Environmental Engineering Reference
In-Depth Information
κ
e
and
σ
e
is proportional to the temperature
T
κ
e
σ
e
=
L
0
T
.
(3.1)
L
0
, known as the Lorenz numberof free electrons, is equal to
σ
e
T
=
π
2
k
B
3
e
2
K
e
=
2.44
×
10
−
8
W
K
−
2
.
L
0
=
(3.2)
When the size of materials decreases to nanoscale, thermal
conductionismainlycontributedbyphonons.Forinstance,previous
experiments found that thermal conductivity of carbon nanotubes
(CNTs) (including both metallic and semiconducting ones) is domi-
nated by phonons at all temperatures [1]. This is partially because
in nano-materials, confinement effects induce energy splitting of
both electrons and phonons. The energy splitting is of the order
of eV for electrons, while it is much smaller, of the order of meV,
for phonons. Consequently, phonons are much easier to be excited
by temperature than electrons, thus dominate thermal conduction
in nano-materials. In the following sections, we focus on thermal
conduction contributed byphonons.
3.1.2
Fundamental Length Scales of Thermal Transport
Thermal transport (i.e., phonon transport) has two fundamental
length scales [2].
3.1.2.1 The characteristic wavelength of phonon
λ
Assuming a Debye temperature of
θ
D
and a lattice constant of
a
;
at high temperatures (
T
>θ
D
),
λ
∼
a
; and at low temperatures
(
T
<θ
D
),
λ
∼
θ
D
a
/
T
.Asthetemperaturedecreases,
λ
increasesand
thus quantum effects become more and more important in thermal
transport. That is why the quantum phenomena are observable
only at low temperatures. Note that the carbon materials have very
high Debye temperature (diamond: 2230 K [3]). Therefore room
temperature actually corresponds to low-temperature region for
carbon systems.
3.1.2.2 The phonon mean free path
l
Phonon mean free path is the average transport distance of phonon
between two successive scatterings.
l
is determined by two types
