Volume averaged phonon Boltzmann Transport Equation for simulation of heat transport in composites
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Heat transfer in nano-composites is of great importance in a variety of applications, including in thermoelectric materials, thermal interface and thermal management materials, and in metamaterials for emerging microelectronics. In the past, two distinct approaches have been taken to predict the effective thermal conductivity of composites. The first of these is the class of effective medium theories, which employs Fourier conduction as the basis for thermal conductivity prediction. These correlate composite behavior directly to volume fraction, and do not account for inclusion structure, acoustic mismatch, and sub-continuum effects important in nanocomposites. More recently, direct numerical simulations of nanoscale phonon transport in composites have been developed. Here the geometry of the inclusion or the particulate phase is represented in an idealized way, and the phonon Boltzmann Transport Equation (BTE) solved directly on this idealized geometry. This is computationally intensive, particularly if realistic particle composites are to be simulated. Here, we develop, for the first time, a volume-averaged formulation for the phonon BTE for nanocomposites, accounting for the complex particle-matrix geometry. The formulation is developed for a nanoporous domain as a first step and then a nanocomposite domain is considered. The phonon BTE is written on a representative elemental volume (REV) and integrated formally over the REV using the laws of volume averaging. Extra integral terms resulting from the averaging procedure are approximated to yield extra scattering terms due to the presence of inclusions or holes in the REV. The result is a phonon BTE written in terms of the volume-averaged phonon energy density, and involving volumetric scattering terms resulting from both bulk scattering and scattering at the interfaces of the inclusions in the REV. These volumetric scattering terms involve two types of relaxation times: a volume-averaged bulk scattering relaxation time resulting from phonon scattering in the bulk matrix material, and an interface scattering relaxation time resulting from volume-averaging scattering due to interfaces within the REV. These relaxation times are determined by calibration to direct numerical simulations (DNS) of the particle or pore-resolved geometry using the phonon BTE. The additional terms resulting from the volume-averaging are modeled as in-scattering and out-scattering terms. The scattering terms are written as a function of a scattering phase function, and the interface scattering relaxation time. The scattering phase function represents the redistribution of phonon energy upon scattering at the interface. Both interface scattering relaxation time and scattering phase function matrix are functions of the interface geometry and the phonon wave vector space. The scattering phase function in the model is evaluated in the geometric optics limit using ray tracing techniques and validated against available analytical results for spherical inclusions. The volume-averaged bulk scattering relaxation time takes in to consideration the effects of the pores on the effective thermal conductivity of the composite. It is calibrated using a Fourier limit solution of the nanoporous domain. The resulting governing equations are then solved using a finite volume discretization and the coupled ordinates method (COMET). In the gray limit, the model is applied to nanporous geometries with either cylindrical or spherical pores. It is demonstrated to predict effective thermal conductivity across a range of Knudsen numbers. It is also demonstrated to be much less computationally intensive than the DNS. This model is extended to include non-gray effects through the consideration of both polarization and dispersion effects. For non-gray transport, the bulk and interface scattering relaxation times are now wave-vector dependent. Two different models are proposed for determining the interface scattering relaxation times, one assuming a constant value of interface scattering relaxation time, and another which accounts for variation with wave vector. As before both bulk and interface relaxation times are calibrated with the DNS solution in the Fourier and ballistic limits. The scattering phase function developed for gray transport in the geometric limit is expanded to consider the appropriate energy exchanges between different phonon modes assuming elastic scattering. The non-gray volume-averaged BTE is compared to the DNS for a range of porosities at the limits of bulk average Knudsen number and for intermediate average Knudsen numbers. The model with variable interface scattering relaxation times is found to better predict the variation of effective thermal conductivity with wave vector, though both models for interface scattering are less accurate than the gray model. Further, the volume-averaged BTE is extended for two material composites. We solve the volume-averaged BTE model for particle sizes comparable to the phonon wavelength in the composite matrix. We employ analytical scattering phase functions in the Mie scattering limit for particles to include wave effects. The calibration of model relaxation time parameters is conducted similar to that in the gray volume-averaged BTE model for nanoporous materials. The composite domain is solved in the Fourier limit to calibrate the volume-averaged bulk relaxation time. This relaxation time parameter considers the material properties of both the host material and particle. For small particle sizes, calibration in the ballistic limit is conducted using a nanoporous domain. This is possible as the interface scattering relaxation time is driven primarily by the travel time of the phonons between particles, and not by the residence time inside the particle. The scattering phase function is computed considering properties of both the host material and the particle scatterers. We solve the volume-averaged BTE for the two-material composite for a silicon host matrix with spherical germanium particles. We demonstrate the gray two-material composite domain for varying porosities over a range of Knudsen numbers. The present work creates a pathway to model thermal transport in nanocomposites using volume-averaging which can be used in arbitrary geometries, accounting for both bulk scattering and boundary scattering effects across a range of transport conditions. The model accounts not only for the volume fraction of particulates and inclusions, but also their specific shape and spacing. It also accounts for sub-continuum effects. Furthermore, the volume-averaging method also allows inclusion of wave effect through the scattering phase function so that particles on the order of the phonon wavelength or smaller can be considered. The formulation is also generalizable to the limit when the particles are large compared to the wavelength; in this limit, geometric optics may be employed to compute the scattering phase function. Overall, the volume averaging approach offers a computationally inexpensive pathway to including composite microstructure and subcontinuum effects in modeling nanoporous materials and composites.
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