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ELASTIC PROPERTIES OF NANOTUBES

Explore experimental and modeling approaches to study elastic properties of nanotubes, with comparisons to ab initio calculations. Comparison with steel, micro-mechanical manipulations, Raman effect basics for beginners, and graphene phonons are discussed.

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ELASTIC PROPERTIES OF NANOTUBES

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  1. ELASTIC PROPERTIES OF NANOTUBES Nanotube learning seminar series SZFKI B.Sas, T. Williams 12 September 2005

  2. HOW TO LEARN ABOUT ELASTICITY OF CNT • Approaches 1. Experimental: i) “Macroscopic” mechanical measurements ii) Microscopic spectroscopic measurements 2. Modelling: i) Continuum elasticity ii) Phonon dispersion and anharmonicity 3. Comparison with ab initio calculation

  3. GPa σzz 30 - E=350GPa δL/L % 0 - 0 10

  4. Tensile Loading of Ropes of SWNTs (Yu et al PRL 2000) E=1000GPa

  5. L y σxx=f/2πR x x 2-D ELASTICITY δR Uniaxial force F=0 2R F=f L+δL L δWp=δWel 2πR fδL=∫σxxuxxdS= 2 πRLσxxδL/L σxx=f/2πR Euxx= σxx uyy=δR/R= -σP uxx σP=(K-μ)/(K+μ) E=4K μ/(K+ μ) E3-D=E2-D /wall thickness

  6. (2-D Elasticity) BENDING MODEL F=0 F=f 2R d L compression [E3-DE2-D/wall thickness] dilatation

  7. d [nm] 4 - 2 - F [nN] E3-D=1000GPa E2-D=300Nm-1

  8. COMPARISON WITH STEEL STEEL 100 0.1 0.1 100 8 10 CNT 1000 10 100 50 0.6 5000 Young mod E3-D Strain limit stress limit filling factor density stress limit cable [GPa] [%] [GPa] [%] [g cm-2] [Kg force mm-2] Hung by Φ500μm CNT thread

  9. Micro-Mechanical Manipulations • Rotational actuators based on carbon nanotubes (Nature, 2003.) Electrostatic motor.

  10. (ω0- ωPn)ω0(ω0+ωPn) RAMAN EFFECT FOR BEGINNERS I ω0 ω0 ω0-ωph α,ωph dipole emission cosω0t , cos(ω0±ωPn)t excitation Eincosω0t ω0 24000cm-1 25000cm-1

  11. Graphene phonons

  12. ħω0 RAMAN FOR BEGINNERS II CLASSICAL QUANTUM m,e u x κ=mω2el ħωel Eincosω0t g Dipole:

  13. RAMAN III APPLIED STRESS ustatic≠0 ⇒intensity change by δωel ⇒ reveals by δωPn ⇒ lifts phonon mode degeneracies by symmetry reduction Ustatic=0 ustatic≠0

  14. m = 0 1 2 3 4 Eg A2u Sanchez-Portal et al.

  15. L σyy 2πR y σxx x Hydrostatic pressure Capped ends 2-D ELASTICITY δR 2R L L-δL P=0 P=p δWp=δWel πR2pδL=∫σxxuxxdS= 2 πRLσxxδL/L 2πRLpδR=∫σyyuyydS=2 πRL σyyδR/R σxx=pR/2 σyy=pR uyy/uxx=2 if σP=0

  16. L 2πR σyy y σxx x 2-D ELASTICITY Hydrostatic pressure P=p Capped ends σxx=pR/2 ; σyy=0 uxx=pR/2E ; uyy=-σPpR/2E + σxx=0 ; σyy=pR uxx=-σPpR/E; uyy=pR/E = σxx=pR/2 ; σyy=pR uxx=(1-2σP)pR/2E ; uyy=(2-σP)pR/2E uyy/uxx = (2-σP)/(1-2σP)

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