Environmental Engineering Reference
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where the thermal conductivity
λ
indicates the ability to conduct
heat in materials. Fourier's law aptly describes the thermal-
transport phenomena of a bulk material whose dimension is much
larger than the mean free path and/or coherent length of heat
carriers such as electrons and phonons. However, this law is no
longer valid when the sample dimension reduces down to the mean
free path of carriers. The problem of “breakdown of Fourier's law”
originating from the reduction of sample size has been a long
standing theoretical interest that goes back to Peierls' early work in
the 1920s [1].
Overthepastfewdecades,ourunderstandingofphonon-derived
thermal transport in such a small system has been advancing at
a remarkable pace. Various striking phenomena that cannot be
explained by Fourier's law have been discovered: the quantization
of thermal conductance [2-5], the universal phonon-transmission
fluctuation [6, 7], and the Anderson localization [8-10]. In this
chapter, we present an overview of phonon transport or phonon-
derived thermal transport in small objects. In Section 2.2, we give
an account of a basic theory and simulation techniques of phonon-
derived thermal transport that makes it possible to explain novel
phonon transport phenomena of small objects with the breakdown
of Fourier's law. In Section 2.3, we give a review of the non-Fourier
thermaltransportincarbonnanotubesastypicalexamplesofactual
nanomaterials.
2.2 Theory of Nanoscale Phonon Transport
In this section, we describe theory of thermal transport by coherent
phonons in a quasi-one-dimensional (quasi-1D) system whose
lengthismuchshorterthanthecoherencelength,whichisbounded
by phonon scattering events. We now introduce the Landauer
approach for describing the coherent phonon transport in a quasi-
1D system.
2.2.1
Landauer Theory of Phonon Transport
Let us consider the situation depicted in Fig. 2.1. A coherent
phonon conductor in the absence of many-body interactions, such
