Nonlinear Mechanics of Monolayer Graphene Rui Huang

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1 Nonlinear Mechanics of Monolayer Graphene Rui Huang Center for Mechanics of Solids, Structures and Materials Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin

2 Acknowledgments Qiang Lu (postdoc) Zachery Aitken (undergraduate student) Prof. Marino Arroyo (Spain) Prof. Li Shi (UT/ME) Funding: DoE and NSF (ARRA)

3 What is special about graphene? So much about carbon nanotubes, let s play with graphene now. Can two-dimensional crystals exist in our 3D space? Rippling or not? Uniqueness of D: electron transport,,q Quantum Hall effect, etc. Any interesting mechanics? Novoselev et al. 005

4 Graphene Based Devices Patterned graphene on oxide for bipolar p-n-p junctions Suspended nanoribbons for NEMS devices Ozyilmaz et al., 007. Garcia-Sanchez, et al., 008.

5 Mechanical properties of monolayer graphene Young s modulus? Ultimate tensile strength? Bending modulus? What is the thickness of a graphene monolayer? Interlayer spacing in graphite: nm C-C covalent bond length: 0.14 nm Mechanically reduced thickness: nm We consider monolayer graphene as a D membrane with no thickness.

6 Nonlinear Continuum Mechanics of D Sheets D-to-3D deformation gradient: F ij X x i J X dx FdX x 3 x X 1 x 1 In-plane deformation: D Green-Lagrange strain tensor E JK 1 ( F F δ ) ij ik JK Bending: D curvature tensor (strain gradient) FiI xi Κ ni ni X X X J I J No need to define any thickness! Strain energy (hyperelasticity): ( E, ) U Φ Κ da A Lu and Huang, Int. J. Applied Mechanics 1, (009).

7 Stresses and Moments in D nd Piola-Kirchoff stress and moment (work conjugates) S Φ E M Φ Κ Tangent moduli: C KL S E KL Φ E E KL D KL M Κ KL Φ Κ Κ KL Λ KL S Κ KL M E KL Φ E Κ KL Intrinsic coupling between tension and bending An incremental form of the generally nonlinear and anisotropic behavior: ds ds ds 11 C C C 11 1 C C C 1 C C C 13 3 de11 Λ11 Λ1 Λ13 dκ de + Λ1 Λ Λ3 dκ de 1 Λ 31 Λ 3 Λ 33 dκ dm11 D11 D1 D13 dκ11 Λ11 Λ1 Λ31 de dm D1 D D3 dκ + Λ1 Λ Λ3 de dm 1 D31 D3 D33 dκ1 Λ13 Λ3 Λ33 de 1 11 Lu and Huang, Int. J. Applied Mechanics 1, (009).

8 Units for D quantities Φ( E,Κ) strain energy density function: J/m S D stress: N/m Φ E C KL E S Φ E E D in-plane modulus: N/m KL KL M Φ Κ moment intensity: (N-m)/m D KL M Κ KL Φ Κ Κ KL bending modulus: N-m Λ KL S M KL Κ KL E E Κ KL p g Φ coupling modulus: N Analogous to the D plate/shell theories.

9 Atomistic Modeling of Graphene Rectangular unit cell in an arbitrary direction, subject to in-plane stretching (, ):a c a and bending. (n,n): armchair nd -generation REBO potential (Brenner et al., 00) Bond angle effect (second-nearest nearest neighbors) Dihedral angle effect (third-nearest neighbors) Radical energetics (defects and edges) α (n,n) (n,0): zigzag Energy minimization (molecular mechanics) for equilibrium states. Stress and moment calculations Energy derivation Virial stress calculations Direct force evaluation

10 Pure Bending of Graphene Roll up a rectangular graphene sheet into a cylindrical tube without in-plane stretch (E 11 0) F 1 πr X cos π L L 0 πr X sin π L L κ E n 11 1 R 1 πr 1 L Str rain energy per atom (ev/atom m) B1, armchair B1, zigzag B, armchair B, zigzag B*, armchair B*, zigzag Bending curvature (1/nm) per length (nn N) Ben nding moment B1, armchair B1, zigzag B, armchair B, zigzag B*, armchair B*, zigzag Bending curvature (1/nm)

