Microscopic Effective Medium Model for Thermal Conductivity of Two Dimensional Nano-Porous and Micro-Porous Media

Two-dimensional Nano-porous (NP) and micro-porous (MP) materials are currently used in a variety of applications which require the knowledge of the thermal conductivity (k). In NP and MP materials, two pertinent length scales determine the phonon thermal conductivity: 1) the ratio of inter-pore distance, δ, and the mean free path (m.f.p.) of phonons, 1, 2) the ratio of the pore diameter, d, and the m.f.p. of phonons. This is schematically shown in Fig. 1. In the traditional diffusion-approximation (macroscopic models) based models (1 ≪ d, δ) for the thermal conductivity of porous materials, the effective thermal conductivity, keff , of the porous material for a given shape of the pores and direction of the heat flow is only a function of the volume fraction (φ) of the pores. Therefore, in the diffusion approximation, keff can be written as

where km is the thermal conductivity of the host medium. Our focus is on cylindrical pores. We only consider the heat flow in the transverse direction as shown in Fig. 1a. For example, if φ <40%, Maxwell-Garnett effective medium model (MG EMM) can be used. f(φ) for MG EMM is given by

Experimental data on two-dimensional micro-porous silicon made of cylindrical pores have shown that the macroscopic model given by Eq. (1) grossly over predicts keff of the porous materials. In MP and NP materials, the phonon transport is ballistic in nature because of the dominant scattering of phonons from the pore boundaries. Ballistic transport becomes dominant when m.f.p is comparable to or larger than d and δ. In this regime, the Boltzmann Transport Equation (BTE) must be solved without invoking the diffusion approximation. Solving the BTE for such a complex network of pores is a challenging task, and a few previous works exist where BTE was solved numerically under various simplifying assumptions regarding the geometry and the arrangement of the pores. Both of these investigations assumed rectangular pores for two-dimensional composite or cubical pores for three-dimensional composites; however, in reality, these pores are never so simple in their geometry. Typically, these pores are cylindrical in shape for two-dimensional composites and nearly spherical in shape for three-dimensional composites. The solution of BTE for the multitude of non-planar pores, although achievable, will be a very tedious task.