Size and porosity effects on thermal conductivity of nanoporous material with an extension to nanoporous particles embedded in a host matrix

Abstract

Abstract.A formula for the effective thermal conductivity of nanoporous media is derived, following a thermodynamic approach. An extension to nanocomposites composed of a homogeneous matrix wherein porous nanoparticles are dispersed is proposed as well. The originality of the model is that it is based on extended irreversible thermodynamics, a theory specifically designed for sub-scaled systems. Two different situations are discussed, in the first one, nanoporous silicon with spherical porous inclusions of micro, meso and macro-dimension respectively is considered. The description is validated by comparison with experimental data and five other models. Analysis of the results shows an excellent agreement of our theoretical approach with experiments in the whole range of porous radii, from 2 to 100 nm. In the second part of the work, thermal conductivity of porous silicon nanoparticles embedded in a germanium host matrix is investigated. The coupled influence of the pore and nanoparticles sizes is emphasized.Keywords: nanoporous media, thermal conductivity, phonon scattering, extended thermodynamics IntroductionThe effect of porosity is to decrease considerably the thermal conductivity of crystalline materials, as is observed, for instance, in nanoporous silicon [1][2][3][4][5]. This lowering property has been widely exploited with the objective to reinforce thermal insulation in several systems as microsensors, integrated circuits, semiconductor devices. It has therefore fostered an increasing interest during the last decade. Thermal properties of nanoporous media are generally investigated by referring to, for instance, kinetic theory of phonons [3], molecular simulations [5], Monte Carlo simulations [6], two-component materials [7] or hydrodynamic models [8]. In this work, we take a different route and prefer a thermodynamic approach. To be explicit, we will base our study on extended irreversible thermodynamics (EIT) [9,10], which is a formalism specifically dedicated to the treatment of micro and nano systems and highfrequency processes. For the sake of completeness, the present analysis will be compared to other theoretical models and experimental data.Nanoporous materials are generally made out of a homogeneous matrix wherein nanopores are dispersed. In that respect, such systems are similar to nanocomposites, wherein nanoparticles are embedded in a matrix, the difference being that nanoparticles are replaced by nanopores. A great deal of papers have been devoted to the problem of heat transport in nanofluids and nanocomposites, see for instance the review paper by Michaeledis [11]. In particular, the problem of the overall heat conductivity of the system as a function of the volume fraction of the nanoparticles and their size has drawn much attention. Our purpose in the present work is twofold. Firstly, it is the purpose to study the role of porosity and the size of the pores on the heat conductivity of porous systems, like nanoporous silicon, via our thermodynamic formalism. Secondly, we examine, using t...

Thermal rectifier efficiency of various bulk–nanoporous silicon devices

“…This is understandable since at larger pore sizes, the ballistic contribution becomes negligible with respect to the diffusive one ( ≪ 1) and Fourier's heat transfer is dominating. The effect of the porosity on the thermal conductivity is not to be investigated further, since it has been thoroughly studied in a previous work [22]. It is generally admitted that porosity reduces the thermal conductivity.…”

“…1. The corresponding values in the porous segment are obtained from the volumetric average of the silicon and air data [22]. Making use of these numerical values, it is found that the thermal boundary resistance coefficient and the Kapitza length are given by R=2.2*10 -9 m 2 K/W and ≈ 2 * 10 m. The latter result is significant as it justifies that the effect of interfacial thermal boundary resistance can be neglected for device lengths L longer than 2 µm, which represents a reasonable value within the context of the present work.…”

“…The present analysis uses instead for the heat conductivity a relation proposed by Sumirat et al [21], which emphasizes more particularly the effect of phonon scattering by pores. The Sumirat model has been validated with results derived by Machrafi and Lebon [22] in the framework of extended irreversible thermodynamics [23], it is mathematically simpler than the hydrodynamic model used by Criado et al and allows for a more exhaustive exploration of several aspects of the problem.…”

Analytical model for phonon transport analysis of periodic bulk nanoporous structures

“…In previous studies (Jean et al, 2014), aMonte Carlo (MC) technique have been employed to extract geometry-determined for nanoporous bulk materials with selected periods and porosities. In other studies (Minnich and Chen, 2007;Machrafi and Lebon, 2015), simple expressions have been proposed to compute . However, some divergence can often be found between lattice thermal conductivities predicted by phonon MC simulations and by analytical models using .…”

“…For larger nanoporous structures, phonon Monte Carlo (MC) simulations are employed, in which the exact structure geometry and frequency-dependent phonon mean free paths (MFPs) can both be incorporated [3,[27][28][29]. Other approaches include the discrete ordinate method [12,30,31], hybrid lattice dynamics-continuum mechanics technique [13,32], and various analytical models [13,24,[33][34][35][36].…”