Thermal conductivity (k) of CNTs

Have a look at C60

Heat transfer media : phonons and electrons

C60 has a perfect symmetry and -electrons move freely On C60 surface, so it has a high k = 3000-4000 W/mK

-e-heat

Heat resistance at interball

Strong inter-ball interaction gives better thermal conduction

Thermal conductivity of C60 crystal (=0.4 W/mK at room temp)

C60 crystal

T260 K

f.c.c phases.c phase

C60 rotation within crystal begins at 260 K

所以 F.C.C 相之 orientational disordering 比 S.C. 相高

What is orientation ordering?

N

S

N

S

N

S

N

S

Stronger interaction between C60

so better thermal conduction.

Weaker interaction between C60, so poor thermal conduction

S.C phase

f.c.c phase

ordering

disordering

K (W/mk)

T (K)

0.4

26085

Temp dep

Temp indep0.4 + 0.4(25 %)

Thermal conductivity of C60 crystal

1.Why K increases by 25% at 260 K,?2.Why K becomes temp indep above 260 K?3.Why K becomes temp dep at 85 K?

Thermal conductivity, K

K = 1/3Cv..lCv: specific heat: sound speedl: phonon mean free path

260 K Why K increases by 25 %

85 K

f.c.c

s.c

K = 1/3Cv..linvariables

增加

phonon

phonon

scattering

propagation

> 260 K, l = 50 Å, Cv = 50 j/kmol

260 K

Why K is temp indep above 260 K

phonon

Strong scattering above 260 K, so phonons only get luck with propagation, and temp makes no difference here!

Debye temp D of C60 crystal ~ 75 K (very low)because weak inter-ball interaction (or low density of inter-ball phonon mode).

D = hνm/KK: boltzmann constanth: planck constantνm: Debye frequency

Debye temp: the temp of a crystal’s highest normal mode of vibration

Normal mode: an oscillation in which all particles move with the same frequency and phase.

νm = (3N/4V)1/2·VsN/V: number density of atomVs: effective speed of sound

Why K becomes temp dep at 85 KThere still has some mis-orientated C60

Duration cooling, needs a time for C60 alignement.

260 K 85 K

Thermal conductivity of CNTsDiamond (C-sp3 bond): stiff (faster conduction: K = 1500 W/mK) soft bonding (slow conduction)

p.s. diamond has perfect lattice, less defects, so phonon density is high !

CNTs are sp2 bonds (C-C=C-C),stronger than diamond

Double bonds

So higher K is expected for CNTs?Yes, but only when CNTs zero defects and perfect crystallized

K of a single graphite layeris very high (>2000 W/mK),phonon only moves on in-plane

phonon

Layer spacing (0.335 nm)

Weak inter-layer interaction weakens the thermal conductivity K. Phonon also has to move along the c-axis

CNTs have higher inter-layer spacing (0.34 nm),so K is similar to a single graphene sheet

c-axis

In-plane direction (a-axis)

Another calculations of K

1. (1/A)(dQ/dt) = - K(dT/dz)A: cross section areadQ: thermal energydt: time intervaldT/dz: temp gradient along z direction

2. K = (1/3vkBT2) <J(t)·J(0)>dt0

v: volumekB: boltzmann constantT: sample temp<J(t)·J(0)>: mean value of heat flux vector

40000 W/mK

K

Temp (k)

(10, 10) tube (ideal model)

100 K 400 K50 K

6000 W/mK

1. Between 50-400 k, K is temp dep 2. Why a change-over emerges at 100 k?

K = Cv··l <100 k, l constant, so K is dominated by Cv.

>100 k, Cv constant, so K is dominated by l, and l decreases as temp increase, due to umklapp process,

What is umklapp process ?

umklapp process 熱阻 ( 類似電阻 )

k1k2

k3

1st BZ

k1 + k2 = k3 + Gk1: wave vector of phonon 1

k2: wave vector of phonon 2

k3: k1 and k2 碰撞後合向量

G = 0, no heat resistance, and phonon wave vector moving forward(normal process)

G 0, heat resistance, and phonon moving backward(umklapp process)

G: reciprocal lattice vector

k1 k2

k3 k3+Gk3+G

1st BZ

Thermal expansion and contraction of CNTs

Zero thermal expansionexpansioncontraction

compensation

e.g. Invar alloy (Fe65Ni35 ,Zero thermal expansion)

Polymers: rubber, polyethyleneLayered mateirals: graphite and BN3D oxides: NaTi2P3O12, ZrW2O8

Thermal contraction materials

Low dimension system (nanowires, nanotubes…)

Thermal expansion or contraction is determined by competition between internal energy and entropy.

高溫區中低溫區Entropy dominates, harmonic regimecontraction

Internal energy plays crucial role, anharmonic regime,expansion

Vo V1

V = V1-Vo

V/Vo < 0, contractionV/Vo > 0, expansion

PRL, 92,015901, 2004

260 K

200 cm-1

ellipsoidal deformationSoftest vibration mode of C60

Thermal contraction of CNTsTube length

Tube volume

: thermal linear expansion coefficient

: thermal volumetric expansion