11 Bending Modulus of Graphene 0.5 Bending modulus (nn nm) B1, armchair B1, zigzag B, armchair B, zigzag B*, armchair θ jil l i k θ ijk j θ q3 θ q1 θ q4 θq Θ q Θ q1 Θ q3 Θ q4 B*, zigzag Bending curvature (1/nm) Bending moment-curvature is nearly linear, with slight anisotropy. Including the dihedral effect leads to higher bending energy and bending modulus. Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 4, 1000 (009).

12 Physical Origin of Bending Modulus Bending modulus of a thin elastic plate: D dm dκκ 3 ~ Yh h For monolayer graphene, bending moment and bending stiffness result from multibody interatomic interactions (second and third nearest neighbors). D VA( r ) b 14T θijk 3 σ π 0 ij 0 D 0.83 ev (0.133 nn-nm) by REBO-1 D 1.4 ev (0.5 nn-nm) by REBO- D 1.5 ev (0.38 nn-nm) by first principle Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 4, 1000 (009).

13 Coupling between bending and stretching Strain energy per atom (ev V/atom) W ( κ, ε ) Dκ + M ( κ κ ) + σε + D( κ κ ) Cε R nm R Strain ε R/R 0 1 The tube radius increases upon relaxation, leading to simultaneous bending and stretching. Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 4, 1000 (009).

14 Uniaxial Stretch of Monolayer Graphene Uniaxial i stretch: t λ L L 0 Nominal strain: ε λ 11 Green-Lagrange strain: E ( 1) 1 11 λ E E1 0 per atom (ev) Energy (ev) Energy A A B B Nominal strain ε C C D D Lu and Huang, Int. J. Applied Mechanics 1, (009).

15 40 Biaxial Stresses under Uniaxial Stretch (Poisson s effect) 30 Nominal stress (N N/m) 35 S P S 10 P S P 1 and P Nominal strain ε Green Lagrange strain E 11 (N/m) Membrane stress ( Energy derivation: S 11 dφ de 11 P 11 dφ dε ( ( n) ( m) ) Virial stress calculation for nominal stress: P X X ij 1 A m n J J F ( mn) i Relationship between nominal stresses and nd P-K stress: P11 P1 S 11, S P, S1, S1 1+ ε 1+ ε P 1

16 Anisotropic tangent moduli ds11 C11 C1 C13 de ds C1 C C3 de ds1 C31 C3 C33 de 1 (n,n): armchair 11 C 11 (N/m) α 0 (zigzag) α o α o α 30 o (armchair) C 1 (N/m) 50 α (n,n) (n,0): zigzag Graphene is linear and isotropic under infinitesimal deformation, but becomes nonlinear and anisotropic under finite deformation. Lu and Huang, Int. J. Applied Mechanics 1, (009). C 31 (N/m) Coupling between tension and shear Green Lagrange strain E 11

17 Fracture strength under uniaxial stretch (N/m) No ominal stress P α 0 (zigzag) α o Nominal strain ε α o α 30 o (armchair) Nominal fracture strain Chiral angle (degree) 3 Nom minal fracture stre ess (N/m) Fracture occurs as a result of intrinsic instability of the homogeneous deformation: P P ε Φ 0 ε The nominal stress and strain to fracture depend on the direction of uniaxial stretch.

18 Graphene Nanoribbons: Edge Effects Garcia-Sanchez, et al. (008). Koskinen, et al. (008) Various edge states: zigzag/armchair/mixed, H-terminated, reconstructed Edge effect leads to size-dependent electronic properties of GNRs. Are mechanical properties of GNRs size dependent?

