

#all PEMo

compute PEMoa GMo pe/atom

compute PEMo GMo reduce sum c_PEMoa

#all PES

compute PESa GS pe/atom

compute PES GS reduce sum c_PESa

#fix fixz all setforce NULL NULL 0.0

fix data all ave/time 2000 1 2000 v_L1X v_L1Y v_L2X v_L2Y v_R1X v_R1Y

v_R2X v_R2Y v_LX v_LY c_PE c_PEMo c_PES file relax.mos2_w3.data.txt

# ---------- Energy Minimization ----------------------

timestep 0.0005

dump 1 all custom 1000 relax.mos2_w3.lammpstrj id mol type x y z fx fy fz

thermo 500

thermo_style custom step temp pe ke etotal v_LX v_LY ly

thermo_modify lost ignore

# ---------- Energy Equilibration --------------------

min_style cg

minimize 1.0e-12 1.0e-12 1500 1500

#fix fxnvt all nvt temp 0.1 0.1 0.1

fix fxnpt all npt temp 8.0 8.0 0.1 iso 0.0 0.0 1.0

run 400000

undump 1

unfix fxnpt

write_restart relax.mos2_w3.restart

# ---------- Set Simulation --------------------

reset_timestep 0

unfix data

103

fix data all ave/time 2000 1 2000 v_L1X v_L1Y v_L2X v_L2Y v_R1X v_R1Y

v_R2X v_R2Y v_LX v_LY c_PE c_PEMo c_PES file load.mos2_w3.data.txt

dump 2 all custom 1000 load.mos2_w3.lammpstrj id mol type x y z fx fy fz

variable Ly equal "ly"

print "Initial Length, Ly=${Ly}"

fix fxnvt all nvt temp 8.0 8.0 0.1

variable srate equal 0.05

variable erate equal "v_srate/1.0e3"

fix fxdeform all deform 1 y erate ${erate} units box

run 2000000

write_restart load.mos2_w3.restart

# ---------- Simulation Done ---------------------------

print "All done"