PHYS466 ProjectKyoungmin Min, Namjung Kim and Ravi Bhadauria

ContentsContents

• INTRODUCTION

• CURRENT RESEARCH

• OBJECTIVES

• SIMULATION SETUP

• RESULT AND DISCUSSION

• CONCLUSION

• REFERENCES

• INTRODUCTION

• CURRENT RESEARCH

• OBJECTIVES

• SIMULATION SETUP

• RESULT AND DISCUSSION

• CONCLUSION

• REFERENCES

Graphene: IntroductionGraphene: Introduction

• A single layer of sp2 hybridized carbon atoms in a honeycomb lattice

• Multiple use in areas of Electronics, Material Science and Mechanical Engineering

• Extraordinary mechanical properties (fracture strength is 200 times greater than steel) [2]

• A single layer of sp2 hybridized carbon atoms in a honeycomb lattice

• Multiple use in areas of Electronics, Material Science and Mechanical Engineering

• Extraordinary mechanical properties (fracture strength is 200 times greater than steel) [2]

Graphene Graphene

Current researchCurrent research

Approach Sheet description

Young’s modulus(TPa)

Experimental observation [6]

Graphene 1.000

Structural mechanics:Stiffness matrix[6]

Zigzag sheeta=9.847nm / b=1.969nm

1.024

Structural mechanics:Stiffness matrix[6]

Zigzag sheeta=20.178nm /

b=4.184nm

1.033

molecular structural mechanics[5]

Zigzag sheeta=6.3945nm / b=4.1841nm

1.040

molecular structural mechanics[5]

Armchair sheeta=6.1531nm / b=4.2630nm

1.042

Young’s modulus Young’s modulusChirality Shear

modulus(TPa)

Zigzag sheet[5]a=6.3945nm / b=4.1841nm

0.213

Armchair sheet[5]a=6.1531nm / b=4.2630nm

0.228

Chiral sheet[5]a=4.7130nm / b=3.2559nm

0.233

Zigzag sheet[7]# of cells from 4 to 10

0.667~0.490

Armchair sheet[7]# of cells from 4 to 10

0.482~0.447

Shear modulus Shear modulus

Chirality Poisson’s ratio

Zigzag sheet[7]# of cells from 4 to 10

0.129~0.190

Armchair sheet[7]# of cells from 4 to 10

0.261~0.0.216

Poisson’s ratio Poisson’s ratio

Approach Remarks

Indentation load Critical wrinkling load

Wrinkling behavior Wrinkling behavior

ObjectivesObjectives

• Investigate the shear properties of finite sized graphene under various

shear loading conditions.

• Chirality effects and size dependence

• Investigate the wrinkling behavior under shear conditions.

• Investigate the shear properties of finite sized graphene under various

shear loading conditions.

• Chirality effects and size dependence

• Investigate the wrinkling behavior under shear conditions.

Simulation SetupSimulation Setup

• Adaptive Intermolecular Reactive Empirical Bond Order(AIREBO)[11] extends

the interaction range by changing cutoff function based on REBO potential [13].

• AIREBO potential allows for covalent bond breaking and creation with

associated changes in atomic hybridization within classical potential.

• Adaptive Intermolecular Reactive Empirical Bond Order(AIREBO)[11] extends

the interaction range by changing cutoff function based on REBO potential [13].

• AIREBO potential allows for covalent bond breaking and creation with

associated changes in atomic hybridization within classical potential.

POTENTIAL FUNCTION POTENTIAL FUNCTION

• Large-scale Atomic/Molecular Massively Parallel Simulator(LAMMPS) is used

as molecular dynamics engine.

• Large-scale Atomic/Molecular Massively Parallel Simulator(LAMMPS) is used

as molecular dynamics engine.

LAMMPS[12] LAMMPS[12]

Simulation SetupSimulation Setup

BOUNDARY CONDITIONS BOUNDARY CONDITIONS

2 4 6 8 10 12 14 16 18

2

4

6

8

10

12

14

16

18

Original

Deformed

Move and Fixtop and bottomMove and Fixtop and bottom

0 5 10 15 20 25

2

4

6

8

10

12

14

16

18

Fix top and bottomafter simple shearFix top and bottomafter simple shear

2 4 6 8 10 12 14 16 18

2

4

6

8

10

12

14

16

18

Original

Deformed

Apply velocity ontopApply velocity ontop

1) Apply strain directly 2) Displace and relax top atoms 3) Apply velocity BC

• Similar to 2), takes more time

• Time Step: 0.1 fs• Relaxed 10000 steps before simulation• Displaced 0.05 Å and relaxed 100000 steps

• Easily unstable

Results and DiscussionResults and Discussion

• After 400 steps, temperature, pressure and potential energy reached at the stable state.• After 400 steps, temperature, pressure and potential energy reached at the stable state.

