The Macroscopic transport equations of phonons in solids




Fryer, Michael

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There has been an increasing focus on using nanoscale devices for various applications ranging from computer components to biomechanical sensors. In order to effectively design devices of this size, it is important to understand the properties of materials at this length scale and their relevant transport equations. At everyday length scales, heat transport is governed by Fourier’s law, but at the nanoscale, it becomes increasingly inaccurate. Phonon kinetic theory can be used to develop more accurate governing equations. We present the moment method, which takes integral moments of the phonon Boltzmann kinetic equation to develop a set of equations based on macroscopic properties such as energy and heat flux. The advantage of using this method is that transport properties in nanodevices can be approximated analytically and efficiently. A number of simplifying assumptions are used in order to linearize the equations. Boundary conditions for the moment method are derived based on a microscopic model of phonons interacting with a surface by scattering, reflection or thermalization. Several simple, one dimensional problems are solved using the moment method equation. The results show the effects of phonon surface interactions and how they affect overal properties of a nanoscale device. Some of these effects were observed in a recent experiment and are replicated by other modeling techniques. Although the moment method has described some effects of nanoscale heat transfer, the model is limited by some of its simplifying assumptions. Several of these simplifying assumptions could be removed for greater accuracy, but it would introduce non-linearity into the moment method.



Phonon, Non-Equilibrium Thermodynamics, Heat Transport, Nanotechnology, Transport Equations