Nonlocal and nonlinear effects in hyperbolic heat transfer in a two-temperature model
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Nonlocal and nonlinear effects in hyperbolic heat transfer in a two-temperature model A. Sellitto , I. Carlomagno
and M. Di Domenico
Abstract. The correct analysis of heat transport at nanoscale is one of the main reasons of new developments in physics and nonequilibrium thermodynamic theories beyond the classical Fourier law. In this paper, we provide a two-temperature model which allows to describe the different regimes which electrons and phonons can undergo in the heat transfer phenomenon. The physical admissibility of that model is showed in view of second law of thermodynamics. The above model is applied to study the propagation of heat waves in order to point out the special role played by nonlocal and nonlinear effects. Mathematics Subject Classification. 74Jxx, 80Axx. Keywords. Electron and phonon temperature, Thermal wave propagation, Acceleration wave, Nonlocal and nonlinear effects.
1. Introduction Modern ultrafast laser-assisted manufacturing technology is empowering the fabrication of miniaturized nano/microscale devices for electronics, optics, medicine and energy applications [1,2]. A continuous control of the temperature’s rise should be required to avoid any possible problem in the laser-assisted nanoscale manufacturing. The analysis of heat transport in those situations (which may be also labeled as extreme, since they are “very far from equilibrium”), indeed, requires the use of theoretical models which go beyond the classical Fourier law; the heat transport, in fact, might be no longer diffusive (and therefore describable by the classical Fourier law), but ballistic, or hydrodynamic [3–10], since miniaturized nano/microscale devices usually show characteristic dimensions which are comparable to (or smaller than) the mean free path of the heat carriers. This paper deals, therefore, with the heat transport phenomenon at nanoscale which is a very hard, but compelling and fashionable research playground. In modeling that phenomenon, indeed, one should also observe that: i. although with a different importance, in common materials used at nanoscale both the electrons and the lattice vibrations (i.e., the phonons) are the heat carriers [11–14]. Both heat carriers are not in an equilibrium state during the heat transfer; ii. thermodynamic constitutive equations containing nonlocal and nonlinear spatial terms are needed when the attention is put on systems subjected to important spatial gradients [4,15–18]. According with the above observations, by regarding the electrons and the phonons as a mixture of heat carriers flowing through the crystal lattice, and assuming that they are endowed with their own temperatures [19,20], here we propose a theoretical model based on the following equations which allow to take into account memory, nonlocal and nonlinear effects: 0123456789().: V,-vol
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A. Sellitto, I. Carlomagno and M. Di Domenico
qi,e i =0 cev e Qeij,j 2qje qj, λe θ,ei qe q˙ie + i + − e ei − =0 τ1e τ1e cv θ τ1e 2e qi,e j Qeij Q˙ ei
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