ELASTIC PROPERTIES OF NANOTUBES

Nanotube learning seminar series

SZFKI

B.Sas, T. Williams 12 September 2005

HOW TO LEARN ABOUT ELASTICITY OF CNT

• Approaches

1. Experimental:i) “Macroscopic” mechanical measurementsii) Microscopic spectroscopic measurements

2. Modelling:i) Continuum elasticityii) Phonon dispersion and anharmonicity

3. Comparison with ab initio calculation

GPa

δL/L %0

0 -

30 -

10

σzz

E=350GPa

• Tensile Loading of Ropes of SWNTs (Yu et al PRL 2000)

E=1000GPa

F=0 F=f

δWp=δWel

fδL=∫σxxuxxdS= 2 πRLσxx δL/L

2-D ELASTICITY

2πR

L L+δL

2R

δR Uniaxial force

L

x σxx=f/2πR

y

x

σxx=f/2πR

σP=(K-μ)/(K+μ)E=4K μ/(K+ μ)

Euxx= σxx

uyy= δR/R= -σP uxx

E3-D=E2-D /wall thickness

BENDING MODEL

d2R

F=0 F=f

dL

R

24

Ef

3

D2

2d

Lρ

2

compression

dilatation

L

(2-D Elasticity)

[E3-DE2-D/wall thickness]

F [nN]

d [nm]

2 -

4 -

E3-D=1000GPaE2-D=300Nm-1

COMPARISON WITH STEEL

Young mod E3-D

Strain limitstress limitfilling factordensity stress limit cable

[GPa][%][GPa][%][g cm-2][Kg force mm-2]

STEEL1000.10.1100810

CNT100010100500.65000

Hung byΦ500μmCNT thread

Micro-Mechanical Manipulations

• Rotational actuators based on carbon nanotubes (Nature, 2003.) Electrostatic motor.

RAMAN EFFECT FOR BEGINNERS I

α,ωph

ω0 ω0

ω0-ωph

excitation Eincosω0t

tcosωEtcosωuu

ααp 0inPnPn

0dipole emissioncosω0t , cos(ω0±ωPn)t

(ω0- ωPn) ω0 (ω0+ωPn)ω0

25000cm-124000cm-1

Graphene phonons

RAMAN FOR BEGINNERS II

xm,e

Eincosω0t κ=mω2el ħω0

ħωel

tm

ee

el02

02

2

cos

inExpDipole:

g

u

tm

ee

Egxu

egexuueExg

elel

elel

00

22

0

2

2

0

cos

22

inEp

p

elmxgxu

222

tuu

uuu

uu

staticPn

PnPnstatic

el

0cos

CLASSICAL QUANTUM

RAMAN III

APPLIED STRESS

ustatic≠0 ⇒ intensity change by δωel

⇒ reveals by δωPn

⇒ lifts phonon mode degeneracies by symmetry reduction

2

2

u

rV

u

Ustatic=0

ustatic≠0

Eg

A2u

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

L L-δL

2R

δR

P=0 P=p

δWp=δWel

πR2pδL=∫σxxuxxdS= 2 πRLσxx δL/L

2πRLpδR=∫σyyuyydS=2 πRL σyyδR/R

σxx=pR/2 σyy=pR

uyy/uxx=2 if σP=0

2-D ELASTICITY Hydrostatic pressureCapped ends

2πR

L

x

y

σxx

σyy

σxx=pR/2 ; σyy=0uxx=pR/2E ; uyy=-σPpR/2E

σxx=0 ; σyy=pRuxx=-σPpR/E; uyy=pR/E

+

=

σxx=pR/2 ; σyy=pRuxx=(1-2σP)pR/2E ; uyy=(2-σP)pR/2E

x

y

σxx

σyy

2πR

L

2-D ELASTICITY Hydrostatic pressure P=pCapped ends

uyy/uxx = (2-σP)/(1-2σP)