Environmental Engineering Reference
In-Depth Information
3.1.3
Different Transport Regions
Phonon mean free path is the key quantity for classifying transport
regions. Thermal transport is diffusive for systems with sizes much
larger than phonon mean free path. This is the classical case where
Fourier's law issatisfied.
=−
κ
x
T
,
j
(3.4)
where
j
is the heat flux,
κ
is material's thermal conductivity, and
x
T
is the temperature gradient along the transport direction (
x
direction).
When the system size is much smaller than phonon mean free
path, phonons experience almost no scatterings during transport.
In this case, thermal transport is quasi-ballistic or ballistic. Ballistic
thermal transport exhibits a lot of unexpected, interesting features.
For instance, there is no temperature gradient for ballistic thermal
transport, thus Fourier's law fails. In such systems, thermal con-
ductivity is not a constant but linearly dependent on the transport
length. Instead, thermal conductance is a well-defined quantity and
does notchange asthe system length increases.
The diffusive region, that is, the classical limit, has been well
studied until now. Benefitting from the development of mesoscopic
transport theory, ballistic transport can be well described, for
instance, by the Landauer formalism to be shown later. However,
much less is known for the transition region between ballistic and
diffusiveregions.Therewillbeatransitionfromballistictodiffusive
transport when increasing system size. Increasing the temperature
wouldalsoinduceaballistic-diffusivetransition,sincephononmean
free path is temperature-dependent. To understand how thermal-
transport changes from the quantum limit to the classical limit,
we need to investigate the transition region. The corresponding
method development will be a promising direction for future
research.
3.1.4
The Landauer Formalism
The Landauer formalism [4], first proposed to describe mesoscopic
electronic transport, can be generally used for studying mesoscopic