RELAXATION RELAXATION

0 2000 4000 6000 8000 10000 120000

200

400

600

Timestep

Tem

perature, [K

]

0 2000 4000 6000 8000 10000 12000-1500

-1000

-500

0

Timestep

Pressure, [B

ar]

0 2000 4000 6000 8000 10000 12000-6.88

-6.86

-6.84

-6.82

Timestep

Potentia

l E

nergy

Tem

pera

ture

Tem

pera

ture

Pres

sure

Pres

sure

Pote

ntial

EPo

tenti

al E

Relaxation point

Results and DiscussionResults and Discussion

• Temperature, pressure and potential energy show discontinuity to corroborate the fracture.• Temperature, pressure and potential energy show discontinuity to corroborate the fracture.

UNDER SHEAR LOAD UNDER SHEAR LOAD

Tem

pera

ture

Tem

pera

ture

Pres

sure

Pres

sure

Pote

ntial

EPo

tenti

al E

Fracture

Shear 1 (Armchair) Shear 2 (Armchair)

Shear 1 (Zigzag) Shear 2 (Zigzag)

Armchair Armchair

Zigzag Zigzag

Results and DiscussionResults and Discussion

• As the size of structure increases, more shear stress and strain can be hold.• For small atoms case, chirality affects more on shear modulus comparing to large atoms case.• Shear modulus increases as the size of structure increases.

• As the size of structure increases, more shear stress and strain can be hold.• For small atoms case, chirality affects more on shear modulus comparing to large atoms case.• Shear modulus increases as the size of structure increases.

SHEAR STRESS-STRAIN RELATIONSHIP SHEAR STRESS-STRAIN RELATIONSHIP

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

10

20

30

40

50

60

Shear Strain

She

ar S

tres

s, [

GP

a]

Zigzag, 72atoms

Armchair, 72atomsZigzag, 576atoms

Armchair, 576atoms

Results and DiscussionResults and Discussion

• As the size is increased, shear properties are converged to bulk value(shear modulus = 480GPa(zigzag) and 450GPa(armchair), shear strength=60GPa)•The edge effect from different chiralities influences the behavior of the structure, especially in small size.

• As the size is increased, shear properties are converged to bulk value(shear modulus = 480GPa(zigzag) and 450GPa(armchair), shear strength=60GPa)•The edge effect from different chiralities influences the behavior of the structure, especially in small size.

SHEAR PROPERTIES SHEAR PROPERTIES

0 100 200 300 400 500 600 700 800 900250

300

350

400

450

Number of Atoms

Shear

modulu

s,

[GP

a]

0 100 200 300 400 500 600 700 800 90030

40

50

60

Number of Atoms

Shear

str

ength

, [G

Pa]

0 100 200 300 400 500 600 700 800 9000.12

0.14

0.16

0.18

Number of Atoms

Fra

ctu

re s

hear

str

ain

Zigzag

Armchair

Shea

r M

odul

usSh

ear

Mod

ulus

Shea

r St

reng

thSh

ear

Stre

ngth

Frac

ture

sh

ear

stra

in

Frac

ture

sh

ear

stra

in

Shear modulus in bulk case = 480Gpa(zigzag) and 450GPa(armchair)

Shear strength in bulk case = 60 GPa

Results and DiscussionResults and Discussion

• The ratio of amplitude and half wave length relationship [10]

• The ratio of amplitude and half wave length relationship [10]

WRINKLING BEHAVIOR WRINKLING BEHAVIOR

2(1 )A

: Amplitude

: Half-wavelength

: Poisson's ratio

: Shear strain

A

010

2030

4050

0

10

20

30

40

50

-4

-2

0

2

4

010

2030

4050

0

20

40

60

-4

-2

0

2

4

0 10 20 30 40 500204060

-4

-3

-2

-1

0

1

2

3

4

ACAC

ZZZZWrinkle viewWrinkle view

MD result, 836 atoms MD result, 836 atoms

# of Atoms ν (Armchair) ν (Zigzag)

72 0.270766 0.270766

180 0.272553 0.256468

364 0.257362 0.239489

576 0.243957 0.225191

836 0.240383 0.224298Bulk case [9] 0.21

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Strain

Am

plitu

de /

Wav

elen

gth

Zigzag, 72 atoms

Zigzag, 836 atoms

Armchair, 72 atoms

Armchair, 836 atoms

γγ

A /

λA

/ λ

THEORETICAL RESULTS THEORETICAL RESULTS

ConclusionConclusion

• Optimized boundary condition for shear stress was explored.

