IWNET12
IWNET12
Generalized Transport Equations for Heat Transfer in Nanosystems:
Nanowires and Plane Sheets
David Jou1, Xavier Alvarez1 and Antonio Sellitto1
1 Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra, Catalonia, Spain
Abstract
Fourier's law for heat conduction is valid for experimental times longer than the average collision time and
for length scales longer than the mean-free path. Here, we present some generalized heat transport equations
incorporating non-vanishing collision time and mean-free path, and we discuss several aspects of those equations.
We apply them to two physical situations: nanowires (silicon nanowires and metallic nanowires) and plane sheets
(silicon layers and graphene sheets). In nanowires, those equations lead to a Fourier's equation with a size-
dependent thermal conductivity: thus, in this situation Fourier's law is apparently very successful, but at the
condition of suitably redening the thermal conductivity, which is one of the main topics in heat transport in
nanosystems. In contrast, in plane sheets, in radial heat transfer from a central heat source (a topic of interest
for the refrigeration of nanodevices), the inuence of non-local terms cannot be reduced to an eective Fourier's
transport equation. Therefore, we deal also with this radial situation because it may contribute to clarify about
the limits of Fourier's law. In particular, we deal with: a) inuence of boundary conditions (smooth walls or
rough walls) and their role on the eective thermal conductivity of nanowires; b) relative role of phonon and
electron heat transfer in metallic nanowires and the conditions of validity of Wiedeman-Franz's law in metallic
nanowires; c) compatibility of those equations with the second law, especially in a size domain comparable to
the mean-free path; d) dynamical aspects related to thermal wave propagation.
References
[1] G. Lebon, D. Jou and J. Casas-Vázquez, Understanding non-equilibrium thermo-dynamics, Springer, 2010
[2] A. Sellitto, D. Jou and X Alvarez, "Heat waves and phonon-wall collisions in nano-wires", Proc Royal Society
A 467 (2011) 2520-2533
[3] A. Sellitto, J. Bafaluy and D. Jou, "Nonlocal eects in radial heat transport in silicon thin layers and
graphene sheets", Proc Roy Soc A (2011) (DOI 10.1098/rspa.2011.06584)
[4] D. Jou, A. Cimmelli and A. Sellitto, Nonlocal heat transport with phonons and electrons: application to
metallic nanowires, Intel J Heat and Mass Transfer 55 (2012) 23382344
[5] A. Sellitto, F. X. Alvarez, D. Jou, Geometrical dependence of thermal conductivity in elliptical and rect-
angular nanowires, Int J Heat Mass Transf 55 (2012) 31143120
E-mail: David.Jou@uab.cat

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