1. PHYS466 Project Kyoungmin Min, Namjung Kim and Ravi Bhadauria

2. Contents INTRODUCTION CURRENT RESEARCH OBJECTIVES SIMULATION SETUP RESULT AND DISCUSSION CONCLUSION REFERENCES INTRODUCTION CURRENT RESEARCH OBJECTIVES SIMULATION SETUP RESULT AND DISCUSSION CONCLUSION REFERENCES

3. Graphene: Introduction A single layer of sp 2 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 sp 2 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

4. Current research ApproachSheet description Young’s modulus(TPa) Experimental observation [6] Graphene 1.000 Structural mechanics: Stiffness matrix[6] Zigzag sheet a=9.847nm / b=1.969nm 1.024 Structural mechanics: Stiffness matrix[6] Zigzag sheet a=20.178nm / b=4.184nm 1.033 molecular structural mechanics[5] Zigzag sheet a=6.3945nm / b=4.1841nm 1.040 molecular structural mechanics[5] Armchair sheet a=6.1531nm / b=4.2630nm 1.042 Young’s modulus ChiralityShear 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 ChiralityPoisson’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 ApproachRemarks Indentation loadCritical wrinkling load Wrinkling behavior

5. Objectives 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.

6. Simulation 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 Large-scale Atomic/Molecular Massively Parallel Simulator(LAMMPS) is used as molecular dynamics engine. LAMMPS[12]

7. Simulation Setup BOUNDARY CONDITIONS Move and Fix top and bottom Move and Fix top and bottom Fix top and bottom after simple shear Fix top and bottom after simple shear Apply velocity on top Apply velocity on top 1) Apply strain directly2) Displace and relax top atoms3) 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

8. Results and Discussion After 400 steps, temperature, pressure and potential energy reached at the stable state. RELAXATION Temperature Pressure Potential E Relaxation point

9. Results and Discussion Temperature, pressure and potential energy show discontinuity to corroborate the fracture. UNDER SHEAR LOAD Temperature Pressure Potential E Fracture

10. Shear 1 (Armchair)Shear 2 (Armchair) Shear 1 (Zigzag)Shear 2 (Zigzag) Armchair Zigzag

11. Results 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

12. Results 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 Modulus Shear Strength Fracture shear strain Fracture shear strain Shear modulus in bulk case = 480Gpa(zigzag) and 450GPa(armchair) Shear strength in bulk case = 60 GPa

13. Results and Discussion The ratio of amplitude and half wave length relationship [10] The ratio of amplitude and half wave length relationship [10] WRINKLING BEHAVIOR AC ZZ Wrinkle view MD result, 836 atoms # of Atoms ν (Armchair)ν (Zigzag) 720.270766 1800.2725530.256468 3640.2573620.239489 5760.2439570.225191 8360.2403830.224298 Bulk case [9]0.21 γ γ A / λ THEORETICAL RESULTS

14. Conclusion 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.

15. References [1][1] F. Schendin et al., Nature Materials, Vol. 6, pp. 652-655 (2007) [2][2] C. Lee et al., Science, Vol. 321, No. 5887, pp. 385-388 (2008) [3] [3] I. W. Frank et al., Journal of Vacuum Science and Technology B, Vol. 25, No. 6, pp. 2558-2561 (2007) [4][4] J. Meyer et al., Nature, Vol.446, pp. 60-63 (2007) [5][5] A. Sakhaee-Pour, Solid State Communications, Vol. 149, No. 1-2, pp. 91-95 (2009) [6][6] C. Li and T. W. Chou, International Journal of Solid and Structures, Vol. 40, No. 10, pp. 2487-2499 (2003) [7][7] R. Faccio et al., Journal of Physics: Condensed Matter, Vol. 21, No. 28, pp. 5304-5310 (2009) [8][8] H. Bu et al., Physics Letters A, Vol. 373, No. 37, pp. 3359-3362 (2009) [9][9] H. Zhao et al., Nano Letters, Vol. 9, No. 8, pp. 3012-3015 (2009) [10][10] Y. W. Wong and S. Pellegrino, Journal of Mechanics of Materials and Structures, Vol. 1, No. 1, pp. 27-61 (2006) [11][11] S. Stuart et al., Journal of Chemical Physics, Vol. 112, No. 14, pp. 6472-6486 (2000) [12] [12] S. J. Plimpton, Journal of Computational Physics, Vol. 117, pp. 1-19 (1995) [13][13] D. W. Crenner et al., journal of Physics: Condensed Matter, Vol. 14, No. 4, pp. 783-802 (2002) [14][14] K. Min and N. R. Aluru, In preparation (2010) [1][1] F. Schendin et al., Nature Materials, Vol. 6, pp. 652-655 (2007) [2][2] C. Lee et al., Science, Vol. 321, No. 5887, pp. 385-388 (2008) [3] [3] I. W. Frank et al., Journal of Vacuum Science and Technology B, Vol. 25, No. 6, pp. 2558-2561 (2007) [4][4] J. Meyer et al., Nature, Vol.446, pp. 60-63 (2007) [5][5] A. Sakhaee-Pour, Solid State Communications, Vol. 149, No. 1-2, pp. 91-95 (2009) [6][6] C. Li and T. W. Chou, International Journal of Solid and Structures, Vol. 40, No. 10, pp. 2487-2499 (2003) [7][7] R. Faccio et al., Journal of Physics: Condensed Matter, Vol. 21, No. 28, pp. 5304-5310 (2009) [8][8] H. Bu et al., Physics Letters A, Vol. 373, No. 37, pp. 3359-3362 (2009) [9][9] H. Zhao et al., Nano Letters, Vol. 9, No. 8, pp. 3012-3015 (2009) [10][10] Y. W. Wong and S. Pellegrino, Journal of Mechanics of Materials and Structures, Vol. 1, No. 1, pp. 27-61 (2006) [11][11] S. Stuart et al., Journal of Chemical Physics, Vol. 112, No. 14, pp. 6472-6486 (2000) [12] [12] S. J. Plimpton, Journal of Computational Physics, Vol. 117, pp. 1-19 (1995) [13][13] D. W. Crenner et al., journal of Physics: Condensed Matter, Vol. 14, No. 4, pp. 783-802 (2002) [14][14] K. Min and N. R. Aluru, In preparation (2010)