• The shear properties and wrinkling behavior of finite size graphene have been studied in this work.

• Chirality effect and size dependency of shear modulus, shear strength and fracture shear strain are observed.

• Wrinkles were observed and needed further investigation.

• Optimized boundary condition for shear stress was explored.

• The shear properties and wrinkling behavior of finite size graphene have been studied in this work.

• Chirality effect and size dependency of shear modulus, shear strength and fracture shear strain are observed.

• Wrinkles were observed and needed further investigation.

ReferencesReferences

[1] F. Schendin et al., Nature Materials, Vol. 6, pp. 652-655 (2007)

[2] C. Lee et al., Science, Vol. 321, No. 5887, pp. 385-388 (2008)

[3] I. W. Frank et al., Journal of Vacuum Science and Technology B, Vol. 25, No. 6, pp. 2558-2561 (2007)

[4] J. Meyer et al., Nature, Vol.446, pp. 60-63 (2007)

[5] A. Sakhaee-Pour, Solid State Communications, Vol. 149, No. 1-2, pp. 91-95 (2009)

[6] C. Li and T. W. Chou, International Journal of Solid and Structures, Vol. 40, No. 10, pp. 2487-2499 (2003)

[7] R. Faccio et al., Journal of Physics: Condensed Matter, Vol. 21, No. 28, pp. 5304-5310 (2009)

[8] H. Bu et al., Physics Letters A, Vol. 373, No. 37, pp. 3359-3362 (2009)

[9] H. Zhao et al., Nano Letters, Vol. 9, No. 8, pp. 3012-3015 (2009)

[10] Y. W. Wong and S. Pellegrino, Journal of Mechanics of Materials and Structures, Vol. 1, No. 1, pp. 27-

61 (2006)

[11] S. Stuart et al., Journal of Chemical Physics, Vol. 112, No. 14, pp. 6472-6486 (2000)

[12] S. J. Plimpton, Journal of Computational Physics, Vol. 117, pp. 1-19 (1995)

[13] D. W. Crenner et al., journal of Physics: Condensed Matter, Vol. 14, No. 4, pp. 783-802 (2002)

[14] K. Min and N. R. Aluru, In preparation (2010)

[1] F. Schendin et al., Nature Materials, Vol. 6, pp. 652-655 (2007)

[2] C. Lee et al., Science, Vol. 321, No. 5887, pp. 385-388 (2008)

[3] I. W. Frank et al., Journal of Vacuum Science and Technology B, Vol. 25, No. 6, pp. 2558-2561 (2007)

[4] J. Meyer et al., Nature, Vol.446, pp. 60-63 (2007)

[5] A. Sakhaee-Pour, Solid State Communications, Vol. 149, No. 1-2, pp. 91-95 (2009)

[6] C. Li and T. W. Chou, International Journal of Solid and Structures, Vol. 40, No. 10, pp. 2487-2499 (2003)

[7] R. Faccio et al., Journal of Physics: Condensed Matter, Vol. 21, No. 28, pp. 5304-5310 (2009)

[8] H. Bu et al., Physics Letters A, Vol. 373, No. 37, pp. 3359-3362 (2009)

[9] H. Zhao et al., Nano Letters, Vol. 9, No. 8, pp. 3012-3015 (2009)

[10] Y. W. Wong and S. Pellegrino, Journal of Mechanics of Materials and Structures, Vol. 1, No. 1, pp. 27-

61 (2006)

[11] S. Stuart et al., Journal of Chemical Physics, Vol. 112, No. 14, pp. 6472-6486 (2000)

[12] S. J. Plimpton, Journal of Computational Physics, Vol. 117, pp. 1-19 (1995)

[13] D. W. Crenner et al., journal of Physics: Condensed Matter, Vol. 14, No. 4, pp. 783-802 (2002)

[14] K. Min and N. R. Aluru, In preparation (2010)