19 Excess Edge Energy and Edge Force Zigzag edge: f Z f Z atom (ev) Energy per armchair, un relaxed armchair, 1 D relaxation zigzag, un relaxed zigzag, 1 D relaxation Eq. (6), γ 11.1 ev/nm Eq. (6), γ 10.6 ev/nm Armchair edge: f A Lu and Huang, arxiv: f A /W (nm 1 ) Edge energy (ev/nm ) Edge force (ev/nm) r 0 Armchair Zigzag Armchair Zigzag (nm) DFT [17] (GPAW) DFT [18] (VASP) DFT [] (SIESTA) MM [0] (AIREBO) MD [1] MM (REBO)

20 Edge buckling of GNRs Zigzag GNR Intrinsic wavelength ~ 6. nm Excess en nergy (ev/nm) y e 05*x *x *x 0.054*x Buckle wavelength λ (nm) Armchair GNR Intrinsic wavelength ~ 8.0 nm Ex xcess energy (ev/n nm) y.e 06*x *x *x 0.014*x Buckle wavelength λ (nm) The wavelengths for edge buckling do not scale with D/f. Lu and Huang, arxiv:

21 F GNRs under Uniaxial Tension U ( ε ) Φ ( ε ) WL + γ ( ε )L F δu FLδε σ F 1 du dφ + W WL dε dε W dγ dε Zigzag GNRs Armchair GNRs

22 Φ Interior Energy Function ( ε ) a 0 + a 1ε + a ε + a 3ε + a 4ε + a 5ε + a 6ε + a 7ε + a 8ε Zigzag Armchair a Φ( (ε ) a a a a a a a a dφ dε d Φ dε

23 γ Edge energy function ( ε ) b 0 + b1ε + bε + b3ε + b4ε + b5ε + b6ε + b7ε + b8ε Zigzag Armchair b b γ (ε ) b b b b b b b dγγ dε d γ dε

24 D Young s Moduli of GNRs σ ( ε ) dφ + dε W dγ dε d Φ E + dε W d γ dε ( ε ) Young s modulus under ifiit infinitesimal i strain: 4b E 0 a + W

25 Fracture of graphene ribbons Zigzag edge Armchair edge

26 MD simulations of fracture (300 K) Zigzag edge Armchair edge Zigzag GNRs: fracture starts away from the edges; fracture strain same as that for an infinite graphene (PBC); Armchair GNRs: fracture starts near the edges; fracture strain lower than that for an infinite graphene (PBC).

27 Graphene on Oxide Substrates HR-STM image (Stolyarova a et al., 007) RMS Ozyilmaz et al., 007. The 3D morphology is important for the transport properties of graphene-based devices. Correlation length Ishigami et al., 007.

28 Van der Waals Interaction Lennard-Jones potential for particle-particle interactions: 6 ( ) r0 r0 V r V0 r r 1 Monolayer-substrate interaction (energy per unit area): U U ( h) U h 0 1 h0 h h r h 0 h 0 h -U 0

29 Van der Waals Thickness and Energy Gupta et al., 006. Sonde et al., 009. Interlayer spacing in graphite ~ 0.34 nm; AFM measurements of h 0 for graphene on oxide range from 0.4 to 0.9 nm; The adhesion energy (U 0 )h has not tbeen measured ddirectly; Theoretically estimated values for U 0 range from 0.6 to 0.8 ev/nm.

30 Strain-Induced Corrugation h h 0 + Asin kx U U + U total vdw graphene Linear perturbation analysis: U vdw U graphene 1 λ A U + f 0 1 h h π π 4D + Cε A λ λ ε c ~ λ ~ 3 nm

31 Graphene on Rough Substrates Conformal or non-conformal?

32 Summary Nonlinear continuum mechanics for D graphene monolayer Atomistic modeling of graphene under bending and stretching Excess edge energy, edge forces, and induced edge buckling Graphene nanoribbons under uniaxial tension: edge effects on elastic modulus and fracture Graphene on oxide: van der Waals interaction p and corrugation

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