elastic properties of nanotubes nanotube learning seminar series szfki b.sas, t. williams 12...
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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)